Costly Public Transfers in Repeated Cooperation Under Imperfect - - PowerPoint PPT Presentation

Costly Public Transfers in Repeated Cooperation Under Imperfect Monitoring

Mikhail Panov and Sergey Vorontsov

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Research Question

Question:

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Research Question

Question: How can cartel members sustain collusion?

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Research Question

Question: How can cartel members sustain collusion? By a threat of going into a price war. Under imperfect monitoring happens with probability 1; punishes everybody.

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Research Question

Question: How can cartel members sustain collusion? By a threat of going into a price war. Under imperfect monitoring happens with probability 1; punishes everybody. In this paper How can transfers among cartel members help additionally sustain collusion?

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Model (based on Sannikov 2007)

Two-player game in continuous time.

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Model (based on Sannikov 2007)

Two-player game in continuous time. Imperfectly observable productive actions A1

t P A1 and A2 t P A2

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Model (based on Sannikov 2007)

Two-player game in continuous time. Imperfectly observable productive actions A1

t P A1 and A2 t P A2

Public Brownian signal about these actions dXt “ µpA1

t , A2 t q dt ` dZt

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Model (based on Sannikov 2007)

Two-player game in continuous time. Imperfectly observable productive actions A1

t P A1 and A2 t P A2

Public Brownian signal about these actions dXt “ µpA1

t , A2 t q dt ` dZt

Expected profits under the play of A “ pA1, A2q W i

t pAq “ Et

” r

8

ż

t

e´rps´tqgipA1

t , A2 t qds

ˇ ˇ ˇAs, s P rt, 8s ı

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Model (based on Sannikov 2007)

Observable costly transfers k P r0, 1s — passthrough ratio;

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Model (based on Sannikov 2007)

Observable costly transfers k P r0, 1s — passthrough ratio; dΓi

t is sent by Player i at t;

k ¨ dΓi

t is received by Player ´i at t;

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Model (based on Sannikov 2007)

Observable costly transfers k P r0, 1s — passthrough ratio; dΓi

t is sent by Player i at t;

k ¨ dΓi

t is received by Player ´i at t;

Γ1

t and Γ2 t are cumulative transfers until t.

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Model (based on Sannikov 2007)

Observable costly transfers k P r0, 1s — passthrough ratio; dΓi

t is sent by Player i at t;

k ¨ dΓi

t is received by Player ´i at t;

Γ1

t and Γ2 t are cumulative transfers until t.

Total expected payoffs under the play of pA, Γq “ pA1, A2, Γ1, Γ2q W i

t pA, Γq “ Et

” r

8

ż

t

e´rps´tq´ gipA1

t , A2 t qds ´dΓi t`kdΓ´i t

¯ˇ ˇ ˇAs, s P rt, 8s ı

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Stage game and minimax payoffs

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Feasible profits

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Sannikov 2007

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Costly transfers for k “ 0.5

Player 1 sends $100

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This paper

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Supporting the best agreement, k=0

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Related Literature

Collusion under imperfect monitoring Green and Porter 1984; Abreu, Pearce and Stachetti 1986; Fudenberg, Levine, and Maskin 1994; Sannikov 2007. Collusion with asymmetric punishments Harrington and Skrzypacz 2007. Repeated games with transfers Levine 2002; Goldluecke and Kranz 2012. Continuous games with observable actions Simon and Stinchcombe 1989; Bergin and MacLeod 1993; Hackbarth and Taub 2015.

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Main idea

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Main idea

SPNE “ self-enforcing agreement.

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Main idea

SPNE “ self-enforcing agreement. An agreement specifies productive actions and transfers after any history; players can publicly deviate in transfers; specifies continuations after observed deviations.

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Main idea

SPNE “ self-enforcing agreement. An agreement specifies productive actions and transfers after any history; players can publicly deviate in transfers; specifies continuations after observed deviations. Strategies are defined only after an agreement is proposed!

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Agreement

Money Burning case: k “ 0.

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Agreement

Money Burning case: k “ 0. Fix a small ǫ ą 0.

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Agreement

Money Burning case: k “ 0. Fix a small ǫ ą 0. An agreement recommends productive actions and transfer processes after any public history possible under it.

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Agreement

Money Burning case: k “ 0. Fix a small ǫ ą 0. An agreement recommends productive actions and transfer processes after any public history possible under it. Each Player at any time can choose her hidden productive action; announce a deviation when she is required to transfer a positive amount of money, i.e. when EtpΓi

sq increases.

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Agreement

Money Burning case: k “ 0. Fix a small ǫ ą 0. An agreement recommends productive actions and transfer processes after any public history possible under it. Each Player at any time can choose her hidden productive action; announce a deviation when she is required to transfer a positive amount of money, i.e. when EtpΓi

sq increases.

Inertia Restriction: after a public deviation at pt, Xtq, an agreement can not require transfers from the players until the first time s when |ps, Xsq ´ pt, Xtq| ě ǫ

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Agreement

H is the set of public histories possible under the agreement. Inertia Restriction implies that at any history Ht P H there will be

• nly finitely many observed deviations.

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Agreement

H is the set of public histories possible under the agreement. Inertia Restriction implies that at any history Ht P H there will be

• nly finitely many observed deviations.

Let LpHtq be the subhistory of Ht until the last public deviation.

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Agreement

H is the set of public histories possible under the agreement. Inertia Restriction implies that at any history Ht P H there will be

• nly finitely many observed deviations.

Let LpHtq be the subhistory of Ht until the last public deviation. An agreement is a possibly infinite tree of leafs: each leaf is labeled by an element from LpHq; each leaf specifies recommended productive action plans assuming there will be no further deviations.

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Agreement

H is the set of public histories possible under the agreement. Inertia Restriction implies that at any history Ht P H there will be

• nly finitely many observed deviations.

Let LpHtq be the subhistory of Ht until the last public deviation. An agreement is a possibly infinite tree of leafs: each leaf is labeled by an element from LpHq; each leaf specifies recommended productive action plans assuming there will be no further deviations. Stategy: Given an agreement, a strategy for Player i specifies within each leaf a productive action plan and a stopping time at which the player deviates.

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Promise keeping

Within each leaf, promised continuation values satisfy dW i

t “ rpW i t ´ gipAtqqdt ´ rdΓi t ` rβi t

` dXt ´ µpAtqdtq for some vector-process βi.

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Profitable Deviations

Instead of defining a payoff for each strategy, check only that there are no profitable deviations.

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Profitable Deviations

Instead of defining a payoff for each strategy, check only that there are no profitable deviations. Sufficient to look only on finitely many observed deviations.

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Profitable Deviations

Instead of defining a payoff for each strategy, check only that there are no profitable deviations. Sufficient to look only on finitely many observed deviations. When evaluating the gain, use upper Lesbegue integral.

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Incentive compatibility

An agreement is self-enforcing if and only if the following two conditions hold

1 (One-stage DP in hidden actions) in all leafs and at all

times @a1

i P Ai, gipAtq ` βi tµpAtq ě gipa1 i, A´i t q ` βi tµpa1 i, A´i t q

2 (One-stage DP in observable actions) in all leafs at all

times when a player can publicly deviate, her promised continuation value in the leaf following the deviation is no greater than on path.

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Solution

Kpǫq is the set of payoffs attainable in self-enforcing agreements. Let S be the set of payoffs attainable in Sannikov PPEs. Suppose that r is small enough, so that S has interior. Lemma 1: S Ă Kpǫq.

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Solution

Kpǫq is the set of payoffs attainable in self-enforcing agreements. Let S be the set of payoffs attainable in Sannikov PPEs. Suppose that r is small enough, so that S has interior. Lemma 1: S Ă Kpǫq. Lemma 2: Kpǫq is convex and is above the minimax lines.

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Solution

Kpǫq is the set of payoffs attainable in self-enforcing agreements. Let S be the set of payoffs attainable in Sannikov PPEs. Suppose that r is small enough, so that S has interior. Lemma 1: S Ă Kpǫq. Lemma 2: Kpǫq is convex and is above the minimax lines. Lemma 3: For all sufficiently small ǫ, Kpǫq is comprehensive.

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Solution

Lemma 4: For all sufficiently small ǫ, at each point of BKpǫq that is not a static NE and that lies strictly above the minimax lines, Sannikov optimality equation is satisfied.

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Solution

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Solution

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Solution

Lemma 5: For all sufficiently small ǫ, the curved part of BKpǫq meets the minimax lines either at a static NE or at a 90 degree angle.

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Solution

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