HW3

Submit a single .pdf via Canvas by the date of the deadline (11:59 pm)

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

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1) Proposition 5.1: Show that if s* is a strictly dominant pure-strategy equilibrium in a game, then s* is the unique pure-strategy Nash equilibrium in the same game. (5 points)

Hints: You need to show both (a) implication and (b) uniqueness.

2) Public Good Contribution: Three players live in a town, and each can choose to contribute to fund a streetlamp. The value of having the streetlamp is 3 for each player, and the value of not having it is 0. The mayor asks each player to contribute either 1 or nothing. If at least two players contribute then the lamp will be erected. If one player or no players contribute then the lamp will not be erected, in which case any person who contributed will not get his money back.

a. Write out each player's best-response correspondence. (5 points)

b. What outcomes can be supported as pure-strategy Nash equilibria? (5 points)

3) Hawk-Dove: The following game has been widely used in evolutionary biology to understand how fighting and display strategies by animals could coexist in a population. For a typical Hawk-Dove game there are resources to be gained (e.g., food, mates, territories), denoted as v. Each of two players can choose to be aggressive, as Hawk (H), or compromising, as Dove (D). If both players choose H then they split the resources but lose some payoff from injuries, denoted as k. Assume that k > v/2. If both choose D then they split the resources but engage in some display of power that carries a display cost d, with d < v/2. Finally, if player i chooses H while j chooses D then i gets all the resources while j leaves with no benefits and no costs.

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

b. Assume that v = 10, k = 6, and d = 4. What outcomes can be supported as pure-strategy Nash equilibria? (5 points)

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

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

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

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

8) 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)

9) 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)

10) 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)

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