Page 194 - DCOM303_DMGT504_OPERATION_RESEARCH

Unit 9: Game Theory

9.4 Two-Person-Zero-Sum Games of 2 x m and n x 2 Notes

A game in which one of the two players has 2 strategies whilst opponent has more than 2 (say m)
is called 2 × m or n × 2 game depending upon whether columns or rows offer more choices.

For example, The following payoff matrices are of 2 × 3 and 3 × 2 size
 
 
    
   
 
In such cases we have to find the saddle point. If it exists give the optimal strategies of the
players. If no saddle point exists for (2 x m) or (x x 2) game then we have to reduce the game to
(2 x 2) which can be done by using (1) Dominance method, (2) Sub-game method and (3) Graphic
Method.

Dominance Method

Dominance method specifies the rules to be followed to eliminate strategies without affecting
the final solution.

Situations arise when some strategies dominate others in the payoff matrix. A strategy is said to
dominate another strategy if all its possible outcomes are preferable. In case of player A, row is
said to be dominating row if every element of that row is higher than or equal to the corresponding
entries of the another row called the dominating row. A column is said to be dominating if
every element of that column is lower than or equal to corresponding elements of another
column called the dominating column. Since the dominated strategies will never be chosen,
they can be removed from the payoff matrix to simplify the game. Here, we have to proceed row
by row and column by column and identify dominated rows and columns and delete them and
reduce the size of the payoff matrix. If it is reduced 2×2 matrix, use algebraic method or minimax
method to find the optimal solution. If not, use sub-game method or graphical method to reduce
the matrix further:

Example:

 
 
  
 
 
Step 1: All the entries of 1 row are higher than that of III row. Hence, III row is dominated by 1
row. Hence eliminate III row.

 
  
 
 
Step 2: All the elements of I column are less than that of II column. Hence, eliminate II column.

 
  
 
 

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