The Nash Equilibrium - A Unique Pure Strategy

Part A 

The unique pure strategy based on the Nash Equilibrium is low . Setting a low price is thus the dominant strategy that will be used by both of the players. The payoffs with the low price will be $57,000 for each of the players. The prisoner’s dilemma for the given scenario is unique and the outcome of the Nash Equilibrium can be defined as being inefficient. This is because both companies would be better in case they decide to set high prices but they would have to cooperate on this. If one company defects and sets a low price the high-priced firm would stand to lose. 

The analysis would be given as follows: 

Part B 

In case the two firms are to play the game for a finite number of periods such as 4 years which is given as 4 rounds, the last round will be a one-shot game, the play would be thus deterministic and the cooperative strategy of setting a high price will not be sustained. Both firms will thus opt for the low-price strategy at round 4. With this knowledge in mind, round 3, which is the next-to-last period, will also be deterministic and they will charge a low price. Round 2 and round 1 will also have a low price when one follows the same logic. The total profit for each firm would thus be given as: 

Total Profit = $57,000 * 4 = $228,000 

Part C 

The one-time gain when a firm chooses to defect against an opponent by setting a low price while the other would have a high price would be $8,000. The firm would earn $72,000 instead of $64,000 with setting the low price. This is calculated as: 

$72,000 - $64,000 = $8,000 

The firm will earn $57,000 in the grim strategy and the loss in each future period would be $7,000 and is calculated given as: 

$64,000 - $57,000 = $7,000 

For δ = 0.8, the loss in all future periods will be $28,000 as shown in the calculation. A one-time gain of $8,000 does not exceed the loss of $28,000 and thus it will be worthwhile to cooperate. 

δ/ (1- δ) *$7000 = (0.8/1-0.8) *$7,000 = $28,000 

Additionally, the firm will choose to deviate when: 

15,000 <35,000 and the firm does not have the incentive to deviate at δ = 0.8 

The range of values for cooperation to be sustained would be given by the arbitrary rate 

The statement will only be true when δ ≥ 8/15 

Part D 

For the given scenario, the end of the game may not have been initially known or have been anticipated. The probability that the game would have continued would have been 1.0. For the given case, both firms would have made use of the grim trigger strategy to be able to get profits and sustain the high price outcome. The profit realized would have been: 

$64,000 * 4 = $256,000 

This is different from part b where the two firms were operating under a finite number of periods and thus used a low-price strategy.