Bargaining Game

Much of economic theory is concerned with the process and conditions under which individuals or firms maximize their own benefits or minimize their own costs in markets in which their individual actions do not materially influence others (perfect competition). There are, however, many cases in which economic decisions are made in situations of conflict, where one party's actions induces a reaction from others. An example is wage bargaining between employers and unions. A more simple case is the of duopoly, in which the price set by one seller will be based on his view of that set by the other in reply. The mathematical theory of games has been applied to economics to help elucidate problems of this kind.

The Nash bargaining game

The Nash bargaining game is a two-player noncooperative game where two players attempt to divide a good, say a cake, between them. Each player requests an amount of the cake. If their requests are compatible, each player receives the amount requested; if not, each player receives nothing. The simplest form of the Nash bargaining game assumes the utility function for each player to be a linear function of the amount of cake they get. (We may assume the utility functions of the players are equal since utility functions are determined only up to a nonnegative multiplicative constant and a constant term.) According to traditional game theory, an infinite number of Nash equilibria exist for this game. Given any request, the corresponding strategy of the equilibrium pair simply requests the remainder of the cake. If the first person did not request the entire cake for herself, we have a strict Nash equilibrium. If the first player did request the whole cake, the equilibrium is not a strict Nash equilibrium since the second player receives the same amount regardless of what she demands. (If she makes her equilibrium demand of 0, then player 2 receives nothing. However, if player 2 makes any nonzero demand, she will still receive the same amount, namely nothing, because any nonzero demand will push the total sum of demands greater than the amount of cake available.) If both players act to maximize expected utility, traditional game theory dictates each should demand half. Intuitively, this appears not only as the rational thing to do (`rational' meaning maximizing personal expected utility), but also as the "fair" thing to do.

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.