# A Perfect Bayesian Bargaining problem

Thought I'd post interesting questions that come my way to try and boost activity!

A seller and a buyer negotiate over trade of a single indivisible good. The good is either of low quality (in which case its value is 0 to the seller and $0.5$ to the buyer) or high quality (in which case its value is $1$ to the seller and $1.5$ to the buyer). The seller knows the quality but 2 the buyer knows only that the probability that the quality is high is $θ$ (0 < $θ$ < 1).

Consider a two-period bargaining game. If the buyer’s first- period offer is rejected, the buyer makes a second price offer in period 2. If the second offer is rejected, there is no trade. Both players discount the future slightly.

Assume that in each period the only offers permitted are $0.5$, $1$ and $1.5$ (the buyer could also decide not to make any offer).

a) Define Perfect Bayesian Equilibrium in this context.

b) Does there exist a pure strategy perfect Bayesian equilibrium in which the first price offer is $0.5$?

c) Under what conditions does a pure strategy equilibrium exist?

• This looks an awful lot like a homework question with no effort shown. May 4 '15 at 20:58
• Two questions, under what conditions does the seller accept an offer (equal to or greater than, or strictly greater than his value)? and Do buyer and seller know each others' values? May 6 '15 at 15:00
• I) Greater than or equal to. II) Yes, I think we should assume that the payoffs of the game are perfect information. May 6 '15 at 15:58