Editing Mechanism design (section)

===Revelation principle===
{{main|Revelation principle}}

A proposed mechanism constitutes a Bayesian game (a game of private information), and if it is well-behaved the game has a [[Bayesian Nash equilibrium]]. At equilibrium agents choose their reports strategically as a function of type
:<math>\hat\theta(\theta)</math>

It is difficult to solve for Bayesian equilibria in such a setting because it involves solving for agents' best-response strategies and for the best inference from a possible strategic lie. Thanks to a sweeping result called the revelation principle, no matter the mechanism a designer can<ref>In unusual circumstances some truth-telling games have more equilibria than the Bayesian game they mapped from. See Fudenburg-Tirole Ch. 7.2 for some references.</ref> confine attention to equilibria in which agents truthfully report type. The '''revelation principle''' states: "To every Bayesian Nash equilibrium there corresponds a Bayesian game with the same equilibrium outcome but in which players truthfully report type."

This is extremely useful. The principle allows one to solve for a Bayesian equilibrium by assuming all players truthfully report type (subject to an [[incentive compatibility]] constraint). In one blow it eliminates the need to consider either strategic behavior or lying.

Its proof is quite direct. Assume a Bayesian game in which the agent's strategy and payoff are functions of its type and what others do, <math>u_i\left(s_i(\theta_i),s_{-i}(\theta_{-i}), \theta_{i} \right)</math>. By definition agent ''i'''s equilibrium strategy <math>s(\theta_i)</math> is Nash in expected utility:
:<math>s_i(\theta_i) \in \arg\max_{s'_i \in S_i} \sum_{\theta_{-i}} \ p(\theta_{-i} \mid \theta_i) \ u_i\left(s'_i, s_{-i}(\theta_{-i}),\theta_i \right)</math>

Simply define a mechanism that would induce agents to choose the same equilibrium. The easiest one to define is for the mechanism to commit to playing the agents' equilibrium strategies ''for'' them.
:<math>y(\hat\theta) : \Theta \rightarrow S(\Theta) \rightarrow Y </math>

Under such a mechanism the agents of course find it optimal to reveal type since the mechanism plays the strategies they found optimal anyway. Formally, choose <math>y(\theta)</math> such that
: <math>
\begin{align}
\theta_i \in {} & \arg\max_{\theta'_i \in \Theta} \sum_{\theta_{-i}} \ p(\theta_{-i} \mid \theta_i) \ u_i\left( y(\theta'_i, \theta_{-i}),\theta_i \right) \\[5pt]
& = \sum_{\theta_{-i}} \ p(\theta_{-i} \mid \theta_i) \ u_i\left(s_i(\theta), s_{-i}(\theta_{-i}),\theta_i \right)
\end{align}
</math>