L21 - Collusion

Question 1

What are the conditions necessary for oligopolists to collude?

Answer 1

C(1) –> Sellers must be aware of each other’s strategies

• If a firm is not aware, how can it punish a deviation?
• Firms may form cartels to check that each other are conforming to the agreement

C(2) –> Sellers must interact repeatedly

• Incentive “to deviate” must be counteracted by a credible long-term punishment
• Punishments usually mean a price war (period of low prices)

Question 2

What type of well known game are duopolies effectively competing in?

Answer 2

• They are effectively in a prisoner’s dilemma –> like Bertrand’s model of Oligopolies cause of the nature of the game both players would defect or use the minimum level of prices
• in the Cournots model of oligopoly if they both chose to restrict their output –> they could both enjoy the monopoly profits
• but in the short term –> both firms have a short term incentive to produce more that it said it would
• in Bertrand’s model if they both colluded to raise their prices they both would be better off at the monopolist price –> but if they were to do so they would have the short term incentive to undercut each other to maximise their profit

Question 3

How do you solve a finitely repeated prisoner’s dilemma?

Answer 3

• Notice that the game is a prisoner’s dilemma
  The “one-shot” Nash equilibrium (if game played once) is (deviate, deviate)
  Is (cooperate, cooperate) a Nash equilibrium of the repeated game
• NAÏVE: could be, if firms threaten to play (deviate, deviate) benefit from deviating in period 1 can be eliminated in the 4 remaining periods…
• CORRECT: use backward induction to find a Nash equilibrium in each subgame
• looking from the final game –> as even in the last game regardless how they played before both firms still have an incentive to deviate
• As this incentive can appear in all previous games their is a sub game Nash equilibria to (deviate, deviate) in all periods

Question 4

How do you solve an infinitely repeated Prisoner’s Dilemma?

Answer 4

• In an infinitely repeated game, backwards induction cannot be used
  (Same is true for an indefinitely repeated game)
  Instead we ask whether a certain strategy is a Nash equilibrium
• Grim trigger strategy –> Suppose each firm plays cooperate, as long as the other has always done so
  But if a player chooses deviates: –> - firms revert to playing the one-shot Nash equilibrium (deviate, deviate) FOREVER

Question 5

When is a Grim Trigger Strategy defined as a Nash equilibrium?

Answer 5

Does this strategy define a Nash equilibrium?
Yes, if no incentive to deviate from it

To analyse this, we first need to be able to construct the firms’ payoffs:

• i) short-term benefit: playing deviate when other plays cooperate
• ii) Long-term punishment: both playing deviate instead of cooperate

Question 6

How do you calculate the short-term benefits in the Grim Trigger Strategy?

Answer 6

To calculate the short-term benefits you must work out the difference between each firms (Deviate, Cooperate) and their (Cooperate,Cooperate)

Question 7

How do you calculate the long-term punishment in the Grim Trigger Strategy?

Answer 7

• First find the difference between (Cooperate, Cooperate) and (Deviate, Deviate) –> for the first period
• however, firms are punished forever
• Therefore we need to get future payoffs in terms of present value
• Therefore to calculate the long-term punishment you divide the original answer by the discount rate r

Question 8

How do you calculate the present value of future payoffs?

Answer 8

present value = X x 𝛿
𝛿 = 1/(1+r)
- where X is the value of the money in the base year
- where 𝛿 is the future payoff and 
- r is the interest rate

Question 9

How do you calculate Discounted Payoffs?

Answer 9

• Suppose a player expects to receive a payoff of in every future period
• In the present period, the player values the payoff of:
• period 1 = X x 𝛿 =X𝛿
• period 2 = X x 𝛿 x 𝛿 =X𝛿^2

The expected present discounted value of this stream of payoffs is:
- X(𝛿+𝛿^2+𝛿^3+…+𝛿^n…)~= X𝛿/(1-𝛿)

If you substitute in 𝛿=1/(1+r), then X𝛿/(1-𝛿) =X/r

Question 10

How do you solve an infinitely repeated prisoner’s dilemma?

Answer 10

• By comparing short-term benefits and the long-term punishment
• he strategy (cooperate, cooperate) forever is a Nash equilibrium if:
• short-term benefit is less than long-term punishment
• which is true if interest rate is sufficiently low