gameL11

# gameL11 - Myerson and Satterthwaite, "Efficient Mechanisms...

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Myerson and Satterthwaite, "E cient Mechanisms for Bilateral Trading," JET 1983 In many bilateral bargaining situations with asymmetric information, ex post e ciency is inconsistent with incen- tive compatibility and individual rationality. One can compute the highest expected surplus consistent with IC and IR. If the equilibrium to a bargaining game yields that surplus, then the game constitutes an optimal mechanism. Player 1 (seller) owns an indivisible object and has valua- tion distributed according to the continuous and positive density function, f 1 ( v 1 ) , over the support [ a 1 ,b 1 ] .A l so denote the distribution function as F 1 ( v 1 ) . Player 2 (buyer) has valuation distributed according to the continuous and positive density function, f 2 ( v 2 ) ,over the support [ a 2 ,b 2 ] . Also denote the distribution function as F 2 ( v 2 ) . Types are assumed to be

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The two (risk neutral) players observe their valuations and enter into a bargaining game. Assume that partici- pation is voluntary (IR). The outcome speci f es whether the object is sold and if so, at what price. An implicit as- sumption is that the players cannot credibly "prove" their valuations to the other player. You can always pretend to have a di f erent valuation (type). What kinds of mechanisms are most e cient? A direct bargaining mechanism is a game in which each player reports his type to the referee, and the referee chooses whether the object is transferred, and how much the buyer must pay. As opposed to an indirect mechanism that might be closer to what we think of as bargaining. For example, the buyer and seller name a price; if the bid price is above the ask price, the object is sold for the average of the two prices, and if the ask price is above the bid price, the seller retains the object.
Let p ( v 1 ,v 2 ) denote the probability that the object is transferred, given the seller reports v 1 and the buyer re- ports v 2 . Let x ( v 1 ,v 2 ) denote the expected payment from the buyer to the seller, given the seller reports v 1 and the buyer reports v 2 . Because of risk neutrality and separability, it does not matter whether the buyer sometimes pays without pur- chasing. A direct mechanism is Bayesian incentive compatible (IC) if honest reporting forms a Bayesian equilibrium.

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The Revelation Principle: For any Bayesian equilibrium of any bargaining game, there is an equivalent incentive- compatible direct mechanism yielding the same outcome (when the honest equilibrium is played). In this sense, it is without loss of generality to restrict attention to direct mechanisms. "proof" Start with an equilibrium of the indirect mech- anism. De f ne the direct mechanism as follows. Each player reports his type, and the referee computes what actions would have been chosen in the indirect mecha- nism, and what the resulting outcome would have been. This outcome is selected in the direct mechanism. Since
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## This note was uploaded on 07/17/2008 for the course ECON 817 taught by Professor Peck during the Fall '07 term at Ohio State.

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gameL11 - Myerson and Satterthwaite, "Efficient Mechanisms...

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