Problem Set 1
a1
b1
a2
b2
10, 10
3, 12
12, 3
5, 5
Notice the above payoff matrix represents a one-shot prisoner’s dilemma (one-shot PD); where
the four payoffs (T, R, P, S) = (12, 10, 5, 3), and action ai means player i cooperates (C) and
action bi means player i defects (D), for i = 1, 2.
1. Draw a graph of player 2's expected payoff line that results from -choosing action a
(cooperating), and the expected payoff line from choosing action b (defecting); as the other
player 1’s probability of choosing action a varies from 0 to 1. Explain why this diagram implies
it is still a dominant or guaranteed best strategy for player 2 to always defect even when player 1
can choose to randomly cooperate with any probability p1 from 0 to 1.
C= cooperate D= defect
Player 2
Player 1
C
a1
D
b1
C
a2
10, 10
R, R
12, 3
T, S
D
b2
3, 12
S, T
5, 5
P, P
It is the best strategy for Player 2 to defect regardless of whether player 1
cooperates because if player 2 defects and player 1 cooperates then player 2 will
equal 12 (temptation) and player 1 will have 3 (sucker payoff) therefore player 2
has the advantage and dominates in this situation (player 2 > player 1). So player
2 is better off defecting if player 1 cooperates. If player 2 and player 1 defect then
both have 5 (penalty) which is an equilibrium (player 1 = player 2), if player 2
cooperates while player 1 defects, player 2 would have had less and would have
had 3 the suckers payoff, therefore it is much better for player 2 to defect.
a2= a1;
a2 < b1;
b2 > a1;
b2 = b1
1