The Revelation Principle

The strategic interactions among the designer of a mechanism and its participants can be modeled as a multi-stage game. The designer moves first by selecting a mechanism. The participants observe the mechanism, and move thereafter. For ease of exposition, we restrict our present attention to a two-stage game, in which the participants play a simultaneous-move (i.e., one-shot) game in the second stage. After the designer announces their choice of mechanism, the agents face a game of incomplete information.1 Formally, such a 1 Consist with our earlier comment, we restrict our present attention to one-shot games of incomplete information. That said, we aim to apply our methodology to extensive-form games of incomplete information. game is denoted by Γ = [Ω, [n], {Ai}i∈n, {Ti}i∈n, g, {ui}i∈n], where [n] is the set of players (or agents), of size n; Ai is the set of actions available to player i ∈ [n], with A = ∏ i=1 Ai as the joint action space; and Ti is the set of types (private information) available to player i ∈ [n], with T = ∏ i=1 Ti as the joint type space. Additionally, a joint distribution F over types is assumed to be common knowledge, known to both the players and the designer. We define a strategy of a player i as a function si : Ti → Ai, and use st to denote the vector (s1(t1), . . . , sn(tn)). The function g : A → Ω maps a joint action profile into a space Ω of possible outcomes; that is, g(s(t)) is the outcome when player i of type ti plays strategy si and the remaining players of type t−i play strategy s−i. Finally, 2 For example, the outcome function of an auction maps a profile of bids to a pair comprised of an allocation and a payment rule. player i’s utility ui : Ω × T→ R depends on both the outcome of the game and (in general) all players’ types. Given a game Γ, its solution is a joint strategy profile s∗ that the players are predicted to play under certain assumptions. For example, dominant-strategy equilibria when they exist, and otherwise the Bayes-Nash equilibrium solution concept, are often applied to solve a game. We define both concepts in our current formalism, presently.