Mechanism Design (New Developments)

Abstract

Game theory provides methods to predict the outcome of a given game. Mechanism design concerns the reverse question: given some desirable outcome, can we design a game which produces it? Formally, the environment is <A, N, Θ>, where A is a set of feasible and verifiable alternatives or outcomes, N = {1,...,n} is a set of agents, and Θ is a set of possible states of the world. Except where indicated, we consider private values environments, where a state is θ = (θ₁,..., θ _n ) ∈ × _i Θ _i = Θ, each agent i knows his own ‘type’ θ _i ∈ Θ _i , and his payoff u _i (a, θ _i ) depends only on the chosen alternative and his own type. (This does not rule out the possibility that the agents know something about each others’ types.) If values are not private, then they are said to be interdependent. A mechanism or contract Γ = (S, h) specifies a set of feasible actions S _i for each agent i, and an outcome function h : S A. An outside party (a principal or social planner), or the agents themselves, want to design a mechanism which produces optimal outcomes. These are often represented by a social choice rule (SCR) F : Θ → A. A social choice function (SCF) is a single-valued SCR. Implicitly, it is assumed that the mechanism designer does not know the true θ, and this lack of information makes it impossible for her to directly choose an outcome in F(θ). Instead, she uses the more roundabout method of designing a mechanism which produces an outcome in F(θ), whatever the true θ may be.