Game Theory

Graphical Method

The method discussed in the previous section is feasible when the value of n is small, because the larger value of n will yield a larger number of 2 X 2 sub-games. In this section, we discuss another method for solving 2 X n games. This method can only be used in games with no saddle point, and having a pay-off matrix of type n X 2 or 2 X n.

Example

Consider the following pay-off matrix

Solution.

The game does not have a saddle point as shown in the following table.

First, we draw two parallel lines 1 unit distance apart and mark a scale on each. The two parallel lines represent strategies of player B.
If player A selects strategy A1, player B can win –2 (i.e., loose 2 units) or 4 units depending on B’s selection of strategies. The value -2 is plotted along the vertical axis under strategy B₁ and the value 4 is plotted along the vertical axis under strategy B₂. A straight line joining the two points is then drawn.
Similarly, we can plot strategies A₂ and A₃ also. The problem is graphed in the following figure.

The lowest point V in the shaded region indicates the value of game. From the above figure, the value of the game is 3.4 units. Likewise, we can draw a graph for player B.

The point of optimal solution (i.e., maximin point) occurs at the intersection of two lines:

E1 = -2p₁ + 4p₂ and
E2 = 8p₁ + 3p₂

Comparing the above two equations, we have

-2p₁ + 4p₂ = 8p₁ + 3p₂

Substituting p₂ = 1 - p₁
-2p₁ + 4(1 - p₁) = 8p₁ + 3(1 - p₁)
p₁ = 1/11
p₂ = 10/11

Substituting the values of p₁ and p₂ in equation E1

V = -2 (1/11) + 4 (10/11) = 3.4 units