HW2

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) eBay: Hundreds of millions of people bid on eBay auctions to purchase goods from all over the world. Despite being carried out online, in spirit, these auctions are similar to those that have been conducted for centuries. Is an auction a game? Why or why not? (5 points)

2) Penalty Kicks: Imagine a kicker and a goalie who confront each other in a penalty kick that will determine the outcome of a soccer game. The kicker can kick the ball left or right, while the goalie can choose to jump left or right. Because of the speed of the kick, the decisions need to be made simultaneously. If the goalie jumps in the same direction as the kick, then the goalie wins and the kicker loses. If the goalie jumps in the opposite direction of the kick, then the kicker wins and the goalie loses. Model this as a normal-form game and write down the matrix that represents the game you modeled. (5 points)

3) Matching Pennies: Players 1 and 2 each put a penny on a table simultaneously. If the two pennies come up the same side (heads or tails) then player 1 gets both pennies; otherwise, player 2 gets both pennies. Represent this game as a matrix. (5 points)

4) 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. Write down the normal form of this game and write down the matrix that represents the game. (5 points)

5) Prove Proposition 4.1: If the game has a strictly dominant strategy equilibrium, then it is the unique dominant strategy equilibrium. (5 points)

6) Weak Dominance: A weakly dominant strategy equilibrium is defined similarly to a strictly dominant strategy equilibrium but the > is replaced with >=.

(a) Provide an example of a game in which there is no weakly dominant strategy equilibrium. (5 points)

(b) Provide an example of a game in which there is more than one weakly dominant strategy equilibrium. (5 points)

7) Discrete First-Price Auction: An item is up for auction. 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 pays his bid, while the loser does not pay. If both players bid the same amount then a coin is tossed to determine who the winner is, and the winner gets the good and pays his bid while the loser pays nothing.

(a) Write down the game in matrix form. (5 points)

(b) Does any player have a strictly dominated strategy? (5 points)

8) Iterated Elimination: In the following normal-form game, which strategy profiles survive iterated elimination of strictly dominated strategies? (5 points)

9) 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.

10) 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)

11) 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)

Problem 8