Belief-Based Refinements in Signalling Games

Interim Bayesian Persuasion: First Steps

“…To gain space, I define them more formally and prove the result in the appendix. The first of these refinements, R 1 , is a variant on the notion of undefeated equilibrium (Mailath et al, 1993;Umbhauer, 1994). The concept of undefeated equilibrium relies on the idea that a deviation of the sender from her equilibrium action π to an information system π should be interpreted as an attempt to indicate that she would prefer to play the equilibrium outcome associated to π in P(p, S).…”

“…10 The second notion, R 2 , is an adaptation of the notion of core mechanisms of Myerson (1983) to my framework so as to take into account the fact that the designer chooses an information system rather than a full mechanism.The third notion, R 3 refines the notions of neologism proofness of Farrell (1993) and of perfect sequential equilibrium of Grossman and Perry (1986). 11 For exposition purposes, suppose as in Farrell (1993) that a deviating player 10 Contrary to the original definitions, I do not assume that all types using π in the new equilibrium must prefer either the new equilibrium (Mailath et al, 1993), or some best response of the receiver to the belief generated by π in the new equilibrium, to the original equilibrium (Umbhauer, 1994).…”

“…If the beliefs of the players in the original equilibrium do not anticipate this, the original equilibrium should be discarded. This idea was developed independently in Mailath et al (1993) and Umbhauer (1994) with small differences. Formally, in a sender-receiver game with general action set for the sender, if T is the type set of the sender and e is the original equilibrium, consider an equilibrium e and an action of the sender a which is on the equilibrium path of e but off the equilibrium path of e .…”

“…Unlike Mailath et al (1993) or Umbhauer (1994), I do not require all types who use a in e to prefer e to e, or even some best response of the receiver to her belief in following a in e .…”

“…My main result is to show that three different refinement concepts lead to the selection of the high type optimal equilibria. The first refinement is a version of the notion of undefeated equilibria (Mailath et al, 1993;Umbhauer, 1994) which was developed in the context of signaling games. The second refinement is an adaptation of the notion of core mechanism of Myerson (1983), developed for the analysis of informed principal problems, to my framework.…”