Sustaining collusion in a prisoners’ dilemma

Sustaining collusion in a prisoners’ dilemma • oligopoly context

Implications of one-shot PD analysis • Collusion between oligopolists is undesirable but it is also unlikely to be stable • firms are likely to be involved in a ‘prisoners’ dilemma’ – especially given that collusion is illegal and subject to punishment • they can agree to collude BUT this still leaves problem of enforcement. • So regulators don’t have to worry? • Problem is that collusion is sometimes stable e.g. if there is some possibility of enforcement or cooperation can be legally enforced

Possibility of Enforcement through Punishment Strategies • If the situation is one that is played out over a reasonable period of time (i.e. repeated –) then there may be scope for some kind of punishment for breaking the agreement • repeated contracting • long established, long-term relationships • agreements made about pricing/output over multiple time-periods • scope for serious penalties to be imposed

Maintaining collusion with punishment strategies: repetition • A situation where there is some probability that the scenario is repeated indefinitely • Payoffs for the string of repetitions = sum of payoffs from each repetition • and firms aim to maximise these expected value of payoffs • Intuition: If payoffs from collusion are high enough, a string of high payoffs from colluding could be higher than the one-off gain from ‘cheating’ followed by a string of low payoffs due to the likely break down of the collusive agreement

Possibility of Enforcement through meta-punishment-strategies • If the situation is one that is played out over repeated time periods then firms need a long-term, meta-strategy for the repeated prisoners’ dilemma • The meta-strategy should include a response i.e. punishment for cheating • Whether gains from collusion outweigh one-off gains of cheating plus costs of breakdown (=punishment for cheating) depends on: • The meta-strategies of the firms. • The probability of repetition

Possible meta-strategies in a repeated prisoners’ dilemma • e.g. (1) each firm announces that should the other cheat on the low-output agreement then in the next time period it will react by raising its own output – Tit-for-Tat. • e.g. (2) each firm announces that should the other cheat on the low-output agreement then in the next time period and all time periods thereafter it will react by raising its own output - GRIM.

Example: repeated PD with ‘grim’ meta-strategies • In the indefinitely repeated game: probability of repetition = P • Assume that if one of the firm’s cheats then it expects the other firm will cheat thereafter; the collusive agreement breaks down altogether - a ‘grim’ strategy • But if neither cheats they both expect the other to continue to collude

Expected payoffs • If one of the firms cheats then its expected payoff is: • 3 then 1 in every repetition (P = probability of repetition < 1) • 3 + 1P + 1P2 + 1P3+.........+1Pn+........... • an infinite series which converges and sums to: • = 3 + 1P/(1 - P)

Expected payoffs • If one of the firms cheats then its expected payoff is: • 3 then 1 in every repetition (P = probability of repetition) • 3 + 1P + 1P2 + 1P3.................... • = 3 + 1P/(1 - P) • As long as the firm cooperates the other cooperates so the firm’s payoff is a string of 2s until the game ends • = 2 + 2P + 2P2 + 2P3+........2Pn+............ • Which converges and sums to: 2/(1- P)

A decision rule for the firms • Both firms should collude (cooperate) i.e. no incentive to cheat as long as; • Payoff from colluding > Payoff from cheating • 2/(1- P) > 3 + 1P/(1- P) • 2 > 3(1- P) + 1P • 2 – 3 > P(1-3) ……….(divide through by -2) • -1/-2 < P i.e. ½ < P or P > ½ • So if P (= probability of repetition) is large enough both firms have an incentive to collude for as long as they have the opportunity to do so

Generalising to any PD scenario If c > a > d > b check that this is a PD

Expected payoffs • If cheat gain c followed by d till end; • c + dP + dP2 + dP3+.......dPn+....... • = c + dP/(1 -P) • If cooperate payoff is ‘a’ till game ends • = a + aP + aP2 + aP3+..........aPn+....... • = a/(1- P)

Expected payoffs • If cheat gain c followed by d till end; • c + dP + dP2 + dP3+.......dPn+.......= c + dP/(1 -P) • If cooperate payoff is ‘a’ till game ends • = a + aP + aP2 + aP3+..........aPn+....... = a/(1- P) • So the firms should collude as long as: • a/(1- P) > c + dP /(1- P) • a > c(1- P) + dP • a – c > P(d-c) ………dividing by (d-c) which is negative leads to the condition: (a-c)/(d-c) < P …..P needs to be > than (a-c)/(d-c)

Decision rule • Cooperate if: • P > (a-c)/(d-c) • If this condition is satisfied the players have an incentive to cooperate in all repetitions • In the previous numerical example: • c = 3, a = 2, d =1, b = 0 • So condition P > (a-c)/(d-c) is: • P > (2-3)/(1-3) = -1/-2 = ½ • ½ = critical value of P such that players have the incentive to cooperate

Class exercise • What is the critical value of P that ensures collusion in this game • Why and how does the value of P differ from the previous example?

Exercise • In this example c = 6, a = 3, d = 2, b = 1 • Gains from cheating: • 6 + 2P/(1- P) • Benefits from cooperation: • 3/(1- P) • Firms should collude if: • 3/(1- P) > 6 + 2P/(1- P) • (3-6)/(2-6) < P • P > ¾

Thinking about the probability of sustaining collusion • Critical value of P (the probability the game is repeated one more time) is ¾ in this new situation compared to ½ in the previous example – why the difference? • Hint: compare the two sets of payoffs

Thinking about the probability of sustaining collusion • Critical value of P (the probability the game is repeated one more time) is ¾ in this situation compared to ½ in the previous example • In first scenario c = 3, a = 2, d =1, b = 0 • In the second example c = 6, a = 3, d =2, b=1 • the gains from cheating (c) are much higher relative to the loss when both players cheat (d) - there is more incentive to cheat.

But..... • This ‘Grim’ strategy combination is not the only one that can sustain cooperation • The FOLK theorem says that there are an infinite number of strategies that can enforce any given outcome in a indefinitely or infinitely repeated game e.g. more refined carrot and stick strategies such as tit-for-tat • Too many alternative strategies and no way to predict which will be used

Class problem • Why might repetition a fixed number of times not be sufficient to sustain collusion in a PD scenario e.g. repetition 100 times (or n times where n is any number less than infinity)? • Hint – think about the last repetition e.g. if there are 20 repetitions, think of the 20th repetition, and then think of the 19th and so on (this is backward induction)

Implications • Oligopoly collusion will be sustained in some circumstances e.g. where (i) the probability of repetition is high enough (ii) short term gains from cheating relatively low (iii) costs from also being cheated on are relatively high • but new entrants to the sector also have to be kept out

Test your understanding • Collusion: Explain why game theorists predict that collusion between oligopolists maybe less fragile if there is some possibility of repetition in the longer term.

Oligopoly Exercise 2 2. Under what circumstances is collusion between firms more likely to be sustained? Illustrate your answer with reference to the FT article on cooperation between DVD makers

Exercise 2: possible answer 2. Collusion is more likely to be sustained if it can be enforced over the long term i.e. through repetition. In this case, the one-off gains from breaking the agreement (followed by repeated non-agreement) could be outweighed by longer-term (repeated) gains from commitment to the agreement. This appears to be the determining factor that induced Sony and Toshiba to agree on a common platform for DVDs; the potential gains from becoming the market leader were outweighed by the losses that would be sustained by maintaining a market divide. In the long-term, collusion is also more likely to be maintained if the participants can make credible threats of punishment that will be enforced against any party that breaks the collusive agreement (e.g. by imposing penalties for default). A collusive agreement may also be sustained if the parties to the agreement share norms of commitment.

Sustaining collusion in a prisoners’ dilemma