Dynamic Oligopoly

Dynamic OligopolyWeek 4: Tacit collusion. Martin ch 3. Belleflamme and Pietz 14.2Week 5: Dynamic competition with price rigidities. Maskin and Tirole (1988): A Theory of Dynamic Oligopoly: II: Price Competition, Kinked Demand Curves and Edgeworth Cycles, EconometricaDynamic Oligopoly: Supergames*

Dynamic Oligopoly: Supergames

Tacit CollusionCollusion among firms occurs when firms act in unison to set prices or divide up market to exploit consumers.In practice formal agreements between firms are difficult to enforce (and often illegal).How can informal agreements, i.e. unaccompanied by any binding contract, be self-enforcing?Repeated interaction introduces scope for self-enforcing tacit collusion. Not necessarily a formal agreement, just a common understanding about collusive behaviour and how deviations from it will be punished.Dynamic Oligopoly: Supergames*


Tacit CollusionTheoretical analysis of tacit collusion often draws on theory of repeated games.We look at case in which firms interact repeatedly and can condition their behaviour on the history of their interaction.Well require that strategies form a subgame perfect equilibrium.Well look at conditions under which equilibrium behaviour allows firms to collude.Dynamic Oligopoly: Supergames*


Repeated GamesIn a repeated game a stage game is played repeatedly, with players observing the outcome of the stage game after each stage.A players payoff from the repeated game is the discounted sum of payoffs from the individual stage games: i.e. player is repeated game payoff is i = t t1it, where [0, 1] is a discount factor and it is is stage game payoff in period t.What can be achieved in an equilibrium of the repeated game depends on the structure of the stage game.

Dynamic Oligopoly: Supergames*


Repeated GamesExamples of stage games: i) Prisoners Dilemma. Key features: cooperative outcome (c, c) yields payoffs 1 = 2 = 2 2s best-response to cooperation by 1 is defect yielding payoff 2 = 3unique equilibrium (d, d) yielding payoffs 1 = 2 = 1Dynamic Oligopoly: Supergames*

Player 2 dcPlayer 1d1, 13, 0c0, 32, 2


Repeated Gamesii) Cournot duopoly with zero costs and p = 1 Q.Players: {1, 2}Strategies: q1 0, q2 0Payoffs: 1 = (1 q1 q2)q1, 2 = (1 q1 q2)q2 Key features: joint-payoff maximising quantities q1 = q2 = yielding payoffs 1 = 2 = 1/8 best-response to q1 = is q2 = 3/8 yielding payoff 2 = 9/64 unique equilibrium q1 = q2 = 1/3 yielding payoffs 1 = 2 = 1/9 Dynamic Oligopoly: Supergames*


Repeated Gamesiii) Bertrand duopoly with zero costs and p = 1 Q Players: {1, 2}Strategies: p1 0, p2 0Payoffs: 1 = (1 p1)p1, 2 = 0 if p1 < p2 1 = (1 p1)p1, 2 = (1 p2)p2 if p1 = p2 1 = 0, 2 = (1 p2)p2 if p1 > p2 Key features: joint-payoff maximising prices p1 = p2 = yielding payoffs 1 = 2 = 1/8 best-response to p1 = is p2 = yielding payoff 2 unique equilibrium p1 = p2 = 0 yielding payoffs 1 = 2 = 0Dynamic Oligopoly: Supergames*


General ResultsFinitely repeated game: the stage game is played in periods t = 1, 2, , T.A general result: If the stage game has a unique Nash equilibrium, then the finitely repeated game has a unique subgame perfect equilibrium (SPE) where the Nash equilibrium of the stage game is played in every period.Dynamic Oligopoly: Supergames*


General ResultsReason: SPE strategies must specify that players play Nash equilibrium strategies of the stage game in period T. Given this (and because there is a unique Nash equilibrium in the stage game), nothing a player does in T 1 can influence SPE outcome of period T. Therefore in T 1 in order to maximise the sum of discounted payoffs from T 1 and T a player should maximise profits in stage T 1, i.e. play T 1 as if it were last period. This means they must play Nash equilibrium strategies of stage game in T 1. Etc., etc.Dynamic Oligopoly: Supergames*


General ResultsInfinitely repeated game: stage game is played in t = 1, 2, 3, i.e. no final period. Two general results:The infinitely repeated game has a SPE where the Nash equilibrium of the stage game is played in every stage.There are many other SPE (so-called "Folk theorems"). In particular, if sufficiently close to 1 the joint-payoff maximising outcome can be attained in a SPE.Dynamic Oligopoly: Supergames*


General ResultsA formal statement of 2: Let siN denote a Nash equilibrium strategy of the stage game for player i, and let itN = it(s1N, ,snN) denote the corresponding Nash equilibrium payoff of the stage game for player i.Consider a profile of stage game strategies (s1C, , snC) yielding each player i a payoff of itC = it(s1C, , snC) > itN.If is sufficiently close to one, then there exists a SPE of the infinitely repeated game where player i receives a repeated game payoff i = itC/(1 ).Dynamic Oligopoly: Supergames*


General ResultsThe SPE is supported by strategies which promise to play sC, but threaten to revert to sN:Play siC in period 1. In period t, if the outcome of all preceding periods has been (s1C, , snC) then play siC; otherwise play siN. If all players adopt this strategy this results in (s1C, , snC) being played in every period and each player receiving a repeated game payoff of itC/(1 ). Such strategies variously called trigger, grim, or Nash reversion strategies.



General ResultsThese strategies will form a Nash equilibrium if no player can do better than follow this strategy, given that all the other players are following this strategy.Suppose player i assumes all the other players follow trigger strategy, and is considering whether to use it as well: If player i uses trigger strategy she gets i = itC/(1 ). How could she improve on this?Dynamic Oligopoly: Supergames*


General ResultsIf player i deviates from siC, all other players will revert to their stage game equilibrium strategies in subsequent periods, and so in subsequent periods the best player i can do is choose siN, getting itN in each period. In the period when she deviates she should choose the best response to the cooperative strategy, i.e. maximise it (s1C, si1C, si, si+1C, , snC). Denote the resulting stage payoff by itD .Optimal deviation yields i = itD + itN/(1 ).Dynamic Oligopoly: Supergames*


General ResultsPlayer i has no incentive to deviate if itC/(1 ) itD + itN/(1 )or (itD itC)/(itD itN)If (itD itC)/(itD itN) for all i, then the trigger strategies form a Nash equilibrium.Note itD itC > itN and so (itD itC)/(itD itN) < 1. Thus there exists a < 1 such that the condition holds for all i.Dynamic Oligopoly: Supergames*


General ResultsTrigger strategies form an equilibrium if sufficiently high, specifically (itD itC)/(itD itN)Note that this condition can be written (itD itC)/(itD itC + itC itN)or 1/(1 + (itC itN)/(itD itC))So in order to sustain collusion in equilibrium using trigger strategies we need benefit of collusion (itC itN) to be large enough relative to temptation to deviate (itD itC) Dynamic Oligopoly: Supergames*


General ResultsTo see that the Nash equilibrium will be subgame perfect consider a subgame beginning in period t. This subgame is itself an infinitely repeated game where in each period the players play the stage game, receive a stage game payoff, and the payoff from this subgame is the discounted sum of the stage payoffs.That is, the subgame is just like the original repeated game. Dynamic Oligopoly: Supergames*


General ResultsThere are two possibilities:If a previous period resulted in an outcome other than (s1C, ..., snC) the players plan to play the Nash equilibrium of the stage game in every stage. This constitutes a Nash equilibrium for the subgame.If the outcome of all previous periods is (s1C, ..., snC) the players plan to play the trigger strategy. This also constitutes a Nash equilibrium for the subgame.Dynamic Oligopoly: Supergames*


Example: The Infinitely Repeated Prisoners DilemmaLets return to our symmetric Prisoners Dilemma. The trigger strategy is:In period 1 play c. In period t > 1, if the outcome of all preceding periods has been (c, c) then play c; otherwise play d.Dynamic Oligopoly: Supergames*

Player 2 dcPlayer 1d1, 13, 0c0, 32, 2


Example: The Infinitely Repeated Prisoners DilemmaSuppose Player 1 adopts this strategy. If Player 2 also uses this strategy the outcome will be (c, c) in every period: 2 = 2 + 2 + 2 2 + = 2/(1 ).If Player 2 plays d in period 1, then Player 1 will choose d in periods 2, 3, . Player 2s optimal deviation results in (c, d) in period 1 and (d, d) in all subsequent periods.Payoff from deviation: 2 = 3 + 1 + 2 1 + = 3 + /(1 ).No incentive to deviate if 2/(1 ) 3 + /(1 )or . Dynamic Oligopoly: Supergames*


Example: The Infinitely Repeated Prisoners DilemmaIf then it is a Nash equilibrium for both players to use the trigger strategy.In fact, this Nash equilibrium is subgame perfect.Thus, if is sufficiently high (i.e., if players are sufficiently patient) cooperation can be sustained as a subgame perfect equilibrium in the infinitely repeated Prisoners Dilemma.Dynamic Oligopoly: Supergames*


Example: Basic Cournot Gameii) Cournot duopoly with zero costs and pt = 1 Qt.Can joint-profit maximising outcome be sustained as an equilibrium?From earlier slide: q1tC = q2tC = , q1tN = q2tN = 1/3, 1tC = 2tC = 1/8, 1tD = 2tD = 9/64, 1tN = 2tN = 1/9.The trigger strategy isIn period 1 produce 1/4. In period t > 1, if the outcome of all preceding periods has been (1/4, 1/4) then produce 1/4; otherwise produce 1/3. This is a subgame perfect Nash equilibrium as long as (itD itC)/(itD itN) = 9/17



Example: Basic Cournot GameNote that this result can be generalised to the case with n firms: itC = 1/(4n), 1tD = (n+1)2/(4n)2, itN = 1/(n + 1)2Joint-profit maximising outcome can be sustained in equilibrium as long as (itD itC)/(itD itN) = (n+1)2/(n2 + 6n +1) The expression (n+1)2/(n2 + 6n +1) increases with n. In this sense, collusion is harder to sustain the larger the number of firms in the market.E.g. if = 0.6 can sustain collusion between 2 firms but not between 5 firms. Dynamic Oligopoly: Supergames*


Example: Basic Bertrand Gameiii) Bertrand duopoly with zero costs and Qt = 1 pt Can joint profit maximising outcome be sustained as an equilibrium?From earlier slide: p1tC = p2tC = , p1tN = p2tN = 0, 1tC = 2tC = 1/8, 1tD = 2tD = 1/4, 1tN = 2tN = 0.The trigger strategy isIn period 1 charge . In period t > 1, if the outcome of all preceding periods has been (, ) then charge ; otherwise charge 0. This is a subgame perfect Nash equilibrium as long as (itD itC)/(itD itN) = 1/2Dynamic Oligopoly: Supergames*


Example: Basic Bertrand GameWith n firms: itC = 1/(4n), 1tD = 1/4, itN = 0Joint-profit maximising outcome can be sustained in equilibrium as long as (itD itC)/(itD itN) = 1 1/n The expression 1 1/n increases with n. Again, collusion is harder to sustain the larger the number of firms in the market. Dynamic Oligopoly: Supergames*


Example: Price versus Quantity CompetitionIn previous lectures we found that in markets for homogeneous goods and constant average and marginal costs, price competition is fiercer than quantity competition. Equilibrium prices lower when firms compete in prices.These lectures looked at static models. What can we say about price versus quantity competition from perpective of repeated game theory?Is it easier to sustain collusion under price or quantity competition?Dynamic Oligopoly: Supergames*


Example: Price versus Quantity CompetitionCompared to quantity competition, under price competition benefit of collusion is larger, but so is temptation to deviate.With n = 2 the first effect dominates and it is easier to sustain collusion using grim trigger strategies under price competition.With n > 2 the second effect dominates and it is easier to sustain collusion using grim trigger strategies under quantity competition.Dynamic Oligopoly: Supergames*


not at all only under price under either price or competition quantity competition Example: Price versus Quantity CompetitionCan joint profit maximisation be sustained (by trigger strategies)? If n = 2:If n = 3: 0 9/17 0 4/7 2/3 not at all only under qty under either price or competition quantity competition Dynamic Oligopoly: Supergames*


Stick and Carrot StrategiesWe found that trigger strategies sustain collusion in a linear Cournot duopoly if > 9/17 0.53.What if firms less patient? E.g. if = ?Trigger strategies cannot sustain full collusion in this case.But while trigger strategies are simple, perhaps they are too grim?Perhaps there are some other repeated game strategies that might sustain collusion?Dynamic Oligopoly: Supergames*


Stick and carrot strategies allow players to go back to cooperative play (carrot) after a deviation from cooperation, but only after some punishment has been meted out (stick).The presence of the carrots give players an incentive to use the stick.Even if the punishment involves choices that are not part of a Nash equilibrium of the stage game, the carrot might make such punishment credible.This gives possibility of more severe punishments that deter deviation from cooperation.Dynamic Oligopoly: Supergames*Stick and Carrot Strategies


We will present stick and carrot strategies for Cournot duopoly with zero costs and pt = 1 QtRecall joint profit maximisation: q1tC = q2tC = , 1tC = 2tC = 1/8Game can be in either cooperative or punishment phase depending on history of playS&C strategy specifies in cooperative phase produce , in punishment phase produce Players keep track of which phase the game is in as followsDynamic Oligopoly: Supergames*Stick and Carrot Strategies


Begin in period one in cooperative phasei) If game was in cooperative phase in previous period and outcome was (, ) then game stays in cooperative phaseii) If game was in cooperative phase in previous period and outcome was not (, ) then game switches to punishment phaseiii) If game was in punishment phase last period and outcome was (, ) then game switches to cooperative phaseiv) If game was in punishment phase in previous period and outcome was not (, ) then game stays in punishment phaseDynamic Oligopoly: Supergames*Stick and Carrot Strategies


If both players follow this strategy it will result in (, ) in every period, and so joint profits maximised.But is this a subgame perfect Nash equilibrium?We will assume player 1 follows stick and carrot strategy and check whether player 2 has an incentive to deviate.Note 2t(, ) = 1/8, 2t(, ) = 0. Player 2s best-response function is q2 = (1 q1)/2 so best response to is 3/8: 2t(, 3/8) = 9/64. Best response to is : 2t(, ) = 1/16.Dynamic Oligopoly: Supergames*Stick and Carrot Strategies


To check whether these strategies form an equilibrium we need to check that punishment is credible: having entered punishment phase players do not have incentive to deviate from producing collusion is optimal: no incentive to deviate from Dynamic Oligopoly: Supergames*Stick and Carrot Strategies


First consider a subgame in which a player has deviated in previous period (i.e. we are in punishment phase)By complying with stick and carrot strategy player 2 gets 0 + (1/8)/(1 ) = VPSuppose player 2 decides to deviate this period, then comply with stick and carrot strategy next period. Her best deviation gives 1/16 + VPDynamic Oligopoly: Supergames*Stick and Carrot Strategies


Optimal to comply now rather than later if VP 1/16 + VP (1 )VP 1/16 /8 1/16 (Note: if its optimal to comply now rather than later, its also optimal to comply now rather than never. i.e. VP 1/16 + VP VP (1/16)/(1 ).)

Dynamic Oligopoly: Supergames*Stick and Carrot Strategies


Next consider a subgame where stick and carrot strategy requires producing q2 = (i.e. cooperative phase)By complying with stick and carrot strategy player 2 getsVC = (1/8)/(1 )By deviating for one period then complying she gets 9/64 + VPOptimal to comply if VC 9/64 + VP (1/8)/(1 ) 9/64 + 2 (1/8)/(1 ) 1/8



The stick and carrot strategies form a subgame perfect Nash equilibrium as long as . Thus in the infinitely repeated Cournot quantity setting game, full collusion may be sustainable with stick and carrot strategies even when it is not sustainable by grim trigger strategies.Dynamic Oligopoly: Supergames*Stick and Carrot Strategies


In general, a stick and carrot strategy specifies qC and qP. Let itC = it(qC, qC) denote stage game payoff in cooperative phase (when both players comply)itP = it(qP, qP) denote stage game payoff in punishment phase (when both players comply) itD denote stage game payoff when j complies and i deviates optimally in cooperative phaseitE denote stage game payoff when j complies and i deviates optimally in punishment phaseDynamic Oligopoly: Supergames*Stick and Carrot Strategies


For these to form a subgame perfect Nash equilibrium qC and qP must satisfyPunishment is credible:itE itP (itC itP)(C)Collusive output is sustainable:itD itC (itC itP)(S)Note that if these two conditions are met punishment does not occur in equilibrium. The threat of punishment disciplines players and allows them to achieve qC in every period.



The best collusive outcome that can be sustained occurs when qP threatens the harshest credible punishment. i.e. condition C binds:itE itP = (itC itP)ii)Full collusion is sustainable if condition (S) can then be met when qC = . If S cannot be met when qC = , the best that can be achieved is the lowest output satisfying itD itC = (itC itP)Dynamic Oligopoly: Supergames*Stick and Carrot Strategies


Optimal stick-and-carrot strategy in the linear Cournot duopoly (based on Belleflamme and Peitz Figure 14.1, with p = 1 Q, c = 0)Dynamic Oligopoly: Supergames*Stick and Carrot Strategies


delta = 1/(1 + r)*******

Date post:	15-Nov-2015
Category:	Documents
Upload:	akash-bhatia
View:	21 times
Download:	0 times

Dynamic Oligopoly

Documents