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.

Bargaining games refer to situations where two or more players must reach agreement regarding how to distribute an object or monetary amount. Each player prefers to reach an agreement in these games, rather than abstain from doing so; however, each prefers that agreement which most favours his interests. Examples of such situations would be the bargaining involved in a labour union and the directors of a company negotiating wage increases, the dispute between two communities about the distribution of a common territory or the conditions under which two countries can start a programme of nuclear disarmament. Analyzing these kinds of problem looks for a solution specifying which component in dispute will correspond to each party involved.

Players in a bargaining problem can bargain for the objective as a whole at a precise moment in time. The problem can also be divided so that parts of the whole objective become subject to bargaining during different stages.

In a classical bargaining problem the result is an agreement reached between all interested parties, or the status quo of the problem. It is clear that studying how individual parties make their decisions is insufficient for predicting what agreement will be reached. However, classical bargaining theory assumes that each participant in a bargaining process will choose between possible agreements, following the conduct predicted by the rational choice model. It is particularly assumed that each player's preferences regarding the possible agreements can be represented by a von Neumann-Morgenstern utility function.

Nash [1950] defines a classical bargaining problem as being a set of joint allocations of utility, some of which will correspond to that the players would obtain if they reach an agreement, and another which represents what they would get if they failed to do so.

A bargaining game for two players is defined as a pair (F,d) where F is the set of possible joint utility allocations (possible agreements), and d is the disagreement point.

For the definition of a specific bargaining solution is usual to follow Nash's proposal, setting out the axioms this solution should satisfy. Some of the most frequent axioms used in the building of bargaining solutions are efficiency, symmetry, independence of irrelevant alternatives, scalar invariance, monotonicity, etc.

The Nash bargaining solution is the bargaining solution which maximizes the product of agent's utilities on the bargaining set.

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.