Question 1
Consider a bilateral double auction with the buyer and seller types distributed independently of each other and uniformly on [0; 1]. Their no-trade payoff is zero. Suppose that prior to the auction they have to decide whether to participate. There is no cost of participation, but the traders have lexicographic preferences: First of all they care about expected payoffs, but in case of indifference they prefer not to participate in the auction.
1.) Assume that all the traders enter and then coordinate on playing the linear equilibrium derived in class/textbook (Gibbons). Is this a Perfect Bayesian Equilibrium? Is this a Bayesian Equilibrium? Why or why not?
2.) Now assume that those types who would not trade in the linear equilibrium with all types present do not enter. Does the auction with this reduced set of participants still have a linear equilibrium? If yes, describe it. If not, prove why not. (Hint: You need not derive the equilibrium from scratch. Just study the original linear strategies with care.)
3.) Disregarding the existence of types who do not participate, how does efficiency - measured (inaccurately!) as the proportion of type pairs who trade, divided by the proportion that "should" trade - change relative to the game where everyone enters?
4.) Show that for the sellers in [0, x) and the buyers in (y, 1] entering, the linear equilibrium bidding strategies will be the same, for any x and y in [0, 1].
5.) Hence, for what values of x and y will we get full efficiency (disregarding types who did not enter)?
6.) To what extent does this result agree/disagree with the basic intuition about the effect of asymmetric information on efficiency in the basic double auction model analysed in the book?