Understanding Poker Errors Through Prospect Theory - Part II

Part I of this column introduced Prospect Theory, an economic and psychological theory that helps explain people's behavior in risk situations. It also introduced my co-author, Dr. Rachel Croson, who is an associate professor at the Wharton School of the University of Pennsylvania and studies these theories and their relationship to poker players, among others.



To recap, Prospect Theory explains three types of errors that people in general and poker players in particular make:



• Loss Aversion (the pain of a loss is about twice as much as the joy of a gain)



• Asymmetric Risk Preferences (preferring not to gamble when ahead, but to gamble more when behind)



• Misestimation of Probabilities (thinking that low-probability events are more likely than they are)



Part I looked at loss aversion in depth. This column will look at the other two error types and summarize how this information can help you to become a better player.



In Part I, we introduced a couple of players. Here, we will look more closely at Player B, who raised preflop with pocket jacks and was called by the big blind. When an ace fell on the flop, Player B was check-raised, and was sure the big blind had an ace. He called, hoping to hit a jack, which is a terrible play mathematically. He folded to a bet on the turn when a jack did not come.



We now return to an examination of Prospect Theory and how it relates to poker.



Asymmetric risk preferences: When it comes to gains, people are risk-averse. That is, when they are given the chance to win $100 for sure or flip a coin for $200, they tend to choose the sure win. However, when it comes to losses, individuals are risk-loving. That is, they prefer a fifty-fifty chance of losing $200 to a sure loss of $100. In a more striking example, researchers asked 95 subjects which they would prefer:



A. Winning $3,000 for sure



B. Winning $4,000 with an 80 percent chance

Of course, the expected value (EV) of B is higher than the expected value of A ($3,200 is larger than $3,000), yet 80 percent of the subjects chose option A. Could it be that these subjects just did not like to gamble? Not really, because later in the experiment, these same subjects were offered the following choice:



C. Losing $3,000 for sure



D. Losing $4,000 with an 80 percent chance



Here, the expected value of C is higher than the expected value of D (-$3,000 is larger than -$3,200), yet 92 percent of the subjects chose option D. Now these same subjects are risk-loving, willing to sacrifice the $200 of expected value in order to take the gamble. In other words, people would rather take a sure profit than a good gamble, but would rather take a bad gamble than a sure loss.



To seasoned gamblers, this is a very surprising result, in that the vast majority chose the proposition with the worse EV each time. However, at the poker table we see this behavior all the time, as the following examples show.



Player B (the one with pocket jacks who was check-raised after an ace hit the board) showed such a tendency. If he believed his opponent had an ace, there was no reasonable mathematical reason to call the raise. If he did not believe the opponent had an ace, he should have reraised on the flop, or at least called the turn. His play is completely inconsistent, but it is also very common. If he folded, he took a certain loss. Prospect Theory tells us that people would rather take a losing gamble than a certain loss, which he did.



You also see this behavior when a player limps in preflop from middle position. Now, there is a raise and a reraise, making it two more bets to the original limper. How can he possibly have a hand that is not good enough to raise, but suddenly is good enough to call two extra bets from opponents representing very strong hands? Almost certainly, he can't. But again, he would rather take a bad gamble than a sure loss, especially when the gamble might result in a big gain (although not enough to compensate for his long-term losses by consistently making this call). In fact, one way to recognize a good player is by observing that he can put in money and fold on the same street. This behavior represents a player who has overcome his own biased decision-making.



Let's look at an example of poker players being risk-averse in gains. Say Player B again holds pocket jacks, raises, and is called by the big blind. Now the flop comes 9-8-4. Again, he bets, and is check-raised. Most often, he will be up against a 9, an 8, or a draw, with his opponent hoping he has two big cards. He should reraise and make the most of the fact that he is frequently ahead. But doubts creep into his mind. What if the big blind flopped two pair or a set? If so, Player B will lose more money. He frequently simply calls the raise and calls all the way, taking what he hopes is a win rather than gamble for a bigger one with higher expected value.



One final example every poker player can relate to is leaving the table. When players are winning, they are willing to leave even if the game is good (sure profit instead of a good gamble). If they are losing, they are more willing to stay even if the game is bad (taking a bad gamble rather than a sure loss). Players who keep excellent session records will probably notice that their average hours per win is smaller than their average hours per loss.



Misestimating probabilities: Many players find probabilities difficult to handle, especially in the heat of battle. But psychologists have demonstrated consistent errors in how people misperceive probabilities. In particular, individuals tend to overweight low probabilities, less than about 20 percent. This effect is especially strong if the outcome caused by the low-probability event is one for which you're hoping (sometimes called wishful thinking).



Returning to Player B with the pocket jacks and an ace on the flop, when deciding by gut rather than calculation the likelihood of hitting the third jack, he is likely to overestimate his chances.



You see it happen every day. A seat-of-the-pants player flops a draw, and misses. He cries, "Unbelievable. I can't believe I missed this hand." A while later, the same player flops a hand and gets drawn out on, perhaps even by the same draw he missed earlier. You would think that he would expect it, since he was so amazed when his own draw missed. But that's not the case. "Unbelievable," he shrieks. "I can never get a hand to stand up."



How else can a player play a small pocket pair to the turn and river in a small to medium pot when there are several overcards and opponents, hoping to hit a set? Because memories are selective, he remembers his infrequent but large wins, but forgets his much more frequent but smaller losses.



How can all of this help you? Prospect Theory shows us that most people have a natural tendency to weigh losses more heavily than gains, to gamble more in loss-related situations and less in gain-related ones, and to misestimate probabilities in a systematic way.



We can look at recommendations in two areas:



• Exploiting these errors in others



• Overcoming these errors in ourselves



Exploiting these errors in others: In Part I, we discussed exploiting loss-averse opponents by calling them less and reading their hands more accurately. Players who are risk-loving in losses and risk-averse in gains can be expected to play tighter when they are ahead and looser when they are behind. Pay attention to their stack size and how they stand (whether they are winning or losing). Trust their play more when they are winning, and a lot less when they are losing. Sometimes, these players can be convinced not to leave by reminding them that they are stuck and need only one or two wins to get even. They may stay and gamble it up, hoping for a miracle.



Players who misestimate small probabilities tend to stay in too long. Value bet these players regularly on the flop and turn, even with thin values, as they could easily be drawing to unlikely hands. Trust their river raises a bit more, since pretty much any card might make their hand. Congratulate them when they draw out, and emphasize the size of the pot they just won. This will make them happy they stayed and chased.



Overcoming these errors in ourselves: Are you making any of these errors? If so, you need to work to overcome them. You can't just coast and hope you play better. Winning poker requires understanding and overcoming your own natural biases.



Remind yourself to recognize situations with positive expected value, whether you are ahead or behind. Play in games in which your bankroll allows you to exploit these situations. Remind yourself that poker is a long-run game, and putting yourself in position to make money is the key, even if that requires short-term losses.



Treat each decision independently from the one before, rather than taking more risks when you're behind in a pot (or for the evening) than when you're ahead. And evaluate gains and losses evenly, rather than overweighting the losses relative to the gains. After all, both contribute equally to your bankroll.



Don't just estimate your outs and pot odds; calculate them to avoid overweighting small probabilities. And when the results of your calculation tell you to fold, do it.

Barry offers poker lessons tailored to each student's specific strengths and weaknesses. Please visit his website at www.barrytanenbaum.com or e-mail him at [email protected].



Rachel is always on the hunt for new sources of data regarding gambling behaviors. Her website can be found at http://www.wharton.upenn.edu/faculty/crosonr.html, and she can be reached at [email protected]. You can write to her for detailed references regarding Prospect Theory.