Page 184 - DCOM303_DMGT504_OPERATION_RESEARCH

Unit 9: Game Theory

Maximin = Maximum of Row Minimum =3 Notes
Minimax = Minimum of Column Maximum = 3
Therefore Saddle point = (a , b ), and
2 3
Value of the game = 3

Thus pay-off matrix is drawn from as point of view. A positive pay-off indicates that firm A has
gained the market share at the expense of firm B and negative value indicates B’s gain at A’s
expense. The problem now is to determine the best strategy for A and B. With the assumption
that each one is not aware of the move the other is likely to take reference to the pay off matrix
if firm A employs strategy a , then firm B employs strategy b in order to maximize its gain.
1 2
Similarly, if A’s strategies are a and a , then B’s strategies are b and b respectively. Now, firm
2 3 3 1
A would like to make the best use of the situation by choosing the maximum of these minimal
pay-offs. Since the minimal pay-offs corresponding to a , a and a are respectively -8, 3 and -10,
1 2 3
firm A would select a as its strategy. The decision rule here is Maximin strategy. Similarly, if firm
2
chooses b , then A will prefer a and if B uses b and b then firm A uses the strategy a . To
1 1 2 3 2
minimize the advantage occurring to A, firm B would select a strategy a . To minimize the
2
advantage occurring to A, firm B would select strategy a . To minimize the advantage occurring
2
to A, firm B would select a strategy that would yield the least advantage to its competitor, i.e., b .
3
The decision rule here is Minimax strategy.
Here, Minimax value = Maximin value = 3, which the value of the game and corresponds to the
saddle point.
Saddle point can be easily obtained for 2-person pure strategy games. We shall deal with
2-person mixed strategy games.
Here, the players play more than one strategy and no saddle point exists. To determine the
optimal strategies, the analyst needs to evaluate the probabilities (the proportion of time for
which each strategy is played). For doing so we have 3 methods namely:
1. Algebraic Method
2. Iterative Method for Approximate Solution

3. Linear Programming Method.
9.2.1 Algebraic Method

Let A and B be any two players with the following pay-off matrix; a , a and b , b denote the
1 2 1 2
strategies of A and B respectively. Let Pij denote the elements of pay-off matrix i, j = 1,2

Player B
Player A Strategies b1 b2
a1 P11 P12
a2 P21 P22

Player A has only 2 strategies namely, a and a . If probability that he chooses a is x then
1 2 1
probability that he chooses a is 1 – x. Similarly, if probability that player B chooses b and b is
2 1 2
y and 1 – y respectively. Let us consider the expected gain which is the weighted average of the
possible outcomes and is the product of payoff and the probabilities of the strategies. If player
B plays b throughout, then the gain to A is equal to
1
xP + (1–x)P ………………… (1)
11 21

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