Methods Of Solving Games

Principle of Dominance in Games:

• In case there is no saddle point the given game matrix (m × n) may be reduced to m × 2 or 2 × n or 2 × 2 matrix, which will help us to proceed further to solve the game. To discuss the principle of dominance, let us consider the matrix given below:

The row minimums and column maximums show that the problem is not having saddle point. Hence we have to use method of dominance to reduce the size of the matrix.

(i)Consider the first and second strategies of B. If B plays the first strategy, he loses 2 units of money when A plays first strategy and 4 units of money when A plays second strategy. Similarly, let us consider B’s second strategy, B gains 4 units of money when A plays his first strategy and gains 3 units of money when A plays second strategy. Irrespective of A’s choice, B will gain money. Hence for B his second strategy is superior to his first strategy. In other words, B's second strategy dominates B's first strategy. Or B’first strategy is dominated by B's second strategy. The reduced matrix is:

(ii)Consider B’s III and IV strategy. When B plays IV strategy, he loose 4 units of money when A plays his first strategy and 2 units of money when A plays his second strategy. Whereas, when B plays his III strategy, he gains 3 units of money and 4 units of money, when A plays his I and II strategy respectively. Hence B’s IV strategy (pure strategy) is dominating the third strategy. Hence we can remove the same from the game. The reduced matrix is:

•        Whenelements of a column, say ith are less than or equals to the corresponding elements of jth column, then jth column is dominated by ith column or ith column dominates jth column. Consider the matrix given below

•   Let A play his first strategy, then he looses 2 units of money and looses 4 units of money when B plays his second strategy. But when A plays his second strategy, he gains 1 unit of money for B’s firststrategy and gains 2 units of money, for B’s second strategy. Hence, A's second strategy (purestrategy) is superior to A's first strategy or A's second strategy dominates A's first strategy or
• A's first strategy is dominated by A's second strategy. We can closely examine and find that elements of A's second strategy are greater than the elements of first strategy. Hence we can formulate general rule of dominance for rows. When the elements of rth row are greater than or equals to elements of sth row, then rth row dominates sth row or sth row is dominated by rth row.