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{{short description|Analog of Pareto efficiency for situations with incomplete information}}
{{Bayesian statistics}}
'''Bayesian efficiency''' is an analog of [[Pareto efficiency]] for situations in which there is [[incomplete information]].<ref name="implementation">Palfrey, Thomas R.; Srivastava, Sanjay; Postlewaite, A. (1993) ''[https://books.google.com/books?id=lZTls-JJSxgC&pg=PA13 Bayesian Implementation.]'' Pg. 13-14. {{ISBN|3-7186-5314-1}}</ref> 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.<ref name="implementation"/> 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.<ref name="implementation"/> Often, the players have types that are hidden from the other player.<ref name="implementation"/>

==Overview==
The lack of complete information raises a question of when the efficiency calculation should be made.<ref name="implementation"/> 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.<ref name="implementation"/> 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.<ref name="implementation"/>

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.<ref name="implementation"/><ref>Baltagi, Badi Hani. (2001) ''[https://books.google.com/books?id=qjBF3Z3xcWYC A Companion to Theoretical Econometrics.]'' Blackwell Publishing. {{ISBN|1-4051-0676-X}}</ref>

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

''Ex ante efficiency'': <math>x</math> is incentive compatible, and there exists no incentive compatible allocation rule <math>y:T\to A</math> that

:<math>\int U^i(y(t),t)dG^i(t) \geq \int U^i(x(t),t)dG^i(t)</math>

for all <math>i</math>, with strict inequality for some <math>i</math>.

''Interim efficiency'': <math>x</math> is incentive compatible, and there exists no incentive compatible allocation rule <math>y:T\to A</math> that

:<math>\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)</math>

for all <math>i</math> and <math>t_i</math>, with strict inequality for some <math>i</math> and <math>t_i</math>.

''Ex post efficiency'': <math>x</math> is incentive compatible, and there exists no incentive compatible allocation rule <math>y:T\to A</math> that

:<math>U^i(y(t),t) \geq U^i(x(t),t)</math>

for all <math>i</math>, with strict inequality for some <math>i</math>.

Here, <math>G^i</math> are beliefs, <math>U^i</math> are utility functions, and <math>i</math> are agents. An ex ante efficient allocation is always interim and ex post efficient, and an interim efficient allocation is always ex post efficient.<ref name="implementation"/>

==References==
{{reflist}}

{{game theory}}

[[Category:Bayesian statistics]]
[[Category:Pareto efficiency]]
[[Category:Game theory]]
[[Category:Law and economics]]
[[Category:Mathematical optimization]]
[[Category:Optimal decisions]]
[[Category:Electoral system criteria]]
[[Category:Welfare economics]]

Bayesian efficiency - Revision history

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