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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