Charles Roddie Nuffield College, Oxford

 Link between what an agent has done in past and what he is expected to do in future  Two approaches:  Exact ▪ Do x repeatedly to establish reputation for x ▪ Mainly behavioral type models (Fudenberg & Levine (’89) etc.)  Directional ▪ Choose higher x now and you will be expected to choose higher x in future ▪ Mainly signaling game models

 Basic results:  exist increasing separating equilibria  including a dominant (Riley) separating equilibrium  this is selected by the equilibrium refinement D1  for a continuum of types it is the unique separating equilibrium  Main condition: Single crossing  Higher types are willing to take higher signals than lower types in exchange for better beliefs

 So signaling game satisfies single crossing  Separating equilibria, dominant sep. eq. selected by D1 refinement, etc.  Reputational effects in 1 st stage only  But if second stage is not final, there will be signaling then too  I.e. repeated signaling  This will affect 1 st stage signaling

o Holmstrom (‘99): reputation for productivity o Mester (‘92): 3-stage Cournot duopoly o Vincent (‘92): trading relationship o Rep. for tough bargaining by signaling low value o Mailath & Samuelson (‘01): rep. for product quality  We will approach question in general 1. Without functional forms & specific application 2. Allowing for general type spaces, not just 2 types 3. Allowing for arbitrary time horizon  2. and 3. give a new qualitative result  A commitment property with long game and continuum of types

Supermodularity (of payoffs) Supermodularity (of value function) Signaling game satisfying single crossing. Dominant separating equilibrium.

 Idea Supermodular signaling payoff Supermodular value function Supermodular value function Supermodular value function … Period n Period n-1 Period n-2 Supermodular signaling payoff Supermodular signaling payoff

Complete inf. static NE Complete inf. Stackelberg

 Stackelberg leadership property characteristic of behavioral type approach  Dynamic signaling model:  Tractable directional model ▪ Model calculable in and out of limits ▪ Reputation also in short and very long run  Normal types as appropriate to setting; no use of non-strategic types  Extends results to impatience

 Markov equilibrium of infinite game  Exists as fixed point  Continuity of value function iterator important  Need to tidy up value function first to get compact space  Equilibrium continuous in parameters  So study limit game directly  In limit game, IC conditions from Stackelberg game hold (see below)  Use IC and uniqueness results for continuum of types  IC pins down strategy, up to initial condition  Deal with edge cases

 Theory of Signaling Games Theory of Signaling Games Generalize the theory Find comparative statics & continuity properties  Signaling and Reputation in Repeated games  Part 1: Finite Games Part 1: Finite Games Construct & solve repeated signaling game Equilibrium selection (recursive D1 refinement)  Part 2: Stackelberg Limit Properties Part 2: Stackelberg Limit Properties ▪ Formalize argument above

 Signaling theory  Riley (‘79), Mailath (’87), Cho & Kreps (‘87), Mailath (‘88), Cho & Sobel (‘90), Ramey (‘96), Bagwell & Wolinsky (‘02)  Repeated signaling games  Mester (‘92), Vincent (‘98), Holmstrom (‘99), Mailath & Samuelson (‘01), Kaya (‘08), Toxvaerd (‘11)