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**QUESTION 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**

- 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. | [S_{1}, S_{2}] where S_{1} and S_{2} 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**

- 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 **

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**QUESTION 4**

- Mark the correct sequence for the following statements:
- Finite games have at least one Nash equilibrium in pure strategies.
- 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**

- 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**

- 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. |