Equilibrium Refinement in Finite Action Evidence Games

TL;DR

This paper examines equilibrium refinements in evidence games with finite actions, introducing a disturbed game approach to ensure the existence of truthful equilibria and characterizing their properties.

Contribution

It extends the concept of truth-leaning equilibria by introducing disturbed games with uncertainty, ensuring existence and providing a simple characterization of purifiable truthful equilibria.

Findings

Purifiable truthful equilibria exist and are characterized simply.

These equilibria are receiver optimal and match the payoff of optimal mechanisms.

The approach restores equilibrium existence in finite action settings.

Abstract

Evidence games study situations where a sender persuades a receiver by selectively disclosing hard evidence about an unknown state of the world. Evidence games often have multiple equilibria. Hart et al. (2017) propose to focus on truth-leaning equilibria, i.e., perfect Bayesian equilibria where the sender discloses truthfully when indifferent, and the receiver takes off-path disclosure at face value. They show that a truth-leaning equilibrium is an equilibrium of a perturbed game where the sender has an infinitesimal reward for truth-telling. We show that, when the receiver's action space is finite, truth-leaning equilibrium may fail to exist, and it is not equivalent to equilibrium of the perturbed game. To restore existence, we introduce a disturbed game with a small uncertainty about the receiver's payoff. A purifiable truthful equilibrium is the limit of a sequence of truth-leaning equilibria in the disturbed games as the disturbances converge to zero. It exists and features a simple characterization. A truth-leaning equilibrium that is also purifiable truthful is an equilibrium of the perturbed game. Moreover, purifiable truthful equilibria are receiver optimal and give the receiver the same payoff as the optimal deterministic mechanism.