Monday, February 28, 2011

Optimal usage of Iron Man medals

It's almost the first of the month, where Full Tilt players will all be getting their monthly cache of Iron Man medals to blow through. Relative exchange rates for Iron Man medals are slightly different than that of the Full Tilt Points, so in the same way I optimized expected utility from Full Tilt Points, with the same assumptions as for the Full Tilt Points, I made the utility-adjusted table for Iron Man medals. Short and sweet.

A few quick notes:
  • You can trade Iron Man medals for Full Tilt Points, but the rate is terrible and you are always better off directly spending the medals on whatever you'd get with the points, be it bonus or tournament ticket.
  • For some reason, with Iron Man medals, purchasing tournament tickets does not deduct against rakeback, though it does for Full Tilt Points. So there is no MGR hit, but playing the tournament will earn you a little bit of rakeback, thus negative values of "Lost RB" in the table.
  • Since we're not directly considering the value of the Full Tilt Points we get by entering a tournament here, we'll ignore them entirely. We'll see that tournaments are still the best even without accounting for this.

ItemCost in medalsMedals You GetBase ValueLost RBCE (after tax)¢/medal (CE)% of best
$600 Bonus30000$600.00$162.00$304.3010.143376.2%
$209+$7 Super Turbo11000$209.00-$1.89$146.3513.3043100.0%
$200+$16 FTOPS10500$200.00-$4.32$122.6411.679587.8%
$207+$9 FTOPS Super Turbo10500$207.00-$2.43$135.9812.950397.3%

Compared to the results for Full Tilt Points, here we notice that:
  • The Super Turbo is a better deal than the FTOPS events; with Iron Man medals, the Step 5 ticket is only priced 4.8% higher than the FTOPS ticket, whereas with Full Tilt Points, it's 9.1% higher.
  • At these relative rates, you need to be roughly a millionaire before the certainty equivalent of the FTOPS event catches up with that of the single-table tournament. Of course, if you're going to play an FTOPS event anyway, you should just get the ticket for it.
  • The bonus is an absolutely terrible rate with Medals, whereas it was close to full value for the Full Tilt Points. This is a consequence of the bonus being a deduction against rakeback while the tournament entries are not. If you don't have rakeback, the $600 bonus is actually 5% better than the tournament tickets.

Conclusion

All players should spend their Iron Man medals on single-table tournament tickets, regardless of levels of wealth and risk aversion.

Sunday, February 27, 2011

All-in adjusted standard deviation, and what you should be using it for

The statistic frequently referred to as all-in adjusted winnings is designed to cancel out some "luck" effects in measuring winrates, and it differs from realized winnings in hands where all remaining players were all-in on the same pre-river street. For such hands, all-in adjusted winnings is defined as replacing realized winnings from the all-in pot with the the player's pot equity upon becoming all-in. In other words, this statistic gives winnings if every eligible pot were "run infinity times" (a la "run it twice").

Many are quick to point out that the all-in adjustment captures only a small portion of the "luck" in poker. It only captures "handwise" luck rather than "rangewise" luck, and only for the portion of times when players got all-in. Different play styles may lead to fewer all-in pots with certain types of hands. Nonetheless, it still always produces a more accurate measure of true underlying winrate.

All-in adjusted winrate are an unbiased estimator of true winrate

Though the intuition behind this is usually pretty clear, some people trick themselves into thinking that all-in adjusted winrate might be biased in one direction or another. I don't know how they manage to do so. It's easy to show that it's not.

Let X be the actual realized results in a given hand. Let Y be the results ignoring any money that came from a (main or side) pot that involved two or more players being all-in with cards still to come. Let Z be the results of such all-in pots. Then, clearly,
If we let A denote the all-in adjusted winnings from the hand, then, by definition,
therefore
and thus all-in adjusted winrate is an unbiased estimator of true winrate.

It's also intuitively clear and easily demonstrated that all-in adjusted winnings have lower variance than realized winnings. edit: It's actually not quite as simple as I first thought, since Y and Z are not independent. Since only one of Y and Z is positive in any given hand, we can condition on the event E that no all-in occurs, and let p be the probability of this event:
So all-in adjusted winnings are a better estimator for true winrate than realized winnings. Again, most people know this and do in fact use all-in adjusted winnings as an improvement over realized winnings in estimating their winrates.

All-in adjusted standard deviation

Here's something you probably don't know — Holdem Manager can calculate the standard deviation of all-in adjusted results. It's not included in the software, but it can be coded as a custom stat (credit due to nofolmholdm on the Holdem Manager forums).

Copy this text into a file called "customstats.txt" and put it into your Holdem Manager\Reports folder:

<Stat GroupName="Default" ColumnName="EVStdDevbigblind" ValueExpressions="10*stddev((case when EV.SklanskyBucks <> 0 then EV.SklanskyBucks else ph.NetAmountWon end)/GT.BigBlind) as EVStdDevbigblinds; 1 as DenomOfOne;" Evaluate="EVStdDevbigblinds/DenomOfOne" ColumnHeader="EV Std Dev\nbb" ColumnFormat="0.00" ColumnWidth="*" Tooltip="EV Standard Deviation" />

You can then add this to Holdem Manager reports like any other stat. You'll notice that the Holdem Manager summary output for the old data I used last time includes this.


The standard deviation of all-in adjusted winnings are 69-85% of the true winnings in this data, depending on game structure and how often one gets all-in. This is a bigger reduction than I expected! So, even though results of all-in pots is only a portion of the "luck" in poker results, it turns out to be a pretty significant portion. I'd say it's big enough that all of us should go through the trouble of using this statistic.

Be careful — all-in adjusted standard deviation does not reflect our true expected standard deviation like all-in adjusted winnings reflect our true expected winnings. All-in adjusted standard deviation reflects our true standard deviation in hypothetical poker games where we would always "run it infinity times". It does give us a glimpse of how much we can lower our "variance" by running it as many times as we can. But, until we can all start doing pure equity chops in the poker games we play, we don't get to use all-in adjusted standard deviation in any bankroll formulas.

However, in estimating our long-term winrates from observed data, all-in adjusted standard deviation will give us a better idea of how close our realized winrate is to our true winrate. When forming confidence intervals, if all-in adjusted standard deviation is, for example, 69% of the size of unadjusted standard deviation, the confidence interval it yields will be 69% of the size, which is a great narrowing of our interval for any given level of confidence.

For example, for the $1/$2 CAP data, if we ignored both all-in adjusted earnings (EV bb/100) and all-in adjusted standard deviation (Ev Std Dev bb) and simply used the unadjusted, realized earnings and standard deviation, assuming approximate normality and no Bayesian prior (I believe this is discussed in The Mathematics of Poker by Chen/Ankenman), our 95% confidence interval for our winrate in bb/100 is about [-1.77, 8.53].

If we replace realized winnings by all-in adjusted winnings, which improves accuracy, the 95% confidence interval is [-2.94, 7.36]... sadly, we had been running above all-in EV, so this interval paints a more accurate picture of what we should expect long-term.

If we then make the second change of replacing standard deviation with all-in adjusted standard deviation, we get to narrow the interval back down to [-1.51, 5.93]. This is our most accurate and most narrowed-down interval.

Conclusions

Put the all-in adjusted standard deviation stat into your Holdem Manager and use it, along with all-in adjusted winnings, when estimating your winrate. Easy game.

Caveat - card removal effects

In practice, database and analysis software such as Holdem Manager can only calculate the all-in adjustment assuming that all unrevealed cards are still in the deck. That is, in a situation where the cards that players have already folded are more or less likely to be of a certain rank or suit, since there is no way to know the identities of the folded cards, any practical all-in adjustment will be off by a little and will not use probabilities that accurately reflect the true pot equities.

It's never been established what the net effect of this is for different types of players. Since poker is zero-sum aside from rake, any all-in adjustment biases would cancel out entirely if the entire poker population were added together. Nonetheless, it is possible that a player with certain style of strategy amid certain game dynamics should expect a bias. Situations which bias it one way should be mostly cancelled out by situations which bias it the other way, though it's possible that the net effect would be nonzero.

Some forum discussions in the past have speculated that the net effect might cause all-in adjusted winnings to overestimate true winnings for winning players, but it is impossible to draw a firm conclusion from informal data sampling such as this, as those who "ran really bad" may be more likely to participate.

The bias it would introduce is certainly small, and we will ignore it. We will also assume that Holdem Manager and similar software are computing this accurately, though there have been issues in the past.

Tuesday, February 22, 2011

How far from Gaussian (normal) are poker results?

In a perfect world, as far as mathematical modeling and easy closed-form manipulation are concerned, we'd be thrilled if every random variable we ever dealt with had the Gaussian (a.k.a. Normal) Distribution. The most important of its many desirable properties is that the average of any finite-variance random variable converges to a Gaussian distribution, due to the Central Limit Theorem.

Poker results, on a per-hand or per-tournament basis, of course, do not have the Gaussian distribution. For one, the probability distribution of poker results are discrete, rather than continuous. Beyond that, we would generally expect much more weight on the extreme outcomes in a poker result distribution than the corresponding Gaussian distribution of the same mean and variance.

So, how "close" to Gaussian are the distributions of poker results for different games? If the distribution of the results of a single hand of a certain game of poker are not close to Gaussian, how many hands must be played before the average becomes close to Gaussian via the Central Limit Theorem? And what are the practical implications for bankroll management?

Data

I ran some simulations on some old data from my own play, assuming that the approximate probability distribution of per-hand results in each game was simply the empirical distribution based on my historical data. There are a number of problems with this that will keep this from being anywhere near a perfect assumption, but it's the best we can do. With large enough sample sizes in (hypothetical) constant game conditions, it would be fine.


$1/2 CAP NL play is 6-handed NL Holdem with 30bb (30 big blind) stacks. The betting cap ensures that no hands are ever played deeper than 30bb, though some are played shallower against shortstacked opponents.

$1/2 RUSH NL is exclusively 9-handed NL Holdem at Rush tables, with 100bb starting stacks (auto-reloading to 100bb every hand) and frequent deeper stacks. In the current online poker marketplace, the Rush games seem to be the best opportunity for gathering data on deeper-stacked play due to their high liquidity and speed of play.

$1/$2 PL is 6-handed PL Omaha, mostly on shallow or cap tables, around 40bb stacks.

Visualizations of Normality

For the sake of developing an intuition for "how long the long run is" for achieving approximate normality, I looked at histograms of the empirical probability distribution of poker hand outcomes, plotted against the Gaussian distribution of matching parameters (the thin blue curve).

I plotted the behavior of these distributions after 1 hand (top-left), 10 hands (top-right), 1,000 hands (bottom-left), and 100,000 hands (bottom-right). The x-axis is in big blinds, rather than dollars.

For the $1/2 CAP data:
As will be the case for each of the data sets, after only 1 hand, the distribution of results is not close to normal, as it is much too clustered around points near zero and puts much more weight on tail outcomes (for example, +30 and -30, though it's hard to see in the picture). After 10 hands, there are some interesting small "bumps" in the frequencies around +/- 30bb, probably an artifact of so many 30bb CAP hands involving getting stacks in against one other player.

After 1,000 hands, I was surprised to see just how close to Gaussian the distribution already was. There isn't that even that much visual improvement going up to 100,000 hands. The shortstacked nature of the cap games seems to produce fast convergence to approximate normality.

For the deeper-stacked $1/2 RUSH data:
Similar results here. The convergence is smoother, in that there are no "bumps" after 10 hands caused by frequent +/- 30bb pots, as in the $1/2 CAP data. Though it is not easily visible, the fit after 100,000 hands is better in the tails here than after 1,000 hands, as opposed to the $1/2 CAP data. Still, the results are not too far from Gaussian after even 1,000 hands... less than an hour of 4-tabling Rush!

For the $1/$2 PLO data:
Not too different than the $1/2 RUSH data, though perhaps a little slower to converge. The distribution of PLO results should have significantly higher variance than comparable NL Holdem data, but here, my only PLO data is in shallow-stacked games. Even so, we can visually observe that the convergence of these 40bb PLO hands is similar to that of 30bb NL Holdem hands. The higher-variance effects of PLO might only really start to matter with deeper stacks.

As a final point of comparison, I thought it would be interesting to look at heads-up tournaments, where the mass of the probability distribution would be only on the two extreme possibilities of -1 and + ~1 (adjusting for rake). I used some fictitious data based the rake structures of $55+$2.50 HU SNGs and on a winrate of 55%, with the buyin size normalized to 100:
Here it takes at least 100,000 hands (tournaments) before the distribution begins to look anything like a continuous Gaussian. Though I haven't tried anything with multitable tournaments (much higher skew), we'd expect those to be even less Gaussian.

So, as we've seen here, and as the Central Limit Theorem guarantees, once we play enough hands or tournaments, our results will be very close to Gaussian. Consistency of yearly results might be the #2 concern for profit-minded players, and as far as this goes, it looks like Gaussian approximations should be great here for cash game players putting in any amount of volume. Live players, however, will be suffering from both fewer hands/year and higher variance from deeper stacks... my guess would be that a live player would want to be putting in at least 500 hours/year to be able to assume approximate normality or annual results.

Most online cash game players, full-time or part-time, will be reasonably accurate by projecting their annual results to follow a Gaussian distribution with appropriate mean and variance.

However, the #1 concern for a poker player is the ability to sustain one's bankroll. It turns out that a popular and effective bankroll management formula is derived from an assumption of perfect normality — is this assumption a good fit?

Accuracy of Gaussian ruin probabilities

While we've seen that year-end results should be quite close to Gaussian, a poker player who goes broke in the middle of the year due to some perhaps non-Gaussian sudden downswings is not going to be able to reach the end of the year to achieve his nice, nearly-Gaussian result. So, in terms of the probabilities that the path of one's bankroll would cross a certain lower bound (usually zero), does this data behave similarly enough to that of perfectly Gaussian data?

Note that the true Gaussian distribution is unbounded, so a very small percentage of paths will have quick movements of very large magnitude. Actual poker distributions are bounded, so in this way, we would expect the Gaussian paths to fall to zero more often. On the other hand, actual poker distributions have higher kurtosis (a.k.a. fatter tails, that is, more likelihood is put on extreme results than in the Gaussian distribution), which would counteract this effect. Which of these effects will dominate?

If poker hands were perfectly Gaussian, then we could very closely approximate our (discrete) bankroll path by the (continuous) stochastic process of Brownian Motion — essentially, a continuous extension of Gaussian random variables. In this case, the ruin probability, the probability of ever hitting 0 from a given starting point and a winrate with a given mean and variance, would follow this formula, widely-known from Chen and Ankenman's The Mathematics of Poker but also an easily-derived property of Brownian Motion:

where B is (starting) bankroll, μ is the mean of the poker results process, and σ is its standard deviation.

Of course, we don't have a nice formula for the actual ruin probabilities given that our poker results are not perfectly Gaussian, but we can run Monte Carlo simulations to approximate these ruin probabilities for different starting bankrolls and compare them to the exact result for the Gaussian approximation.

The left column is starting bankroll, in big blinds (for the tournament, a value of 100 represents one tournament buyin).

For the $1/2 CAP data:

For the $1/2 RUSH data:

For the $1/$2 PLO data:

For the fictitious HU tournament distribution:

We first notice that the risk of ruin for the $1/2 PLO data set is 1 in all cases, as of course will always be the case when one's winrate is negative... your author is still working on his PLO game and has a very limited sample size so far. Whoops.

We then notice that, across the board, the Gaussian approximation to the ruin probability is higher than the simulated ruin probability with the empirical distribution. The difference appears to be decreasing in bankroll size, as we would expect from the Central Limit Theorem. The first few lines are for excessively small bankrolls, so they are not of any particular practical interest. The difference appears to be largest for the tournaments, as we would expect.

It looks like the boundedness of the true distribution is a bigger effect than the higher kurtosis, so the result is that actual ruin probabilities are lower than the easy formula suggests, which is great! Moreover, since the errors are small (especially for reasonable bankroll sizes which yield reasonably low practical risks of ruin), at least for cash games, we'll be a little bit extra-conservative by simply using the easy Gaussian formula.

The formula for Gaussian ruin probabilities is a very close approximation to true ruin probabilities for poker results for reasonable bankroll sizes.

Conclusions and Implications

Basic cash game results seem to be close enough to Gaussian for both terminal results and probabilities of hitting sufficiently far-away lower bankroll bounds along the way. Therefore we can rely on the easy Gaussian ruin probability formula for modeling purposes, and we will approximate long-term results (such as when evaluating expected utility over one year) with Gaussian distributions, at least outside of tournament play.

We should be careful about drawing definitive broad conclusions from these simulations. We have treated only a few different poker games, at only one moment in time for the poker economy, and only of one player's particular strategy. We should expect that the results may be different for poker games with deeper stacks, or with looser players. Games with higher variance or games with less continuous one-hand result distributions (such as limit games) should be further from Gaussian.

Thursday, February 17, 2011

The irrelevance of the Griffin/Qureshi prop bet to poker, and why there might be an abundance of gambling culture around our game

The Ashton Griffin/Haseeb Qureshi prop bet story has been making its rounds throughout the poker media for the last week or so. The short version of the story is that these two high-stakes poker players made a six-figure prop bet on whether or not Ashton could run 70 miles in 24 hours, risking not only Ashton's physical health but also their friendship. Haseeb's firsthand account of his side of the bet and his emotional struggle with it, in case anyone has somehow not read it yet, is a fascinating and well-written story (Part 1, Part 2).

I don't have any meaningful reactions to the actual events of this story that haven't already been expressed by plenty of others already. Among those who discussed the story, I thought crAAKKer and Bellatrix did a great job, and I generally share their views.

The story itself is perhaps less interesting than the poker community's fascination with it. Perhaps it was a slow news week, but many bloggers and podcasters seemed eager to draw conclusions from this story about the poker world as a whole, including an entire article from PokerNews about whether or not crazy prop bets like these are good for poker.

My concerns began when Haseeb himself was the first to claim to tie his degenerate experience to "the world of poker" (from Part 2 of his blog):
Something that I've come to think about is that perhaps there's something about the world of poker players that's fundamentally unhealthy. This generation of online poker players and its culture has existed for less than ten years, yet I've always had some assumption lodged deep in my psyche that if I'm not finding happiness through poker that it's just something wrong with me. And yet, there are so many people at every level of poker who are so deeply unhappy. It leaves me wondering.

And perhaps that's what really is the most difficult challenge for this generation of poker players. To infiltrate a world that is at its root, deeply unhealthy and imbalanced...
Did I miss the part of this story that was about poker?

The Griffin/Qureshi story does not involve poker at all. Sure, Griffin and Qureshi are both poker players. But it seems they are also, to use a phrase which gives them the benefit of the doubt, active high-stakes proposition bettors.

It's pretty clear that Haseeb's intent in the above quote is that there's something about the world of crazy prop bets or the overall baller lifestyle that's fundamentally unhealthy. But this world is not fundamentally related to poker.

Poker is a competitive strategy game, and many of its players enjoy it solely on that basis. They make rational investment decisions with respect to their poker career, and the financial risk associated with the random elements of a poker game is merely an unavoidable cost of playing. These players will never play casino games, bet sports, or make impulsive prop bets. They are not "gamblers" in any meaningful sense of the word; they would be much better described as long-term investors.

I would suggest that the Griffin/Qureshi story is a gambling story and NOT a poker story. With all due respect to the personal experience Haseeb was willing to share, the conclusions he has drawn in the quote above are not conclusions about the world of poker players. They are conclusions about the tiny world of gamblers who happen to have made it to the top of the world of poker players.

Accordingly, I don't see this story as having any sort of implications for the poker community. I don't see why it should be either good or bad for poker. And, beyond the novelty value of it being a story about two poker players, I don't see why it should be carrying nearly as much content in poker-related publications as it has been.

Is prop betting part of poker culture?

I admit that, holed up in my ivory tower, I am likely quite out of touch with the "real-world" poker culture, centered upon the fascinating antics of our most successful players, and I would not be so silly as to suggest that the culture of a particular game is confined entirely to the game itself. If poker players as a group are actually characterized by not only playing poker, but also by making crazy prop bets, then indeed that is part of the poker culture.

However, I would dispute the idea that prop bets are indeed characteristic of poker players. Rather, they are characteristic of a certain subgroup of poker players: gamblers who play poker. And it just so happens that the highest-stakes, highest-profile poker players in our world are probably especially likely to be gamblers. I can think of three reasons why.

1) Self-selection of gamblers to the game of poker

We'll keep the math light today, and we'll make up most of the numbers, just for the sake of illustration.

Let's separate society into those who like to gamble and those who do not. Pulling a number out of thin air, let's say that 5% of society enjoys gambling, in the sense of perhaps house-edge casino games or of unique wagers on uncertain-probability sporting events/prop bets (but NOT "gambling" merely in the sense of playing competitive strategy games for money, which is distinctly different from each of these).


Now let's separate society into those who like to play poker and those who do not. Using the PPA's estimate of 55 million American poker players of various levels, we might guess that around 17% of society likes to play poker. Some portion of this 17% likes poker because it's a competitive strategy game, some portion likes poker because it's a fun way to gamble, and some portion likes it for a combination of those reasons.


What does the overlap between these two groups look like? It seems reasonable to expect that gamblers should be more likely to get into poker than non-gamblers, for a few reasons:
  • Unlike most other competitive strategy games, poker is very popular to play for money at all levels of stakes. Therefore, if someone who is predisposed towards enjoying gambling is going to pick a strategy game to play, they are more likely to choose the one where they will easily find millions of opponents who are willing to play it for their desired level of stakes.
  • Much of society perceives poker as a gambling activity. Regardless of whether or not this is meaningful or accurate, this should compel people who like gambling to try poker.
  • Casinos strengthen this perception by offering poker alongside their degenerate house-edge games. Gamblers who frequent casinos are likely to try poker at some point.
So let's say that 50% of all gamblers are also poker players, putting about 15.2% of non-gamblers as poker players. Then the overlap looks like this:


The result is that about 14% of poker players are gamblers, versus 5% of the population as a whole. Therefore poker players as a group become more identified with gambling than other groups, despite the fact that no actual aspect of the game of poker has contributed to these effects.

Play with the numbers and the effects can be more drastic. If you take a broader view of "gambling" and let the percentage of gamblers exceed the percentage of poker players, the percentage of poker players that are also gamblers can get as high as you want.

2) Self-selection of gambling-predisposed poker players to higher-stakes play

We have seen in our analysis of the WSOP Main Event that high-stakes poker games demand either a high level of wealth or a high level of risk tolerance (or even risk-seeking-ness). Realistically, though, in the age of $100k-buyin tournaments and nosebleed cash games, the WSOP Main Event is pretty small. If we were to perform any sort of reasonable risk analysis on the highest-stakes games in the world, we'd find that tremendous levels of wealth are needed for a risk-averse individual to participate, and yet the general consensus is that very few players actually have a proper bankroll for these games.

Among rational players who aren't millionaires to begin with and who manage their poker career like a long-term, low-risk investment, few of them will ever be able to climb all the way up through the stakes and reach the highest-stakes games.

On the other hand, gamblers with the same skill sets as these rational players are likely to throw caution to the wind and move up too quickly, or impulsively play in stakes that are outright too high for them. Among this group, some of them will succeed and then maintain themselves at the higher stakes. Therefore, the highest-stakes games should have a significantly higher percentage of degenerate gamblers than the poker community as a whole.

3) The focus of media on interesting personalities

Finally, now that we've got our high-stakes players, the media will inevitably focus their attention on those players who produce the best stories. Gamblers (of various degrees) produce better stories than those who are more conservative with risk. Therefore we should expect an even higher percentage of the high-stakes players we hear about to have tendencies towards gambling than the high-stakes community as a whole.

All told, as we restrict our attention to the biggest poker players, we might expect as many as half of them to also be gamblers. Their gambling activity, which is at best tangential to the game of poker, will not be representative of the world of poker.

Conclusion: the REAL poker culture

The most talked-about members of the poker community are an extremely small sample of the entire poker community, and we should expect this sample to also be strongly biased in favor of people who are predisposed to degenerate gambling. The vast majority of the members of the world of poker would never dream of ever making a wager as ridiculous as that of Griffin and Qureshi. Even among successful, competitive, professional poker players, I expect that few of them share the view of poker culture that these prop bettors have.

Since this behavior is not characteristic of the vast majority of poker players, and since the behavior is completely unrelated to the game of poker (but probably related to a self-selecting subgroup of poker players), the culture of these crazy prop bets is a separate culture from that of poker.

Stories such as these should be neither good nor bad for poker. Suggesting that stories such as these are somehow inherent to or representative of poker, however, is almost certainly bad for poker, as well as incorrect.

Friday, February 11, 2011

Optimal usage of Full Tilt Points, featuring more expected utility analysis of MTTs

With online poker rooms putting more of a focus on promotions and loyalty programs that involve player loyalty points, it's becoming more important for the bottom line of a serious players to optimize his or her points usage.

For the tl;dr crowd, THIS is the table you should probably be using:

Best Table: Accounting For Tax and Risk Aversion
ItemCost in pointsPoints You GetBase ValueLost RBCE (after tax)¢/point (CE)% of best
Carte Blanche2500$1.00$0.27$0.510.202985.4%
$5k Bonus11000000$5,000.00$1,350.00$2,535.840.230597.0%
$209+$7 Super Turbo47500280$209.00$49.41$110.040.233098.0%
$200+$16 FTOPS43500640$200.00$42.66$89.210.208287.6%
$207+$9 FTOPS Super Turbo43500360$207.00$44.55$102.530.2377100.0%

We will now go through the underlying assumptions and the steps of building up to it.

What Most People Do

While there are a variety of tournament tickets available in the Full Tilt Store, for small-stakes players, the tournament options for lower buyins are not as good as those at the $216 level. We'll restrict our attention to these $216 tournaments and account for a case of extra risk aversion through smaller wealth for small-stakes players. We'll look at three different tournaments:
  • $209+$7 Super Turbo — A 6-handed single-table tournament (STT) that fires quickly and takes only a few minutes to play, thanks to a 10BB starting stack... a great setting for the assumption that we're a break-even player!
  • $200+$16 FTOPS — A generic NL multi-table tournament (MTT), with broad appeal and a large field. Number of entrants (6607) and prize structure is based off of FTOPS XIX Event #1.
  • $207+$9 FTOPS Super Turbo — A Rush Super Turbo MTT, with niche appeal and limited opportunities to exercise skill. On the plus side, it has a smaller field and a very short running time and thus is an appealing way to roll through points quickly. Number of entrants (1726) and prize structure is based off of FTOPS XIX Event #12.

It's not too hard to look at the offerings in the Full Tilt store and calculate the pure EV per point of each item.

Naïve Table #1: No Rakeback
ItemCost in pointsValue¢/point% of best
Carte Blanche250$1.000.484.1%
$5k Bonus1100000$5,000.000.45454545595.5%
$209+$7 Super Turbo47500$209.000.4492.5%
$200+$16 FTOPS43500$200.000.45977011596.6%
$207+$9 FTOPS Super Turbo43500$207.000.475862069100.0%

The above table is for players without rakeback. Most quick analyses also factor in the effects of rakeback:

Naïve Table #2: 27% Rakeback
ItemCost in pointsBase ValueLost RBTotal Value¢/point% of best
Carte Blanche250$1.00$0.27$0.730.292078.2%
$5k Bonus1100000$5,000.00$1,350.00$3,650.000.331888.9%
$209+$7 Super Turbo47500$209.00$49.41$159.590.336090.0%
$200+$16 FTOPS43500$200.00$42.66$157.340.361796.9%
$207+$9 FTOPS Super Turbo43500$207.00$44.55$162.450.3734100.0%

The final column, "% of best", lets us know how much value we're losing if we went with an option other than the best option.

For example, if we have rakeback and we can break even at tournaments, under the incomplete methods of this table, we should be saving our points and waiting for the FTOPS to come around. If we instead want to roll through our points faster, we're getting only 90% of our maximum value by playing the Super Turbos. If we instead want a sure thing, we're getting only 88.9% of our maximum value by taking the bonuses. If we don't want to spend time clearing the bonus and instead go with Carte Blanche or electronics, we're getting 78.2% of our maximum value.

Carte Blanche and the $5k (and other, worse-value) cash bonuses are available only to Black Card members; players with less than Black Card are limited to the tournaments or to electronics, the value of which all seem to be worse than the Carte Blanche rate (in the 70-90% range, checking a few against Amazon.com). Luckily, Full Tilt has chosen their direct points-to-cash Carte Blanche rate to be awful compared to the bonuses. While clearing a bonus does take time, we will ignore that for the purposes of this article. So we can look at the Carte Blanche rate as an upper bound to the points-to-cash value for non-Black Card members, and the $5k bonus rate as an upper bound (some people might need to choose smaller bonuses at worse rates, depending on their volume) to the points-to-cash value for Black Card members.

So, rakeback or not, the basic value hierarchy seems to go:

Carte Blanche & Electronics < $5k Cash Bonus < Tournament Tickets < FTOPS Tournament Tickets But, as we've seen with the WSOP, even a break-even tournament player loses value in tournaments due to taxes and risk aversion, particularly in large-field tournaments with high payouts for the top places.

Will adjusting to certainty equivalents make the fixed-value cash bonus preferable to the tournaments?

Will the high risk in FTOPS events versus 6-handed $209+$7 Super Turbos make us prefer the latter?



If You're A Tournament Player

First, we should notice that, if we are tournament players and would be playing these particular tournaments anyway, the tickets are obviously the best value. We need not consider the utility effects; we were going to spend $216 cash to enter that STT or MTT, so the ticket is worth exactly $216 (minus rakeback adjustments) to us. We're done.

If you play tournaments anyway, obviously just get the tournament tickets.

For The Rest Of Us

From here on out, we make these assumptions:
  • We don't regularly play tournaments, but we are willing to in order to roll through our points. We'll ignore the opportunity cost of our time spent playing the tournament.
  • We are an approximately break-even player in any tournament we choose to play, so we will finish in each place with equal probability.
  • Therefore our expected value (NOT expected utility!) of each tournament is equal to the value of the tournament after the rake. For example, the expected value of the $209+$7 Super Turbo is $209, not $216.
  • We have rakeback and are the highest tier of Black Card, receiving a 4x points multiplier on points we earn (like all elements of the Black Card program, the points multiplier turns out to have a very negligible effect on the numbers we'll get).
If we're going to be playing a tournament that we wouldn't have played anyway, we need to adjust the above table to account for two benefits of playing this extra tournament: points and rakeback that we gain from paying the rake on the tournament.

Less Naïve Table: Accounting For Points+Rake From Tournaments
ItemCost in pointsPoints You GetBase ValueLost RBTotal Value¢/point% of best
Carte Blanche2500$1.00$0.27$0.730.292077.5%
$5k Bonus11000000$5,000.00$1,350.00$3,650.000.331888.1%
$209+$7 Super Turbo47500280$209.00$49.41$159.590.338089.8%
$200+$16 FTOPS43500640$200.00$42.66$157.340.367197.5%
$207+$9 FTOPS Super Turbo43500360$207.00$44.55$162.450.3766100.0%

The effects are pretty minimal. Much more important are the effects of taxes and risk aversion.

Certainty Equivalents Under Tax and Risk Aversion

In the style of our WSOP analysis, we will start by making similar assumptions on our wealth, income, and risk tolerance:
  • Our preferences for different levels of wealth and risk are governed by our usual utility function (isoelastic with ρ=0.8). We are subject to both U.S. Federal and New Jersey State income tax. again, NJ has higher income tax than most other states, but the effect on the certainty equivalents we produce is not large.
  • Our net worth is $80,000.
  • Our income for the year aside from our points usage is $40,000. (We assume that we will almost certainly be winning poker players for the year, but since the loss associated with entering a tournament and losing is simply the rakeback hit, we don't really need the distinction between poker and non-poker income this time.)
We also make a few other small assumptions about the tournaments:
  • FTOPS events will run with no overlay. If there ends up being an overlay, we get more EV than our base buyin amount, but we also get a rakeback hit (I think). I don't think this happens very often in NL events, historically.
  • FTOPS events actually have extra value that we're ignoring because there are bounties for eliminating a red pro. This probably adds somewhere around 1% to our EV for an FTOPS event if 1% of the field are red pros. We'll ignore it here rather than try to model it.
With these assumptions, we calculate the certainty equivalents for each of the tournaments under consideration:

Best Table: Accounting For Tax and Risk Aversion
ItemCost in pointsPoints You GetBase ValueLost RBCE (after tax)¢/point (CE)% of best
Carte Blanche2500$1.00$0.27$0.510.202985.4%
$5k Bonus11000000$5,000.00$1,350.00$2,535.840.230597.0%
$209+$7 Super Turbo47500280$209.00$49.41$110.040.233098.0%
$200+$16 FTOPS43500640$200.00$42.66$89.210.208287.6%
$207+$9 FTOPS Super Turbo43500360$207.00$44.55$102.530.2377100.0%

Observe that:
  • Now that we're accounting for tax, the cents per point is lower across the board, but some options suffer more than others.
  • The relative value of the riskless options has risen, due to the discounting of the values of the tournaments from taxes and risk aversion.
  • The low-rake FTOPS event is still the best, but now the Super Turbo STT has overtaken the $200+$16 FTOPS. Keep in mind that the "variance" is much lower in a 6-person tournament than a tournament with over 1,000 players, and accordingly, the reduction to the certainty equivalent is lessened.
  • The difference in value between the three best options ($5k bonus, Super Turbo, and FTOPS Super Turbo) is now quite small. This average player should probably just go with whichever option is convenient.
Other cases
Let's look at the numbers for a few other cases of wealth and income. To model a small-stakes grinder who would be out of his comfort zone playing $216 tournaments, let's look at a case of very small wealth and income:

Small-Stakes Grinder ($5k net worth, $2k annual income)
ItemCost in pointsPoints You GetBase ValueLost RBCE (after tax)¢/point (CE)% of best
Carte Blanche2500$1.00$0.27$0.650.258788.0%
$5k Bonus11000000$5,000.00$1,350.00$3,233.900.2940100.0%
$209+$7 Super Turbo47500280$209.00$49.41$137.000.290198.7%
$200+$16 FTOPS43500640$200.00$42.66$59.570.139047.3%
$207+$9 FTOPS Super Turbo43500360$207.00$44.55$70.810.164155.8%
The variance-free bonus has become the best option, though the small-stakes player will have trouble clearing it quickly. Surprisingly, even someone with this small of a net worth is pretty close to indifferent between the guaranteed cash and the 6-handed $216 tournament. The large-field, high-risk FTOPS have become very unfavorable, however. A player wealthier than our default player has lesser tax consequences and less risk aversion to the higher "variance" of the FTOPS events:

Affluent Pro ($500k net worth, $100k annual income)
ItemCost in pointsPoints You GetBase ValueLost RBCE (after tax)¢/point (CE)% of best
Carte Blanche2500$1.00$0.27$0.480.191678.9%
$5k Bonus11000000$5,000.00$1,350.00$2,395.500.217889.6%
$209+$7 Super Turbo47500280$209.00$49.41$104.710.221791.3%
$200+$16 FTOPS43500640$200.00$42.66$96.370.224892.5%
$207+$9 FTOPS Super Turbo43500360$207.00$44.55$104.820.2430100.0%
The wealthier player should probably save up his points for the FTOPS.

Conclusions
A non-tournament player of average wealth/income ($80k/$40k) is close to indifferent between spending points on $5k bonuses, $216 single-table tournaments, and $216 FTOPS events.

A player with less wealth/income should probably choose the $5k bonus or the $216 single-table tournament.

A player with more wealth/income should probably save his points up for the FTOPS.

Wednesday, February 9, 2011

How much "rake" are we really paying in the WSOP Main Event?

It's time to take our utility function out of the realm of hypothetical coin flips and into the real-world, life-changing arena of the World Series of Poker Main Event, where dreams are crushed and careers are made.

We are told that every poker player's dream is to win this particular tournament. The only obstacle standing between them and their share of an eight-figure prize pool is the enormous field of competitors that it attracts. We're told that the glory of the bracelet is what we should care about, and the possibility of a huge payday and becoming an instant legend in the game keeps emotional motivations at a high and rational risk management at a low.

While the non-financial value of becoming the world champion may be immense, the 1st-place winner doesn't really do that much better than the 2nd-place winner in terms of expected utility on the purely financial value of the prizes. In terms of sudden windfalls in individual wealth, how different are $8.9 million and $5.5 million? As we have discussed, diminishing marginal utility of wealth suggests: not very. Even for the smaller prizes, the real, enjoyable differences between various 5- and 6-figure scores are not nearly as large as they may seem on the payout table, thanks to risk-adjusted utility and its partner in crime, progressive income tax rates. .

Somehow, I feel like it might actually be the case that this article is the first time that anybody has taken poker's flagship event and performed a simple expected utility analysis on it. In the unlikely event that any WSOP Main Event participant is willing to temporarily look beyond his or her dreams of poker immortality and actually consider the real cost of the tournament, here's a starting point. After all, the $10,000 entry fee is a significant amount of money for the vast majority of the competitors. Any $10,000 investment merits some careful analysis.

Assumptions

Let's consider a generic WSOP Main Event competitor with the following characteristics:
  • His preferences for different levels of wealth are governed by our usual utility function (isoelastic with ρ=0.8) and he is subject to both U.S. Federal and New Jersey State income tax. NJ has higher income tax than most other states, but the effect on the certainty equivalents we produce is not large.
  • His net worth is $80,000, which affects his preferences for risky opportunities through the above utility function. [estimated roughly from 2007 median family net worth (source)]
  • His non-poker salary for the year is $40,000, which figures into his wealth for utility as well as affects his income tax bracket for the year. [estimated roughly from 2009 median income for 25-and-older males (source)]
  • His poker winnings for the year (aside from this tournament) are $10,000. This is significant, as it gives him the full tax deduction if he loses the tournament. Since $10,000 is the most he can lose, we can capture higher levels of poker winnings by just modifying his salary, which will have the same effect.
  • His poker skills are such that he is a break-even player and will finish in each possible position with equal probability. If you prefer, you can consider every competitor in the event to have the same level of skill or to be employing the same strategy. In terms of aggregate money lost to taxes and to risk aversion, we don't care which players are actually superior if they all come from the same income situation, and our break-even player here certainly doesn't care about anybody else's utility.
After taxes and risk aversion, on average, how much of the $10,000 does our generic competitor get to enjoy?

Results

We use the payout data from the 2010 World Series of Poker and apply the above assumptions.
  • Rake — Based on the number of players and the total prize pool, it looks like Harrah's kept $600 of each $10,000 entry in 2010. The $417 net rake accounts for the effects of taxes; since the rake comes out of the $10,000 entry fee, it's effectively a tax deduction, so its real cost is discounted. I've heard some tournament players complain that this rake is too high. That very well may be, but that might be the least of their problems.
  • Tax on Winnings — Based on our individual's tax situation, assuming that he is completely risk-neutral (i.e. temporarily ignoring the discounting of random payoffs under the utility function), he is paying an average of $637 more in taxes by playing in the tournament than he is if he were to not play. If each of the 7,319 players were Americans with similar (low) income as our generic player, income taxes would automatically take a guaranteed 3rd place in the tournament with a $4.6 million dollar payday. Note that this is an average over all possible finishes and includes the overwhelmingly likely outcome that our player gets a $10,000 tax deduction by not finishing in the money! Since he moves into a higher tax bracket when he wins a big prize, the net tax effect is positive on average.
  • Utility Loss (Risk Aversion) — Now we add his risk aversion into consideration and look at his expected utility after rake and taxes. This is where he really gets hurt. $2,380 could buy lots of nice things, but that's how much must be thrown into the consuming flames of "variance" in order for our generic competitor to take his shot at the big game. Diminishing marginal utility of wealth is a big deal when we're looking at seven-figure payoffs. For example, the after-tax utility of the 1st-place prize of $8.9 million is only twice as much as that of the 82nd-place prize of $79,806! That means that a player with this income level and risk tolerance would be indifferent between taking $79,806 for certain and taking a 50% chance at $8.9 million. About 30% of the prize pool goes towards payouts in excess of $1 million to the top 8 finishers, but the additional utility of these dollars is quite small compared to that of the first $1 million.
  • Certainty Equivalent — After enjoying his triple-scoop of various flavors of rakes (actual, government, and risk adjustments), our break-even competitor is left with only $6,566 of his $10,000 entry in actual, consumable, after-tax equity. It's up to him to decide whether or not the thrill of competing in the championship is worth his $3,434. (To be specific, $3,434 is the difference between the certainty equivalent of playing in the tournament and the after-tax utility of skipping the tournament.)

Other Cases

While our assumptions provide a strong caution for the amateur player who has satellited into the event, how about players with different income situations?

Let's take the assumptions on net worth, income, and level of risk aversion and vary them one at a time while holding the others constant.

Net Worth Certainty Equivalent
$0 $5,713
$80,000 $6,566
$250,000 $7,154
$500,000 $7,533
$1,000,000 $7,879
$5,000,000 $8,554

Changing the player's net worth affects only the blue slice of the pie, the utility loss due to risk aversion. While the player with $80,000 is losing a lot of equity, wealthier players do significantly better.

The player with a $5 million net worth is losing only $392 due to risk adjustments; this might be about the realistic level of wealth where it "makes sense" to play a large-field $10,000 tournament.

People near or below median levels of income should realize that the level of risk of playing in the WSOP Main Event is essentially too much for them from a purely financial standpoint, as the size of the blue slice of the pie reflects.

Poker Winnings1 Certainty Equivalent
$0 $3,718
$5,000 $5,145
$10,000 $6,566
$100,000 $7,078
$250,000 $7,663
$500,000 $8,178
$1,000,000 $8,453
$5,000,000 $9,108
1 annual poker winnings outside of this tournament, interchangeable with changes in non-poker salary above $40k base

Increasing income from poker has the same tax effects as increasing non-poker salary (as long as at least $10,000 of income is from poker), so we can look at them together.

Higher levels of income for the year help increase the certainty equivalent in two ways: higher wealth leads to lower loss of utility due to risk aversion, and being in a higher tax bracket already helps reduce the negative tax effects of a big WSOP score. For this reason, a higher income for the year makes the tournament more profitable than a similarly-higher net worth, as seen by comparing the two tables.

In fact, if his salary is above $510,000, the player is already in the highest tax bracket and no outcome in the tournament will change the player's tax rate. The effect of this is that the tax-adjusted rake (red slice of the pie) is at its minimum of $336, and the average tax on winnings (yellow slice of the pie) is $0.

If poker/gambling income is reduced below $10,000, the tax effects become enormous as the player no longer gets a tax deduction for losing the tournament. For the player with no poker winnings to deduct against, the average amount of tax he pays for playing in the tournament is a whopping $3,178.

ρ (risk aversion) Certainty Equivalent
0 $8,946
0.2 $7,978
0.5 $7,094
0.8 $6,566
0.9 $6,435

For risk aversion, I found that the value ρ=0.8 for the parameter in the utility function was the best fit for what I expect a normal person's risk preferences to be, but everybody is different and should experiment with different values of ρ for themselves. Lower values of ρ have significant effects on the value of the tournament, but values of ρ that are too low are probably not realistic. As noted earlier, you may not pass up on a 50% chance of winning $8.9 million in favor of a guaranteed $79,806, but when your net worth is as low as our generic competitor, I doubt you'd really require very much more.

Conclusions
  • The typical Average Joe (or Chris, as it were), poster child of the broad appeal of the World Series of Poker, is losing a surprising amount of equity to tax and to "variance" by playing the Main Event.
  • If you're American (or under any other tax system that gives you no deduction or carryover for a net "gambling" loss on the year), you probably just shouldn't play the Main event if there's a significant chance that you won't make at least $10,000 this year in poker otherwise. If you choose to play anyway, realize that you are paying a tremendous premium. And, incidentally, it might help your chances in the tournament to be good enough at poker to have consistent winning years outside of tournaments, but that's another story.
  • Even if you're a successful poker player, consider the fact that, thanks to both taxes and relative risk aversion, your equity in the tournament is significantly higher in years where you have higher income. It might be worth skipping the Main Event during a bad year.
  • Hopeful amateurs, particularly those with net worth under $1 million, should strongly consider whether or not the experience is worth these costs.
Of course, for many, it is worth the costs. The experience of competing on poker's biggest stage with players from around the world and from all levels of the game is unique and perhaps priceless to some.

None of this analysis should be seen as a slight against the WSOP in particular — the huge fields and prizes are what create the appeal of the event, and any other large-field big-buyin tournament would annihilate collective utility in a similar fashion, though perhaps to a lesser extent. Much of the value of the World Series of Poker Main Event is non-financial, and there's nothing wrong with that. Just be sure to consider this against the real hit to your bank account and your economic happiness.

Future Plans
  • How could we quantify the external benefits and costs of various outcomes? For example, it would be easy to incorporate sponsorship bonuses, dealer tips, and travel expenses into the payouts. We could even try to quantify intangibles such as the expected future marketing value of a strong finish, or even the personal value of becoming world champion.
  • How much better can our equity get if we sell shares of ourselves? What if we had a backer? I expect that these hedges can easily provide very significant reductions in the loss of utility due to risk aversion.
  • And, of course, the 10,000-pound donkey in the room: Our skills might be better than that of the rest of the field. As our pure expected value in the tournament increases, so will our expected utility. In this article, we've considered only the player who has an equal chance of finishing in any of the 7,319 places. I am working on creating some realistic probability distributions on the place finished in the tournament as a function of a player's EV-edge on the field. Once we have a reasonable function, we can look at some interesting and extremely important related problems. For a given projected skill edge, what's our certainty equivalent of playing the event? For varying personal income situations, what's the minimum skill edge needed to be able to profit from playing the Main Event?

(06/08/2011): I look at the effect of skill edges and optimal hedging ratios in my followup to this post, located here: WSOP Utility Analysis revisited, part 2: How many shares should a WSOP Main Event player sell off?

Sunday, February 6, 2011

Utility, Part 3: Tax Effects

Last time, we found a simple utility function that reasonably approximates realistic levels of risk aversion. Now, we'll wrap up the series by incorporating the effects of U.S. taxes, which will result in our final, ready-to-use, practical utility function.

Restricting Utility to One Year

While traditional expected utility analyses are implicitly made on an ongoing basis, with no particular regard to the flow of time, U.S. taxes are paid based on yearly income and will require us to change our focus to year-by-year realizations of wealth. Accordingly, we can shift our utility function to account for our preexisting wealth and focus on the change in wealth that occurs during the upcoming year.

Instead of using the standard ρ=0.8 isoelastic utility function (defined on our ongoing wealth), we can shift the function to the left by the amount of our preexisting wealth and thus have a utility function defined on the change in wealth we realize over the upcoming year. If we let the variable w represent our preexisting wealth and x our change in wealth during the upcoming year, then our shifted utility function is:
That is, if our current net worth is $100,000 and we are considering flipping a coin for $10,000, instead of comparing the utility of having a net worth of $90,000 or $110,000 on this graph:
... we instead compare the utility of having an annualized change in net worth of -$10,000 or $10,000 on this graph:
So, in order to define utility on the change in our wealth in the upcoming year, we shift the original utility function to the left by our net worth at the end of the prior year. The results, of course, will always be the same regardless of any shift. This is just a change in methodology, but it allows us to introduce a tax function which operates on each year's income.

This approach offers some other benefits. The shifted utility function captures our preferences for different yearly salaries (perhaps useful when comparing different non-random job opportunities) just as well as it captures our preferences for risky year-long opportunities. Also, annnualized poker returns are more easily compared to alternatives in traditional investments such as stocks and bonds.

Using the Shifted Function Within the Year

Now that our utility function is defined over an entire year's results, we need to be careful about how we use this function throughout the year to evaluate risky opportunities we come across. It will be necessary to incorporate both year-to-date results and future plans.

If the random variable X represents our gains and losses for the entire year, it can be broken up into its parts:
For example, let's say it's June 1st, we've made $5,000 so far for the year, and we come across the opportunity to flip a coin for $1,000 with a 55% chance of winning. Then:
  • Xpast = $5,000; our year-to-date earnings are constant.
  • Xpresent is a random variable representing the opportunity we are facing, taking value +$1,000 with probability 0.55 and -$1,000 with probability 0.45.
  • Xfuture is a random variable representing all possible outcomes of our future results after this coin flip, from June 1st through December 31st of this year. This includes our usual "daily grind" along with probabilistically accounting for other opportunities we may encounter (such as this special coin flip).

In particular, this approach allows our within-year risk tolerances to change during the year. For example, on January 1st, for our given level of wealth and our projections of our winrates and volatility across different games, it might be the case that we maximize our expected utility by planning to play $1/$2 for the entire year. But then if we happen to run well or poorly in the beginning of the year, even if our risk tolerance hasn't changed, a reexamination of our utility function could cause us to move up or down to $2/$4 or $0.50/$1 during the year based on our year-to-date results.

Therefore, despite the fact that we are now defining our utility function on an entire year's results, we are still able to use this function throughout the year to make rational individual decisions while conditioning on the year-to-date results we have realized.

The Tax Function

Now we're ready to consider the effects of U.S. income taxes.

Everybody's tax situation is different, and you should make the necessary changes to the tax brackets in your own analyses, but here we assume that the taxpayer is single and subject to New Jersey state income tax in additional to federal income tax.

We assume that the player is a hobbyist poker player, rather than a professional poker player, though filing as a professional poker player would not change this method by much; The professional poker player must pay an additional percentage of his income in self-employment tax, but this percentage is a flat rate across all levels of income, so it will not impact any expected utility decisions except possibly those that risk causing negative yearly earnings.

With these assumptions, the 2011 federal tax brackets are (source):
Taxable IncomeMarginal Tax Rate:
$0-$8,50010%
$8,500-$34,50015%
$34,500-$83,60025%
$83,600-$174,40028%
$174,400-$379,15033%
$379,150+35%
and the 2010 NJ tax brackets (can't find any info on 2011 yet) are (source):
Taxable Income
Marginal Tax Rate:
$0-$20,000
1.4%
$20,000-$35,000
1.75%
$35,000-$40,000
3.5%
$40,000-$75,000
5.525%
$75,000-$500,000
6.37%
$500,000+
8.97%

We also assume that all negative effects of artificially-high Adjusted Gross Income (the sum of all winning sessions, prior to deducting all losing sessions) for non-professional poker players are ignored. This is a big assumption — there are several ways that this "phantom income" can cause at least a little bit of extra tax to be owed, but that's a story for another article. These effects are simply too complicated to capture in a simple model. Attempting to track these would also require that our utility function be defined on the sum of individual winning sessions as well as the sum of individual losing sessions, rather than just the net.

With these tax rates, the function t(x) that maps pre-tax earnings to our after-tax earnings looks like this:
As expected, the difference between pre-tax and after-tax dollars becomes larger at higher income levels. Notice that this after-tax function satisfies the usual properties of utility functions: it is increasing, continuous, and nonincreasing in slope. Since U.S. tax rates are progressive, the slope becomes lower at higher levels of income. Also notice that, for negative yearly earnings, the pre-tax and after-tax earnings agree; poker players of either filing status get no deduction or carryover for losing years.

Poker vs. Non-Poker Income

The above all holds only when our only income is from sources that the IRS classifies as "gambling", which includes poker. However, since we all usually have income from non-"gambling" sources, we need to modify the tax function to account for the case where we have a net loss on "gambling" for the year, as gambling losses cannot be deducted against other income. That is, if we make $40,000 at our job and lose $5,000 at poker for the year, we are taxed on $40,000, not $35,000. Let t(x) be the basic tax function (graphed above), let s be our non-"gambling" (salaried) income for the year, and let g be our "gambling" income for the year. Then the generalized tax function is:

Combining the Tax Function with the Utility Function

As we assume that we gain no satisfaction or dissatisfaction from the amount of tax revenue we produce for our governments, our utility is realized on after-tax dollars. So our final utility function comes from combining the shifted utility function with the tax function:
This final tax-adjusted utility function (plotted in red below against g, with s=0) effectively adds additional risk aversion to our original isoelastic utility function. Compare it to the non-tax-adjusted utility function from earlier (blue) and notice that there is additional concavity.
Since the tax function is piecewise-linear, taxes will induce no additional risk aversion for risks that cannot move us out of the tax bracket we would otherwise be in. However, for any risky opportunity that could move us up or down into a new tax bracket, the progressive nature of the tax rates will always create some additional risk aversion.

Progressive income tax creates additional risk aversion when there is any probability of risky opportunities moving an individual into a higher or lower tax bracket.

So, finally, we've got our utility function. It no longer has nice analytical mathematical properties, but is still easy to evaluate numerically. Now we can move on to tackling interesting practical problems such as those mentioned at the end of part 1.
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