The ultimatum game

The ultimatum game is another a two-player noncooperative game where two players attempt to divide a good, again, say a cake, between them. However, we assume that one player (the proposer) has sole possession of the cake and offers a certain amount of the cake to the second player (the receiver), keeping the rest for himself. The second player has only two choices: take the offer or leave it. If player two takes the offer, each player receives the amount of cake due. If player two chooses to leave it, each player receives nothing. Compared to the Nash bargaining game, the ultimatum game has a significantly larger strategy space. Each strategy has two components, prescribing what demand the player will make as a proposer and what demands the player will accept as a receiver. If the cake divides into N pieces and we forbid purely altruistic behavior (demanding nothing) and completely greedy behavior (demanding everything) the game has 2^(N-1)*(N-1) possible strategies. Most treatments of the ultimatum game consider only a small subset of the possible strategies.

According to von~Neumann-Morgenstern game theory, if the good can divide into infinitely many pieces, an infinite number of Nash equilibria exist. When talking about the ultimatum game, though, it proves fruitful to use another solution concept, that of subgame perfection. We say an equilibrium is subgame perfect if the strategies present in that equilibrium are also in equilibrium when restricted to any subgame. Consider a population of players where all make fair offers (half of the cake) and only accept fair offers, a strategy typically called "Fairman." Although this strategy is a Nash equilibrium (no player can do better by changing her strategy), it is not subgame perfect: in a mixed population containing players of all strategies, Fairman does not do as well as the strategy which makes a fair offer but accepts any offer. Consequently, if one thinks a credible equilibrium of a game must be subgame perfect, the number of credible equilibria shrink. If players act to maximize expected utility, then proposers should demand the entire cake minus epsilon (if the cake is infinitely divisible) or N-1 pieces (if the cake has N pieces). Receivers, on the other hand, should accept any nonzero offer.