The image shows a two-person zero-sum game with the following matrix:

The problem asks to solve this game using the Simplex Method.

Steps to Solve Using Simplex Method:

1. Convert the Payoff Matrix: To apply the Simplex method, we usually aim to maximize the expected value for player A (row player) and minimize it for player B (column player). We will adjust the matrix so that all elements are non-negative by adding a constant value to the entire matrix.

2. Set Up the Linear Programming Problem:

   • The objective is to find the optimal mixed strategies for players A and B.
   • We assign probabilities $p_1, p_2, p_3$ to player A's strategies and $q_1, q_2, q_3$ to player B's strategies.
   • For player A, we maximize the expected payoff subject to the constraints of player B.
   • For player B, we minimize their maximum loss subject to the constraints of player A.

3. Find Optimal Strategies:

   • The minimum of the row maxima and the maximum of the column minima should converge to the same value, which is the value of the game.
   • The solution will lie in finding these values using the Simplex algorithm or by solving the linear programming formulation.

Summary of Information:

• The matrix is already arranged with row minimums and column maximums.
• The value of the game $V$ lies between -2 and 2, i.e., $-2 \leq V \leq 2$.

Would you like me to go through the full simplex method step-by-step for this game or focus on the key steps? Let me know if you have any specific questions.

Relative Questions:

1. What is the difference between pure strategy and mixed strategy in game theory?
2. How does the simplex method work in solving linear programming problems?
3. Why is it important to convert all matrix elements to non-negative values in game theory?
4. What is the significance of the value of the game in zero-sum games?
5. How do you interpret the results of a two-person zero-sum game?

Tip: In a zero-sum game, the sum of the payoffs for both players is always zero, so maximizing the payoff for one player automatically minimizes the payoff for the other player.