I was a guest on the Thinking Poker Podcast, a fantastic podcast which I highly recommend. On this episode, I joined Andrew and Nate to chat about my poker life, discuss the most interesting pieces from this blog, and help delve into some game theory and strategy analysis.
Check it out here: Thinking Poker Podcast, Episode 45: Mike Stein of Quantitative Poker.
Showing posts with label skill vs. luck. Show all posts
Showing posts with label skill vs. luck. Show all posts
Tuesday, August 27, 2013
Tuesday, August 20, 2013
QuadJacks article: Poker Still Needs the Skill Game Argument
I wrote a response article for QuadJacks defending the ongoing value of spreading awareness of poker as a skill game in the current political environment:
Check it out here: Poker Still Needs the Skill Game Argument.
Check it out here: Poker Still Needs the Skill Game Argument.
Saturday, September 29, 2012
20 thoughts on skill vs. chance in poker, part 20: Summary and assessment of approaches to predominance
<-- Part 19: The German "predominantly chance" study
To bring our long journey to a close, it is worthwhile to condense the wide array of different perspectives we've treated on predominance of skill and chance into one central list. If predominance is a question with no perfect answer, which answer is the best one?
The approaches are presented roughly in order of least-consistent to most-consistent by my assessment, with the tables are broken up by the type of approach to predominance being used:
Emotional Approaches
Outcome-Based Approaches
Fundamental Approaches
While a fundamental, non-outcome-based approach is warranted for the quantitative question of predominance of factors within a game, we see that most of these fundamental approaches are based on superficial elements of games. With that in mind, in a desire for consistency across different game durations and different player populations, I think that the most appropriate approach to predominance is:
There's simply no other well-defined way to handle games of mixed skill and chance. There can't be anything in the middle of the axis of relative influence of skill and chance without arbitrarily locking in a specific game duration and ignoring the possibility of legally redefining games to be longer matches. Furthermore, as we've seen, the approaches that do implicitly specify a game duration find themselves relying on further assumptions about the type of player to measure, the specific skill/chance metric to use, and what it means to "control" an outcome. There's simply no clear alternative.
The approach that I recommend is a simple, clean, readily-applicable cut across the set of all games. In particular, it classifies all conceivable games within the broad family of "poker" the same way, which very few other approaches do.
Moreover, I believe this is consistent with the social intentions of treating games of chance and games of skill separately in the first place. Games of chance offer no personal or social benefits other than the pure entertainment value for participants who choose to play them. This shouldn't be ignored, but games of strategy offer that and much more. Strategy games ("mind sports") are the mental analog of physical sports and thus are valuable to society for the same reasons that physical sports are, perhaps moreso in an era where more and more careers demand mental acuity rather than physical prowess. Strategy games with random components teach valuable and broadly-applicable lessons to a human brain which is very poorly-equipped to handle randomness without training.
Economic concerns about wealth changing hands and desire to regulate activities which might attract problem gamblers are a bit of a red herring as they might relate to predominance, as these can certainly both take place within games of skill. (While this is outside the scope of this essay, nothing I say here should be taken as ignoring the existence of problem gamblers in poker.) It is clear to me that casino games, lotteries, and markets deserve different regulation and treatment than that of strategy games.
One possible practical tweak would be to limit this approach to games with nontrivial strategic depth, i.e. no betting on a tic-tac-toe where ties are resolved with a coinflip. This could be handled by a case-by-case observation of the strategy differentials between players of a game as it is commonly played. This leaves the question of how much of a strategy differential would be needed, but that's an easier question than predominance in general. In practice, any game with real-world players who don't all adopt the same strategy should be suitable.
As a final mathematical note, in the formal structure of applying a metric upon skill and chance simultaneously as we discussed earlier, classifying all games with any amount of skill as games of skill is analogous to a lexicographic ordering over the dual inputs of skill and chance. To map games into the axis of relative influence of skill and chance, the lexicographic ordering first sorts all games according to their degree of skill and only sorts by the degree of chance to break exact ties in skill, much as how words are ordered in a dictionary by first sorting by the first letter of the word and only using the second letter to sort among words with common first letters. For example, chess and rando chess are an exact tie in skill, but rando chess has more chance, so chess would be sorted to have a higher relative influence of skill over chance than rando chess. More practically, a poker cash game where players could run it twice (or infinite times) would have a higher relative influence of skill over chance than a cash game without this option.
Closing thoughts
The most important idea to keep in mind when handling any aspect of skill vs. chance or gambling vs. not gambling in poker is that poker is very similar to most other strategy games with random elements such as bridge, Magic: the Gathering, and Scrabble. No logical approach to predominance could classify poker as predominantly chance in a way that doesn't also necessarily classify some other strategy games or variants thereof as predominantly chance. There is nothing fundamental to poker that isn't also fundamental to other competitive strategy games. Poker is unique in its popularity, the stakes for which it is currently played, and its marketing as and historical treatment as gambling, but is not unique in any structural way.
Really, in light of this, it's not fair that the onus is on the poker community to prove that poker is a game of skill. The onus should instead be on those who wish to classify poker as a game of chance and, as we've seen here, it's much more straightforward to find issues in those approaches than it is to blindly stab in the dark with a dozen different and imperfect intuitive arguments as to why poker is a game of skill. In many ways, the entire suggestion of there being a "debate" over skill and chance in poker is a bit of a farce.
One final issue that I have with the undefined notion of predominance is that there is no record or case history (that I'm aware of) of the test ever being actually quantitatively applied to any game. We are living in an era where games in general are proliferating and overlapping new and different parts of our lives. Don't game designers deserve clear guidance as to what they can and can't do with chance in their games without falling into classifications as gambling?
Again, it is worth acknowledging that I have no expertise in law and have significantly simplified the legal definitions of predominance, which vary from state-to-state and across countries. I also have been generous in my assumption that mathematics has any direct bearing on interpreting law, as much of this depends on politics and intuition over logic. It'd be great if we could get a more formal, consistent definition of the legal statutes of predominance, but, in the real world, the PPA is doing a great job of managing these different arguments in making real progress for our game, despite the lack of complete logical consistency of most of these arguments. Nothing I have said here should be seen as undermining the excellent efforts of these parties.
Thanks for reading. I'd be happy to expand upon any of these ideas for any effort which aims to improve public awareness and understanding of poker or for anything which could contribute to legal arguments. Really, though, I'd love to continue the discussion with anyone at all, so if you're interested in talking with me about any aspect of this essay, please contact me!
To bring our long journey to a close, it is worthwhile to condense the wide array of different perspectives we've treated on predominance of skill and chance into one central list. If predominance is a question with no perfect answer, which answer is the best one?
The approaches are presented roughly in order of least-consistent to most-consistent by my assessment, with the tables are broken up by the type of approach to predominance being used:
- Emotional Approaches — These are common intuitively-appealing arguments which consider neither the way the game is played nor the fundamental structure of the game. The only role such arguments might play is in quickly illustrating a rough idea to an audience which is not interested in understanding any deeper details of the game in question.
- Outcome-Based Approaches — A type of approach frequently discussed throughout this essay, these attempt to measure predominance between skill and chance solely by looking from different angles at real-world game outcomes. These approaches usually either don't dig into the nuances of the fundamental structure of the game or ignore them completely. Keep in mind that all outcome-based approaches fail on populations of similarly-skilled players and depend on duration of play and thus have significant consistency problems than would be acceptable from an approach of pure logic. These arguments can be quite illustrative and convincing, moreso than purely emotional approaches, but that they may happen to apply to various real-world forms of poker is a coincidence. Therefore, we should be uncomfortable with making these our primary arguments, but they can be solid supplemental arguments.
- Fundamental Approaches — These are based entirely on the fundamental structure of the game and thus will be consistent over time and over different player populations. Ideally, this is the only type of approach on which long-term legal classifications would be based.
Emotional Approaches
| Approach | Used by | see part # | Issues |
|---|---|---|---|
| Games with historical or social associations to gambling are games of chance. | Almost everyone you've ever met | n/a | This has no bearing on a scientific discussion, but is a deeply-entrenched perspective in most of society which underlies the default framing of poker in this discussion. |
| Games which involve many skills or personal characteristics to succeed at are games of skill. | Many individuals and judges | 10 | Depth of skill in a game is not necessarily related to the number of personal characteristics that inform proper strategy, and chance is not related at all. |
Outcome-Based Approaches
| Approach | Used by | see part # | Issues |
|---|---|---|---|
| Games for which positive-expectation players may not exist in a given population are games of chance. | Those who believe that rake makes poker into gambling | 3, 15 | This is an improper logical conclusion; the existence of positive-expectation players implies skill, but not the converse. |
| Games in which the average player's skill level and play habits will principally control game outcomes are games of skill. | German law | 4, 19 | Highly sensitive to the chosen population of players, thus inconsistent; faces definitional issues of "control". |
| Games whose outcomes resolve themselves using the random elements less than half of the time are games of skill. | Cigital | 14 | Not inherently related to depth of skill in a game. In poker, ignores size of wins and losses, which are crucial elements to real outcomes. |
| Games for which positive-expectation players exist in some population are games of skill. | Levitt & Miles, Fiedler and Rock (CRF) | 3 | Can identify when a game is high-skill, but cannot distinguish between low-skill games and games where all of the observed players are equally skilled, so it only works sometimes. |
Fundamental Approaches
| Approach | Used by | see part # | Issues |
|---|---|---|---|
| Games with lots of overt and salient random components (e.g. cards, dice) are games of chance. | Many individuals | 2, 9 | Poker's overt randomness is not necessarily greater than that of other gamess. Non-overt randomness is a significant source of randomness in most games. |
| Games which appear to be very complex are games of skill. | Many individuals, at least one judge | 9 | Obviously unrelated to chance and only slightly related to skill; that this is true for most non-poker games in our world is purely coincidence. |
| Any game which does not always offer its players equal challenges is a game of chance. | Pennsylvania V Dent, proponents of duplicate poker | 13, 16 | Not related to depth of skill in a game. Strategic and complexity-based uncertainty exist even in games with even challenges. Very few games truly do offer equal challenges, even among games commonly accepted as skill. |
| Games in which a player can deliberately lose are games of skill. | Sklansky | 5 | Non-robust for game modifications where guaranteed-loss options are added, unless we consider avoiding that option a meaningful exercise of skill. |
| Games which are played in a tournament structure are games of skill. | Alfred Denning, Swedish law, many individuals | 7 | Games can have arbitrarily large or small skill or chance components while still adhering to a tournament-based structure. |
| One-player games offered openly by companies must not be games of skill, otherwise skilled players would bankrupt the company. | McCrory, not Judge Weinstein | 11 | This is an intuitive way of separating orthogames from house games, which is important. |
| Any closed, symmetric, strategy game is a game of skill. | Me | 20 | A refinement of the above, this is the best practical solution to a logically-incomplete problem. |
While a fundamental, non-outcome-based approach is warranted for the quantitative question of predominance of factors within a game, we see that most of these fundamental approaches are based on superficial elements of games. With that in mind, in a desire for consistency across different game durations and different player populations, I think that the most appropriate approach to predominance is:
Any closed, multiplayer game with any amount of skill should be classified as a game of skill.
There's simply no other well-defined way to handle games of mixed skill and chance. There can't be anything in the middle of the axis of relative influence of skill and chance without arbitrarily locking in a specific game duration and ignoring the possibility of legally redefining games to be longer matches. Furthermore, as we've seen, the approaches that do implicitly specify a game duration find themselves relying on further assumptions about the type of player to measure, the specific skill/chance metric to use, and what it means to "control" an outcome. There's simply no clear alternative.
The only consistent solution is to push every game in the middle to one end or the other. It is clear that skill is, historically and practically, the more important property.
The approach that I recommend is a simple, clean, readily-applicable cut across the set of all games. In particular, it classifies all conceivable games within the broad family of "poker" the same way, which very few other approaches do.
Moreover, I believe this is consistent with the social intentions of treating games of chance and games of skill separately in the first place. Games of chance offer no personal or social benefits other than the pure entertainment value for participants who choose to play them. This shouldn't be ignored, but games of strategy offer that and much more. Strategy games ("mind sports") are the mental analog of physical sports and thus are valuable to society for the same reasons that physical sports are, perhaps moreso in an era where more and more careers demand mental acuity rather than physical prowess. Strategy games with random components teach valuable and broadly-applicable lessons to a human brain which is very poorly-equipped to handle randomness without training.
Economic concerns about wealth changing hands and desire to regulate activities which might attract problem gamblers are a bit of a red herring as they might relate to predominance, as these can certainly both take place within games of skill. (While this is outside the scope of this essay, nothing I say here should be taken as ignoring the existence of problem gamblers in poker.) It is clear to me that casino games, lotteries, and markets deserve different regulation and treatment than that of strategy games.
One possible practical tweak would be to limit this approach to games with nontrivial strategic depth, i.e. no betting on a tic-tac-toe where ties are resolved with a coinflip. This could be handled by a case-by-case observation of the strategy differentials between players of a game as it is commonly played. This leaves the question of how much of a strategy differential would be needed, but that's an easier question than predominance in general. In practice, any game with real-world players who don't all adopt the same strategy should be suitable.
As a final mathematical note, in the formal structure of applying a metric upon skill and chance simultaneously as we discussed earlier, classifying all games with any amount of skill as games of skill is analogous to a lexicographic ordering over the dual inputs of skill and chance. To map games into the axis of relative influence of skill and chance, the lexicographic ordering first sorts all games according to their degree of skill and only sorts by the degree of chance to break exact ties in skill, much as how words are ordered in a dictionary by first sorting by the first letter of the word and only using the second letter to sort among words with common first letters. For example, chess and rando chess are an exact tie in skill, but rando chess has more chance, so chess would be sorted to have a higher relative influence of skill over chance than rando chess. More practically, a poker cash game where players could run it twice (or infinite times) would have a higher relative influence of skill over chance than a cash game without this option.
Closing thoughts
The most important idea to keep in mind when handling any aspect of skill vs. chance or gambling vs. not gambling in poker is that poker is very similar to most other strategy games with random elements such as bridge, Magic: the Gathering, and Scrabble. No logical approach to predominance could classify poker as predominantly chance in a way that doesn't also necessarily classify some other strategy games or variants thereof as predominantly chance. There is nothing fundamental to poker that isn't also fundamental to other competitive strategy games. Poker is unique in its popularity, the stakes for which it is currently played, and its marketing as and historical treatment as gambling, but is not unique in any structural way.
Really, in light of this, it's not fair that the onus is on the poker community to prove that poker is a game of skill. The onus should instead be on those who wish to classify poker as a game of chance and, as we've seen here, it's much more straightforward to find issues in those approaches than it is to blindly stab in the dark with a dozen different and imperfect intuitive arguments as to why poker is a game of skill. In many ways, the entire suggestion of there being a "debate" over skill and chance in poker is a bit of a farce.
One final issue that I have with the undefined notion of predominance is that there is no record or case history (that I'm aware of) of the test ever being actually quantitatively applied to any game. We are living in an era where games in general are proliferating and overlapping new and different parts of our lives. Don't game designers deserve clear guidance as to what they can and can't do with chance in their games without falling into classifications as gambling?
Again, it is worth acknowledging that I have no expertise in law and have significantly simplified the legal definitions of predominance, which vary from state-to-state and across countries. I also have been generous in my assumption that mathematics has any direct bearing on interpreting law, as much of this depends on politics and intuition over logic. It'd be great if we could get a more formal, consistent definition of the legal statutes of predominance, but, in the real world, the PPA is doing a great job of managing these different arguments in making real progress for our game, despite the lack of complete logical consistency of most of these arguments. Nothing I have said here should be seen as undermining the excellent efforts of these parties.
Thanks for reading. I'd be happy to expand upon any of these ideas for any effort which aims to improve public awareness and understanding of poker or for anything which could contribute to legal arguments. Really, though, I'd love to continue the discussion with anyone at all, so if you're interested in talking with me about any aspect of this essay, please contact me!
Friday, September 28, 2012
20 thoughts on skill vs. chance in poker, part 19: The German "predominantly chance" study
<-- Part 18: The relationship between level of stakes and degree of chance
I had always imagined that one of the points I'd write about in this essay would be how no academic studies have ever concluded that poker is predominantly chance, but the recently-published paper out of Germany, Is Poker a Game of Skill or Chance? A Quasi-Experimental Study, changed that1.
The paper is not made public freely anywhere, but the lead author is willing to share it with interested parties who contact him directly and has been polite and receptive to constructive feedback. Additionally, some other summary of and commentary on the paper has been published openly:
First, a brief overview of the study's methods. 300 real poker players were recruited and, through self-identification, were sorted into two groups, "experts" and "average". Each participant played 60 hands of 6-handed Holdem, split between No Limit and Limit, against other participants via computer in a duplicate poker format where the deal of each hand was rigged. The rigging was done in such a way as to attempt to measure the effects of chance by giving better-than-average, average, and worse-than-average cards to each of the players in a predetermined way. The participants were not aware of any rigging of the deck and believed that they were playing a typical poker game. After play was complete, the winrates of both the expert and average players across each of the three rigged card conditions were computed. The two core findings were:
Before we delve into the flaws in these conclusions and in the methodology and assumptions that lead to them, it's worth noting that the seemingly very-small sample size of 60 hands per player is not really the problem here. Many poker players are inclined to laugh the study away at the first mention of this sample size, but the authors are actually quite self-aware and acknowledging of the shortcomings and limitations of their study. The 60-hand cutoff was likely important for practically gathering the needed data with real human volunteers.
However, the authors also argued in favor of the 60-hand timescale due to their interpretation of German law, which attempts to measure whether or not game outcomes depend
From the paper:
Regardless, even if the sample size is low, I don't think that's the problem with this study. I would expect the results to be the same even with a large sample size; the issue is with the rest of the methodology and the biases it introduces. The two core findings each result from problems in the approach taken.
Finding #1
The authors concluded that chance predominates skill in poker because being dealt into a seat rigged to get dealt the winning hand more often than usual increases one's winrate by more than the difference in winrates between strong and weak players. This suggests that the interpretation of predominance being used is one in which any game in which there is some nonzero probability of random game elements causing a player to be unable to win despite their skill is a game of predominantly chance.
This is a theoretically-interesting definition of predominance, but it would classify almost all games with random elements as games predominated by chance as long as the deck (or other source of randomness) is fixed to make the probabilities extreme enough. The average player would almost surely win in a backgammon game where his rolls were much more likely to come up 6-6 and the expert’s were much more likely to come up 1-2. The average player would almost surely win in a Magic: the Gathering game (or any other card game) where the expert’s deck was rigged to deliver a very skewed, unplayable mix of card types. An average player would likely win a Scrabble game where the probabilities were altered so that his expert opponent received very few consonants.
To properly take this sort of perspective on predominance, a quantitative refinement is needed. If you go far enough into the tails of the random distributions, any game of skill with random components would be concluded as a game of chance, so the real question here is what p-value of randomness is necessary for the average player to overcome the expert's skill advantage. If the average player beats the expert poker player with the aid of only 51st-percentile random in-game outcomes in his favor, then I think that would be a reasonably intuitively convincing argument as to a game being predominantly chance. However, if the rigging has to push the favor of the cards into the 99th percentile for the average player to beat an expert, that doesn’t really show anything. I expect that no expert at any game would beat a weaker player over 99% of the time.
In my correspondence with the lead author, he acknowledged that this would be a challenging target for future work, but found it to be unnecessary for the scope of this study since, regardless of the p-value of the extreme randomness given in the rigging condition, each player received this rigging an equal number of times among the 60 hands. However, the degree of good fortune given in this chance-shifting condition will certainly affect the conclusion. If less-extreme randomness were given in the "better-than-average cards" condition of this study, i.e. if it were a "only-very-slightly-better-than-average cards" condition, then the skill edge of the experts would dominate. The specific nature of the rigging seems to suggest a rather extreme perturbation of the randomness in poker.
Finding #2
The study found that weaker players outperformed expert players in the "better-than-average cards" condition, particularly in Limit Holdem. This should be a direct consequence of experts making proper poker folds that, unknowingly, turn out to be really bad folds in rigged poker when you're artificially more likely to win the hand with whatever cards you happen to be holding. (In case it's not obvious, you shouldn't fold very often in that game.)
The particular nature of the rigging, while still not quite clear to me, favors the naïve tendencies of the novice player. As described in the study:
This means that the typical amateur mistake of continuing with what was once a strong starting hand after a bad flop or turn will go on to be rewarded. For example, an expert may prudently fold 7♦7♣ on a Q♣K♥2♥ flop, or A♦K♦ on a Q♣J♣6♦9♣ turn. These may be correct moves in poker, but are pretty terrible moves in rigged poker where the game has controlled for the fact that you're going to spike your card on the river or that none of your opponents will make their draws. Meanwhile, the average players will incorrectly chase their draws and be rewarded on the river much more often than regular poker probabilities would dictate. This effect should be exacerbated in Limit Holdem, where the stronger players will find the right folds on flops and turns with overcards despite high pot odds.
The study does take note of the fact that average players call more often than experts and that experts fold more. The study does acknowledge that this bias may exist and may impact results:
The only way I can see around introducing some sort of bias is to not rig the deck at all, which would dictate an approach that doesn't gather its own data and instead uses a large real-world database of hands provided by a commercial internet poker site, as some other studies have done. These real hands could be filtered to find which hands involved "good hands" by whatever metric was desirable, and this would prevent manipulated probabilities from favoring one player type over another. The question of what metric to use would still be difficult.
The author defended this part of the methodology, again believing that it was fair because both expert and average players had the same conditions. The fundamental issue here, though, is that changing the probabilities of the random elements in poker changes the game to something other than poker.
The subjects were essentially lied to (not maliciously) in that they were not playing the game they thought they were. The strategies for rigged poker are different than the strategies for poker, and if the subjects had known about the methodology and when they were in the rigged conditions, then the expert poker players probably would have properly picked up on the strategic adjustments and continued to outperform the weaker players.
Also, I expect that expert poker players would be more willing to trust that an ostensibly normal poker game in an experimental setting is being run fairly. In contrast, weaker players may be guided by instincts to "play a rush" or to otherwise irrationally manipulate their assessment of what are supposed to be independent probabilities, which could benefit them in this rigged poker game. Regardless, that the subjects are misled as to the probabilities of the game outcomes means that the impact of proper strategy will be obfuscated, as the skilled players are trying to apply skills from a different game.
Overall, the authors approach the task of measuring predominance in poker from a reasonably sophisticated scientific perspective with no evidence of anything but an earnest effort. If one was tasked with attempting to produce a formal, science-based argument that poker is predominantly chance, these authors have done so fairly well. Perhaps a lack of practical poker experience led them to overlook or underestimate the impact of these methodological biases on the results. I don't think they deserve the ire of our community, but I also don't think there is any meaningful validity to their conclusions.
While I don't believe that this was the motivation of the study, it would be wrong to omit the observation that German poker players owe personal income tax on their poker winnings only if poker is considered to be a game of skill, but owe nothing if poker is considered to be gambling. This sort of tax rule could certainly shape a cultural and social willingness among poker players in Germany to want to keep poker treated as gambling — an amusing (or depressing) contrast with the interests of players in the U.S. to have poker seen as a game of skill.
Still, a study in an academic journal has global impact. Even if German poker players would be better off if poker were treated as predominantly chance and gambling, this is not the case in most of the rest of the world, and it's also a classification that I feel to be intellectually dishonest and fundamentally wrong.
Part 20: Summary and assessment of approaches to predominance -->
(back to index)
1In fact, the lead author of the German paper brought my attention to two of its cited papers which also contend that poker is predominantly chance, at least under some conditions:
I had always imagined that one of the points I'd write about in this essay would be how no academic studies have ever concluded that poker is predominantly chance, but the recently-published paper out of Germany, Is Poker a Game of Skill or Chance? A Quasi-Experimental Study, changed that1.
The paper is not made public freely anywhere, but the lead author is willing to share it with interested parties who contact him directly and has been polite and receptive to constructive feedback. Additionally, some other summary of and commentary on the paper has been published openly:
- Neuroskeptic was the first to break the story to the poker community with a short but mostly-complete summary of the paper.
- Poker's own Short-Stacked Shamus at Hard-Boiled Poker wrote about his impressions from his read.
- Jennifer Ouellette of the Scientific American blog network took a thorough look at this paper, the DiCristina ruling, and a variety of other perspectives on poker from a scientific standpoint. I found this to be a solid read, and it included a great quote for a mainstream article:
“Good poker requires that you make sound game-theoretic decisions but there is still plenty of freedom to try and outsmart your opponents,” [Vonk] said. “Other casino games miss that second element. All you can do in blackjack or roulette is make the best possible mathematical decisions, and even then, you will still lose in the long run. I have never been attracted to those games. It’s the fact that you play against other people that makes poker so interesting, and that makes it possible to actually be a winner at the game.”
- In Is skill in poker – and elsewhere – just one great big bluff?, Tom Chivers of The Telegraph uses the study as a basis for extrapolating some broader ideas from behavioral economics, though I find his take on the study to be a bit shallow and tangential... and the headline to be pretty close to unforgivably sensationalist, inaccurate, and unrelated to the content.
First, a brief overview of the study's methods. 300 real poker players were recruited and, through self-identification, were sorted into two groups, "experts" and "average". Each participant played 60 hands of 6-handed Holdem, split between No Limit and Limit, against other participants via computer in a duplicate poker format where the deal of each hand was rigged. The rigging was done in such a way as to attempt to measure the effects of chance by giving better-than-average, average, and worse-than-average cards to each of the players in a predetermined way. The participants were not aware of any rigging of the deck and believed that they were playing a typical poker game. After play was complete, the winrates of both the expert and average players across each of the three rigged card conditions were computed. The two core findings were:
- The rigged-card conditions for receiving better cards had more of an impact on winnings than the skill of the players, therefore "chance clearly dominates skill; thus, poker should be classified as gambling".
- While experts outperformed average players overall and were able to lose less money with worse-than-average cards, it turns out that, in the "better-than-average cards" rigging condition, average players outperformed expert players, as illustrated in the below graph. The authors take this to support the conclusion that the cards are what primarily affect outcomes and that players' strategies are much less impactful.
Before we delve into the flaws in these conclusions and in the methodology and assumptions that lead to them, it's worth noting that the seemingly very-small sample size of 60 hands per player is not really the problem here. Many poker players are inclined to laugh the study away at the first mention of this sample size, but the authors are actually quite self-aware and acknowledging of the shortcomings and limitations of their study. The 60-hand cutoff was likely important for practically gathering the needed data with real human volunteers.
However, the authors also argued in favor of the 60-hand timescale due to their interpretation of German law, which attempts to measure whether or not game outcomes depend
"solely or principally on chance rather than on the players' abilities... under [the conditions] which the game is typically initiated and played, which depends on the skills and experience of the average player... an individual who is generally interested in playing the game, has learned the fundamental rules and has had some practice playing."
From the paper:
On the one hand, it could plausibly be argued that the influence of strategy and skill would be more prominent in longer poker sessions and would entail a stronger impact on the game’s outcome. On the other hand, it could be assumed that with longer play periods, the difference in players’ level of skill would decrease. This would lead to a greater contribution of chance to the outcome and a need for new, inexperienced players to reduce the effect of chance.I think this is a dubious, hand-waving argument to justify such a short time horizon. I'm not sure how I'd estimate how many hands of play in a poker game are necessary for an average or amateur player to start to closely-approximate the skill level of an expert, but I'm sure it's way, way more than a few hundred.
Regardless, even if the sample size is low, I don't think that's the problem with this study. I would expect the results to be the same even with a large sample size; the issue is with the rest of the methodology and the biases it introduces. The two core findings each result from problems in the approach taken.
Finding #1
The authors concluded that chance predominates skill in poker because being dealt into a seat rigged to get dealt the winning hand more often than usual increases one's winrate by more than the difference in winrates between strong and weak players. This suggests that the interpretation of predominance being used is one in which any game in which there is some nonzero probability of random game elements causing a player to be unable to win despite their skill is a game of predominantly chance.
This is a theoretically-interesting definition of predominance, but it would classify almost all games with random elements as games predominated by chance as long as the deck (or other source of randomness) is fixed to make the probabilities extreme enough. The average player would almost surely win in a backgammon game where his rolls were much more likely to come up 6-6 and the expert’s were much more likely to come up 1-2. The average player would almost surely win in a Magic: the Gathering game (or any other card game) where the expert’s deck was rigged to deliver a very skewed, unplayable mix of card types. An average player would likely win a Scrabble game where the probabilities were altered so that his expert opponent received very few consonants.
To properly take this sort of perspective on predominance, a quantitative refinement is needed. If you go far enough into the tails of the random distributions, any game of skill with random components would be concluded as a game of chance, so the real question here is what p-value of randomness is necessary for the average player to overcome the expert's skill advantage. If the average player beats the expert poker player with the aid of only 51st-percentile random in-game outcomes in his favor, then I think that would be a reasonably intuitively convincing argument as to a game being predominantly chance. However, if the rigging has to push the favor of the cards into the 99th percentile for the average player to beat an expert, that doesn’t really show anything. I expect that no expert at any game would beat a weaker player over 99% of the time.
In my correspondence with the lead author, he acknowledged that this would be a challenging target for future work, but found it to be unnecessary for the scope of this study since, regardless of the p-value of the extreme randomness given in the rigging condition, each player received this rigging an equal number of times among the 60 hands. However, the degree of good fortune given in this chance-shifting condition will certainly affect the conclusion. If less-extreme randomness were given in the "better-than-average cards" condition of this study, i.e. if it were a "only-very-slightly-better-than-average cards" condition, then the skill edge of the experts would dominate. The specific nature of the rigging seems to suggest a rather extreme perturbation of the randomness in poker.
Finding #2
The study found that weaker players outperformed expert players in the "better-than-average cards" condition, particularly in Limit Holdem. This should be a direct consequence of experts making proper poker folds that, unknowingly, turn out to be really bad folds in rigged poker when you're artificially more likely to win the hand with whatever cards you happen to be holding. (In case it's not obvious, you shouldn't fold very often in that game.)
The particular nature of the rigging, while still not quite clear to me, favors the naïve tendencies of the novice player. As described in the study:
During the game, one expert player and one average player received (a) the winning hand 15 times and the losing hand 5 times (winner’s box condition), (b) the winning hand 10 times and the losing hand 10 times (neutral box condition) and (c) the winning hand 5 times and the losing hand 15 times (loser’s box condition)and
Using this computer-based method of playing, the hands of individual players and the flop, turn and river cards were manipulated to produce a standardized ranking order for each hand in terms of the probabilities of winning (cf. the standardized sequence of play of "duplicate poker"). It was established in advance that the cards of the opening hands, and the associated distribution of chances of winning, were reflected in the river in the same order for the first three places. In contrast, places were allowed to vary with respect to the flop and turn.So the rigging is done in a way which controls whether or not the player's hand goes on to be the best hand among all six players by the time the river is dealt, and this is further controlled so that the best preflop hands end up being the best river hands, even though the flops or turns could be unfavorable.
This means that the typical amateur mistake of continuing with what was once a strong starting hand after a bad flop or turn will go on to be rewarded. For example, an expert may prudently fold 7♦7♣ on a Q♣K♥2♥ flop, or A♦K♦ on a Q♣J♣6♦9♣ turn. These may be correct moves in poker, but are pretty terrible moves in rigged poker where the game has controlled for the fact that you're going to spike your card on the river or that none of your opponents will make their draws. Meanwhile, the average players will incorrectly chase their draws and be rewarded on the river much more often than regular poker probabilities would dictate. This effect should be exacerbated in Limit Holdem, where the stronger players will find the right folds on flops and turns with overcards despite high pot odds.
The study does take note of the fact that average players call more often than experts and that experts fold more. The study does acknowledge that this bias may exist and may impact results:
It is unclear whether the described advantage of average players with good cards under "fixed limit" conditions, due to their less purposeful [meaning continuing too often with weaker hands] style of play, is an artifact of the applied design or a phenomenon that can also be detected in the reality of poker play.I expect that this effect and the bias it introduces is indeed highly significant, enough so to fully explain the outperformance of average players in the "better-than-average cards" condition.
The only way I can see around introducing some sort of bias is to not rig the deck at all, which would dictate an approach that doesn't gather its own data and instead uses a large real-world database of hands provided by a commercial internet poker site, as some other studies have done. These real hands could be filtered to find which hands involved "good hands" by whatever metric was desirable, and this would prevent manipulated probabilities from favoring one player type over another. The question of what metric to use would still be difficult.
The author defended this part of the methodology, again believing that it was fair because both expert and average players had the same conditions. The fundamental issue here, though, is that changing the probabilities of the random elements in poker changes the game to something other than poker.
The subjects were essentially lied to (not maliciously) in that they were not playing the game they thought they were. The strategies for rigged poker are different than the strategies for poker, and if the subjects had known about the methodology and when they were in the rigged conditions, then the expert poker players probably would have properly picked up on the strategic adjustments and continued to outperform the weaker players.
Also, I expect that expert poker players would be more willing to trust that an ostensibly normal poker game in an experimental setting is being run fairly. In contrast, weaker players may be guided by instincts to "play a rush" or to otherwise irrationally manipulate their assessment of what are supposed to be independent probabilities, which could benefit them in this rigged poker game. Regardless, that the subjects are misled as to the probabilities of the game outcomes means that the impact of proper strategy will be obfuscated, as the skilled players are trying to apply skills from a different game.
Overall, the authors approach the task of measuring predominance in poker from a reasonably sophisticated scientific perspective with no evidence of anything but an earnest effort. If one was tasked with attempting to produce a formal, science-based argument that poker is predominantly chance, these authors have done so fairly well. Perhaps a lack of practical poker experience led them to overlook or underestimate the impact of these methodological biases on the results. I don't think they deserve the ire of our community, but I also don't think there is any meaningful validity to their conclusions.
While I don't believe that this was the motivation of the study, it would be wrong to omit the observation that German poker players owe personal income tax on their poker winnings only if poker is considered to be a game of skill, but owe nothing if poker is considered to be gambling. This sort of tax rule could certainly shape a cultural and social willingness among poker players in Germany to want to keep poker treated as gambling — an amusing (or depressing) contrast with the interests of players in the U.S. to have poker seen as a game of skill.
Still, a study in an academic journal has global impact. Even if German poker players would be better off if poker were treated as predominantly chance and gambling, this is not the case in most of the rest of the world, and it's also a classification that I feel to be intellectually dishonest and fundamentally wrong.
Part 20: Summary and assessment of approaches to predominance -->
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1In fact, the lead author of the German paper brought my attention to two of its cited papers which also contend that poker is predominantly chance, at least under some conditions:
- Best Hand Wins: How Poker is Governed by Chance by Vincent Berthet — This study references the Cigital study but takes the further step of considering that the cards dealt to players affect their actions, and hence that a hand that does not end in showdown is not necessarily a hand where chance played no role at all. This study finds that 72.8% of hands which end in no showdown were nonetheless won by the player who held the best hand at the point when the hand ended. This study comes off as fairly informed and aware of poker and is worth a quick read. It's a reasonable different perspective than the Cigital study of what it means for cards to "control" an outcome, but the logical misstep in concluding that this means that poker is predominantly chance comes from assuming that the threshold of predominance occurs halfway between 1 and 1/N, where 1 is the probability of the best cards winning if poker were determined entirely by the deal of the cards and where 1/N is the probability of the best cards winning if poker were determined entirely by skill (i.e. if the cards don't matter at all). There is no quantitative basis for assuming that the threshold would be at the midpoint, especially when player actions and strategies do directly impact this statistic. One could easily design different games which have varied levels of this best-hand-winning rate statistic which have high skill, low skill, high chance, or low chance. Since this approach admittedly focuses on winners of pots rather than winners of money and thus ignores betting, all of my observations of the shortcomings of showdown-based studies would apply here.
- The work of Ingo Fiedler — The work by Fiedler and Rock is commonly used as support of the predominance of skill in poker, as their statistically-minded critical repetition frequency shows that, by their measure, skill overtakes chance in poker after about 1,000 hands. Meyer and this German study, however, note that a 2011 paper by Fiedler, unrelated to the topic of skill and chance in poker, finds that the median online poker account in a large database played less than 1 hour/month of poker, causing these players to not quite reach the CRF threshold within a year. Considering the median online poker account is pretty close to considering the median human who has ever played poker even once and will almost certainly include far too many former or very infrequent players. Again, this might be a reasonable interpretation of these particular German statutes, but I think it's unrealistic for other uses.
Thursday, September 27, 2012
20 thoughts on skill vs. chance in poker, part 18: The relationship between level of stakes and degree of chance
<-- Part 17: Why does poker need randomness?
In addition to the practical reasons that poker needs randomness as discussed in the previous thought, many players will say that another reason that poker "needs" randomness is for various reasons of attracting people to the game. This isn't quite a need, but it's certainly a desired property, but not only for the obvious reason of allowing weak players to overrate their skills.
To a skilled poker player, yes, the most blatant upside of poker having randomness is that it obfuscates the relative skill of the players. If a player were hoping to ascertain his skill ranking within a group of players solely by observing game results, a lot of volume would have to be put in before the statistics confidently revealed who the best players were. The winning players will make money over this period and, especially at the lower levels, one's opponents are also otherwise likely to misevaluate their own skills due to a long list of foibles of the human mind. Meanwhile, serious players who study game strategy away from the table are more readily able to quickly assess the skill ranking within a group of players by observing the mistakes that the players make, rather than merely waiting for the results to converge over time.
The more subtle marketing advantage of poker's randomness is that it leads towards bigger prize pools, which helps attract mainstream attention, make television coverage more exciting, and creates more higher-utility interactions between serious players and casual players. The core idea here is as follows: assuming that the required expected value for a winning player were held constant, the higher the influence of chance in a game, the higher the stakes must be for the flow of money from weaker to stronger players to be of this desired size.
For example, if poker had less chance, then, for any type of poker tournament, the buyin need not be as high as $10,000 or $1,000,000 to sufficiently incentivize the best players in the world to compete. If poker tournaments were usually won by the best player in the field, then the $10,000 World Series of Poker Main Event would have a seven-digit expected value and an absurdly high ROI for the world's best player. Games such as Scrabble and Magic: the Gathering, where the influence of chance over skill is usually smaller (over the timescale of, say, a weekend) are typically played with prize pools several orders of magnitude smaller than those that are common in poker, which perhaps contributes to them being less interesting for onlookers and curious would-be players. If poker had less chance in its outcomes, then the pull of the economic equilibrium would reduce the buyin size of poker tournaments until the best players' expected values were closer to what they are in the current poker market. This is a concern not only for the incentivizing of players, but also in allowing for a stable poker economy; even if all players had rational assessments of their abilities, a faster expected flow of money between weaker and stronger players would cut through the entertainment budgets of the casual players more quickly.
This isn't to say that exciting, big-money poker events need to be marketed alongside gambling as they so often are. The fact that the poker which reaches the television airwaves and mainstream consciousness is so often taking place in casinos and cut together to highlight the emotions and swings of the players is, in my opinion, a significant detrimental factor in cementing the ignorant onlooker's misperceptions of poker as homogeneous with traditional gambling. Poker could be presented to the public with a more toned-down, analytical, competitive framing. The big-dollar appeal would still attract interest even if the game were not conflated with gambling. A Scrabble tournament with a million-dollar first prize would get plenty of attention. For better or for worse, people are compelled to follow games which award big prizes.
All that aside, an academic approach to analyzing poker ends up largely ignoring these potentially financially-lucrative facts about poker's appeal and ability to produce consistent winnings for its best players. From the broader perspective of game design, as noted by Richard Garfield in his lecture, an important advantage of using chance in games is that it allows a wider set of players to enjoy playing the game with each other. Deterministic games such as chess are rarely fun when played between people of vastly different abilities, but poker can be entertaining to all parties involved even when their skills greatly vary. This isn't because the weaker players are necessarily interested in gambling or that they necessarily enjoy games of chance in general. It's because a game with random elements will usually be more dynamic and interesting in its gameplay than a deterministic game. Not only might the weaker player win sometimes, but, even when they lose, they won't always be losing in the same way each time they play.
Even if the money were entirely taken out of poker, it would still be an excellent game for social gameplay. That any fundamentally mathematics-based game could appeal to as broad an audience as poker does today is a testament to the compelling diversity of play that its randomness generates. Poker manages to be at just about the sweet spot of balance between skill and chance, both for casual play and for competitive play, and that should not be taken for granted. Taking the chance out of poker, if it were even possible to do so, might be hurt the game.
Part 19: The German "predominantly chance" study -->
(back to index)
In addition to the practical reasons that poker needs randomness as discussed in the previous thought, many players will say that another reason that poker "needs" randomness is for various reasons of attracting people to the game. This isn't quite a need, but it's certainly a desired property, but not only for the obvious reason of allowing weak players to overrate their skills.
To a skilled poker player, yes, the most blatant upside of poker having randomness is that it obfuscates the relative skill of the players. If a player were hoping to ascertain his skill ranking within a group of players solely by observing game results, a lot of volume would have to be put in before the statistics confidently revealed who the best players were. The winning players will make money over this period and, especially at the lower levels, one's opponents are also otherwise likely to misevaluate their own skills due to a long list of foibles of the human mind. Meanwhile, serious players who study game strategy away from the table are more readily able to quickly assess the skill ranking within a group of players by observing the mistakes that the players make, rather than merely waiting for the results to converge over time.
The more subtle marketing advantage of poker's randomness is that it leads towards bigger prize pools, which helps attract mainstream attention, make television coverage more exciting, and creates more higher-utility interactions between serious players and casual players. The core idea here is as follows: assuming that the required expected value for a winning player were held constant, the higher the influence of chance in a game, the higher the stakes must be for the flow of money from weaker to stronger players to be of this desired size.
For example, if poker had less chance, then, for any type of poker tournament, the buyin need not be as high as $10,000 or $1,000,000 to sufficiently incentivize the best players in the world to compete. If poker tournaments were usually won by the best player in the field, then the $10,000 World Series of Poker Main Event would have a seven-digit expected value and an absurdly high ROI for the world's best player. Games such as Scrabble and Magic: the Gathering, where the influence of chance over skill is usually smaller (over the timescale of, say, a weekend) are typically played with prize pools several orders of magnitude smaller than those that are common in poker, which perhaps contributes to them being less interesting for onlookers and curious would-be players. If poker had less chance in its outcomes, then the pull of the economic equilibrium would reduce the buyin size of poker tournaments until the best players' expected values were closer to what they are in the current poker market. This is a concern not only for the incentivizing of players, but also in allowing for a stable poker economy; even if all players had rational assessments of their abilities, a faster expected flow of money between weaker and stronger players would cut through the entertainment budgets of the casual players more quickly.
This isn't to say that exciting, big-money poker events need to be marketed alongside gambling as they so often are. The fact that the poker which reaches the television airwaves and mainstream consciousness is so often taking place in casinos and cut together to highlight the emotions and swings of the players is, in my opinion, a significant detrimental factor in cementing the ignorant onlooker's misperceptions of poker as homogeneous with traditional gambling. Poker could be presented to the public with a more toned-down, analytical, competitive framing. The big-dollar appeal would still attract interest even if the game were not conflated with gambling. A Scrabble tournament with a million-dollar first prize would get plenty of attention. For better or for worse, people are compelled to follow games which award big prizes.
All that aside, an academic approach to analyzing poker ends up largely ignoring these potentially financially-lucrative facts about poker's appeal and ability to produce consistent winnings for its best players. From the broader perspective of game design, as noted by Richard Garfield in his lecture, an important advantage of using chance in games is that it allows a wider set of players to enjoy playing the game with each other. Deterministic games such as chess are rarely fun when played between people of vastly different abilities, but poker can be entertaining to all parties involved even when their skills greatly vary. This isn't because the weaker players are necessarily interested in gambling or that they necessarily enjoy games of chance in general. It's because a game with random elements will usually be more dynamic and interesting in its gameplay than a deterministic game. Not only might the weaker player win sometimes, but, even when they lose, they won't always be losing in the same way each time they play.
Even if the money were entirely taken out of poker, it would still be an excellent game for social gameplay. That any fundamentally mathematics-based game could appeal to as broad an audience as poker does today is a testament to the compelling diversity of play that its randomness generates. Poker manages to be at just about the sweet spot of balance between skill and chance, both for casual play and for competitive play, and that should not be taken for granted. Taking the chance out of poker, if it were even possible to do so, might be hurt the game.
Part 19: The German "predominantly chance" study -->
(back to index)
Wednesday, September 26, 2012
20 thoughts on skill vs. chance in poker, part 17: Why does poker need randomness?
<-- Part 16: Does duplicate poker have higher skill relative to chance?
Some competitive games other than poker have randomness that would ideally be removed and could be removed without changing the strategy and appeal of the game. The toy game example of rando chess, traditional chess with a random component tacked onto the end, is probably a worse game than traditional chess for both casual and competitive play; if rando chess were the game that was invented and popularized first, it would likely be replaced by traditional chess rather quickly. Similarly, physical sports affected by factors such as wind and weather might end up being better, more engaging games in a world where these games were not influenced by these external factors and where raw individual physical prowess and game strategy were the only factors in the outcomes.
Poker, however, could not practically remain the same game if the random deal of the cards were removed.
Consider the number of different situations that could take place in a game of poker. This number accounts for every possible decision point for every combination of holecards, boardcards, and betting lines. Even for as simple of a game as Heads-Up Limit Holdem, this number looks to be about 290 quadrillion — I'm not going to double-check this, but the point is, it's big. Let's design a way to play Heads-Up Limit Holdem without its intrinsic random components and call it deterministic poker. Playing deterministic poker would involve each player sitting down and specifying the decisions they would make in each of these 290 quadrillion game situations. After the players (and their ancestors) finished writing out this full strategy specification, a supercomputer could compare the players' strategies against each other and decide the winner by playing through each of these situations. Alternatively, each of the 290 quadrillion game situations could be played through in sequence by the players. Neither of these are particularly efficient, but, with a long enough amount of time and infinite human longevity, the game of poker could indeed be played fully deterministically. There's certainly no intrinsic randomness in deterministic poker, and it would never be treated as gambling.
Instead, in the real world, the game of poker ends up being essentially the same game as deterministic poker (with the added bonus of being realistic to play through) by having its players play through a randomly-chosen subset of these 290 quadrillion game situations for whatever period of time they find comfortable. Poker and deterministic poker are equal in expectation for the players and admit the same strategies and skills. Just as rando chess and chess have the same strategies and would eventually produce the same ranking of their players, poker and deterministic poker are, in this sense, the same game. The same is true for other strategy games with intrinsic random components, such as bridge, backgammon, Scrabble, and Magic: the Gathering. The game trees for these games are too large to play through in one sitting or in one lifetime, but, by randomly shuffling between these game situations, the depth of strategic diversity of these games can be obtained and enjoyed in a reasonable amount of time.
Thus is the miracle of randomization, enabling an activity that could not exist without it, and yet most of our society fears this type of uncertainty. In poker, since the variances are always finite, risk management principles can allow a player to control and ultimately ignore the effects of the intrinsic random components of the game. From this perspective, the presence of the intrinsic random components in real poker does not make the game meaningfully different from deterministic poker.
This is in stark contrast to other games which are commonly treated as gambling. One-player casino games without randomness would just be machines or tables that a customer could walk up to and, after specifying a full strategy for any potential strategic inputs, immediately be handed some smaller portion of his or her bet back. Casinos would be much less popular in this world.
Poker without its intrinsic random components would still be a strategically-rich and compelling game.
Games of chance without their intrinsic random components would be nothing.
Part 18: The relationship between level of stakes and degree of chance -->
(back to index)
Some competitive games other than poker have randomness that would ideally be removed and could be removed without changing the strategy and appeal of the game. The toy game example of rando chess, traditional chess with a random component tacked onto the end, is probably a worse game than traditional chess for both casual and competitive play; if rando chess were the game that was invented and popularized first, it would likely be replaced by traditional chess rather quickly. Similarly, physical sports affected by factors such as wind and weather might end up being better, more engaging games in a world where these games were not influenced by these external factors and where raw individual physical prowess and game strategy were the only factors in the outcomes.
Poker, however, could not practically remain the same game if the random deal of the cards were removed.
Consider the number of different situations that could take place in a game of poker. This number accounts for every possible decision point for every combination of holecards, boardcards, and betting lines. Even for as simple of a game as Heads-Up Limit Holdem, this number looks to be about 290 quadrillion — I'm not going to double-check this, but the point is, it's big. Let's design a way to play Heads-Up Limit Holdem without its intrinsic random components and call it deterministic poker. Playing deterministic poker would involve each player sitting down and specifying the decisions they would make in each of these 290 quadrillion game situations. After the players (and their ancestors) finished writing out this full strategy specification, a supercomputer could compare the players' strategies against each other and decide the winner by playing through each of these situations. Alternatively, each of the 290 quadrillion game situations could be played through in sequence by the players. Neither of these are particularly efficient, but, with a long enough amount of time and infinite human longevity, the game of poker could indeed be played fully deterministically. There's certainly no intrinsic randomness in deterministic poker, and it would never be treated as gambling.
Instead, in the real world, the game of poker ends up being essentially the same game as deterministic poker (with the added bonus of being realistic to play through) by having its players play through a randomly-chosen subset of these 290 quadrillion game situations for whatever period of time they find comfortable. Poker and deterministic poker are equal in expectation for the players and admit the same strategies and skills. Just as rando chess and chess have the same strategies and would eventually produce the same ranking of their players, poker and deterministic poker are, in this sense, the same game. The same is true for other strategy games with intrinsic random components, such as bridge, backgammon, Scrabble, and Magic: the Gathering. The game trees for these games are too large to play through in one sitting or in one lifetime, but, by randomly shuffling between these game situations, the depth of strategic diversity of these games can be obtained and enjoyed in a reasonable amount of time.
Thus is the miracle of randomization, enabling an activity that could not exist without it, and yet most of our society fears this type of uncertainty. In poker, since the variances are always finite, risk management principles can allow a player to control and ultimately ignore the effects of the intrinsic random components of the game. From this perspective, the presence of the intrinsic random components in real poker does not make the game meaningfully different from deterministic poker.
This is in stark contrast to other games which are commonly treated as gambling. One-player casino games without randomness would just be machines or tables that a customer could walk up to and, after specifying a full strategy for any potential strategic inputs, immediately be handed some smaller portion of his or her bet back. Casinos would be much less popular in this world.
Poker without its intrinsic random components would still be a strategically-rich and compelling game.
Games of chance without their intrinsic random components would be nothing.
Part 18: The relationship between level of stakes and degree of chance -->
(back to index)
Tuesday, September 25, 2012
20 thoughts on skill vs. chance in poker, part 16: Does duplicate poker have higher skill relative to chance?
<-- Part 15: Does rake matter?
Duplicate poker is a poker variant which attempts to reduce the impact on game outcomes of the intrinsic randomness from the shuffle of the cards. The general principle is that two poker games take place on two separate tables, with the order of the shuffled cards controlled to be the same at both tables for each hand. The payoffs to each player, rather than being the standard number of chips won or lost in that hand, are based on a function of the difference between the net chip performance between that player and his or her duplicate at the duplicated table. In this sense, some of the salient chance features are mitigated by playing duplicate poker instead of regular poker. A player dealt a good starting hand isn't automatically being given a profitable opportunity, as he must outperform a duplicate player who has also been dealt the same hand. In the sense that regular poker "is skill in the long run" once the randomness of the cards has evened out, duplicate poker attempts to speed up this timescale by keeping the hands even throughout any interval of play. It's ok that "no amount of skill can change a deuce into an ace", as this will impact both of the paired players.
While duplicate poker has never gathered much popularity, its cause has been taken up by various companies over the years. The eponymous DuplicatePoker launched the first real-money online duplicate poker room several years ago, closing its doors and disappearing in 2008 without much public statement. More recently, SkillBet has launched a variant of duplicate poker in which there is only one human player at each table with the rest of the players being bots, which removes the possibility of particularly-exploitive collusion between conspiring players sitting at different seats that duplicate poker would provide. Perhaps most relevant is that the International Federation of Poker, an organization seeking to spread awareness of poker as a mind sport, has adopted a duplicate poker format for a major tournament it held last year in London.
In all three of these cases, the parties involved have marketed their ideas using claims that duplicate poker "has more skill" than regular poker, or, in some cases, asserting that it "is all skill" and has no chance elements at all. There's a strong intuitive appeal here, but are these claims accurate?
At this point in the essay, we quickly recognize the idea that duplicate poker has more skill than regular poker to be nonsense. Removing chance from a game is not the same as adding skill to a game. The depth of strategy in most duplicate poker games should be roughly the same as regular poker. A duplicate poker cash game with a typical payoff function on the differential between player performances would have the nice property of the correct EV-maximizing strategies being the same as they would be for a regular poker cash game, as every incremental chip won or lost contributes equally to the players' payoffs. Duplicate poker tournaments, however, will necessarily involve different considerations for any payoff function due to the nonlinearity of chip values, which will lead to different strategies than both cash game poker and traditional tournament poker. The scoring system for the IFP's London tournament and for SkillBet's tournament games, essentially rewarding only the players who get the highest scores at their tables, would induce players who are behind in the standings to take increasingly large risks as the end of the tournament period nears — and to hope their opponents were not making the same plays, which would eliminate the possibility of earning or losing any points in that hand. These strategies will not necessarily be more or less intricate in their strategic depth than the strategies for corresponding cash game, just different, perhaps in the same way that a regular poker cash game and a regular poker tournament have different strategic considerations due to nonlinear chip values.
If moving to duplicate poker has any impact on the relative influence of skill and chance in poker, it would certainly be to reduce the chance in the game, but even that is not formally clear. The rules of the duplicate variant do certainly eliminate the role of chance in the random deal of the cards, but that is only one source of randomness in poker. Randomness in poker also comes from the unknown or simultaneous strategic choices of one's opponent, as discussed in part 2 of this essay. When it comes to that source of chance, not only does duplicate poker not reduce it, but it actually increases it. The outcome of a hand or game will depend not only on the traditional strategic interplay between a player and his opponents at the same table, but also on the strategies of his duplicate, and the sensitivity of the outcome to this additional source of randomness can be immense. For example, if a good player is in the fortunate position of playing a duplicate poker cash game against a very poor partner (opponent) at the duplicated table, the good player might properly fold a speculative, slightly -EV hand, but when his duplicate decides to play it, the good player will often get a large positive or negative swing depending on whether or not his duplicate hits his draw. In a situation where folding and calling might be close in expected value, regular poker would let the player choose to fold and take no further risks in the hand, but this situation in duplicate poker is precisely the situation most likely to lead the two paired players to make different choices and lead to a large payoff or loss. It is totally plausible that a player's outcomes in duplicate poker could have higher variance than that of regular poker. Even if we don't focus on the spread of a player's outcome distributions, the duplicate poker variant certainly introduces an additional source of randomness to the game while it eliminates the salient source of the shuffle of the cards. It's not clear that one outweighs the other.
A similar take on duplicate poker by Nick Jones at Pokerfuse drew parallels between this fallacy of duplicate poker and the incompleteness of Sklansky's Fundamental Theorem of Poker. In the sense of my own generalizations to the Fundamental Theorem of Poker, duplicate poker could only hope to mitigate some of the randomness that lies between Level 0 and Level 3, leaving all of the strategywise uncertainty untouched (and adding the very impactful extra source of strategywise uncertainty from the duplicate player).
For duplicate poker to truly have no chance elements at all, as some of its proponents have claimed, is to accept that the object of a poker game is to predict what cards will fall, that an expert poker player will have the prescience to know to properly continue with a speculative hand only when it will end up hitting its draw. In reality, poker accepts that future random events are unpredictable and instead has the objective of maximizing expected value over these future uncertain events, and there is plenty of uncertainty remaining in duplicate poker.
On the whole, from any measure of the relative influence of skill and chance in games, it's not clear that duplicate poker is any further along the theoretical scale of predominance than regular poker is.
While it has never become popular or had many strong endorsers, thus suggesting that it's not a particularly good variant of poker, duplicate poker is a bona fide variant of poker. The game is still poker, it's just poker under a perturbation to the payoffs and strategies that I (and presumably others) feel is an unnecessary departure from the purity of the game.
The salient randomness is convincing, though. Among other implications, duplicate poker does gives each of its players much more of an "equal challenge" than regular poker. So, while I wouldn't discourage entrepreneurs and poker enthusiasts from trying to pass off duplicate poker in a region or social context which discriminates against regular poker for various informal reasons relating to salient randomness, please don't claim that duplicating out the shuffle of the cards makes the game all skill. Just focus on the honest fact that the impact on the outcomes of the random deal of the cards is mitigated and that the remaining uncertainty in the game is more obviously similar to that of other games commonly accepted as skill.
Part 17: Why does poker need randomness? -->
(back to index)
Duplicate poker is a poker variant which attempts to reduce the impact on game outcomes of the intrinsic randomness from the shuffle of the cards. The general principle is that two poker games take place on two separate tables, with the order of the shuffled cards controlled to be the same at both tables for each hand. The payoffs to each player, rather than being the standard number of chips won or lost in that hand, are based on a function of the difference between the net chip performance between that player and his or her duplicate at the duplicated table. In this sense, some of the salient chance features are mitigated by playing duplicate poker instead of regular poker. A player dealt a good starting hand isn't automatically being given a profitable opportunity, as he must outperform a duplicate player who has also been dealt the same hand. In the sense that regular poker "is skill in the long run" once the randomness of the cards has evened out, duplicate poker attempts to speed up this timescale by keeping the hands even throughout any interval of play. It's ok that "no amount of skill can change a deuce into an ace", as this will impact both of the paired players.
While duplicate poker has never gathered much popularity, its cause has been taken up by various companies over the years. The eponymous DuplicatePoker launched the first real-money online duplicate poker room several years ago, closing its doors and disappearing in 2008 without much public statement. More recently, SkillBet has launched a variant of duplicate poker in which there is only one human player at each table with the rest of the players being bots, which removes the possibility of particularly-exploitive collusion between conspiring players sitting at different seats that duplicate poker would provide. Perhaps most relevant is that the International Federation of Poker, an organization seeking to spread awareness of poker as a mind sport, has adopted a duplicate poker format for a major tournament it held last year in London.
In all three of these cases, the parties involved have marketed their ideas using claims that duplicate poker "has more skill" than regular poker, or, in some cases, asserting that it "is all skill" and has no chance elements at all. There's a strong intuitive appeal here, but are these claims accurate?
At this point in the essay, we quickly recognize the idea that duplicate poker has more skill than regular poker to be nonsense. Removing chance from a game is not the same as adding skill to a game. The depth of strategy in most duplicate poker games should be roughly the same as regular poker. A duplicate poker cash game with a typical payoff function on the differential between player performances would have the nice property of the correct EV-maximizing strategies being the same as they would be for a regular poker cash game, as every incremental chip won or lost contributes equally to the players' payoffs. Duplicate poker tournaments, however, will necessarily involve different considerations for any payoff function due to the nonlinearity of chip values, which will lead to different strategies than both cash game poker and traditional tournament poker. The scoring system for the IFP's London tournament and for SkillBet's tournament games, essentially rewarding only the players who get the highest scores at their tables, would induce players who are behind in the standings to take increasingly large risks as the end of the tournament period nears — and to hope their opponents were not making the same plays, which would eliminate the possibility of earning or losing any points in that hand. These strategies will not necessarily be more or less intricate in their strategic depth than the strategies for corresponding cash game, just different, perhaps in the same way that a regular poker cash game and a regular poker tournament have different strategic considerations due to nonlinear chip values.
If moving to duplicate poker has any impact on the relative influence of skill and chance in poker, it would certainly be to reduce the chance in the game, but even that is not formally clear. The rules of the duplicate variant do certainly eliminate the role of chance in the random deal of the cards, but that is only one source of randomness in poker. Randomness in poker also comes from the unknown or simultaneous strategic choices of one's opponent, as discussed in part 2 of this essay. When it comes to that source of chance, not only does duplicate poker not reduce it, but it actually increases it. The outcome of a hand or game will depend not only on the traditional strategic interplay between a player and his opponents at the same table, but also on the strategies of his duplicate, and the sensitivity of the outcome to this additional source of randomness can be immense. For example, if a good player is in the fortunate position of playing a duplicate poker cash game against a very poor partner (opponent) at the duplicated table, the good player might properly fold a speculative, slightly -EV hand, but when his duplicate decides to play it, the good player will often get a large positive or negative swing depending on whether or not his duplicate hits his draw. In a situation where folding and calling might be close in expected value, regular poker would let the player choose to fold and take no further risks in the hand, but this situation in duplicate poker is precisely the situation most likely to lead the two paired players to make different choices and lead to a large payoff or loss. It is totally plausible that a player's outcomes in duplicate poker could have higher variance than that of regular poker. Even if we don't focus on the spread of a player's outcome distributions, the duplicate poker variant certainly introduces an additional source of randomness to the game while it eliminates the salient source of the shuffle of the cards. It's not clear that one outweighs the other.
A similar take on duplicate poker by Nick Jones at Pokerfuse drew parallels between this fallacy of duplicate poker and the incompleteness of Sklansky's Fundamental Theorem of Poker. In the sense of my own generalizations to the Fundamental Theorem of Poker, duplicate poker could only hope to mitigate some of the randomness that lies between Level 0 and Level 3, leaving all of the strategywise uncertainty untouched (and adding the very impactful extra source of strategywise uncertainty from the duplicate player).
For duplicate poker to truly have no chance elements at all, as some of its proponents have claimed, is to accept that the object of a poker game is to predict what cards will fall, that an expert poker player will have the prescience to know to properly continue with a speculative hand only when it will end up hitting its draw. In reality, poker accepts that future random events are unpredictable and instead has the objective of maximizing expected value over these future uncertain events, and there is plenty of uncertainty remaining in duplicate poker.
On the whole, from any measure of the relative influence of skill and chance in games, it's not clear that duplicate poker is any further along the theoretical scale of predominance than regular poker is.
While it has never become popular or had many strong endorsers, thus suggesting that it's not a particularly good variant of poker, duplicate poker is a bona fide variant of poker. The game is still poker, it's just poker under a perturbation to the payoffs and strategies that I (and presumably others) feel is an unnecessary departure from the purity of the game.
The salient randomness is convincing, though. Among other implications, duplicate poker does gives each of its players much more of an "equal challenge" than regular poker. So, while I wouldn't discourage entrepreneurs and poker enthusiasts from trying to pass off duplicate poker in a region or social context which discriminates against regular poker for various informal reasons relating to salient randomness, please don't claim that duplicating out the shuffle of the cards makes the game all skill. Just focus on the honest fact that the impact on the outcomes of the random deal of the cards is mitigated and that the remaining uncertainty in the game is more obviously similar to that of other games commonly accepted as skill.
Part 17: Why does poker need randomness? -->
(back to index)
Monday, September 24, 2012
20 thoughts on skill vs. chance in poker, part 15: Does rake matter?
<-- Part 14: Does rate of showdown matter?
Most of our analysis so far has treated poker on its fundamental merits, ignoring or dismissing its social association with casino gambling. Whatever poker actually has in common with casino games is largely historical and incidental rather than being due to the inherent similarities with those games, whereas poker is fundamentally of the same structure as games which are almost universally seen as predominantly skill. Nonetheless, this association with gambling easily persists due to some salient similarities between poker and casino games, the most potentially-convincing trap being that the "house always wins" is also true in poker due to rake. If we accept that most real-world poker is raked, does this make poker meaningfully similar to house games?
Let's take a look at how the rake in poker can make poker resemble a casino game from some perspectives. Indeed, a raked poker game is no longer zero-sum but negative-sum, therefore the average player will lose. The house will take some portion of every "bet" in both poker and in casino games, be it from a fee such as a rake or from a built-in edge (which could be reframed as a rake on an otherwise payout-shifted zero-sum version of a one-player casino game). It's possible that a certain poker game could have so high of a rake that no player could have a positive expected value. If so, the player payoffs in this highly-raked poker game could resemble those of a casino game such as craps or video poker; players who make more proper decisions will lose less on average, but will still lose. Alternatively, the odds in casino games with player inputs could be shifted so that the best players do win on average, as is the case with blackjack. So why doesn't a rake "make poker into a casino game"?
Well, this has mostly been answered in part 11 during our discussion of why expected value doesn't dictate whether or not a game is a game of skill. As discussed there, while it's true that winning players can only exist in raked poker when the rake is not set to be exceedingly high, the lack of existence of winning players in a symmetric game does not necessarily imply anything about the nature of the underlying game. Just as shifting the player payoffs of a slot machine due to a promotion doesn't add any skill or reduce any chance in the play of that slot machine, adding a rake doesn't reduce any skill or increase any chance in poker.
For completeness, there are a few other reasons why rake specifically is not a component of the fundamental structure of a game.
The fact that the house makes money by offering poker does not make it any different than any other business. Of course a casino acting as a third-party operator for a competitive, symmetric game like poker will charge a fee to operate the game, just as a theater will take in money from everyone who sees a movie and just as the grocery will be making a profit from the sale of a gallon of milk. This is certainly the case for a commercial organizer of any game of any type.
The government's expert witness in the DiCristina case took a perspective which also implicitly involved rake, basing his arguments upon the perspective that, if poker were a game of skill, more of its players should be long-term winners. Games of skill don't need to have a certain percentage of players that win. This approach's focus on how many players end up making money also conflates the amount of skill in a game with the number of players participating in it. As the ruling pointed out, a large, multiplayer tournament in any game will have few winners and many losers.
If the rake imposed by a third-party operator were to be a factor in the classification of games, then this would apply equally to any game, regardless of the degree of chance present. A for-money chess, backgammon, bridge, or Scrabble tournament would cease to be predominantly skill when the tournament organizer takes out a fee of almost the entire prize pool. This rule would effectively create caps on rakes that would be dynamic and based on the player populations, rather than on the inherent structure of the game in question. Practical consistency would be difficult, as players might face variable external costs relating to the playing of the game, such as the costs of materials necessary to play (as in golf or in trading card games) or the costs of transportation for the players to reach the venue. It is a farce to try to build a world in which fees charged to run a game should legitimately affect the legal classification of the game.
The motivation that would presumably underlie such a classification would be concern over players going broke. Perhaps society would want to restrict the opportunities for players to lose money at games. Mathematically, however, in terms of the long-term probability of going broke, there is no difference between a negative expected value game and a zero expected value game. In each case, the probability of losing any finite bankroll over an infinite time horizon is 1. Since there can be no symmetric game where all players have positive expected value, this would logically lead to all symmetric games for money being prohibited or being treated similarly to casino games. Even without this fact, the divergence between player skill levels at any game is enough to create negative expected value situations for the weaker players. A generalized strategy game of any skill and chance structure could reproduce the player win probabilities of any given poker game if the set of players' strategies were chosen to be appropriately divergent. If we're searching for a sensible distinction between games of skill and games of chance, this doesn't get us there.
From a social, regulatory, or practical standpoint, rake cannot be seen as intrinsic to poker. While economic policies of price controls on certain goods and services may be useful in various contexts, they do not change the underlying nature of the activity. Similarly, macro-poker skills such as bankroll management and table selection are not direct evidence of skill in poker, though such concepts couldn't exist in a symmetric game with no skill. External or market forces do not have any bearing on the roles of skill and chance in a game.
Part 16: Does duplicate poker have higher skill relative to chance? -->
(back to index)
Most of our analysis so far has treated poker on its fundamental merits, ignoring or dismissing its social association with casino gambling. Whatever poker actually has in common with casino games is largely historical and incidental rather than being due to the inherent similarities with those games, whereas poker is fundamentally of the same structure as games which are almost universally seen as predominantly skill. Nonetheless, this association with gambling easily persists due to some salient similarities between poker and casino games, the most potentially-convincing trap being that the "house always wins" is also true in poker due to rake. If we accept that most real-world poker is raked, does this make poker meaningfully similar to house games?
Let's take a look at how the rake in poker can make poker resemble a casino game from some perspectives. Indeed, a raked poker game is no longer zero-sum but negative-sum, therefore the average player will lose. The house will take some portion of every "bet" in both poker and in casino games, be it from a fee such as a rake or from a built-in edge (which could be reframed as a rake on an otherwise payout-shifted zero-sum version of a one-player casino game). It's possible that a certain poker game could have so high of a rake that no player could have a positive expected value. If so, the player payoffs in this highly-raked poker game could resemble those of a casino game such as craps or video poker; players who make more proper decisions will lose less on average, but will still lose. Alternatively, the odds in casino games with player inputs could be shifted so that the best players do win on average, as is the case with blackjack. So why doesn't a rake "make poker into a casino game"?
Well, this has mostly been answered in part 11 during our discussion of why expected value doesn't dictate whether or not a game is a game of skill. As discussed there, while it's true that winning players can only exist in raked poker when the rake is not set to be exceedingly high, the lack of existence of winning players in a symmetric game does not necessarily imply anything about the nature of the underlying game. Just as shifting the player payoffs of a slot machine due to a promotion doesn't add any skill or reduce any chance in the play of that slot machine, adding a rake doesn't reduce any skill or increase any chance in poker.
For completeness, there are a few other reasons why rake specifically is not a component of the fundamental structure of a game.
The fact that the house makes money by offering poker does not make it any different than any other business. Of course a casino acting as a third-party operator for a competitive, symmetric game like poker will charge a fee to operate the game, just as a theater will take in money from everyone who sees a movie and just as the grocery will be making a profit from the sale of a gallon of milk. This is certainly the case for a commercial organizer of any game of any type.
The government's expert witness in the DiCristina case took a perspective which also implicitly involved rake, basing his arguments upon the perspective that, if poker were a game of skill, more of its players should be long-term winners. Games of skill don't need to have a certain percentage of players that win. This approach's focus on how many players end up making money also conflates the amount of skill in a game with the number of players participating in it. As the ruling pointed out, a large, multiplayer tournament in any game will have few winners and many losers.
If the rake imposed by a third-party operator were to be a factor in the classification of games, then this would apply equally to any game, regardless of the degree of chance present. A for-money chess, backgammon, bridge, or Scrabble tournament would cease to be predominantly skill when the tournament organizer takes out a fee of almost the entire prize pool. This rule would effectively create caps on rakes that would be dynamic and based on the player populations, rather than on the inherent structure of the game in question. Practical consistency would be difficult, as players might face variable external costs relating to the playing of the game, such as the costs of materials necessary to play (as in golf or in trading card games) or the costs of transportation for the players to reach the venue. It is a farce to try to build a world in which fees charged to run a game should legitimately affect the legal classification of the game.
The motivation that would presumably underlie such a classification would be concern over players going broke. Perhaps society would want to restrict the opportunities for players to lose money at games. Mathematically, however, in terms of the long-term probability of going broke, there is no difference between a negative expected value game and a zero expected value game. In each case, the probability of losing any finite bankroll over an infinite time horizon is 1. Since there can be no symmetric game where all players have positive expected value, this would logically lead to all symmetric games for money being prohibited or being treated similarly to casino games. Even without this fact, the divergence between player skill levels at any game is enough to create negative expected value situations for the weaker players. A generalized strategy game of any skill and chance structure could reproduce the player win probabilities of any given poker game if the set of players' strategies were chosen to be appropriately divergent. If we're searching for a sensible distinction between games of skill and games of chance, this doesn't get us there.
From a social, regulatory, or practical standpoint, rake cannot be seen as intrinsic to poker. While economic policies of price controls on certain goods and services may be useful in various contexts, they do not change the underlying nature of the activity. Similarly, macro-poker skills such as bankroll management and table selection are not direct evidence of skill in poker, though such concepts couldn't exist in a symmetric game with no skill. External or market forces do not have any bearing on the roles of skill and chance in a game.
Part 16: Does duplicate poker have higher skill relative to chance? -->
(back to index)
Thursday, September 20, 2012
20 thoughts on skill vs. chance in poker, part 14: Does rate of showdown matter?
<-- Part 13: On control and equal challenges
A simple, popular outcome-based approach to predominance considers how often real-world hands of a particular poker variant end in a showdown. Intuitively, the argument here is that, when a hand ends in a showdown, the cards held by the players have "controlled" the outcome, and when a hand instead ends in all but one player folding, since the cards were never shown, the "cards didn't matter" and thus the players' decisions to fold are what "controlled" the outcome.
This idea was most famously formalized in a 2009 study by security firm Cigital which computed showdown rates using a 100 million hand database of real Holdem hands, supplied by PokerStars. The study itself is a short read with a simple approach. It immediately discloses in its summary that it doesn't attempt to quantify the effect of chance, merely to provide "compelling statistics about the way that the outcomes of games are largely determined by players' decisions rather than chance", which is a terrific self-awareness. So how compelling are its arguments?
The primary finding of the Cigital analysis is that, 75.7% of the time, in a real-world Holdem hand there's no showdown. A secondary finding is that, of the remaining 24.3% of hands that do go to showdown, only about half of them are won by the hand that, had no players have folded, would have gone on to be the best hand. Therefore the random deal of the cards only controls the outcome about 12% of the time. This echoes the ideas of the toy game "Luck Holdem", where players place bets on their hands prior to the deal and then run out the cards and see who wins, as described by Howard Lederer in a rather solid short essay on skill and chance in poker (written many years ago, when he was still a respected member of our community).
Showdowns, however, do not directly correlate with either skill or chance. Once again, I can't put it better than Ike Haxton already has:
The ability to fold is only one of various strategic options, not necessarily of any greater worth than any other. It is possible that there might someday be a type of poker game which doesn't allow folding but which nonetheless involves more skill than the forms of poker that we play today. Chance would probably be higher as well, but with enough strategic depth, the skill could be increased moreso than the chance was under any desired measure. An approach such as that of the Cigital study would not capture the role of skill in this type of poker game. Similarly, this sort of approach is also dependent on the underlying population and thus could change over time if player tendencies shift due to development of the game strategy. It doesn't seem likely for multiplayer NL or Limit Holdem, but it's theoretically possible that, someday, they might be played using strategies that led to more showdowns and that these strategies, unknown to us currently, turn out to be better strategies exhibiting a deeper probing into the skill of the game. More realistically, the Cigital study may have yielded different results if play-money or micro-stakes games were observed.
Another study, published by sociologist Kyle Siler, explored the relationship between player strategies and payoffs using real hands of poker. The study contains some interesting observations on the relationship between social psychology and players' strategies at poker, but the most memorable "finding" of the study, thanks to the media attention this point received, was one relating to showdown rates. Siler found that players who won pots more often were more likely to be losing players, unsurprising to any experienced poker player but generally misinterpreted in the media. Naturally, overly-loose strategies will lead to winning more pots, but will lose money overall. Siler suggests the idea that inexperienced or poor players may conflate the ideas of winning pots and winning chips, causing them to skew their strategic decisions towards trying to win hands, which is an interesting behavioral hypothesis.
While I don't think that Siler meant for this conclusion about losing players winning more pots to be taken as a statement on the role of skill and chance in poker, to do so properly would nonetheless illustrate that players' strategies control the outcome, whether one is considering the outcome as the winner of the hand or as the magnitude of the amounts won and lost by the players in the hand. A player can choose a loose strategy and will win more pots by doing so, which shows that a player's strategic choices exhibit control over how often they win hands. Similarly, if the player who chooses a loose strategy ends up losing more money by doing so, this shows that the player's strategic choices also exhibit control over how many chips they win — which is, of course, the object of poker, rather than trying to win every pot. The various mainstream treatments of Siler's study seemed to generally miss this distinction, instead drawing unwarranted and illogical conclusions like "in gambling, quit while you're ahead".
Returning to the core idea of the Cigital study, there is some potential philosophical trouble with the role that the random components play in guiding player decisions. If we are to say that the players and their decisions have fully controlled the outcome within a hand that did not go to showdown, are we ignoring the fact that the cards dealt to the players played a role in their decisions? Has the random deal of the cards "controlled" the decisions of the players, thus transitively controlling the outcome of the hand? The key distinction here may be that compelling, suggesting, or guiding the players' decisions is not the same as controlling them; the players are endowed with free will and can choose to call that river bet with their nut-low missed flush draw if they want to. The random elements can guide strategic choices in a way that is occasionally forced... but it's only forced if the player wants to play well. If the player were truly controlled by the cards, there would be no strategic option available. That bad moves exist within a game is emblematic of the existence of diverse strategies within that game.
For either consideration of what an outcome is in poker, despite the fact that folding and bluffing aren't necessary conditions for skill to exist in a card game, they certainly serve well as the most obvious and salient ways in which players can control outcomes. In a context where we are forced to defend poker on the basis of the winner of a hand being the outcome, with no consideration given to the fact that superior players will win bigger pots and lose smaller ones, arguments such as that of the Cigital study play an important role. It's at least a little intellectually dishonest, as it's in some sense only a coincidence that the most popular forms of real-life poker happen to involve enough folding for showdowns to occur less than half of the time, but, given the ambiguity and indeterminacy of the question of predominance, some degree of hand-waving is often necessary.
Part 15: Does rake matter? -->
(back to index)
A simple, popular outcome-based approach to predominance considers how often real-world hands of a particular poker variant end in a showdown. Intuitively, the argument here is that, when a hand ends in a showdown, the cards held by the players have "controlled" the outcome, and when a hand instead ends in all but one player folding, since the cards were never shown, the "cards didn't matter" and thus the players' decisions to fold are what "controlled" the outcome.
This idea was most famously formalized in a 2009 study by security firm Cigital which computed showdown rates using a 100 million hand database of real Holdem hands, supplied by PokerStars. The study itself is a short read with a simple approach. It immediately discloses in its summary that it doesn't attempt to quantify the effect of chance, merely to provide "compelling statistics about the way that the outcomes of games are largely determined by players' decisions rather than chance", which is a terrific self-awareness. So how compelling are its arguments?
The primary finding of the Cigital analysis is that, 75.7% of the time, in a real-world Holdem hand there's no showdown. A secondary finding is that, of the remaining 24.3% of hands that do go to showdown, only about half of them are won by the hand that, had no players have folded, would have gone on to be the best hand. Therefore the random deal of the cards only controls the outcome about 12% of the time. This echoes the ideas of the toy game "Luck Holdem", where players place bets on their hands prior to the deal and then run out the cards and see who wins, as described by Howard Lederer in a rather solid short essay on skill and chance in poker (written many years ago, when he was still a respected member of our community).
Showdowns, however, do not directly correlate with either skill or chance. Once again, I can't put it better than Ike Haxton already has:
The notion that skill is limited if you can't fold and unlimited if you can is hand-wavy nonsense that is superficially compelling only because it coincidentally aligns well with our intuitions about which games are more or less skillful out of the class of commonly played gambling games. There's no folding in bridge or gin but they are obviously highly skillful games. Heads up limit holdem with no folding allowed would be another. One can conceive of poker variants that allow folding but have essentially no room for skill (NLHE with face up hands for instance).Indeed, poker without folding is still a game which fits a broader definition of poker, as long as checking and betting are still allowed. Heads-Up Limit Holdem without folding, as Ike mentions, still has plenty of skill, though it is probably less skill than regular poker. To that extent, it's reasonable to say that adding the option of folding back into a poker game should generally serve to increase the depth of skill of the game — but introducing any new strategically-nontrivial move to a game would increase the strategic space of the game.
The ability to fold is only one of various strategic options, not necessarily of any greater worth than any other. It is possible that there might someday be a type of poker game which doesn't allow folding but which nonetheless involves more skill than the forms of poker that we play today. Chance would probably be higher as well, but with enough strategic depth, the skill could be increased moreso than the chance was under any desired measure. An approach such as that of the Cigital study would not capture the role of skill in this type of poker game. Similarly, this sort of approach is also dependent on the underlying population and thus could change over time if player tendencies shift due to development of the game strategy. It doesn't seem likely for multiplayer NL or Limit Holdem, but it's theoretically possible that, someday, they might be played using strategies that led to more showdowns and that these strategies, unknown to us currently, turn out to be better strategies exhibiting a deeper probing into the skill of the game. More realistically, the Cigital study may have yielded different results if play-money or micro-stakes games were observed.
Another study, published by sociologist Kyle Siler, explored the relationship between player strategies and payoffs using real hands of poker. The study contains some interesting observations on the relationship between social psychology and players' strategies at poker, but the most memorable "finding" of the study, thanks to the media attention this point received, was one relating to showdown rates. Siler found that players who won pots more often were more likely to be losing players, unsurprising to any experienced poker player but generally misinterpreted in the media. Naturally, overly-loose strategies will lead to winning more pots, but will lose money overall. Siler suggests the idea that inexperienced or poor players may conflate the ideas of winning pots and winning chips, causing them to skew their strategic decisions towards trying to win hands, which is an interesting behavioral hypothesis.
While I don't think that Siler meant for this conclusion about losing players winning more pots to be taken as a statement on the role of skill and chance in poker, to do so properly would nonetheless illustrate that players' strategies control the outcome, whether one is considering the outcome as the winner of the hand or as the magnitude of the amounts won and lost by the players in the hand. A player can choose a loose strategy and will win more pots by doing so, which shows that a player's strategic choices exhibit control over how often they win hands. Similarly, if the player who chooses a loose strategy ends up losing more money by doing so, this shows that the player's strategic choices also exhibit control over how many chips they win — which is, of course, the object of poker, rather than trying to win every pot. The various mainstream treatments of Siler's study seemed to generally miss this distinction, instead drawing unwarranted and illogical conclusions like "in gambling, quit while you're ahead".
Returning to the core idea of the Cigital study, there is some potential philosophical trouble with the role that the random components play in guiding player decisions. If we are to say that the players and their decisions have fully controlled the outcome within a hand that did not go to showdown, are we ignoring the fact that the cards dealt to the players played a role in their decisions? Has the random deal of the cards "controlled" the decisions of the players, thus transitively controlling the outcome of the hand? The key distinction here may be that compelling, suggesting, or guiding the players' decisions is not the same as controlling them; the players are endowed with free will and can choose to call that river bet with their nut-low missed flush draw if they want to. The random elements can guide strategic choices in a way that is occasionally forced... but it's only forced if the player wants to play well. If the player were truly controlled by the cards, there would be no strategic option available. That bad moves exist within a game is emblematic of the existence of diverse strategies within that game.
For either consideration of what an outcome is in poker, despite the fact that folding and bluffing aren't necessary conditions for skill to exist in a card game, they certainly serve well as the most obvious and salient ways in which players can control outcomes. In a context where we are forced to defend poker on the basis of the winner of a hand being the outcome, with no consideration given to the fact that superior players will win bigger pots and lose smaller ones, arguments such as that of the Cigital study play an important role. It's at least a little intellectually dishonest, as it's in some sense only a coincidence that the most popular forms of real-life poker happen to involve enough folding for showdowns to occur less than half of the time, but, given the ambiguity and indeterminacy of the question of predominance, some degree of hand-waving is often necessary.
Part 15: Does rake matter? -->
(back to index)
Wednesday, September 19, 2012
20 thoughts on skill vs. chance in poker, part 13: On control and equal challenges
<-- Part 12: Why not sports betting and investing?
One common approach to considering the nature of the role of chance in a game is to focus on the degree to which the players and their actions or decisions exhibit control over the outcome. Certainly, in a game with close to no intrinsic random elements such as chess, ignoring the coinflip to start the game, the actions of the players will be essentially the only factor which controls the outcome, whereas in games of pure chance such as roulette, the player clearly has no control at all over the outcome. When considering whether or not market-based activities such as sports betting or investing might be games of skill, this approach would focus on the fact that the player cannot influence the result of the bet, so it draws a distinction between betting on an external event and playing a game like poker, as we have in the previous thought.
The Pennsylvania Supreme Court once used this approach to rule on the predominance of chance in a house game called "Electro-Sport", apparently some bastardized non-poker derivative of poker. Thankfully, the court recognized this key difference between video poker and poker. This ruling was cited in Pennsylvania V Dent as follows:
There is a danger in this sort of approach when it comes to poker with the distinction between "improving an outcome" and "determining an outcome". If the "outcome" in poker were considered to be which player wins a single hand, then this creates a focus on who wins at showdown rather than on how many chips are won or lost in each pot, which will be treated in more detail in a future thought. Still, I can think of no games of skill that don't involve the players having significant control over the outcome, though often the player will not have full control, as in poker. It's pretty good as a rule of thumb. Every strategy game where the strategies have a nontrivial impact are games where players have some degree of control over the outcome; players choose strategies, and strategies contribute to outcomes in games of skill. What I feel this approach does best is to cleanly separate strategy games from advantage gambling without strategic interaction (e.g. market-based activities, video poker such as this Electro-Sport), and it does so in an intuitive way. Without the strategic interaction of a multiplayer game, there is generally no means for a player to control the outcome in a one-player game.
A related but different approach to predominance focuses on the nature of the role of skill in a game by focusing on whether players of a game are always presented with "equal challenge" as in whether or not the particular game positions and strategic opportunities available to one player during a game will always be available to the other players as well. This was famously captured in a North Carolina Court of Appeals ruling, also cited in Pennsylvania V Dent:
The premise here is shaky. While the possibility of players being presented with unequal challenges does necessarily imply the presence of chance in a game, it has no bearing on the depth of skill in a game. Really, this approach presupposes that a fair card game where each of the players have the same probabilities of getting dealt each possible hand is an "unequal challenge" as soon as the players are dealt different hands. Competitive games with intrinsic and overt random elements, such as poker, bridge, backgammon, and Scrabble, are still symmetric games as long as the probabilities are fair, even if players end up getting given different cards each time they play. The focus should be on the structural symmetry, rather than on the presence of random components.
Overall, the presence of equal challenge doesn't seem to correlate very well with the influence of either skill or chance in a game. Rando chess, with any probability of reversal, does happen to present its players with an equal challenge, but can be modified to reach any desired point in the fallacious skill-chance continuum. More importantly, duplicate poker, as we'll discuss soon, is a poker variant that manages to present its players with more of an equal challenge but does not necessarily eliminate (or even reduce!) the role of chance in the game. All that this approach captures is that the game contains a nonzero element of chance, but, despite this fact being overlooked by the ruling, this is true even for almost-fully-deterministic games such as chess and billiards, where the random selection of the turn order does indeed present the players with an unequal challenge.
Of these two approaches, both are fairly narrow. As the treatment of the many thoughts in this essay have shown, there is much more relevant detail and nuance to the relationship between skill and chance in games than can be captured by a single argument alone. That being said, as far as simple arguments go, the idea of control is a reasonably good measure of the presence of skill in a game, though it doesn't measure chance well. On the other hand, the idea of equal challenge must necessarily discriminate against games whose random components generate a variety of gameplay, despite this being a common component of many high-skill games.
Part 14: Does rate of showdown matter? -->
(back to index)
One common approach to considering the nature of the role of chance in a game is to focus on the degree to which the players and their actions or decisions exhibit control over the outcome. Certainly, in a game with close to no intrinsic random elements such as chess, ignoring the coinflip to start the game, the actions of the players will be essentially the only factor which controls the outcome, whereas in games of pure chance such as roulette, the player clearly has no control at all over the outcome. When considering whether or not market-based activities such as sports betting or investing might be games of skill, this approach would focus on the fact that the player cannot influence the result of the bet, so it draws a distinction between betting on an external event and playing a game like poker, as we have in the previous thought.
The Pennsylvania Supreme Court once used this approach to rule on the predominance of chance in a house game called "Electro-Sport", apparently some bastardized non-poker derivative of poker. Thankfully, the court recognized this key difference between video poker and poker. This ruling was cited in Pennsylvania V Dent as follows:
While appellee has demonstrated that some skill is involved in the playing of Electro-Sport, we believe that the element of chance predominates and the outcome is largely determined by chance. While skill, in the form of knowledge of probabilities, can improve a player’s chances of winning and can maximize the size of the winnings, chance ultimately determines the outcome because chance determines the cards dealt and the cards from which one can draw — in short, a large random element is always present. That the skill involved in Electro-Sport is not the same skill which can indeed determine the outcome in a game of poker between human players can be appreciated when it is realized that holding, folding, bluffing and raising have no role to play in Electro-Sport poker. Skill can improve the outcome in Electro-Sport; it cannot determine it.
There is a danger in this sort of approach when it comes to poker with the distinction between "improving an outcome" and "determining an outcome". If the "outcome" in poker were considered to be which player wins a single hand, then this creates a focus on who wins at showdown rather than on how many chips are won or lost in each pot, which will be treated in more detail in a future thought. Still, I can think of no games of skill that don't involve the players having significant control over the outcome, though often the player will not have full control, as in poker. It's pretty good as a rule of thumb. Every strategy game where the strategies have a nontrivial impact are games where players have some degree of control over the outcome; players choose strategies, and strategies contribute to outcomes in games of skill. What I feel this approach does best is to cleanly separate strategy games from advantage gambling without strategic interaction (e.g. market-based activities, video poker such as this Electro-Sport), and it does so in an intuitive way. Without the strategic interaction of a multiplayer game, there is generally no means for a player to control the outcome in a one-player game.
A related but different approach to predominance focuses on the nature of the role of skill in a game by focusing on whether players of a game are always presented with "equal challenge" as in whether or not the particular game positions and strategic opportunities available to one player during a game will always be available to the other players as well. This was famously captured in a North Carolina Court of Appeals ruling, also cited in Pennsylvania V Dent:
[W]hile all games have elements of chance, games which can be determined by superior skill are not games of chance. For example, bowling, chess, and billiards are games of skill because skill determines the outcome. The game itself is static and the only factor separating the players is their relative skill levels. In short, the instrumentality for victory is in each player’s hands and his fortunes will be determined by how skillfully he use (sic) that instrumentality. Poker, however, presents players with different hands, making the players unequal in the same game and subject to defeat at the turn of a card. Although skills such as knowledge of human psychology, bluffing, and the ability to analyze odds make it more likely for skilled players to defeat novices, novices may yet prevail with a simple run of luck. No amount of skill can change a deuce into an ace. Thus, the instrumentality for victory is not entirely in the player’s hand. In State v. Taylor, our Supreme Court noted this distinction. 111 N.C. 680, 16 S.E. 168 (1892).Ah, yes, no amount of skill can change a deuce into an ace. Indeed, this approach to predominance turns out well for mostly-deterministic games of skill (modulo coinflips) but not for many other games of skill such as poker, bridge, backgammon, or Scrabble. (Also note that this ruling, like many others, treats "game of skill" as "game involving skills".)
The premise here is shaky. While the possibility of players being presented with unequal challenges does necessarily imply the presence of chance in a game, it has no bearing on the depth of skill in a game. Really, this approach presupposes that a fair card game where each of the players have the same probabilities of getting dealt each possible hand is an "unequal challenge" as soon as the players are dealt different hands. Competitive games with intrinsic and overt random elements, such as poker, bridge, backgammon, and Scrabble, are still symmetric games as long as the probabilities are fair, even if players end up getting given different cards each time they play. The focus should be on the structural symmetry, rather than on the presence of random components.
Overall, the presence of equal challenge doesn't seem to correlate very well with the influence of either skill or chance in a game. Rando chess, with any probability of reversal, does happen to present its players with an equal challenge, but can be modified to reach any desired point in the fallacious skill-chance continuum. More importantly, duplicate poker, as we'll discuss soon, is a poker variant that manages to present its players with more of an equal challenge but does not necessarily eliminate (or even reduce!) the role of chance in the game. All that this approach captures is that the game contains a nonzero element of chance, but, despite this fact being overlooked by the ruling, this is true even for almost-fully-deterministic games such as chess and billiards, where the random selection of the turn order does indeed present the players with an unequal challenge.
Of these two approaches, both are fairly narrow. As the treatment of the many thoughts in this essay have shown, there is much more relevant detail and nuance to the relationship between skill and chance in games than can be captured by a single argument alone. That being said, as far as simple arguments go, the idea of control is a reasonably good measure of the presence of skill in a game, though it doesn't measure chance well. On the other hand, the idea of equal challenge must necessarily discriminate against games whose random components generate a variety of gameplay, despite this being a common component of many high-skill games.
Part 14: Does rate of showdown matter? -->
(back to index)
Tuesday, September 18, 2012
20 thoughts on skill vs. chance in poker, part 12: Why not sports betting and investing?
<-- Part 11: Why not blackjack or other casino games?
Similarly to how games like blackjack are not quite of the same fundamental structure as poker, activities like sports betting, horse racing, prop betting, and stock investing, which I will collectively refer to as market-based activities, are not symmetric, closed, competitive games. They are quite a bit closer to this than advantage games played against the house, in that they are generally symmetric, multiplayer, and can offer much more sizable edges to experts.
The biggest difference between these pursuits and poker is that market-based activities are not closed games, a term I've used in describing the class of games of which poker is a member. A closed game is a game with clearly-identified players and a distinct beginning and end.
Why does this matter? And how is this meaningfully different from poker, which is sometimes philosophically seen as "one big session" over one's entire career? Here are the implications I've thought of for what it means for a game to be closed:
None of this is any statement against market-based activities involving skill, or against them involving more or less skill than poker does. Certainly there is tremendous skill in both, but the skills involved in non-closed games are more like "skills involved" in finding +EV opportunities as they arise, rather than strategic outperformance of a competitor in a game with structured, unchanging rules and underlying dynamics.
Despite the favorable, accessible, and practical comparisons that can be made between investing and poker, which I absolutely support, I don't think it's fully intellectually honest to classify market-based activities as games of skill. It's certainly not that they don't involve skill. It's that they aren't games in the closed sense.
Part 13: On control and equal challenges -->
(back to index)
Similarly to how games like blackjack are not quite of the same fundamental structure as poker, activities like sports betting, horse racing, prop betting, and stock investing, which I will collectively refer to as market-based activities, are not symmetric, closed, competitive games. They are quite a bit closer to this than advantage games played against the house, in that they are generally symmetric, multiplayer, and can offer much more sizable edges to experts.
The biggest difference between these pursuits and poker is that market-based activities are not closed games, a term I've used in describing the class of games of which poker is a member. A closed game is a game with clearly-identified players and a distinct beginning and end.
Why does this matter? And how is this meaningfully different from poker, which is sometimes philosophically seen as "one big session" over one's entire career? Here are the implications I've thought of for what it means for a game to be closed:
- To whatever extent something like sports betting is a game, it's not a game that two competitors can sit down and play against each other over any pre-specified moment in time. Investments and bets contingent on real-world events happen in realtime.
- The marketplace in any betting or investment market is arbitrarily large and changing. Poker might be a 2-player, 6-player, 9-player, or 2000-player game depending on how it's played, but it's always established that way before the game begins. Meanwhile, financial markets are "games" with often many many thousands of participants. The amount of interest in the market can change rapidly, and the influx or outflux of participants can significantly impact the mechanics of way the price (or betting line) moves.
- Since the opportunities to "play the game" (profitably bet or invest) arrive at random time intervals, a "player" could go for a long time without actually doing anything. Moreover, a player who has found something to bet on is done making decisions and done thinking as soon as the bet is placed. Contrast this to closed strategy games like poker, where players must make decisions constantly as they play. This arguably has meaningful social and legal implications in a world that values competition and mental exercise; the ongoing strategic engagement of closed strategy games and the demand to be mentally involved in determining the outcome could be a positive enough effect to distinguish such activities from those where a bettor or investor is external to outcomes.
- It isn't possible to replicate a set of conditions from the past to have a rematch of the same "game" with the same rules of sports betting or investing. As market conditions and the real world continue to change, the relative expected performances within any set of players will fluctuate even while the players keep their strategies fixed. This is in contrast to poker's evolution, which is a deeper strategy development into a fixed game with unchanging rules. Market-based activities are analogous to a continuum of slowly-changing poker variants, perhaps where the probabilities of certain cards being dealt are slightly different from day-to-day.
- Thus there's no way to structure a traditional tournament or competition to determine the best players at these games, unless artificial market events were designed. In poker, a long enough tournament would indeed (eventually) rank the players according to their skills, but in market-based activities, the "game" will have changed before enough of a sample size could be captured to identify the best investor over a month, year, or lifetime.
- Not only are the underlying probabilities and "rules of the game" changing continuously in market-based activities, but, even at any given moment in time, as noted in part 9, the probabilities are of a qualitatively different type than those within poker. The randomness underlying investments can be fat-tailed or of arbitrarily large spikiness (i.e. not only high variance, but high variance of variance), whereas the probabilities present in closed strategy games are unchanging and always lead to easily-modeled, bankroll-friendly thin-tailed randomness. This has clear practical implications for the financial planning of players and thus could conceivably be a distinction that is of regulatory or social interest.
None of this is any statement against market-based activities involving skill, or against them involving more or less skill than poker does. Certainly there is tremendous skill in both, but the skills involved in non-closed games are more like "skills involved" in finding +EV opportunities as they arise, rather than strategic outperformance of a competitor in a game with structured, unchanging rules and underlying dynamics.
Despite the favorable, accessible, and practical comparisons that can be made between investing and poker, which I absolutely support, I don't think it's fully intellectually honest to classify market-based activities as games of skill. It's certainly not that they don't involve skill. It's that they aren't games in the closed sense.
Part 13: On control and equal challenges -->
(back to index)
Monday, September 17, 2012
20 thoughts on skill vs. chance in poker, part 11: Why not blackjack or other casino games?
<-- Part 10: Games of skill vs. "involving skills"
We've generally restricted our focus so far to symmetric, closed, competitive strategy games (orthogames). Traditional casino gambling almost exclusively consists of one-player "games" played against the house, structurally dissimilar from poker and from all other traditional games of skill. To me, the presence of strategic interaction is the essence of skill when it comes to non-physical games, which is why I feel that the categorical divide between one-player and multiplayer games should be central to the approach of classifying games. However, there are some subtleties to the dividing line between house games and competitive games that are worth considering further before we assume them all away.
The argument that I find to best illustrate the ideal legal distinction between house games and poker was stated well in the law review article Video Poker and the Skill Versus Chance Debate. Unfortunately, the article does not seem to be freely available anymore, but I had noted its summary of its main idea:
This definition is clearly an easy fit for most -EV casino games, all of which are more or less identical to a series of simple bets on random events. If the player inputs in a game such as (most) video poker or craps only serve to change what type and degree of fixed-odds -EV bet is made, then the "game" is essentially a slot machine variant with a few different levers to pull. It is less obvious that this definition applies to blackjack and other games that can, under some conditions, be +EV for the player, but I believe that a careful analysis shows no meaningful theoretical distinctions between blackjack and strictly -EV house games.
It is first worth acknowledging that blackjack exists as somewhat of a special case, an anomaly, a market inefficiency, a triumph or travesty (depending on your perspective) of human irrationality and behavioral tendencies. The casinos make enough profit off of suboptimal play in blackjack that they make the most money by continuing to offer the game and spending resources to identify and block people who play correctly. The fact that the average person knows that blackjack can be beaten presumably drives a lot of traffic to that game. But even blackjack being a potentially beatable game doesn't make it a game of skill.
The expected value (EV) of the player in a house game is essentially just a cost of playing that the casino sets. Any casino game could have its odds or payoffs shifted so that it had zero EV for the player, with the casino instead charging some sort of external fee or rake, and the game itself, in terms of its mechanics and decisions, would be the same. Even if the casino chose to set the payoffs and fees so that the player had a positive EV on something like a slot machine, perhaps as a promotion, there is no sense in which the playing of the slot machine has had any skill or strategy injected into it. Identifying the profitability of the promotion would be a skill involved in being successful at making money at casino games, but this skill is external to the game itself and, for the reasons I've argued in the last thought, not of any particular bearing on the fundamental nature of the game.
It is potentially worthy of note that, despite being stylized to resemble a game played against a human dealer, blackjack is still a one-player game. The house is not a player in the sense of game theory, since it has no ability to make decisions and instead follows a pre-specified strategy. The optimization problem posed to the player of blackjack is one of decision theory, not game theory.
There are some potentially-interesting implications of this perspective. A symmetric, competitive two-player variant of blackjack could be created where players compete against each other instead of playing against the house. The players might alternate positions with each other and each follow the same set of rules (note that traditional blackjack is not actually symmetric since the dealer cannot split or double down), and this adaptation of blackjack would make it into a game in the same family as poker, though it may not have much of a depth of skill. Similarly, a symmetric, competitive, multiplayer game could be made into a one-player game played against a casino by offering electronic or dealer-operated instances of it in which the house played the role of one of the players with a fixed strategy, much like the heads-up limit holdem machines introduced onto casino floors in Vegas.
The key distinction here is that a "player" adhering to a fixed strategy is not a player at all in the sense of game theory. When one knows with certainty that one's opponent will adhere to a certain strategy, there is no room for strategic interaction. Note that this is different than situations in simple games where each player would likely know the opponent's strategy or at least what the opponent should be doing, such as when two experts play a heads-up No Limit Holdem endgame with 5bb effective stacks. Each will almost certainly closely approximate the known Nash equilibrium of this subgame. However, whether it be as a mistake or as an attempt to exploit perceived weaknesses in his opponent, each player could change their strategy if they wanted to. The opportunity for strategic deviation does exist. In contrast, in something like blackjack, the house's decisions are fixed by definition of the rules of the game. Sure, a trivial game such as 5bb heads-up No Limit Holdem will not have much of a depth of skill, but its structure as a game where each party has strategic choices keeps it distinct from casino gambling, even though the outcomes will closely resemble coinflips as they will in any low-skill game.
This highlights some issues with what many have adopted as a quick and easy approach to predominance. This common but imperfect approach classifies any game in which a player can sometimes have a positive EV as a game of skill and a game in which a player can only have a zero or negative EV as a game of chance. There's a good intuitive appeal here, but this would classify some clear pure-chance activities as games of skill under conditions, perhaps promotional, where the payouts or odds were shifted to favor the player, completely independently of the depth of skill or strategy within the activity. More troubling would be that a game such as poker could be classified as a game of chance if the rake were too high, an idea which will be treated further in a later part of this essay. Categorizing games purely on their EV under various conditions may be of practical financial importance to a player trying to earn a living by playing a game, but does not necessarily measure any structural presence of skill in the games. It's possible to be a professional gambler by making a living from exploiting profitable casino game opportunities, and doing this does involve skill, but this is not the same as being a professional player of poker or any other symmetric, competitive game.
Logically, there one direction of the relationship between EV and the existence of skill in a game that holds up in some cases. If we assume that we're only looking at symmetric games, symmetry being key here, then the existence of a +EV player/strategy for any given game under some conditions indicates that there must be some component of skill in the game, i.e. that nontrivial strategies exist and impact the game outcome. However, the converse is not true; it is not the case that the lack of existence of +EV players/strategies imply that there is no skill in the game. A clear game of skill such as chess could fail to demonstrate +EV players if the rake taken on the game was too high or, alternatively, if the observed population was all of expert players of similar ability. The argument which focuses only on EV is essentially an outcome-based approach to predominance and thus has shortcomings in each of the ways we've explored already.
In terms of actual legal interpretations of predominance, unfortunately, most of the time, house games and multiplayer strategy games are lumped in together and treated as a homogeneous group. Another part of the mostly-wonderful the DiCristina ruling that disappointed me was a very succinct dismissal of the possibility of treating one-player "games" separately from multiplayer games, though I acknowledge that the historical precedent probably does work this way:
Part 12: Why not sports betting and investing? -->
(back to index)
We've generally restricted our focus so far to symmetric, closed, competitive strategy games (orthogames). Traditional casino gambling almost exclusively consists of one-player "games" played against the house, structurally dissimilar from poker and from all other traditional games of skill. To me, the presence of strategic interaction is the essence of skill when it comes to non-physical games, which is why I feel that the categorical divide between one-player and multiplayer games should be central to the approach of classifying games. However, there are some subtleties to the dividing line between house games and competitive games that are worth considering further before we assume them all away.
The argument that I find to best illustrate the ideal legal distinction between house games and poker was stated well in the law review article Video Poker and the Skill Versus Chance Debate. Unfortunately, the article does not seem to be freely available anymore, but I had noted its summary of its main idea:
In order to be profitable to the casinos, these games [video poker] must be based on chance. If skilled gamblers could win regularly, the casinos (or bar or restaurant owners) would begin to lose money, and the machines would soon be removed from the gambling floor.This is a sensible approach to the definitions of skill, chance, and predominance; any 1-player gamble played against the house that a for-profit corporate entity offers openly must be a game of chance, because, in theory, it wouldn't make sense for a game freely and openly offered by a business to be a game of skill, as adverse selection and a rational customer base would ensure that the profitable customers would bankrupt the company. Games of skill could only exist between two or more players competing in a symmetric game, where perhaps a company could take a fee for facilitating the game.
This definition is clearly an easy fit for most -EV casino games, all of which are more or less identical to a series of simple bets on random events. If the player inputs in a game such as (most) video poker or craps only serve to change what type and degree of fixed-odds -EV bet is made, then the "game" is essentially a slot machine variant with a few different levers to pull. It is less obvious that this definition applies to blackjack and other games that can, under some conditions, be +EV for the player, but I believe that a careful analysis shows no meaningful theoretical distinctions between blackjack and strictly -EV house games.
It is first worth acknowledging that blackjack exists as somewhat of a special case, an anomaly, a market inefficiency, a triumph or travesty (depending on your perspective) of human irrationality and behavioral tendencies. The casinos make enough profit off of suboptimal play in blackjack that they make the most money by continuing to offer the game and spending resources to identify and block people who play correctly. The fact that the average person knows that blackjack can be beaten presumably drives a lot of traffic to that game. But even blackjack being a potentially beatable game doesn't make it a game of skill.
The expected value (EV) of the player in a house game is essentially just a cost of playing that the casino sets. Any casino game could have its odds or payoffs shifted so that it had zero EV for the player, with the casino instead charging some sort of external fee or rake, and the game itself, in terms of its mechanics and decisions, would be the same. Even if the casino chose to set the payoffs and fees so that the player had a positive EV on something like a slot machine, perhaps as a promotion, there is no sense in which the playing of the slot machine has had any skill or strategy injected into it. Identifying the profitability of the promotion would be a skill involved in being successful at making money at casino games, but this skill is external to the game itself and, for the reasons I've argued in the last thought, not of any particular bearing on the fundamental nature of the game.
It is potentially worthy of note that, despite being stylized to resemble a game played against a human dealer, blackjack is still a one-player game. The house is not a player in the sense of game theory, since it has no ability to make decisions and instead follows a pre-specified strategy. The optimization problem posed to the player of blackjack is one of decision theory, not game theory.
There are some potentially-interesting implications of this perspective. A symmetric, competitive two-player variant of blackjack could be created where players compete against each other instead of playing against the house. The players might alternate positions with each other and each follow the same set of rules (note that traditional blackjack is not actually symmetric since the dealer cannot split or double down), and this adaptation of blackjack would make it into a game in the same family as poker, though it may not have much of a depth of skill. Similarly, a symmetric, competitive, multiplayer game could be made into a one-player game played against a casino by offering electronic or dealer-operated instances of it in which the house played the role of one of the players with a fixed strategy, much like the heads-up limit holdem machines introduced onto casino floors in Vegas.
The key distinction here is that a "player" adhering to a fixed strategy is not a player at all in the sense of game theory. When one knows with certainty that one's opponent will adhere to a certain strategy, there is no room for strategic interaction. Note that this is different than situations in simple games where each player would likely know the opponent's strategy or at least what the opponent should be doing, such as when two experts play a heads-up No Limit Holdem endgame with 5bb effective stacks. Each will almost certainly closely approximate the known Nash equilibrium of this subgame. However, whether it be as a mistake or as an attempt to exploit perceived weaknesses in his opponent, each player could change their strategy if they wanted to. The opportunity for strategic deviation does exist. In contrast, in something like blackjack, the house's decisions are fixed by definition of the rules of the game. Sure, a trivial game such as 5bb heads-up No Limit Holdem will not have much of a depth of skill, but its structure as a game where each party has strategic choices keeps it distinct from casino gambling, even though the outcomes will closely resemble coinflips as they will in any low-skill game.
This highlights some issues with what many have adopted as a quick and easy approach to predominance. This common but imperfect approach classifies any game in which a player can sometimes have a positive EV as a game of skill and a game in which a player can only have a zero or negative EV as a game of chance. There's a good intuitive appeal here, but this would classify some clear pure-chance activities as games of skill under conditions, perhaps promotional, where the payouts or odds were shifted to favor the player, completely independently of the depth of skill or strategy within the activity. More troubling would be that a game such as poker could be classified as a game of chance if the rake were too high, an idea which will be treated further in a later part of this essay. Categorizing games purely on their EV under various conditions may be of practical financial importance to a player trying to earn a living by playing a game, but does not necessarily measure any structural presence of skill in the games. It's possible to be a professional gambler by making a living from exploiting profitable casino game opportunities, and doing this does involve skill, but this is not the same as being a professional player of poker or any other symmetric, competitive game.
Logically, there one direction of the relationship between EV and the existence of skill in a game that holds up in some cases. If we assume that we're only looking at symmetric games, symmetry being key here, then the existence of a +EV player/strategy for any given game under some conditions indicates that there must be some component of skill in the game, i.e. that nontrivial strategies exist and impact the game outcome. However, the converse is not true; it is not the case that the lack of existence of +EV players/strategies imply that there is no skill in the game. A clear game of skill such as chess could fail to demonstrate +EV players if the rake taken on the game was too high or, alternatively, if the observed population was all of expert players of similar ability. The argument which focuses only on EV is essentially an outcome-based approach to predominance and thus has shortcomings in each of the ways we've explored already.
In terms of actual legal interpretations of predominance, unfortunately, most of the time, house games and multiplayer strategy games are lumped in together and treated as a homogeneous group. Another part of the mostly-wonderful the DiCristina ruling that disappointed me was a very succinct dismissal of the possibility of treating one-player "games" separately from multiplayer games, though I acknowledge that the historical precedent probably does work this way:
2. Gambling Not Limited to House-Banked GamesCertainly, we must keep in mind that both some legal systems and some of the general public will refuse to appreciate the relevance of the categorical distinction between multiplayer strategy games and simple gambling. So, if we are in a situation where we must consider both types of games together, we can't classify the relative influence of skill and chance in a game by EV alone, as that might be affected by the particular nature of an operator's chosen fee structure or the population of opponents under observation.
There is no evidence in the record that Congress considered whether a game was housebanked was a relevant characteristic in determining whether it constituted gambling under the IGBA. Neither dictionary nor common law definitions of gambling distinguish between games based on that factor. Nor does any federal gambling statute other than the IGRA rely on whether a game is house-banked to define its scope.
Part 12: Why not sports betting and investing? -->
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