University of Kansas, Fall 2007
Philosophy 666: Rational Choice Theory
Ben Eggleston—eggleston@ku.edu
test on game theory: answer key
(October 24, 2007)
Instructions:
 Answer all of the following questions on the answer sheets provided. You can write on this
list of
questions, but credit will be awarded only for answers written on answer
sheets.
 Do not access any book, notebook, newspaper, calculator, computer, cell
phone, or other possible source of inappropriate aid during the test, do not
leave the room before you are finished taking the test, and be sure to finish
the test within this 50minute testing period—no credit will be given for any
work done after you access any possible source of inappropriate aid, after you
leave the room for any reason, or after the end of the testing period.
 When you are finished, be sure your name is written on each of your answer
sheets, and turn them in. You do not need to turn in this list of questions.
Questions:
The matrices in questions 1–7 involve zerosum games, with each cell containing a number that
represents the row player’s utility and the negation of the column player’s
utility.
 Analyze the following game using dominance considerations and write the strategy pair(s) corresponding to its solution(s). When writing the strategy pair(s),
write each of them in the form (R_{x}, C_{y}),
where x = 1, 2, or 3 and y = 1, 2, 3, or 4.

C_{1} 
C_{2} 
C_{3} 
C_{4} 
R_{1} 
2 
4 
7 
3 
R_{2} 
7 
8 
6 
5 
R_{3} 
2 
9 
1 
4 
Answer:
(R_{2}, C_{4}).
Explanation:
—C_{2} is dominated by C_{1} (and by C_{4}).
—What remains of R_{3} is dominated by what remains of R_{2}.
—What remains of C_{3} is dominated by what remains of C_{4}.
—What remains of R_{1} is dominated by what remains of R_{2}.
—What remains of C_{1} is dominated by what remains of C_{4}.
 Write the equilibrium strategy pair(s) for the following game. Again, for each pair
you write, use the form (R_{x}, C_{y}),
where x = 1, 2, or 3 and y = 1, 2, 3, or 4.

C_{1} 
C_{2} 
C_{3} 
C_{4} 
R_{1} 
5 
4 
4 
4 
R_{2} 
7 
3 
6 
2 
R_{3} 
2 
2 
1 
4 
Answer:
(R_{1}, C_{2}) and (R_{1}, C_{4})
 Suppose that the expected utility of the following game for
the column player is a formula involving q (where q is her
probability of playing C_{1} as opposed to C_{2}) in which the
coefficient of q ends up being 0, because of the strategy (p R_{1},
(1 – p) R_{2}) that the row player happens to be playing.
Derive the value of p that makes this the case, by starting with (1) a
formula expressing the expected utility of the game for the row player,
(2) a formula expressing the expected utility of the game for the column player, or (3)
a formula expressing the value
of the game, and simplifying whatever formula you start with until you can isolate an expression
involving p that you can then use to solve for p. (Show your
work, of course.)

C_{1} 
C_{2} 
R_{1} 
3 
9 
R_{2} 
8 
5 
Answer:
Start with the expected value utility of the game from the column player’s
point of view:
EU_{Column}(game)
= q((p)(3) + (1 – p)(8)) + (1 – q)((p)(9) +
(1 – p)(5))
= q(3p + 8 – 8p) + (1 – q)(9p + 5 – 5p)
= q(–5p + 8) + (1 – q)(4p + 5)
= q(–5p + 8) + (4p + 5) – q(4p + 5)
= q(–5p + 8 – 4p – 5) + (4p + 5)
= q(–9p + 3) + (4p + 5)
Now make the coefficient of q equal to 0:
–9p + 3 = 0
–9p = –3
p = 3/9
p = 1/3
 What values of p and q make [(p R_{1}, (1 –
p) R_{2}), (q C_{1}, (1 – q) C_{2})]
an equilibrium strategy pair for the following game? (You do not have to show your
work. An answer of the form ‘p = __, q = __’ can earn full
credit.)

C_{1} 
C_{2} 
R_{1} 
8 
4 
R_{2} 
1 
7 
Answer:
p = 6/10 (or, reducing the fraction, p = 3/5)
q = 3/10
 Suppose that the row player believes that the column player is going to
play the following game with the strategy (½ C_{1}, ½ C_{2}).
What is the row player’s best response?

C_{1} 
C_{2} 
R_{1} 
4 
8 
R_{2} 
6 
2 
Answer:
EU_{Row}(game)
= p((1/2)(4) + (1/2)(8)) + (1 – p)((1/2)(6) + (1/2)(2))
= p(2 + 4) + (1 – p)(3 + 1)
= p(6) + (1 – p)(4)
= 6p + 4 – 4p
= 2p + 4
The row player wants to maximize this value, within the constraint that 0 ≤
p ≤ 1, so he should set p = 1 (its maximum possible value),
which is to say that he should simply play pure strategy R_{1}.
 Suppose that, in a twoperson zerosum game with row strategies R_{1}
and R_{2} and column strategies C_{1} and C_{2} (as in
the games in the previous several questions), the row player is playing
strategy (½ R_{1}, ½ R_{2}), and the column player computes
(correctly) that if she, too, plays a mixed strategy giving equal weight to
each of her two pure strategies (i.e., plays strategy (½ C_{1}, ½ C_{2})),
then she gets as high an expected utility as she can against the row player's
strategy (½ R_{1}, ½ R_{2}). Does this mean that [(½ R_{1},
½ R_{2}), (½ C_{1}, ½ C_{2})] is an equilibrium
strategy pair for the game under consideration? Why or why not?
Answer:
No, it does not. We know that if the row player is playing (½ R_{1},
½ R_{2}) and the column player is playing (½ C_{1}, ½ C_{2}),
the column player has no incentive to unilaterally deviate, since (½ C_{1},
½ C_{2}) is a best response for her to the row player’s playing of (½
R_{1}, ½ R_{2}). But we do not know the analogous fact about
the row player (i.e., that (½ R_{1}, ½ R_{2}) is a best
response to (½ C_{1}, ½ C_{2})), and we would need to know
that in order to conclude that [(½ R_{1}, ½ R_{2}), (½ C_{1},
½ C_{2})] is an equilibrium strategy pair.
 Suppose that, for the following game, the strategy pair [(¼ R_{1},
¾ R_{2}), (⅓ C_{1},
⅔ C_{2})] is an equilibrium pair. What is the value of the game?

C_{1} 
C_{2} 
R_{1} 
a 
5 
R_{2} 
6 
a 
Answer:
value
= weighted average of possible outcomes
= (¼)(⅓)a + (¼)(⅔)(5) + (¾)(⅓)6 + (¾)(⅔)a
= a/12 + 10/12 + 18/12 + 6a/12
= (7/12)a + 28/12
 Write the purestrategy equilibrium pair(s) for the following game (or, if
there aren’t any, write ‘no purestrategy equilibrium pairs’). For each strategy pair
you write, use the form (R_{x}, C_{y}),
where x = 1 or 2 and y = 1 or 2. Is the game a coordination
game (also known as a battle of wills), a prisoner’s dilemma, or neither?

C_{1} 
C_{2} 
R_{1} 
1, 7 
7, 5 
R_{2} 
2, 8 
5, 6 
Answer:
(R_{2}, C_{1}) is the purestrategy equilibrium strategy pair.
The game is neither a coordination game nor a prisoner’s dilemma.
 Follow the instructions for the previous question, but for the following
game:

C_{1} 
C_{2} 
R_{1} 
5, 9 
2, 2 
R_{2} 
4, 5 
8, 8 
Answer:
(R_{1}, C_{1}) and (R_{2}, C_{2}) are the
purestrategy equilibrium pairs.
This game is a coordination game.
 Write a matrix like the following and fill it in with eight numbers (two
in each cell) so that it represents a prisoner’s dilemma. Then (1) name the
two important solution concepts that do not coincide in a prisoner’s dilemma,
(2) state the definition, criteria, or identifying characteristics of each of those two
solution concepts, and (3) indicate, for each of those two solution concepts,
which strategy pair(s) in your matrix is (or are) associated with that
solution concept. As above, for each strategy pair
you write, use the form (R_{x}, C_{y}),
where x = 1 or 2 and y is 1 or 2.

C_{1} 
C_{2} 
R_{1} 
, 
, 
R_{2} 
, 
, 
Answer:

C_{1} 
C_{2} 
R_{1} 
2, 2 
0,3 
R_{2} 
3, 0 
1, 1 
The two concepts are equilibrium and Pareto optimality. An equilibrium
outcome is one from which neither player has an incentive to unilaterally
deviate. A Pareto optimal outcome is one such that neither player can do better
(than in it) without the other player doing worse. In the above matrix, (R_{2},
C_{2}) is the equilibrium outcome, while the other three—(R_{1},
C_{1}), (R_{1}, C_{2}), and (R_{2}, C_{1})—are
the Pareto optimal ones.
Instructions, revisited:
As stated in item 3 of the instructions, you do not need to turn in this
list of questions.