use dominance to slove the game with following pay-off matrices: ((2 3 1/2) (3/2 2 0) (1/2 1 1)).
ANS: A=(1/4,0,3/4); B=(1/4,0,3/4); v=7/8.
i got answer, but how to do this? i need solution
To solve the game using dominance, you need to compare the payoffs of each strategy for both players and eliminate dominated strategies.
Let's denote the pay-off matrix as follows:
A = ((2, 3, 1/2), (3/2, 2, 0), (1/2, 1, 1))
To solve the game, follow these steps:
Step 1: Dominance by row elimination
- We check if any strategy for player A is strictly dominated by another strategy.
- In row 1: 2 is greater than 1/2, and 3 is greater than 3/2, so strategy A1 weakly dominates strategy A3.
- In row 3: 1/2 is less than 1 and 1 is greater than 0, so strategy A3 is strictly dominated by strategy A2.
- After eliminating dominated strategies, the reduced pay-off matrix becomes:
A = ((2, 3), (3/2, 2))
Step 2: Dominance by column elimination
- We check if any strategy for player B is strictly dominated by another strategy.
- In column 1: 2 is greater than 3/2, so strategy B1 weakly dominates strategy B2.
- In column 2: 3 is greater than 2, so strategy B1 also strictly dominates strategy B3.
- After eliminating dominated strategies, the reduced pay-off matrix becomes:
A = ((2,), (3/2,))
Step 3: Calculate mixed strategies
- Since there is only one strategy left for player A, player A will always choose strategy A1.
- To calculate the mixed strategy for player B, we need to find the probabilities p1 and p2 for B1 and B2, respectively.
- We set up the equations for expected payoffs:
2p1 + (3/2)(1 - p1) = v
where v is the expected payoff for player B.
- Solving the equation, we find p1 = 1/4 and p2 = 0.
- Therefore, player B will choose strategy B1 with a probability of 1/4 and strategy B3 with a probability of 3/4.
Step 4: Calculate the value of the game
- We substitute the mixed strategies into the pay-off matrix:
((2, 3), (3/2,))
- The expected payoff for player B, v, is the average of the payoff for each of B1 and B3:
v = (2)(1/4) + (3)(3/4) = 7/8.
Final solution:
Player A will always choose strategy A1, while player B will choose strategy B1 with a probability of 1/4 and strategy B3 with a probability of 3/4. The value of the game is 7/8.