Risk Preferences:
From EV to EU

From an early stage in their development, winning poker players learn to evaluate decisions based on expected value and not results. This leads to some players assessing their decisions based only on expected value. Although expected value is an important factor, making it the only consideration overstates its importance. For a rational poker player risk should also be considered.

There are three ways we can feel towards risk: risk seeking, risk neutral, and risk averse. Risk seekers are commonplace at the poker table; if given the choice between taking a gamble or taking the gamble’s expected value, they will always prefer the uncertainty of the gamble. If their love of gambling is strong enough, these individuals will be willing to take on bets with a negative expected value — just for the thrill of it. This is a fair description of most losing players. They are paying for the privilege of being at the poker table; some combination of the uncertainty and a chance of winning is enough for them to be there. From the professional’s perspective, such behavior is irrational, but fortunately, enough people engage in it to provide him with a living.

Risk neutrality is the next category. Given the choice between a gamble or its expected value, a risk neutral individual will be strictly indifferent between the two. The only item of interest to the risk neutral player is a gamble’s expected value, with more being preferred to less. When poker books or poker players talk about maximizing expected value, without mention of risk, they are effectively advocating risk neutral preferences.

The trouble with risk neutrality is that it leads to some anomalous behavior. This can be demonstrated with a thought experiment that dates back to the 18th century. The so-called St.Petersburg paradox owes its name to Daniel Bernoulli. The game is as follows:

A coin is tossed repeatedly until it first lands heads, when the game stops. Your payoff will be 2ⁿ, where n equals the total number of tosses. If the first toss is heads, you would get paid 2 ducats; with a sequence of tails and then heads, you would receive 4 ducats, for tails-tails-heads, you would receive 8 ducats, and so on. So how much would you pay to play this game?

The expected value is calculated as follows. With probability ½, the coin will land heads and you will receive 2 ducats. With probability ¼, the coin will land tails then heads and you will receive 4 ducats. With probability 1/8, you will receive 8 ducats. In theory an infinite sequence of flips is possible, and the expected value of the sequence is infinite.

Since a risk neutral individual ranks gambles based solely on expected value, he would be willing to pay any finite sum of money for the privilege of playing — he would instantly, and without any regret, invest his entire net worth in the game; Bill Gates would be willing to fork over a cool $53 billion. But this is incredibly bizarre behavior since paying even a two-digit sum for the gamble is likely to lose you money.

The trouble with risk neutrality is that it only looks at the expected value of a payoff. Therefore, the same weight is attached to winning 2 ducats with probability ½ as winning 1,048,576 ducats with probability 0.000000095367 even though the very high payoffs are unlikely to be realized in a lifetime of repeatedly playing this game, let alone in a one-shot deal. Figure I shows the payoff and probability for the first 10 flips:

Figure I: Truncated Payoff
Distribution for the St. Petersburg Gamble

St. Petersburg paradox payoffs

The key takeaway from the St. Petersburg paradox is that we cannot rely solely on expected value as a paradigm for making optimal decisions in risky situations. We must also account for risk, but showing a preference for risk, via risk seeking behavior, is not right either. This leads to the last category of risk preferences: risk aversion.

A risk averse individual has a strict preference for a gamble’s expected value over the risky gamble itself. As professional poker players, we would very much like to perfectly earn our expected value every session, but if the game was like that, the risk seekers would have no interest in playing.

The key to risk aversion is that expected value maximization is not the only criterion for making decisions, variability, commonly denoted by the variance, is also a factor. The sweet spot would be a gamble with high expected return and low variability; unfortunately, this is seldom the case, as high risk and high return are often inextricably linked.¹⁰⁰ In order to choose between a high return/high risk gamble or a low return/low risk gamble, we need a systematic method of making decisions, known as expected utility theory. ¹⁰¹

This theory is very elegant, and should come easily to people who are used to thinking about expected value. The act of taking the average over the set of possibilities, “taking the expectation,” is exactly the same. The key difference is that instead of averaging monetary payoffs, we replace these with another numerical quantity known as utils.

Utils are a quick way of summarizing all of the features of a certain situation that make it desirable, or not. For example, you might prefer chocolate flavored ice cream to vanilla, this would be represented by chocolate ice cream having a larger payoff denoted in utils than vanilla. If you were presented with a risky gamble involving the two, say, receiving chocolate with probability 0.5, or vanilla with probability 0.75, then you could represent the decision as an expected utility maximization problem. First, decide how many utils to ascribe to chocolate and vanilla; then, multiply by the probabilities to get the expected utilities — your optimal choice will have the highest numerical figure.

In theory, there are an infinite number of factors that could affect our utility. Personality factors are at work here, people high on agreeableness, the tendency to empathize and be considerate to others, are likely to derive utility from other people’s payoffs. (See “Personality” starting on page 315.) At a poker table, these people are likely to play for the social aspects of poker; they are not playing to win. Such a person’s expected utility maximization would predominately focus on these interpersonal concerns instead of issues of expected return and risk.

Returning to the notion of risk aversion, the key aspect of the relevant utility function will be that of diminishing marginal utility. More of something, be it ducats, dollars, or diamonds, is always better than less of it, but additional (marginal) units are worth less satisfaction (utility) than earlier units. For example, the second million dollars of wealth gives you a lot of satisfaction, but not as much as the first million — making an all-or-nothing coin flip for a risk averse millionaire would be a very bad bet.

We can represent risk aversion mathematically using what is known as a utility function. This is a simple equation of the form y = f(x), where y represents utility and x represents money. f(x) means we take some function of money, x, such as taking the square root, in order to convert to the corresponding utility level, y.

There are two conditions that a function must meet in order for it to equate to risk aversion:

Utility should always increase as money does — having more is always preferred to less.

As money increases, utility should increase at a decreasing rate — diminishing marginal utility.¹⁰²

Taken together these two conditions imply that our risk averse utility function should always be positively sloped, but that it should gradually flatten out, approaching the horizontal, as money on the x-axis increases. One such function is the natural logarithmic function y = ln(x):¹⁰³

Figure II: The Natural Logarithm

This is the function that Daniel Bernoulli used to resolve the St. Petersburg paradox in 1738. Converting money into utility and then averaging over all possibilities, the St. Petersburg gamble is only worth a finite amount of expected utility. The paradox is now resolved,¹⁰⁴ as a risk averse individual will only pay a relatively small finite sum to play the game.

The other classes of risk preferences can also be represented graphically using the same method. Risk neutrality can be expressed as any positively sloped straight line (constant marginal utility):¹⁰⁵

Figure III: A Risk Neutral Utility Function: y = x Utility

Risk seeking can be represented by a positively sloped function that increases at an increasing rate (increasing marginal utility):

Figure IV: A Risk Seeking Utility Function: y = x2

Returning to risk aversion, the utility function y = ln(x) has a very special link with the topic of the last chapter: the Kelly criterion. The Kelly-optimal bet size is equivalent to the bet that maximizes the expected utility of this function. In order to incorporate Kelly betting into an expected utility framework, all we need to do is substitute y = ln(x) into the utility function.

Figure II only shows a portion of the whole graph. In fact, wealth (x-axis) never reaches zero, as zero wealth would represent a utility level of minus infinity. This represents the property that Kelly betting will never lead to you going entirely broke, as the optimal Kelly bet size reduces in absolute value as your bankroll does.

As you can recall, the Kelly bet will maximize the growth rate of our bankroll, therefore, it represents the upper limit for rational risk taking. As professionals, we will likely want to take less risk (be more risk averse) than the Kelly criterion says, so our utility functions will show greater curvature (they will flatten out quicker) than the logarithmic function.

One important feature of risk aversion is that it can be welfare increasing to voluntarily enter contracts with a negative expectation. Insurance is the classic example. Insurance companies need to write contracts that can ensure them a profit, so that the insurance premiums paid by the customer must be greater than the expected payout from liabilities. For the individual, buying insurance will decrease his expected wealth; but the reduction in uncertainty over future states of the world can lead to increases in his expected utility.

The other side of risk aversion is that you should avoid some high risk positive expectation gambles. Multi-table tournaments are the best example of this in poker. These tournaments can have hundreds or even thousands of entrants, but the bottom 90 percent of finishers go home with nothing, and the majority of the prize-pool goes to a small number of people with the eventual winner pocketing up to 30 percent of the total.¹⁰⁶

This is not a good distribution for a downward sloping (risk averse) utility function. The majority of the time you’ll enter the tournament and no matter how big your edge could realistically be, you’ll go home with nothing, underperforming your expectation. Furthermore, the utility of the lost entry fee is relatively high since it will occur in a state where your total wealth is relatively low.

Occasionally, you’ll make it to the final table, or even win the whole tournament, pocketing a large payday, sometimes a life changing sum, and now you’ll have lots of money. Say you won 200 times your initial buyin. However, because you now have such a surplus of cash, there will be relatively little utility difference between winning 200 or 199 times your buyin. That last unit of wealth, “marginal unit,” would have provided you with much more utility in the state of the world where you busted out of the tournament with nothing.

Working out the sums to produce an expected utility calculation, it could easily be that many of these high buyin, large field multi-table tournaments could result in a negative expected utility for risk averse individuals with a positive expectation. As a result, these individuals should prefer not to enter the tournament despite having an edge in it.

There are a couple of explanations for the enduring popularity of this form of poker. It could well be that the ancillary benefits of winning a tournament, such as fame, prestige, or sponsorship could be sufficiently large to tip a high risk tournament towards having positive expected utility.

Or, it could be that as imperfectly rational humans, we tend to overestimate the probability, such as winning a large tournament, of rare events occurring. (This is part of the reason why lotteries are so popular.)¹⁰⁷ I know of this fact myself; when running well on the first day of a large field tournament, I’ll start daydreaming about getting to the later stages despite there being a lot of high variance poker still to be played.

Figure V: All Three Risk Attitudes

For the risk averse function, y = 1n(x), utility grows at a slower rate than wealth, indicating that a risk averse individual requires bets to be sufficiently +EV before they will acept them.

The risk seeking function, y = x2, gets more utility from each extra unit of wealth than the last unit, indicating that a risk seeking individual can accept -EV bets in an attempt to seek out higher utillity levels.

The risk neutral function, y = x, treats each unit of wealth in exactly the same way, leading to the St. Petersburg paradox, where an infinitely small chance of winning an infinitely large amounf of money is treated the same as winning 2 units of wealth with probability 0.5.

Risk Preferences in The Brain

Levels of risk taking vary enormously over the population. Some people will go out of their way to seek risks, either through extreme sports, being an entrepreneur, or gambling, while others will do whatever they can to avoid unnecessary risk. These are two extremes, and as poker players we must straddle between them: avoiding unnecessary risks in our chosen career, but finding some way to deal with the daily swings.

A clever experimenter, advertising executive, or financial advisor can often alter our risk taking behavior through the subtle arrangement of how a decision is presented.¹⁰⁸ However, there are certain neurological factors which effect the cross section of risk taking over the population. Specifically, attitudes towards risk seem to be controlled by a specific part of the brain. When this part becomes damaged, patients will take significantly more risk in either plus- or minus-EV gambling experiments. When the gamble is minus-EV, the patients significantly underperforms normal participants.¹⁰⁹

Extreme pathological gambling can be seen as an addiction, similar to drug or alcohol abuse, caused by a dysfunction of the brain’s reward mechanism.¹¹⁰ Pathological gambling is just one extreme of a whole spectrum of behavior, with problem gambling being a less severe but still debilitating version. It has been shown that both problem and pathological gambling can have their root causes in the brain, and are caused by the lack of certain chemicals known as neurotransmitters. The problem gambler attempts to increase the level of these chemicals in his brain by engaging in risk taking.

These findings are extremely interesting, and they corroborate with my experience as a member of the professional poker community. I know many skilled players who have repeatedly gone through a cycle of diligently building a bankroll at the lower limits, only to lose it all taking stupid risks in games they shouldn’t be playing. These are otherwise intelligent individuals who really should know better — without the self-destructive cycle they could be huge lifetime winners. So it would be beneficial for any aspiring pros to truthfully self-examine their behavior for any clues of this pattern; doing so could save high risk individuals from a very unproductive career choice.

Loss aversion is at the other extreme to pathological gambling. The term loss aversion refers to the situation where a loss of a particular size is felt far stronger than an equivalent gain. Losses looming larger than equivalent gains is a fundamental part of rational risk aversion; however, some people display an avoidance of losses that is impossible to reconcile with a realistic utility function.

The key empirical finding that led to the creation of loss aversion is that many people will turn down extremely plus-EV low stakes bets, such as turning down an even chance of winning $200 or losing $100.¹¹¹ Although this bet is worth $50 in expectatiuon, the pain of losing $100 is too great for many of us to bear.

The rejection of such bets is not consistent with a plausible utility function. Since a risk averse utility function must always be downwards sloping, if we reject this bet at such a low wealth level then we must be incredibly risk averse when it comes to large bets which leads to the rejection of unbelievably plus-EV high stakes gambles. One example is that a person with total wealth of $340,000 who rejects an even chance gamble between winning $105 or losing $100, will also reject an even chance gamble between winning $635,670 or losing $4,000.¹¹² This is not plausible behavior.

If a bet is small enough relative to your bankroll, then you should effectively be behaving in a risk neutral manner, willing to take zero-EV plays, since the risk of ruin is (due to the law of large numbers) implausibly small. You would be able to see this by zooming in on a section of the graph for a risk-adverse utility function: take two points close to each other on the x-axis, corresponding to your wealth after winning or losing a zero-EV play. Since these points are so close, the part of the graph that connects them will show essentially no curvature, and as we saw, a straight line corresponds to risk neutrality. And when playing poker, if you systematically avoid zero-EV plays, this will have negative knock-on effects for the strength of your overall strategy, as it would cause too much timidity on your part and allow aggressive opponents to run over you.

Prospect Theory

The problem is that people do not commonly act in accordance with expected utility theory. This theory tells us that wealth should be the key driver of our utility, and with a small stakes gamble the difference in wealth level between winning and losing is negligible. Therefore, we may as well take the small plus-EV gamble, or accept the even smaller zero-EV gamble. In reality, people are instead driven by changes in wealth: losing hurts, even if it’s just a trivial sum since we feel that taking on the gamble has been a “mistake.”

We can explain this aversion to small scale gambles with an alternative to the expected utility paradigm known as prospect theory. The key difference in this theory is that changes in wealth are the drivers of our utility, and losses feel worse than gains. See Figure VI for a graphical illustration of a prospect theory utility function.

Figure VI: A Prospect Theory Utility Function Utility

Decisions (or “prospects”) are evaluated around a reference point, our current wealth level, which is reflected by the zero point on the x-axis (wealth). When it comes to evaluating gains (the positive part of the x-axis) the utility function is downward sloping which indicates risk aversion.

The line changes slope in the south-west quadrant of the graph: instead of sloping downward, the graph is now upwards sloping and much steeper. Because of this curvature, even a small monetary loss will result in a large negative utility shock. This reaction to losses is the key difference between expected utility and prospect theory where losses lead to a much larger decrease in utility than an equivalent gain (loss aversion).

This varying reaction to losses and gains allows prospect theory to explain a greater variation of plausible human behavior. Due to the asymmetric treatment of gains and losses, the rejection of small scale plus-EV bets does not lead to ridiculous restrictions on large scale betting.

In terms of my own play, I know that prospect theory is much better at explaining my involuntary emotional reaction when it comes to losses. If I’m having a small losing day, then winning enough money to get me back to breakeven will make me much happier than if I won an equivalent amount after I was already up a significant amount that day. Since my reference point is always the amount of money I had at the start of the day, winning a small amount will make me feel much better when it gets me out of the domain of losses.

There is evidence that loss aversion is a biologically hardwired trait in all of us. It has been shown that capuchin monkeys display the same behavior of disliking losses in risky situations.¹¹³ Clearly, loss aversion has been with us for a very long time.¹¹⁴ Compare the curvature of the risk seeking utility function in Figure IV with the domain of losses in Figure VI. You’ll notice that the two graphs are identically sloped — they both shoot upwards at an increasing rate as you move to the right. This means that loss aversion causes us to be risk seeking in the domain of losses. How many times have you seen a player rapidly move up stakes in an attempt to win back a loss? And in doing so, increase the size of their loss by many times? This is classic risk seeking behavior — induced by their aversion to losses.

Being in the domain of losses is so painful to many people that they are willing to increase their risk exposure just to get out of this — they would actually prefer a risky bet to its expected value, just to give themselves a better chance of digging themselves out of a hole. This is a strong psychological response that nature has endowed us with; it’s also bad news, as the best way to grow a bankroll over time is to always be risk averse.

Comparing Expected
Utility and Prospect Theory

The two theories are very different in the way they seek to explain the world around us. Prospect theory is what is known as a “positive” theory, it attempts to explain the way people actually make decisions. Expected utility theory is a “normative” theory, it attempts to explain how people should rationally make decisions.

When applying this knowledge to poker, we should attempt to act in accordance with expected utility theory since this is the theory that can best improve our results — it shows us the “rational” way to behave. We should still be aware of prospect theory since it provides an excellent description of the most common traps and pitfalls that real people fall into when dealing with risky situations, especially when dealing with losses.¹¹⁵

Returning to poker, the key takeaway from expected utility theory is that we should act with a cool indifference to small scale plus-EV bets. We should gladly take them on, but we should be unmoved if they result in losses since that is an unavoidable facet of gambling for a living. In these situations, we need to fight the instinctive evolutionary response that we have been programmed with. For bets that are sufficiently small, we should be willing to take zero-EV plays as this will improve out metagame without increasing our risk of ruin. But when it comes to larger bets, such as in game selection, shot selection, or high-buyin tournaments, we should require a bet to be sufficiently plus-EV to take it on.

Prospect theory shows us the danger of overly focusing on our current results: “Am I winning today?” is a question that many of us fret over, when in actual fact, if our bankroll is large in relation to the stakes we are playing, we should care little for the answer to it.

Aversion to loss prevents a lot of people from realizing their full potential in poker. As we saw in the last chapter, when a favorable game is available, it can be rational to take shots at higher games with a much shorter bankroll than conventional wisdom dictates. However, a lot of people turn down this sort of opportunity as the pain of losing would be too great to bear. From a personal perspective, I turned down many potential opportunities to move up in stakes during my early career because of this even though aggressive shot taking at juicy higher stakes games is perfectly consistent with rational behavior.

Loss aversion has another pernicious effect on poker players: it induces risk seeking behavior while in the domain of losses. As we saw earlier, risk seeking is not consistent with profit maximizing poker — it causes people to take unwarranted risks and can lead to the acceptance of minus-EV bets. Be very cautious with your play on big losing days since loss aversion can cause you to take on a string of subsequent bad bets. If this behavior in any way describes you, consider setting up a specific daily stop-loss; limiting the amount of money in your poker account; or find a friend whose judgement you can trust who will tell you when to stop gambling for the day. However, be careful here. As you approach the stop-loss, a loss limit can cause you to change strategy and put you in too much of a survival mode which can reduce, or even make negative, your overall expectation.

When it comes to the field of finance, prospect theory fits the empirical data much better than expected utility theory. According to expected utility, the average return from stocks has been implausibly high — their combination of expected return and standard deviation imputes an astounding level of risk aversion for the average investor.¹¹⁶ For more plausible levels of risk aversion, investors should overwhelmingly prefer stocks to any other asset based on historical returns. If you combine the assumptions that investors are loss averse, and that they evaluate the gains and losses in their portfolio over a set evaluation period, then the average return of stocks over safe assets becomes realistic.¹¹⁷

Conclusion

When poker books talk about maximizing the EV of your decisions, they are implicitly assuming that you are entirely neutral in your preference towards risk. As the St. Petersburg paradox shows, the correct way to gamble is to be risk averse. This links with the last chapter on the Kelly criterion; together, they show the minimum amount of risk aversion we should employ when attempting to grow a bankroll over time. Expected value can only be a sufficient statistic to guide your decisions if the gamble on offer is sufficiently small relative to your bankroll — this should be the case in the limits you most typically play. Larger gambles always require a trade-off between expectation and risk.

People can have vastly different perceptions and attitudes towards risk. Excessive gambling is as addictive as drugs or alcohol; people with a predisposition towards this sort of thrill seeking will be fighting against strong headwinds if they choose professional poker as a career — how many alcoholics make good bar managers? This love of gambling means these people are also the most likely to attempt poker as a vocation. Be wary if this in any way describes you — just under 1 percent of the population is thought to suffer from gambling issues. If you are reading this book, the chances of you being a problem gambler are no doubt higher.

A version to risk is the key to long term survival in the competitive poker economy. However, an irrational fear of incurring small losses can be just as debilitating for an aspiring pro as the other extreme form of behavior.

These two restrictions create a narrow band of risk attitudes that I believe is absolutely necessary to have in order to succeed at poker. We must be risk averse, otherwise we will overbet our bankroll or play in minus-EV games. But we must also find a way of coping with the constant volatility while reducing opportunity costs by moving up in stakes when our bankroll and skill level are sufficient to challenge at a higher level.

Psychological Biases

“When the facts change, I change my mind. What do you do, sir?”
— John Maynard Keynes

When playing poker, our goal is to make good decisions. The first part of this is playing well, and I believe the simplest framework for this is the game theory paradigm — using a combination of exploitive and unexploitable play. The second part involves making good risk/reward tradeoffs. This has been covered in the previous two chapters.

Even if you know how to make good decisions, the implementation of this knowledge can be sidetracked by a number of psychological biases. These biases can subtly affect our judgement, undermining our ability to make the best decisions.

Forewarned is forearmed.

Overconfidence

This bias is among the most common, and poker players are among the worst offenders. When asked to rate their abilities relative to the average population at some task, the vast majority of people believe themselves to be above average. For example, 90 percent of drivers think that they are above average, implying at least 40 percent of the population must be wrong!¹¹⁸People are also overconfident with respect to their level of knowledge;

answers that a respondent is 100 percent certain of are only correct 80 percent of the time.¹¹⁹

I know of no formal survey of poker players’ confidence in their abilities, but if one was done, I’m sure at least 99 percent of those asked would put themselves in the upper half. We’ve all seen rank amateurs, who barely know the rules of the game, win a couple of pots and start acting as if they’re thinking, “Hey, this game isn’t so difficult!” At least on the surface, very few losing players appear to be aware that they are among the worst players at the table. Since everyone has their winning days, these players can always point to sessions where they have won. Their frequent losing days can be blamed on external factors such as forgetting to wear their lucky socks or the fact that, “that guy always get there on the river.”

One explanation for the prevalence of overconfidence is rooted in terms of evolutionary advantage. When scarce resources implied a fight for survival, having an enduring confidence in one’s own abilities allowed our ancestors to take the necessary risks to survive. This trait might have helped our ancestors hundreds of thousands of years ago, but when it comes to playing poker today, overestimating our ability with respect to our opponents can be a big error.

Overconfidence is what makes the poker world go around. If the worst player at a table could always rationally be aware of this fact and not be playing for fun, it’s likely that he would choose to exit the game. Then the second worst player would become the worst, and so he would also refuse to play. This would continue until the best player at the table is left sitting all on his own, with nobody left to play.

This bias is at its most insidious after a sustained period of winning. Nothing else validates our egos quite like it. And it’s natural to feel confident after winning a lot, but it’s all too easy to stray over the border into overconfidence, where our beliefs exceed reality.

If you have recently been winning more than ever before, be on the look out for overconfidence creeping into your self-assessment. Have you made any recent changes to your style that you could attribute to an increased win rate? How large a factor has running above EV been in your results? Have you considered broadening your sample by considering how players similar to you are faring in the game?

Poker-based overconfidence can spill into other areas of life.Success often leads some players into thinking they can beat another zero-sum game: actively managed investments.¹²⁰ There are a number of parallels between poker and investing, but there are also many differences.

For one thing, investing is a game with poorly defined rules — some hereto unforeseen event can always occur, drastically altering the investment landscape. That is, risk can appear from nowhere. Notice that this is different from poker where a flush will always beat a straight.

The truth is that many successful poker players probably are capable of also earning superior risk-adjusted returns in an actively managed investment portfolio. However, doing so would still require an enormous amount of dedication and vigilance. The trouble with overconfidence is it makes people believe in the mirage that such success is easy. See “Investing” starting on page 335 for an introduction to investing for poker players.

Overconfidence is such a massive factor in our personalities it’s difficult to counter. Studies show that the only people able to rationally assess their abilities with respect to others are the clinically depressed.¹²¹ For the rest us, the bias is so deeply ingrained that there is little we can do. My best advice for playing poker would be to refuse to play in a game without a clear edge —if your advantage appears to be slight at best, your natural overestimation of your own abilities might mean that you are in fact the sucker.

This is similar to the margin of safety concept among value investors.¹²² These investors typically own a relatively concentrated portfolio of a few investments. In order to protect themselves against downside risk, they require a “margin of safety” — their estimate of a security’s intrinsic value must be sufficiently above the market price. This procedure protects them against any biases or errors in their valuation method.

Framing

This bias reflects the fact that our choices are often determined by the style of the presentation of decisions over the substance. By “framing” a decision in terms of either losses or gains, an experimenter can manipulate our choices among identical prospects.

Here is a classic example, courtesy of Daniel Kahneman and Amos Tversky.¹²³ An infectious disease is spreading, 600 people have been affected, you must choose a course of treatment:

Options A and B are of course identical in either frame. (In Frame 1, the emphasis is on lives saved, in Frame 2, the choice is presented in terms of the number of deaths). In the first frame, 72 percent choose option A, while in the second frame, 78 percent choose option B. Just by altering the emphasis from lives saved to lives lost, there is a full 50 percent swing of respondents choosing A despite no actual difference in the consequences (from 72 percent to 22 percent).

Kahneman and Tversky explain this intriguing phenomenon with their prospect theory of risky choice. (See “Risk Preferences: From EV to EU” starting on page 278.) The first frame emphasizes gains, therefore, individuals perceive themselves to be in the domain of gains, and thus are risk averse. The second choice is framed in terms of losses, and so individuals are in the risk seeking domain.

In the first frame, choosing B feels risky since you will probably kill the 200 people you could have saved for definite. In the second frame, choosing A will consign 400 people to death, so you may as well gamble on the chance of saving everyone.

This is all highly irrational. In theory, we should be able to pick whichever choice out of A and B that we prefer and stick to it in whichever frame we are presented with. The result with practical value from all of this is that by altering our frame of reference from losses to gains, we can make the variance of poker a little bit more bearable.

As shown in the last chapter, the psychic pain of losing money can be enough to make people seek out high variance gambles as a way of getting even. In the long run, this mode of gambling will at best be unproductive, and at worst it can cost you your entire bankroll.

The key to reducing this disutility is to get yourself back into the frame of gains. Sure, you might be down this session, but if you broaden your frame of reference you will be up over some other time period: be it the last two days, the last week, month, or even year.

Breaking even for months on end is no fun, but the truth is you are making a good living at a game of cards. No doubt you have many friends envious of your take-home pay last year.

There is less mental reframing required on winning days. In the domain of gains, individuals will be risk averse: This is the appropriate attitude to take. However, some people take this too far: They will quit for the day whenever they have won a certain amount just so they can “book” a winning day.

For these people, being in the domain of gains makes them inappropriately risk averse. I find this behavior is relatively less common, although it’s usually found in people who also suffer from acute loss aversion. Most high stakes pros that I know are capable of “going in for the kill” on a big winning day. This has much to do with the scarcity of prey at those limits and the probable fact that when winning, some poor players will begin to fear you and thus begin to play in a more predictable manner —they may quit bluffing if you’re in the pot.

Note that the people who start upping the stakes when losing and quit the game when winning are getting things completely backward. If you are winning, it’s more likely than average that you are in a high-edge game, and vice versa. Additionally, the Kelly criterion tells us to scale our bets down after a period of losses, scaling them up after a winning period — the exact opposite of this behavior. In addition, this tendency to chase losses and cut gains short is also prevalent in the stock market where it’s a well known empirical fact known as the disposition effect. ¹²⁴

Poker sites can use framing to their advantage in the way they describe their frequent player incentives. Extra rewards for high volume players can be presented in two ways: as additional rakeback deposited straight into players’ accounts, or as free consumer items and gift cards. From a rational perspective, poker players should prefer rakeback, as this money can then be used for whatever purpose the player wants. However, this doesn’t account for the poor framing that rakeback has in players’ heads.

That is, rakeback is perceived as just a rebate on a loss, and hence it doesn ’t provide much utility. Why should I be happy about getting $1,000 in rakeback deposited into my account if my 20 percent rakeback deal implies that I’ve lost the much larger sum of $4,000 in rake? Furthermore, money deposited into a poker account, even if it’s a large amount, can often be seen as no more than a rounding error of the account balance: just another day’s swing and nothing to get excited about.

Casinos are well aware of the power of framing. High volume players are often given “free comps:” free hotel rooms, free food, and free show tickets. These comps are framed as gains, and gamblers act incredibly favorably to them, often responding by showing a lot of brand loyalty to their casino.

However, it’s the way these items are framed that are the key to their popularity. If instead they were framed as cash rebates on their expected losses, then they would probably be less popular. This is because they would be the domain of losses, reminding the casino-goers that they are effectively paying the casino for a service. This is, of course, highly irrational. If you were presented with either some free comps or an equivalent cash value, then you should always go for the cash because you can spend the cash on the comps if that’s what you want most of all (and if not, you can buy something that you like even more).¹²⁵

The Importance of Emotion

As studies of individuals with a certain type of brain damage show, emotions are a large part of the decision making process.¹²⁶ Even if we are consciously attempting to make rational decisions, our innate emotional responses to stimuli are likely more of a factor than we can imagine.

Our emotions can be harnessed by individuals with unscrupulous aims. Advertisers will influence us into buying their products by creating positive emotional associations with them.

When did you last see a model in a catalogue frowning?
Emotions can bias our decisions at the table because our emotional response to luck will vary enormously, depending on whether we are the beneficiary or the benefactor of fortune.Losing a big pot that we were ahead in creates a surge of intense emotion, we might express thoughts such as, “how could this happen... to me... again?” This heightened emotional respons imprints the hand vividly in our memory; we will remember the hand for at least the rest of the session, and possibly for some time after.¹²⁷

The feeling of fluking a win in a big pot is incredibly different. It’s hard to describe, mainly because there is so little of an emotional reaction. It can be hard to feel overjoyed, mostly because we feel an acute sense of embarrassment at winning a pot we didn’t really deserve. We might think, “Oh well, at least I sometimes get them my way, I was beginning to think I never would.” This low emotional reaction makes this aspect of luck intensely forgettable.

These varying emotional responses mean that we will systematically underestimate the importance of luck when winning; while overestimating the impact of luck while losing. I believe that this bias could be responsible for part of the process whereby losing players delude themselves into thinking they actually are winners.

Dealing with the bias is again difficult. First, we should temper our enjoyment on winning days. No doubt we hit a lot of lucky draws, we just can’t recall them since they didn’t leave a distinct emotional impression on us. Second, we should feel less down about losing: taking bad beats happens to everyone.

This bias can also lead to an overestimation of our abilities. Suppose we win a lot of money in a heads-up session. In order for this to occur, you will certainly need to win more than your fair share of lucky hands. But this isn’t how you’ll remember it: your main memory of the session will be how well you played, not how many cards you hit.

Suppose instead that you lose a lot of money against a similar opponent. Your chief memory of the session will be just how different things could have been, “If only he hadn’t hit every single flop for hours on end.” In summary, when we win, it’s because we played great, when we lose, it’s because we got unlucky. This is otherwise known as attribution bias: crediting wins to our own abilities, while losses are due to some external factor. I think that this is largely caused by our varying emotional reaction to wins and losses causing us to systematically misstate the importance of luck.

The practical response to these factors is the same as for overconfidence: You should be less willing to play in games without a clear edge.

Cognitive Dissonance

Cognitive dissonance refers to the internal struggle we feel between two clashing sets of ideas.¹²⁸The feeling leads to discomfort, and so people usually resolve it by rejecting the information contrary to their actions or prior beliefs. An example would be making a large and irreversible purchase. In order to reduce feeling of regret, people will ignore the negative aspects of their choice; focusing only on the positives. When the dissonance is finally resolved, they will have convinced themselves that the action they took was the correct one.

This is closely related to another common error known as confirmation bias — the act of seeking out information in accordance with our prior beliefs. As Nassim Nicholas Taleb very eloquently points out in The Black Swan, we should instead test our hypotheses by attempting to reject them — seeking out disconfirmatory evidence, if any. For example, while being a strong advocate of raising-or-folding when entering a pot first-in, I should still strongly evaluate the evidence for calling. (See “Appendix to Rules for Preflop Limit Hold ’em: Polaris's Non-Standard Preflop Plays” starting on page 60 in “Part One: Strategy.”) That’s because if only the evidence against calling is evaluated, then I can fall prey to the confirmation bias. (Note that the hypothesis being tested here is raise-or-fold, and evidence against calling confirms my prior hypothesis.)

This bias can hamper our progress as poker players. Winning money at this game tends to provide us with validation for our style of play and approach to the game. However, there are often different ways to play a hand, or different styles of playing poker. If you can keep an open mind about different approaches to the game, it can ensure that you continue to grow and learn as a player.

I fear some of the old school players are victims of cognitive dissonance. The games were much easier some years ago, and so it was possible to win a lot of money just as long as you made smaller mistakes than everyone else. But the game has moved on, and now some of these players are doing much worse than before; they are unable to come to terms with the fact that “their” way of playing might not be the “best” way.

Cognitive dissonance can be a powerful force in poker as it can be difficult to acknowledge that our poor decision making is to blame for losses that could run into the hundreds of thousands or millions of dollars. I’ve heard that some high stakes players claim their losses at the highest level are due to being cheated —of course it has nothing to do with the fact that someone else might be better than them or that the variance, relative to the win rate, is quite high at these stakes.

The best way to deal with this bias is to discuss poker with a number of different players, ideally people with differing approaches to the game. When you disagree on something, focus on reasons why they might be right, rather than why they might be wrong. Additionally, when you see a player you respect play a hand in an unexpected way, focus on the benefits and not the disadvantages of their decisions.

Projection Bias

This bias doesn’t lead us into making bad decisions at the table, but it does lead to us making bad predictions about the future. Specifically, projection bias¹²⁹occurs when we make faulty predictions about our future tastes, failing to anticipate the ways in which they will differ from the present. This bias often appears in the medical area. People assume that major injuries such as blindness or losing the ability to walk will have a major impact on their quality of life. In actual fact, people become acclimatized to their new situation extremely quickly, reporting satisfaction levels close to what they were before their injury.

More generally, the bias manifests as such: people assume that some major event, such as being diagnosed with illness, getting a promotion at work, or winning lots of money at poker, will significantly alter their level of life satisfaction. In reality, their level of satisfaction does alter briefly after the event, but in a short space of time, they become accustomed to the new situation and their happiness levels revert close to their long term average. (This is known as hedonic adaptation.¹³⁰) If happiness is related to wealth, then satisfaction levels would have increased markedly over the twentieth century; this has not been the case.

Take the example of a promotion at work. People imagine that the extra prestige and financial rewards of being one rung higher up the corporate ladder will make them happier. The truth is that most often it won’t. The extra cash will make them happier for a while, but soon it will become “normal,” losing its significance. In addition, the new job will bring longer hours and greater stress — two factors that can notably lower happiness levels.

In poker, the only yardstick of success is winning money, and this is highly susceptible to projection bias. We spend hours dreaming about how winning x amount of money will really increase our utility. The trouble is, after winning that amount, our frame of reference will shift, and suddenly it won’t seem like so much. Now you’ll get a new, higher, figure in your head and the cycle will repeat.

I know of this from personal experience. When I was a $3/6 pro, I imagined that making it to the middle limits, say $30-$60, would make me really happy about my poker career. However, my frame of reference shifted, now $30-$60 is the new normal, and playing that limit feels less good than I had projected.

Poker is a multi-layered society, and you’ll tend to compare yourself with the others on your stratum. When you make a move up, you will place yourself amongst a new group of peers. No matter how well you’ll do, there’ll always be someone better off to compare yourself with, and as you move up, your frame of comparison will shift to a better group of players. According to surveys, many people would be willing to take a large cut in absolute income, if by doing so, they placed higher in the relative income scale. Most people would prefer to earn $50,000 when the average is $25,000; rather than earning $100,000 in a group where the average is $200,000.¹³¹

This preference for high relative compensation instead of high absolute compensation helps to explain how pay levels can steadily spiral upwards in certain industries such as banking and professional sports. Making a million dollars a year is a great haul in the ordinary world, but it can feel positively unfair when a colleague has been paid more for similar work. When it comes to team work, people also systematically overestimate their contribution to the final product which means that even an equal pay packet for equal partners can leave some people upset at their level of compensation.

Another issue related to projection bias is that we fail to anticipate how we will come to feel about a sequence of outcomes. People have an incredibly strong preference for an income profile that increases over time.¹³² This is the reason why workers tend to receive year on year pay rises, even if their productivity doesn’t increase. Making $100,000 in a year of Internet gambling might sound great to you at the outset of your career, but if you made $200,000 last year, then that $100,000 will probably feel like a failure.¹³³

As the poker economy becomes more competitive, this is something many pros will have to deal with. The issue should be largely an academic one as win rates in poker are likely to remain above many other jobs, but there’s nothing quite as deflating for your ego to work harder than ever before on your game, only to do less well than the year before. This is something I have personally struggled with.

No doubt projection bias serves another useful evolutionary purpose. If we could fully appreciate how little working hard and moving up in the world would truly change our level of satisfaction, we might not go to all the bother. It’s also my opinion that this bias is particularly prevalent in poker: the hierarchy is well defined, meaning there is always a higher level we can aspire to reach, and success can be incredibly transitory, getting to the top is no good if you can’t stay there.

Conclusion

Nature has endowed us with a great number of psychological biases; in this chapter, I have sampled just a selection of these errors, focusing on the ones most important to poker.

A number of these biases seem like understandable quirks of the evolutionary process: daily life would be much harder if we knew how average we are in many respects; how often we choose based on the presentation of a decision; how influenced by emotion our recollections and choices are; how often we are in error; and how similar the future is likely to feel relative to the present.

The process of attempting to reduce these errors is known as debiasing. Becoming more cognisant of these biases can only improve your long-term results, and I think the best place to start is to try noticing biases in others around you. My favorite examples of repeatedly biased decision making is in the TV program Property Ladder which has been shown on both sides of the Atlantic. The premise for this show is for people to buy, renovate, and sell on property to make a profit.

The people on this show are incredibly overconfident, even, or perhaps especially, if they’ve never bought or sold a house before. This confidence is definitely irrational, and it grows with each profitable transaction they go through, even if their profit on the deal was caused by some external factor (the British housing market was steadily rising during much of this show’s life).

Second, if the renovation isn’t a success, people will never sell the house at a loss. This is the disposition effect at work (caused by loss aversion). Again, this is irrational. People are aware of the out of pocket cost from selling at a loss, but they fail to appreciate the opportunity cost of holding onto a losing property — the equity in the house, if sold, could be invested elsewhere. Usually, these deals are done with a high level of leverage, and to prevent them from losing even more money by not selling, the property needs to appreciate at least as much as the interest rate on their mortgage.

Third, cognitive dissonance reigns supreme. The presenter on the show is there in the guise of providing help and advice to the developers. The developers should find this help useful as the presenter is an experienced property developer. But the vast majority of the time the presenter’s advice is ignored as people cling irrationally to their prior beliefs. If they had less of a psychic investment in these beliefs, they would make much better use of this additional information, and, ultimately, would make more profitable deals.

The trouble with having a biased perception of the world is that even though these traits might be readily apparent in others, they can remain incredibly hard to detect in your own personal behavior — “I’m not overconfident, I really am that good. ” So I think the key here is to develop a greater sense of your own fallibility, even if that goes against your nature.

Personality

When I tell people about my line of work, the second most asked question, behind the classic, “But how much did you lose?” is, “Can you teach me how to play?” I used to answer this latter question with an automatic yes. And by playing on sites with attractive bonus schemes, I thought pretty much anyone could grind out a minimum wage style existence at poker.

However, over time my opinion on this has reversed. Not only has the standard of play increased at all levels of the game, but also, I’ve seen a number of intelligent people who, perplexingly, just couldn’t hack it.

Part of this is the issue of risk preferences (covered in “Risk Preferences: From EV to EU” starting on page 278): Some people either have an overbearing love of gambling, or hate losing so much that professional poker is simply an unwise career choice. Psychological biases are another factor. (See “Psychological Biases” starting on page 299.) People behave in predictable ways, making systematic deviations from theoretically correct unbiased decision making.

I also believe that personality traits are a significant predictor of success: The right mix of factors can really improve your chances of making it. Furthermore, this also explains why people get drawn to different specialities in poker; it’s rare to see players capable of playing more than one game at the very highest level — barring the odd notable exception, of course. In addition, I think certain traits are of varying importance depending upon game type; what makes a high stakes no-limit player great may not help that individual become a strong limit player.

This chapter’s discussion will be based on what is known as the five-factor model which seeks to break personality down into just a few independent factors. Each of these can be broken down further, with some researches using up to 18 factors. The five factor model has the virtue of simplicity and has been the most empirically tested.¹³⁴ These factors vary through the population due to some combination of nurture and genetics, perhaps indicating that there is an extra dimension across which some people are better suited to professional poker than others.

Conscientiousness

This trait measures our ability to carefully pursue our goals, acting in accordance with personal plans over spontaneous whims. Highly conscientious people often achieve well at school or other areas that require self-organized behavior. People low on conscientiousness are often unable to control impulses, and can be susceptible to attention-deficit hyperactivity disorder (ADHD).

I think a very low score on conscientiousness would make poker a much more difficult profession. These people are often unable to motivate themselves to perform tasks they would like to accomplish. In poker, it’s important to motivate yourself into putting in the necessary hours of play and study, and there is nobody outside of yourself pushing you to accomplish this.

Low conscientiousness is also a predictor of drug, alcohol, or gambling addictions, as these people are often unable to resist short-term gratification. In “Risk Preferences,” I pointed out that gambling addicts can never really expect to succeed at this career. A predilection towards problem gambling is in part due to this unchangeable personality factor of low conscientiousness. People high on conscientiousness often still enjoy the stimuli involved in causing addiction, it’s just that they have the internal fortitude to resist over-indulgence.

High levels of conscientiousness are not unequivocally good, as this sort of behavior taken to an extreme is also debilitating. Highly conscientiousness people are often unable to work outside of a very rigid routine, and they are incapable of dealing with new or novel situations.

I also believe that limit players should have higher levels of conscientiousness than no-limit players. This is because certain plays are bad no matter the circumstances in limit: cold-calling a suited connector in an aggressive high-stakes short-handed game is a mistake for anybody. The high conscientiousness individual has the self-restraint to avoid this sort of play. In no-limit, there’s a wider degree of possible situations, and having a slightly lower level of conscientiousness will enable you to appreciate possible unconventional plays by thinking outside the box.

Neuroticism

Neuroticism refers to the tendency to experience negative emotions such as stress, anxiety, guilt, or depressed moods. People high on neuroticism will react negatively towards setbacks; events which a person high on emotional stability (the opposite to neuroticism) would shrug off with hardly any notice. Emotionally stable people are calmer, more even-tempered, and less likely to feel stressed.

Neuroticism can of course be context dependent; for example, many successful players would experience more anxiety in social situations than in their preferred environment at the poker table.

Highly neurotic individuals will struggle at poker. During a session, you will constantly be bombarded with opportunities for negative reaction — lost pots, missed bets, and ever-present swings. Having enough emotional stability to ride out these upsets is really important. Ideally, mistakes should be capitalized on as opportunities to learn and improve; whereas a neurotic individual will react to slip-ups with negative, and ultimately fruitless, emotions.

High emotional stability is not a panacea — an even keel comes with its own dangers. People low in neuroticism are amongst the most likely to suffer from the behavioral biases such as overconfidence. (See “Overconfidence” starting on page 299 in “Psychological Biases.”) The worrying that neurotics constantly subject themselves to actually gives them a fairer evaluation of their abilities than the less anxious. In this respect, a little bit of neuroticism can keep you on your toes, reducing the probability of a serious blow up by keeping your ego in check.

A small dose of neuroticism can mix well with a high score in conscientiousness. The strong work ethic and fear of failing can propel these individuals far. The only price is that downward swings will be accompanied with all of the negative emotions neuroticism brings about.

Extraversion

People high on extraversion enjoy seeking out new experiences and are more outgoing than introverts. Extraversion doesn’t only measure your predisposition towards meeting new people and dealing with social situations, it also measures your gratification from new experiences.

Introverts, on the other hand, need less outside stimulation. Their drive is more self-oriented, and they are happy to go about doing things by themselves. If also high on conscientiousness, introverts can be highly driven in their pursuit of goals, it’s just they are less likely to speak up about it. This trait is less clear-cut than the others, I think both extraverts and introverts can succeed in poker, but often via radically different career paths.

Players who pursue a stratospheric rise through the ranks are likely to be extraverts. This trait increases their reward from seeking out new experience, so an individual high on extraversion will derive much pleasure from testing his abilities at each higher level in the game. It’s likely that a great number of the famous professionals score high on this trait.

On the other hand, extraversion causes a great percentage of these individuals to win and then proceed to lose back multi-million dollar bankrolls. The once prolific individuals who you think, “But where is he now?” are likely all extraverts. Unfortunately, their high reward from new stimuli caused them to take too much risk and blow their entire bankroll away. Incidentally, there is a high correlation between extraversion and pathological gambling.

Introverts are likely to pursue their climb in poker methodically and without as much notice from the wider world. They are likely to move up one step at a time, quietly and without fuss, steadily building a bankroll. Introverts are unlikely to lose significant chunks of their bankroll back to the poker economy. Given enough time, they will eventually fulfill their true potential.

Both extraverts and introverts can succeed, but both groups are prone to making diametrically opposed errors. Extraverts will occasionally take too much risk at some stage, ruining all of their hard work up until then. A famous online player, known by the alias Isildur1, fits this bill. At one stage he was up around $6 million in the highest games online, but proceeded to lose all of that and more. During one infamous session, he played eight heads-up tables simultaneously against some of the best players in the world: four against Tom Dwan, two against Phil Ivey, and two against Patrik Antonius. Isildur1 was clearly a very talented player, but high levels of extraversion caused him to take on a ruinous amount of risk.

Introverts will often take on too little risk, spending time at lower levels when their bankroll and skill level would be sufficient to win more higher up. While this error is seemingly less hazardous, it can cost introverts a lot in terms of missed opportunities. There are many benefits to taking the occasional shot at a higher game. Most importantly, if successful, you can then give your win rate a permanent boost. Second, at higher limits you will come into contact with a senior tier of professional; by playing amongst these skilled players and observing their style, you can take your own understanding of the game to a higher level.

In the early stages of the poker boom, the error caused by extraversion was much less significant. The games up to even the highest levels spread online were all much easier than the standard of play today. In that environment, taking risks and moving up quickly was clearly the optimal strategy. The reward to implanting yourself amongst the elite was huge; the downside was also smaller as the highest game running for a while was the $25-$50 no-limit — compared to the $500-$1,000 today.

The environment as I write this is much changed. Poker is more competitive and more predatory in nature. The player pool at the very top has become very small, and even the smallest errors are seized upon by the professionals in those games. High levels of extraversion can cause a problem, as there is a great temptation and risk to play beyond your means. The online culture has also changed, winning and then losing a huge amount of money can be considered “brag-worthy” by many.

This setting could well favor the introverts. That is the ones who have been diligently building their bankroll and skill level over the past few years might be the most fit for survival in the new poker economy.

Openness

The factor openness relates to two quite distinct things: intelligence (such as measured by IQ) and openness to new ideas and experiences. These two facets of openness are best viewed as distinct attributes which are weakly positively correlated (since intelligent people will likely have a wider range of interests than others).

Openness as a trait is unrelated to the other factors such as conscientiousness and neuroticism. For example, one could be open to new ideas, highly diligent, and prone to neuroticism even though this may seem like an unusual mix of characteristics. I don’t think an openness towards new ideas should ever be seen as a negative in poker. As we saw in earlier discussion, we are all prone to psychological biases, so keeping an open mind about things and avoiding whatever conventional thinking is popular should never be a bad thing.

Saying that, I believe that openness adds more value in big bet poker, just as how conscientiousness was more of a factor in limit. There is a much wider range of possible situations and spots in big bet — the factor of bet sizing allows for more opportunities in the play. With this being the case, the ability to make unusual connections between separate ideas, and the inclination to experiment with different lines and styles will be of greater value. I think this is particularly so at the highest levels where play in big bet poker can be extremely fluid — the advantage accruing to whoever can assess a constantly changing environment quicker.

This fits with at least my general impression of players at the top of the game today. Limit players are more willing to diligently work on one thing for a long time, such as creating an unexploitable strategy, much like a scientist. Whereas no-limit players will have a more diffuse intellect, capable of combining a variety of qualitative information — much like an artist.

Agreeableness

This trait measures our attitudes towards others. People high in agreeableness will be considerate and compassionate towards others, and they will tend to perceive others in a positive light. People low on agreeableness will tend to be more suspicious or wary of people, at least until they have got to know them a bit better. Extremely disagreeable people will have almost no regard for other people.

For poker, it’s my opinion this factor is the least important. Yes, people high on agreeableness are unlikely to have the ruthless streak required to dispose strangers of their hard earned cash, but I think these people are unlikely to be drawn to the career. When very agreeable people play poker, they see it as more of an opportunity to make friends and socialize with others, not as an opportunity for profit. The enforced solitude in the online game of working from home for long hours is also unlikely to appeal; in terms of live play, people with gambling habits are unlikely to be the most attractive company for a highly agreeable person.

Since people who are strong with this trait are unlikely to become professionals, you could see poker as being an industry filled with ruthless self-interested manipulators. Jack Straus uttered the famous line, “I’d bust my own grandmother if she played poker with me.”

However, I think this is an exaggerated view. Most professionals are not psychopaths, they just lie towards the middle and one end of a range of normal behavior. In fact, psychopaths would make poor professionals as these individuals are lacking in the conscientiousness required to thoroughly master a subject.¹³⁵ Poker pros can succeed with a perfectly average level of agreeableness, combined with the required competitiveness and intelligence. This could help explain why most pros are men, as women tend to have a higher average level of agreeableness across the population.

Conclusion

The five factors are not on a binary scale, you don’t exist on either one end or the other of each; instead, there’s a continuum of possible values on each trait. A low score on conscientiousness or a high score on agreeableness or neuroticism will make professional poker a difficult career. Extraversion will determine your path in poker, with extraverts surfing more frequently on the high and low waves. Big bet poker will reward individuals high on openness; whereas, limit poker stresses conscientiousness —explaining to some extent the general lack of overlap between the professional communities in each game type. When a well known big bet pro joins a high stakes limit table, it’s usually as the mark the game builds around, rather than one of the sharks (and vice versa).

Valuing
Your Poker Business

Part of being an intelligent poker player is making rational decisions about how much to play. Since the poker boom, this has been a fairly elementary decision since most players have been able to make more money multitabling at the small stakes than in many professional careers, with even greater rewards accuring to those who play higher stakes. In such a situation, cramming as many poker hours into the week can be the most profitable use of your time.

But as the global economic crisis has been slowly unfolding across the world, the situation has been changing: games are slowly drying up, major legislative action around the world has threatened many pros’ livelihoods, and the poker boom is, if not over, definitely slowing. This has created the realization for many that being a professional poker player is, in fact, a highly risky and unstable career.

On an intuitive level, this risk would imply that you need to be earning more from poker than an alternative in order to make going pro a plus-EV decision. In this chapter, I will attempt to precisely quantify this intuition using the tools of finance.

Stocks are often valued using a technique called discounted cash flow analysis: future cash flows are estimated and then converted into current-day currency to produce their present value (PV).¹³⁶ This value can then be compared with the current market price, and if the two differ substantially, the stock will be a candidate to buy or sell. (If the stock is overpriced, you can attempt to profit from this even if you don’t own the stock by using a technique called short selling.) Note that in this context, value refers to what you actually get from a cash flow, (its “intrinsic worth”), which, in the stock market is not necessarily equal to the price you pay — this is how professional investors find underpriced stocks.

The same technique can be used to compute the present value of any cash flow stream, such as the money you expect to make from poker. Valuing a prospective poker career and comparing it to what else is on offer is beneficial for both pros and aspiring amateurs. A low present value from poker would imply the amateur should stick to the day job, and perhaps the pro should rejoin the real world. When a career is being valued, as opposed to a stock, the resulting value is known as your human capital. ¹³⁷

Consider cash flow streams A and B in the table below:

Which do you prefer? Stream B yields an additional $100,000 over the entire period, so that could be your choice. But if you are impatient, or if you can earn a high rate of return on the early cash flows, you might prefer Stream A since it produces a higher return over the first two years. This question can only be answered by doing some math.

For Stream B, the $100,000 received in Year 1 is worth more today than any other years’ $100,000 cash flow. Money received earlier in time can be put to receive interest, and is hence worth more than the same amount received at a later date — this principle is known as the time value of money. If interest rates are at 5 percent, then $100,000 received today will be worth $127,628 in Year 5.

$127,628=(1.05 ) ⁵($100,000 )

This is known as the future value of the cash flow (the future value of $100,000 today is $127,628 in Year 5). But we can also do the reverse calculation, called discounting, and compute the present value (today, otherwise known as Year 0) of $100,000 received in Year 5 which is $78,353.

This means that for Stream B, receiving $100,000 in Year 5 is equivalent to $78,353 of today’s income (because this amount will, over time, earn exactly enough interest to be worth $100,000 by Year 5). Also notice that the present value of a cash flow depends on the length of time and the interest rate. Further away cash flows and a higher interest rate both result in a lower present value.

To compare the two streams, you must discount each future cash flow to Year 0, putting them all on a comparable basis by adjusting for the time value of money, then add them together to produce the total present value.

Now choosing the most valuable stream is a matter of picking the highest present value. Here is the calculation, discounting all cash flows at 5 percent:

Stream B is better since it’s worth an additional $64,000 in present value. At some discount rate much higher than 5 percent, Stream A would become the most valuable, as a high discount rate increases the importance of receiving money sooner rather than far off in the future.

Stream A was set up to look similar to the cash flows a fledgling poker pro might expect in the current environment. He’s confident of having at least a couple of high earning years, but beyond that he can’t predict, due to the uncertainty about the future in the current environment, a high average wage. This is how I’ve chosen to adjust the value from poker for the high level of risk involved in the game. You could argue that this was done far too aggressively, but it’s just a stylized example.

In a real life decision, it’s my opinion that a lot of people would choose the online poker career since it’s currently much more profitable. However, considering the future trajectory of your income is important, and as in this case, the long term view might imply that although poker puts more money in your pocket today, it actually creates less long-term present value. However, bear in mind that this poker career is not bad per se, it just creates less in terms of overall present value than the alternative. A lot of poker pros are young students, people without a highly profitable alternate career to pursue, in which case poker will easily beat earning the minimum wage in unskilled labor or what you might be paid at an entry level position.

Valuing a stock is not much more complicated; estimate the future free cash flows generated by the business (those not required for reinvestment to keep the business running), and discount them to the present day. If the per-share discounted cash flows are greater than the price, then the stock is undervalued.

Also, choosing the right discount rate is important. In my earlier example, the rate of 5 percent was used. If banks offered 5 percent interest as the going rate, and the future cash flows were known to be received with certainty, then 5 percent would be the correct discount rate to use. In finance, this is known as the risk free rate, which is a key measure in the economy — as of the time of this writing the risk free rate is much lower than this.

The government is the only institution able to borrow money at the risk free rate since a key assumption in finance is that it can always print more currency to pay off its lenders to avoid bankruptcy. Any riskier cash flows, such as from corporations, must be discounted at a higher rate.

According to an academic study,¹³⁸ the U.S. government has on average over the 20th century been able to borrow money at 4.8 percent, whereas the average U.S. corporation has returned 10.1 percent per year to its stock owners. The difference is an estimate of the equity risk premium. Since companies are risky, they have had to give away a higher average return to compensate investors — few would own a risky stock if it only produced the same return as a risk free investment. In addition, the 10.1 percent return is split into two parts: the risk free rate, which accounts for the time value of money, and the risk premium as a compensation for the possibility of loss.

10.1 percent would be a reasonable discount rate to value an average risk U.S. stock with, but what discount rate should be used when valuing your future poker income? It should be obvious that being an online poker pro is a risky business, much riskier than, say, Exxon Mobile, the largest U.S. stock trading today. This extra risk would require choosing a higher rate than this to discount your poker income when computing present value.

There is one type of business venture that more closely resembles poker: venture capital. In this area of the market, savvy

private investors put money into fledgling start-up companies —at a high risk of failure. To compensate for this risk, discount rates from anywhere in the range 20 to 100 percent are used, and these high rates will shrink the present value of a cash flow much faster. For example, a discount rate of 100 percent would shrink the present value in half each period.

So what is an appropriate rate to discount your poker income at? My answer is: At the very least, 25 percent, and possibly as high as 50 percent. Of course, this depends a lot on the specifics, with long-term winners able to use rates on the low end. I justify these high rates since the total risk in poker can be decomposed into two parts: Your payoff is risky, win rates are always backward looking, never guaranteed in the future, and second, there is the inescapable risk in the poker economy — legislative factors, substitute forms of gambling, and the balance between good and bad players in the system.

In addition, some players will have much less win rate uncertainty than others, usually due to a combination of extreme multitabling and a good rakeback deal. These players should use a much lower discount rate, perhaps 20 percent, due to their lower cash flow risk. However, the systematic poker risk is unavoidable and must still be accounted for.

Let’s take another look at the table from earlier, and I’ll take the liberty of renaming the two rows:

In this case, the risk of poker has already been accounted for by assuming profits will follow a downward trajectory. It would also be wrong to combine that assumption with a high discount rate since this double-counts risk.

The calculation of present value can be done much quicker than in my earlier example. If you have the correct discount rate and assume the cash flows will remain constant over time, then all that is needed is to forecast the first cash flow. Then, dividing this cash flow by the discount rate gives you the present value of the entire future stream of cash.¹³⁹

Forecasting $200,000 in Year 1 income from poker, with a discount rate of 25 percent (0.25 in decimal form), results in a present value of:

This is a much simpler procudure than having to forecast and discount each future years’ individual cash flow. And now that we have incorporated risk via the discount rate, we don’t have to assume that our profits will decline over time since the high discount rate shrinks the present value of far off cash flows for us. Because of my choice of discount rate, the present value of 800,000 is much higher than the figure calculated earlier. A discount rate of approximately 54 percent would be consistent with PV of $368,204 and Year 1 income of $200,000.

Let’s compute the present value of our alternative career choice now labelled “Other.” What is an appropriate discount rate for valuing a conventional career? This will depend on the riskiness, (investment bankers would need a higher rate than doctors), and your age (younger people have more years ahead of them, implying a lower discount rate). Additionally, in a conventional career, your earnings will likely grow over time as you learn new skills, and this can be represented by a reduction in the discount rate.

Two sources suggest discounting career earnings at a range of 3 to 3.5 percent.¹⁴⁰ This seems too low to me and my preference would be a range of 5 to 10 percent. So assuming a discount rate of 10 percent, the present value of the alternate career is $1,000,000.

which is $200,000 more in present value than the poker career. And as you can see, despite poker having double the initial cash flow, because the discount rate for poker is 2.5 times as large, the present value from the other career is higher.

This illustrates an important relationship between the discount rate and the present value of an income stream: The value of a stream, in terms of the multiple of Year 1 earnings, is just the inverse of the discount rate times the Year 1 earnings. (A discount rate of 10 percent makes the present value of a stream ten times Year 1 earnings.)

The same concept applies in the stock market: Due to their high required return, risky stocks are worth a low multiple of future earnings (called a P/E multiple, think of Goldman Sachs), while safe stocks have high P/E’s (think of Google). Goldman Sachs has a low P/E because its profits are based on having an informational edge trading in the markets which could easily be competed away at any time, whereas Google has a high P/E since its business is built on such a strong franchise.

The parallels with poker are obvious. The poker economy is constantly changing and there is no guarantee that your edge will continue in future. This risk makes poker worth a lower multiple of current earnings than most conventional careers.

But what about non-money matters? Say your day job is becoming stagnant and boring, and you really want to give poker a shot. If your preference for poker is strong, then there’s nothing wrong with choosing it over a higher PV alternative, but it’s still valuable to go through this sort of analysis to help figure out the cost of your preferences. In the example above, the PV of poker is $200,000 less than your current job, so you could make a rational decision to play poker if you would be willing to pay this much to quit your job. This is quite a price for freedom!

Taxes and any non-salary benefits need to be considered in the analysis. Cash flows should always be stated net of tax. This will make poker a much more attractive option in countries such as the U.K. which don’t tax gambling winnings. If your employer will offer you a matching contribution in your 401(k) or other defined-benefit retirement plan, then the value of this should be added to your after-tax salary.

Real options¹⁴¹ are prevalent in poker. The study of financial options shows that having the right, but not the obligation, to take certain actions in the future has value. For example, a financial put is a derivative security that increases in value when an underlying security decreases in value; if the market price falls below the agreed exercise price, the owner can choose to receive the difference between the two. The put has value to the holder since he can exercise the option if the underlying falls in price, but if this doesn’t occur, he’s not obliged to do so.

The option to abandon your poker career is similar to a put option where the underlying security is the value of your future poker earnings and the exercise price is the value of your outside career option. A higher exercise price increases the value of this option, it effectively creates a floor for your earnings, since if poker becomes unprofitable, you can exercise the option by returning to work and still make a decent living. Due to the presence of this option, people with in-demand skills have the ability to take more risk with poker than others.

Poker also has the nice feature that you are free to play as much as you like. If things are going particularly well, you have the option to spend more time playing. This is known as a real call option (call options increase in value as their underlying security appreciates). In fact, everywhere I look in poker, valuable options can be seen. These include changing stakes, learning new games, and varying hours played. The bottom line is that owning these options has value — reducing the required PV from poker to make it a rational career choice.

To sum up, the key to valuing your career is to understand the interaction between your current earnings and the discount rate —a high rate results in your career being worth a relatively low multiple of current earnings. Discount rates for poker could vary from 25 to 50 percent, and for a conventional career from 5 to 10 percent. This would imply that to make poker your profit maximizing choice, you would need to earn in the initial earning period between 2.5 and 10 times as much than in your best alternative career. Adding in the optionality present in poker would allow you to accept a slightly lower multiple.

As a rule of thumb, I suggest to only play poker full-time ifyou can expect to earn at least twice as much as in any other available job. Depending on your outside career option, you might require an even greater multiple from poker. For example, tenured professors have close to a job for life, implying a discount rate close to the risk free rate, and would require a much higher premium to give that up for full time poker.

This required premium from poker has comfortably been there for high achievers. The question of whether we can expect it to persist will be treated in “Predicting Future Poker Returns” starting on page 347. There are two effects which could make poker less attractive for these superstars. First, the games at the highest levels are more competitive and less likely to run. This means the high stakes players will need to value their earnings using a high discount rate. Second, high achievers are likely to have enough skills and intelligence to command a high wage in the conventional job market. These two factors will result in a very high required wage from poker — something which may disappear if the games continue to get worse.

My personal opinion is that online poker will only become a more difficult career as time goes by, but the future is uncertain. So it’s possible that average win rates will converge to a level where many players are not receiving adequate compensation for the risk involved, and because of this, I encourage players moving up the ranks today to retain as many outside options as possible —reducing their dependency on poker if things turn sour. It should also be noted that the online boom has been somewhat of an anomaly in poker history. Prior to this, some poker writers strenuously argued that poker was much better as a profitable hobby than a full time career.¹⁴² But they were in the minority.

Once properly valued, your poker business is as much an asset on your personal balance sheet as a house, car, stock, or bond.¹⁴³ With one difference: It cannot be traded in. This makes choosing an optimal investment portfolio more of a challenge: It’s best to find assets that diversify the risks poker exposes you to.

Investing

More than any career I can think of, in poker, success is defined solely by making lots of money. In other careers, successful people will add value to a corporation, and as a result of this, they can command a higher wage in the jobs market. In poker, your success is defined intrinsically by your net earnings —wealth is not just a fortunate side effect of being skilled.

Due to the meritocratic nature of poker, there are many young twenty-somethings who have amassed sizeable fortunes. In a conventional career, many of these people would no doubt be on the road to success, but their current tangible net worth would be almost always lower.

This makes learning how to invest well a profitable sideline for poker players. In fact, investing could easily be a source of more lifetime dollars earned for these players than their initial poker success — at a 7.2 percent compound return, your invested wealth will double every 10 years. Additionally, there isn’t a lot of job security in poker, So developing passive income will provide a safety net against any downturns.

In a normal career, your earnings would start off much lower — slowly increasing over a number of years. In the early years, this leads to less of a saveable surplus after consumption, meaning their dollar-weighted investments will be in the market for a shorter period of time prior to retirement — reducing the power of compound interest.

It’s also common for new investors to make mistakes in their initial investment approach. Part of this is unavoidable as we tend to learn best by making mistakes and learning from them. However, young poker players frequently have a lot more money at stake, meaning any errors can result in a far greater dollar loss.

Consumption Smoothing

In “Risk Preferences: From EV to EU” starting on page 278, the importance of risk aversion was introduced. Risk aversion is the consequence of another important fact: diminishing marginal utility of wealth. If additional units of wealth provide less of an increase to our standard of living than earlier units, we should avoid significant zero expected value gambles since the losses are felt more than the gains.

Diminishing marginal utility of wealth has implications for how we should spend and save our money over time: If our current earnings are low, we should dip into our savings, or perhaps borrow, in order to maintain an appropriate living standard. If our earnings are high, we should save some of the excess for a future state where our income is lower. This principle is known as consumption smoothing. ¹⁴⁴

Figure I: Consumption Smoothing in Practice consumption

Figure I is a graphical illustration of consumption smoothing. It shows two alternative consumption plans where an individual is endowed with 100 units of consumption to spread over 10 separate time periods. The left-hand image is a totally flat consumption profile: 10 units are consumed every period. On the right-hand side, consumption peaks towards the middle years; the resulting deficit is made up with a reduction in the latter periods. In this profile, the dotted triangle is gained relative to the flat profile; whereas the dashed triangle is lost. Due to diminishing marginal utility of wealth, the dotted triangle will provide less satisfaction than is lost from the dashed triangle. Therefore, the flat consumption profile is the optimal choice.

Let’s reconsider a concept from the previous chapter: human capital. Consider an individual who intends to make playing poker his lifetime work. Due to the high risk in poker, his human capital (the present value of his life’s earnings) is a relatively low multiple of current earnings. In other words, his expected lifetime earnings will be clustered predominantly in the next few years’ work. In order to smooth his lifetime’s consumption, he will have to save a large fraction of his winnings from the immediate future intending to draw down on those savings much later. Specifically, I think a pro earning six-figures or more a year should be saving at least 50 percent of his income.

This is the academic argument for why saving and investing decisions should be such an important part of optimizing your well-being from the poker lifestyle. It’s merely tangential to the fact that investing money over a long time period can compound your wealth exponentially. If your money were to double every 10 years, after 30 years, you would end up with eight times your initial portfolio. For a rich 21 year old poker player, such a time horizon is easily achievable. Even longer horizons will provide much greater compound returns still.

Behavioral Errors

There are two main reasons why someone might fail to take the long-term approach I advocate. First, they might not want to, and in this case we will just have to agree to disagree. Second, they might fully intend to do it, but like an unpleasant homework assignment, they procrastinate over the implementation until it’s too late.

This second reason has been an area of research for a number of behavioral economists who are interested in why people make poor choices over time, and how they can be helped. Let’s explore this a little more.

Would you rather receive $10 today or $15 tomorrow? When asked this question, many people will opt for the smaller immediate reward. If this person has any personal wealth, or any access to borrowing facilities, this is a poor decision. Rather than take the $10 today, you could borrow $10 from the bank, spend it, and then repay the loan with the $15 you receive tomorrow. This would be the optimal choice at any interest rate below 50 percent per day — well above even loan shark rates.

What if your choice was between $10 in 365 days, or $15 in 366 days? In this case, the vast majority of people who selected the $10 earlier are able to wait the extra day to earn the additional $5. Notice that this pair of decisions is inconsistent with rational choice; in 365 days time the choice will be identical to the other of $10 today or $15 tomorrow, in which case the preference is for the $10. Such reversals of preference based solely on the passage of time are theoretically incorrect.¹⁴⁵

The preference reversal is caused by a differing treatment of immediate and delayed rewards, and it has been used to explain both addictive behavior¹⁴⁶ and serial procrastination.¹⁴⁷ We really want to implement our new diet and exercise regime, but we will do so tomorrow. When tomorrow comes, we will find a reason to delay another day, and so on...

The same effect explains why some people mean to reign in the spending or to adjust their portfolio, but never manage to. If this in any way describes you, the trick is to make irreversible decisions while in a cool frame of mind. Richard Thaler and Cass Sunstein¹⁴⁸ have developed a system of automatically deducting a portion of employees’ wages into an investment account prior to them receiving their pay. The deductions start in the future, and gradually increase over time (below the rate of wage increases) so employees never notice a reduction in their observed pay. By delaying the payment until the future, employees are able to use the cool-thinking part of their brain that is not tempted by immediate gratification.

Poker players with impulse control issues could use similar devices.¹⁴⁹Automatic payments to an investment account could be set up, and they could choose to invest in illiquid assets, such as certificates of deposit or direct investment in property, which cannot be immediately sold without incurring a loss.

Beating the Market

Certain analogies can be drawn between playing poker and investing in the market. Some say the stock market is the greatest casino of all, and poker players might be tempted to try to transfer their prowess from beating one zero-sum game to the other. Although the stock market is a positive-sum game, (investors make money), on average, trying to achieve higher risk adjusted returns is a zero-sum game (before costs): for every winner there is a loser.

This proposal ignores the law of comparative advantage, the proposal that people should stick to producing the goods they are relatively best at. If your hourly rate is higher playing poker than actively managing your investments, then you should stick to playing poker. People with a higher rate of return from investing should, in fact, be professional investors, not professional poker players. Poker has the advantage that you will reach the long-run much quicker — losing months are a rarity for many. But in investing, it’s common for skilled active managers to underperform their benchmark for years at a time.

However, investing being a positive-sum game allows you to earn expected profits without possessing any edge. Since companies are, in effect, bidding for scare capital, investors in the markets will be provided with a competitive rate of return. For instance, investors in the U.S. market have received an average return of 5.3 percent above the risk free rate.

The market portfolio is of special significance in academic finance. This comprises the entire universe of risky assets in existence, bundled together, and weighted by their relative market capitalization — larger stocks have a higher percentage weight. For good reasons, the market portfolio is the recommended holding for many investors.¹⁵⁰

If you have no valuable information about the prospects of risky assets, then investing in the market portfolio allows you to take full advantage of others’ private information. Every time an active manager buys or sells a stock in an attempt to beat the market, the supply/demand balance for shares alters feeding through into a change in the market price. This price change will be fully reflected in a change in the relative holding of that stock in the market portfolio. Each time an active manager makes an attempt to beat the market, this gets reflected in the market portfolio, enhancing its efficiency. The result that the aggregation of private information through a market mechanism leads to good predictions about the future is known as the wisdom of crowds. ¹⁵¹

The market portfolio is the recommended holding in the capital asset pricing model (CAPM). The CAPM is very similar to the result of the Nash equilibrium in game theory. The Nash equilibrium shows us the way two or more perfectly rational agents will play a game; the CAPM shows us the way markets should behave in a world of perfect markets and information. The CAPM is also an equilibrium theory because it tells us what would happen in a world where each investor’s portfolio maximizes his risk/reward ratio, given that this is true for every investor.

The CAPM is just a theoretical construct, and like the Nash equilibrium for poker, it cannot exist perfectly in the messy world around us. However, it’s still a relatively good approximation of reality, and you can do well following its simple advice.¹⁵² With the wonders of modern finance, you can invest in the U.S. stock market portfolio at a cost of just 0.07 percent per year.¹⁵³

Once the market portfolio has become your vehicle for investing in risky assets, the only investment choice remaining is to select the proportion of your wealth to invest in risk free assets.¹⁵⁴ I have recommended two books on asset allocation in “Appendix A: Further Reading,” starting on page 377, to help you make this choice.

Labor Income Risk

As poker players, we are unavoidably exposed to an idiosyncratic source of risk.¹⁵⁵We have already examined one way in which this alters our investment decision — we should precautiously invest a large portion of our excess earnings. The risk from poker will also alter the composition and overall target risk level of our portfolio.

Diversification is often referred to as the one free lunch in economics. By combining a selection of many unrelated risky investments, you can create an overall portfolio that is less risky than the sum of its parts. Correlation is the statistic typically used to measure the relation between two assets: Assets with a high positive correlation will tend to move in the same direction as each other, providing less of a diversification benefit. Negatively correlated assets provide the greatest diversification benefit as these tend to move in opposite directions — these assets are naturally in high demand and are hard to find. Uncorrelated assets move independently of one another and provide good diversification benefits.

Our labor income is positively correlated with the gambling sector as a whole. When gambling becomes less popular, poker players suffer from fewer and tougher games. For this reason, your financial assets will diversify your labor income the most when you avoid a high exposure to assets that are also positively correlated with gambling.

Internet gambling stocks, such as PartyPoker, will have the highest correlation with online poker. These are the stocks that novice poker playing investors are often drawn to, but they are the stocks they should want to hold the least. Many poker players enthusiastically bought shares in PartyPoker and Neteller when these two stocks listed on the market in 2005 and 2004 respectively. But they nosedived when the Unlawful Internet Gambling Enforcement Act became law in the U.S. in 2006. Online gamblers who had bought these stocks saw the value of their assets, and their careers, plummet in unison.

The positive correlation between these stocks and online poker can be used to your advantage: by shorting them you can profit from any falls in the market price. This would be best done with a deep out-of-the-money put option. This security has a one-off upfront cost, and it will expire worthless unless the underlying stock falls heavily in price, in which case the option will profit enormously. These put options would act as an (imperfect) insurance policy on your poker career, paying off in states of the world where online gambling becomes deeply unpopular.¹⁵⁶ As I showed in “Risk Preferences,” risk aversion can make even negative expected value insurance attractive to hold.

Las Vegas real estate is another asset class which is positively correlated with poker. Gambling is the only industry of any note in Las Vegas, so if gambling becomes unpopular, there will be fewer jobs in Las Vegas, making it a less attractive place to live. If this is where you live and you intend to for a long time, I don’t think there is anything wrong with buying a house there as your primary residence. On the other hand, I would avoid unnecessary speculation in this asset class. (For instance, don’t buy any rental properties in the area or buy any property to “flip.”)

The correlation between poker and the overall stock market is also of concern. The higher the correlation, the more safe assets you will want to own (they will diversify both your labor income and risky investments).¹⁵⁷ I know of two competing points of view. One is that gambling is a “sinful” activity, like alcohol or tobacco, where dependent consumers will maintain their purchases no matter the point in the market cycle. The other is that gambling is a “luxury” good, our survival is not dependent on it so people will cut back when times are difficult. The luxury good argument implies that poker income will be positively correlated with the stock market — providing less of a diversification benefit.

I think the sin argument dominates at the lowest levels. When being a losing player is relatively inexpensive, people can keep it up even without a job, in which case they might devote even more time to the activity. But being a losing player is an incredibly expensive habit anywhere from middle stakes and up: Losers can expect to pay amounts up to thousands of dollars for an average night’s entertainment. The turnover of losing players at these limits is high, and I know of few fish who have been able to afford playing regularly for years on end.

The correlation between poker income and the market should therefore be stake dependent: the high stakes will be positively correlated, with the low stakes roughly uncorrelated. This view is in accordance with the evidence that the high stakes Las Vegas live poker scene was much affected by the dot com bubble bursting in the early noughties.¹⁵⁸ Furthermore, the online high stakes economy has been on a slow downward trajectory since the the credit crunch began (and this probably isn’t a coincidence).

This implies that, all else equal, a high stakes player should invest a greater proportion of his portfolio in risk free assets (since his labor capital behaves similarly to the market). However, high stakes players will, in general, have a higher ability to take risk —they can withstand investment losses without it immediately affecting their standard of living. Your individual level of risk aversion is another important factor — less risk averse people can own more risky assets.

Some economists advocate high risk leveraged portfolios to young investors.¹⁵⁹The rationale being that young investors have a large holding in the intangible asset of their human capital even if they have not accumulated much wealth yet. Therefore, a leveraged portfolio is not as risky as it might seem as any losses can be quickly replenished (as long as they have a job or some other source of revenue). This allows investors to take less risk later on in life, leading to more efficient lifetime risk taking.

This argument doesn’t really apply to poker players. Poker players convert their human capital into financial wealth much quicker than most ordinary careers. An established high stakes pro, although he might be in his early twenties, will have a total capital profile more like some someone in his fifties. Players moving up the poker ladder will have a more typical profile and could take the leveraged approach. However, their cash would provide a higher return in their poker bankroll.

Conclusion

This chapter has been my attempt to condense the vast field of investing into a short primer for poker players. Knowledgeable investors might disagree with some of my claims; for the sake of brevity I have left out a lot of information and arguments. But hopefully, this chapter has helped to whet you appetite with the field, and I encourage you to do further research into areas that have caught your interest. And just as in poker, there is no single path to success: Many different approaches and strategic variations can do equally well, and my focus has been on what I see as the path of least resistance by highlighting an approach which will serve you safely and easily.

The key takeaway from this chapter is that taking some time to invest your winnings well can be very useful. In terms of hourly rate, spending the short amount of time to educate yourself and implement a rational plan will dwarf your poker win rate.

The same behavioral issues that cause people to underperform in poker can also affect their ability to invest optimally. For people like this, buying the house they live in can be an especially rewarding investment — reducing the temptation from liquid cash on hand and providing a roof over their heads for the worst case scenario.

Other Topics

Introduction

Bankroll Management
and Shot Selection

Risk Preferences:
From EV to EU

Psychological Biases

Personality

Valuing
Your Poker Business

Investing

Other Topics

Introduction

Bankroll Management and Shot Selection

Risk Preferences: From EV to EU

Psychological Biases

Personality

Valuing Your Poker Business

Investing

Bankroll Management
and Shot Selection

Risk Preferences:
From EV to EU

Valuing
Your Poker Business