Signaling and Reputation in Repeated Games

1. Signaling and Reputation in Repeated Games Charles Roddie Nuffield College, Oxford

2. What is reputation? • 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

3. Signaling and reputation • In literature, many 2-stage repeated games with signaling in 1ststage • E.g. 2-Stage Cournot competition / limit pricing • If signaler takes higher in 1st stage • Signals lower • Higher expected in 2nd stage • Competitors’ lower in 2ndstage • higher than complete inf. static NE • Reputational incentives in 1st period

4. Signaling game basics • Signaler has type , takes signal • Is subsequently believed to be • May generate response, resulting in… • Payoff , increasing in • Separating equilibria • Type takes , injective • IC: • IR:

5. What makes a tractable game? • 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 • If , and • Then

6. Supermodular signaling games • This single crossing is: • Weaker than usual Spence-Mirrlees • Implied by supermodularity of • Makes it easy to construct signaling games is supermodular if: Taking any two variables ,; fixing others: If and Then If , equivalent to:

7. Application: 2-stage Cournot duopoly • Profit where • For signaler , supermodular in • For , supermodular in • In 2nd stage, lower signaled lower • Value fn. for 2nd period supermodular in , so in , where • Given in 1st stage, overall profit supermodular in

8. Application: 2-stage Cournot • So signaling game satisfies single crossing • Separating equilibria, dominant sep. eq. selected by D1 refinement, etc. • Reputational effects in 1st stage only • But if second stage is not final, there will be signaling then too • I.e. repeated signaling • This will affect 1st stage signaling

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

10. Parameterized supermodular signaling payoffs • Parameterized signaling payoff • Parameterized by • E.g. duopoly stage 1, depends on P2’s quantity • Suppose is supermodular • Riley equilibrium , increasing in y • Value function • Then 𝑉 is supermodular (See appendix for intuition)

11. Supermodularity as input and output Supermodularity (of payoffs) Signaling game satisfying single crossing. Dominant separating equilibrium. Supermodularity (of value function)

12. Application to repeated signaling • Idea Period n Period n-1 Period n-2 Supermodular signaling payoff Supermodular signaling payoff Supermodular signaling payoff … Supermodular value function Supermodular value function Supermodular value function

13. Model • Signaler: • Type • varies according to Markov process , monotonic • Action • Supermodular payoff , increasing in • Discount factor • Respondent: • Action , simultaneous with • Best response: increasing fn. • Implied by supermodular payoff • discount factor will not matter

14. Recursive solution • Value function for signaler • Value at time when beliefs are , type is • Suppose is supermodular, inc. in • Generates value of signaling in period • Takes into account discounting, type change

15. Recursive solution, cont. • Suppose is expected in period . • Then signaling payoff is: • Supermodular; take Riley eq. • Depends on : strategy • Value fn. is supermodular, increasing in • To find • best response to and strategy • Take fixed point. Increasing in .

16. Recursive solution, cont. • Then value function is supermodular, increasing in • Allows value function iteration • Gives “Dynamic Riley equilibrium” • Signaler’s strategy

17. What is happening? • Continual separation of types • Continual incentive to signal • Benefit of signaling: improve in next period • Reputational motive: • Take higher • Thought to be higher and so • Expected to take higher in future • Can be additional pure signaling motive • Respondent rewards higher

18. Equilibrium selection • Dynamic Riley equilibrium is just one equilibrium • Must justify choice of Riley equilibrium in each derived signaling game • Equilibrium refinement D1 selects Riley equilibrium in a signaling game • Provided initial type-beliefs have full support • In repeated signaling game, belief about type always has full support • If always full support for all • Recursive application of D1 selects dynamic Riley equilibrium

19. Calculations: work incentives : ability : productivity Complete inf. static NE Complete inf. Stackelberg

20. Calculations: dynamic Cournotduopoly

21. Stackelberg property in limit • Stackelberg signaling game: stage game with Signaler moving 1st • Limit , continuum of types, becoming persistent • Signaler takes Riley equilibrium of Stackelberg game • If respondent does not care about type directly, this is just the Stackelberg complete inf. action • Subject to separating from the lowest type • Any , provided • Result above holds but in Stackelberg game use payoff:

22. Stackelberg properties: comments • 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

23. Proving the Stackelberg result • 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

24. Limit Incentive Compatibility • Limit: , (same idea for ) • Let • What does when believed to be • Suppose signaler has just signaled • In equilibrium, he signals true type • Gets some outcome O in period t • In next period, does and gets best response to this and • What if he signals instead? • At t, does , gets best response to this and • Postpones O to next period; afterwards no difference • Better to signal • Since , prefers to • I.e. satisfies IC conditions from Stackelberg game

25. Papers • Theory of Signaling Games • Generalize the theory • Find comparative statics & continuity properties • Signaling and Reputation in Repeated games • Part 1: Finite Games • Construct & solve repeated signaling game • Equilibrium selection (recursive D1 refinement) • Part 2: StackelbergLimit Properties • Formalize argument above

26. Related Literature • 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)

27. Appendix: Parametric supermodular signaling payoffs • Assume continuum types, differentiability (Not necessary) • Value fn. • For sep. eq., IC implies • Suppose is supermodular • Signaling payoff parameterized by • E.g. duopoly stage 1, depends on P2’s quantity • Can show increasing in y • , so V is supermodular