Page 182 - DCOM303_DMGT504_OPERATION_RESEARCH

Unit 9: Game Theory

Play Notes

A play occurs when each player selects one of his available strategies. Two basic assumptions in
a play are:
(a) The choices of courses of action by players are made simultaneously.

(b) No player knows the choice of his opponents until he has decided on his own.

Outcome

Every combination of strategies of players determines an outcome called pay-off, where pay-off
is nothing but a gain to a player. A loss is considered as a negative gain.

Pay-off Matrix

The gains resulting from a game is presented in the form of a table called “ pay-off matrix”. A
pay-off matrix comprises n rows and m columns. Where n and m indicate the number of strategies
of first player and second player respectively. The pay-offs of each combination of the strategies
of players are placed as elements of matrix. A positive element shows the gain to the first player
(i.e., payment from II to I) and negative entry indicates the loss to the I player (i.e., payment from
I to II).
For instance, consider the following pay-off matrix:

1 2 3 4 5
1 4 8 -2 6 4
2 3 6 5 3 2
3 2 -9 1 7 10

If the player chooses the first strategy and the II player uses second strategy, then the I player
gains 8 units and the II player pays 8 units and similarly, if the 1 player chooses the third strategy
and second player uses II strategy, then the 1 player loses 9 units and pays it to the II player and
II player gains 9 units.
Strategies are classified into two types, namely, Pure strategy and Mixed Strategy.
1. A pure Strategy is a decision of the player to always select the same strategy.

2. A Mixed Strategy is a decision of the player to select more than one strategy with fixed
probabilities. A mixed strategy is advantageous since the opponent is always kept guessing.

Value of the Game

The value of the game is the “expected gain to a player” if he and his opponent use their best
strategies.
Saddle Point

A saddle point in a pay-off matrix corresponds to that element of the matrix which represents
the ‘Maxmin’ value of a player and Minimax value of his opponent.
For this we find Maximum element of each column and then find the Minimum value of column
Maxima known as Minimax. Similarly, we identify minimum element of each row and then find
the Maximum of those entries known as Maximin. If Minimax = Maximum of those entries

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