The Myerson - Satterthwaite Theorem

The Myerson-Satterthwaite Theorem (MS) is an impossibility result for bilateral bargaining. The original cite is Myerson and Satterthwaite (1983). The proof here most closely follows Krishna (2009) and Milgrom (2004). One person, the seller, owns one unit of an indivisible object. The seller values the object at c, which I refer to as the seller’s type. The other person, the buyer, values the object at v, which I refer to as the buyer’s type. Each knows her own type but not the other person’s. More generally, one can interpret the environment as one in which two people are bargaining over a proposed change in the status quo, with the “buyer” gaining v from the proposed change and the “seller” losing c. MS says that, subject to restrictions on the joint distribution of c and v, one cannot simultaneously have ex post efficiency, expected budget balance, and interim individual rationality. Ex post efficiency means that trade occurs when v > c (i.e., the buyer values the object more than the seller) but not when v < c. Expected budget balance means that while a third party (e.g., a judge) is allowed to provide a net subsidy for some type profiles (c, v) and collect a net tax for others, the third party’s net transfer must be zero in expectation over (c, v). A stronger requirement would be ex post budget balance, which requires zero net transfers for any (c, v). Interim individual rationality means that the buyer and seller can opt out of the bargaining game after learning own type (e.g., after seeing the good to be traded). Put differently, MS says that if one demands expected budget balance and interim individual rationality, then bargaining cannot be ex post efficient. This does not say that the outcome will be inefficient for all values of c and v, only that it will be inefficient for some values. For many standard bargaining models, the expected inefficiency can be large, on the order of a loss of 25% of expected surplus. One can construct games with equilibria that satisfy ex post efficiency and expected budget balance, and one can construct games with equilibria that satisfy ex post efficiency and interim