How Economists Define the Revelation Principle

A Look at the Revelation Principle in Game Theory and Bayesian Games

Pinky promise
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The revelation principle of economics is that truth-telling, direct revelation mechanisms can generally be designed to achieve the Bayesian Nash equilibrium outcome of other mechanisms; this can be proven in a large category of mechanism design cases. Put into other words, the revelation principle holds that there is a payoff-equivalent revelation mechanism that possess an equilibrium in which players truthfully report their types to any Bayesian game.

Game Theory: Bayesian Games and Nash Equilibrium

A Bayesian game has the most relevance in the study of economic game theory, which is essentially the study of strategic decision-making. A Bayesian game in one in which the information about the characteristics of the players, otherwise known as the player's payoffs, is incomplete. This incompleteness of information means that in a Bayesian game, at least one of the players is uncertain of the type of another player or players.

In a non-Bayesian game, a strategic model is considered a if every strategy in that profile is the best response, or the strategy that produces the most favorable outcome, to every other strategy in the profile. Or in other words, a strategic model is considered a Nash equilibrium if there exists no other strategy that a player could employ that would produce a better payoff given all the strategies chosen by the other players.

A Bayesian Nash equilibrium, then, extends the principles of the Nash equilibrium to the context of a Bayesian game which has incomplete information. In a Bayesian game, Bayesian Nash equilibrium is found when each type of player employs a strategy that maximizes the expected payoff given the actions of all the types of other players and that player's beliefs about the types of the other players.

Let's see how the revelation principle plays into these concepts.

Revelation Principle in Bayesian Modelling

The revelation principle is relevant to a modelling (that is, theoretical) context when there exists:

  • two players (usually firms)
  • a third party (usually the government) managing a mechanism to achieve a desirable social outcome
  • incomplete information (in particular, the players have types that are hidden from the other player and from the government)

Generally, a direct revelation mechanism (in which telling the truth is a Nash equilibrium outcome) can be proven to exist and be equivalent to any other mechanism available to the government. In this context, a direct revelation mechanism is one in which the strategies are just the types a player can reveal about himself. And is it the fact that this outcome can exist and be equivalent to other mechanisms that comprise the revelation principle. The revelation principle is used most often to prove something about the whole class of mechanism equilibria, by selecting the simple direct revelation mechanism, proving a result about that, and applying the revelation principle to assert that the result is true for all mechanisms in that context.