HOMEWORK ASSIGNMENT #4 [25 POINTS]. DUE IN CLASS ON WED. JAN. 28.

1. [10 Points] Rewrite Taylor's partial decision matrix for the PD Supergame, Table 3 on p. 78 of PC, filling in the average utlity per play of the sub-game to each player of the pure strategy combinations represented in the matrix, assuming the sub-game pay-offs in the standard PD Matrix (Colman p. 115). (Using average utilities allows you to ignore discounting for payoffs received in the future. In calculating the averages, for example, if a strategy's payoff approaches an average of 2 from above, indicate its average pay-off as "2+"; if it approaches an average of 2 from below, indicate its average pay-off as "2-") Circle all Nash equilibrium combinations of pure strategies. Assume that you can ignore mixed strategy Nash equilibria. Apply the three tests in Step 4 on page 2 of Handout #5 to determine if there is an individually rational solution to this matrix.

2. [10 Points] Modify the matrix from part 1 above, as follows: (a) Enter discounted utilities rather than average utilities into the cells of the matrix, assuming a discount rate of .01 (i.e., a discount factor (Taylor) or discount parameter (Axelrod) of .99). See Axelrod, p. 13, for a useful formula for calculating discounted utilties. [NOTE: If you are unable to calculate discounted utilities, MAKE SURE THAT YOU FIND OUT DURING CLASS ON MONDAY, JAN. 26] (b) Add one further strategy for each player--FRIEDMAN (Axelrod, p. 36).

Circle all Nash equilibrium combinations of pure strategies. Assume that you can ignore mixed strategy Nash equilibria. Apply the three tests in Step 4 on page 2 of Handout #5 to determine if there is any individually rational solution to this matrix.

3. [5 Points] Expand the matrix from part 2 above to include the following two additional strategies for both players:

EXPLOITER = Play the pattern C,C,D repeatedly so long as opponent cooperates on every move. If opponent ever fails to cooperate, play D for the rest of the supergame.

EXPLOITED = Cooperate on every move, so long as opponent plays the pattern
C,C,D repeatedly. If opponent ever departs from that pattern, play D for the rest of the supergame.

Enter the discounted utilities of each combination of pure strategies. Circle all Nash equilibrium combinations of pure strategies. Assume that you can ignore mixed strategy Nash equilibria. Apply the three tests in Step 4 on page 2 of Handout #5 to determine if there is any individually rational solution to this matrix.

NOTE: On this homework assignment and throughout the course you may use the following approximations:

(.99)2 = .98

(.99)3 = .97