Saturday, 7 March 2009

Rationality & Game Theory - The Centipede Game & Chess Players



"Game theory is a branch of applied mathematics that is used in the social sciences (most notably economics), biology, engineering, political science, international relations, computer science (mainly for artificial intelligence), and philosophy." (Wikipedia),

The centipede game (introduced by Rosenthal, 1981) is one mathematical game which tests the conflicts between self-interest and mutual benefit in human interactions. The traditional centipede game had a limit of 100 rounds (hence the name). (see Wikipedia)

Ignacio Palacios-Huerta and Oscar Volij have recently published a study looking at the behaviour of skilled chess players playing the centipede game. (pdf link here)
The centipede game is perhaps the best example of what is known as “paradoxes of backward induction.”These paradoxes involve sequential games all of whose correlated equilibria, and a fortiori all its Nash equilibria, imply a very counterintuitive play.

A particular instance of the centipede game can be described as follows. A pile of $4 and a pile of $1 are lying on a table. Player I has two options, either to “stop” or to “continue.”If he stops, the game ends and he gets $4 while Player II gets the remaining dollar. If he continues, the two piles are doubled to $8 and $2, and Player II is faced with a similar decision: either to take the larger pile ($8), thus ending the game and leaving the smaller pile ($2) for Player I, or to let the piles double again and let Player I decide. The game continues for at most six rounds. If by then neither of the players have stopped, Player I gets $256 and Player II gets $64. Figure 1 depicts
this situation.

Palacios-Huerta and Volij found that the chess players were far more likely to play optimally than non-chess playing students. Amongst chess players, grandmasters always played optimally and took the $4. This increased rationality (hyper-rationality) amongst chess players has the disadvantage that they are less likely to share a larger reward through cooperation (suboptimal game strategy). I find this study very interesting. One interpretation of the results is that chess grandmasters may be more sociopathic or selfish than the average person (see chart above), i.e. by being more rational and less morally empathic to others.


CHESS PLAYERS PROPORTION OF OBSERVATIONS AT EACH TERMINAL NODE
BY TYPE OF PLAYER I IN THE PAIR

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