Bayesian efficiency
Bayesian efficiency
- Bayesian efficiency** is an analog of [[pareto-efficiency]] for situations in which there is incomplete information. 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.
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.