Cash Game Tax Planning Calculator - 2013 Update

I have updated the Cash Game Tax Planning Calculator for the new year, containing some brand-new potential negative tax effects which will impact certain amateur poker players.

If you haven't used the Cash Game Tax Planning Calculator before and you're a US poker player, especially if you're only playing part-time and don't file your taxes as a professional, now is the perfect time to start. The purpose of this spreadsheet is to calculate a player's after-tax expected value based on their intended poker play for the year. This is important to do in advance, as a few different negative tax effects can easily turn a winning player into a losing player. You'll also probably want to check out the older posts on the Cash Game Tax Planning Calculator, as I'll only discuss the 2013 changes here.

The new spreadsheet is freely available here, with continued thanks to pokerfuse.com for hosting:

Download Here (Excel)

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

Runtime and efficiency

Note that this version of the sheet takes about 3-4 minutes per row for the default recommended loop size of N=1,000,000 when you hit the Calculate button. It's not frozen or broken! If you'd like to test it out to make sure it's working before you embark on a longer run, set N=100 in cell K6 for faster calculation, but keep in mind that the results will likely have significant error for N<1,000,000. Excel is really not an ideal platform for these calculations, but I find that it's not too much trouble to run them overnight. This certainly isn't commercial-grade software, it's merely something I've built for myself that I'm making freely available for the benefit of the poker community. (i.e. bear with it and thanks for your patience!) New tax considerations for 2013

The recent fiscal cliff compromise included a few ways to penalize high-income tax payers... but, as with many pieces of the tax code which attempt to assess income, these income thresholds look at Adjusted Gross Income rather than net income. AGI is a before-deductions figure which includes "phantom income" for amateur poker players resulting from the sum of all losing sessions they play throughout the year. Amateurs report the sum of their winning sessions as income and deduct the sum of their losing sessions (to the extent of their winnings) later as an itemized deduction.

Net taxable income is generally what ends up affecting the tax a player owes, but AGI is overinflated along the way, and that's what these three new rules consider. Thus these new effects will be triggered by many amateur players whose actual incomes are not high at all. (If you need a refresher on poker tax basics, check out the 2+2 tax sticky.)

1. Phaseout of itemized deductions for "high-income taxpayers"
   
   The first new rule triggers for taxpayers whose AGI exceeds $250k for single taxpayers, $275k for heads of household, and $300k for those who are married and filing jointly. Once this threshold is exceeded, most itemized deductions are reduced ("phased out") by 3% of the amount of the excess of AGI to the threshold, up to a maximum phaseout of 80% of these deductions. Luckily, the deductions which are phased out do not include the gambling loss deduction, which would have resulted in a very costly direct surtax on gross amateur poker earnings, but essentially all other common types of itemized deductions are included. Further discussion and links can be found in this thread.
   
   This will create a negative tax effect on those amateur poker players whose phantom poker income pushes them over the $250k threshold. Professional players (who have no phantom income) and all other Americans should be mindful of this new threshold as well, but they get the privilege of having their AGI reasonably approximate their true income. Naturally, the biggest impact here will be on those with many non-poker itemized deductions.
2. Phaseout of personal exemption for "high-income taxpayers"
   
   Similarly, using the same thresholds as the above effect, a taxpayer's personal exemptions are reduced by 2% for every $2,500 of AGI in excess of the threshold. This, again, impacts all amateur poker players with high gross winnings, and will hit the hardest on those with dependents who would take multiple personal exemptions.
   
   Personal exemptions were not considered at all in the 2012 build of this spreadsheet, but are accounted for in the new version to allow for this effect to be captured.
3. 3.8% surtax on investment income for "high-income taxpayers"
   
   The final noteworthy change is a 3.8% surtax to fund Obamacare, applied to the excess of AGI over a certain threshold, but not exceeding the amount of one's investment income. For this rule, the threshold is $200k for single taxpayers, $250k for those who are married and filing jointly, and $125k for those who are married and filing separately.
   
   Poker winnings, be they amateur or professional, are not considered investment income, so this effect will only matter on players who happen to have significant income from traditional investments, but it hits very hard for those who do.

Other changes in this version include an update of the default tax brackets for 2013 and a bug fix regarding the way deductions were treated in some cases where itemized deductions and amateur gambling losses were less than the standard deduction.

New inputs & instructions

To account for these new tax considerations, there are 4 new inputs in the 2013 version, all of which are located in the Tax Rates tab at the bottom of the sheet. All other inputs are the same as the previous version, and everything else about the sheet, its backend, and its runtime should essentially be familiar.

• Personal Exemption — The total amount of your personal exemptions. For 2013, this is $3,800 for a single person with no dependents. If you have dependents such that you take more than one exemption, this should be changed to $3,800 times the number of total exemptions you take.
• Portion of salary which is investment income — The approximate amount of investment income you expect to have this year. Note that your Annual Salary figure on the main tab should also include this amount in its total, as the Annual Salary figure is intended to include all non-poker sources of income. This new field simply requests the portion of that total non-poker income which comes from investments so that the effect of the new investment surtax can be computed.
• Phaseout Threshold — The threshold for the phaseout of itemized deductions and personal exemptions as described above. Set this equal to $250k if you are single, $275k for head of household, and $300k for married filing jointly.
• Investment Tax Phaseout Threshold — The threshold for the phaseout of itemized deductions and personal exemptions as described above. $200k for single taxpayers, $250k for those who are married and filing jointly, and $125k for those who are married and filing separately.

2013 new trouble case #1: High-Volume Amateur

Each of these new effects will only unduly impact amateur players, due to the AGI inflation from their phantom poker winnings; high-income professional players will be paying higher taxes as well, but only when their legitimate winnings exceed the income thresholds. With that in mind, we first look at the case of an amateur player with high phantom poker winnings and thus significant overinflation of AGI.

Consider a high-volume amateur poker player, perhaps a full-time student playing poker on the side as his primary source of income. He has $80k net worth, non-poker income of $3k per year, all of which is from investments, and $3k in non-poker itemized deductions (state taxes paid last year, perhaps). Let's say that he plays in certain poker games such that he has an hourly rate of $40 and a standard deviation of $1,800 per hour, and that he typically plays for 2 hours per session. I'm envisioning a solid winner at multitabling small/mid stakes online here, which is not currently relevant to very many American taxpayers, but I've chosen a higher-variance and higher-volume profile to highlight the potential impact of the new rules.

This player will certainly put in enough volume to avoid the classic negative tax effects of the risk of a losing year or the risk of being unable to deduct losses due to not hitting the standard deduction, so we don't worry about that in the graph. However, once his number of sessions increases to about 250 for the year, his expected AGI from the sum of his winning sessions starts to hit the threshold and the effects of the investment surtax and exemption/deduction phaseout start to cut into his profits, leveling off at a maximum effective loss of $2,500 when these are all fully phased out.

The new tax rules won't turn this high-volume amateur from a winning player into a losing player, but they will take a rather high percentage of his average after-tax profit at certain volume levels.

2013 new trouble case #2: Wealthy Amateur

Some low-volume amateur players will already be at or above the income thresholds from their non-poker income, in which case even a single poker session will produce phantom income that will cut directly into the new tax rules. These players are in much worse shape than those in the first case.

Consider a upper-middle-class person with a successful career outside of poker who enjoys occasionally playing poker for fun, but still cares about winning, or at least not losing money in her poker career. Let's say she has $1M net worth, and non-poker income of $250k per year, $20k of which is from investments, and $15k in non-poker itemized deductions. For simplicity, let's say that her choice of game and her winrate profile is the same as that of the high-volume amateur, except that the wealthy amateur fits in far fewer of these 2-hour sessions each year.

The impact on expected after-tax winnings is dramatic. In 2012, she would have had to play about 40 sessions per year to break even (due primarily to the classic negative tax effect of having a losing year). In 2013, she now must play three times as much volume to break even after-tax, and her earnings are increasingly divergent from what they would have been in 2012. The gap between the earnings lines would continue to grow until all investment earnings were met with the 3.8% surtax and all deductions and exemptions were phased out to their maximum.

Any amateur poker player whose income is at, over, or near the thresholds for the new tax rules will have to be very careful. Depending on their winrate profile and the amount of investment income and deductions they expect to have, many will be forced out of the game.

A brief note on a third potential trouble case (it could happen to you)

Many amateur poker players will not fall into either of the above two cases. However, if such a player happens to run well enough during the first portion of 2013, perhaps by spiking a big tournament score, such that the likelihood of hitting the thresholds goes on to become significant, they will suddenly find themselves impacted by these effects.

The spreadsheet is designed to be used on an ongoing basis after every session is played, or at least after any big wins or losses. As another example, a large early realized loss will mitigate the negative tax effect of having losses which, along with other deductions don't exceed the itemized deduction, which will benefit the profitability of future sessions during the year. Running this calculator often will make sure you always know the costs of these new negative tax effects as well as the ones that have always been around.


Everybody's situation will be different, particularly under these new rules, since they are sensitive to the particular amounts of investment income and non-poker itemized deductions you expect to have. If you're a responsible player, you should check out the calculator and to use it to compute the after-tax expectations for your individual situation. Here's hoping that you'll still be +EV after-tax, but if not, it's better to determine that now rather than putting in your year of play and getting hit with an excessive tax bill at the end.

As usual, I am eager to hear your feedback, suggestions, questions, bug reports, etc.

I'm also working on coding a version of the calculator in [R], mostly for my own edification as I learn the language and for my own use. It won't be as pretty or user-friendly, and I am not planning to ever be able to compile it into something palatable to the general public, but it will be faster and more flexible if you're comfortable with code. Feel free to email me if you want to mess around with it.

See also:
Cash Game Tax Planning Calculator - Instructions (2012)
Examples, charts, and general results

Cash Game Tax Planning Calculator - Examples, charts, and general results

If you haven't read the first post about this calculator, containing the download link and the instructions, you should do so first here.

Today, we will do a few more runs of some different plans and conditions for our example player and see how his bottom line is affected. I strongly recommend that you download the spreadsheet and play along at home with your own personal situation to see how it compares to the cases discussed here. The examples I go through will illustrate some important overall trends, but there should be substantial differences in the magnitude of the impacts from player to player.

Our example player

From the first post, we recall that our example player has the following characteristics and pays 2012 federal and New Jersey income taxes as a single taxpayer.

We had found that playing his home game ten times and making a trip to a cardroom to play $1/$2 five times led this player to effectively lose $74 on the year. How are different circumstances and decisions contributing towards this loss, and what can the player do differently to try to mitigate it?

We've seen that the negative tax effects hit the hardest on the very first sessions played in a year. Let's see how the costs look when looking at even fewer sessions than we first considered for this player.

The cost of social play, and overview of charts

Let's assume that our player is totally committed to playing his home game at least occasionally. The social value is worth it to him, even if the tax effects do end up making it cost him money on average. How much is he losing?

We run the spreadsheet for different session counts of the home game (with zero live cardroom trips) and look at the three charts produced by the file, located in the three yellow tabs at the bottom of the spreadsheet.

The first chart shows the year-end certainty equivalent as a function of the number of sessions played. The slope is more negative for the first few sessions as the player starts cutting into the standard deduction and risks a losing year. As more sessions are played, the slope increases (though it nonetheless stays negative here), which will generally be true for any game and for any set of inputs.

The second chart plots the true year-end winrate against the raw year-end winrate (the before-tax information provided by the player), again plotted against the number of sessions on the x-axis. The red line, representing true winrate, can never exceed the dotted green line of the raw winrate, though it will approach it as volume of play increases. For this particular home game, we see that it's not even close.

The third chart is similar to the second, but instead plots the marginal winrate, the per-session winrate for each additional session. We see through the jaggedness of the red line that there is more simulation error in the marginal winrate than the true winrate. A higher value of N and thus a much longer runtime would be necessary to get a perfect handle on this. Nonetheless, the trend is clear, and the overall effect here is:

In most situations, after a certain number of initial sessions to cover the biggest negative tax effects, the marginal winrates should increase as the number of sessions increases.

So, yeah, ouch. The variance is so much greater than the tiny winrate of the home game that it's a losing proposition even if 26 sessions of it are played. It turns out that our player, if playing only this home game, would have to play the game almost 200 times just to break even in certainty equivalent for the year! The loss of the standard deduction and the effect of a losing year is absolutely brutal for smaller-stakes players and where edges are thin relative to variance.

This would not be a good game to play to attempt to derive any profit from, but, if the primary motivation is having fun, it's only going to cost $114 in certainty equivalent to play this game ten times for the year. Frustrating, but probably worth it.

Adding cardroom play

Let's assume that our player will play his ten sessions of the home game no matter what. Then one of the only variables that he has any meaningful control over is the number of trips he makes to the local live cardroom, where we presume that profit is a bit more of a goal here. How many trips will he have to make before he ends up effectively ahead for the year?

Well, here's one answer — our player will, on average, effectively make money for the year if he's able to make 11 trips to play $1/$2.

Unfortunately, his hourly rate is still quite low compared to his raw winrate, showing that the negative tax effects are having quite an impact.

While the player needs to play 11 sessions of $1/$2 to make a profit for the year, it turns out that, given that he's already playing ten sessions of the home game, even a single session of $1/$2 has a (barely) positive certainty equivalent. If he weren't already playing the home game, the first $1/$2 session would be a loser (-$35 in certainty equivalent), as it would begin to delve into the standard deduction, but the home game has done that already here. This illustrates a rule that should hold true in general:

The effective profitability of any given session goes up as the number of other sessions increases.

Higher stakes

What if he were to play $2/$5 instead? His raw winrate goes from $8/hr to $15/hr, but the standard deviation increases by 180%. How will the interplay of higher winrate and higher risk affect decisions for the part-time player?

Here, I've plotted the $2/$5 results in a yellow line against the $1/$2 results in blue.

The shape of the $2/$5 curve is quite different. Playing $2/$5 instead of $1/$2 demands 21 trips to the cardroom to break even, rather than 11. However, $2/$5 will be more profitable than $1/$2 if the player ends up playing more than 23 cardroom sessions.

We also look at the marginal per-session winrates for $2/$5, plotted here without the results for $1/$2:

Comparing this chart to the prior marginal winrate chart for $1/$2 shows the difference between the different stakes. The marginal winrate is quite negative for $2/$5 at first, but reaches a much higher peak after a few dozen sessions, even coming quite close to reaching the raw winrate. This illustrates another general result:

Games with higher winrates but higher variance will take more sessions to become profitable after tax, but will converge to raw winrates faster.

The value of putting in longer sessions

Since the effect of the loss of the standard deduction is a byproduct of the necessary session-by-session accounting for amateur players, it would always be desirable to group more of a player's poker results into a single session. Since there's currently no acceptable argument that a week/month/year of poker play can be considered a single session, the only way to achieve this is to actually play longer hours in each session.

So, what if our example player had the stamina to be able to put in 16-hour-long sessions at the live cardroom instead of twice as many 8-hour-long sessions?

For $1/$2 play, we plot the original 8-hour session plan in blue and the 16-hour session plan in yellow, where we count each 16-hour session as two spots on the x-axis so that the two lines correspond to the same amount of hours played.

And, similarly, for $2/$5 play,

Considering that the length of sessions has no effect on raw winrates and does not commonly fit into how most poker players make their decisions, the positive effects of condensing one's poker hours into longer sessions are quite dramatic. The improvement is more substantial on the $1/$2 play, where the negative impact of the standard deduction issue was greater in the first place. In each case, the longer sessions help bring the true hourly rate closer to the raw hourly rate. Overall, the conclusion here is an important one for part-time amateur players getting hit by the standard deduction effect:

Playing a longer session instead of multiple smaller sessions can substantially reduce the negative impact of the potential loss of the standard deduction.

The effect of losing sessions and other itemized deductions

Restricting our attention now to the original case of ten home game sessions and five $1/$2 sessions, how much better off would our player be if he had more possible itemized deductions?

In reality, taxpayers rarely have control over their non-poker itemized deductions, but it should also be noted that this includes year-to-date losing poker sessions. For example, if the player has $4,000 in other itemized deductions for 2012 and has already booked $2,000 in losing sessions so far in 2012, the tax effects for the purposes of future decisions are the same as if he had $6,000 in itemized deductions and no losing poker sessions.

Here, the x-axis is the amount of itemized deductions that the player has already realized for the year:

We see that the cost of this tax effect is approximately constant for most smaller values of itemized deductions, as this player will be unlikely to accumulate enough losing sessions to make up the distance to the $5,950 standard deduction, but that the after-tax value of playing poker is recovered quickly as the deductions approach the amount of the standard deduction. The rightmost point on this graph, where itemized deductions exceed the standard deduction, is a case where the standard deduction tax effect is completely gone. The raw certainty equivalent here is $440, and without the impact of the standard deduction tax effect, the player is able to come much closer to fully realizing this than he does when he has no other deductions. Overall, in cases where the standard deduction tax effect would otherwise be significant:

An amateur who has accumulated enough losing sessions and other itemized deductions to come close to or exceed the standard deduction is able to retain much more of the value of his future play for the year.

Conclusion

These different cases illustrate how significant the effective costs and expected after-tax payoffs from a part-time, amateur poker career can differ based on individual facts and circumstances and as losing sessions are accumulated throughout a given year. The rules of thumb in this article can shape your intuition for understanding some overall effects, but intuition is still not going to be a reliable means of approaching this calculation. It's best to continually update and use the spreadsheet yourself.

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

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.

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

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

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

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

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

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

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

Model inputs

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

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

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

Parameter choices

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

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

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

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

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

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

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

Algorithm

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

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

Some results

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

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

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

---

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

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

---

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

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

Conclusions

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

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

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

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

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

How far from Gaussian (normal) are poker results?

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

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

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

Data

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

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

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

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

Visualizations of Normality

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

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

For the $1/2 CAP data:

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

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

For the deeper-stacked $1/2 RUSH data:

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

For the $1/$2 PLO data:

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

As a final point of comparison, I thought it would be interesting to look at heads-up tournaments, where the mass of the probability distribution would be only on the two extreme possibilities of -1 and + ~1 (adjusting for rake). I used some fictitious data based the rake structures of $55+$2.50 HU SNGs and on a winrate of 55%, with the buyin size normalized to 100:

Here it takes at least 100,000 hands (tournaments) before the distribution begins to look anything like a continuous Gaussian. Though I haven't tried anything with multitable tournaments (much higher skew), we'd expect those to be even less Gaussian.

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

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

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

Accuracy of Gaussian ruin probabilities

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

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

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

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

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

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

For the $1/2 CAP data:

For the $1/2 RUSH data:

For the $1/$2 PLO data:

For the fictitious HU tournament distribution:

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

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

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

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

Conclusions and Implications

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

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