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Expected Utility

Utility of Money
Earlier we discussed the concept of expected value, or the amount of chips you expect to gain from making a certain move. In almost all ring-game situations, taking into account expected value and implied odds will give you enough information to make the right decision. But, underlying the concept of expected value, there is an important assumption: The assumption that every dollar has the same value.
This assumption is not always true. Let's say that that your bankroll is a million dollars. Someone offers to bet you a million dollars on a coin flip. Would you take that bet? Some people would, but most wouldn't. The reason for that is that, if you have a million dollars in the bank, then earning a second million won't bring you much utility (happiness). After all, if you have that extra million, what could you do with it? The utility (happiness) gained from that second BMW and house isn't quite equal to the loss you would experience if you were suddenly made penniless.

The reason for this is that your Utility of Money changes depending on how much money you have. Each level of worth or income has associated with it a certain level of utility. That utility is not necessarily going to increase uniformly. For most people, the second million is not worth as much as the first. This is called Diminishing Marginal Utility. People who fall into this category are called risk-averse:

Diminishing Marginal Utility
Other people would take the bet, because the second million is worth exactly as much to them as the first million. These people are called risk-neutral:

Constant Marginal Utility
Now, there are many people who wouldn't take the million dollar bet at even-money odds, but if they had a 55% chance of winning instead of 50%, they might take it. Of course, many people still wouldn't take it, but some would.

Now, what if the tables were turned? What if you were offered the million-dollar bet with 45% chance of winning? Believe it or not, there are some people who would take the bet. This is because the marginal utility they get from the second million dollars exceeds the utility they get from the first million. Perhaps they need exactly $2 million to save their business, and anything less would result in bankruptcy.

This condition (increasing marginal utility) also partially explains why people buy lottery tickets. In some states, you will get around 30 cents of EV on every dollar you play on the lottery. These are terrible odds! And yet, some people who realize this still play, because the utility they get out of becoming a multi-millionaire (or even just thinking about it) is worth that 70 cents to them. People who fall into this category are called risk-loving:

Increasing Marginal Utility
Risk-loving people are the most susceptible to becoming compulsive gamblers. This is because they enjoy taking bets with bad expected value.

Expected Utility and Poker
You might ask, what does all this have to do with poker? Well, there are a few things. First of all, the type of person your opponent is (risk-averse, risk-neutral, or risk-loving) will affect how he plays at the table. Many players, when they try to move up limits, play scared. This is because the increased amount of money they have at the table is high enough that they are no longer at a constantly sloping point in their utility curve. This helps to explain why you should not play at a limit which is over your head. If you are risk-averse at that limit, you will be conceding alot of small edges. The opposite situation is also true: you may do well at the $10-$20 because you like precisely that amount of risk. If you played $5-$10, or $2-4, you may now be risk-loving, and be playing too loose!

At first glance, one might think that risk-loving is the same as loose, and risk-averse is the same as tight. This is not true: a person may be playing tight simply because that is the best strategy given the type of opponents he is facing. A strong player in a No-Limit game will vary between playing tight and loose, but he is risk-neutral. He will neither turn down a bet with a slight edge, nor lay a slight edge to his opponent. The condition that allows him to vary between tight and loose is that the play of his opponents requires him to.

It's important to make sure that you yourself are playing risk-neutral, and also to read your opponent correctly. If your opponent is playing tightly, don't simply assume that he is risk-averse, and start bluffing like crazy at him. It is important to learn quickly whether he is truly risk-averse or simply playing tightly to take advantage of the other players.

Bankroll Risk vs. Stack Risk
So if you should be risk-neutral with your stack, what about your bankroll? We advise that you be risk-averse with your bankroll. The amount of money that you win or lose in any one session or even a few sessions shouldn't matter too much to your bankroll. Otherwise, your emotions will come into play too greatly, and playing scared is always a bad idea. But how does this make sense? How can you be risk-neutral with your stack and risk-averse with your bankroll at the same time? The answer is that the shape of your utility curve changes as the area you are looking at changes. If the area is small enough, the risk-averse curve will start to look more like a straight line. This is kind of like how the earth is round, but the area we can see with the naked eye is so small that it looks flat to us.

Utility of Tournament Chips
Expected Utility Theory also explains why tournament play is different from ring game play. You may have noticed that people are more risk-averse in tournaments than in ring games. This is because of expected utility: if you have 1,000 in chips early in a tournament, it is most likely a bad idea to take an all-in on a 50-50. This is because getting that second 1,000 in chips is not worth that much to you, but if you lose your first 1,000, you're out of the tournament.

On the other hand, in the middle of a tournament, it may become a good idea to take that very same bet! This is because having a big stack will give you some extra utility because you can now steal blinds. Also, if you are down to a short stack later in the tournament, you would probably gladly take that 50-50 all-in. This is because the utility of having that first 1,000 is much diminished because of the blinds.

Your utility curve in a tournament is dynamic, or changing over time. A winning tournament player is aware of what utility he gets from the chips he is betting, and how his utility curve changes throughout the tournament. He doesn't settle for a positive expected value in chips; if he talks of a positive expected value, it means a positive expected value in prize money. In other words, he demands a positive expected utility on all his bets.

Utility of Deals
Towards the end of tournaments, it is common for poker players to strike deals. Often, the prize structure is very top-heavy, giving first place nearly twice as much money as second. However, the blinds are so high at this point in the tournament that luck will be the primary factor determining the victor. Instead of battling it out, poker players often would rather strike a deal and give each player a proportion of the prize pool. These deals take into account the utility curves of the players. If a poker player is a billionaire and is playing in a $200 buy-in tournament, he probably does not care that much about the variance involved with winning. However, a player who barely scraped together that much money to enter the tournament is likely to be eager to strike a deal. When making a deal, take into account your opponents' and your utility curves. Do not let them take advantage of you because they suspect you are risk-averse. For example, it is speculated that there was no deal in the 2003 WSOP because Sammy Farha thought he could bully Chris Moneymaker at the final table. Farha, a multi-millionaire, knew that the money involved intimidated Moneymaker, an average player. Ultimately, Moneymaker kept his cool, played solid poker, and won the World Series.

Next Article: Expected Utility, Part 2

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