Non-cooperative bargaining when T is finite

To understand what will happen in this model of bargaining, consider the version of the game which has a finite terminal period, T, and suppose that the factory and the residents have identical discount factors, denoted by S. Suppose that T is even. The game can be solved by backward induction. Suppose that the parties enter the final period, period T, with no agreement locked in. Then, since T is even, the factory has the right to make an offer in this final period. Since the game ends in period T, the factory's offer must be accepted by the residents. Thus the factory has all of the bargaining power, and would propose the split that is most advantageous to it, which is nF = n,nR = 0. Since this occurs in the final round, the offer must be accepted by the residents, since they have no choice according to the rules of the game.

Now roll back to period T - 1. The residents know that they must offer the factory at least Sp, otherwise the factory will reject the offer and wait until the last period, where they receive p. Thus the residents offer [,(1 -S)k and the factory accepts. Knowing this, and rolling back to period T - 2, the factory will know that the residents will only accept an offer at least as big as 5(1 -8)n. Thus the factory offers {[1 - 5(1 - 5)]n, 5(1 - 5)n} and this is accepted by the residents.

The pattern is now clear. In period 0, the factory gets to make an offer. It constructs an offer which gives the residents the discounted value of the payoff that they expect to get in the next period, which is:

and it keeps for itself:

To express these formulae more compactly, note that:

Define SE = 1+52 + 54 +... + 5T and So = 5 + 53 + 55 +... + 5T-1. Then S = SE + SO. Finally, note that:

and so:

Now the factory's payoff is: and the residents' payoff is:

Since these payoffs have been constructed in such a way that the factory and the residents are willing to accept these payoffs as soon as the game begins, the equilibrium also involves both parties agreeing immediately.

Even though the bargaining rules allow the parties to 'haggle', the parties do not actually do any haggling in equilibrium. This does not mean, however, that rules governing haggling do not matter. The fact that offers and counteroffers are permitted under the rules of the game, combined with the fact that both parties are impatient, leads to an equilibrium in which an offer is accepted immediately. But the payoffs in (11.5) and (11.6) obviously depend on the length of the bargaining process. For example, if the bargaining rules stipulated that the game must be a 'one-shot' offer and acceptance game, with no possibility of counteroffers, then the party that is allowed to make the initial offer will obtain the entire surplus. Clearly, then, bargaining institutions and rules matter for bargaining outcomes.

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