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.

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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.

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.

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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.

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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.

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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.

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?