HW3

Submit a single .pdf (or handcopy) for the written problems to me via email (or in person) by the date of the deadline (11:59 pm)

(1) Please show ALL of your work!!!!

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1) Write the definition of expected utility/payoff. Provide examples and compute the expected utility of a player step by step.

2) Battle of the Sexes: Find all the pure-strategy and mixed-strategy Nash equilibria in Battle of the Sexes. (5 points)

3) Penalty Kicks: Find all the pure-strategy and mixed-strategy Nash equilibria in Penalty Kicks. (5 points)

4) Rock–paper–scissors: Find all the pure-strategy and mixed-strategy Nash equilibria in Rock-paper-scissors. (5 points)

5) Monitoring: An employee (player 1) who works for a boss (player 2) can either work (W) or shirk (S), while his boss can either monitor the employee (M) or ignore him (I). As in many employee-boss relationships, if the employee is working then the boss prefers not to monitor, but if the boss is not monitoring then the employee prefers to shirk. The game is represented by the following matrix (see below)

Find all the pure or mixed-strategy Nash equilibria of this game. (5 points)

6) Cops and Robbers: Player 1 is a police officer who must decide whether to patrol the streets or to hang out at the coffee shop. His payoff from hanging out at the coffee shop is 10, while his payoff from patrolling the streets depends on whether he catches a robber, who is player 2. If the robber prowls the streets then the police officer will catch him and obtain a payoff of 20. If the robber stays in his hideaway then the officer’s payoff is 0. The robber must choose between staying hidden or prowling the streets. If he stays hidden then his payoff is 0, while if he prowls the streets his payoff is −10 if the officer is patrolling the streets and 10 if the officer is at the coffee shop.

a. Describe the game in a matrix. (5 points)

b. Find all the pure or mixed-strategy Nash equilibria of this game. (5 points)

7) Discrete All-Pay Auction: Each bidder submits a bid. The highest bidder gets good, but all bidders pay their bids. Consider an auction in which player 1 values the item at 3 while player 2 values the item at 5. Each player can bid either 0, 1, or 2. If player i bids more than player j then i wins the good and both pay. If both players bid the same amount then a coin is tossed to determine who gets the good, but again both pay.

a. Describe the game in a matrix. (5 points)

b. Find all the pure or mixed-strategy Nash equilibria of this game. (5 points)

8) Does a pure-strategy Nash equilibrium always exist in a game? If so, why? If not, why? (5 points)

9) Imagine an extensive-form game in which player i has K information sets.

a. If the player has an identical number of m possible actions in each information set, how many pure strategies does he have? (5 points)

b. If the player has m_k actions in information set k ∈ {1, 2, . . . , K}, how many pure strategies does the player have? (5 points)

10) Strategies and Equilibrium: Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A while it continues to player 2 if he chooses B. Player 2 can then choose C or D, with the game ending after C and continuing again with player 1 after D. Player 1 can then choose E or F, and the game ends after each of these choices.

a. Model this as an extensive-form game tree. Is it a game of perfect or imperfect information? (5 points)

b. How many terminal nodes does the game have? How many information sets (for each player)? (5 points)

c. How many pure strategies does each player have? (5 points)

d. Imagine that the payoffs following choice A by player 1 are (2, 0), those following C by player 2 are (3, 1), those following E by player 1 are (0, 0), and those following F by player 1 are (1, 2). What are the Nash equilibria of this game? Does one strike you as more “appealing” than the other? If so, explain why. (5 points)

11) Centipedes: Imagine a two-player game that proceeds as follows. A pot of money is created with $6 in it initially. Player 1 moves first, then player 2, then player 1 again, and finally player 2 again. At each player’s turn to move, he has two possible actions: grab (G) or share (S). If he grabs he gets 2/3 of the current pot of money, the other player gets 1/3 of the pot, and the game ends. If he shares then the size of the current pot is multiplied by 3/2 and the next player gets to move. At the last stage at which player 2 moves, if he chooses S then the pot is still multiplied by 3/2, player 2 gets 1/3 of the pot, and player 1 gets 2/3 of the pot.

a. Model this as an extensive-form game tree. Is it a game of perfect or imperfect information?

b. How many terminal nodes does the game have? How many information sets (for each player)?

c. How many pure strategies does each player have?

d. Find the Nash equilibria of this game. How many outcomes can be supported in equilibrium?

e. Now imagine that at the last stage at which player 2 moves, if he chooses to share then the pot is equally split among the players. Does your answer to part (d) change?