Transcript Signaling, Repeated Signaling, and Reputation
Charles Roddie Nuffield College, Oxford
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
In literature, many 2-stage repeated games with signaling in 1 st stage E.g. 2-Stage Cournot competition / limit pricing If signaler takes higher 𝑞 𝑃1 in 1 st stage Signals lower 𝑐 𝑃1 Higher expected 𝑞 𝑃1 in 2 nd stage Competitors’ 𝑞 𝑃2 lower in 2 nd stage ⇒ higher 𝑞 𝑃1 than complete inf. static NE Reputational incentives in 1 st period
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′ 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 𝑦 𝑈(𝜃, 𝑦, 𝜃 𝑚𝑖𝑛 )
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 𝜃 1 < 𝜃 2 , 𝑥 1 < 𝑥 2 Then 𝑈 𝜃 1 , 𝑥 1 , 𝜃 1 ′ ⇒ 𝑈 𝜃 2 , 𝑥 1 , 𝜃 1 ′ and 𝜃 1 ′ ≤ 𝜃 2 ′ ≤ 𝑈 𝜃 1 , 𝑥 2 , 𝜃 2 ′ < 𝑈 𝜃 2 , 𝑥 2 , 𝜃 2 ′
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This single crossing is: Weaker than usual Spence-Mirrlees Implied by supermodularity of 𝑈 𝑓 is supermodular if: Taking any two variables 𝑥 , 𝑦 ; fixing others: If 𝑥 1 ≤ 𝑥 2 and 𝑦 1 ≤ 𝑦 2 Then 𝑓 𝑥 1 , 𝑦 1 + 𝑓 𝑥 2 , 𝑦 2 ≥ 𝑓 𝑥 2 , 𝑦 1 𝜕 2 𝑓 If 𝑓 ∈ 𝐶 2 , equivalent to: 𝜕𝑥𝜕𝑦 + 𝑓(𝑥 ≥ 0 1 , 𝑦 2 ) Makes it easy to construct signaling games
Profit 𝑃 𝑞 𝑃1 + 𝑞 𝑃2 − 𝑐 𝑝 𝑞 𝑝 where 𝑃′′ ≤ 0 For signaler 𝑃1 , supermodular in (−𝑐 𝑃1 , 𝑞 𝑃1 , −𝑞 𝑃2 ) For 𝑃2 , supermodular in (𝑞 𝑃1 , −𝑞 𝑃2 ) In 2 nd stage, lower signaled 𝑐 𝑃1 Value fn. for 2 nd ⇒ lower period supermodular in 𝑞 𝑃2 (−𝑐 𝑃1 , −𝑞 𝑃2 ) , so in (𝜃, 𝜃′) , where 𝜃 = −𝑐 𝑃1 Given 𝑞 𝑃2 in 1 st stage, overall profit supermodular in (𝜃, 𝑞 𝑃1 , 𝜃′)
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
o 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 Mailath & Samuelson (‘01): rep. for product quality We will approach question in general 1.
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Without functional forms & specific application 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
Parameterized signaling payoff 𝑈 𝑦, 𝜃, 𝑥, 𝜃 ′ Parameterized by 𝑦 E.g. duopoly stage 1, depends on P2’s quantity 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) Supermodularity (of value function) Signaling game satisfying single crossing.
Dominant separating equilibrium.
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
Signaler: Type 𝜃 ∈ Θ ▪ varies according to Markov process 𝜓 , monotonic Action 𝑥 ∈ 𝑋 Supermodular payoff 𝑢 𝑃1 (𝜃, 𝑥, 𝑦) , increasing in 𝑦 Discount factor 𝛿 𝑃1 Respondent: Action 𝑦 ∈ 𝑌 , simultaneous with 𝑥 Best response: increasing fn. Δ Θ × 𝑋 → ℝ ▪ Implied by supermodular payoff ▪ discount factor will not matter
Value function for signaler 𝑉 𝑡 ( Value at time 𝑡 when beliefs are , type is 𝜃 Suppose 𝑉 𝑡+1 is supermodular, inc. in Generates value of signaling 𝑉 𝑡 ′ in period 𝑡 Takes into account discounting, type change
Suppose 𝑦 is expected in period 𝑡 .
Then signaling payoff is: 𝜃, 𝑥, 𝜃′ ↦ 𝑢 𝑃1 𝜃, 𝑥, 𝑦 + 𝑉 𝑡 ′ (𝜃 ′ , 𝜃) Supermodular ; take Riley eq.
Depends on 𝑦 : strategy 𝑠 𝑌 (𝑦, 𝜃) Value fn. 𝑤(𝑦, 𝜃) is supermodular , increasing in 𝑦 To find 𝑦 = 𝑠 𝑃2 (𝑡)( 𝑦 = best response to 𝜃 and strategy 𝑠 𝑌 (𝑦,⋅) Take fixed point. Increasing in 𝜃 .
Then value function 𝑉 𝑡 = 𝑤(𝑠 𝑃2 ( supermodular, increasing in 𝜃 is Allows value function iteration Gives “Dynamic Riley equilibrium” Signaler’s strategy 𝑠 𝑃1 (𝑡) = 𝑠 𝑌 (𝑠 𝑃2 (
Continual separation of types Continual incentive to signal 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
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 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
Stackelberg signaling game: stage game with Signaler moving 1 st Limit 𝛿 𝑃1 → 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 Any 𝛿 𝑃1 , provided 𝑢 𝑃1 = 𝑢 𝑋 𝜃, 𝑥 + 𝑢 𝑌 (𝜃, 𝑦) Result above holds but in Stackelberg game use payoff: 𝑢 𝑋 𝜃, 𝑥 + 𝛿 𝑃1 ⋅ 𝑢 𝑌 (𝜃, 𝑦)
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
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: 𝜓 𝜃 = [𝜃] , 𝛿 𝑃1 Let 𝜎 𝜃 ≔ 𝑠 𝑃1 𝜃 , 𝜃 = 1 (same idea for What 𝜃 does when believed to be 𝜃 Suppose signaler has just signaled 𝜃′ In equilibrium, he signals true type 𝜃 𝛿 𝑃1 Gets some outcome O in period t In next period, does 𝜎 𝜃 What if he signals 𝜃′ and gets best response 𝑦 instead?
< 1 ) to this and 𝜃 At t, does 𝜎 𝜃′ , gets best response 𝑦′ to this and 𝜃′ Postpones O to next period; afterwards no difference Better to signal 𝜃 Since 𝛿 𝑃1 = 1 , 𝜃 prefers (𝜎 𝜃 , 𝑦) to (𝜎 𝜃 ′ , 𝑦′) I.e. 𝜎 satisfies IC conditions from Stackelberg game
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
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)
Assume continuum types, differentiability (Not necessary) Value fn. 𝑉 𝜃 = 𝑈 𝜃, 𝑥 𝜃 , 𝜃 For sep. eq., IC implies 𝑑𝑉 𝑑𝜃 Suppose 𝑈(𝑦, 𝜃, 𝑥, 𝜃′) 𝜕𝑈 = 𝜕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 𝜕𝑦𝜕𝜃 𝑉 𝑦, 𝜃 = 𝜕 𝜕𝑦 𝜕 𝜕𝜃 𝑈 𝑦, 𝜃, 𝑥 𝑦, 𝜃 , 𝜃 > 0 , so V is supermodular = 𝜕 2 𝑈 𝜕𝑦𝜕𝜃 𝜕 2 𝑈 𝜕𝑥 + 𝜕𝑦𝜕𝑥 𝜕𝑦