Math Problem Statement
Determine whether or not the following games are fair. (a)
4 2
−1 6
This is a fair game. This is not a fair game.
Solution
To determine if a game is fair, we need to analyze whether the expected value (or average payoff) for each player is equal to zero. A fair game ensures that neither player has an advantage over the other.
Given the payoff matrix:
4 & 2 \\ -1 & 6 \end{bmatrix}$$ ### Steps to Analyze: 1. **Define probabilities for rows (player 1)**: Let $$p_1$$ and $$p_2$$ be the probabilities for rows $$1$$ and $$2$$, respectively. Since $$p_1 + p_2 = 1$$, we can represent $$p_2 = 1 - p_1$$. 2. **Define probabilities for columns (player 2)**: Let $$q_1$$ and $$q_2$$ be the probabilities for columns $$1$$ and $$2$$, respectively. Similarly, $$q_1 + q_2 = 1$$, so $$q_2 = 1 - q_1$$. 3. **Calculate expected payoff for Player 1**: Using probabilities $$p_1, p_2, q_1, q_2$$, the expected payoff $$E$$ for Player 1 is: $$E = p_1 (q_1 \cdot 4 + q_2 \cdot 2) + p_2 (q_1 \cdot -1 + q_2 \cdot 6)$$ 4. **Fairness Condition**: For the game to be fair, $$E = 0$$ for any choice of strategies. This requires that the game matrix must balance out such that no player consistently has a positive payoff. #### Testing Fairness: Let’s examine whether the matrix satisfies this condition. Without loss of generality, we solve $$E = 0$$ with equal probabilities: - Assign $$p_1 = p_2 = 0.5$$ and $$q_1 = q_2 = 0.5$$. - Substitute into the equation for $$E$$: $$E = 0.5 \left(0.5 \cdot 4 + 0.5 \cdot 2\right) + 0.5 \left(0.5 \cdot -1 + 0.5 \cdot 6\right)$$ $$E = 0.5 \left(2 + 1\right) + 0.5 \left(-0.5 + 3\right)$$ $$E = 0.5 \cdot 3 + 0.5 \cdot 2.5 = 1.5 + 1.25 = 2.75$$ Since $$E \neq 0$$, **this is not a fair game**. --- Would you like a detailed explanation of solving similar problems or additional examples? ### Related Questions: 1. What is the definition of a fair game in game theory? 2. How can probabilities be adjusted to make a game fair? 3. What role does the expected value play in determining fairness? 4. Can a game matrix ever be fair if it has unequal payoffs? 5. How do mixed strategies affect fairness in games? **Tip:** In analyzing fairness, always confirm that the expected payoff is zero for all strategies, not just specific cases!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Game Theory
Probability
Expected Value
Formulas
E = p1 * (q1 * a11 + q2 * a12) + p2 * (q1 * a21 + q2 * a22)
p1 + p2 = 1
q1 + q2 = 1
Theorems
Fairness Condition in Game Theory
Suitable Grade Level
Grades 10-12
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