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 the given two-person zero-sum game, Player A and Player B have different strategies to choose from, and each player wants to maximize their own payoff while minimizing their opponent's payoff.
To reduce the game by dominance property, we evaluate each row and column and eliminate dominated strategies. A dominated strategy is one that is always worse than another strategy, regardless of the opponent's choice.
Let's go through each row and column one by one:
Row I: 1 3 2 7 4
Row II: 3 4 1 5 6
Row III: 6 5 7 6 5
Row IV: 2 0 6 3 1
Analyzing each column:
Column 1: 1 3 6 2
Column 2: 3 4 5 0
Column 3: 2 1 7 6
Column 4: 7 5 6 3
Column 5: 4 6 5 1
In Row I, we can see that the payoff for Player A when choosing strategy I dominates strategy II. Therefore, we eliminate strategy II from Row I.
In Column 4, the payoff for Player B when Player A chooses strategy II dominates the payoff when A chooses strategy IV. Therefore, we eliminate strategy IV from Column 4.
After eliminating dominated strategies, the reduced game becomes:
Player B
1 2 3 5
Player A I 1 3 2 4
III 6 5 7 5
Now, to find the optimal strategies for each player and the value of the game, we need to solve for the saddle point. A saddle point occurs when the maximum value in a row is equal to the minimum value in the corresponding column.
In Row I, the maximum value is 4, which occurs in column 4.
In Row III, the maximum value is 7, which occurs in column 3.
In Column 3, the minimum value is 2, which occurs in Row I.
In Column 4, the minimum value is 4, which occurs in Row I.
Therefore, the optimal strategy for Player A is to choose strategy I, and the optimal strategy for Player B is to choose strategy 3.
The value of the game is the payoff when both players play their optimal strategies. In this case, the payoff is 4, which means Player A can expect to win 4 units if both players play optimally.