Signaling and Reputation in Repeated Games

Charles RoddieNuffield College, Oxford

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

Signaling and reputation

In literature, many 2-stage repeated games with signaling in 1st stage

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 2nd stage

higher than complete inf. static NE Reputational incentives in 1st period

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:

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

Supermodular signaling games This single crossing is:1. Weaker than usual Spence-Mirrlees2. 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:

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

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

Repeated signaling models of reputationo Holmstrom (‘99): reputation for productivityo Mester (‘92): 3-stage Cournot duopolyo Vincent (‘92): trading relationship

o Rep. for tough bargaining by signaling low valueo Mailath & Samuelson (‘01): rep. for product quality

We will approach question in general1. Without functional forms & specific application2. Allowing for general type spaces, not just 2 types3. Allowing for arbitrary time horizon

2. and 3. give a new qualitative result A commitment property with long game and continuum of

types

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 is supermodular𝑉

(See appendix for intuition)

Supermodularity as input and output

Supermodularity

(of payoffs)

Supermodularity

(of value function)

Signaling game satisfying single

crossing.Dominant separating equilibrium.

Application to repeated signaling Idea

Supermodular

signaling payoff

Supermodular

value function

Supermodular

value function

Supermodular

value function

…

Period n Period n-1 Period n-2Supermodula

rsignaling

payoff

Supermodular

signaling payoff

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

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

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 .

Recursive solution, cont.

Then value function is supermodular, increasing in

Allows value function iteration

Gives “Dynamic Riley equilibrium” Signaler’s strategy

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

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

Calculations: work incentives

: ability: productivity

Complete inf. static NE

Complete inf. Stackelberg

Calculations: dynamic Cournot duopoly

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:

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

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

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

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: Stackelberg Limit Properties▪ Formalize argument above

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)

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