12 Game Theory

8/3/2019 12 Game Theory

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Operations Research Unit 12

Sikkim Manipal University 199

Unit 12 Game Theory

Structure

12.1. Introduction

12.2. Competitive situations

12.2.1. Marketing different brands of a commodity

12.2.2. Campaigning for elections

12.2.3. Fighting military battles

12.3. Characteristics of competitive games

12.3.1 n-Person Game

12.3.2 Zero-Sum Game

12.3.3 Two-Person Zero-Sum Game (Rectangular Game).

12.3.4 Strategy

12.3.5 Pure Strategy

12.3.6 Mixed Strategy

12.4. Maximin Minimax principle

12.4.1. Saddle Point

12.4.2. Solution to a Game with Saddle Point12.5. Dominance

12.5.1 Solving Games Using Dominance

12.6. Summary

Terminal Questions

Answers to SAQs and TQs

12.1 Introduction

Game theory was developed by John Von Newman. He worked on game theory right from 1928.

But, it gained prominence only after 1944 when he published (along with Mrogenstren) the work

Theory of games and economic behaviour. This field of study is fast developing and it is highly

resourceful.


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Learning Objectives

After studying this unit, you should be able to understand the following

1. The competitive situations

2. Application of Game Theory to such situations

3. Find a solution using Saddle point and Dominance principle

12.2 Competitive Situations

Competitive situations occur when two or more parties with conflicting interests operate. The

situations may occur as follows.

12.2.1. Marketing different brands of a commodity.

Two (or more) brands of detergents (soaps) try to capture the market by adopting various

methods (courses) such as advertising through electronic media, providing cash discounts to

consumers or offering larger sales commission to dealers.

12.2.2. Campaigning for elections.

Two (or more) candidates who contest an elections try to capture more votes by adopting various

methods (courses) such as campaigning through T.V., door to door campaigning or

campaigning through public meetings.

12.2.3. Fighting military battles.Two forces fighting a war try to gain supremacy over one another by adopting various courses of

action such as direct ground attack on enemy camp, ground attack supported by aerial attack or

playing defensive by not attacking.

We consider each of the above situations to be a competitive game where the parties (players)

adopt a course of action (play the game).

Self Assessment Questions 1

Fill in the blanks

1. Competitive situation occur when ______ or ________ parties with ______ _________

operates.

2. In competitive game prayers have _________ number of courses of action available to them.

12.3 Characteristics of a Competitive Game


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A competitive game has the following characteristics.

1. The number of players (competitors) is finite.

2. Each player has finite number of courses of action (moves).

3. The game is said to be played when each player adopts one of his course of action.4. Every time the game is played, the corresponding combination of courses of action leads to a

transaction (payment) to each player. The payment is called pay-off (gain). The pay-off may

be monetary (money) or some such benefit as increased sales, etc.

5. The players do not communicate to each other.

6. The players know the rules of the game before starting.

12.3.1 n-Person Game

A game in which n players participates is called n-person game.

A game in which two players participate is called 2-person game (two-person game).

12.3.2 Zero-Sum Game

If a game is such that whenever it is played the sum of the gains (pay-off) of the players is zero, it

is called zero-sum game.

A zero-sum game which has two players is called two-person zero-sum game. It is called

rectangular game.

In a two-person zero-sum game, the gain of the one player is equal to the loss of the other.

12.3.3 Two-Person Zero-Sum Game (Rectangular Game).

A two-person zero-sum game is a game in which

i) two players participate

ii) the gain of one player it the loss of the other.

In a two-person zero-sum game, let the players be A and B. Let m21 A,...A,A be the m courses

of action for player A. Let n,2,1 B...BB be the n courses of action for player B. Let

)n....,2,1jm.....,2,1i(a ij == be the pay-off (gain) of player A when he plays the course of action,

iA and player B plays the course of action jB . Then, thefollowing matrix is the pay-off (gain)

matrix of player A.


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Player B

n21 B....................BB

Player A

m

2

A

.

.

.

A

A

This is a nm (read as m by n) game.

Here, ija is the gain of A. Also, ija is the loss of B. Therefore, (- ija ) is the gain of B. And so, the

pay-off matrix of B is obtained by writing (- ija ) in the place of ija in the above matrix and then

writing the transpose of the matrix.

12.3.4 Strategy

In a game, the strategy of a player is the predetermined rule by which he chooses his course of

action while playing the game.

The strategy of a player may be pure strategy or mixed strategy.

12.3.5 Pure Strategy

While playing a game, pure strategy of a player is his predecision to adopt a

specified course of action (say rA ) irrespective of the strategy of the opponent.

12.3.6 Mixed Strategy

While playing a game, mixed strategy of a player is his predecision to choose his course of action

according to certain pre-assigned probabilities.

Thus, if player A decides to adopt courses of action 21 AandA with perspective probabilities 0.4

and 0.6, it is mixed strategy.

Example 1: (2-finger morra game).

Two persons A and B play a game they should simultaneously raise their hand and exhibit

either one finger or two fingers. If both of them show one finger or if both show two fingers, A

should pay Rs. 10 to B. On the other hand, if one player shows one finger and the other player

mn2m1m

n22221

n11211

a....................aa

...

...

...

a....................aa

a....................aa


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shows two fingers, B should pay Rs. 5 to A.

Here, the pay-off matrix of A is

Player B

B1 (one finger) B2 (twofingers)

Player A

A1 (one finger)

A2 (two fingers)

-10 5

5 -10

Here, suppose player A decides to show one finger )A( 1 , his strategy is pure strategy. On the

other hand, suppose A decides to play 1A with probability 0.5 and 2A with probability 0.5, his

strategy is mixed strategy. (This means, if he is to play repeatedly, sometimes he should play 1A

and at other times he should play2

A . He should mix1

A and2

A randomly almost equal

number of times.)


Write one line answer

1. State any one characteristics of a competitive game.

2. When do we call a game as Zero sum game.

3. What is a rectangular game?

4. What is pure strategy?

12.4 Maximin Minimax Principle

(of solving a two-person zero-sum game)

Suppose player A and player B are to play a game without knowing what the

other player would do. However, player A would like to maximize his profit and player B would like

to minimize his loss. And thus, each player would expect his opponent to be calculative.

Suppose playerA plays 1A . Then, his gain would be n11211 a,...a,a according

as Bs choice isn21

B,...B,B . Let }a,...a,a{minn112111

=a . Then,1

a is the minimum gain of A

when he plays 1A . (Here, 1a is the minimum pay-off in the first row.) Similarly, if A plays 2A , his

minimum gain is 2a which is the least pay-off in the second row. Thus proceeding, we find that

corresponding to As play m21 A,......A,A , the minimum gains are the row

minimums m21 ,.., aaa . Suppose A chooses that course of which ia is maximum. This maximum


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of the row minimum in the pay-off matrix is called maximin. The maximin is

= )a(

j

min

i

maxija

Similarly, when B plays, he would minimize his maximum loss. The maximum loss to B when jB

is )a( ijmax

ij=b . This is the maximum pay-off in the

thj column. The minimum of the column

maximums in the pay-off matrix is called minimax. The minimax is

= )a(i

max

j

minijb

For some games, the maximin and the minimax are equal. That is,

)say(v==ba . Such games are said to have saddle point.

On the other hand, if ba< , the game does not have saddle point.

(Note thata cannot be greater thanb ).

12.4.1. Saddle Point

In a two-person zero-sum game, if the maximin and the minimax are equal, we say that the game

has saddle point.

Saddle point is the position where the maximin (maximum of the row

minimums) and minimax (minimum of the column maximums) coincide.

If the maxmin occurs in theth

r row and if the minimax occurs in theth

s column, the position (r, s)

is the saddle point. Here, rsav= is the common value of the maximin and the minimax. It is called

the value of the game.

The value of a game is the expected gain of player A when both the players adopt optimal

strategy.

Note 1: If a game has saddle point, and if (r, s) is the saddle point, suggested solution to boththe players is pure strategy. For player A, the suggested solution is rA . For player B, the

suggested solution is sB .

Note 2: If a game does not have saddle point, the suggested solution is mixed strategy.

Note 3: A game is said to be fair if its value is zero.


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12.4.2. Solution to a Game with Saddle Point

Consider a two-person zero-sum game with players A and B. Let m21 A,...A,A be the courses of

action for playerA. Let n21 B,.....B,B be the courses of action for player B.

The saddle point of the game is found as follows.

1. The minimum pay-off in each row of the pay-off matrix is circled (marked with )

2. The maximum pay-off in each column is boxed (marked with )

3. In the above process, if any pay-off is circled as well as boxed, that pay-off is the value of the

game. The corresponding position is the saddle point.

Let (r, s) be the saddle point. Then, the suggested pure strategy for player A is rA . The

suggested pure strategy for playerB is sB . The value of the game is rsa .

Note: However, in the above procedure, if none of the pay-off is circled as well as boxed, the

game does not have saddle point. And so, the suggested solution for the players is mixed

strategy.

Example 2:

Verify whether the 2-finger morra game explained earlier has saddle point. If so, write down the

solution for the game.

Solution:

The pay-off matrix of player A is

Player B

B1 (one finger) B2 (two fingers)

A1 (one finger)

Player A

A2 (two fingers)

1. The minimum pay-off in each row is circled.

2. The maximum pay-off in each column is boxed.

3. Since none of the pay-off is circled as well as boxed, the game does not have saddle point.

And so, the solution for the game is mixed strategy for both the players.

-10

-10

5

5


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Example 3:

Two persons A and B, without showing each other, place a coin each on the

table. If the coins happen to be of the same denomination, player A will take both of them. If they

happen to be of different denominations, player B will take both of them.Suppose player A has afew one-rupee coins and two-rupee coins. And suppose player B has a few one-rupee, two-

rupee and five-rupee coins.

i) Write down the pay-off matrix of A. Does the game have saddle point? If so, write down the

solution.

ii) What happens to the game if both the players play only with one-rupee and two-rupee

coins?

Solution:

i) The pay-off matrix of A is -

Player B

B1 (one Rs.) B2 (two Rs) B3 (five Rs.)

A1 (one rupee)

Player A

A2 (two rupees)

1. The minimum pay-off in each row is circled. (Here, the minimums repeat.)2. The maximum pay-off in each column is boxed.

3. The pay-off -1 is circled as well as boxed. Therefore, the game has a saddle point. It is

the position (1, 3).

The solution to the game is

a) Strategy for A is1A .

b) Strategy for B is3

B .

c) Value of the game is v = -1 rupees.

ii) If both the players play only with one-rupee and two-rupee coins, the pay-off matrix of A is

-22

1

-2

-1 -1


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Player B

B1 B2

A1

Player A A2

Here, the game does not have saddle point. Therefore, the suggested solution is mixed strategy

for the players.

Example 4:

A labour union of a firm is negotiating a new 5-year settlement regarding payments with the

management. The options the union has are 1A : Aggressive bargaining, 2A : Bargaining with

reasoning and 3A : Conciliatory approach. The likely mode of response from the management

are 1B : Aggressive bargaining, 2B : Bargaining with reasoning, 3B : Legalistic approach and

4B : Conciliatory approach. The gains to the union in each case are as follows.

Union Management

B1 B2 B3 B4

A1 20 15 12 35

A2 25 14 8 10

A3 -5 4 11 0

What strategy would you suggest for the two sides? What is the value of the

game?

- 2

-1

2

1


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Solution:

Union Management

B1 B2 B3 B4

A1 20 15 12 35

A2 25 14 8 10

A3 -5 4 11 0



3. The value 12 is circled as well as boxed. And so, the game has a saddle point. It is the

position (1, 3).

Therefore, the solution to the game is

a) Strategy for the union is 1A : Aggressive bargaining.

b) Strategy for the management is 3B : Legalistic approach.

c) Value of the game is v = 12.

Example 5:

Solve the game

B1 B2 B3

A1 6 12 7

A2 7 9 8

Is the game fair?

Solution:

B1 B2 B3

A1 6 12 7

A2 7 9 8


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-7



3. The value 7 is circled as well as boxed. And so, the game has a saddle point. It is the position

(2, 1).


a) Strategy for A is2A .

b) Strategy for B is1B .


d) The game is not fair because v is not equal to zero.

Example 6:

A two-person zero-sum game, has the following pay-off matrix. Solve the game.

--

-

-

47

20

03

18

Is the game fair?

Solution: B1 B2

A1 8 -1

A2 3 0

A3 0 -2

A4 -4


2. The maximum pay-off in each column is boxed.3. The value 0 is circled as well as boxed. And so, the game has a saddle point. It is the position

(2, 2).


a) Strategy for the player is2A (second course of action).


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b) Strategy for the opponent is2B (second course of action).


d) Since the value of the game is 0, the game is fair.


Are the following statements True or False

1. Saddle point occurs at row minimum and column maximum.

2. If the value of the game is zero then it is called 0 sum game.

3. The pay of matrix represents the gain for top player.

12.5 Dominance

In a rectangular game, suppose in the pay-off matrix of player A, each pay-off in one specific row

)rowr(th

exceeds the corresponding pay-off in another specific row )rows(th

. This means,

whatever be the course of action adopted by player B, for A, the course of action rA yields greater

gains than the course of action sA . Therefore, rA is a better strategy than sA irrespective of the

strategy of B. And so, we say that rA dominates sA .

On the other hand, suppose each pay-off in a specific column )columnp(th

is less than the

corresponding pay-off in another specific column )columnq(th

. This means, for player B,

strategy pB has lesser loss than strategy qB irrespective of strategy of A. And so, we say that

pB dominates qB . Thus,

a) In the pay-off matrix, if each pay-off in the rowrth

is greater than (or equal to ) the

corresponding pay-off in the rowsth

, rA dominates sA .

b) In the pay-off matrix, if each pay-off in the columnpth

is less than

(or equal to) the corresponding pay-off in the columnqth

, pB dominates qB .

Sometimes, a convex combination of two or more courses of action may dominate another

course of action.

Whenever a course of action (say sA or qB ) is dominated by others, that course of action

( sA or qB ) can be deleted from the pay-off matrix. Such deletion will not affect the choice of

the solution.


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Such deletion of courses of action reduces the order of the pay-off matrix.

Successive reduction of the order using dominance property helps us in solving games.

12.5.1 Solving Games Using Dominance(Solving two-person zero-sum game with saddle point)

Consider a two-person zero-sum game with players A and B. Let m21 A,......A,A be the courses

of action for player A. Let n,2,1 B...BB be the courses of action for player B.

Suppose the game has saddle point. Then, using dominance property, it is possible to

successively delete the courses of action of A as well as B such that ultimately the pair

comprising the saddle point alone remains. The procedure in this regard is as follows.

a) In the pay-off matrix, if each pay-off in therowr

this greater than (or equal to ) the

corresponding pay-off in the rowsth

, rA dominates sA . And so, sA is deleted.

b) In the pay-off matrix, if each pay-off in the columnpth

is less than (or equal to) the

corresponding pay-off in the columnqth

, pB dominates qB . And so, qB is deleted.

c) The above steps are repeated in succession until the saddle point is reached. And

hence, the solutions is written down.

Note: Sometimes, a convex combination of two or more courses of action may dominate another

course of action.

Example 8:

Solve the following game using dominance property.

B1 B2 B3 B4

A1 20 15 12 35

A2 25 14 8 10

A3 -5 4 11 0

Solution:

In the pay-off matrix, each pay-off in the first row exceeds the corresponding pay-off in the third

row. Therefore, 1A dominates 3A . And so, 3A is deleted. The reduced matrix is


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B1 B2 B3 B4

A1 20 15 12 35

A2 25 14 8 10

Here, each pay-off in the third column is less than corresponding pay-off in the first column.

Therefore, 3B dominates 1B . Similarly, 3B dominates 2B . Also, 3B dominates 4B . Thus, the

matrix reduces to

B3

A1 12

A2 8

Here, since 12>8,1A dominates 2A . And so, finally the matrix reduces to

B3

A1 ( 12 )

Thus, (1, 3) is the saddle point. And so, the solution to the game is

a) Strategy forA is 1A .

b) Strategy forB is 3B .

c) Value of the game is v = 12

Example 9:

Solve the following zero-sum game and find its value.

Company Y

P Q R S

A 6 2 4 1

B 6 1 12 3

Company X C 3 2 2 6

D 2 3 7 7

Solution:

In the pay-off matrix, each pay-off in the second column is less than (or equal to) the

corresponding pay-off in the third column. And so, the course of action Q dominates R.


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Similarly, Q dominates S.

After deleting R and S, the reduced matrix is

P Q

A 6 2B 6 1

C 3 2

D 2 3

Here, pay-off in the second row is greater than (or equal to) the corresponding pay-offs in the first,

third as well as fourth rows. Therefore, B dominates A, C and D.

After deleting A, C and D, the reduced matrix is

P Q

B [ 6 1]

Here, 1


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Solution:

The pay-off matrix of B is

Player A

A1 A2

B1 -4 1

Player B

B2 -7 -3

B3 0 -6


Fill in the blanks

a. The row whose elements are less than the corresponding elements of another row is

__________.

b. If the average of any 2 columns is less than or equal to the corresponding elements of

another column is _________.

12.6. Summary

In this unit of game theory we studied the concept of competitive situations where the

characteristics of competitive game and its strategy we considered. The maximum minimum

principle is discussed briefly saddle point and Dominance is explained with clear cut examples.

Terminal Questions

1.Solve the following rectangular game.

---

--

--

23450

24031

22123

85022


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2. In a rectangular game, pay-off matrix of player A is

Player B

B1 B2

A1 5 7Player A A2 4 0

i) Solve the game.

ii) Write down the pay-off matrix of B and then, solve the game.

3. Briefly describe characteristics of competitive game.

4. Explain MAXIMIN-MINIMAX principle

Answers To Self Assessment Questions


1. Two, more, with conflicting, interest

2. Finite


1. Players do not communicate to each other

2. When the value of the game is 0.

3. Two person zero sum game.

4. When a player always play only one strategy irrespective of Opponents move


1. True 2. True c) False


1. Deleted 2. Deleted

Answer for Terminal Questions

1. Value of the game is 1. As Strategy (0, 1, 0, 0) Bs Strategy (0, 0, 1, 0, 0)

2. Value is 5. As Strategy (1, 0) Bs Strategy (1, 0)

3. Refer Section 12.34. Refer Section 12.4

.

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12 Game Theory