  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.
