Chapter 18: Modeling reputations

1. Chapter 18: Modeling reputations Atsushi Iwasaki

2. 18.1 An Alternative Model of Reputations • A single long-lived player, or the firm • A continuum of small and anonymous players, or consumers, indexed by • In each period t, the firm chooses an effort level • Each consumer is long-lived and observes an idiosyncratic realization of a signal. • The two possible values: z (good) and z (bad), with marginal distribution • In each period t and for each group of consumers having experienced a common history of signals, a proportion of this group receives the good signal.

3. Payoffs of the game • The (normal) firm's stage-game payoff depends on • its revenue: a function p(F) of consumer expectations about effort • its costs: Low effort is costless; high effort is c. • A consumer receives payoff 1 from signal z- (good) and 0 from z (bad). • Consumer expectations: • a distribution function F(x) • The proportion of consumers who expect the firm to exert high effort with probability less than or equal to x. • The revenue function p(F) is strictly increasing, so that higher expectations of high quality lead to higher revenue;

4. 完全価格差別モデル(perfect price discrimination) • 各消費者は毎期毎に1単位の財を購入 • 企業は各消費者にその留保価格を支払わせる：各消費者が支払いたい価格？ • p(1): 消費者が確率1でhigh effortを予想しているときの収入 • p(0):消費者が確率0でhigh effortを予想しているときの収入 • を仮定することで，high effortがefficient • 同様に　　　　　　　　　　を仮定することでhigh effortがthe pure Stackelberg action for the firm

5. Firm’s types and replacements • In the repeated game, the normal firm maximizes the discounted sum of expected profits, with discount factor δ. • Two types of firm • Normal: The firm choose high or low effort. • Inept: The firm can only choose low effort. • Before play begins, nature determines the original type of the firm, choosing normal with probability and inept . • The firm learns its type, but consumers do not. • In each subsequent period, there is a probability λ that the firm is replaced • With probability of the new firm being normal. • Consumers cannot observe whether a replacement has occurred. • For example, the ownership of a restaurant might change without changing the restaurant's name and without consumers being aware of the change.

6. Flow of the game • At the beginning of period t, each consumer i is characterized by her posterior probability that the firm is normal. • Her posterior probability that the firm will exert high effort, denoted . • If the firm is normal, it makes its (unobserved) effort choice. • The firm receives revenues that depend • on F() of consumers' beliefs about the firm's effort, • but not on the firm's type or action in that period. • Consumers observe their own signals and update beliefs about the type of firm. • Finally, with probability λ, the firm is replaced.

7. History and belief functions • For consumer i, • a period t history is a t-tuple of signals in • the payoffs in periods 0 through t-1 • A belief function for consumer i is a function • is the probability consumer i assigns to the firm exerting high effort in period t, given history . • For firm, given a period t history , , there is an induced probability measure on , . • Then, given v and , • The revenue in period t after the history h1 is given by

8. A pure strategy for a normal firm • The pair will be an equilibrium if • is maximizing for normal firms after every effort history • Consumers' beliefs about effort choice, , are (correctly) determined by Bayes' rule.

9. Posterior probability of consumers • The normal firm always chooses high effort. • The posterior probability: • : a prior probability that the firm is normal and that the normal firm chooses high effort. • : the posterior belief of a consumer who had observed

10. Pure-strategy equilibria • Definition 18.1.1 (High-effort equilibrium) • 1: Firm’s strategy is sequential rational. • 2: Consumers’ belief is consistent. • Low-effort equilibrium

11. Proposition 18.1.1 • 企業が入れ替わる (replacement) 可能性がある場合，high-effort equilibriumが存在する企業のコストの上限が存在する • 企業がhigh effortするコストがそれほど大きくなければ high-effort equilibriumが存在する • しかし，常にlow effortを選ぶような企業に入れ替わる可能性がないと，消費者の企業のタイプに対する事後確率が１になる（企業のタイプが確実にわかってしまう）ため， high-effort equilibriumが存在しなくなる．

12. 18.2: The Role of Replacements • Replacementsがなければ，high-effort equilibriumは存在しない • Replacementsがなく，企業のタイプがnormalとわかっている（　　　　　） • 企業が努力すると想定しているとき，bad signalを観測した消費者は，企業は努力したが，たまたま間違った観測が起きたと考える． • それぞれの消費者が異なるシグナルを観測しうる場合，企業はlow effortを選ぶ誘因を持つ．

13. Incomplete information case • 企業のタイプが完全にはわからない不完備情報　　　　の場合でも同じ議論が成立 • The posterior probability of consumers • αは企業がとる純粋戦略 • 同様に，事後の信念も定義できる

14. Proposition 18.2.1 • 企業が消費者に「製品の品質を落とすかもしれないよ」と脅すことがよい均衡を達成することを助ける • ここで「評判」の目的は消費者に企業がnormalで，high-effortを選ぶと納得してもらうこと． • このとき，replacementsが企業にとってhigh-effortを選ぶインセンティブを与える． • もちろんreplacementsの代わりにcompetitionも同様の効果を与える（Section 18.4.6）

15. 18.3: Good Types and Bad Types • Product-choice game • A long-lived player 1 facing a short-lived player 2 • Normal or Bad (inept) type • Bad type commits to action L • The lower bound on player 1 's payoff • Under perfect monitoring, player 1 must earn at least his minmax payoff of 1 (Prop. 15.3.1).

16. Proposition 18.3.1 • Any payoff in the interval (1, 2] is also an equilibrium payoff for a sufficiently patient player 1 in the game of incomplete information. • A tighter bound is not available, and the possibility of an inept type has no effect on the set of payoff possibilities for player 1.

17. A belief-free equilibrium with complete information collapses • Player 1 plays in each period • Player 2はhでもlでも各期の期待利得は1.5 • Suppose the normal player 1 chooses L with probability • is the period t posterior of player 2 that player 1 is bad type. • However, bad signal pushes upward the posterior. • The probability that player 1 chooses L decreases • The posterior will be pushed above 1/2, at which point the equilibrium collapses.

18. Good Types • Product-choice game • A long-lived player 1 facing a short-lived player 2 • Normal or Good type • Good type commits to the pure Stackelberg action H • An equilibrium in the perfect monitoring game • The normal player 1 plays H in every period, supported by the threat that any deviation to L prompts the perpetual play of Ll. • If we add replacements to this model, such a equilibrium is no longer a sequential equilibrium, when player 1 is • sufficiently patient: • replacements sufficiently unlikely:

19. 18.4: Reputations with Common Consumers • The model so far assumes that the players receive idiosyncratic signals. • In the absence of replacements, consumers who receive bad signals do not punish the firm. • 「badはたまたまだ！」 • If the consumers receivecommon signals, there is no difficulty in using bad signals to trigger punishments.

20. Belief-Free Equilibria with Idiosyncratic Consumers • Consider a version of the private monitoring product-choice game analyzed in section 12.5. • 以下のような均衡を構成できることがわかっているが，belief-free 以外の均衡についてはほとんどわかっていない． • An belief-free equilibrium • player 1 plays in every period • player 2 chooses h with • probability a2’ after good signal and • probability a2after bad signal, • where a2’ = a2+ 1/(2d(p – q))

21. Common Consumers • We retain the model of section 18.1, except that • in each period, either all consumers receive a common good outcome or all receive a common bad outcome. • We restrict attention to public strategy profiles. • Hence after any history, every consumer holds the same expectation of high effort. • The pricing function from section 18.1 • There exist equilibria in which the normal firm oftenexerts high effort.

22. An equilibrium • Firm’s strategy • Initially exert high effort and continue to do so as long as good signal is realized. • Bad signal prompts L > 1 periods of low effort and low price (punishment) • An equilibrium as long as the cost c is sufficiently small. • これまではincomplete informationやreplacementsがないと達成出来なかったhigh-effort equilibriaがcommon signalの導入で達成可能になる

23. Markov strategies • common consumer modelをidiosyncratic consumer modelに合わせて理解するためにMarkov strategiesに着目 • 消費者の信念をcommon signalに合わせて更新 • Definition 18.4.1

24. Proposition 18.4.1 • Markov strategyの概念を使って，企業がhigh effortを実行するマルコフ均衡が存在するコストの上限を導ける． • The value function of the normal firm • The payoff from exerting low effort • を計算すると以下の不等式を得る

25. Remaining • 18.4.4 Replacements • 企業のタイプが入れ替わる • 18.4.5 Continuity at the Boundary and Markov Equilibria • Prop. 18.4.2の一般化 • 18.4.6 Competitive Markets • 競争によるhigh-effort equilibriumの達成

26. 18.5: Discrete Choices • ここまでは消費者の信念の変化に対する反応（consumer choice）は連続的に表現 • 本節ではこれを離散的に表現することを考える． • Consumer chooses h or l. • これまで扱ってきたproduct-choice gameで, ならば消費者はhを，そうでなければlを選ぶようになる． • Proposition 18.5.1

27. 18.6: Lost Consumers • 18.1の消費者は企業からどんな悪いシグナルを受け取ろうが，企業から商品を購入し続ける． • 本節では，sufficiently pessimistic consumerが購入を止めるoutside optionを導入

28. The Purchase Game • If the consumer buys (chooses b), high effort produces a good outcomewith and low effort a good outcome with probability • The consumer values • a good signal at • a bad signal at • If the consumer does not buy (chooses d), then no signal is observed

29. The Purchase Game (contd.) • Normal firms can exert either high or low effort, and inept firms inevitably exert low effort. • The firm is replaced in each period with probability • With the replacement being normal with probability • The essential message of the previous sections continues to hold in the presence of the outside option.

30. Proposition 18.6.1 • 証明は • Prop. 18.4.3 (2) • Prop. 18.5.1

31. 18.6.2 Bad Reputations: The Stage Game • 消費者の事後確率が１／２を下回ると彼らは企業から購入しなくなる． • Normalがhigh effortを実行するインセンティブはno-trade zoneを避けることから生じる． • 消費者は企業を雇ってサービスを提供させる • 企業は医者でアスピリンを処方するか心臓移植するかを決める • 企業はPCサポートでハードディスクをフォーマットするか新しいPCを進めるかを決める • どちらの判断がよいかは状態（ランダム）によって決まる

32. 18.6.2 Bad Reputations: The Stage Game (contd.) • 自然が状態をランダムに決める． • ステージゲームは展開型ゲームとなる． • 消費者がHireを選べば，企業は提供するサービスのレベルを決める • このゲームは一意の逐次均衡をもち，そこで，企業は状態に合わせたサービスを提供する．

33. 18.6.3 The Repeated Game • 企業はlong-run player, 消費者はshort-run player • 各期に新しい消費者プレイヤがやってくる． • 自然はその度に状態を決定し，企業にだけ事後の状態を伝える． • 消費者は企業を雇うか否かを決定し，企業はサービスレベルを決定する． • その期の終わりに公的シグナルYを観測する • X: 企業が雇われない • H: High effort service が提供された • L : Low effort service が提供された • 企業が常に雇われて，適切なレベルのサービスを提供するのがtrivialな均衡 • 一方で，企業がminmax payoffを与えられる均衡も存在する． • 企業が絶対に雇われることがない（利得はゼロ）

34. 18.6.4 Incomplete Information • 不完備情報の場合は企業の利得が著しく低い均衡が実現する • 確率 で，企業はnormal. • 確率 で，企業はbad • 毎期，独立かつ同一の分布から確率γでH, 1-γでLを選択する（ランダム）． • ただし，γはnormalと振舞いが異なるよう以下の制約をつける • Hを観測することで企業がbadである事後確率が増加する． • Prop. 18.6.4 • この設定の元，均衡におけるnormalの利得の上限は０になる

35. 18.6.5 Good Firms • Normal: H or L • Bad: L only • Stackelberg: H only • Only Stackelberg and bad types • Consumers will enter iff • η: the probability of the Stackelberg type

36. Equilibrium with three tyeps • Region B: • If Prob(B) > 1 - η*, the consumerwill never hire the firm. • Region S: • If Prob(S) is atleast η*, consumers will always hire the firm. • Other region: • カーブより下の部分で，normalが得る利得は全て均衡になる • normalが十分patientなら利得は０に収束

37. 18.6.6 Captive Consumers • Consumerにタイプを導入 • Normal: 1-ε • Captive: εの確率で企業の履歴に関わらず購入するconsumer • Prop. 18.6.6 • δが1に，εが0に近づく限りは企業の均衡利得は0に収束 • Prop. 18.6.7 • δが1に近づき，εがある程度大きいと，企業の利得はuに近づく．

38. 18.7: Markets for Reputations • 評判を売買することを考える • 商品を購入することは売り手の評判を購入することとみなす． • 世代重複経済 (an overlapping generations economy) の2期間のスナップショット • 無限期間への一般化も可能 • 消費者と企業の１回の取引終了後， • ２期過ごした古い企業 はいったん全て消える • １期過ごした新しい企業は古い企業になる． • このとき，元の名前を維持するか， • 元の名前を放棄して，新しい名前にするか， • 元の名前を放棄して，古い名前を購入する． • Prop. 18.7.1 • どんな均衡でも古い名前の取引が起こる． • Prop. 18.7.2 and 18.7.3 • Reputation equilibriumの様々な特徴づけを示している．