Mechanism Design

Mechanism Design under Quasi-Linear Preferences

Green & Laffont (ECMA, 1977), Characterization of Satisfactory Mechanisms for the Revelation of Preferences for Public Goods

Proved that Groves Mechanism under monotonic preference and separable utility is the ONLY mechanism that can satisfy incentive compatibility and reaches pareto optimum. Show that well-defined mechanisms which select Pareto optimal outcomes (referred to as satisfactory mechanisms), independently of the question of truthful revelation, are isomorphic to the mechanisms proposed by Groves (solution sets are the same solution to the same equation).

Reduced Form Implementability

Implementability: Given reduced form auction Q (overall prob. that agent of type t gets the object), determine whether there exists an symmetric auction q (prob. of agent i gets the object given others’ type)

Matthews (ECMA, 1984) On the implementability of reduced form auctions

For i.i.d. environments with symmetric auctions, Matthews (1984) conjectured that the only restriction on for implementability was the necessary condition that the probability that the “winner” had a type in the a subset A could not exceed the probability that there was a bidder with type in A (the MRM condition).

Maskin & Riley (ECMA, 1984) Optimal auctions with risk averse buyers

Theorem 7 proved something like this result for increasing step functions on the unit interval. Their proof is long, tedious, and unintuitive. Matthews extended their result to general increasing functions on the unit interval, and conjectured this form of the theorem.

Border (ECMA, 1991) Implementation of Reduced Form Auctions: A Geometric Approach

Proved Mattewss’ conjecture for general abstract measure spaces of types, which need not have an order, so the notion of increasing need not be defined.

Reduced form links bidder’s type to probability of winning in the gamble, ignoring the details of the auction (order, bid, etc).

Direct form: strategy of bidders. Full form: bid + deciding rule + payment.

Hart & Reny (ETB, 2015) Implementation of Reduced Form Auctions: A Simple Approach and a New Characterization

Introduced the criterion of stochastic dominance as a criterion of implementability, based on Border, K. (2007).

Border (ET, 2007) Reduced Form Auctions Revisited

If P is a reduced form, then it satisfies the MRM (Maskin-Riley-Matthews-Border) condition.

total expected allocation weight (or winning probability) across all N bidders whose types belong to the subset A <= probability that at least one bidder has a type in A.

Total winning probability for bidders in A does not exceed the available “allocation capacity” which depends on the likelihood of at least one bidder having a type in A.

This constraint ensures that the auction is feasible and respects the rules of probability and allocation.

Bayesian Incentive Compatibility (BIC) and Dominant Strategy Incentive Compatibility (DIC)

*Manelli & Vincent (ECMA, 2010) Bayesian and Dominant Strategy Implementation in the Independent, Private Values Model

Under private, independent valuation and linear utility, BIC mechanism can be achieved by DIC mechanism, generating the same probability of transaction and expected payment.

DIC: reporting truthfully is unconditionally optimal (always better than any other report, no matter how others behave), and Weakly Dominant Strategy Equilibrium.

BIC: conditional on others report truthfully, reporting truthfully maximizes agent’s expected utility given true value.

*Moldovanu et al. (2011) Bayesian and Dominant Strategy Implementation Revisited

Matching

Risk

Chiappori & Reny (TE, 2016) Matching to Share Risk: Matching to Share Risk

The paper proves that, under nontransferable utility, a stable matching always exists, it is generically unique, and it is negatively assortative. It complements the general assumption that utility can be transfered between individuals, but not necessarily at a constant “exchange rate” as implied by TU. The result can be interpreted that, high-skill individuals may be attracted to lowskill individuals and vice versa not only for risk-sharing reasons, but also because specialization makes it possible to avoid counterproductive effort. When selecting a mate, a female must take into account two conflicting considerations. On the one hand, she may prefer a less risk averse partner because he is willing to accept a larger share of the income risk. On the other hand, she may prefer a more risk averse partner because he is willing to accept a smaller amount of joint income as long as it is not too uncertain.