Bayesian efficiency

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Bayesian efficiency addresses an appropriate economic definition of Pareto efficiency where there is incomplete information.^[1] Under Pareto efficiency, an allocation of a resource is Pareto efficient if there is no other allocation of that resource that makes no one worse off while making some agents strictly better off.^[1] A limitation with the concept of Pareto efficiency is that it assumes that knowledge about other market participants is available to all participants, in that every player knows the payoffs and strategies available to other players so as to have complete information.^[1] Often, the players have types that are hidden from the other player.^[1]

The lack of complete information raises a question of when the efficiency calculation should be made.^[1] Should the efficiency check be made at the ex ante stage before the agent sees their types, at the interim stage after the agent sees their types, or at the ex post stage where the agent will have complete information about their types? Another issue is incentive.^[1] If a resource allocation rule is efficient but there is no incentive to abide by that rule or accept that rule, then the revelation principle asserts that there is no mechanism by which this allocation rule can be realized.^[1]

Bayesian efficiency overcomes problems of the Pareto efficiency by accounting for incomplete information, by addressing the timing of the evaluation (ex ante efficient, interim efficient, or ex post efficient), and by adding an incentive qualifier so that the allocation rule is incentive compatible.^[1]^[2]

Bayesian efficiency separately defines three types of efficiency: ex ante, interim, and ex post. For an allocation rule $x:T\to A$ :

Ex ante efficiency: $x$ is incentive compatible, and there exists no incentive compatible allocation rule $y:T\to A$ that

$\int u^{i}(y(t),t)dG^{i}(t)\geq \int u^{i}(x(t),t)dG^{i}(t)$

for all $i$ , with strict inequality for some $i$ .

Interim efficiency: $x$ is incentive compatible, and there exists no incentive compatible allocation rule $y:T\to A$ that

$\int U^{i}(y(t),t)dG^{i}(t_{-i}|t_{i})\geq \int U^{i}(x(t),t)dG^{i}(t_{-i}|t_{i})$

for all $i$ and $t_{i}$ , with strict inequality for some $i$ and $t_{i}$ .

Ex post efficiency: $x$ is incentive compatible, and there exists no incentive compatible allocation rule $y:T\to A$ that

$U^{i}(y(t),t)\geq U^{i}(x(t),t)$

for all $i$ , with strict inequality for some $i$ .

Here, $G^{i}$ are beliefs, $U^{i}$ are utility functions, and $i$ are agents. An ex ante allocation is always interim and ex post efficient, and an interim efficient allocation is always ex post efficient.^[1]

References

1 2 3 4 5 6 7 8 9 Palfrey, Thomas R.; Srivastava, Sanjay; Postlewaite, A. (1993) Bayesian Implementation. Pg. 13-14. ISBN 3-7186-5314-1
↑ Baltagi, Badi Hani. (2001) A Companion to Theoretical Econometrics. Blackwell Publishing. ISBN 1-4051-0676-X

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 equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium 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 Screening 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 Potential 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 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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