Two-person zero-sum games involve two players where the total gains and losses sum to zero. Such games can be represented using a payoff matrix which shows the payoffs for each combination of strategies. The value of the game is the expected payoff when both players use optimal strategies. If a saddle point exists in the matrix, where the maximin and minimax values are equal, it represents the optimal strategies and value of the game.

1. Mrs.G.Nagalakshmi, M.Sc.,M.Phil.,
Mrs.K.Muthulakshmi, M.Sc.,M.Phil.,
Ms.S.Swathi Sundari, M.Sc.,M.Phil.,

2. Two Person Zero Sum Games
When there are two competitors(players) playing a game is
called a two-person game.
The algebraic sum of gains and losses of all the players is
zero are called zero-sum games.
Zero sum games with two players are called two-person zero
sum game.
Strategies (nothing but a set of rules)
It may be of two
types :

3. Optimum Strategy : A course of action which puts the
player in the most preferred position, irrespective of the
strategy of his competitors, is called an optimum strategy.
Payoff Matrix : When players select their particular
strategies, the payoff’s (gains or losses) can be represented
in the form of a matrix called the payoff matrix.
Value of the Game : It is expected payoff of play when all
the players of the game follow their optimum strategies.
Fair & Unfair Game : If the value of the game is zero, then
its called an fair game and if it is non-zero then it is called
unfair game.

4. In a two person game, if saddle point exist it is solved by
using pure strategies, otherwise mixed strategies decide
results.
Maximin Value : Minimum value in each row denoting
minimum possible gain from each strategy of A. Maximin
value is the maximum of these minimum values.
Minimax Value : Maximum value in each column denoting
maximum possible loss from each strategy of B. Minimax
value is the minimum of these maximum values.
Saddle Point : If the maximin value and the minimax
value are equal, the corresponding cell is the saddle point,
and the value is value of the game and the corresponding
strategies called an optimum strategies.

5. Therefore, (i) Value of the game is 2
(ii) saddle point is (1,3)
(iii) Optimum Strategy for Company A is ‘3’.