Math  /  Discrete

QuestionRefer to the diagram, where the numerical data show profits in millions of dollars. Beta's profits are shown in the northeast corner and Alpha's profits in the southwest corner of each cell. Which cell represents a Nash equilibrium? (A) celld (B) cell A (C) Cells B and C both represent a Nash equilibrium. (D) There is no Nash equilibrium in this game.

Studdy Solution

STEP 1

1. A Nash equilibrium occurs when no player can benefit by changing their strategy while the other player's strategy remains unchanged.
2. We need to analyze each cell to determine if it represents a Nash equilibrium.

STEP 2

1. Define Nash equilibrium in the context of the problem.
2. Analyze each cell to check for Nash equilibrium conditions.
3. Determine which cell(s) represent a Nash equilibrium.

STEP 3

A Nash equilibrium is a situation in a game where each player's strategy is optimal given the strategy of the other player. No player has anything to gain by changing only their own strategy unilaterally.

STEP 4

Analyze Cell A: - If Alpha changes their strategy from Cell A to Cell B, their profit decreases from 20to20 to 10. - If Beta changes their strategy from Cell A to Cell C, their profit decreases from 20to20 to 10. - Therefore, neither player benefits by changing their strategy unilaterally from Cell A.

STEP 5

Analyze Cell B: - If Alpha changes their strategy from Cell B to Cell A, their profit increases from 10to10 to 20. - If Beta changes their strategy from Cell B to Cell D, their profit decreases from 30to30 to 15. - Therefore, Alpha benefits by changing their strategy unilaterally from Cell B.

STEP 6

Analyze Cell C: - If Alpha changes their strategy from Cell C to Cell D, their profit decreases from 30to30 to 15. - If Beta changes their strategy from Cell C to Cell A, their profit increases from 10to10 to 20. - Therefore, Beta benefits by changing their strategy unilaterally from Cell C.

STEP 7

Analyze Cell D: - If Alpha changes their strategy from Cell D to Cell C, their profit increases from 15to15 to 30. - If Beta changes their strategy from Cell D to Cell B, their profit increases from 15to15 to 30. - Therefore, both players benefit by changing their strategy unilaterally from Cell D.

STEP 8

Since neither player benefits by changing their strategy unilaterally in Cell A, Cell A represents a Nash equilibrium.
The cell that represents a Nash equilibrium is:
Cell A \boxed{\text{Cell A}}

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