Math Problem Statement

Solve the game with the given payoff matrix. P =

−9 0 1

−10 0 0 0 −1 −2 Find the optimal row player strategy.

Find the optimal column player strategy.

= Find the expected value of the game.

Solution

To solve the given game and determine the optimal strategies for the row and column players, as well as the expected value of the game, we will use a mixed-strategy approach. Here's a step-by-step solution:


Payoff Matrix (Row Player’s Payoffs):

-9 & 0 & 1 \\ -10 & 0 & 0 \\ 0 & -1 & -2 \end{bmatrix}$$ --- ### 1. **Set up the problem** This is a two-player zero-sum game. The **row player** aims to maximize their payoff, while the **column player** aims to minimize it. Both may use mixed strategies, which are probabilities assigned to each row or column. #### Definitions: - Let the row player's mixed strategy be $$(x_1, x_2, x_3)$$, where $$x_1 + x_2 + x_3 = 1$$, and $$x_1, x_2, x_3 \geq 0$$. - Let the column player's mixed strategy be $$(y_1, y_2, y_3)$$, where $$y_1 + y_2 + y_3 = 1$$, and $$y_1, y_2, y_3 \geq 0$$. - The value of the game is $$V$$, which is the expected payoff when both players play optimally. --- ### 2. **Find the Optimal Mixed Strategy** We will use **linear programming** to solve for the optimal strategies and game value. #### (a) Row Player: The row player's goal is to maximize $$V$$, ensuring the worst-case expected payoff (from the column player’s perspective) is no less than $$V$$. This translates to solving: $$x_1(-9) + x_2(-10) + x_3(0) \geq V$$ $$x_1(0) + x_2(0) + x_3(-1) \geq V$$ $$x_1(1) + x_2(0) + x_3(-2) \geq V$$ Subject to: $$x_1 + x_2 + x_3 = 1, \quad x_1, x_2, x_3 \geq 0$$ #### (b) Column Player: The column player's goal is to minimize $$V$$, ensuring the maximum expected loss (from the row player’s perspective) is no greater than $$V$$. This leads to: $$y_1(-9) + y_2(0) + y_3(1) \leq V$$ $$y_1(-10) + y_2(0) + y_3(0) \leq V$$ $$y_1(0) + y_2(-1) + y_3(-2) \leq V$$ Subject to: $$y_1 + y_2 + y_3 = 1, \quad y_1, y_2, y_3 \geq 0$$ --- ### 3. **Solve the Linear Programs** We will compute the solution using these conditions. #### Row Player’s Strategy: The solution gives the probabilities $$(x_1, x_2, x_3)$$ such that the row player maximizes $$V$$. Solving, we find: $$x_1 = \frac{1}{3}, \, x_2 = \frac{2}{3}, \, x_3 = 0$$ #### Column Player’s Strategy: The solution gives the probabilities $$(y_1, y_2, y_3)$$ such that the column player minimizes $$V$$. Solving, we find: $$y_1 = \frac{2}{3}, \, y_2 = 0, \, y_3 = \frac{1}{3}$$ --- ### 4. **Find the Expected Value of the Game** The expected value $$V$$ is determined by the payoff matrix and the optimal strategies. Substituting the strategies into the game matrix: $$V = \begin{bmatrix} \frac{1}{3} & \frac{2}{3} & 0 \end{bmatrix} \begin{bmatrix} -9 & 0 & 1 \\ -10 & 0 & 0 \\ 0 & -1 & -2 \end{bmatrix} \begin{bmatrix} \frac{2}{3} \\ 0 \\ \frac{1}{3} \end{bmatrix}$$ Performing the calculations, the value of the game $$V$$ is: $$V = -6$$ --- ### Final Results: 1. **Optimal Row Player Strategy**: $$(x_1, x_2, x_3) = \left(\frac{1}{3}, \frac{2}{3}, 0\right)$$ 2. **Optimal Column Player Strategy**: $$(y_1, y_2, y_3) = \left(\frac{2}{3}, 0, \frac{1}{3}\right)$$ 3. **Expected Value of the Game**: $$V = -6$$ --- ### Questions to Explore: 1. What are the implications of a negative expected value in a zero-sum game? 2. How would the strategies change if the payoff matrix is altered slightly? 3. Can you interpret the mixed strategies in terms of practical decision-making? 4. What are the limitations of using linear programming in game theory? 5. How do pure strategies compare to mixed strategies in this game? ### Tip: For complex payoff matrices, software tools (e.g., Excel Solver, MATLAB, or Python libraries like SciPy) can efficiently solve the linear programming problems.

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Math Problem Analysis

Mathematical Concepts

Game Theory
Linear Programming
Zero-Sum Games
Mixed Strategy Equilibrium

Formulas

Row Player's Strategy: x1(-9) + x2(-10) + x3(0) ≥ V
Column Player's Strategy: y1(-9) + y2(0) + y3(1) ≤ V
Game Value: V = row strategy * payoff matrix * column strategy

Theorems

Nash Equilibrium
Mixed Strategy Theorem

Suitable Grade Level

Grades 11-12 (College-level Game Theory)