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# AMS 335/ECO 355 PROBLEM SET 1: Penalty Kicks, Meeting Up and Price Competition

USStony Brook UniversityAMS 335ECO 355ECO355AMS335Game TheoryPenalty Kicks Meeting UpPrice Competition

# AMS 335/ECO 355

PROBLEM SET 1

Solve the following exercises from the textbook:

3.2 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 normalform game and write down the matrix that represents the game you modeled.

3.3 Meeting Up: Two old friends plan to meet at a conference in San Francisco, and they agree to meet by “the tower.” After arriving in town, each realizes that there are two natural choices: Sutro Tower or Coit Tower. Not having cell phones, each must choose independently which tower to go to. Each player prefers meeting up to not meeting up, and neither cares where this would happen. Model this as a normal-form game and write down the matrix form of the game.

3.6 Price Competition: Imagine a market with demand p(q) = 100 − q. There are two firms, 1 and 2, and each firm i has to simultaneously choose its price pi. If pi < pj , then firm i gets all of the market while no one demands the good of firm j . If the prices are the same then both firms split the market demand equally. Imagine that there are no costs to produce any quantity of the good. (These are two large dairy farms, and the product is manure.) Write down the normal form of this game.

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