# Situations where revelation principle may not hold

Revelation principle is a powerful statement regarding Bayesian Nash equilibrium. However it may not hold always, as where players do not fully know their preferences, or when preference elicitation involves cost.

What are other situations where revelation principle is not applicable? Are there papers which have circumvented these issues with alternate versions of the principle?

You might want to be a little more precise about what you mean by "Revelation Principle" as there are many formulations of the "Revelation Principle" out there, some of which are stronger than others. Each of these formulations makes a different claim and relies on a particular set of assumptions. Of course, the claim will often fail to be true if some of the assumptions are false.

(The following is from notes I got from a microeconomics class.)

Consider for instance the following version of the revelation principle, which is from Repullo (1985), Review of Economic Studies:

Repullo's Revelation Principle : Let $g$ be a dominant strategy mechanism for the game $\Gamma \equiv ( g, U_1, \dots, U_n)$, where $g$ is some game form. For each equilibrium selection function $s : \Theta \rightarrow S$, there exists an equivalent direct dominant strategy mechanism $h$ to $g$ (where $\Theta$ is the set of types). If in addition the equilibrium selection function $s^* : \Theta \rightarrow S$ is surjective, then the dominant equilibrium outcome under $h$ are a subset of the dominant equilibrium outcome under $g$ for all $\theta \in \Theta$.

The bold part is important. If it is not satisfied, there may still be a non-truthtelling equilibrium in the equivalent direct mechanism. An example is provided in Repullo (1985), Review of Economic Studies pp 223-229.

$$A \equiv \{a,b,c,d\}$$ $$\Theta_1 \equiv \{ \theta_1', \theta_1''\}$$ $$\Theta_2 \equiv \{ \theta_2', \theta_2''\}$$

$$\begin{array}{c |c c c c} & a & b & c & d \\ \hline u_1(\cdot, \theta_1') & 2 & 4 & 2 & 4\\ u_1(\cdot, \theta_1'') & 1 & 0 & 2 & 4\\ u_2(\cdot, \theta_2') & 2 & 2 & 4 & 4\\ u_2(\cdot, \theta_2'') & 1 & 2 & 0 & 4\\ \end{array}$$

$$S_1 \equiv \{s_1',s_1'',s_1'''\}$$ $$S_2 \equiv \{s_2',s_2'',s_2'''\}$$

The game form is

$$\begin{array}{c |c c c} & s_2' & s_2'' & s_2''' \\ \hline s_1' & a & b & b \\ s_1'' & c & d & c \\ s_1''' & c & b & a \\ \end{array}$$

On can check than the following is an equivalent direct mechanism

$$\begin{array}{c |c c } & \theta_2' & \theta_2'' \\ \hline \theta_1 & a & b \\ \theta_1 & c & d \\ \end{array}$$

Yet, when the types are $(\theta_1',\theta_2')$, although telling the truth is a dominant strategy, any other report of preferences is also a dominant strategy. This may be quite bothersome as it means that for some configuration of the types, saying the truth is only one equilibrium among others. As a consequence, we have no real guarantee that "telling the truth" will be played (it may even be that truth-telling is Pareto dominated by another equilibrium. Using a focal point argument, this may further undermine the relevance of the truth-telling equilibrium).

The above issue is due to the fact that in the original game, some strategies are never played, which is ruled out if $s^*$ is surjective. So Repullo's version of the Revelation Principle (requiering that every dominant strategy equilibrium outcome of the equivalent game be among the equilibrium outcome of the initial game for every possible configuration of the types) only holds if the equilibrium selection function be surjective, and fails otherwise.