Legal disputes and non-cooperative bargaining theory
The non-cooperative bargaining approach models a situation in which offers and counteroffers can be made, as they would be in a 'real world' bargaining situation between parties haggling with each other during a legal dispute. The standard non-cooperative setup is as follows.
Suppose that there are two parties to a legal dispute. Time is discrete, and is indexed by t = 0,1,2,...,». The size of each time period (the time between offers and counteroffers) is denoted by A > 0, which can be interpreted as the absolute size of the frictions in the bargaining process, or the time taken between each round of the bargaining process. As Д approaches zero, these frictions disappear.
Bargaining proceeds as a game of alternating offers. To apply the framework, let us return to our earlier two-party example from Chapter 3, where the two parties are the factory (F) and the residents (R). The parties bargain over a surplus, which is the dollar value of the gains from trade. For convenience, we denote this surplus by n > 0. Dispute resolution is assumed to take place according to the following procedure. In even periods t = 0,2,4,...,~ the factory makes an offer to the residents, where an offer is a proposal pair (nF ,n-nF) to share the gains from surplus. The residents R can either accept this offer, in which case the game ends; or they can reject F's offer, in which case the game moves to the next period. In odd periods t = 1,3,5,...,» the roles of the players are reversed. If a player receives a share of the surplus xt in period t, then the benefit from that share is:
where rt > 0 is the discount rate of player i. Let St = e~hA be player i's discount factor. This reflects the value that the party places on future dollars, relative to current dollars. A high discount factor (or a low discount rate) means that the party places a relatively high value today on dollars to be received in the future.