Consider the bilateral trade problem in which the seller's cost c and the buyer's valuation v are independent random draws from a distribution that is uniform on the interval [0, 1]. The goal of this exercise is to determine if there exists a direct mechanism that implements the efficient outcome in (Bayes) Nash equilibrium.
More precisely, buyer and seller play a game in which they report their valuation. The rules of the game are as follows: if the seller reports c' and the buyer reports v', the object is transferred from seller to buyer with probability p(c',v'), and the buyer makes the transfer t(c',v'). This transfer is to be paid regardless of whether or not the object changes hands. This type of game is called a revelation game or a direct mechanism.
The question at hand is then whether it is a Nash equilibrium for the buyer and the seller to report their valuation truthfully in a revelation game where p(·, ·) is the efficient trading rule.

(a) Define the efficient trading rule p(c, v).

(b) Compute the expected probability of trade of seller type c under this trading rule.