a seller S and a buyer B. S owns a used car that B would like to acquire. A car of quality q is worth q to S and q + v to B, with 0 < v < 1. S knows the quality of her car, while B only knows that q is drawn from a uniform distribution on [0, 1]. All of this is common knowledge. Bargaining between the players takes the following simple form: B makes a take-it-or-leave-it offer b which S can accept or reject. If S accepts, trade takes place at price p = b. If S rejects, no trade takes place. If a car of quality q is traded at price p, then the seller’s payoff is uS = p and the buyer’s payoff is uB = q + v - p. If no trade takes place, then uS = q and uB = 0.
Model this situation as a Bayesian game. What are the type spaces and action spaces of the two players?
Define what a strategy is for each player.
Derive the seller’s optimal strategy. Show that it does not depend on the buyer’s strategy.
Determine the Bayesian Nash equilibria of the game