Download - Signaling and Reputation in Repeated Games
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