Centipede game
- For the video game, see Centipede (video game)
In game theory, the Centipede Game is a game with 100 turns. Each player in turn has the chance to stop the game and take the slightly larger share of a slowly increasing pot, or risk a short-term loss by passing the turn to his opponent. The payoffs are arranged so that if one passes the game to the opponent and the opponent takes the pot one receives slightly less than if one had taken the pot at that stage. Although the original centipede game had 100 stages (hence the name), any game with this structure but a different number of moves is called a centipede game. The unique Nash equilibrium of these games is for the first player to take the pot on the very first turn of the game; however in empirical tests relatively few players do so.
Equilibrium analysis
Determining that defection by the first player is the unique Nash equilibrium can be established by backward induction. Suppose we reach the final stage of the game, the last player will do better by defecting (and taking a slightly larger amount of the pot). Since we suppose the second player will defect, the first player does better by defecting in the second to last turn (giving him a slightly higher payoff than he would have received by allowing the second player to defect on the next move). But knowing this, the second player ought to defect on the third to last move and take slightly more than if she would have allowed the first player to defect on the next turn. This reasoning proceeds backwards through the game tree until one concludes that the best action is for the first to defect.
In the example pictured above, this reasoning proceeds as follows. If we were to reach the last stage of the game, Player 2 would do better by choosing d instead of r. However, given that 2 will choose d, 1 should choose D in the second to last move (since 1 would receive 3 instead of 2). However, given that 1 would choose D in the second to last round 2 should choose d in the third to last round since she would receive 2 instead of 1. But given this Player 1 should choose D in the first round giving him 1 instead of 0.
Empirical results
Several studies have demonstrated that the Nash equilibrium play is rarely observed. For examples see McKelvey and Palfrey (1992) and Nagel and Tang (1998). As in many other game theoretic experiments, scholars have investigated the effect of increasing the stakes. As with other games, for instance the ultimatum game, as the stakes increase the play approaches (but does not reach) Nash equilibrium play.
Significance
Like the Prisoner's Dilemma, this game presents a conflict between self-interest and mutual benefit. Since, if it could be enforced, both players would prefer that they both cooperate throughout the entire game. However, a player's self-interest or players' distrust can interfere and create a situation where both do worse than if they had blindly cooperated. Although the Prisoner's Dilemma has received substantial attention for this fact, the Centipede Game has received relatively less.
See also
References
- McKelvey, R. and T. Palfrey (1992) "An experimental study of the centipede game," Econometrica 60(4), 803-836.
- Nagel, R. and F.F. Tang (1998)."An Experimental Study on the Centipede Game in Normal Form - An Investigation on Learning," Journal of Mathematical Psychology, 42, 356-384.
External links