L21 - Collusion

Question 1

What are the conditions for Oligopolies to collude?

Answer 1

1) Sellers must be aware of each other’s strategies
(May form cartels to see if deviations occur)

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) (Illustration of Price War in notes)

Question 2

What is the Grim Trigger Strategy? (For infinitely repeated games)

Answer 2

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!

This is done since in infinite games we cannot use backward induction.

Question 3

What do we use to find out the Nash for finite games?

Answer 3

Backwards Induction.

Explanation covered in notes.

Question 4

Does the Grim Trigger strategy define a Nash Equilibrium?

Answer 4

Yes, if no incentive to deviate from it.

Only need to calculate SR Benefit and LR Punishment.

Question 5

Using Diagram 1.4 in notes how do you calculate the Short Term Benefit ( playing deviate when other plays cooperate)?

Answer 5

a firm’s payoff of (deviate, cooperate) = 200
a firm’s payoff of (cooperate, cooperate) = 150

short term benefit is the difference between the two:
firm 1’s short-term benefit: 200 – 150 = 50
firm 2’s short-term benefit: 200 – 150 = 50

Question 6

Using Diagram 1.4 in notes How do you calculate the Long Term punishment in one period ( both playing deviate instead of cooperate) ?

Answer 6

a firm’s payoff of (cooperate, cooperate) = 150
a firm’s payoff of (deviate, deviate) = 75

The punishment in one period is the difference between the two:
firm 1’s per-period punishment: (150-75)
firm 2’s per-period punishment : (150-75)
- To get punishment in one period

However, since Firms punished forever, need to get future payoffs in terms of present value

Question 7

How do you calculate the present value of future pay offs for £100 today in tomorrow’s value?

Answer 7

Suppose the rate of interest is 5% , so r = 0.05
If £100 is invested for 1 period, then the payoff in the next period is:

100+ 100 x 0.05 = 100(1+0.05) = 105

Question 8

How do you calculate the present value of future pay offs for £100 tomorrow in today’s value?

Answer 8

If £X is invested for 1 period, then next period’s payoff:
X + X * 0.05 = X (1+ 0.05)

So, X(1 + 0.05) = 100
X = 1/1.05(100) = 95.24

More generally : £100 tomorrow is equal to δ (100) today where δ = 1/(1+r)

Question 9

How do you calculate discounted payoffs? Suppose a player expects to receive a payoff of X in every future period

Answer 9

In the present period, the player values the payoff of:

period 1 = X * δ = X δ
period 2 = X * δ * δ = X δ2
period 3 = X * δ * δ * δ = X δ3
period n = X * δ * δ * δ * δ……. = X δn

The expected present discounted value of this stream of payoffs is:
X (δ + δ2 + δ3 + δn ….) ≅ Xδ / (1- δ)

Question 10

How do you calculate the overall long term punishment with discounted payoffs and present value calculations?

Answer 10

a firm’s payoff of (cooperate, cooperate) forever = 150/r
a firm’s payoff of (deviate, deviate) forever = 75/r

The long-term punishment is the difference between the two:
firm 1’s long-term punishment: (150-75)/r = 75r
firm 2’s long-term punishment : (150-75)/r = 75r

Question 11

Under Infinitely Repeated Games of Diagram 1.4 what is the Nash Equilibrium?

Answer 11

The strategy (cooperate, cooperate) forever is a Nash equilibrium if:
short-term benefit is less than long-term punishment 

50 < 75/r

r < 75/50 = 1.5

Question 12

EVERYTHING ELSE IS IN NOTES

Answer 12

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