Monday, May 30, 2011

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

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

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

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

Connecting skill edge to a finish probability distribution

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

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

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

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

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

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

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

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

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

To reiterate:

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

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

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

Adding this finish probability distribution to the WSOP expected utility analysis

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

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

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

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

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

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

Coming up next...

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

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

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

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

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

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

Saturday, May 7, 2011

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

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

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

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

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

Model inputs

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

Parameter choices

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

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

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

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

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

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

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


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

Some results

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

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

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

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

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

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

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


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

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

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

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

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