Recall the bargaining game from part (b) of this chapter's Guided Exercise, where player 1 is interested in selling a television to player 2. Player 2's value of owning the television, v, is privately known to player 2. Player 1 only knows that v is uniformly distributed between 0 and 1. Consider a variation of the game in which player 1 makes the offer in both periods, rather than having alternating offers as in the Guided Exercise. That is, in the first period, player 1 offers a price p1 .
If player 2 rejects this offer, then play proceeds to the second period, where player 1 makes another offer, p2 . Assume, as before, that the players discount second-period payoffs by the factor d.
Calculate the perfect Bayesian equilibrium of this game. Here are some hints to help you with the analysis.
In the perfect Bayesian equilibrium, player 2 uses a cutoff rule in the first period. The cutoff rule is characterized by a function c : [0, 1] S [0, 1], whereby player 2 accepts an offer p1 if and only if v Ú c(p1).
Thus, whatever player 1 offers in the first period, if the offer is rejected, then player 1's updated belief about player 2's type is that it is uniformly distributed between 0 and c(p1). In other words, because player 1 knows that any type v Ú c(p1) would have accepted p1 , that the offer was rejected tells player 1 that player 2's valuation is between 0 and c(p1).
This is Bayes' rule in action. Begin your analysis by determining player 2's optimal behavior in the second period and then calculating player 1's optimal second-period price offer when facing the types in some interval [0, x]. Then try to determine the function c. In this regard, the key insight is that type c(p1) will be indifferent between accepting p1 and waiting for the offer in period 2.