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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In literature, many 2-stage repeated games with
signaling in 1st stage
E.g. 2-Stage Cournot competition / limit pricing
 If signaler takes higher 𝑞𝑃1 in 1st stage
 Signals lower 𝑐𝑃1
 Higher expected 𝑞𝑃1 in 2nd stage
 Competitors’ 𝑞𝑃2 lower in 2nd stage
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⇒ higher 𝑞𝑃1 than complete inf. static NE
Reputational incentives in 1st period
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Signaler has type 𝜃, takes signal 𝑥
 Is subsequently believed to be 𝜃 ′
▪ May generate response, resulting in…
 Payoff 𝑈(𝜃, 𝑥, 𝜃′), increasing in 𝜃′
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Separating equilibria
 Type 𝜃 takes 𝑥(𝜃), injective
 IC: 𝑈 𝜃, 𝑥 𝜃 , 𝜃 ≥ 𝑈 𝜃, 𝑥 𝜃 ′ , 𝜃 ′
 IR: 𝑈 𝜃, 𝑥 𝜃 , 𝜃 ≥ max 𝑈(𝜃, 𝑦, 𝜃𝑚𝑖𝑛 )
𝑦
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Basic results:
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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 𝜃1 < 𝜃2 , 𝑥1 < 𝑥2 and 𝜃1′ ≤ 𝜃2′
Then 𝑈 𝜃1 , 𝑥1 , 𝜃1′ ≤ 𝑈 𝜃1 , 𝑥2 , 𝜃2′
⇒ 𝑈 𝜃2 , 𝑥1 , 𝜃1′ < 𝑈 𝜃2 , 𝑥2 , 𝜃2′
This single crossing is:
1. Weaker than usual Spence-Mirrlees
2. Implied by supermodularity of 𝑈
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𝑓 is supermodular if:
Taking any two variables 𝑥, 𝑦; fixing others:
If 𝑥1 ≤ 𝑥2 and 𝑦1 ≤ 𝑦2
Then 𝑓 𝑥1 , 𝑦1 + 𝑓 𝑥2 , 𝑦2 ≥ 𝑓 𝑥2 , 𝑦1 + 𝑓(𝑥1 , 𝑦2 )
If 𝑓 ∈
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𝜕2 𝑓
𝐶2 , equivalent to:
𝜕𝑥𝜕𝑦
≥0
Makes it easy to construct signaling games
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Profit 𝑃 𝑞𝑃1 + 𝑞𝑃2 − 𝑐𝑝 𝑞𝑝 where 𝑃′′ ≤ 0
 For signaler 𝑃1, supermodular in
(−𝑐𝑃1 , 𝑞𝑃1 , −𝑞𝑃2 )
 For 𝑃2, supermodular in (𝑞𝑃1 , −𝑞𝑃2 )
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In 2nd stage, lower signaled 𝑐𝑃1 ⇒ lower 𝑞𝑃2
Value fn. for 2nd period supermodular in
(−𝑐𝑃1 , −𝑞𝑃2 ), so in (𝜃, 𝜃′), where 𝜃 = −𝑐𝑃1
Given 𝑞𝑃2 in 1st stage, overall profit
supermodular in (𝜃, 𝑞𝑃1 , 𝜃′)
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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
o
o
o
Holmstrom (‘99): reputation for productivity
Mester (‘92): 3-stage Cournot duopoly
Vincent (‘92): trading relationship
o Rep. for tough bargaining by signaling low value
o
Mailath & Samuelson (‘01): rep. for product quality
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We will approach question in general
1.
2.
3.
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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
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Parameterized signaling payoff 𝑈 𝑦, 𝜃, 𝑥, 𝜃 ′
 Parameterized by 𝑦
 E.g. duopoly stage 1, depends on P2’s quantity
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Suppose 𝑈 is supermodular
Riley equilibrium 𝑥 𝑦, 𝜃 , increasing in y
Value function 𝑉 𝑦, 𝜃 = 𝑈 𝑦, 𝜃, 𝑥 𝑦, 𝜃 , 𝜃
Then 𝑉 is supermodular
(See appendix for intuition)
Supermodularity
(of payoffs)
Signaling game satisfying
single crossing.
Dominant
separating equilibrium.
Supermodularity
(of value function)
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Idea
Period n
Supermodular
signaling payoff
Period n-1
Supermodular
signaling payoff
Period n-2
Supermodular
signaling payoff
…
Supermodular
value function
Supermodular
value function
Supermodular
value function
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Signaler:
 Type 𝜃 ∈ Θ
▪ varies according to Markov process 𝜓, monotonic
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 Action 𝑥 ∈ 𝑋
 Supermodular payoff 𝑢𝑃1 (𝜃, 𝑥, 𝑦), increasing in 𝑦
 Discount factor 𝛿𝑃1
Respondent:
 Action 𝑦 ∈ 𝑌, simultaneous with 𝑥
 Best response: increasing fn. Δ Θ × 𝑋 → ℝ
▪ Implied by supermodular payoff
▪ discount factor will not matter
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Value function for signaler 𝑉𝑡 (𝜃, 𝜃)
 Value at time 𝑡 when beliefs are 𝜃 ∈ ΔΘ, type is 𝜃
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Suppose 𝑉𝑡+1 is supermodular, inc. in 𝜃
′
Generates value of signaling 𝑉𝑡 in period 𝑡
 Takes into account discounting, type change
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Suppose 𝑦 is expected in period 𝑡.
Then signaling payoff is:
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Supermodular; take Riley eq.
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𝜃, 𝑥, 𝜃′ ↦ 𝑢𝑃1 𝜃, 𝑥, 𝑦 + 𝑉𝑡′ (𝜃 ′ , 𝜃)
 Depends on 𝑦: strategy 𝑠𝑌 (𝑦, 𝜃)
 Value fn. 𝑤(𝑦, 𝜃) is supermodular, increasing in 𝑦
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To find 𝑦 = 𝑠𝑃2 (𝑡)(𝜃)
 𝑦 = best response to 𝜃 and strategy 𝑠𝑌 (𝑦,⋅)
 Take fixed point. Increasing in 𝜃.
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Then value function 𝑉𝑡 = 𝑤(𝑠𝑃2 (𝜃), 𝜃) is
supermodular, increasing in 𝜃
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Allows value function iteration
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Gives “Dynamic Riley equilibrium”
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Signaler’s strategy 𝑠𝑃1 (𝑡) = 𝑠𝑌 (𝑠𝑃2 (𝜃), 𝜃)
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Continual separation of types
 Continual incentive to signal
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Benefit of signaling: improve 𝑦 in next period
 Reputational motive:
▪ Take higher 𝑥
▪ Thought to be higher 𝜃 and so
▪ Expected to take higher 𝑥 in future
 Can be additional pure signaling motive
▪ Respondent rewards higher 𝜃
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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
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In repeated signaling game, belief about type
always has full support
 If 𝜓(𝜃) always full support for all 𝜃
 Recursive application of D1 selects dynamic
Riley equilibrium
𝜃: ability
𝑥: productivity
Complete inf. static NE
Complete inf. Stackelberg
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Stackelberg signaling game: stage game with
Signaler moving 1st
Limit 𝛿𝑃1 → 1, 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
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Any 𝛿𝑃1 , provided 𝑢𝑃1 = 𝑢𝑋 𝜃, 𝑥 + 𝑢𝑌 (𝜃, 𝑦)
 Result above holds but in Stackelberg game use
payoff:
𝑢𝑋 𝜃, 𝑥 + 𝛿𝑃1 ⋅ 𝑢𝑌 (𝜃, 𝑦)
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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
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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
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 IC pins down strategy, up to initial condition
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Deal with edge cases
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Limit: 𝜓 𝜃 = [𝜃], 𝛿𝑃1 = 1 (same idea for 𝛿𝑃1 < 1)
Let 𝜎 𝜃 ≔ 𝑠𝑃1 𝜃 , 𝜃
 What 𝜃 does when believed to be 𝜃
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Suppose signaler has just signaled 𝜃′
In equilibrium, he signals true type 𝜃
 Gets some outcome O in period t
 In next period, does 𝜎 𝜃 and gets best response 𝑦 to this and 𝜃
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What if he signals 𝜃′ instead?
 At t, does 𝜎 𝜃′ , gets best response 𝑦′ to this and 𝜃′
 Postpones O to next period; afterwards no difference
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Better to signal 𝜃
 Since 𝛿𝑃1 = 1, 𝜃 prefers (𝜎 𝜃 , 𝑦) to (𝜎 𝜃 ′ , 𝑦′)
 I.e. 𝜎 satisfies IC conditions from Stackelberg game
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Theory of Signaling Games
• Generalize the theory
• Find comparative statics & continuity properties
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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
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Signaling theory
 Riley (‘79), Mailath (’87), Cho & Kreps (‘87), Mailath
(‘88), Cho & Sobel (‘90), Ramey (‘96), Bagwell &
Wolinsky (‘02)
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Repeated signaling games
 Mester (‘92), Vincent (‘98), Holmstrom (‘99),
Mailath & Samuelson (‘01), Kaya (‘08), Toxvaerd
(‘11)
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Assume continuum types, differentiability
Value fn. 𝑉 𝜃 = 𝑈 𝜃, 𝑥 𝜃 , 𝜃
 For sep. eq., IC implies
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𝑑𝑉
𝑑𝜃
=
(Not necessary)
𝜕𝑈
𝜕𝜃
Suppose 𝑈(𝑦, 𝜃, 𝑥, 𝜃′) is supermodular
 Signaling payoff parameterized by 𝑦
▪ E.g. duopoly stage 1, depends on P2’s quantity
 Can show 𝑥 𝑦, 𝜃 increasing in y

𝜕2
𝑉
𝜕𝑦𝜕𝜃
𝑦, 𝜃 =
𝜕 𝜕
𝑈
𝜕𝑦 𝜕𝜃
𝑦, 𝜃, 𝑥 𝑦, 𝜃 , 𝜃
 > 0, so V is supermodular
=
𝜕2 𝑈
𝜕𝑦𝜕𝜃
𝜕2 𝑈 𝜕𝑥
+
𝜕𝑦𝜕𝑥 𝜕𝑦