Revelation principle

The revelation principle is a fundamental principle in mechanism design. It can be summarized as follows:

If a social-choice function can be implemented by an arbitrary mechanism,

then the same function can be implemented by an incentive-compatible-direct-mechanism.

Moreover, the equilibrium payoffs in both cases are the same.

.^[1]^:224–225

Example

Consider the following example. There is a certain item that Alice values as $v_A$ and Bob values as $v_B$ . The government needs to decide who will receive that item and in what terms.

A social-choice-function is a function that maps a set of individual preferences to a social outcome. An example function is the utilitarian function, which says "give the item to a person that values it the most". We denote a social choice function by Soc and its recommended outcome given a set of preferences by Soc(Prefs).
A mechanism is a rule that maps a set of individual actions to a social outcome. A mechanism Mech induces a game which we denote by Game(Mech).
A mechanism Mech is said to implement a social-choice-function Soc if, for every combination of individual preferences, there is a Nash equilibrium in Game(Mech) in which the outcome is Soc(Prefs). Two example mechanisms are:
- "Each individual says a number between 1 and 10. The item is given to the individual that said the lowest number; if both say the same number, then the item is given to Alice". This mechanism does NOT implement the utilitarian function, since for every individual that wants the item, it is a dominant strategy to say "1" regardless of his/her true value. This means that in equilibrium the item is always given to Alice, even if Bob values it more.
- First-price sealed-bid auction is a mechanism which implements the utilitarian function. For example, if $v_B>v_A$ , then any action profile in which Bob bids more than Alice and both bids are in the range $(v_A,v_B)$ is a Nash-equilibrium in which the item goes to Bob. Additionally, if the valuations of Alice and Bob are random variables drawn independently from the same distribution, then there is a Bayesian Nash equilibrium in which the item goes to the bidder with the highest value.
A direct-mechanism is a mechanism in which the set of actions available to each player is just the set of possible preferences of the player.
A direct-mechanism Mech is said to be Bayesian-Nash-Incentive-compatible (BNIC) if there is a Bayesian Nash equilibrium of Game(Mech) in which all players reveal their true preferences. Some example direct-mechanisms are:
- "Each individual says how much he values the item. The item is given to the individual that said the highest value. In case of a tie, the item is given to Alice". This mechanism is NOT BNIC, since a player who wants the item is better-off by saying the highest possible value, regardless of his true value.
- First-price sealed-bid auction is also NOT BNIC, since the winner is always better-off by bidding the lowest value that is slightly above the loser's bid.
- However, if the distribution of the players' valuations is known, then there is a First-price sealed-bid auction#Incentive-compatible variant of first-price sealed-bid auction which is BNIC and implements the utilitarian function.
- Moreover, it is known that Second price auction is BNIC (it is even IC in a stronger sense - dominant-strategy IC). Additionally, it implements the utilitarian function.

In mechanism design, the revelation principle is of utmost importance in finding solutions. The researcher need only look at the set of equilibrium characterized by incentive compatibility. That is, if the mechanism designer wants to implement some outcome or property, he can restrict his search to mechanisms in which agents are willing to reveal their private information to the mechanism designer that has that outcome or property. If no such direct and truthful mechanism exists, no mechanism can implement this outcome/property. By narrowing the area needed to be searched, the problem of finding a mechanism becomes much easier.

Proof

Suppose we have an arbitrary mechanism Mech that implements Soc.

We construct a direct mechanism Mech' that is truthful and implements Soc.

Mech' simply simulates the equilibrium strategies of the players in Game(Mech). I.e:

Mech' asks the players to report their valuations.
Based on the reported valuations, Mech' calculates, for each player, his equilibrium strategy in Mech.
Mech' returns the outcome returned by Mech.

Reporting the true valuations in Mech' is like playing the equilibrium strategies in Mech. Hence, reporting the true valuations is a Nash equilibrium in Mech', as desired. Moreover, the equilibrium payoffs are the same, as desired.

History and Variants

There are different revelation principles for different solution concepts:

For implementation in dominant strategies and truthfulness in dominant strategies, the revelation principle was introduced by Allan Gibbard.^[2]
Later this principle was extended to the broader solution concept of Bayesian Nash equilibrium, described above, by Dasgupta, Hammond and Maskin,^[3] Holmstrom,^[4] and Myerson^[5] "To any Bayesian Nash equilibrium of a game of incomplete information, there exists a payoff-equivalent revelation mechanism that has an equilibrium where the players truthfully report their types."^[6]^:165

In correlated equilibrium

The revelation principle says that for every arbitrary coordinating device a.k.a. correlating there exists another direct device for which the state space equals the action space of each player. Then the coordination is done by directly informing each player of his action.

References

↑ Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.
↑ Gibbard, A. 1973. Manipulation of voting schemes: a general result. Econometrica 41, 587–601.
↑ Dasgupta, P., Hammond, P. and Maskin, E. 1979. The implementation of social choice rules: some results on incentive compatibility. Review of Economic Studies 46, 185–216.
↑ Holmstrom, B. 1977. On incentives and control in organizations. Ph.D. thesis, Stanford University.
↑ Myerson, R. 1979. Incentive-compatibility and the bargaining problem. Econometrica 47, 61–73.
↑ Robert Gibbons, Game theory for applied economists, Princeton University Press, 1992

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian-Nash Perfect Bayesian Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock-paper-scissors Pirate game Dictator game Public goods game Blotto games War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Screening game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey

See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons Tyranny of small decisions

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