January 16, 2018
###### How should the concept of distributive justice influence the response of the United States to evidence of global warming? Why?
January 16, 2018

QUESTION 1

1. The expected payoff of player 1 from the mixed strategy is the __________ of the expected payoff for each of her _________strategies, where the weights are the probabilities given by _____.
 A. product; pure; B. weighted average; dominant; player 2 C. weighted sum; pure; D. weighted difference; mixed; nature

15 points

QUESTION 2

1. Consider a two-person game, where player 1 has two strategies and player 2 has three strategies. Which of the following express a mixed strategy equilibrium? [mark all that apply]
 A. where  is the probability distribution of player 1 over her pure strategies, similar for   . B. [(1,0,0), (1,1/2)] C. [(p, 1-p,0), (q, 1-q)] and p and q are in the interval [0,1] D. [S1, S2] where S1 and S2 are the sets of pure strategies of player 1 and 2. E. [(p1, p2,1-p1-p2), (1, 0)] and p1 and p2 are in the interval [0,1]

15 points

QUESTION 3

1. Whenever we add uncertainty to a game and a player would like to outguess the others, ________ a NE in _________ strategies. Players would assign ________ to the its strategies in Si, and this gives us the notion of ___________ strategies.
 A. There is not; mixed; values; dominated B. There is; weak; maximin values; maximin C. There is not; pure; weights; dominant D. There is ; pure ;probabilities; mixed

10 points

QUESTION 4

1. Mark the correct sequence for the following statements:
2. Finite games have at least one Nash equilibrium in pure strategies.
3. A pure strategy may be strictly dominated by a mixed strategy, even if this pure strategy is not strictly dominated by any other pure strategy.

III. A pure strategy can be a best response for a mixed strategy if and only if such pure strategy is also best response to any other pure strategy.

 A. TTF B. FFF C. TFT D. FTF

20 points

QUESTION 5

1. Consider a kicker (K) and a goalie (G) in a soccer game. Suppose that if K kicks to the right and G jumps to the right, the probability of a goal is 0.3. If K kicks to the right and G jumps to the left, the probability of a goal is 0.9. If K kicks to the left, the probability of a goal is 0.8 if G jumps to the right and 0.5 if G jumps to the left.
Fill the following matrix with the given probabilities. Assume the kicker is the row player and the goalie is the column player.
 L R L , , R , ,

20 points

QUESTION 6

1. What is correct about Nash equilibrium in mixed strategies:
 A. In a two-player game, mixed strategies   are a Nash equilibrium if each player’s mixed strategy is a best response to the other player s mixed strategy, and none will unilaterally deviate. B. It only exists when all players randomize their pure strategies. C. Graphically, NE is given by the intersection of the player s best response payoffs D. In equilibrium, the mixed strategy of a player must put positive probability on a given pure strategy only if the pure strategy is itself a best response to the mixed strategy of the other player.

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