Saturday, November 19, 2011

Pokerfuse article: "Where and When Should You Buy 'Bubble Insurance'?"

I wrote another piece for pokerfuse, an analysis of "bubble insurance" offerings with a bit of light math, more in line with what I usually write about here.

Check it out here: Where and When Should You Buy “Bubble Insurance”?

Tuesday, November 8, 2011

Pokerfuse article: "DOJ’s Response to Campos and Elie: Summary and Analysis"

I recently had the privilege of writing a guest piece for pokerfuse regarding the recently-published Department of Justice document related to the federal case against former online poker payment processors Campos and Elie.

Check it out here: DOJ’s Response to Campos and Elie: Summary and Analysis.

I do hope to be back to semi-regular writing here soon once I can start finding time for it again. I have some ideas in the pipeline that I may be able to get to in the near future. Thanks for all of your support so far.

Saturday, June 18, 2011

Update/correction to risky site model: Casualty losses and tax effects

This is not the kind of stuff that I want to be spending my time writing about, but I discovered a serious enough practical change to one of my old analyses that I felt it was important to inform you all of it.

In my model for bankroll management and game selection in risky sites, I made the following assumption, which I thought was a reasonable guess as to the tax implications of losing access to poker funds:

...For simplicity, we'll assume that any money lost due to site-specific risks [sites closing down or withdrawals being seized] is treated the same as a poker loss and directly deductible against poker winnings, though I believe this is not exactly true.

It turns out that this is indeed not exactly true. In fact, being able to deduct this money against poker winnings is probably not possible in many cases.

The default classification for such a loss would be as a casualty loss. Even though a player may have been "gambling" on the chance that the site would return his funds, this must be considered a casualty loss rather than a gambling loss, which brings about some serious restrictions in many cases. The effects on the affected player's bottom line are significant.

Tax effects of stolen funds for those who file as professionals

Professional poker players can take such a casualty loss related to his poker business without any limitations. So, if you file as a pro, you can ignore all of this.

The original approach is still accurate for anyone filing as a professional on their taxes.

...but for amateur players...

Amateur players most likely have to take the loss of poker site funds as a casualty loss, which is a deduction that is strictly limited. A casualty loss can only be deducted to the extent that it exceeds 10% of adjusted gross income (AGI) — and, remember, while an amateur player still ends up getting to use his net (positive) gambling winnings for his bottom-line taxes, the AGI is a figure which is calculated after all winning sessions are added, but before losing sessions are deducted.

Amateur players can only take a casualty loss if it exceeds 10% of their AGI, an amount which will always be artificially inflated above actual income due to session-by-session accounting that amateurs must use.

In almost all practical cases, this will mean that losses due to the risks of the current online poker environment will not be deductible at all unless you file as a professional.

Example
Bob, a formerly-winning recreational small-stakes player, decides to deposit $500 on a current risky U.S.-facing poker site. Though he knows he will not make much money, he enjoys playing and wants to keep his poker skills in practice. He averages $3/hr in risky site funds by playing on the site, and after playing for several months, he has run his balance up from $500 to $1,500. The site then has its funds seized, declares bankruptcy, or otherwise absconds with the money. Bob was never able to cash out successfully.

The results:
  • If Bob's AGI for the year is above $15,000, he cannot deduct any of the $1,500 that he lost. This is almost certainly the case, even if Bob does not suffer from undue AGI inflation from his poker sessions.
  • However, Bob still earned $1,000 in gambling winnings from his play on the site, even though he never got his money.
  • Notice that, given that the site ended up disappearing, Bob would have been better off if he lost money on the site. Winning has increased his taxable income, despite not providing any actual money or deductions to Bob.
Effectively losing money by winning at poker on a site that disappears is highly unfortunate, and a pretty big deterrent to playing at all when there is any risk of losing one's online balance to the site-specific risks.

Implications for the model

I modified the old model by making the losses due to unretrievable funds occur after taxes, rather than before.

To keep the charts simple, I considered only the 50NL case this time. See the old article for the other assumptions.


We see that, as one might expect, the effects of this tax correction do not change much based on the per-day probability of site closure (i.e. the average lifetime of the site). If the site is likely to die at some point, the hit of this tax situation will be about the same whenever it happens, at least when cashouts are liquid enough along the way.


Much more important is the liquidity of the site, which will dictate the probability of funds being lost forever when the site disappears. After all, these funds being lost is what causes this fun little tax situation to occur.

Once the probability of losing funds becomes reasonably high, the utility gained by playing drops off quite a bit. Remember, playing and winning will go on to cost money if the site ends up not paying. Even in the extreme case where there is no probability of the site ever allowing a cashout, the "money" won on that site would still be considered gambling winnings for a player who knowingly put money onto the site for the purposes of practicing his game. What a mess.

Overall, I would think that any reasonable estimate of the safety of current risky sites is going to involve more than enough risk to really cut into the expected utility of playing. So, if nothing else, this provides another strong incentive to choose even lower stakes than one otherwise might. The less funds that are at risk, and the less likely a player is to win money and go on to get it stolen from him, the less the effects of these negative tax implications will be.

Other bad news

Americans who don't file as professionals and who currently have funds stuck on Full Tilt Poker (or, more likely, the other, less-reputable sites) will also suffer negative tax effects if those funds end up not being returned to the players. Unless their online bankrolls were more than 10% of their total income for the year (plus the phantom session-by-session income in the AGI), recreational players will get no deductions at all if their money is absconded with.

Strangely, in the event that the funds aren't returned, winning recreational players would have been much better off somehow losing their entire bankrolls prior to April 15.

In particular, if Full Tilt Poker goes on to end up not returning U.S. funds, then of the alleged $150 million in American funds that are stuck there, I would guess that at least $50 million of them will be completely non-deductible, even though much of that sum represents taxable poker winnings.

Possible exceptions

It's possible that, at least in the case of money put on post-Black Friday sites knowing that there would be a chance of not being able to get money out, there might be a way to argue that the losses should be gambling losses rather than casualty losses.

The key factor here is the notion of constructive receipt, which is the rule which causes poker income to be taxable when it is earned rather than when it is cashed out. The underlying principle is that, as soon as a taxpayer is able to undergo actions to have the money in their hand, that income is immediately taxable.

In the case of a poker site that disappears with player funds, one might argue that the player did not actually have the ability to ever receive the money and thus that constructive receipt does not apply. If this approach were deemed to be valid, there would be no taxable income.

It is likely important that the funds weren't accessible at the time they were won, rather than just at some later date when the site closed or when a cashout was attempted. For a hypothetical site which has never processed withdrawals, this would be true. For U.S.-facing sites which are currently not processing U.S. withdrawals, perhaps this is true, though the mere possibility of you being able to successfully withdraw in the time after the money was won might be enough to invalidate this. For pre-Black Friday sites, it's definitely less true, as though it certainly had been difficult to withdraw funds prior to Black Friday, people were able to do it with some degree of regularity. It's hard to guess where this line would be drawn.

Also, it is difficult to find a way to reconcile this sort of accounting rule with the necessary session-by-session accounting, which implicitly assumes constructive receipt.

I find the tax effect described in this article to be quite absurd, even in the context of the other IRS rules that produce various unfair tax situations for poker players. Despite this, there might not be any reasonable way around it. Filing one's taxes as if these losses were gambling losses instead of casualty losses may not be considered appropriate by the IRS, which has a history of interpreting rules for anything related to "gambling" as harshly as they are able to. Fighting to clarify the nature of these losses may be costly.

Thanks to taxdood, Russ Fox, and PokerXanadu for helping me understand this tax situation.

Wednesday, June 8, 2011

WSOP Utility Analysis revisited, part 2: How many shares should a WSOP Main Event player sell off?

Last time, we looked at the relationship between a player's expected utility in the 2010 World Series of Poker Main Event and his skill advantage over the field. Under a particular proposed shape of finish probability distribution, we found that a raw ROI of about 86% (i.e. an average cash of $18,600) was necessary for a player with a typical income, risk aversion, and tax obligation to simply break even in terms of expected utility. Even in a juicy WSOP Main Event field, this is a pretty lofty goal for most, and many near-average players will be forced out of participating on their own dime unless they are willing to effectively pay for the privilege.

Fortunately, backing and staking agreements are common for large-field poker tournaments. Much as the stock market investor would never put a large portion of his capital into a single investment unless it were extraordinarily profitable, the poker player (who "invests in himself" in his poker career) will often benefit from diversifying away some of his risk by hedging his tournament results out to others. If these mediocre winning players players are able to find other parties to put up part of their entry fee in exchange for part of their prize, they will be able to yield a positive expected utility in the event, not only for themselves, but also for their investors.

Shares sold at face value

For the purposes of this analysis, we assume that only option of staking/backing available to the player is to sell off X% of his prize in exchange for X% of the cost of entry (selling shares at 1-to-1, with no markup). The result for the player of such a contract will be the same as if the entire tournament were scaled down by X%. A player can sell off 90% of himself to effectively make the WSOP Main Event a $1,000 buyin tournament for him, with prizes which are exactly proportional to those of the true main event.

For the time being, we are ignoring some other popular forms of staking and backing (listed in the order of likelihood that I might add them to the model in the future):
  • Selling shares at a price other than 100% of face value — If a player with a significant skill edge wanted to sell pieces of his action, in reality, he sould expect to get much better than 1-to-1 from his investors, since he's the one doing the work. The investors would still be left with a very profitable, fast, hands-free investment. Conversely, a -EV player might still be able to gain some expected utility by selling pieces of himself at a discount.
  • Direct backing — One popular form of contract is for the investor to provide all of the player's entry fee in exchange for a payoff equal to a fixed percentage of the player's profit in the event that he cashes. This is a freeroll for the player and will thus always yield him a positive expected utility, and it can still provide the investor with a positive expected utility as well if the player is sufficiently skilled. It is reasonable to expect that this sort of deal may be more favorable than selling shares at face value if the player is very skilled, but also very risk-averse relative to the stakes of the event.
  • Long-term, ongoing backing agreements — Some players have professional backers with whom they enter into long-term deals. The investor pays all of the player's buyins for a series of tournaments in exchange for a percentage of the player's profits, but if the player is already at a net negative from previous tournaments, he must repay that amount to the backer in full before being able to realize any profits from the contract. These agreements have several variables and would be complicated to analyze, and players under such agreements may not have the opportunity to consider other hedging options anyway, since they are often locked into their contracts until they expire.
For now, we consider only the simplest case: shares sold at face value.

Given this opportunity to rescale the stakes of the tournament, assuming that there is an investor willing to buy any amount of shares that the player would offer, how much should the player look to sell off?

Optimizing share-selling for the typical player

Our typical, risk-averse player ($80k net worth, $50k income with at least $10k from poker, risk aversion of 0.8) will realize the following expected utilities based on his skill advantage and the percent of himself he chooses to retain:


Here, the different colored lines represent different several different levels of skill edge, expressed in terms of raw ROI. The horizontal axis shows the percentage of his own action that the player takes; the amount he sells off is equal to 100% minus this number.

In red — For the player of precisely average skill, who has a raw ROI of -6% (due to rake), we see that, regardless of the number of shares he sells, he cannot realize a profitable opportunity in this event. Since he's a break-even player and is risk-averse and experiences tax effects that are negative on average, he's going to lose utility by playing any poker tournament, regardless of how small he makes the stakes.

In orange — When he was forced to take all of his own action, we recall that the small winner (raw ROI of 50%) was forced out of being able to profit from his small skill advantage at all. We showed that the minimum ROI required for a positive expected utility is 86%, and the chart verifies that if this 50% ROI player were to take all of his own action, he would be losing money after taxes and risk aversion. By selling shares, we see that he can realize a small positive expected utility ($121 in certainty equivalent) by playing for about 12% of his own action. The ability to hedge against the entry fee has allowed the skilled, risk-averse player to realize a profitable opportunity where he otherwise could not.

In yellow — This solid winning player (raw ROI of 100%) is a strong enough player that he will realize a positive expected utility even if he takes all of his own action, as we can see by the yellow curve being completely above the x-axis. However, we see that this player will realize an even higher expected utility by selling off some of his action than by paying his own way entirely. He'll improve his certainty equivalent payoff from $532 to $805 by selling off roughly half of his action.

In green — A bigger winner (raw ROI of 150%) turns out to do best by playing for all 100% of his own action. The investment has become so profitable that even a risk-averse individual does best by taking it all on and not hedging it out to others.

In blue — This big winner (raw ROI of 200%) has similar results to the green player above.

Other cases: different risk/tax profiles

If we keep the player's relative risk aversion fixed at ρ=0.8 but increase his wealth from $80k to $500k and his annual income from $50k to $100k, he becomes more able to handle his own risk:


The breakeven player is, of course, still unable to profit, and it turns out the 50% ROI player still benefits slightly from selling off some of his action, but overall, the additional risk tolerance incentivizes this player to hold onto all or most of his own action.



Alternatively, instead of adjusting wealth, we can reduce the player's relative risk aversion from ρ=0.8 to ρ=0.5, representing an individual who is more willing to take on risk (at least for the special occasion of the WSOP Main Event, perhaps):


The nature of the effects is similar. It is worthy to note that the orange curve (raw ROI of 50%) is similar to that of the original analysis for the more risk-averse player, suggesting that a small winner with an average wealth should still be selling off most of his action regardless of his personal preferences for risk. On the other hand, it looks like the yellow curve (raw ROI of 100%) has become roughly the point where the player will prefer to take 100% of his own action for this particular level of risk aversion, so players with significant skill edges should be more inclined to take all of their own action if they have a higher tolerance for risk.

Optimal hedging percentages

Thanks to the complexity of the utility function and the sheer number of different payoffs, there is no simple way to express a formula for the curves we've found above. In order to calculate the optimal hedging percentages (i.e. the percentages of action to take which correspond to the maximum points of these curves), we proceed numerically.

Here, we disregard the ρ=0.5 case treated directly above and consider only the first two cases: the original "typical player" (in yellow below) and the "wealthy player" (in green below):


(Ignore the jaggedness of these curves; the negligible inconsistencies are a consequence of the numerical error of Excel's goal seek solver.)

Rather than only considering five different specific values, this chart looks at every possible value of raw ROI and provides a more comprehensive practical resource.

As we've seen earlier, a player with an ROI less than 0% does best by playing for 0% of his own action, i.e. not playing at all, unless some misinformed or charitable investor were to give him a full stake. For players with positive ROI, we see that there is always some positive percentage of his own action that produces a better profit than not playing at all.

The optimal percentage of his own action that the player should take seems to increase in a convex way; as the player increases his skill edge over the field, the optimal percentage that he should keep increases faster at higher values of ROI. For both the typical player and the wealthy player, there is a "ceiling" level of minimum ROI at which the player should take all of his own action. We see that this is about 136% for the typical player and about 66% for the wealthy player.

Conclusions and comments

These charts should provide a useful guideline for real-world staking and backing decisions for large-field tournaments. Some practical notes:
  • This analysis was done for the 2010 WSOP Main Event. Most other tournaments (likely including the 2011 WSOP Main Event) have much smaller fields, and, accordingly, have less skewed payoffs and have less extreme utility annihilation effects. So, for a $10k tournament with a smaller field, the optimal percentage of one's own action to keep will increase, and the guidelines in this post can be used as a lower bound to this.
  • Similarly, for tournaments with buyins less than $10k, the optimal percentage of one's own action to keep will increase, and for tournaments with buyins greater than $10k, the optimal percentage of one's own action to keep will decrease. In these cases, the guidelines in this post can be used as an upper/lower bound.
  • Note that every aspect of this analysis holds just as true for the person making the investment as it does for the person doing the playing. If a player and his investor(s) all have roughly the same wealth and utility, then each of them will be doing best by taking on the recommended optimal percentage of the player's action as recommended by this model. For example, when the player has a 100% ROI and wants to maximize the total utility among himself and his investor, the parties will both roughly optimize their expected utility by the player keeping 50% of his action while one investor takes the other 50%. In the case of the 50% ROI player, he would take about 12% of his own action while selling off equal pieces of about 12% each to 7 different outside investors.
  • In reality, players can't know their exact ROI in any given tournament. The best that players can do is form something resembling a maximum likelihood estimate based on their assessment of their own ability, the expected field strength of the tournament, and the tournament struture. This could be modeled as a random variable with some uncertainty (likely Gaussian) about the point estimate. That is, if your best guess of your ROI is 50%, a more accurate implementation would involve your ROI being an unknown random variable with mean 50% and some nonzero standard deviation. Close inspection of the first chart shows that the distance between the ROI curves seems to get smaller as ROI gets higher, which means that, in the face of an uncertain ROI, it's best to "round down" a little for the purposes of plugging a fixed ROI into this model. For example, if you estimate your ROI is about 50% but have a lot of uncertainty about this estimate, you will probably get a slightly more accurate result by using something like 45% in these guidelines.
Let me know if you'd like to see me add the considerations of other types of staking and backing contracts to this model. And, as was the case with my risky site bankroll management model, this is a model with many specific variables (wealth, income, risk preferences, ROI, tournament field size, tournament payout distribution) that would ideally all be tailored to each specific player and each specific tournament on a case-by-case basis.

This is a model that should be of tremendous practical value to all tournament players, so if there is enough interest, I might clean up my spreadsheet and make it publicly available in the future.

Monday, May 30, 2011

WSOP Utility Analysis revisited, part 1: How much of an edge is necessary to overcome tax and utility effects in the WSOP Main Event?

A few months ago, I looked at the true costs of playing the WSOP Main Event for the typical player after adjusting for the effects of tax and risk aversion. It's worth a read if you missed it. Overall, we found that there are significant tax and utility effects that a player must overcome, and thus that there is significant extra effective rake in playing the event.

Exactly how much of a skill advantage does a prospective WSOP Main Event player need in order to overcome this effective rake? Can we... quantify it?

The old model was built on the assumption that the average player was equally likely to finish in any of the 7,319 places in the tournament. To allow for our player in question to have a level of skill different from that of the rest of the field, we need a way to map player skill to a specific finish probability distribution, a specification for the probabilities of finishing in each of the 7,319 different places.

Connecting skill edge to a finish probability distribution

If we generalize our scope to a tournament with N players, then we know that, when all players are equally skilled, each player will finish in each place with probability 1/N, as we've stuck to in the old model. If a specific player has a different strategy than the other players (whom we will assume to all still be uniformly skilled), then what does his finish probability distribution look like?

We quantify skill through the typical tournament results convention of raw ROI, which we define as the player's pure return on investment in the absence of tax and utility considerations, but including the rake. For example, in the WSOP main event, since $600 of the $10,000 entry fee goes to rake, the average player's raw ROI is -6%. This is ROI as it is commonly used in discussions and in tracking software; we add "raw" to convey that it is in pure dollar terms, without utility or tax effects. Since this is the measure of skill edge that is commonly used, it is desirable for us to define our correspondence between player strategy and finish probability distribution by creating a mapping between raw ROI and a probability distribution.

My first naïve method of constructing such a family of finish probability distributions was to start from the uniform 1/N distribution, but to perturb the terms such that the probability of finishing in each place was equal to a uniform constant plus the probability of finishing in the prior place. In math, this family of distributions looks like:


In addition to the necessary properties of probability distributions (nonnegative probabilities which sum to 1), this naïve distribution has a basic property that would likely be desirable in any probability distribution for tournament finishes: the distribution is increasing in that, when the player's skill edge is positive, he has a higher probability of finishing in each place than he does the next-worst place. (Similarly, when the player's skill edge is negative, he has a higher probability of finishing in each place than he does the next-higher place.)

However, there is no reason that the increasing increments between each place should be equal throughout the entire distribution, as they are here. Our simple guess may not approximate reality very well. There's also another problem with this family of distributions: it turns out that we can't capture every possible ROI (from -100% to a certain 1st place) through this type of distribution.

Note that this model, as well as all of the forthcoming models, produces the desired result of uniform probabilities of 1/N when we set the player's skill to zero.



A second naïve guess for developing a family of finish probability distributions is to reframe the tournament as a series of heads-up matches, and to give our player a specific probability for winning each heads-up match. This generalizes through logarithms to tournament field sizes that are not powers of 2. In this approach, we get a different probability distribution depending on whether or not we start from the probability of getting 1st place and use conditional probabilities downward through the places, or whether we instead start from the bottom and "condition upwards". If we condition downwards, this distribution looks like:


This is again an increasing distribution, and one that should make sense for a bona fide heads-up tournament with 2^k players, but other than that, there is no intuition as to how it might apply to a non-heads-up tournament.



The most elegant and accurate approach that I've come across is one based on the following foundation: we assume that a single player who is superior to his uniform opponents and who is playing in a tournament with equal starting stacks (as is the case in real poker tournaments) might have a similar finish probability distribution to a player who is not superior to his opponents, but instead starts the tournament with a larger chip stack than the rest of the field. (Thanks to Aaron Brown for this idea for modeling finish probability distributions, and if you enjoy both poker and finance and have not read his excellent book The Poker Face of Wall Street, you are missing out.)

To reiterate:

A superior player's skill edge in a tournament could be approximated in a model that removes his strategic advantage but instead gives him a larger starting chip stack.



From this point, we just need a way of mapping one's chip stack to one's finish probability distribution. The most popular method of this is the Independent Chip Model, i.e. the Malmuth-Harville tournament chip valuation algorithm, and that's what I've chosen to use, as it happens to produce an identical result to instead parameterizing the player's skill edge by starting with his probability of finishing in 1st place and then conditioning downwards. We don't get a closed-form formula, but we get an iterative formula that is easy to implement in Excel:


See The Mathematics of Poker for a discussion of other methods of tournament chip stack valuation; it seems that Malmuth-Weitzman is the only reasonable competing theory to ICM out there, and it only differs from ICM in how a busted player's chips are distributed among the remaining players on average. I haven't spent too much time thinking about the differences between these models. My understanding is that each of them diverges from real historical tournament results, though I do not know if anybody has analyzed this rigorously.

Adding this finish probability distribution to the WSOP expected utility analysis

The hard part's done! Now we can just take this parameterized finish probability distribution and plug it into our good old WSOP Main Event expected utility analysis and see what we get.


We see that, under this model, the relationship between the CE payoff and the player's raw ROI is close to linear. Examining the data confirms that this visual intuition is accurate.

In particular, we see that, for our typical player ($80k net worth, $50k income with at least $10k from poker, risk aversion of 0.8), the ROI needed to simply break even after tax and utility considerations is 86%. So that answers the question we were left with when we first looked at expected utility analysis for the WSOP Main Event.

After correcting for tax effects and risk aversion, the typical player needs to have a raw ROI of 86% in order to break even in certainty equivalent by playing the WSOP Main Event.

Having higher net worth helps alleviate this high threshold. A player with a net worth of $500,000 and a YTD salary of $100,000 needs a 42% ROI to break even. A player with a net worth of $5,000,000 and a YTD salary of $1,000,000 needs only a 7% ROI. Again, we see that the WSOP Main Event is best suited towards the wealthy, despite its "everyman" appeal... though it is precisely the fact that the event appeals to so many everymen that causes the risk to be too great for them.

Being less naturally risk-averse also brings down this threshold. The typical player with $80,000 net worth and a YTD salary of $50,000 needs only a 61% ROI if we reduce his ρ from 0.8 to 0.5. Further reducing ρ to 0.2 still leaves him needing a 33% ROI. Reducing it all the way to 0 (i.e. no risk-aversion, and utility is realized on the full amount of dollars remaining after taxes) brings it down to a 11% ROI. Risk aversion is demanding a higher minimum skill level than tax effects are.

Coming up next...

That's all for today, since building the finish probability distribution took so long. But now that the hard part's out of the way, we can play around with our expanded model in some other practical ways.

Note that I realize that this year's WSOP Main Event is likely to have far fewer than 7,319 competitors. For a smaller tournament, both tax and utility effects will be lessened, and lower skill edge will be necessary to overcome these. It would be interesting to generalize this model to an arbitrary N-player tournament, if there were an algorithmic, standardized payout formula that covered all possible field sizes. (Does one exist for any popular tournament series? Drop me a comment if there is one.)

For now, knowing that a player with typical risk aversion and tax effects will need to have a pure expectation of $8,600 in the WSOP Main Event is a pretty valuable baseline that can guide decisions. This again illustrates how the large field size and high buyin of the WSOP Main Event effectively crowds out the average player form participating — unless that player is willing to pay a high effective cost.

But what if the player could reduce the size of the buyin for this event?

Next time, I look at the possibility of the player being able to sell off some of his action at 1-to-1 (i.e. no markup), which effectively reduces the buyin from $10,000 to something more manageable for less wealthy players. Indeed, doing this can change an unprofitable opportunity into a profitable opportunity for the risk-averse tournament player.

For different levels of raw ROI, we will solve for optimal percentages of one's action to take in the idealized WSOP Main Event. This should be valuable information not only for prospective tournament players, but also those that might buy shares in them. Stay tuned.

Saturday, May 7, 2011

To play or not to play: Optimal game selection with risky operators

There are few certainties in the post-Black Friday online poker world. One fact that we are all still sure of is that nothing has changed with respect to the fact that playing online poker is legal on the federal level in the U.S. If an American player finds a poker site willing to take his business, and if he can manage to get his money to and from that site, he is not violating any federal law by doing so.

So while the remaining U.S.-facing online poker sites are unlikely to be anywhere near as large, reputable, or financially liquid as what players are used to, as long as the risk of losing one's funds is small enough, there may still be some (severely-reduced) profit opportunities for American poker players, or, at the very least, a means of continuing to stay competitive at poker by practicing one's skills at one's leisure.

The remaining U.S. sites should be expected to be different than PokerStars and Full Tilt Poker in at least a few incredibly important ways:
  • Since the subset of former American players which chooses to move to these sites will be highly skewed towards serious, professional players, all players should expect their winrates to drop significantly.
  • Deposits and withdrawals will be more costly and more unstable, and players should account for some nonzero probability of never receiving a cashout.
  • In the event of either voluntary or government-induced site closure, due to liquidity issues and the fact that these sites are less reputable, there is some probability that U.S. players would never get their account balances returned to them.
Not too comforting. To be sure, even if playing on the remaining smaller sites is otherwise acceptable and a reasonable substitute for the experience of pre-Black Friday online poker (which will not be the case for all poker players), these issues are serious and will annihilate a lot of utility. These added risks cause players to have to make tradeoffs involving their profit potential and the amount of money they risk in their account balances at these sites.

Putting it all together, how big are these costs? How bad do these risks have to be before it's not worth even trying to continue to play online? We can quantify this with a model which can estimate the effects of these risks on expected utility through Monte Carlo simulation.

Model inputs

We start by making assumptions about various aspects of the risks and costs of playing on these sites. Some of these are deterministic, and others will be uncertain. Once we have settled upon reasonable estimates of these values for all of these parameters, we can vary the most critical parameters to find break-even thresholds for expected utility, which can guide player decisions.
  • Fees associated with deposits and withdrawals — Deposits and withdrawals at smaller sites are costly, in both inconvenience and fees, so managing account balances carefully will be important to avoid incurring too many of these expenses. Our model will consider fixed fees for depositing or withdrawing, which we can expand to include intangible costs that reflect the inconveniences of these money transfers.
  • Deposit/withdrawal strategies — The simplest way to describe a rule that guides when to cash out or deposit is a set of four numbers. When the player's account balance is below some critical level (such as when it is too low to play his chosen stakes and number of tables), he should redeposit to bring his account balance up to some higher threshold level. Similarly, since he doesn't want to keep an unnecessary amount of money in a risky account, when his account balance hits some upper critical level, he should withdraw to bring his account balance down to some lower threshold level (which might be close to, if not exactly the same as, the threshold level for deposits).
  • Winrate, standard deviation, and play volume — The model will approximate poker results by increments of the appropriate normal distribution. To match the rest of the model, rather than looking at winrate and standard deviation per hand, we can look at these on a per-day basis by factoring in the amount of hands the player plans to play each day. The number of days remaining in the year will be one of the inputs.
  • Per-day probability of site closure — In our model, at the end of each day, we will assume that there is a fixed probability that the site will close forever. Each day will be independent of the last.
  • Probability of getting paid if the site closes — When the site closes, there is a chance that all player balances will be lost.
  • Probability of getting paid on each cashout — For each cashout prior to closure, there is a chance that the player will never see his money (or never be able to cash out in the first place). We can assume that this probability is less than that of the probability of getting paid when the site closes.
  • Tax and utility functions
That's a lot of inputs, and a lot of inputs that we can't necessarily get great estimates of. Since the non-standard parts of this model are those pertaining to the risks of having money with the online site, we can ignore the lesser (or at least more standard) uncertainty on personal game performance parameters and instead assume that we know the player's exact winrate.

Parameter choices

We'll again work with our typical poker player: $80k net worth, $40k non-poker income, $10k year-to-date poker winnings, and isoelsatic utility with ρ=0.8, paying both federal and NJ income taxes. For simplicity, we'll assume that any money lost due to site-specific risks is treated the same as a poker loss and directly deductible against poker winnings, though I believe this is not exactly true.

UPDATE 06/18/2011: The above assumption may only be valid for those who file their taxes as professional poker players. Amateur poker players likely cannot deduct these losses at all in most cases, and this has serious implications for the model. See Update/correction to risky site model: Casualty losses and tax effects.


I'm not entirely familiar with costs of deposit and withdrawal at all of the remaining U.S.-facing poker sites, but for Carbon Poker, it looks like roughly $17 in costs to make a deposit, and about $5 per withdrawal for a medium-volume player.

We'll look at two possible levels of stakes for our player: multitabling $0.25/$0.50 NL at an hourly winrate of $10 with a standard deviation of about $70, or multitabling $0.50/$1.00 NL with an hourly winrate of $15 and a standard deviation of about $140. His $0.50/$1 winrate is higher in dollars, but lower relative to the stakes, which introduces not only the usual higher relative risk to his results, but also a need to keep a higher balance with the site. Keep in mind that he will have a much tougher time achieving these winrates on a remaining U.S.-facing website, since the player pools will be significantly tougher and it may not always be possible to play as many tables as he would normally be accustomed to. In the post-Black Friday market, I would expect winrates to be significantly lower than the good old days, and most players should probably drop down at least a level or two.

We assume that, for $0.25/$0.50 NL, he will redeposit when his balance falls below $800, and he will redeposit up to $1,200. He will withdraw down to $1,200 when his balance hits $2,000. For $0.50/$1 NL, we will double each of these deposit/withdrawal thresholds. Once we settle upon the other parameters, we can play with this to see which withdrawal strategy is optimal.

As a default, based on no science at all and simply my gut assessment of the risks of the current poker market, we'll assume that the per-day probability of site closure is 0.002 (i.e. the site lasts, on average, 500 days), that there is a 50% probability of players getting paid if the site closes, and a 75% probability of getting paid on any given cashout prior to closure.

We'll have him start playing in early May, so there's about N=240 days left in the year.

Algorithm

To calculate the expected utility of playing with these risks, these winrate parameters, and this utility function, we will simulate the system through the following steps:
  1. Initialize the player's starting bankroll by him making his first deposit, tracking the costs of doing so in a running net total starting from his year-to-date winnings. This running net will NOT include his day-to-day poker results; money on the site is not counted as a gain or loss until it is successfully withdrawn.
  2. At the start of each day, the player plays his daily poker session, and we adjust his site balance by a normal random variable with appropriate mean and variance. Since he can't lose more than he has on the site, we truncate this normal distribution on both tails by preventing the magnitude of the swing from exceeding his existing balance.
  3. After his session, if his bankroll is below his deposit threshold, he deposits according to his strategy. We track both the deposit itself and the costs of doing so in the running net.
  4. Then, if his bankroll is above his withdrawal threshold, he withdraws according to his strategy. He ends up receiving the cashout according to the chosen probability, which will be added to the running net, less the costs. If he gets unlucky and misses on the cashout, we assume it is lost forever.
  5. At the end of each day, the site closes down according to the chosen per-day probability. If the site shuts down, the player's balance is returned to him according to the probability of getting paid if the site closes. Either way, if the site closes, we exit the loop and go directly to step 7, as there's no longer any poker to be played (we neglect the possibility of choosing another remaining site for the rest of the year).
  6. Repeat steps 2-5 for each of the remaining N days.
  7. At the end of the year, if the site is still open, for the purposes of evaluating his year-end utility, the player withdraws his balance (and receives it according to the chosen probability). This gives his final net winnings for the year, and we evaluate the after-tax utility of this amount.
  8. Repeat steps 1-7 in a Monte Carlo simulation to simulate the average expected utility.

Some results

For the parameters we chose, we can look at how the results change as we perturb the most uncertain parameters, those related to site-specific risks.

First, if we disregard the assumption of getting paid 50% of the time if the site closes, and instead vary that, we can see what the effects of additional liquidity are and find the break-even liquidity level. Since this liquidity and the probability of successful cashouts are linked, it is desirable to adjust both at the same time, so as we vary the liquidity level, we will assume that the risk of losing a cashout is always half the risk of losing a bankroll in the event of closure.


The rightmost point is when the site is fully secure, which is what decision-makers have been roughly used to in the pre-Black Friday environment. In this case, the player prefers to play his more pure-EV-profitable stake of 100NL. As the site gets less and less secure, the need to keep a higher bankroll at the site for 100NL will create more and more risk, and 50NL becomes better when the site liquidity falls below about 38%. For less than 5% liquidity, the increase in the player's utility by playing either stake is negative, so the player should refrain from playing at all, unless the entertainment value or the value of being able to practice and work on his game is worth the cost.



If we return the liquidity probability to its original default rate of 50%, we can instead look at how expected utility varies with the per-day probability of site closure.


When changing the average lifetime of the poker site, we see that the effect is less linear. Again, the rightmost point corresponds to perfect safety, a site that has no risk of closing. As the probability of closing is increased (and we assume a 50% chance of not being able to get money back after closure), the expected utility drops off significantly. 50NL overtakes 100NL around a per-day closure probability of 0.004 (i.e. a mean lifetime of 250 days). Not playing at all becomes the best choice at a per-day closure probability of around 0.01 (i.e. a mean lifetime of 100 days).



If we return to the original, fixed best estimates of these parameters (50% liquidity, average site lifetime of 500 days), we can tweak the deposit and withdrawal strategies to see which one works best. We'll stick with just 50NL here, since it seems to be the better choice. We need to keep the lower deposit threshold at $800 in order for the player to have enough bankroll to be able to play a bunch of tables of 50NL during his sessions, but we can vary the other three account balance strategy parameters.


We see that we can actually do better than the deposit and withdrawal strategy that we initially chose. It turns out to be better to wait to withdraw until at a higher balance, but still to deposit up to just $1,200. I suppose that this asymmetry comes about due to our utility function; we have a higher relative risk tolerance when we have earned more money on the year. We see that the "tightest" deposit and withdrawal thresholds do not do well here. This is all due to the interplay between the site-specific risks and the costs of moving money, and this behavior will probably change when any of these are modified.

Conclusions

If we compare the middle parts of each of the graphs to the rightmost points (which correspond to perfect safety), we can really see how much our utility suffers under these new risks. Players will need to have significant edges over their competition to overcome this.

In these results, the "shape" of the results is no surprise. We're not seeing anything we wouldn't have been able to guess without doing the math, but quantifying it provides a useful framework for guiding our play decisions under the uncertainty of the site-specific risks.

For our example player here, we see that, under the default assumptions with 50% liquidity and an average site lifetime of 500 days, playing 100NL will be a better risk-adjusted value than 50NL, though it's close enough that the player should move down if he feels the liquidity risk is greater, or that the sites have a significantly shorter expected lifetime. To dissuade him from playing entirely, there would have to be either a very low probability of getting paid back if the site closed, or withdrawals while the site is still operating would have to be less than 50% likely to go through. So, for this winning player, it looks like playing on a remaining U.S.-facing website should be better than quitting entirely as long as he has at least some faith in the current market. Many less skilled players with thinner winrates will be forced out of the market entirely from these additional risks.

This model is well-suited to being dynamically updated as the year goes on. The time horizon and year-to-date winnings will change constantly, which will shape bankroll decisions. Also, it should be useful to update the site-specific risks with better estimates as time goes on and as more information develops in the wake of Black Friday, or as the market positioning of the remaining U.S.-facing sites change.

While it's easy to see results by varying the site-specific parameters in this model when the player-specific variables (utility, income, winrate, etc.) are fixed, it's hard to draw broader conclusions over a more general player base. This is the sort of model that is best applied on a case-by-case basis, with each user's particular play variables fixed and known. If enough people are interested in using the spreadsheet I wrote for this, I might consider cleaning it up and hosting it somewhere.

Wednesday, April 20, 2011

Apocalypse

Last Friday, the webpages at PokerStars.com and FullTiltPoker.com were replaced with the following image. Shortly after that, Americans were no longer able to play poker on these sites. Regardless of the immediate outcomes of this event, and regardless of the eventual future that online poker will have in the U.S. and in the rest of the world, this destructive change marked the sudden and immediate end of the first era of online poker.


After my dozens of hours of wading through online discussions over the past several days, I have been at a loss for what to say regarding what the poker world is calling "Black Friday". I don't have anything valuable to add to the ongoing community conversation for now, so I've erred on the side of staying quiet so as not to add to the chaos.

Nonetheless, I wanted to at least provide a brief summary and a few thoughts, mostly for the benefit of my personal acquaintances who may not be familiar with poker, or who may not have followed this particular issue closely. Since I have this blog, I might as well put it here. It won't be anything new for anyone who has been following the forums.

This is the biggest news in poker history, and the biggest damage ever done to the entire game's economy. It may even be the single highest-impact adverse event in the history of any competitive game.

In as few words as I can:

What happened?

Friday afternoon, the Department of Justice unsealed indictments against individuals associated with the largest U.S.-facing online poker sites, including PokerStars and Full Tilt Poker. These sites served only as a venue for players to play poker against each other, and they did not offer any casino gambling. The charges against the sites include both illegal gambling charges and bank fraud charges. In response to the indictment, these sites blocked U.S. players from depositing, withdrawing, or playing in their games.

Since 2006, there was a reasonable probability that the DOJ would take an action like this at some point. It was more of a matter of when it would happen, rather than if it would happen.

No federal law addresses online poker, though some outdated laws cover sports betting or general gambling games played against the house (rather than between players). The DOJ has felt for years that online poker is illegal under these existing laws, and they have been alone in this assessment.

The UIGEA, passed in 2006, did not change the legality of any form of gambling and failed to provide a framework with which the DOJ could actually prosecute poker sites. However, this law, which targeted banks and other financial intermediaries that dealt with illegal gambling websites, allowed the DOJ to go after the processing of deposits and withdrawals for poker sites. For most major banks, the threat that poker sites might be considered "unlawful online gambling" and that servicing them might attract the costly attention of the DOJ was enough to make it a bad business decision to accept such transactions.

Thus poker sites that continued to serve the U.S. were inevitably destined to work with increasingly less reputable banking partners. At some point, the DOJ was bound to have enough information to make some claim against the poker sites under the broad classifications of money laundering and/or bank fraud. That is the nature of the bank fraud charges in this indictment.

Aside from the alleged bank fraud, which presumably became necessary in order to continue to serve the largest poker market in the world, PokerStars and Full Tilt Poker are reputable, legitimate, global companies. They are explicitly licensed in every country which provides licensing for online poker; the sites would be happy to pay U.S. taxes in exchange for the benefits of U.S. licensing, but the U.S. still has not established a licensing framework. The sites operated in the U.S. under strong legal opinions that peer-to-peer strategy games like online poker do not constitute illegal online gambling. If they can demonstrate that the business of offering online poker to Americans does not constitute illegal online gambling, the bank fraud charges may not apply. It's complicated.

While I generally trust in PokerStars and Full Tilt Poker, I do not support their misrepresentation of their transactions to banks, if the allegations are true.

What are the immediate effects?

Nothing has changed with regard to the legality of online poker for the player. American players are not targeted in any way in this indictment, nor under any federal laws. Even the DOJ agrees that playing online poker does not violate any federal law. All of the laws at play here are those which target only businesses that operate or profit from "illegal gambling".

Despite the fact that the players have broken no laws, their account balances with these sites are currently inaccessible. The sites will attempt to return U.S. players' funds as soon as they are able, but presumably, with an ongoing investigation into allegations of bank fraud in the U.S., the sites are having trouble initiating any further financial transactions in the U.S.

PokerStars and Full Tilt Poker were the only licensed and reputable online poker sites that were willing to accept American players amid the country's ambiguous legal landscape. Other smaller sites continue to serve the U.S., but are neither safe nor liquid enough for the consideration of serious poker players, especially as moving money to and from international poker sites will continue to become more and more difficult until the U.S. changes its laws.

Even if the indicted sites go on to win in court and to clear themselves of all charges, which would allow them to resume their U.S.-facing business, this will take years.

So, basically, for now and for the immediate future, online poker no longer exists in America.

What happens next?

Americans will continue to be unable to access their balances with these sites for some time. While historical precedent and most of the informed legal opinions I've read say that the players will get their money back eventually (possibly years), there's some chance that these funds are permanently seized by the DOJ due to the nature of the fraud charges or otherwise lost due to a future bankruptcy of these one-time giant global poker companies. Having read many different perspectives on this complex legal situation, I think there's at least a 95% chance that U.S. players will eventually get back their money. edit: Just now, a DOJ press release confirmed that the DOJ is looking to allow the sites to return players' money in an expedient manner, so upgrade this to 99%.

Tens of thousands of American online poker pros are essentially out of a job (I do not expect the general public to sympathize with this). Millions more American gamers have lost the opportunity to conveniently and efficiently play the game that they love and responsibly enjoy. Those few that are addicted to gambling on poker will continue to play at the remaining unsafe sites. For both professional and recreational players, replacing online poker with live, brick-and-mortar poker at U.S. casinos is rarely an option, due geographical concerns as well as a variety of economic and efficiency reasons. Serious players will have few options for practicing and improving their game, and the rest of the world will pull ahead of America at competitive poker.

The entire modern global poker industry of the past several years has been built upon PokerStars and Full Tilt Poker, and the damage done to these two big online sites will have effects on the entire world of poker. Some poker tournament circuits and televised poker programs have already been cancelled, and countless more industry and media jobs will be disappearing as the poker economy contracts. I expect that this amounts to thousands of "real-life" jobs lost for Americans. The same negative effects will also carry over into other countries to some extent, as international players will have lost the ability to compete in a fully global player pool.

When the indictments are resolved, hopefully the illegal gambling charges are addressed in a way that leads to a court case that definitively establishes that poker is not unlawful gambling under U.S. law. Depending on who you ask, the illegal gambling charges are somewhere in between a real stretch legally and purely frivolous or just for show. However, the bank fraud charges are severe, and while not necessarily an unwinnable battle for the poker sites, it looks pretty bad for them — though, as I noted earlier, some say that the nature of whether or not fraud was committed does depend in some way upon whether or not the underlying operations were illegal gambling. Nonetheless, because of the severity of the bank fraud charges, the full set of indictments may be settled out-of-court, which would be a tremendous loss for the game of poker.

The silver lining in this catastrophe is the opportunity for poker to finally get its day in court, and I hope that those associated with PokerStars and Full Tilt Poker will push for this.

The other possible silver lining would be if this indictment sped up the process of passing U.S. legislation to allow for domestic licensing of online poker. Some speculate that it will help compel U.S. interests to work towards it faster, as domestic casino companies will no longer need to worry about competing with established international sites for the U.S. player base. Others expect the controversy of this event to dissuade our elected representatives from embracing anything related to online poker. There does not seem to be a consensus.

Personal impact and thoughts

While I knew this day was likely to come at any moment in the past 4.5 years, and while I managed my money and planned my poker career accordingly, that hasn't made it easy to handle.

I play the vast majority of my poker online and will not be able to replace it with live poker to any meaningful degree. Moving to another country to play online poker would be incompatible with the rest of my life. This affects not only my finances, but also my happiness and life balance. Poker has been invaluable to me over the years as a unique outlet for mental exercise, strategic competition, and social interaction, all while being an excellent complement to my academic lifestyle. Over the past few days, it has really sunk in that I truly do value the game on these merits, rather than solely as an income source.

The entire premise of the government's various aggressive actions against online poker as "gambling", as well as society's refusal to properly treat poker the way identically-structured strategy games are treated, is something I have always taken serious issue with. This event is the culmination of a decade of ignorant and misguided policy towards my game, and it really hurts.

I was prepared for this and I'll be okay, but this is life-changing for me — and not in any good ways.

What to do?

If you have any interest in supporting the game of poker, the rights of competitive strategy gamers, or even just in supporting this because it is important to me, I would encourage you to take a look at the PPA's action plan and contact some of our elected representatives. While the charges against these particular poker sites may be legitimate, now that the perceived "bad actors" would be out of the picture anyway, this is an opportunity to gather support for U.S. legislation that will license and domestically regulate online poker. The government needs to hear that millions of its citizens are being negatively impacted by its policies towards online poker, and that its citizens deserve the consumer protections of a safe and explicitly-legal online poker landscape.

The poker world will never be the same as it was prior to Friday, but it will inevitably be rebuilt sometime in the next few years. The second, permanent era of online poker will emerge in a way that suits the interests of U.S. politicians and powerful domestic casino interests. This is discouraging at best, but it's the way laws get changed. As far as the health of poker and its players are concerned: the sooner it happens, the better. Every day in which well-minded, law-abiding, tax-paying Americans don't have access to compete at online poker is an undue intrusion into personal liberties and an insult to the integrity of this great game.

Links to more information

Ongoing 2+2 sticky thread with links to all relevant documents, press releases, and media coverage

Active Twitter folks on the issue and its aftermath: CKrafcik, GaryWise1, Karak2p2, Kevmath, Pokerati, taxdood

PPA's action plan


That's it for now. In general, this blog will continue, as most of my planned future topics were not entirely confined to online poker.

Saturday, April 9, 2011

A logical approach to the skill vs. luck structure of cash games vs. tournaments

In a ruling that, from a practical standpoint, is completely backwards, the Supreme Court in Sweden essentially decided that tournament poker is skill and cash game poker is luck.

Now, there may be more nuance in the ruling than that one article provides. But, if I recall correctly, this is not the first jurisdiction or court ruling that has found tournament poker to be "predominantly skill" or "not gambling" while finding the opposite for cash game poker.

I've even encountered poker players who believe either of these.

After all, the structure of tournament poker strongly resembles that of tournaments in other games that are commonly-accepted as games of skill and are rarely treated as gambling, while cash game poker looks a lot like "table games" in a casino, to the untrained eye.

Perhaps we can even sympathize with those who have made these decisions; perhaps they recognize that skill will always predominate over chance in the long run in any game, and that tournament poker "locks players in" to playing multiple hands of poker. Meanwhile, cash game poker lets players leave after one hand, which might give some gamblers the opportunity to take a quick risk on a hand of poker, just as they might do on the spin of a roulette wheel.

I have a reasoned argument as to why it cannot be the case that all tournaments "are skill" ("are not gambling") while cash games "are luck" ("are gambling").

A logical argument

Assume that it is somehow sensible or consistent that all tournaments are skill while all cash games are luck.

Then a 6-handed, winner-take-all tournament with a $200 entry fee, $200 in starting chips, and fixed (nonincreasing) $1/$2 blinds is skill.

Consider the following modifications to the rules of the tournament:

1) Players can buy in for less than the full amount if they wish, and they receive a proportional amount of chips; since the tournament is winner-take-all, and since each player's winning chances are equal to the ratio of their chips to total chips in play when all players are equally skilled, no inherent strategic advantage is given by this option (modulo short-stack advantages).

2) Players can rebuy when they lose all their chips or whenever they want to add on back up to the maximum.

3) If a player leaves the game by going broke or cashing out, a different player can take the seat and buy in just as the previous player could have rebought.

4) Players can "buy out" of the tournament at any time for their fair share of chip equity, equal to the ratio of their stack to the remaining prize pool. Ignoring the position of the button, again, since the tournament is winner-take-all, this is equal to the expected payoff for the player when he or she is at an equal skill level to his or her opponents, therefore this modification affords no in-game strategic advantage.

Making all four of these changes to the tournament makes it equivalent to a $1/$2 cash game.

So, which of these extra rules changed the fundamental "skill" or "gambling" nature of the game?

Do any of these changes make the game less skillful, or more luck-based?

#1 doesn't reduce skill. While shorter-stacked poker involves less complicated decisions, the level of stack-depth skill at work in any poker game with varying stack sizes will always be at least that of the same poker game if all stack sizes were at the minimum. An appropriate minimum buyin requirement can ensure that this level of skill is suitable.

#2 doesn't reduce skill. It's just a way for players to reenter the game. Notably, it allows a skilled player to continue to participate in the game, even if some bit of "bad luck" caused him to lose a big hand. In the right context, this would only increase skill relative to luck.

#3 doesn't reduce skill, same as above.

#4 doesn't reduce skill. It's just a way for players to leave the game early on any given night. The skill is still exercised over whatever number of hands they chose to play. I get that #4 is the one that Sweden and others might feel does change the skill vs. luck nature of the game, but even if the luck doesn't "even out" until a "long run" is reached, it's important to recognize:
  • If, say, we determine that 1,000 hands is the point where "skill predominates" in a certain poker game, there's no logical or practical difference between that player playing 1,000 hands in one night, or by playing those 1,000 hands in several sessions over the course of his life.
  • Almost every type of poker player will play sufficiently many hands in their life, even casual players.
(And I'm not even bothering to consider the fact that, by any measure, the amount of time in a cash game where "skill predominates" is going to be much, much less than the same necessary amount of time in tournament poker, as any actual poker player knowledgeable of the "LOL donkaments" creed is aware.)

So, with all due respect to naïve judges' attempts to implement an approach to skill vs. luck that recognizes that the duration of play affects the influence of luck on outcomes, a duration-based legal classification has logical as well as practical issues.

Classifying cash game poker as "luck" because it is possible to play it for a very short amount of time per day would be the same as deeming chess as "skill" under normal conditions, but "luck" when the same game of chess is broken up over several days and played a few moves at a time.

Do any of these changes make the game more "gambling"?

#1 doesn't introduce any "gambling". Certainly, giving players the option to play for less than the nominal amount can only reduce the amount of "gambling", by any definition.

#2 doesn't introduce any "gambling". In an environment where all tournaments are "not gambling", a player who has lost in a tournament could very easily enter a new tournament right away, if he were inclined. There's no difference between letting him do that on some table across the room and letting him do that at the same table he started at.

#3 doesn't introduce any "gambling", it just changes the particular players that might be participating at any time.

#4 doesn't introduce any "gambling". Again, this option can only reduce the amount of "gambling", and it does so in an incredibly significant way, especially in big tournaments.

In fact, the effects of #4 are so huge that I would argue that, from any sensible perspective, tournament poker has to be more "gambling" than the equivalent cash game. The primary difference between the two is that tournament poker forces its players to continue taking risks with their money rather than giving them the option to leave when they would like to, and certainly most tournaments will end with the remaining players taking on much more risk than they would be comfortable with, or rationally interested in.

For example, in a $1/$2 cash game, a player can leave if he triples up his stack and doesn't want to risk losing $600. In the equivalent $200-buyin tournament (think the Sunday Million), he can't leave with his $600-equivalent stack, nor any higher stack. If he makes the final table, he will be effectively taking risks for tens of thousands of dollars on every hand.

In almost any multi-table tournament with a reasonable field size, very few of the players who choose to play the tournament are actually well-equipped to rationally tolerate the financial risk they will have to endure if they make it to the end of the tournament and no deal is made. Utility annihilation, etc.

Conclusions

It cannot be logically consistent for all tournaments to be "skill" or "not gambling" while all cash games are otherwise. The differences between the two types of poker are largely practical, in that cash game poker accommodates players' personal schedules and personal risk tolerances much better than tournament poker. As far as game design goes, the additional game flexibility options of the cash game allow for a broader audience to participate, spending time on the game as it is convenient for them. This effect has to be an economic benefit to society from any perspective.

Any factor of cash games that might intuitively seem like enabling "luck" or "gambling" would also exist in tournaments of a certain blind structure, or a series of such tournaments.

Would Sweden really hope to provide regulations over tournament stack/blind structure? After all, it would be easy to define a tournament structure with very large blinds that would be over in fewer hands than any cash game.

Would Sweden also hope to regulate over deal-making in tournaments? After all, if a group of players join a winner-take-all tournament with the intent of chopping the prize pool with a chip-proportional deal after a certain amount of time, that's identical to a cash game.

There's an analogy to finance and investing here. Perhaps a government might decide that it doesn't want its citizens taking any short-term financial risk (a la playing only a few hands of a cash game), and thus only allows long-term investment by letting individuals only deal in instruments whose payoffs are far enough into the future. Consider a bond that pays out $100 in 365 days. Perhaps the current interest rates are such that the value of that bond today is $95. But if somebody buys that bond now, and the interest rates change such that the value increases to $98 the next day, he can't really be stopped from selling that bond to another person and locking up his quick $3 win -- the exact same $3 short-term payoff he could have had by speculating in interest rate futures. Generally, any short-term payoff can be derived synthetically by trading in longer-term payoffs, and settling tournaments early via deal-making achieves the exact same result in poker.

Would Sweden presumably approve of a cash game variant where players were artificially "locked in" to playing a certain number of hours or hands before they could leave? If so, then we're back to the silliness of the example of a game of chess being "skill", but the same game of chess being "luck" when played a few moves at a time and over the course of several days.

From a practical standpoint, looking at the ruling from any of these three natural conclusions should highlight the logical absurdity of the ruling in a manner which I would hope and expect to be apparent to everybody, regardless of knowledge or experience with poker.

Thursday, March 31, 2011

Is it ever rational to pass up on a unique risk because you "can't reach the long run"?

Yes, as it turns out.

More generally, optimal game/stake selection does depend on what future investment opportunities will be available to you. The premise of the question is a little misleading, because, in fact, whether or not to take ANY risky opportunity today (even in choosing a game to play regularly and "reach the long run" with) depends on how many times you intend to play that game. The intuition, however, is clearest when thinking about rare opportunities with abnormally high risk.

For example, is it possible that it's "not worth" playing the WSOP Main Event because it is so high-variance and only occurs once a year (so you "can't reach the long run"), but that the exact same tournament would be worth playing if it occurred more often, such as once a month? It can be tough to reason through this with intuition alone, but it turns out to be pretty easy to show as a consequence of either rational risk aversion and/or progressive tax rates.

An example of a unique opportunity

Let's take the usual example of a "typical" poker player with our usual after-tax utility function, $80k net worth prior to this year, and $40k income on the year (assumed to be all from poker, otherwise tax effects will shift).

Suppose this player were to come across the unique, one-time opportunity on December 31st to risk $10k on a weighted coin flip with a 51% probability of winning $10k and a 49% probability of losing $10k. A +EV opportunity, but a significant percentage of the small-stakes grinder's bankroll. What to do?

If he passes on the opportunity, his total after-tax utility payoff for the year is the utility of his $80k in existing wealth plus what remains of his $40k in income after paying taxes. If he takes the opportunity, his expected after-tax utility is 0.51 * util($80k + tax($50k)), plus 0.49 * util($80 + tax($30k)).

Which of these two outcomes is higher and whether or not the player should take this risky opportunity will depend on both his risk preferences and his tax bracket. We can look at his optimal decision for each different possible value of his year-to-date poker winnings:


The rows denote his current year-to-date poker winnings and the columns denote how many opportunities he will have available to take this coin flip (in this example, only 1).

A red box denotes that he should pass on the opportunity because taking it reduces his expected after-tax utility. A blue box denotes that he should take the opportunity.

Here, since his year-to-date poker winnings are $40k, he should pass on the opportunity. He would need to be up $100k on the year for it to become a profitable opportunity after adjusting for risk aversion and taxes. We see that it again becomes unprofitable in a higher region, when he has $170k or $180k, which we will discuss later.

If the opportunity is not unique

We modify the above situation by making it so that the player will have the option (but not requirement) to take this same coin flip opportunity up to 30 times. For example, maybe it's December 1st, and he knows that he will have the option of taking this coin flip opportunity once per day for the rest of the year.


This chart is simply an expansion of the first one, with additional columns added. The leftmost column agrees with the first chart, and it treats the case where the player will have only one option to take the opportunity. From that, we can calculate the next column over, where he will have 2 separate opportunities to take the coin flip.

If he passes on the opportunity when there are 2 days left, his year-to-date winnings will stay the same, and his expected utility will be the same as his expected utility at that bankroll level of $40k for when there is 1 day left (as he will again have the option of taking or passing on the opportunity). If he takes the opportunity when there are 2 days left, his expected utility will be equal to 0.51 times the expected utility of having year-to-date winnings of $50k with 1 opportunity left, plus 0.49 times the expected utility of having year-to-date winnings of $30k with 1 opportunity left. In this manner, we can "work backwards" iteratively from the known case of 1 day left to find his optimal decisions for all previous days.

The black boxes denote regions which are impossible to reach. Since the player's initial year-to-date winnings are $40k with 30 days left, he'll never be able to reach, say, a $100k bankroll with 29 days left. He'll also never reach year-to-date winnings of -$80k, because that would mean he would have taken the opportunity when his year-to-date winnings were $-70k, but the player would never be able to profitably risk the last $10k of his net worth on any sort of uncertain outcome.

Before we discuss the results, let's first look at what's causing the different shapes between the red and blue regions. Is the particular form of his decision strategy being shaped by risk aversion, tax effects, or both?

Without risk aversion or taxes

If we set our player's risk aversion to zero and remove the effect of taxes, then, as we might expect, he always takes the opportunity:


Without risk aversion or tax effects, all that matters is whether or not the opportunity has a positive expected value. It always does, so he always takes it.

With risk aversion, without taxes

If we return the player's risk aversion level to ρ=0.8 but still ignore taxes, then he starts to pass on the opportunity when his bankroll is too small to afford the risk:


Here, we see that, with only 1 chance at the opportunity, the player will need year-to-date winnings of $130k or more in order for the coin flip to be profitable after adjusting for risk.

If he had only $120k, then he should pass on the coin flip if it were a unique opportunity. However, with more than 1 option to take the coin flip, he should take it when his year-to-date winnings are $120k! This is because, while the immediate expected utility of taking the coin flip might be slightly negative, the player gets additional benefits to expected utility in the future when he happens to win the coin flip and can then take a profitable opportunity the next day. This turns out to be enough to make his expected utility of taking the opportunity positive relative to passing on it.

We see some other effects at the $110k and $100k levels; if there will be enough options to take the same opportunity in the future, that can be enough to turn an unprofitable opportunity into a profitable one.

With taxes, without risk aversion

If we go back to ignoring risk aversion but add in the effects of federal (and the negligible effects of NJ state) income tax, we see some different shapes in the chart:


Without risk aversion, the only factor that would keep the player from taking this +EV coin flip would be adverse tax consequences. Here, with 1 opportunity left (the leftmost column), we see a few regions where the player will pass on the coin flip. These turn out to be the regions where his tax bracket would change based on the outcome of the opportunity.

For example, the player passes on the coin flip when his year-to-date winnings are $0, because he gets no tax deduction if he loses and hits -$10k, but will have to pay some income tax if he wins and hits +$10k, which turns out to be enough to turn the +EV opportunity into a -EV opportunity after taxes.

The effect is similar at $10k, $30k, $40k, and $170k; a win or loss at any of these points will cause a significant jump in the individual's marginal federal tax rate. This is a great illustration of how progressive taxes create additional risk aversion when a risky opportunity has the possibility of changing one's tax bracket (cough cough).

Just as in the previous case, these tax bracket risks are mitigated when there will be additional options to take the opportunity. With enough time left before the end of the year, the player will take the opportunity regardless of his level of wealth. Essentially, the probability of ending up on the threshold of a different tax bracket gets lower the further out from December 31st you go, and the fundamental +EV nature of the opportunity outweighs this risk if there's enough time left.

With both risk aversion and taxes

Combining both risk aversion and taxes, we get back to the first full chart, where both "shapes" of red regions can be seen together:


A few observations:
  • While a pass (red) can turn into a take (blue) when additional options are added (when moving to the right on the chart), the opposite can never happen. Since the additional opportunities are not mandatory, the player could simply blindly pass on some number of them and then be able to realize the full expected utility of the point to the left on the chart. So additional time until utility realization (year-end in our model) can only result in an increase in willingness to take on risk.
  • Tax effects are only eliminated entirely when the player's annual income (poker or otherwise) is high enough that the risky opportunity could never move him out of the highest tax bracket. Otherwise, unless the risky opportunity has a large enough pure expected value, risks will never be taken near the boundaries of tax brackets.
  • The fact that the opportunities are optional does matter. For example, while the player takes the opportunity at a year-to-date winnings level of $90k when there will be 9 total options to take the opportunity available to him, he would not accept the opportunity if he had to be locked in to taking the coin flip all 9 times. If he had to commit to taking all 9 flips, it turns out that he would need $110k to take that opportunity. The optionality lets him quit in the middle if he ends up losing too much. Strategic options always have nonnegative value, and that is as true in this decision theory problem as it is in game theory.

Conclusions

Keep in mind that the implications about unique situations here are about situations with a uniquely high mean/variance tradeoff compared to normal play. All that matters is the shape of the probability distribution, not the particular nature of the opportunity.

For example, if you play $100 heads-up sit-and-goes for a living where you win 55% of the time and lose 45% of the time (ignore rake), and you happened to come across a one-off investment opportunity (perhaps a prop bet) where you could have a 55% chance of winning $100 and a 45% chance of losing $100, that's exactly the same as your usual heads-up sit-and-go, and you should take it. It doesn't matter that this particular opportunity will only happen once; the laws of probability only care about the distribution of the payoffs, and this will "reach the long run" along with your usual results. It doesn't matter that you "can't reach the long run" with a unique opportunity if you'll be able to reach it with other opportunities with similar (or riskier) payoff distributions.

While coding this algorithm in more complicated cases becomes more difficult, there are some powerful prospects for expansion of this model:
  • The discrete nature of this model, while it would be only an approximation for a continuous-time problem in traditional finance, is actually a perfect fit for any poker situation, where the number of sessions or risky opportunities will always be discrete.
  • The same approach can be used to compare any number of different possible games. For example, the player could choose between 4 options: playing his regular medium-stakes game, a lower-stakes game (higher mean relative to variance), or a higher-stakes game (lower mean relative to variance), or not playing at all. Expanding the number of opportunities to a larger number, such as 365 opportunities (playing one session each day for a year), would give a very powerful dynamic stake selection model. Rather than just working off of bankroll "rules of thumb", this model would optimize adaptively based on the running total of year-to-date winnings and provide strong risk- and tax-adjusted mathematical guidelines for decisions such as when to move up, when to "take a shot" in a particularly soft game, when to move down after significant losses, etc.
  • Similarly, this model can answer the question of at what pot size it becomes beneficial to pay $1 to run it twice. The discrete occurrences of pots over a certain threshold size can be plugged into this model as the two choices (one being to not run it twice, the other being to run it twice) and compared.
  • The model could also be expanded to capture randomness in other external investments, such as the player's stock portfolio.
This work so far is quite preliminary and mostly just an illustration of what sort of results a full model would produce. I hope to code the more complicated framework with multiple game choices and normally-distributed payoff distributions in the future — a program that could execute the complete model on an ongoing basis would be very practical.
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