Operation research

Topics: Game theory, Nash equilibrium, Matching pennies Pages: 20 (1665 words) Published: October 16, 2013
Unit 3
GAME THEORY
Lesson 27

Learning Objective:



To learn to apply dominance in game theory.
Generate solutions in functional areas of business and management.

Hello students,
In our last lecture you learned to solve zero sum games having mixed strategies. But...
Did you observe one thing that it was applicable to only 2 x 2 payoff matrices?
So let us implement it to other matrices using dominance and study the importance of

DOMINANCE

In a game, sometimes a strategy available to a player might be found to be preferable to some other strategy / strategies. Such a strategy is said to dominate the other one(s). The rules of dominance are used to reduce the size of the payoff matrix. These rules help in deleting certain rows and/or columns of the payoff matrix, which are of lower priority to at least one of the remaining rows, and/or columns in terms of payoffs to both the players. Rows / columns once deleted will never be used for determining the optimal strategy for both the players. This concept of domination is very usefully employed in simplifying the two – person zero sum games without saddle point. In general the following rules are used to reduce the size of payoff matrix.

The RULES
follow are:

( PRINCIPLES OF DOMINANCE )

you will have to

Rule 1: If all the elements in a row ( say i th row ) of a pay off matrix are less than or equal to the corresponding elements of the other row ( say j th row ) then the player A

will never choose the i th strategy then we say i th strategy is dominated by j th strategy and will delete the i th row.
Rule 2: If all the elements in a column ( say r th column ) of a payoff matrix are greater than or equal to the corresponding elements of the other column ( say s th column ) then the player B will never choose the r th strategy or in the other words the r th strategy is dominated by the s th strategy and we delete r th column . Rule 3: A pure strategy may be dominated if it is inferior to average of two or more other pure strategies.

Now, consider some simple examples

Example 1
Given the payoff matrix for player A, obtain the optimum strategies for both the players and determine the value of the game.
Player B
Player A

6

-3

7

-3

0

4

Solution
Player B
B1

B3

A1

6

-3

7

A2

Player A

B2

-3

0

4

When A chooses strategy A1 or A2, B will never go to strategy B3. Hence strategy

B3 is redundant.
Player B
B1

B2

Row minima

A1

6

-3

-3

A2

-3

0

-3

Column maxima

6

0

Player A

Minimax (=0), maximin (= -3).Hence this is not a pure strategy with a saddle point. Let the probability of mixed strategy of A for choosing Al and A2 strategies are p1 and 1- p1 respectively. We get

6 p1 - 3 (1 - p1) = -3 p1 + 0 (1 - p1) or

p1 =1/ 4

Again, q 1 and 1 - q 1 being probabilities of strategy B, we get 6 q 1 - 3 (1 - q 1 ) = -3 q 1 + 0 (1 - q 1 ) or

q 1 = 1/ 4

Hence optimum strategies for players A and B will be as follows:

A1
SA

A2

1/4

3/4

=

and
B1
SB

B2

B3

1/4

3/4

0

=

Expected value of the game = q 1 (6 p1-3(1- p1)) + (1- q 1 )(3 q 1 + 0(1- q 1 )) = ¾ Example 2
In an election campaign, the strategies adopted by the ruling and opposition party alongwith pay-offs (ruling party's % share in votes polled) are given below:

Opposition Party's Strategies
Campaign
one day in
each city

Ruling Party's Strategies
Campaign one day in each city
Campaign two days in large towns
Spend two days in large rural sectors

Campaign
two days in
large towns

Spend two
days in large.
rural sectors

55
70
75

40
70
55

35
55
65

Assume a zero sum game. Find optimum strategies for both parties and expected payoff to ruling party.
Solution. Let A1, A2 and A3 be the strategies of the ruling party and B1, B2 and B3 be those of the opposition. Then
Player B
B1

B3

A1

55

40

35

A2...
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