Exercise 1.3. Alice (Player 1), Bob (Player 2), and Charlie (Player 3) play the following simultaneous game. They are sitting in different rooms facing a keyboard with only one key and each has to decide whether or not to press the key. Alice wins if the number of people who press the key is odd (that is, all three of them or only Alice or only Bob or only Charlie), Bob wins if exactly two people (he may be one of them) press the key and Charlie wins if nobody presses the key. (a) Represent this situation as a game-frame. Note that we can represent a three- player game with a set of tables. Player 1 chooses the row, Player 2 chooses the column and Player 3 chooses the table (that is, we label the rows with Player 1's strategies, the columns with Player 2's strategies and the tables with Player 3's strategies). (b) Using the game-frame of part (a) obtain a reduced-form game by adding the information that each player prefers winning to not winning and is indifferent between any two outcomes where he/she does not win. For each player use a utility function with values from the set {0,1}. (c) Using the game-frame of part (a) obtain a reduced-form game by adding the information that (1) each player prefers winning to not winning, (2) Alice is indifferent between any two outcomes where she does not win, (3) conditional on not winning, Bob prefers if Charlie wins rather than Alice, (4) conditional on not winning, Charlie prefers if Bob wins rather than Alice. For each player use a utility function with values from the set {0,1,2}.
Exercise 1.3. Alice (Player 1), Bob (Player 2), and Charlie (Player 3) play the following simultaneous game. They are sitting in different rooms facing a keyboard with only one key and each has to decide whether or not to press the key. Alice wins if the number of people who press the key is odd (that is, all three of them or only Alice or only Bob or only Charlie), Bob wins if exactly two people (he may be one of them) press the key and Charlie wins if nobody presses the key. (a) Represent this situation as a game-frame. Note that we can represent a three- player game with a set of tables. Player 1 chooses the row, Player 2 chooses the column and Player 3 chooses the table (that is, we label the rows with Player 1's strategies, the columns with Player 2's strategies and the tables with Player 3's strategies). (b) Using the game-frame of part (a) obtain a reduced-form game by adding the information that each player prefers winning to not winning and is indifferent between any two outcomes where he/she does not win. For each player use a utility function with values from the set {0,1}. (c) Using the game-frame of part (a) obtain a reduced-form game by adding the information that (1) each player prefers winning to not winning, (2) Alice is indifferent between any two outcomes where she does not win, (3) conditional on not winning, Bob prefers if Charlie wins rather than Alice, (4) conditional on not winning, Charlie prefers if Bob wins rather than Alice. For each player use a utility function with values from the set {0,1,2}.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:Exercise 1.3. Alice (Player 1), Bob (Player 2), and Charlie (Player 3) play the
following simultaneous game. They are sitting in different rooms facing a keyboard
with only one key and each has to decide whether or not to press the key. Alice
wins if the number of people who press the key is odd (that is, all three of them or
only Alice or only Bob or only Charlie), Bob wins if exactly two people (he may
be one of them) press the key and Charlie wins if nobody presses the key.
(a) Represent this situation as a game-frame. Note that we can represent a three-
player game with a set of tables. Player 1 chooses the row, Player 2 chooses the
column and Player 3 chooses the table (that is, we label the rows with Player 1's
strategies, the columns with Player 2's strategies and the tables with Player 3's
strategies).
(b) Using the game-frame of part (a) obtain a reduced-form game by adding the
information that each player prefers winning to not winning and is indifferent
between any two outcomes where he/she does not win. For each player use a
utility function with values from the set {0,1}.
(c) Using the game-frame of part (a) obtain a reduced-form game by adding the
information that (1) each player prefers winning to not winning, (2) Alice is
indifferent between any two outcomes where she does not win, (3) conditional
on not winning, Bob prefers if Charlie wins rather than Alice, (4) conditional
on not winning, Charlie prefers if Bob wins rather than Alice. For each player
use a utility function with values from the set {0,1,2}.
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