Thoughts on Game Theory: Part 2

I ended part one of this series on Game Theory by presenting a problem suggested by the economist Richard Thaler. This is the problem:

A group of people is offered a meaningful prize for the most accurate answer. Every member of the group is told to select a number from 0 to 100 that accurately predicts two-thirds of the average number selected by the group. Think about what number you would choose and why you would choose it. Does the belief that everyone in the group is rational affect your choice?

Like many game theoretic situations and questions there is a theoretically correct answer, but there is also an exploitative answer. In many real-world situations, certainly in most poker situations, the exploitative answer will lead to much better results than the answer that game theory says is correct.

The game theorist attacks the problem by thinking along the following lines. Even if everyone guesses 100, two-thirds of their average will be 67. Therefore, it can never be right to guess more than 67. But everyone is rational, so no one will guess more than 67. In that case 2/3 of the average will be less than 45. Similarly, if everyone picks a 45, 2/3 of the average will be 30. You continue this reasoning on to lower and lower picks, eventually realizing that the game theoretic solution is for everyone to guess 0, and tie.

In real life, when this problem has been presented to a group of people for the first time, the average is approximately 20, so the winning choice would have been 13 or 14. The exploitative pick, based on the fact that not everyone is a consummate, rational game player crushes the ‘correct solution.’

Another famous problem in game theory is the Prisoner’s Dilemma. Two men are arrested for a crime, and taken to separate rooms to be interrogated. If both are silent, they will be charged with carrying a concealed weapon and do one year in jail. If both confess, they will both do 10 years for the crime. If one is silent and the other confesses, the rat goes free and the silent one does 15 years. What should they do?

Any two street criminals would be aware that it is bad to rat out your partner. Both would remain silent, and end up doing a year. If they were game theorists, however each would reason as follows. ‘If my partner is silent, and I confess, I’ll do no time instead of the year I would do if remain silent. If my partner confesses, and I’m silent then I’ll do 15 years. If he confesses and I also confess, then I’ll only do 10 years. Confessing is a dominant strategy. No matter what the other person does, I do better to confess.’

Two dumb street thugs would do a year each, but two brilliant game theorists would do 10 years each. Think about that! This problem is often referred to as a paradox, since game theory leads to a such a hopeless outcome.

Let’s take the simple game of rock, paper and scissors. Paper covers (beats) rock, rock smashes (beats) scissors, and scissors cuts (beats) paper. Each choice beats one of the other two and loses to one of them. Unlike the prisoner’s dilemma, there is no simple solution that always leads to your best available outcome. Game theory recommends using a mixed strategy.

Mixed strategies involve choosing some combination of strategies. In this case, the optimal strategy is to randomly select an answer, so you play each of them (rock, paper or scissors) one-third of the time. When we get to the game theories applications to poker, we will see a lot of mixed strategies. Game theory optimal (GTO) players refer to this as having balanced ranges.

Again, a problem arises in both rock-paper-scissors and poker. If your opponents don’t follow a GTO strategy, then you can do much better by trying to exploit them. Let’s say your opponent never plays scissors. He plays only rock and paper. Then you should just play paper. You will win if he picks rock and tie if he picks paper, but never lose since he won’t pick scissors.

In the next column, we will look at a few more famous games that game theorists have studied. We will also discuss the important concepts of pareto optimality and a Nash equilibrium. ♠

Steve ‘Zee’ Zolotow, aka The Bald Eagle, is a successful gamesplayer. He has been a full-time gambler for over 35 years. With two WSOP bracelets and few million in tournament cashes, he is easing into retirement. He currently devotes most of his time to poker. He can be found at some major tournaments and playing in cash games in Vegas. When escaping from poker, he hangs out in his bars on Avenue A in New York City -The Library near Houston and Doc Holliday’s on 9th St. are his favorites.