Chapter 18: Modeling reputations

Atsushi Iwasaki

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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.

]1,0[iH} {L, 1 ta

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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;

•

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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

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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.

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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.

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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

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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.

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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

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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

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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．

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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．

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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

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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 ）

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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).15

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.

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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.

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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:

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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.

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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))

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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.

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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

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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–

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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

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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

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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–

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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

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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

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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.

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Proposition 18.6.1

• 証明は– Prop. 18.4.3 (2)– Prop. 18.5.1

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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31

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．

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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）

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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

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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

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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

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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．

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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．

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