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 1 / 23

Research Question Question: 2 / 23

Research Question Question: How can cartel members sustain collusion? 2 / 23

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. 2 / 23

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? 2 / 23

Model (based on Sannikov 2007) Two-player game in continuous time. 3 / 23

Model (based on Sannikov 2007) Two-player game in continuous time. Imperfectly observable productive actions t P A 1 and A 2 A 1 t P A 2 3 / 23

Model (based on Sannikov 2007) Two-player game in continuous time. Imperfectly observable productive actions t P A 1 and A 2 A 1 t P A 2 Public Brownian signal about these actions dX t “ µ p A 1 t , A 2 t q dt ` dZ t 3 / 23

Model (based on Sannikov 2007) Two-player game in continuous time. Imperfectly observable productive actions t P A 1 and A 2 A 1 t P A 2 Public Brownian signal about these actions dX t “ µ p A 1 t , A 2 t q dt ` dZ t Expected profits under the play of A “ p A 1 , A 2 q 8 ż ” ˇ ı e ´ r p s ´ t q g i p A 1 t , A 2 W i t p A q “ E t t q ds ˇ A s , s P r t , 8s r ˇ t 3 / 23

Model (based on Sannikov 2007) Observable costly transfers k P r 0 , 1 s — passthrough ratio; 4 / 23

Model (based on Sannikov 2007) Observable costly transfers k P r 0 , 1 s — passthrough ratio; d Γ i t is sent by Player i at t ; k ¨ d Γ i t is received by Player ´ i at t ; 4 / 23

Model (based on Sannikov 2007) Observable costly transfers k P r 0 , 1 s — passthrough ratio; d Γ i t is sent by Player i at t ; k ¨ d Γ i t is received by Player ´ i at t ; Γ 1 t and Γ 2 t are cumulative transfers until t . 4 / 23

Model (based on Sannikov 2007) Observable costly transfers k P r 0 , 1 s — passthrough ratio; d Γ i t is sent by Player i at t ; k ¨ d Γ i t is received by Player ´ i at t ; Γ 1 t and Γ 2 t are cumulative transfers until t . Total expected payoffs under the play of p A , Γ q “ p A 1 , A 2 , Γ 1 , Γ 2 q 8 ż ” e ´ r p s ´ t q ´ ¯ˇ ı g i p A 1 t , A 2 t ` kd Γ ´ i W i t q ds ´ d Γ i t p A , Γ q “ E t ˇ A s , s P r t , 8s r ˇ t t 4 / 23

Stage game and minimax payoffs 5 / 23

Feasible profits 6 / 23

Sannikov 2007 7 / 23

Costly transfers for k “ 0 . 5 Player 1 sends $100 8 / 23

This paper 9 / 23

Supporting the best agreement, k=0 10 / 23

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. 11 / 23

Main idea 12 / 23

Main idea SPNE “ self-enforcing agreement. 12 / 23

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. 12 / 23

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! 12 / 23

Agreement Money Burning case: k “ 0. 13 / 23

Agreement Money Burning case: k “ 0. Fix a small ǫ ą 0. 13 / 23

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. 13 / 23

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 E t p Γ i s q increases. 13 / 23

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 E t p Γ i s q increases. Inertia Restriction: after a public deviation at p t , X t q , an agreement can not require transfers from the players until the first time s when |p s , X s q ´ p t , X t q| ě ǫ 13 / 23

Agreement H is the set of public histories possible under the agreement. Inertia Restriction implies that at any history H t P H there will be only finitely many observed deviations. 14 / 23

Agreement H is the set of public histories possible under the agreement. Inertia Restriction implies that at any history H t P H there will be only finitely many observed deviations. Let L p H t q be the subhistory of H t until the last public deviation. 14 / 23

Agreement H is the set of public histories possible under the agreement. Inertia Restriction implies that at any history H t P H there will be only finitely many observed deviations. Let L p H t q be the subhistory of H t until the last public deviation. An agreement is a possibly infinite tree of leafs : each leaf is labeled by an element from L p H q ; each leaf specifies recommended productive action plans assuming there will be no further deviations. 14 / 23

Agreement H is the set of public histories possible under the agreement. Inertia Restriction implies that at any history H t P H there will be only finitely many observed deviations. Let L p H t q be the subhistory of H t until the last public deviation. An agreement is a possibly infinite tree of leafs : each leaf is labeled by an element from L p H q ; 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. 14 / 23

Promise keeping Within each leaf, promised continuation values satisfy dW i t “ r p W i t ´ g i p A t qq dt ´ rd Γ i t ` r β i ` dX t ´ µ p A t q dt q t for some vector-process β i . 15 / 23

Profitable Deviations Instead of defining a payoff for each strategy, check only that there are no profitable deviations. 16 / 23

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. 16 / 23

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. 16 / 23

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 @ a 1 i P A i , g i p A t q ` β i t µ p A t q ě g i p a 1 i , A ´ i t q ` β i t µ p a 1 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. 17 / 23

Solution K p ǫ q 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 K p ǫ q . 18 / 23

Solution K p ǫ q 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 K p ǫ q . Lemma 2: K p ǫ q is convex and is above the minimax lines. 18 / 23

Solution K p ǫ q 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 K p ǫ q . Lemma 2: K p ǫ q is convex and is above the minimax lines. Lemma 3: For all sufficiently small ǫ , K p ǫ q is comprehensive. 18 / 23

Solution Lemma 4: For all sufficiently small ǫ , at each point of B K p ǫ q that is not a static NE and that lies strictly above the minimax lines, Sannikov optimality equation is satisfied. 19 / 23

Solution 20 / 23

Solution 21 / 23

Solution Lemma 5: For all sufficiently small ǫ , the curved part of B K p ǫ q meets the minimax lines either at a static NE or at a 90 degree angle. 22 / 23

Solution 23 / 23