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.
