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(In this and all other questions in which a game matrix is given, Player 1 chooses the row, Player 2 chooses the column, and if there is a Player 3, she chooses the matrix.)  restaurant cricket restaurant  8, X 0, 5 cricket  1, 12 10, 7We say that Players 1 and 2 meet if and only if they choose the same strategy. Suggest a value in the range  such that the above game matrix reflects that- Player 1 prefers being at cricket over being at the restaurant, while Player 2 prefers being at the restaurant over being at cricket, and- each player prefers meeting at a location over being alone at that location, and- meeting is more important to Player 1 than being in their preferred location, and- being at their preferred location is more important to Player 2 than meeting

Question

(In this and all other questions in which a game matrix is given, Player 1 chooses the row, Player 2 chooses the column, and if there is a Player 3, she chooses the matrix.)  restaurant cricket restaurant  8, X 0, 5 cricket  1, 12 10, 7We say that Players 1 and 2 meet if and only if they choose the same strategy. Suggest a value in the range  such that the above game matrix reflects that- Player 1 prefers being at cricket over being at the restaurant, while Player 2 prefers being at the restaurant over being at cricket, and- each player prefers meeting at a location over being alone at that location, and- meeting is more important to Player 1 than being in their preferred location, and- being at their preferred location is more important to Player 2 than meeting

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Solution

The game matrix is as follows:

Restaurant Cricket
Restaurant 8, X 0, 5
Cricket 1, 12 10, 7

Player 1 prefers cricket over the restaurant, so the payoff for cricket should be higher than the restaurant. This is reflected in the matrix as 10 > 8.

Player 2 prefers the restaurant over cricket, so the payoff for the restaurant should be higher than cricket. This is reflected in the matrix as X > 7.

Each player prefers meeting at a location over being alone at that location. This is reflected in the matrix as the diagonal elements (where they meet) are higher than the off-diagonal elements (where they are alone).

Meeting is more important to Player 1 than being in their preferred location. This means that the payoff for meeting at the restaurant (8) should be higher than the payoff for being alone at cricket (1). This is already the case in the matrix.

Being at their preferred location is more important to Player 2 than meeting. This means that the payoff for being alone at the restaurant (X) should be higher than the payoff for meeting at cricket (7).

So, to satisfy all these conditions, X should be a value greater than 7 but less than or equal to 12. Therefore, X could be any value in the range (7, 12].

This problem has been solved

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