Chapter 18: Modeling reputations Atsushi Iwasaki

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.

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;

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

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.

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.

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

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.

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

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

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．

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．

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

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）

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).

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.

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.

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:

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 receive common signals, there is no difficulty in using bad signals to trigger punishments.

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 a2 after bad signal, where a2’ = a2 + 1/(2d(p – q))

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 often exerts high effort.

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

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

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

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

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

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

The Purchase Game If the consumer buys (chooses b), high effort produces a good outcome with 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

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.

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

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

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．

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）

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

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

Equilibrium with three tyeps Region B: If Prob(B) > 1 - η*, the consumer will never hire the firm. Region S: If Prob(S) is at　least 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

18.6.6 Captive Consumers Consumerにタイプを導入 Prop. 18.6.6 Prop. 18.6.7 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．

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．