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.) A B C D a 1,2 1,0 2,1 0,-3 b 1,4 3,5 2,0 1,2 c -1,1 4,1 4,-1 -1,2 d 0,0 -2,-1 5,2 -1,0Select all of the following that are Nash equilibria in the above game. (Note: partial credit is possible for this question.) (1,2) (a,A) (d,C) (b,B) (3,5) (c,B)
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.) A B C D a 1,2 1,0 2,1 0,-3 b 1,4 3,5 2,0 1,2 c -1,1 4,1 4,-1 -1,2 d 0,0 -2,-1 5,2 -1,0Select all of the following that are Nash equilibria in the above game. (Note: partial credit is possible for this question.) (1,2) (a,A) (d,C) (b,B) (3,5) (c,B)
Solution
In a game theory, a Nash equilibrium is a set of strategies, one for each player, such that no player has incentive to unilaterally change her action given the other players' actions. Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less than if she remained with her current strategy.
For each cell, we check if the payoff for each player is the best response to the other player's strategy. If it is, then that cell represents a Nash equilibrium.
Let's check each option:
- (1,2): This is not a Nash equilibrium because it's a payoff, not a strategy.
- (a,A): Player 1 would not change from 'a' as 'a' gives the maximum payoff (1) when Player 2 chooses 'A'. Player 2 would not change from 'A' as 'A' gives the maximum payoff (2) when Player 1 chooses 'a'. So, (a,A) is a Nash equilibrium.
- (d,C): Player 1 would change from 'd' to 'c' as 'c' gives a higher payoff (5) when Player 2 chooses 'C'. So, (d,C) is not a Nash equilibrium.
- (b,B): Player 1 would not change from 'b' as 'b' gives the maximum payoff (3) when Player 2 chooses 'B'. Player 2 would not change from 'B' as 'B' gives the maximum payoff (5) when Player 1 chooses 'b'. So, (b,B) is a Nash equilibrium.
- (3,5): This is not a Nash equilibrium because it's a payoff, not a strategy.
- (c,B): Player 1 would not change from 'c' as 'c' gives the maximum payoff (4) when Player 2 chooses 'B'. Player 2 would change from 'B' to 'A' as 'A' gives a higher payoff (2) when Player 1 chooses 'c'. So, (c,B) is not a Nash equilibrium.
So, the Nash equilibria in the game are (a,A) and (b,B).
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