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

NJ internet gambling bill passes Assembly, awaits governor's signature

I would have liked to have been able to dedicate the first few entries of this blog to some of the ideas I've been working on, but the timing didn't quite work out for that.  Major events moving towards the potential annihilation of the global online poker environment come first.


I will be the first to admit I don't have a great knowledge of politics or of how various legislative processes work.  So I settled in to live-blog the all-but-certain passage of the NJ internet gambling bill from its final committee, not sure whether or not the bill would be debated or just voted on.  Turns out, just voted on.  Also, as it turns out, not the most entertaining session to watch.

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January 10, 2011, NJ Assembly Session
2:55 — The session, scheduled to begin at 1:00, actually begins.
3:28 — Voting begins on "consent bills", those which have been agreed to by both majority and minority sides of the aisles.  There is no debate on consent bills." ... [then they vote on debate bills]
3:56 — Voting begins on nonconsent bills
4:13 — What was supposed to be a brief "time out" break is taken
6:00ish — brief "time out" ends
6:42 — A2570/S490, the NJ internet gambling bill, is brought up as a nonconsent bill, but voted on with no debate.  Passes 63-11-3.

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All that remains now is for Governor Christie to sign it, which could happen tomorrow or within up to 45 days.  The general vibe seems to be that he is very likely to sign it.


In the meantime, no newer or final version of the bill has been uploaded to the bill's page.  So either it hasn't been released yet, or this draft is the final version.  edit: The very knowledgeable PokerXanadu on 2+2 seems confident that this means that the above draft is indeed the final version.


In summary, the bill:

• authorizes casino games and poker to be offered to NJ residents only, and only by companies in Atlantic City.
• establishes that the wagers are deemed to take place in Atlantic City, regardless of where in NJ the player is located.
• sets the minimum age for having an account with one of these businesses to 21, despite the fact that the internet gambling sites will presumably not serve alcohol, and despite the fact that NJ's minimum gambling age is 18.
• makes it a crime for "any person [to offer] games into play or [display] such games through Internet wagering without approval of the commission to do".
• makes no distinctions between poker and casino gambling other than special provisions for dealer tips, which historically are not very common in online poker.

Potential Implications


The signing of this bill into law will be a historic moment in the history of the online gambling industry.  That's super for everyone who cares about online gambling.


For competitive poker players, however, there are a number of potential issues.  If NJ players have the option of playing on NJ-only poker sites with NJ-only player pools as well as existing trusted international poker sites against global player pools, then everything should be great.  If, however, PokerStars and Full Tilt decide that this law makes it illegal for them to serve NJ, then I have a big problem with this law.  


It's also unclear from the current language of the bill which New Jerseyans would be affected here.  If international poker sites are compelled to stop serving NJ residents, will they cease service to those whose permanent residence is in NJ?  What if they have a secondary address in another state?  What if they reside in NJ but log on and play from another state?  What about the converse?  The current language of the bill requires that a customer of an NJ-licensed internet gambling site must have a principal residence in NJ, but the criminal provision makes no mention of residency versus location, so it's not clear to me how international sites might react.


However, at least the current draft of the bill does not put any penalty on players for playing on international sites.  There should also be several potentially-valid legal arguments as to why the criminal provision in this law does not apply to the operations of PokerStars and Full Tilt Poker.  


What will PokerStars and Full Tilt Poker do?


While these international sites did pull out of WA after their state supreme court ruling, the nature of the NJ law is entirely different; WA criminalized ANY online poker play in their state and made it a felony for the player, so PokerStars and Full Tilt would have been facilitating crime by serving Washingtonians.  On the other hand, there is no such provision in the NJ law, and indeed, since the NJ law makes it legal to play online poker with the intended Atlantic City monopoly, I believe there are international trade arguments as to why it HAS to be legal to do the same with international operators.  And if the NJ bill defines that play on NJ sites takes place "in" Atlantic City, then wouldn't play with an international poker site take place "in" that country?


That all assumes that this law even applies to poker sites which do not offer casino games.  Saying nothing about the rest of its merits, the failed federal Reid bill last month at least made an attempt to treat poker properly, while this NJ bill literally treats it the same as casino gambling.  Regardless of whether or not a NJ-only site is able to draw in enough players and sustain enough games to be fun or profitable, the effective-prohibition against NJ players participating in international competition is a tremendous blow to poker as a strategy game.


I sincerely hope that this misguided law will not eliminate international poker sites from the NJ marketplace.  Whatever happens, other states will inevitably follow NJ's lead, and while state-only poker sites should eventually find federally-legal ways to pool their player pools, that could be several years away.  The very integrity of poker as a modern global competition could be at stake.


I can't help but wonder what this law would look like if anybody involved in it actually understood why poker should be treated differently than mindless, nonstrategic gambling "games" against a house.  In a world that routinely mistreats poker by ignorantly classifying it as "gambling", it is perhaps not too much of a twisted surprise that the historic first legal effort by a US state to license and expressly legalize internet gambling is one that could end up really harming the game of poker.


All of the details and implications should fall together over the coming weeks.  In the meantime, I have included a scientific diagram as to why it is bad if NJ residents cannot play with the rest of the world:





Note that the player in NJ is sad and lonely.  His faceless opponents in the rest of the world seem to express no sympathy or concern, but it might be only a matter of time before they are alone as well.

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.