The Two Person Bargaining Problem

Note: This is a only a draft version, so there could be flaws. If you find any errors, please do send email to [email protected]. A more thorough version would be available soon in this space. The Nash bargaining problem represents one of the earliest and most influential results in cooperative game theory. Given two rational and intelligent players and a set of feasible allocations from among which a unique allocation is to be chosen, the Nash bargaining theory proposes an elegant axiomatic approach to solve the problem. This chapter describes the problem and proves the Nash bargaining result. Cooperation refers to coalitions of two or more players acting together with a specific common purpose in mind. Since rationality and intelligence are two fundamental assumptions in game theory, any cooperation between players must take into account the objective of maximizing their own individual payoffs. As we have seen in the previous chapter, the notion of cooperation which is closely tied with the notion of correlated strategies can be developed without abandoning the individual decision theoretic foundations underlying game theory. This has been emphasized by John Nash himself [1, 2]. According to Nash, cooperative actions can be considered as the culmination of a certain process of bargaining among the cooperating players and consequently, cooperation between players can be studied using core concepts of non-cooperative game theory. In this bargaining process, we can expect each player to behave according to some bargaining strategy that satisfies the original utility-maximization criterion as in standard game theory. The ingenious idea of Nash is to define a cooperative transformation that will transform a strategic form game into another strategic form game that has an extended strategy space for each player. The extended strategy set for a player has all the strategies of the original game and also additional strategies that capture bargaining with the other players to jointly plan cooperative strategies. This is on the lines of what we have studied in the previous chapter on correlated strategies. We will illustrate this with the standard example of the prisoner's dilemma problem which provides a classic example 1