Page 183 - DCOM303_DMGT504_OPERATION_RESEARCH

Operations Research

Notes known as Maximin. If Minimax = Maximin, then saddle point exists and the value of the game is
equal to Minimax to Maximin.
If Minimax  Maximin, then no saddle point exists.

If Minimax = Maximin, then the pure strategies are called optimum strategies.
Usually Maximin  value of the game  Minimax. If Maximin = Minimax = 0, the game is fair.
If Maximin = Minimax, the game is strictly determinable. The value of the game is the average
pay-off that would suit the game was played over and over again.

Self Assessment

Give one word for the following:
1. The list of all possible actions that a player takes for every payoff.

2. A strategy that always involves selecting a particular course of action with the probability
of 1.
3. The strategy to choose at least two courses of action with fixed probabilities.

4. A situation where both the players are facing pure strategies.
5. The “expected gain to a player” if the player and his opponent use their best strategies.

9.2 Two Person Zero Sum Game

In a game of two persons if the algebraic sum of the gains of both the players after the game is
zero, then it is called 2-person-zero-sum game.
Assumptions

1. There are 2 players having conflicting interests.
2. Each player has a finite number of strategies.
3. Each strategy selected by a player results into a certain pay-off and algebraic sum of these
pay-offs to both players is zero.
Two-person-zero-sum game with saddle point are called pure strategy games and two-person-
zero-sum game without saddle point are called mixed strategy games.

Let A and B be any 2 firms in an area have been selling a product competing for a larger share of
the market. Let us assume that these firms are considering the same three strategies in a bid to
gain the share in the market: Low advertising, high advertising and quality improvement and
let these firms can employ only one of the strategies at a time. Under these conditions, there are
3 x 3 combinations of the moves possible and the corresponding pay-off is given below. The
strategies of low advertising, high advertising and quality improvement is marked as a , a and
1 2
a for the firm A and b , b and b for the firm B.
3 1 2 3
B’s Strategy
b1 b2 b3 Row Min.
A’s Strategy a1 12 -8 -2 -8
a2 6 7 3 3
a3 -10 -6 2 -10
Col. Max. 12 7 3

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