Cash Game Tax Planning Calculator - Instructions

Happy New Year! I have built a practical and important poker spreadsheet that I hope will make up for a lack of recent content here.

Hopefully 2012 will be a fruitful year for the poker industry — it'd be hard to be worse than 2011 — but in the meantime, while we wait around and attempt to keep our games sharp, us unwitting part-time live poker players in the US need to be mindful of our 2012 income taxes as we plan our play in a year without the volume afforded by stable online poker to help us hit the "long run" by the year's end.

I know I've written about a lot of topics that are interesting, but not quite practical. This is not one of those. This is extremely valuable practical tool that will help you guide real-life decisions and improve your bottom line. The results that you'll find will often be counter to your intuition, especially if you aren't playing very often anymore.

I have made it freely available here (and thanks to our friends at pokerfuse.com for the hosting):

Download Updated 2013 Version Here (Excel)

Download 2012 Version Here (Excel)

(You may have to give permission for macros to run. There's nothing malicious or objectionable.)

What does it do?

This spreadsheet lets you input a plan for your cash game poker play for the year, simulates it, and computes your true bottom-line after-tax winrate.

Why should I care?

This isn't just a simple calculator for how much tax is paid on a certain amount of winnings. It accounts for important and complicated effects of the US income tax rules for poker.

In a perfect world, where poker is taxed in a consistent and fair way and where poker players are easily able to comfortably put in enough volume to get close to the "long run", a poker player would be able to realize the full value of his expected value in a poker game. We do not live in this world, and hence the variance of poker results has a real, quantifiable cost.

Four major forces act to impact a player's bottom-line payoff from a year of poker playing:

1. There is no tax deduction or carryover for a losing year in poker — This affects both amateur and professional players and has a substantial effect on the decisions of which games to play in. For example, upon reaching the end of a year, a poker player who is close to even for the year may have to move down in stakes or stop playing entirely to avoid the "negative tax freeroll" of ending up with a losing year.
2. Progressive tax rates induce extra risk aversion — A similar but lesser effect occurs when a player's poker activity could push them either upwards or downwards into a new tax bracket. Notably, a player in the WSOP risks $10,000 their marginal tax bracket, but will be taxed on their winnings at the highest possible tax rate if he has a big score, which eats into expected after-tax profits. This effect is much weaker at lower-variance pursuits, such as cash games, but can still impact year-end decisions significantly as seen in this model.
3. Personal risk aversion — In my experience, this effect is much smaller than the tax effects, at least for players with reasonable amounts of wealth/bankroll, but is still worth including in the model. Utility theory is a way of approximating and quantifying personal risk aversion, and I've discussed how to construct and apply it to poker decisions it in a series of posts beginning here.
4. Loss of standard deduction for amateur players — Amateur players cannot simply report their net poker winnings on their taxes. Instead, they must take the sum of their losing sessions as an itemized deduction against the sum of their winning sessions. If the player does not have enough other itemized deductions to offset the standard deduction, he will lose out on either the ability to deduct his poker losses or the tax break afforded by the $5,950 standard deduction. This is a very serious tax effect for amateur players who play cash games at reasonable stakes, in many cases effectively introducing a $1,000-$2,000 cost of playing ANY amount of poker during a year.

Previous models I've written about, particularly this one, have focused on effects #1, #2, and #3. I've mostly ignored effect #4 to date, treating its effects as a foregone conclusion that would almost always fully hit any player with a reasonable volume of play.

However, with 2012 being the first full year where many US players will be unable to enjoy the liquidity and convenience of stable online poker, it's going to be hard for many part-time players to put in enough volume to justify the cost induced by effect #4 or to have a sufficiently-low probability of a losing year as effect #1 demands. A winning player who made a solid profit from online poker over the last decade would have happily paid the $1k-$2k yearly "poker license cost" of effect #4, but if he is only going to be able to make a trip to a live cardrooms once a month in 2012, that cost may now exceed his expected profits.

For the suddenly-large group of American low-volume live players who will not be able to get anywhere near as close to the "long run" in 2012 as in years past, these tax effects can completely destroy expected profits. Now, much more than ever, it's necessary to plan ahead for the impact of taxes on one's poker career. This spreadsheet helps guide these decisions.

Setup

Only the cells with the white backgrounds need to be modified with the inputs for your personal circumstances. Let's walk through them all through the example of our classic "typical" player.

• Annual Salary — Your non-poker taxable income for the year, which is treated as nonrandom. Use your best estimate. Professional poker players with no non-gambling income should set this to $0 and reflect their poker income through their poker results.
• Prior Wealth — Your net worth at the start of this year, not including your income for this year. This is only used in calculating the effects of risk aversion, as the utility function depends on your prior wealth. A rough estimate is fine.
• Risk Aversion — Your risk aversion parameter for the utility function built here. If you're convinced that you're completely risk-neutral, feel free to lower this, but, in my opinion, 0.80 should be reasonably accurate for most people. Don't sweat it too much, as the risk aversion effects are usually dominated by the tax effects anyway.
• Other Itemized Deductions — This is the amount of non-poker, non-gambling itemized deductions you will take this year if you were to itemize deductions. This commonly includes state income tax paid in the prior year, mortgage interest, medical expenses, and more. Here, we'll assume that our typical player just has a small itemized deduction for his state income taxes he'll pay during 2012.

• YTD Winning Sessions — Year-to-date winning sessions. This will start at $0 at the beginning of the year, but should be updated on an ongoing basis to give more accurate recommendations as time goes on and as results come in. The power of this spreadsheet is how it makes it convenient to keep up with dynamic reevaluations after each session.
• YTD Losing Sessions — Year-to-date losing sessions. Note that this should be a positive number; if you have $1,000 in losing sessions for the year, put $1,000, not -$1,000.
• File as Pro? — Amateurs will leave as "No", while professional players should change this to "Yes". Filing as a pro removes effect #4, as pros get to report only their net poker income, but pros must pay an additional 15.3% tax for their self-employed income from poker. Keep in mind that most people do not get to choose whether or not they file their taxes as a professional poker player. Consult a tax professional.

The spreadsheet allows you to project calculations based on up to three different types of games. Here, our example player expects to play in a small-stakes home game as well as some typical live $1/$2 and $2/$5 NL games. In each game, you should provide your best estimate of your hourly winrate, your standard deviation (you can draw some rough guidelines for NL holdem games from here), and how many hours you expect to play during each session.

This section is where you input how many times you expect to play each type of game. In our example, our typical player expects to play in his home game ten times this year, and to make a trip to the local cardroom to play $1/$2 five times this year.

It can be useful to run multiple game projections simultaneously to compare them. Fill in additional rows in the table provided and the program will treat them upon hitting the Calculate button. Keep in mind that each extra row increases runtime.

By clicking the Tax Rates tab at the bottom of the spreadsheet, you can modify the state and federal tax brackets to suit your individual situation. The federal tax brackets and standard deduction that I've provided are accurate for taxpayers filing as single in 2012. They should be changed if you are married, see here. You should also change the state brackets to those of your state. The first column contains the increasing tax rates in order, and the second column contains the highest amount of income taxed at that rate. For example, for New Jersey taxes, the first $20,000 of income is taxed at 1.4%, then income between $20,000 and $35,000 is taxed at 1.75%, and so on. The second column of the last row should always be a large number since the program will not properly account for income above that amount. If your state has no income tax, replace all of the percentages in the state income tax table with zeroes.

Calculation and Results

After all of the inputs are properly set, hit the big blue Calculate button to execute the calculation. This should take about 1-2 minutes per row as the program runs through N = 1,000,000 different possible yearly outcomes based on the number of sessions specified. If you just want to test only one possible session plan, leave the unused rows blank to minimize runtime.

Once the calculation is complete, the results appear in the blue cells. Keep in mind that, when you make changes to any of the inputs, the results will NOT be accurate until you've hit the Calculation button again.

• Total Certainty Equivalent — This dark blue column is your bottom-line result. The number reported here is the certainty equivalent of the planned year of play beyond your year-to-date results, that is, it's the amount of additional nonrandom salary that would be equivalent to your planned random poker results. This is not just the after-tax amount of your original expected value; it represents the amount of nonrandom pre-tax money that would be equally preferable to your random poker results.
• True Year-End $/hr — This divides the certainty-equivalent payoff by the number of hours played to return your true, effective average hourly rate over the entire year of play. When considering your bottom line, you should treat this as your true hourly winrate for the year. Due to the four effects detailed above, this will always be less than the raw hourly rates that you provided in your game descriptions, but if these negative tax effects don't end up impacting your results too much (i.e. as if you put in a very high volume of play), your true winrate will approach your raw winrate.
• Marginal True $/hr — If you've run multiple rows, this shows your true winrate for executing the sessions in the current row in excess of the sessions chosen the prior row. This is intended to allow you to see the marginal true winrate over each additional sessionby running multiple rows in which one extra session of a certain game is added in each successive row.

To provide a sense of how much is being lost to the negative tax effects, the raw (that is, the unperturbed, unaffected numbers based on the game information you provided) total $/hr and marginal $/hr are provided for comparison.

In our results, we see that our unfortunate typical player is going to, on average, lose $74 this year by playing his home game ten times and playing $1/$2 in a cardroom five times. His executing this poker plan will end up effectively reducing his salary by $74 versus if he were to not play at all. The negative tax effects have created a cost of playing that exceeds the raw $440 that he would win on average.

Unfortunately, this is not an unusual result. Quite a bit more play is often necessary for a part-time player hit hard by the loss of the standard deduction to be able to break even, let alone profit! If this player doesn't have the time to play poker any more than this, he should consider forming a backing deal which completely eliminates his variance, or, sadly, not playing at all.

This simple case illustrates the need for careful planning through the use of such a calculator. The winning player looking to occasionally stay in practice likely would not expect his poker habit to cost him money, but indeed it might.

Under the hood

The core of the program is a Monte Carlo simulation of the possible year-end poker results, which basically means that the program simulates many random trials and tracks the sample average. Excel, despite having a nice front-end, is not ideal for computations of this magnitude, which is why this runs slowly. The necessary sample size (N = 1,000,000) and associated runtime is higher than I expected; since the utility function maps wide intervals in dollars into tiny intervals in units of utility, a very low standard error on the expected utility is necessary to keep the dollar results accurate.

Playing around with different sets of numbers can take some time, but it's still reasonable to update and run this program after every poker session. Really, though, this should be implemented in a more efficient language than VBA. The methodology is fairly simple.

Limitations

• This isn't a complete solution to the poker planning problem. The truly optimal poker plan for most sets of available games will involve starting at one stake, but moving up or down based on ongoing results throughout the year. Once a decent positive profit is locked up, it becomes safer to move up to a higher-winrate, higher-variance game. I have found that this is too computationally intensive to solve in Excel via backward iteration. Updating and re-running this spreadsheet on an ongoing basis should help. The effect of this simplification to the optimization problem will be to underestimate the true certainty equivalents; when you reserve the right to change stakes in the future rather than lock into your plan, your EV might increase and cannot decrease. So, keep in mind that this spreadsheet essentially forces you to make your plans as if you had to commit in advance to playing a certain number of sessions, while, in reality, you could optimally quit or change games in the middle of the year.
• This doesn't treat tournaments. It'd be conceptually easy to add them, but difficult to program and implement, as tournament finish probability distributions are so much uglier than Gaussians.
• Using anything but a Gaussian distribution for cash game results would be a pain, but the normal approximation to cash game results should be good enough.
• This doesn't currently accommodate the negative tax effects for amateur players of the infamous bad poker tax states, where gambling loss deductions are prohibited or limited for the purposes of state taxes.
• Some other possible negative tax effects of poker that are not treated by this model are the triggering of the Alternative Minimum Tax, the loss of medical deductions due to artifically-high adjusted gross income, and effects on married taxpayers.
• This is designed for US taxes, and I'm not familiar enough with the taxation of poker in other countries to know if this could be useful to non-Americans. However, it should be able to handle any tax system that involves a constant or bracketed percentage tax on poker winnings but disallows deductions or carryover for poker losses. In most cases, I imagine this would involve turning off the state taxes, standard deduction, and self-employment tax in the Tax Rates tab.

I welcome your feedback, suggestions, questions, and bug reports in the comments below. I apologize in advance if my calculator is the bearer of bad news for your part-time poker career, but it's much better to know the costs before you begin playing.


Continued in Part 2: Examples, charts, and general results

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

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

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

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

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

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

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

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

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

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

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

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

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

Tax effects of stolen funds for those who file as professionals

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

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

...but for amateur players...

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

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

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

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

The results:

• If Bob's AGI for the year is above $15,000, he cannot deduct any of the $1,500 that he lost. This is almost certainly the case, even if Bob does not suffer from undue AGI inflation from his poker sessions.
• However, Bob still earned $1,000 in gambling winnings from his play on the site, even though he never got his money.
• Notice that, given that the site ended up disappearing, Bob would have been better off if he lost money on the site. Winning has increased his taxable income, despite not providing any actual money or deductions to Bob.

Effectively losing money by winning at poker on a site that disappears is highly unfortunate, and a pretty big deterrent to playing at all when there is any risk of losing one's online balance to the site-specific risks.

Implications for the model

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

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

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

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

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

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

Other bad news

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

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

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

Possible exceptions

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

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

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

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

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

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

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

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

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

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

Shares sold at face value

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

For the time being, we are ignoring some other popular forms of staking and backing (listed in the order of likelihood that I might add them to the model in the future):

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

For now, we consider only the simplest case: shares sold at face value.

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

Optimizing share-selling for the typical player

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

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

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

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

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

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

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

Other cases: different risk/tax profiles

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

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

---

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

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

Optimal hedging percentages

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

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

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

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

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

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

Conclusions and comments

These charts should provide a useful guideline for real-world staking and backing decisions for large-field tournaments. Some practical notes:

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

Let me know if you'd like to see me add the considerations of other types of staking and backing contracts to this model. And, as was the case with my risky site bankroll management model, this is a model with many specific variables (wealth, income, risk preferences, ROI, tournament field size, tournament payout distribution) that would ideally all be tailored to each specific player and each specific tournament on a case-by-case basis.

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

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.

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

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

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

The remaining U.S. sites should be expected to be different than PokerStars and Full Tilt Poker in at least a few incredibly important ways:

• Since the subset of former American players which chooses to move to these sites will be highly skewed towards serious, professional players, all players should expect their winrates to drop significantly.
• Deposits and withdrawals will be more costly and more unstable, and players should account for some nonzero probability of never receiving a cashout.
• In the event of either voluntary or government-induced site closure, due to liquidity issues and the fact that these sites are less reputable, there is some probability that U.S. players would never get their account balances returned to them.

Not too comforting. To be sure, even if playing on the remaining smaller sites is otherwise acceptable and a reasonable substitute for the experience of pre-Black Friday online poker (which will not be the case for all poker players), these issues are serious and will annihilate a lot of utility. These added risks cause players to have to make tradeoffs involving their profit potential and the amount of money they risk in their account balances at these sites.

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

Model inputs

We start by making assumptions about various aspects of the risks and costs of playing on these sites. Some of these are deterministic, and others will be uncertain. Once we have settled upon reasonable estimates of these values for all of these parameters, we can vary the most critical parameters to find break-even thresholds for expected utility, which can guide player decisions.

• Fees associated with deposits and withdrawals — Deposits and withdrawals at smaller sites are costly, in both inconvenience and fees, so managing account balances carefully will be important to avoid incurring too many of these expenses. Our model will consider fixed fees for depositing or withdrawing, which we can expand to include intangible costs that reflect the inconveniences of these money transfers.
• Deposit/withdrawal strategies — The simplest way to describe a rule that guides when to cash out or deposit is a set of four numbers. When the player's account balance is below some critical level (such as when it is too low to play his chosen stakes and number of tables), he should redeposit to bring his account balance up to some higher threshold level. Similarly, since he doesn't want to keep an unnecessary amount of money in a risky account, when his account balance hits some upper critical level, he should withdraw to bring his account balance down to some lower threshold level (which might be close to, if not exactly the same as, the threshold level for deposits).
• Winrate, standard deviation, and play volume — The model will approximate poker results by increments of the appropriate normal distribution. To match the rest of the model, rather than looking at winrate and standard deviation per hand, we can look at these on a per-day basis by factoring in the amount of hands the player plans to play each day. The number of days remaining in the year will be one of the inputs.
• Per-day probability of site closure — In our model, at the end of each day, we will assume that there is a fixed probability that the site will close forever. Each day will be independent of the last.
• Probability of getting paid if the site closes — When the site closes, there is a chance that all player balances will be lost.
• Probability of getting paid on each cashout — For each cashout prior to closure, there is a chance that the player will never see his money (or never be able to cash out in the first place). We can assume that this probability is less than that of the probability of getting paid when the site closes.
• Tax and utility functions

That's a lot of inputs, and a lot of inputs that we can't necessarily get great estimates of. Since the non-standard parts of this model are those pertaining to the risks of having money with the online site, we can ignore the lesser (or at least more standard) uncertainty on personal game performance parameters and instead assume that we know the player's exact winrate.

Parameter choices

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

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

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

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

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

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

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

Algorithm

To calculate the expected utility of playing with these risks, these winrate parameters, and this utility function, we will simulate the system through the following steps:

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

Some results

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

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

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

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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).

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

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

Conclusions

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

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

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

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

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