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
A two-person zero-sum game is a type of game where the total gains of one player equal the total losses of the other player. In other words, the sum of the payoffs is zero.
To reduce the given game by dominance property, we need to eliminate any dominated strategies. A strategy is dominated if there exists another strategy that always gives a better payoff, regardless of the other player's strategy.
Let's analyze the game matrix:
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
To simplify the game, we first need to identify any dominated strategies. By comparing the payoffs for each strategy for each player, we can determine if any strategy is always worse than another. In this case, there are no dominated strategies.
Next, we need to find the optimal strategies for each player. The optimal strategy is the one that maximizes the payoff for Player A and minimizes the payoff for Player B.
To find the optimal strategies, we can use the concept of saddle points. A saddle point is an element in the game matrix where the maximum value in its row coincides with the minimum value in its column.
In this case, there is no saddle point in the game matrix, so we need to use different methods to determine the optimal strategies.
One common approach is the minimax method. In this method, Player A chooses the strategy that maximizes their minimum payoff, while Player B chooses the strategy that minimizes their maximum payoff.
To find the optimal strategies using the minimax method, we calculate the maximum and minimum payoffs for each strategy for both players:
Player A's minimum payoffs:
I: 1
II: 1
III: 5
IV: 0
Player B's maximum payoffs:
1: 6
2: 5
3: 7
4: 7
5: 6
By comparing the minimum payoffs for Player A, we can see that the optimal strategy for Player A is "IV" (strategy 4).
By comparing the maximum payoffs for Player B, we can see that the optimal strategy for Player B is "III" (strategy 3).
The value of the game can be calculated as the payoff for Player A when they play their optimal strategy against Player B's optimal strategy. In this case, it is 3.