For building any model, certain reasonable assumptions are quite necessary.

**Assumptions
for two-person zero sum game**

For building any model, certain reasonable
assumptions are quite necessary. Some assumptions for building a model of
two-person zero sum game are listed below.

a.
Each player has available to him
a finite number of possible courses of action. Sometimes the set of courses of
action may be the same for each player. Or, certain courses of action may be
available to both players while each player may have certain specific courses
of action which are not available to the other player.

b.
Player A attempts to maximize
gains to himself. Player B tries to minimize losses to himself.

c.
The decisions of both players are
made individually prior to the play with no communication between them.

d.
The decisions are made and
announced simultaneously so that neither player has an advantage resulting from
direct knowledge of the other playerâ€™s decision.

e.
Both players know the possible
payoffs of themselves and their opponents.

**Minimax
and Maximin Principles**

The selection of an optimal
strategy by each player without the knowledge of the competitorâ€™s strategy is
the basic problem of playing games. The objective of game theory is
to know how these players must select their respective strategies, so that they
may optimize their payoffs. Such a criterion of decision making is referred to
as minimax-maximin principle. This principle in games of pure strategies leads
to the best possible selection of a strategy for both players. For example, if player A chooses
his ith strategy, then he gains at least
the payoff min *a*_{ij} , which is minimum of the ith row elements in the payoff matrix. Since his objective is to maximize
his payoff, he can choose strategy *i*
so as to make his payoff as large as possible. i.e., a payoff which is not less
than max min *a**ij* .

Similarly player B can choose jth column elements so as to make his loss
not greater than max min *a**ij* .

If the maximin value for a player is equal to the
minimax value for another player, i.e.

then the game is said to have a saddle point
(equilibrium point) and the corresponding strategies are called optimal
strategies. If there are two or more saddle points, they must be equal. The amount of payoff, i.e., V at
an equilibrium point is known as the *value of the game.* The
optimal strategies can be identified by the players in the long run. *Fair game*

The game
is said to be fair if the value of the game V = 0.

Tags : Operations Management - Game Theory, Goal Programming & Queuing Theory

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