Game theory 9

Homework

For this week we are going to continue to do homework focusing on 6.7 Mixed strategies, 6.8 Mixed Statigies examples and 6.9 Pareto-Optimality and Social Optimality.

7. Chp 6 Exercise 7. In this question we will consider several two-player games. In each payoff matrix below the rows correspond to player A’s strategies and the columns correspond to player B’s strategies. The first entry in each box is player A’s payoff and the second entry is player B’s payoff.

a. Find all Nash equilibria for the game described by the payoff matrix below.

b. Find all Nash equilibria for the game described by the payoff matrix below (include an explanation for your answer).

[Hint: This game has a mixed strategy equilibrium. To find the equilibrium let the probability that player A uses strategy U be p and the probability that player B uses strategy L be q. As we learned in our analysis of matching pennies, if a player uses a mixed strategy (one that is not really just some pure strategy played with probability one) then the player must be indifferent between two pure strategies. That is, the strategies must have equal expected payoffs. So, for example, if p is not 0 or 1 then it must be the case that 5q + 0(1 − q)=4q + 2(1 − q) as these are the expected payoffs to player A from U and D when player B uses probability q.]

8. Consider the two-player game described in the payoff matrix of Figure 6.36:

a. Find all pure-strategy Nash equilibria for this game.

b. This game also has a mixed-strategy Nash equilibrium; find the probabilities the players use in this equilibrium, together with an explanation for your answer.

c. Keeping in mind Schelling’s focal point idea from this chapter, what equilibrium

do you think is the best prediction of how the game will be played? Explain.

9. For each of the following two-player games, find all Nash equilibria. In each of the payoff matrices of Figures 6.37 and 6.38, the rows correspond to player A’s strategies and the columns correspond to player B’s strategies. The first entry in each box is player A’s payoff and the second entry is player B’s payoff.

a.

b.

Rubric:

10. In the payoff matrix of Figure 6.39, the rows correspond to player A’s strategies and the columns correspond to player B’s strategies. The first entry in each box is player A’s payoff and the second entry is player B’s payoff.

a. Find all pure-strategy Nash equilibria of this game

b. Notice from the payoff matrix above that Player A’s payoff from the pair of strategies (U, L) is 3. Can you change player A’s payoff from this pair of strategies to some non-negative number in such a way that the resulting game has no pure-strategy Nash equilibrium? Give a brief (1-3 sentence) explanation for your answer.

(Note that in answering this question, you should only change Player A’s payoff for this one pair of strategies (U, L). In particular, leave the rest of the structure of the game unchanged: the players, their strategies, the payoff from strategies other than (U, L), and B’s payoff from (U, L).)

c. Now let’s go back to the original payoff matrix from part (a) and ask an analogous question about player B. So we’re back to the payoff matrix in which players A and B each get a payoff of 3 from the pair of strategies (U, L). Can you change player B’s payoff from the pair of strategies (U, L) to some nonnegative number in such a way that the resulting game has no pure-strategy Nash equilibrium? Give a brief (1-3 sentence) explanation for your answer.

(Again, in answering this question, you should only change Player B’s payoff for this one pair of strategies (U, L). In particular, leave the rest of the structure of the game unchanged: the players, their strategies, the payoff from strategies other than (U, L), and A’s payoff from (U, L).)

14. Consider the two-player game with players, strategies, and payoffs described in the payoff matrix given in Figure 6.45.

a. Find all of the Nash equilibria of this game.

b. (b) In the mixed strategy equilibrium you found in part (a), you should notice that player 1 plays strategy U more often than strategy D. One of your friends remarks that your answer to part (a) must be wrong because clearly for player 1 strategy D is a more attractive strategy than strategy U. Both U and D give player 1 a payoff of 4 on the off-diagonal elements of the payoff matrix, but D gives player 1 a payoff of 3 on the diagonal while U only gives player 1 a payoff of 1 on the diagonal. Explain what is wrong with this reasoning.