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Charles Roddie Nuffield College, Oxford

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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

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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

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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

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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

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

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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

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Complete inf. static NE Complete inf. Stackelberg

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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

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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

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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

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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)

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Rational Learning Leads to Nash Equilibrium Ehud Kalai and Ehud Lehrer Econometrica, Vol. 61 No. 5 (Sep 1993), 1019-1045 Presented by Vincent Mak

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