poli 118 game theory

PROBLEMS

(1) Let X = {x, y,z} and assume a relation R defined on X such that

R = {(x,x),(y, y),(z,z),(y,x),(x,z),(y,z)}.

(a) What is the preference ordering implied by R? (1 point)

(b) Define two utility functions u ∶ X → R and v ∶ X → R such that u and v represent the relation

R. (3 points)

(2) (Osborne, exercise 5.3.) Person 1 cares about both her income and person 2’s income. Precisely,

the value she attaches to each unit of her own income is the same as the value she attaches

to any two units of person 2’s income. For example, she is indifferent between a situation in

which her income is 1 and person 2’s is 0, and one in which her income is 0 and person 2’s is

2 [formally: (1, 0) ∼ (0, 2), where the first component in each pair is person 1’s income and the

second component is person 2’s income].

(a) Define a utility function u(x, y) consistent with these preferences (let x denote 1’s income

and let y denote 2’s income). (1 point)

(b) How do her preferences order the outcomes (1,4), (2,1), and (3,0)? (1 point)

(c) How does she order the outcomes (1,6) and (3,0)? (1 point)

(3) Recall the matrix for the Prisoner’s Dilemma from lecture (also from Osborne, p. 13). Note that the

12 q t

q 2,2 0,3

t 3,0 1,1

1payoff functions given in the matrix imply the following preferences over action profiles:

(t,q) ≻1 (q,q) ≻1 (t,t) ≻1 (q,t)

(q,t) ≻2 (q,q) ≻2 (t,t) ≻2 (t,q).

(The first action in the pair is player 1’s action, the second action is player 2’s action.)

(a) Determine whether the game in the matrix below differs from the Prisoner’s Dilemma only in

the action labels, or whether it also differs in one or both of the players’ preferences. Briefly

explain your answer. (Hint: consider different ways of replacing X and Y with q and t and

checking the implied preferences against the players’ preferences for the original game.) (2

points)

12 X Y

X 3,-2 1,-1

Y 2,1 0,5

(b) Give one example of a payoff matrix that represents the Prisoner’s Dilemma. It should differ

from any of the examples given in the textbook or the lectures. (2 points)

(4) Consider a situation in which two people can cooperate on a joint project; successful completion

would benefit each of them, but contributing is costly. Each individual’s preferences can be

represented as follows: her payoff is B −c if both she and the other person contribute, −c if she

alone contributes, and 0 if she does not contribute. We assume that 0 < c < B.

(a) Represent this situation using a payoff matrix. (2 points)

(b) Write out player 1’s best response correspondence. (2 points)

(c) Is the game equivalent to either Prisoner’s Dilemma or Stag Hunt? (1 point)