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:

  1. 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.
  2. 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 50-minute 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.
  3. 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 zero-sum games, with each cell containing a number that represents the row player’s utility and the negation of the column player’s utility.

  1. 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 (Rx, Cy), where x = 1, 2, or 3 and y = 1, 2, 3, or 4.
  C1 C2 C3 C4
R1 2 4 7 3
R2 7 8 6 5
R3 2 9 1 4

Answer:

(R2, C4).

Explanation:
—C2 is dominated by C1 (and by C4).
—What remains of R3 is dominated by what remains of R2.
—What remains of C3 is dominated by what remains of C4.
—What remains of R1 is dominated by what remains of R2.
—What remains of C1 is dominated by what remains of C4.

  1. Write the equilibrium strategy pair(s) for the following game. Again, for each pair you write, use the form (Rx, Cy), where x = 1, 2, or 3 and y = 1, 2, 3, or 4.
  C1 C2 C3 C4
R1 5 4 4 4
R2 7 3 6 2
R3 2 2 1 4

Answer:

(R1, C2) and (R1, C4)

  1. 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 C1 as opposed to C2) in which the coefficient of q ends up being 0, because of the strategy (p R1, (1 – p) R2) 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.)
  C1 C2
R1 3 9
R2 8 5

Answer:

Start with the expected value utility of the game from the column player’s point of view:
EUColumn(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

  1. What values of p and q make [(p R1, (1 – p) R2), (q C1, (1 – q) C2)] 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.)
  C1 C2
R1 8 4
R2 1 7

Answer:

p = 6/10 (or, reducing the fraction, p = 3/5)
q = 3/10

  1. Suppose that the row player believes that the column player is going to play the following game with the strategy (½ C1, ½ C2). What is the row player’s best response?
  C1 C2
R1 4 8
R2 6 2

Answer:

EURow(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 R1.

  1. Suppose that, in a two-person zero-sum game with row strategies R1 and R2 and column strategies C1 and C2 (as in the games in the previous several questions), the row player is playing strategy (½ R1, ½ R2), 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 C1, ½ C2)), then she gets as high an expected utility as she can against the row player's strategy (½ R1, ½ R2). Does this mean that [(½ R1, ½ R2), (½ C1, ½ C2)] 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 R1, ½ R2) and the column player is playing (½ C1, ½ C2), the column player has no incentive to unilaterally deviate, since (½ C1, ½ C2) is a best response for her to the row player’s playing of (½ R1, ½ R2). But we do not know the analogous fact about the row player (i.e., that (½ R1, ½ R2) is a best response to (½ C1, ½ C2)), and we would need to know that in order to conclude that [(½ R1, ½ R2), (½ C1, ½ C2)] is an equilibrium strategy pair.

  1. Suppose that, for the following game, the strategy pair [(¼ R1, ¾ R2), (⅓ C1, ⅔ C2)] is an equilibrium pair. What is the value of the game?
  C1 C2
R1 a 5
R2 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

  1. Write the pure-strategy equilibrium pair(s) for the following game (or, if there aren’t any, write ‘no pure-strategy equilibrium pairs’). For each strategy pair you write, use the form (Rx, Cy), 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?
  C1 C2
R1 1, 7 7, 5
R2 2, 8 5, 6

Answer:

(R2, C1) is the pure-strategy equilibrium strategy pair.
The game is neither a coordination game nor a prisoner’s dilemma.

  1. Follow the instructions for the previous question, but for the following game:
  C1 C2
R1 5, 9 2, 2
R2 4, 5 8, 8

Answer:

(R1, C1) and (R2, C2) are the pure-strategy equilibrium pairs.
This game is a coordination game.

  1. 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 (Rx, Cy), where x = 1 or 2 and y is 1 or 2.
  C1 C2
R1 , ,
R2 , ,

Answer:

  C1 C2
R1 2, 2 0,3
R2 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, (R2, C2) is the equilibrium outcome, while the other three—(R1, C1), (R1, C2), and (R2, C1)—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.