Thoughts on Game Theory: Part III

In this column I want to discuss a game referred to by game theorists as Chicken. The basic premise of the game is that two drivers head toward each other at full speed. The first one to swerve is the ‘chicken,’ a cowardly loser. However, if neither swerves the head-on crash is disastrous for both. (For old movie buffs, there is a version of this game played in the classic film Rebel Without a Cause, starring James Dean and Natalie Wood.)

In game theory, it is common to assign numbers to the outcomes, and show the results in a table or matrix format. Let’s say that winning has a score of + 10. Thus, the loser scores – 10. The win occurs when your opponent swerves and you don’t. There are also two possible ties. Both players swerve, and thus are both considered cowards, but neither has the glory of a win. In this case assume both score – 5. If neither swerve there is a terrible crash. Assign this a score of – 500. Both players are left with the following possibilities victory + 10, defeat – 10, tied cowardice – 5, and crash – 500. The payoff table looks like this:

Swerve Straight
Swerve -5,-5 -10,+10
Straight +10,-10 -500,-500

Is there a game theory optimal strategy? You might think that a rational opponent knows there is a huge risk to not swerving, and will swerve. Therefore, you don’t have to swerve and will win. If, however, he thinks the same way, there will be a terrible crash. If the numbers accurately reflected your view of reality, you could calculate a mixed strategy that would include occasionally going straight.

But suppose you decide to use psychology to intimidate your opponent. You tell him you will never swerve. This will work only if he believes you and doesn’t respond by telling the exact same thing. You need to find a way to convince him that you really won’t swerve. One game theorist suggested starting toward him and throwing your steering wheel out the window. Now he knows you can’t swerve. This is known as pre-commitment. Variations of this game have been applied to analyze everything from nuclear war to animal conflicts. Now let’s see how it might be applied to a very common poker situation.

You are at the bubble or approaching a pay jump in a poker tournament. To simplify the discussion, assume the following payouts: First gets 54, second gets 46 and third gets nothing. The players have chip stacks of 100, 100 and 1. Both of the players with 100 want to avoid finishing third, and getting nothing.

The player with 1 chip is on the button, and the blinds are 5 and 10. The short stack acts first, and folds. You are the small blind, what should you do? If you fold, you will be left with 95, while the other players will be left with 105 and 1. The short stack will probably be broke soon. The large stack will have an advantage over you, since he’ll be ahead 105 to 95.

Suppose, however, you decide to play chicken, and raise to 50. Now if he folds, you will have 110 and he will only have 90. But he may decide to see if you are really committed by jamming. Now if you call, one of you will be in the embarrassing position of getting nothing while the player with one chip sneaks into second. If you fold, which is almost certainly better than calling, the chip counts will be 150, 50 and 1.

Now you are a big underdog to finish first. Therefore, you should have made a pre-commitment and jammed yourself. The other big stack should automatically fold. If he actually has aces, and thinks you have suited connectors, he will win less than 80 percent of the time. Thus, by calling he increases his chance to win the extra eight chips, but he will also get nothing around 20 percent of the time. His equity is now .8 times 54 or 43. By calling, he will average less than he would get by finishing second.
In real life situations, with a number of players and a variety of payoffs, calculations get very complicated. The Independent Chip Model can be used to approximate equities for different strategies, but it makes assumptions of equal skill and position that may not be accurate. You probably already knew that the big stack can often jam or make a raise that appears to pre-commit him to going all-in, and steal the pot from intermediate stacks that don’t want to go broke ahead on the short stacks. Now you know you been using game theory. ♠

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.