Utility, Part 3: Tax Effects

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

Restricting Utility to One Year

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

Instead of using the standard ρ=0.8 isoelastic utility function (defined on our ongoing wealth), we can shift the function to the left by the amount of our preexisting wealth and thus have a utility function defined on the change in wealth we realize over the upcoming year. If we let the variable w represent our preexisting wealth and x our change in wealth during the upcoming year, then our shifted utility function is:

That is, if our current net worth is $100,000 and we are considering flipping a coin for $10,000, instead of comparing the utility of having a net worth of $90,000 or $110,000 on this graph:

... we instead compare the utility of having an annualized change in net worth of -$10,000 or $10,000 on this graph:

So, in order to define utility on the change in our wealth in the upcoming year, we shift the original utility function to the left by our net worth at the end of the prior year. The results, of course, will always be the same regardless of any shift. This is just a change in methodology, but it allows us to introduce a tax function which operates on each year's income.

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

Using the Shifted Function Within the Year

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

If the random variable X represents our gains and losses for the entire year, it can be broken up into its parts:

For example, let's say it's June 1st, we've made $5,000 so far for the year, and we come across the opportunity to flip a coin for $1,000 with a 55% chance of winning. Then:

• X_past = $5,000; our year-to-date earnings are constant.
• X_present is a random variable representing the opportunity we are facing, taking value +$1,000 with probability 0.55 and -$1,000 with probability 0.45.
• X_future is a random variable representing all possible outcomes of our future results after this coin flip, from June 1st through December 31st of this year. This includes our usual "daily grind" along with probabilistically accounting for other opportunities we may encounter (such as this special coin flip).

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

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

The Tax Function

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

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

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

With these assumptions, the 2011 federal tax brackets are (source):

Taxable Income	Marginal Tax Rate:
$0-$8,500	10%
$8,500-$34,500	15%
$34,500-$83,600	25%
$83,600-$174,400	28%
$174,400-$379,150	33%
$379,150+	35%

and the 2010 NJ tax brackets (can't find any info on 2011 yet) are (source):

Taxable Income	Marginal Tax Rate:
$0-$20,000	1.4%
$20,000-$35,000	1.75%
$35,000-$40,000	3.5%
$40,000-$75,000	5.525%
$75,000-$500,000	6.37%
$500,000+	8.97%

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

With these tax rates, the function t(x) that maps pre-tax earnings to our after-tax earnings looks like this:

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

Poker vs. Non-Poker Income

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

Combining the Tax Function with the Utility Function

As we assume that we gain no satisfaction or dissatisfaction from the amount of tax revenue we produce for our governments, our utility is realized on after-tax dollars. So our final utility function comes from combining the shifted utility function with the tax function:

This final tax-adjusted utility function (plotted in red below against g, with s=0) effectively adds additional risk aversion to our original isoelastic utility function. Compare it to the non-tax-adjusted utility function from earlier (blue) and notice that there is additional concavity.

Since the tax function is piecewise-linear, taxes will induce no additional risk aversion for risks that cannot move us out of the tax bracket we would otherwise be in. However, for any risky opportunity that could move us up or down into a new tax bracket, the progressive nature of the tax rates will always create some additional risk aversion.

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

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

Utility, Part 2: Finding a Good Utility Function

Last time, we looked at a basic overview of utility functions and their importance as the foundation of any model of preferences over events with uncertain outcomes. Today, we'll take a closer look at specific utility functions and try to find one that is a good enough fit for a typical real-life person's actual risk preferences.

Handling Separation of Bankroll and Personal Wealth

It occurs to me that, before we can model a player's preferences for poker risks, we need to consider the individual player's motivations in playing poker. For example, while most people play poker to maximize their expected utility of wealth, others might play poker for entertainment and never plan to ever withdraw money from their bankroll, and thus never realize any actual utility of wealth from their poker career (such as recreational break-even players). A purely bankroll-driven player such as this would have an unusual sort of utility function that looks something like this:

As long as the player has enough money in his bankroll to be able to play the stakes he wants to play (in this case, $100k), he is indifferent to the size of his bankroll. When he has less than $100k, he can't play his game of choice, and he has no utility.

In practice, these extreme preferences are not too interesting and not too relevant. For one, if the player is willing to move down and play smaller stakes, then this utility function would decrease more gradually to 0 and may end up resembling a typical utility function anyway (perhaps exponential utility). More importantly, it is unlikely that any real-world player is driven purely by being able to play poker with an entirely separate bankroll; most players will be withdrawing money from their bankroll in some fashion and thus realizing traditional utility of wealth.

There may be some parameterization that could capture an individual's degree of "bankroll-driven-ness" in conjunction with his utility of wealth, but simply using the traditional utility function will likely be close enough to this for most people. So, going forward, we will ignore this consideration.

Isoelastic Utility in Practice

From our overview in part 1, the isoelastic utility function exhibited more desirable properties than the exponential utility function. It is worth checking out how good of a fit it might be when we look at opportunities where less than one's entire net worth is at risk.

Among the properties of the isoelastic utility function was that the certainty equivalents for risking one's entire net worth on a coin flip do not depend on the amount of total wealth. It turns out that this easily extends to a useful property of isoelastic utility:

For any risky opportunity, under isoelastic utility, one's preferences do not change under a rescaling of one's wealth along with a proportional rescaling of the amount risked.

With that in mind, let's explore some easy hypothetical scenarios involving simple bets on coin flips. If the results seem consistent with the risk valuations we would expect a typical player to make, then the utility function is a good fit.

Here, we look at the certainty equivalents, expressed as a percentage of expected value, of risking a certain percentage of one's wealth (rows) on a fair coin flip when one has isoelastic utility with parameter ρ (columns). We observe that, for small enough percentages of wealth at risk, the CE of a fair coin flip is almost 100% of the expected value (it's rounded up in the table), which means that the individual is nearly indifferent between taking the coin flip and not. For larger percentages of wealth at risk, the certainty equivalents become significantly lower, especially for larger ρ. This quantifies what we would expect; higher risk aversion means a lower willingness to take large risks.


In addition to looking at the certainty equivalent of a fair flip, we can also look at the minimum probability of winning that would be enough to induce the individual to risk a certain percentage of his wealth on a coin flip biased in his favor:

The behavior is consistent with real-life preferences. When small percentages of the player's wealth is at risk, he is willing to take fairly thin edges, but he demands large edges to risk larger percentages of his wealth.

The specific numbers in the table seem only partially realistic to me. One problem is that the typical person might be inclined to demand even larger edges for the smaller percentages of wealth at risk than those prescribed by ρ=0.8, but as will be noted later, there's not much that can be done about that with isoelastic utility.


For a final approach, we go back to a fair 50-50 flip and look at the minimum overlay needed for the individual to take the risk (that is, an overlay of 2% would mean that he would need to stand to win at least $102 to risk $100 on a fair coin flip):

This is perhaps the most instructive approach to look at, as it is most intuitively matched with one's personal risk preferences. These all look fairly reasonable to me.

Shortcomings

No matter what value of ρ we choose (even if we made ρ very close to 1, as it turns out), the individual with isoelastic utility preferences is willing to risk 1%-25% of his wealth with very small edges. While this seems reasonable for individuals with small levels of wealth, this utility function cannot be fit to a wealthy individual who is still risk-averse for small percentages of his wealth. If we want to use isoelastic utility, we have to concede this point and understand that our model will produce an individual who is willing to accept thin edges for small percentages of his wealth, regardless of his level of wealth.

We might want to have the utility of near-zero wealth (bankruptcy) approach negative infinity if it would be impossible to ever accumulate any more money to start again. Instead, we'll implicitly assume that our individual has some outside source of income that could eventually restart a depleted poker bankroll.

One practical modification that I thought might be important would be to change the utility function to account for various "class levels" of subsistence income. For example, $10,000 might be the minimum amount of wealth where an individual could barely scrape by for a year. Or, as some research suggests, an annual income of $75,000 may be "enough" to be happy in the U.S., and the marginal happiness-return on higher levels of income are greatly diminished, so there might be increased importance on hitting that level of income and less importance on getting any more than that. Either of these effects could suggest that a good, practical utility function needs to be defined piecewise over these different regions of wealth. The isoelastic utility function doesn't capture this.

Ideally, we could construct a utility function which essentially varies the ρ for different levels of wealth, or at least between a few major regions of wealth. All of the above observations exploit the property of isoelastic utility to be able to look only at the percentage of wealth at risk, but in practice, we might expect these risk preferences to vary with the level of wealth. For example, the numbers for ρ=0.6 seem about right for my risk tolerance if my net worth were $10k, whereas something like ρ=0.8 seems like a better fit if my net worth were $100k.

Preferences over having a certain level of risk tolerance as a function of wealth can be captured by varying ρ, and it seems to take a fairly drastic change in personal wealth to create inaccuracies in the choice of ρ. In practice, an individual looking to model his utility with this utility function could find the ρ that fits him best and simply reevaluate and update it every year. This will only be a big issue in cases of sudden very large payouts, such as a small-stakes player winning the WSOP Main Event. So we might expect our utility function to underestimate our risk tolerance for large risks. We can account for this by erring on the side of choosing a higher value for ρ, especially when considering an opportunity with large possible gains or losses.

We're also ignoring all psychological or behavioral effects. For example, prospect theory might dictate that we modify our utility function so that losses hurt more than wins.

Conclusions

Despite the shortcomings, I found that the fit of the isoelastic utility function was better than I expected it to be. It seems to be close enough that it should suffice for modeling purposes, especially when varying ρ as appropriate for different hypothetical levels of wealth. I thought I would have to tweak it a bit to fit different realistic risk valuations for different sizes of risks, but it handled my little tests much better than expected. So we'll take the isoelastic utility function with ρ=0.8 to be our default practical before-tax utility function going forward:

Next time, in Part 3, we conclude by incorporating U.S. income tax considerations.

Utility, Part 1: The Basics

Before we can dive into any models of various poker decisions, we first need to establish the building blocks of models for rational preferences under uncertainty.

In economics, the foundation of any approach to any decision made under uncertainty is expected utility theory, which quantifies risk aversion through establishing a correspondence with diminishing marginal utility of wealth.

Those familiar with the basics of utility can skip to the end of this post, but should stay tuned for the upcoming parts, where I will build upon these fundamentals in ways which are less common in abstract theoretical models, but which are specifically well-suited for practical poker applications.

Motivation

The common heuristic approach to decision-making in poker is to make decisions in order to maximize one's expected value, with volatility ("variance", as it is usually not-quite-accurately termed) an unquantified afterthought, managed through heuristic rules, if at all. People understand that less risk is preferable to more risk as long as expected value remains the same, but there is usually little consideration given to quantifying the value of risk relative to expected value.

How much expected value should a decision-maker be willing to give up in order to reduce the variance of a random payoff by a certain amount? More generally, how do rational decision-makers value the tradeoff between expectation and risk?

General Utility Functions

Preferences over different levels of wealth are quantified by assigning a utility function to each person or entity, a function which maps a level of wealth to a level of overall personal satisfaction derived from that wealth. The usual assumptions on a utility function are that it is:

• Increasing — Everyone prefers more money to less money.
• Continuous — There's no specific amount of wealth that is suddenly much more preferable to a slightly smaller amount of wealth.
• Concave — The slope of the function is decreasing. As one has more wealth, an additional dollar is less valuable, e.g. a poor person is much happier finding $100 than a millionaire is. This is equivalent to the individual being risk-averse, rather than risk-neutral or risk-seeking.

Any function which satisfies these conditions is a potentially reasonable utility function. The precise form of the function will depend on the individual's specific preferences for different levels of wealth and, as we will see, his specific risk preferences.

Isoelastic Utility

One basic example is the isoelastic utility function, given by

Notice that, for ρ=0, this is simply the identity function, which represents no diminishing marginal utility of wealth and no aversion to risk. As ρ increases, the marginal utility of wealth becomes more diminishing, so ρ can be seen as a parameterization of risk aversion. A higher value of ρ means a higher aversion to risk.

For ρ=0.5, this function looks like this:

This function satisfies all of the desired properties. Though the scale of this plot does not indicate it well, the function is always less than that of the identity function, so this utility function can be thought of a means of "discounting" wealth in a way that accounts for diminishing marginal utility of wealth. Note, however, that it is not necessary that the scale of the function match that of the wealth; we shall see that the particular values taken by the utility function are irrelevant for decision-making, as they get mapped back into dollars after accounting for the different random payoffs of an opportunity.

The isoelastic utility function is said to represent constant relative risk aversion (CRRA), as the individual's aversion to risk is always proportional to his wealth. With higher wealth, he is less averse to risk. This is a desirable property and is generally fairly consistent with real-life decisions and the rules of thumb that most poker players use in managing bankroll requirements as they move up in stakes.

Exponential Utility

Another simple example is the exponential utility function, given by

For c=1/150000, this function looks like this:

The exponential utility function is said to represent constant absolute risk aversion (CARA), as the individual's aversion to risk is always constant regardless of his wealth. In practice, few people would exhibit constant absolute risk aversion, as we should expect that most rational individuals' risk aversion should decrease as wealth increases, though perhaps not according to the proportional scale of the CRRA utility function.

The exponential utility function is bounded from above, but that does not mean that an individual with this utility function has any upper bound to the amount of wealth he prefers. We will see, however, that this does make the individual less likely to take risks for large amounts of money.

Utility and Risk Aversion

Let's say an individual who has a net worth of $500,000 and isoelastic utility with ρ=0.5 (defined on his net worth) is given the opportunity to bet all $500,000 on the flip a fair coin, receiving a payoff of $1,000,000 if it comes up heads and being broke if it comes up tails. What is the value to him of taking the bet? The expected value in the amount of wealth he will have after taking the bet is clearly $500,000, but the expected utility of this random payoff is given by

Since the utility function is continuous and increasing, there is a unique dollar value, known as the certainty equivalent, that yields the same expected utility as any random payoff. It is the unique solution of the equation:

Here, the certainty equivalent is $250,000. So while a completely risk-neutral individual should be indifferent between betting his $500,000 net worth on this flip or not, the risk-averse individual with these particular preferences would rather have $250,000 for certain than bet his $500,000 on the flip. Since having $500,000 for certain is even better than having $250,000 for certain, the risk-averse individual of course passes on this opportunity. He would only be willing to spend his net worth to have a 50/50 chance at having either $1,000,000 and $0 if his net worth were less than $250,000.

To get a feel for the practical implications of each of these two basic forms of utility functions, we can look at the certainty equivalents for similar situations of betting one's net worth on a coin flip, for varying values of net worth. The 1st column is the payoff for winning the coin flip (twice the net worth), the 2nd column is the certainty-equivalent value for the individual with isoelastic utility (with parameter ρ=0.5), and the 3rd column is the certainty-equivalent value for the individual with exponential utility (with parameter c=1/150000):

So, for example, an individual with exponential utility (with parameter c=1/150000) would only be willing to spend $4,916.68 on a 50/50 chance of winning $10,000.

A few simple observations:

• For isoelastic utility, the certainty equivalent is always a fixed percentage of the expected value of the coinflip.  This is true regardless of what we set the parameter ρ equal to. So an individual with isoelastic utility is willing to bet his entire net worth on any weighted coinflip with fixed probability of winning (or on any 50/50 coinflip with a fixed percentage overlay, as in the example here), regardless of his wealth. This is unlikely to reflect any real person's preferences for such opportunities, but might be OK when we consider situations where the bet is for less than one's net worth.
• The certainty equivalents under exponential utility decrease significantly when more money is at risk.  While the certainty equivalents in the table above for the smaller flips seem to be roughly in line with what most well-bankrolled poker players (with CARA utility, one's net worth relative to the bet size does not matter) would be willing to pay for these coinflips, most would likely be willing to pay more for the $1M flip.  This disparity can't be rectified by playing with the parameter c; if we reduce c enough that the player would be willing to pay something somewhat closer to $500,000 for the $1M flip, then the certainty equivalents for the smaller flips become extremely close to the pure expected values.

So these two simple utility functions may each be imperfect for capturing real-life risk preferences, at least for individuals risking their entire net worth in the case of isoelastic utility.

These two utility functions are the most commonly-used in mathematical models due to their desirable analytical properties, but for the purposes of making practical poker decisions, where the discrete-time nature of poker opportunities makes it unlikely that the methods of calculus would lead to nice analytical solutions in many models anyway, we should be fine with choosing any admissible utility function that can be evaluated numerically. If we look at more practical situations where only a portion of one's net worth is at risk, we might be able to find a good fit with the isoelastic utility function, or we might be better-served by building some sort of ugly-but-practical "hybrid" utility function.

Eventually, we'll use the methods of utility functions to look at the following questions:

• When players have practical and tax-conscious utility preferences, how much effective rake are we really paying for our chance at the glory of the WSOP Main Event title?
• What sort of approximate hand-by-hand utility functions should the Loose Cannon on the PokerStars Big Game have?
• How can a backer and a player formulate a split of a payoff in a way which is optimal for each of their personal risk preferences?
• Full Tilt takes $1 out of the pot if you want to run it twice; under what conditions would we prefer to pay this fee to reduce volatility?
• Does whether or not we would take a certain risk ever depend on how many opportunities we will be given to play that game? In particular, is it a fallacy to manage the risk in a unique opportunity differently because we are unable to "reach the long run" with it?

But first, coming up next...

• Part 2: Finding or constructing a utility function that accurately represents practical risk preferences for poker players
• Part 3: Effects of taxation — and they're BIG ones

Introductions and Intentions

I'm a student of Mathematics, Statistics, and Economics and am currently working on my PhD in Applied Probability.  For the past six years of my life, I've played poker part-time alongside my full-time studies.  I became involved in poker because I found it to be an excellent and compelling competitive game with immense strategic depth.  I reject the notion that I am a "gambler" in any relevant sense.

As my actual PhD research has become more narrow, focused, and slow-moving, I've found my mind wandering more and more to possible applications of my knowledge and experience to problems that I find more interesting and fun.  Unsurprisingly, most of these ideas are related to poker.

Sweating the details of the failed recent and potentially forthcoming (at least temporary) annihilations of online poker in NJ or the US have also jarred me into no longer taking the great game of poker and my freedoms to compete in it online for granted.  Though online poker is unlikely to become completely dead anytime soon, if it ever did, I'd like to be able to look back on more than just a graph from years of mediocre grinding.  I've got some ideas floating around in my head, they might be good, and I want to get them out there.

So I want to start actively contributing to the poker community, and I've been looking for new outlets to keep my writing skills sharp.  While I've entertained the idea of someday writing a poker book, I don't think I could offer anything new strategically right now.  The ideas that I do feel like writing about are too varied and would lack mainstream appeal.  However, these ideas should be perfect for someday becoming a few dozen potential articles, so blogging seems like the way to go for now.  I'll be putting some rough sketches of my ideas out there, and we'll see where they develop.

I'm not totally sure precisely what the scope of this blog will be, just sure that I want to get started.  It won't all be quantitative, and it probably won't all be poker.

Topics I am planning to discuss on this blog:

• quantifying poker and macropoker situations that perhaps are not always quantified
• game theory and applications
• economic approaches to game structures and rules and their role in the poker ecosystem
• quantitative risk management models accounting for mean-variance tradeoffs in personal utility and tax considerations, beyond simple bankroll rules of thumb
• my take on legislative, legal, and political developments in poker
• scientific and logical approaches to the philosophical and legal questions of whether poker "is gambling" or "is mostly luck"

Topics I might discuss:

• psychology and behavioral economics as they relate to poker
• poker news (beyond legal developments)
• book reviews
• poker tax issues
• non-poker life situations where poker thinking or the lessons of poker are valuable

Topics I am not planning on discussing:

• personal details of my own poker career
• specific game strategy or hand histories
• minutia from my life

So if this sounds interesting, stay tuned.  If you like poker and aren't completely allergic to mathematical thinking, there should be plenty here for you to think about (or at least disagree with), and I'll try to keep it entertaining when I can.