the firm

Chapter 18: Modeling
reputations
Atsushi Iwasaki
1
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 i  [0,1]
• In each period t, the firm chooses an effort level a1t {L, H}
• 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.
2
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;
•
3
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
4
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.
5
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.
6
History and belief functions
• For consumer i,
– a period t history is a t-tuple of signals
– the payoffs in periods 0 through t-1
in
• 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
an induced probability measure on
• Then, given v and ,
,
,
, there is
.
• The revenue in period t after the history h1 is given by
7
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.
8
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
9
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
10
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．
11
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．
12
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
13
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，higheffort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）
14
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.
16
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.
17
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
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.
19
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
– player 2 chooses h with
in every period
• probability a2’ after good signal and
• probability a2 after bad signal,
– where a2’ = a2 + 1/(2d(p – q))
20
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.
21
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higheffort equilibriaがcommon signalの導入で達成
可能になる
22
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
–
23
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
24
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
25
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
–
26
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
27
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
28
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.
29
Proposition 18.6.1
• 証明は
– Prop. 18.4.3 (2)
– Prop. 18.5.1
30
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．
32
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）
33
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
34
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
35
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
36
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．
37
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．
38

Download Report