slove the game with following pay-off 3*3 matrices:((2,3,1/2)(3/2,2,0)(1/2,1,1))
To solve the game with the given payoff matrix, we can use the concept of mixed strategies in game theory. The matrix you provided is a 3x3 payoff matrix where each entry represents the payoffs to the row player (Player A) and column player (Player B) for different strategies chosen.
To find the optimal strategies for both players, we can use the concept of expected payoffs. Player A will try to maximize their minimum expected payoff, while Player B will try to minimize their maximum expected payoff.
Here are the steps to solve the game:
1. Calculate the expected payoffs for Player A:
- To do this, we need to assign probabilities to each of Player A's strategies. Let's assume Player A's strategies are represented by the probabilities p1, p2, and p3 for each row.
- Calculate the expected payoff for each strategy by multiplying the corresponding probabilities with the payoffs and summing them up.
- For example, the expected payoff for Player A's first strategy would be:
Expected payoff for strategy 1 = (2 * p1) + (3/2 * p2) + (1/2 * p3)
- Similarly, calculate the expected payoffs for Player A's other strategies.
2. Calculate the expected payoffs for Player B:
- Assign probabilities q1, q2, and q3 to Player B's strategies (corresponding to the columns of the matrix).
- Calculate the expected payoff for each of Player B's strategies, similarly to how we did it for Player A.
3. Solve the game:
- Now, we need to find the optimal strategies for both players.
- Player A will choose the strategy with the highest minimum expected payoff.
- Equally, Player B will choose the strategy with the lowest maximum expected payoff.
By following the above steps, you can solve the game with the given payoff matrix.