Write short note on Tow-Person Zero Sum Game. Reduce the following game by dominance property and find optimal strategies for each player. Also find the value of game.:
Player B
1 2 3 4 5
Player A I 1 3 2 7 4
II 3 4 1 5 6
III 6 5 7 6 5
IV 2 0 6 3 1
In a two-person zero-sum game, the interests of the two players are completely opposite, meaning that whatever one player gains, the other player loses, resulting in a total sum of zero.
To reduce the game by the dominance property, we compare the payoffs for each player in each strategy. In each row, we look for the maximum payoff for Player A, and then the minimum payoff for Player B in that column. If these payoffs are the same for at least one strategy, then we can eliminate the other strategies.
Using this method, we can reduce the game as follows:
Player B
2 4 5
Player A II 3 4 6
III 5 6 5
IV 0 3 1
By eliminating strategy I for Player A and strategy I for Player B, we can reduce the game.
The optimal strategy for Player A is to choose strategy III, as it has the highest payoff compared to the other strategies. Similarly, the optimal strategy for Player B is strategy IV, as it has the lowest payoff compared to the other strategies.
The value of the game is determined by the payoff at the intersection of the optimal strategies, which in this case is (III, IV), with a value of 5. This means that if both players play optimally, the expected outcome for Player A is 5, while the expected outcome for Player B is -5.