Thoughts on Game Theory: Part Five

John Forbes Nash, Jr. was a fascinating character. He made brilliant contributions to several branches of mathematics including geometry and game theory. He also suffered from mental illness. He is the subject of the biography, A Beautiful Mind, made into a movie with same title.

He is best known in game theory for his work on equilibria in multi-player games. A Nash equilibrium is a situation in which no player can benefit from changing his strategy unilaterally. A Nash equilibrium is frequently not optimal. This occurs when two or more players can simultaneously change strategies producing a better outcome, even though one player alone couldn’t gain by changing.

Let’s look at a simple example. The local poker club has two very strong players, Harry and Pete. Harry is best at hold’em while Pete’s strongest game is PLO. Each does best by playing his best game at one table while the other plays elsewhere.

Harry can average $150 per hour in a hold’em game without Pete, but only $100 if Pete is there. Pete can average $150 per hour in a PLO game without Harry, but only $100 if Harry is there. Worse yet, they can only average $80 playing their worst game at the same table. They can also each average $110 per hour playing without the other in that game. Let’s put this into a table which lists Harry’s results first and Pete’s second:

Pete Harry NLH PLO
NLH 100,80 150,150
PLO 110,110 80,100

One day Pete arrives first, and he starts a hold’em game. He will make $110 per hour. A little later, Harry arrives and helps the club start a PLO game, where he will also make $110. They are in the box on the lower left. This is a Nash equilibrium, since neither player can do better by changing games on his own. If, however, they agree to switch games then they can achieve their optimal outcome of $150 as shown in the box on the top right.

Here is another example of a suboptimal Nash equilibrium. A casino guarantees $1 million in prizes for a tournament with an $1,100 entry fee. Since the casino keeps $100, they reach the guarantee when there are 1,000 or more entries. 1,500 entries are sold. This means that each player is contributing $1100 for $1000 in equity. This is a Nash equilibrium. Nothing any one player does, improves the situation. Strategically the players could do much better by forming pairs, and letting one player play for the pair. Now there would only be 750 entrants. These entrants would have paid the same $1100, but now their equity would be $1330.

Pete and Harry agreeing to switch games in the first example would be perfectly legal. The second strategy (half the players not entering, but sharing profits) would require the players to collude, and might give the casino reasonable grounds for canceling the tournament.

There are a number of other poker spots where illegal collusive behavior would benefit the players colluding. Luckily, there are some areas of life where collusive or cooperative behavior is acceptable, very desirable and leads to a useful Nash equilibrium. The classic example of this is choosing which side of the street to drive on. As long as all local drivers chose the same side, a Nash equilibrium is reached. In America we settled on the right side of the road, while the English decided to drive on the left. Thus, the Nash equilibria are either all players chose right or all chose left.

In the next couple of columns, I’ll look at some aspects of game theory that are extremely useful to poker players. High on this list is situations involving appropriate frequencies for bluffing and value betting. ♠

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.