UConn
University of Connecticut

Self Test

Repeated Prisoner's Dilemmas & Trigger Strategies

Two firms are playing an infinitely-repeated prisoner's dilemma pricing game of the following form:
  Firm 2
  lowhigh
Firm 1low  4 ,  4 11 ,  0
high  0 , 11  6 ,  6
The firms simultaneously set prices at regular intervals. In the equilibrium of this game (if played only once), each firm selects the low price. However, the firms are playing this game repeatedly, for an infinite period of time. The firms utilize trigger strategies in order to try to maintain high prices.

Question 1.

Suppose that both firms adopt the grim trigger strategy. They continue charging the high price unless one of them cheats. Upon one of them charging the low price, they play the equilibrium, low-price strategy for the rest of the game. What has to be true about the interest rate (r) for high prices to be sustainable?

Question 2.

Suppose that both firms adopt a tit-for-tat strategy. They initially charge the high price. In future periods, a firm charges the high price if its competitor did in the previous period, and elects the lower price if its competitor priced low in the previous period. What has to be true about the interest rate (r) for high prices to be sustainable?
Two fishing fleets, Atlantic Harvest and Blue Net Co., share the same fishing grounds. Each season, they independently decide how much to catch. The payoffs below represent seasonal profits (in thousands of dollars):
  Blue Net Co.
  RestrainOverfish
Atlantic HarvestRestrain 10 , 10  0 , 20
Overfish 20 ,  0  5 ,  5
The fleets set their catch levels simultaneously each season. The fleets interact repeatedly and forever, and may use trigger strategies to sustain the cooperative outcome.

Question 3.

Suppose that both fleets adopt the grim trigger strategy. What has to be true about the interest rate (r) for cooperation to be sustainable?

Question 4.

Suppose that both fleets adopt a tit-for-tat strategy. What has to be true about the interest rate (r) for cooperation to be sustainable?
Consider the following game:
  Kid's Corner
  HighLow
Child's PlayHigh 64 , 64 20 , 72
Low 72 , 20 57 , 57

Question 5.

If this game is repeated for a known length of four periods, what will be Child's Play average (per-period) payoff (i.e., total payoffs divided by 4) in the subgame-perfect equilibrium of this game?

Question 6.

If this game is repeated forever, and firms adopt a grim-trigger strategy, what has to be true about the interest rate for cooperation to be sustainable?
For each of the following games, determine if the game is a prisoner's dilemma.

Question 7.

  Player 2
  AB
Player 1X 10 , 10 8 , 4
Y 4 , 8 7 , 7

Question 8.

  Player 2
  AB
Player 1X 5 , 5 8 , 4
Y 4 , 8 7 , 7

Question 9.

  Player 2
  AB
Player 1X 5 , 5 4 , 4
Y 4 , 8 7 , 7

Question 10.

  Player 2
  AB
Player 1X 6 , 6 4 , 8
Y 8 , 4 5 , 5