Game Theory Problem Set

Question 1

Consider the following normal-form game.

                P2 P1.          L.           N. 
    A   8, 11      0, 0
    B  12, −1      2, 5

Assume that the game will be repeated infinitely many times, and that P2 is plays a
grim-trigger strategy.

She plays L in the first period and if (A, L) has realized in the last period.

Otherwise, she plays N.
For which range of δ is the best response of P1 to always play A?

Answer 1

1. Under the grim trigger strategy one profile is the cooperative profile the other profile is the punishment profile.

The expected payoff for not deviating, the cooperative one path profile is:

8 + 8S + 8S^2 +……
8( 1 + S + S^2 + …..)

This is a geometric series 1/ 1-S
8 ( 1/1-S) = 8/1-S

2. Player 1’s Deviation Strategy:

Player one has an incentive to deviate from 8 to 12. In this period, player one gets payoff 12.

The property of grim trigger strategy allows one period of deviation then assumes players play according to there strategies for the rest of the game, playing 2.

12 + 2S + 2S^2 + 2S^3….

This is a geometric series:

12 + 2(1 / 1-S)
12 + 2S/1-2

3. No deviation (one path strategy) yields a higher payoff than the deviation strategy.

8/1-S >= 12 + 2S/1-2
Solve for S
8 >= 12(1-S) + 2S

2/5 <= S

Therefore, the strategy is SPNE if
S >= 2/5

Question 2

Assume that the game will be repeated infinitely many times, and that P2 is plays a
tit-for-tat strategy.

She plays L in the first period and if (A, L) or (B, N) has realized in the previous
period.

Otherwise, she plays N.
P1 also plays a tit-for-tat strategy.
She plays A in the first period and if (A, L) or (B, N) has realized in the previous
period.

Otherwise, she plays B.
For which range of δ does P1 not have an incentive to deviate from this strategy?

                P2 P1.          L.           N. 
    A   8, 11      0, 0
    B  12, −1      2, 5

Answer 2

Tit for tat strategy

At t = 0, start playing, when t > 0 each player will choose the action his opponent took in the previous period, which is (A, L) afterwords

The one path cooperative strategy is the same as before 8/1-S.

The no deviation (one path cooperative profile) strategy yields a higher payoff

8/1-S >= 12 + 2S + 8S^2 + 8S^3……

This is a geometric series:
8/1-S >= 12 + 2S + 8( S^2 + S^3..)

8/1-S >= 12 + 2S + 8(1/1- S^2)

Solve: The strategy is SPNE if S>= 2/3

Question 3

Mixed Strategy Nash Equilibrium Explain.

             P2
            L R P1 T         6, 5     3, 3 B        3, 3      5, 5

Find all Nash equilibria of this game.

Answer 3

Mixed Strategy Nash equilibrium allows us to randomize those strategies. We assign probabilities to each choice and play according to these choices and to play their choices according to their probabilities.

Mixed Strategy:

Let player 2 play L with probability q and R with probability of 1-q.

Player 2 would be indifferent between playing L and R, when the utility of playing L is equal to the utility of playing R

6q + 3(1-q) = 3q+ 5(1-q)
6q + 3 - 3q = 3q + 5 - 5q
6q - 3q - 3q+ 5q = 5 - 3
5 = 2
q = 2/5

1 - 2/5 = 3/5

(q, 1-q) = (2/5, 3/5)

Let player 1 play T with probability p, Let player 2 play B with probability 1-p

5p + 3(1-p) = 3p + 5(1-p)
5p + 3 - 3p = 3p + 5 -5p
5p - 3p - 3p + 5p = 2
4p = 2
p = 1/2

1 - 1/2 = 1/2
(1/2, 1/2)