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Lecture 1: Commitment Payoff Theorem Long-Run Short-Run Models with Perfect Monitoring

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Lecture 1: Commitment Payoff Theorem Long-Run Short-Run Models with Perfect Monitoring Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Lecture 1: Commitment Payoff Theorem Long-Run Short-Run Models with Perfect Monitoring Harry PEI Department of Economics, Northwestern University Spring Quarter, 2020 Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Rules Please ask questions. Let me know if I am going too slow or too fast. Interrupt me if: • I made a mistake (highly likely) • There is something unclear, • There is something you don’t understand. There is no stupid listener, there are only bad speakers. • Also applies when you give talks :) Email me if you have suggestions. Email me if you want to chat about work/life/whatever. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion What is a reputation? Google: A widespread belief that someone or something has a particular habit or characteristic. Two approaches to study reputations: 1. The habit view: Players convince their opponents that they will behave in a particular manner (e.g., always cooperate, tit-for-tat). 2. The characteristic view: Players signal payoff-relevant characteristics over time (e.g., low production cost, high ability, high quality). Similarity: Dynamic games with incomplete information, one informed player facing one/multiple uninformed opponent(s). Difference: Nature of the informed player’s private info. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Overview of this course Mostly focus on the habit view (commitment-type models): • Different versions (how patient the uninformed players are). One lecture on a combination of the two: • Pei (2018): Reputation Effects under Interdependent Values. One lecture on the characteristic view (rational-type models): • Pei (2020): Reputational Payoffs and Behaviors without Commitment. One lecture on reputational bargaining: • Cool application of reputation models to bargaining problems. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Intellectual History: Chainstore Paradox • A monopolist has branches in T 2 N towns. He faces one potential competitor in each town. • In period s 2 f1; 2; :::; Tg, monopolist plays against the competitor in the s-th town. C In Out M F A (2,0) (0, −1) (1,1) • Monopolist’s total payoff is the sum of payoffs in T markets. • Every competitor perfectly observes all actions chosen before. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Intellectual History: Chainstore Paradox Unique subgame perfect equilibrium: • Every competitor chooses In and monopolist chooses Accommodate. What is wrong with this prediction? • No matter how long the time horizon is, the monopolist never fights. • Even if a competitor observes the monopolist fighting the past 1000 entrants, he still believes that he will be accommodated with prob 1. Something is missing in complete information game models. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Intellectual History: Commitment Type Models How to fix this? • Kreps and Wilson (1982), Milgrom and Roberts (1982). Known as the Gang of four. Idea: Perturb the game with a small prob of commitment type. • With probability " > 0, the monopolist is irrational, doesn’t care about payoffs, and mechanically fights in every period. • With probability 1 − ", the monopolist is rational, maximizes the sum of his payoffs across periods. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Result: Gang of Four Theorem: Gang of Four For every " > 0, there exists T∗ 2 N such that when T ≥ T∗, on the equilibrium path of every sequential equilibrium, • Strategic monopolist chooses F & each potential entrant chooses Out in the initial T − T∗ periods Another way to say this: Monopolist fights (i.e., builds his reputation for being aggressive) in all except for the last T∗ periods. Takeaway from this result: The option to build reputation dramatically affects forward-looking agents’ incentives. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Proof: Take Home Exercise Bonus Question: Figure out why sequential equilibrium Proof Idea: Characterize the equilibrium via backward induction. Equilibrium Behavior: • In the first T − T∗ periods, rational incumbent plays F and entrant stays out. No learning takes place. • In the last T∗ periods, entrant enters with positive prob, strategic incumbent mixes between F and A. Learning happens gradually. Probability of entry makes the strategic incumbent indifferent, and strategic incumbent’s mixing probability makes the entrant indifferent. Establish Uniqueness of Sequential Equilibrium Outcome: Pin down the entrants’ on-path beliefs in the last few periods. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Robustness of the Gang of Four Insight? Gang of four result requires: • Finite horizon, backward induction. • A particular stage-game. • Entrant can perfectly observe monopolist’s action. • Sequential equilibrium. More important concern: Does it rely on the nature of incomplete info? • Fix G = (N; A; u), and let w 2 Rn be a stage-game NE payoff. Folk Theorem under Incomplete Information: Fudenberg and Maskin (1986) For any " > 0 and any payoff vector v > w, there exists T∗ 2 N such that for ∗ any T > T , there exists a strategy si for each player i such that in the T-fold repetition of G where each player i is rational with probability 1 − " and committed to si with probability ", there is a sequential equilibrium where players’ average payoff vector is within " of v. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Fudenberg and Levine (1989, 1992) How robust is the gang of four insight? • both finite and infinite horizon models. • general stage game payoffs. • imperfect monitoring. • weaker solution concepts. • not sensitive to the details of incomplete info. Will present all results in infinite horizon. Robust to long finite horizon. • Proof: Not constructive but captures the logic behind reputations. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Infinitely Repeated Game with One Long-Run Player • Time: t = 0; 1; 2; ::: • Long-lived player 1 (P1), vs an infinite sequence of short-lived player 2s (P2). • Players simultaneously choose their actions a1 2 A1 and a2 2 A2. * Actions in period t: a1;t and a2;t. • Stage-game payoffs: u1(a1;t; a2;t), u2(a1;t; a2;t). P1 t * P1’s discounted average payoff : t=0(1 − δ)δ u1(a1;t; a2;t). • Public signal: y 2 Y, with ρ(yja1; a2) the prob of y under (a1; a2). * yt: public signal in period t. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Introducing Commitment Types P1’s type space Ω ≡ f!sg S Ωm, type is perfectly persistent. 1. !s is the strategic type. Can flexibly choose actions in order to maximize payoffs. ∗ m m 2. Each α1 2 Ω represents a commitment type, with Ω ⊂ ∆(A1). ∗ Does not care about payoffs and plays α1 in every period. P2’s prior belief: π 2 ∆(Ω). What can players observe? t t t • Player 1’s history: h1 2 H1 ≡ Ω × fA1 × Yg . t t t • Player 2’s history: h2 2 H2 ≡ fA2 × Yg . Important: P2 in period t observes (y0; :::; yt−1). m Assumptions: A1; A2; Y and Ω are finite, π has full support. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Fudenberg and Levine (1989) Commitment Payoff Theorem (Pure Commitment & Perfect Monitoring) Let’s make two simplifying assumptions: 1. Perfect monitoring: Y = A1 × A2 and ρ(a1; a2ja1; a2) = 1. 2. There exists a commitment type that plays a pure action. ∗ m ∗ For every commitment action a1 2 Ω , P1’s commitment payoff from a1 : v∗(a∗) ≡ min u (a∗; a ): 1 1 ∗ 1 1 2 a22BR2(a1 ) Let u1 be P1’s lowest stage-game payoff. Commitment Payoff Theorem: Fudenberg and Levine (1989) For every " > 0, there exists T 2 N, ∗ m such that when π attaches prob more than " to commitment type a1 2 Ω , strategic P1’s payoff in any Bayes Nash Equilibrium is at least: T T ∗ ∗ (1 − δ )u1 + δ v1 (a1 ): Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion What does the commitment payoff theorem tell us? Patient P1’s payoff lower bound (applies to all BNEs): T T ∗ ∗ (1 − δ )u1 + δ v1 (a1 ): What happens when the informed player is patient, i.e., δ ! 1? ∗ ∗ • P1’s payoff lower bound ! v1 (a1 ). ∗ • Patient P1 receives at least his commitment payoff from a1 . Does it depend on the details of the type space? ∗ • No. We just need type a1 to occur with positive prob. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Refinement for Repeated Complete Info Games Let A2 ≡ fα2 2 ∆(A2)jα2 best replies against some α1 2 ∆(A1)g Patient P1’s lowest equilibrium payoff: min max u1(a1; α2): α22A2 a12A1 P1’s highest equilibrium payoff: max min u1(a1; α2): α22A2,α1 that α2 best replies against a12supp(α1) In many games of interest, the option to build a reputation selects a subset of high payoffs for P1. Sometimes, it selects P1’s highest equilibrium payoff. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Commitment Payoff Theorem in Product Choice Game A firm (P1) and a sequence of consumers (P2s). – T N H 2; 1 −1; 0 L 3; −1 0; 0 Repeated complete information game: • P1’s payoff can be anything within [0; 2]. Positive prob of commitment type that mechanically plays H. • Patient strategic incumbent guarantees payoff ≈ 2 in every BNE. The option to build a reputation for playing H selects the highest equilibrium payoff for the firm. Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion Commitment Payoff Theorem in Entry Deterrence Game An incumbent (P1) and a sequence of entrants (P2s). – O I F 2; 0 −1; −1 A 3; 0 0; 1 Repeated complete information game: • P1’s payoff can be anything within [0; 2]. Positive prob of commitment type that mechanically plays F. • Patient strategic incumbent guarantees payoff ≈ 2 in every BNE. The option to build a reputation for playing F selects the highest equilibrium payoff for the incumbent.
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