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h3answ08

# h3answ08 - Department of Economics The Ohio State...

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Department of Economics The Ohio State University Econ 805—Homework #3 Answers Prof James Peck Winter 2008 1. Mas-Colell, exercise 7.D.1. Answer: A strategy must specify what actions are taken at each of a player’s information sets. Thus, the number of strategies is the product, N Y n =1 M n . 2. Consider the following game. The players are three voters, who must vote either YES or NO (no abstentions). The referendum will succeed if at least two of the voters vote YES, and it will fail otherwise. Suppose that player 1 receives a payo ff of 2 if the referendum succeeds, and a payo ff of 2 if it fails. Players 2 and 3 receive a payo ff of 1 if the referendum succeeds, and a payo ff of 1 if it fails. Assume that we have a private ballot, where players cannot see how the other players vote until the ballots are counted. (a) What is the normal form of this game? That is, for i = 1 , 2 , 3 ,write out the strategy sets, S i , and the payo ff functions, π i . (b) Express the normal form game using matrices. (For a three player game, you can write it as a collection of matrices, one matrix for each strategy of player 3. Player 3 chooses the matrix, player 1 chooses the row, and player 2 chooses the column. Each box of each matrix has three numbers, corresponding to the payo ff s of the three players.) (c) Represent this game in extensive form. Answer: (a) For i = 1 , 2 , 3 , we have S i = { yes, no } . The payo ff functions are given by π 1 ( yes, yes, yes ) = 2 π 1 ( yes, yes, no ) = 2 π 1 ( yes, no, yes ) = 2 π 1 ( yes, no, no ) = 2 π 1 ( no, yes, yes ) = 2 π 1 ( no, yes, no ) = 2 π 1 ( no, no, yes ) = 2 π 1 ( no, no, no ) = 2 1

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and π 2 ( yes, yes, yes ) = 1 π 2 ( yes, yes, no ) = 1 π 2 ( yes, no, yes ) = 1 π 2 ( yes, no, no ) = 1 π 2 ( no, yes, yes ) = 1 π 2 ( no, yes, no ) = 1 π 2 ( no, no, yes ) = 1 π 2 ( no, no, no ) = 1
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