Games and Economic Behavior
Privacy and trust affect our strategic thinking, yet they have not been precisely modeled in mechanism design. In settings of incomplete information, traditional implementations of a normal-form mechanism -by disregarding the players' privacy, or assuming trust in a mediator-may fail to reach the mechanism's objectives. We thus investigate implementations of a new type. We put forward the notion of a perfect implementation of a normal-form mechanism M: in essence, a concrete extensive-form
... nism exactly preserving all strategic properties of M, without relying on a trusted mediator or violating the privacy of the players. We prove that any normal-form mechanism can be perfectly implemented by a verifiable mediator using envelopes and an envelope-randomizing device (i.e., the same tools used for running fair lotteries or tallying secret votes). Differently from a trusted mediator, a verifiable one only performs prescribed public actions, so that everyone can verify that he is acting properly, and that he never learns any information that should remain private. * We would like to thank participants to numerous seminars, and especially Ran Canetti, Dino Gerardi, Sergiu Hart, Bart Lipman, and Dave Parkes, for their comments. We are grateful to the NSF grant SES-0551244 for financial support. The Goal of Maximum Privacy (and Minimum Trust) Among auctions in the private-value setting, the famous second-price sealed-bid mechanism is a normal-form mechanism whose reports consist of numerical bids -one for each player-and whose outcome function returns the player with the highest bid as the winner, and the value of the second-highest bid as the price. The characteristic property of this mechanism is economic efficiency achieved in dominant strategies. Indeed, because it is optimal for each player to report his true valuation no matter what the others may report, one can expect that the player with the highest valuation wins the item for sale. We shall use this mechanism to illustrate the problems of privacy and trust when abstract normal-form mechanisms are concretely implemented with the help of a mediator. Trusted vs. Verifiable Mediators. Consider the following two mediated implementations, M 1 and M 2 , of the second-price sealed-bid mechanism. M 1 (With a Trusted Mediator): The players seal their bids into envelopes and hand them to the mediator, who then privately opens them, privately evaluates the outcome function, and finally publicly announces the winner and the price. M 2 (With a Verifiable Mediator): The players seal their bids into envelopes and hand them to the mediator, who then publicly opens all of them, so as to enable everyone to compute the winner and the price. Notice that the mediators of these two implementations are asked to perform different types of actions. The mediator of M 1 is asked to perform private actions, so that only he knows whether he has performed the right actions. By contrast, the mediator of M 2 only performs public actions, so that everyone can verify that the right actions have been performed. Accordingly, these two implementations are at opposite ends with respect to privacy and trust. On the one extreme, implementation M 1 reveals nothing more than the announced outcome, but requires total trust in the mediator. Indeed, the players cannot verify whether the mediator announces the correct outcome, nor whether he will keep all bids secret after the auction is over.