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Prof. Gaoquan Liu Queens College, CUNY

Business Economics Bus247, Spring 2013

Solution to Homework 1

1. (Basic Game Theory) Time magazine and Newsweek are considering which cover story to publish next week. Both of them have two alternatives to choose from: “Impeachment” or “Financial Crisis”. Below there are three versions of game. A. (Time and Newsweek are equally matched.) This corresponds to the case when the game is symmetric. (Time and Newsweek get the same payoff when both magazines cover the same story.)

Time

●

Impeachment Financial Crisis

Newsweek Impeachment 45, 45 40, 90

Financial Crisis 90, 40 20, 20

Does the game have dominant strategies for both players? If yes, describe them. Can the game be solved by dominant strategies? Answer: Yes, Time’s dominant strategy is “impeachment,” and Newsweek’s dominant strategy is also “impeachment.” So the game can be solved by dominant strategies: both Time and Newsweek will choose “impeachment.” Analysis: For Time, who is the first player and the row player, we compare its rows: (45, 90) to (40, 20). 45>40, 90>20, the first row (impeachment) is always the best, so impeachment is the dominant strategy for Time. For Newsweek, who is the second player and the column player, we compare its columns: (

45 40 ¿ . 45>40, 90>20, the first column ¿ to ( 20 90

(impeachment) is always the best, so impeachment is the dominant strategy for Newsweek. ●

Determine all Nash equilibria. Newsweek Impeachment Financial Crisis Time Impeachment 45, 45 90, 40 Financial Crisis 40, 90 20, 20 Answer: Using the double check method that I showed you in the class, from the above table, (impeachment, impeachment) has the double check and so it is the Nash equilibrium.

B. (Time is less popular than Newsweek.) This version of game corresponds to the assumption that Time is a less popular magazine (Time’s payoff is less than Newsweek’s when both magazines cover the same story.)

Time

Impeachment Financial Crisis

Newsweek Impeachment 30, 60 40, 90

Financial Crisis 90, 40 10, 30

1

Prof. Gaoquan Liu Queens College, CUNY

●

Business Economics Bus247, Spring 2013

Can the game be solved by dominant strategies? If yes, describe them.

Answer: No, because Time doesn’t have a dominant strategy, although Newsweek has “impeachment” as its dominant strategy. If the game can be solved by dominant strategies, it must be the case that every player has a dominant strategy. Analysis: For Time, now we compare its two rows (30, 90) to (40, 10). No row is always the best: 3010. For Newsweek, we compare its columns (

60 40 ¿ to ( ¿ . The first 90 30

column is always the best, and the second column is always the inferior: 60>40 and 90>30. So for Newsweek, impeachment is the dominant strategy and financial crisis is the dominated strategy. ●

Can the game solved by elimination of dominated strategies? If yes, describe them.

Answer: Yes. In the first step: we eliminate Newsweek’s dominated strategy “financial crisis.” Now with the reduced game, impeachment is the dominated strategy for Time: 30...