Bayesian efficiency

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Bayesian efficiency is an analog of Pareto efficiency for situations in which 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]

Overview

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 efficient allocation is always interim and ex post efficient, and an interim efficient allocation is always ex post efficient.^[1]

References

^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ 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

v t e Topics of game theory
Definitions	Congestion game Cooperative game Determinacy Escalation of commitment Extensive-form game First-player and second-player win Game complexity Graphical game Hierarchy of beliefs Information set Normal-form game Preference Sequential game Simultaneous game Simultaneous action selection Solved game Succinct game Mechanism design
Equilibrium concepts	Bayes correlated equilibrium Bayesian Nash equilibrium Berge equilibrium Core Correlated equilibrium Coalition-proof Nash equilibrium Epsilon-equilibrium Evolutionarily stable strategy Gibbs equilibrium Mertens-stable equilibrium Markov perfect equilibrium Nash equilibrium Pareto efficiency Perfect Bayesian equilibrium Proper equilibrium Quantal response equilibrium Quasi-perfect equilibrium Risk dominance Satisfaction equilibrium Self-confirming equilibrium Sequential equilibrium Shapley value Strong Nash equilibrium Subgame perfection Trembling hand equilibrium
Strategies	Appeasement Backward induction Bid shading Collusion Cheap talk De-escalation Deterrence Escalation Forward induction Grim trigger Markov strategy Pairing strategy Dominant strategies Pure strategy Mixed strategy Strategy-stealing argument Tit for tat
Classes of games	Auction Bargaining problem Global game Intransitive game Mean-field game n-player game Perfect information Large Poisson game Potential game Repeated game Screening game Signaling game Strictly determined game Stochastic game Symmetric game Zero-sum game
Games	Go Chess Infinite chess Checkers All-pay auction Prisoner's dilemma Gift-exchange game Optional prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Lewis signaling game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Electronic mail game Rock paper scissors Pirate game Dictator game Public goods game Blotto game War of attrition El Farol Bar problem Fair division Fair cake-cutting Bertrand competition Cournot competition Stackelberg competition Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Induction puzzles Trust game Princess and monster game Rendezvous problem
Theorems	Aumann's agreement theorem Folk theorem Minimax theorem Nash's theorem Negamax theorem Purification theorem Revelation principle Sprague–Grundy theorem Zermelo's theorem
Key figures	Albert W. Tucker Amos Tversky Antoine Augustin Cournot Ariel Rubinstein Claude Shannon Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin John Conway Jean Tirole Jean-François Mertens Jennifer Tour Chayes John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Axelrod Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey
Search optimizations	Alpha–beta pruning Aspiration window Principal variation search max^n algorithm Paranoid algorithm Lazy SMP
Miscellaneous	Bounded rationality Combinatorial game theory Confrontation analysis Coopetition Evolutionary game theory Glossary of game theory List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons

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