Two retailers A and B are planning to open two stores in two towns, X and Y. Town X contains 60% of the population of the two towns, and town Y contains the other 40%. If both stores open in the same town, they will split the total business of both towns equally. If each store opens in a different town, each will take the total business of that town. Where should each store locate?
The payoff matrix for retailer A is:
The entries above represent the percentage of the total business of the two towns that retailer A gets in each case.
A game defined by a payoff matrix is said to be strictly determined if and only if there is an element of the matrix that is the smallest element in its row and at the same time the largest element in its column. This entry is called the saddle point, and is the value of the game. Each player chooses the strategy corresponding to that row and column respectively. The smallest elements in rows X and Y are 50 and 40 respectively, and the largest element in columns X and Y are 50 and 60 respectively. 50 is the smallest element in row X and the largest element in column X so the value of the game is 50 and the element in row X, column X is a saddle point. Both players will locate in town X.
Suppose there is a choice of three towns, X, Y and Z, and that the payoff matrix is:
| A\B |
X |
Y |
Z |
| X |
50 |
50 |
80 |
| Y |
50 |
50 |
80 |
| Z |
20 |
20 |
50 |
The smallest element in each row is 50, 50 and 20 respectively, and the largest element in each column is 50, 50 and 80 respectively. Now there is no single element that is at the same time the smallest element in its row and the largest element in its column. Instead there are four such entries, and each is a saddle point. The game is strictly determined with value 50, and each retailer choosing to locate in towns X or Y.
