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Home > Strategy > Expected Utility

Poker Strategy - Applying the Concept of Expected Utility

Utility of Money

We have discussed the concept of Expected Value, or the amount of money/chips you expect to gain from making certain moves. In almost all ring game situations, taking into account Expected Value (EV) and implied odds will give you enough information to make the right decision. But, underlying the concept of Expected Value, is an important premise: The assumption of every dollar having the same value.

This assumption is not totally true. Let's say that your bankroll is $1,000,000. Someone offers to bet you $1,000,000. on a coin flip. Would you take on that bet? Some people would, but most wouldn't. The reason for that is this, if you have 1 million dollars in the bank, then earning a second mill 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 Bently and house isn't quite equal to the loss you would feel if you were suddenly made flat broke.

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

poker training-utility

Other folks would take the bet, because the second million is worth equally as much to them as the first million. These folks are called risk-neutral:

poker training-utility

Now, there are many people who wouldn't take the million dollar bet at even-money odds, but if they had a 55-60% chance of winning instead of 50%, they might.

Now, what if the tables were turned? What if you were offered the million-dollar bet with a 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, known as (increasing marginal utility) also partially explains why people buy lottery tickets. In some states, you will get around 30 cents of EV on every buck you play on the lottery. Horrible odds!  Yet, some people who understand this continue to play, why? Because the utility they get out of becoming a millionaire (or even dreaming about it) is worth that 70 cents to them. People who fall into this category are called risk-lovers:

poker training-utility

Risk-loving people are the most susceptible to becoming compulsive gamblers. This is because they enjoy taking bets with bad expected value for the excitement of a possible win. They may never take the time to manage and grow their bankroll responsibly over time.

Expected Utility and Poker

So you ask, what does all this have to do with poker? Well, there are a few things to consider. First of all, the type of person your opponent is (risk-averse, risk-neutral, or risk-loving) will affect how he or she plays at the table. Many players, when they try to move up in limits, get scared. This is because the increased amount of money they have at the table is high enough that they are no longer at a 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 giving up a lot of small edges. The opposite situation is also true: you may play well at the $10-20 because you like precisely that amount of risk. If you played at the $5-10, or the $2-4, you may now be risk-loving, and be playing way too loose.

This is not to say that risk-loving is the same as loose, and risk-averse is the same as tight. See, 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 lay 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 he gauges by the play of his opponents.

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 just assume that he is risk-averse, and start bluffing like crazy at him. It is important to quickly learn whether he is truly risk-averse or simply playing tightly to take advantage of the other players.

Bankroll risk vs. ChipStack 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 very much to your bankroll. Otherwise, your emotions will come into play too greatly, and playing scared (on tilt) should always be avoided. 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.

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 1000 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 1000 in chips is not worth that much to you, but if you lose your first 1000, 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 1000 is much diminished because of the increased blinds.

Your utility/chips 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 winner. 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 multi-millionaire 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 Chris, an average player (who earned his seat through internet wsop qualifiers) kept his emotions cool, played solid poker all the way, and won the World Series of Poker.