# 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 *(n _{F} ,n-n*

_{F}) 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

*x*

*in period t, then the benefit from that share is:*

_{t}

where *r** _{t} >* 0 is the

*discount rate*of player i. Let

*S*

_{t}=*e*

*~*be player i's

^{hA}*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.