Mst541 Unit i

Theory of Games

Unit-I

Prof. Narinder Verma

School of Business Management prof Narinder Verma2

Strategic Behavior- Uncertain

A lot of what we do, involves optimizing against various alternatives:

What should I do after 10+2?

Should I do MBA after graduation?

What should be my major specialization?

What kind of job be my dream job?

Should I join family business?


Strategic Behavior- Uncertain

Or at times against nature in general:

Should I take an umbrella today?

What crops should I plant this season?

How do we treat this disease or injury?

How do I fix my out of order car?

We sometimes imagine it as a game against opponents


Prisoners DilemmaTwo suspects arrested for a crime were put into

prison separately.

prisoners decide whether to confess or not.

If both confess, both sentenced to 5 years of jail

If neither confesses, both will be sentenced to 1 year of jail

If one confesses and the other does not, then the confessor goes free (no jail) and the non-

confessor sentenced to 9 years of jail

What should each prisoner do?


Battle of Seaes

A couple deciding how to spend the evening

Wife would like to go for a movie

Husband would like to go for a cricket match

Both however want to spend the time together

Scope for strategic interaction


Understanding the Game

Game refers to a situation of conflict

and competition in which two or more

competitors are involved in the

decision-making process in anticipation

of certain outcomes over a period of

time.

1. Competitors are called players

2. Action is called Strategy

3. Anticipated outcome is the payoff


Understanding the Game

1. pricing of products is affected by

the price of the competitor.

2. Success of TV programme

depends upon the presence of

competing TV programmes in the

same time slot.

A basic feature here is that the final

outcome depends primarily upon

the combination of strategies


Game theory is a series of mathematical

models that deal with interactive

decision-making situations under the

conditions of conflict and competition.

The study of oligopolies

The study of cartels; e.g. OPEC

The study of military strategies.

Study of effect of promotional campaigns

Defining Game Theory


Game Theory

Game theoretic notions go back thousands of years

Talmud and Sun Tzu's writings.

Modern theory credited to John von Neumann and Oskar Morgenstern 1944.

Theory of Games and Economic Behavior.

John Nash (A Beautiful Mind fame) generalized these results and provided the basis of the modern field.


Classification of Games

2-person Game:

A game that involves exactly two players

n-person Game

A game that involves exactly n players

Zero-sum Game:

When the sum of gains to one player is exactly equal to the losses to other player, so that the sum of the gains and the losses is equal to zero(0).

Non-zero sum Game:

A game whose sum is not equal to zero


Classification of Games

Simultaneous Move Game:

Each player has to take action simultaneously like Racing, Stone-paper-Scissors, Chidiya Ud etc

Sequential Move Game

One player moves

Second player observes and then moves

Like Chess, Tic-Tac-Toe (naughts and crosses), Discuss Throw, promotional campaigns by competitors


Elements of a Game

A game consists of:

A set of players

A set of strategies for each player

The payoffs to each player for every possible strategy

Strategy is a course of action that a

player adopts for every payoff


Strategies of Games

Optimal Strategy:

The particular strategy that optimizes a players gains or losses, without knowing the competitors courses of action

Pure Strategy:

A particular strategy that a player chooses

to play again and again regardless of other

players strategies

It is a deterministic situation


Strategies of GamesMixed Strategy:

A set of strategies that a player chooses on a particular move of the game with some fixed probabilities

It is a probabilistic situation

Value of the Game:

It is the expected gain or loss in a game

when a game is played a large number of

times

It is represented by V


payoff matrix to player A

a11 a12 a1n

a21 a22 ........ a2n

.

. aij

am1 am2 . amn

A1

A2

.

.

Am

Two-person Zero-sum Game

B1 B2 Bn

Strategies of player BS

tra

teg

ies

of

pla

yer

A


A1,A2,..,Am are the strategies of player A

B1,B2,...,Bn are the strategies of player B

aij is the payoff to player A (by B) when the player A plays strategy Aiand B plays Bj (aij is ve means B got |aij| from A)



Two boutiques A and B in Solan compete

such that each gets 50% of clientele. As gain would be Bs loss. B moots an idea to gain market share by:

1. Discount coupons - B1

2. Decreasing price - B2

3. TV Advertisement - B3

4. 10% Cash Back - B4



A has an idea that B might do this. So A

also moots an idea to gain market share

by:

1. Discount coupons - A1

2. TV Advertisement - A2

3. 10% Cash Back - A3

The payoff matrix in Rupees for boutique

A is given as:



8 -6 2 1

4 9 4 5

7 -5 3 -7

B1 B2 B3 B4

A1

A2

A3

Strategies of player B

Str

ate

gie

s o

f

pla

yer

A

1. Find the optimal strategies for both A and B.

2. Find the value of the game.


Maximin and Minimax principle

Maximin principle:

For each row, find the minimum, then

The maximum out of these minimums is the Maximin value for A

Minimax principle:

For each column, find the maximum, then

The minimum out of these maximums is the Minimax value for B


Maximin and Minimax principle

Saddle point:

When Maximin value is equal to the Minimax value then the game is said to have an equilibrium point.

Equilibrium point is called the Saddle point.

The corresponding strategies are called optimal strategies. These are the pure ones.

At saddle point,

Maximin = Minimax = Value of Game (V)


Maximin and Minimax principleA game may have more than one saddle point.

A game may not have a saddle point at all.

In general,

Maximin Value V Minimax Value

A game is said to be a fair game if the

Maximin = Minimax = 0 (zero)

A game is said to be strictly determinable if

the Maximin = Minimax = Value of the game (V)


8 -6 2 1

4 9 4 5

7 -5 3 -7

B1 B2 B3 B4

A1

A2

A3

MaxCol 8 9 4 5

Row min

-6

4

-7

minimax

maximin

Illustration Continues


Solution is based on the principle of securing

the best of the worst for each player. If the

player A plays strategy A1, then whatever

strategy B plays, A will get at least -6 (loses

at most Rs. 6).

Thus to maximize its minimum returns, A

should play strategy A2.

If A plays strategy A2, then whatever B

plays, will get at least 4. and if A plays

strategy A3, then he will get at least -7(loses

at most Rs. 7) whatever B plays.


Now if B plays strategy B1, then whatever

A plays, he will lose a maximum of 8.

Similarly for strategies B2,B3,B4. (These

are the maximum of the respective

columns).

Maximin = Minimax = 4 = Saddle point

Thus here, 4 is value of the game and

appropriate strategies are A2, B3

Thus to minimize this maximum

loss, B should play strategy B3.


Two boutiques A and B in Solan compete

such that each gets 50% of clientele. As gain would be Bs loss. B moots an idea to gain market share by:

1. Discount coupons - B1 2. Decreasing price - B23. TV Advertisement - B34. 10% Cash Back - B45. Newspaper Inserts - B5

Illustration 2


A has an idea that B might do this. A also

moots an idea to gain market share by:

1. Discount coupons - A12. Decrease price - A23. TV Advertisement - A34. 10% Cash Back - A4

The payoff matrix in Rupees for boutique

A is given as:

Illustration 2


3 -1 4 6 7

-1 8 2 4 12

16 8 6 14 12

1 11 -4 2 1

B1 B2 B3 B4

A1

A2

A3

A4


Str

ate

gie

s o

f

pla

yer

A

1. Find the optimal strategies for both A and B.

2. Find the value of the game.

B5


maxCol 16 11 6 14 12

Row min

-1

-1

6

-4

minimax

maximin

Illustration 2

3 -1 4 6 7

-1 8 2 4 12

16 8 6 14 12

1 11 -4 2 1

B1 B2 B3 B4 B5

A1

A2

A3

A4


Illustration 3

The following game gives As payoff. Determine p, q that will make the entry a22 a saddle point.

1 q 6

p 5 10

6 2 3

A1

A2

A3

B1 B2 B3

Col max 6 5 10

Row min

1

5

2


Since a22 must be a saddle point,

There for p has to be at least as large

as 5, so

And also q has to be at least as small

as 5, so

5p

5q

Illustration 3

Find row minimums and column

maximums with considering the

values of unknowns


Specify the range for the value of the

game in the following case assuming

that the payoff is for player A.

3 6 1

5 2 3

4 2 -5

A1

A2

A3

B1 B2 B3

Col max 5 6 3

Row min

1

2

-5

Illustration 4


maximin minimax (2 3)

Hence the game has no saddle point.

When there is no saddle point, then

Maximin Value V Minimax Value i.e., 2 V 3.

Thus the value of the game lies

between 2 and 3.

Illustration 4


Principles of DominanceThese are used to reduce the size of the payoff matrix by deleting certain inferior rows and or columns

Dominance rules are especially used for the evaluation of the two-person zero-sum games without a saddle point

1. For player B who is assumed to be a loser, if each element in a column Cr is greater than or epual to the corresponding element in another column Cs then the column Cr is said to be dominated by Cs (or inferior to) and there column Cr can be deleted


Principles of Dominance

2. For player A who is assumed to be a gainer, if each element in a row Rr is less than or equal to the corresponding element in another row Rs then the row Rr is said to be dominated by Rs (or inferior to) and there row Rr can be deleted

3. A strategy k can also be dominated if it is inferior to an average of two or more other pure strategies

Dominance rules are framed assuming that the payoff matrix is a profit matrix for player A


8 6 2 8

8 9 4 5

7 5 3 5

B1 B2 B3 B4

A1

A2

A3

Strategy A3 is dominated by the

strategy A2 and so can be eliminated.

Eliminating the strategy A3 , we get:

Principles of Dominance:Ex1


8 6 2 8

8 9 4 5

B1 B2 B3 B4A1

A2

Eliminating the strategies B1 , B2, and

B4 we get the reduced payoff matrix:

For player B, strategies B1, B2, and B4 are

dominated by the strategy B3.



2

4

B3

A1

A2

Now , for player A, strategy A1 is dominated

by the strategy A2

Eliminating the strategy A1



4

B3

A2

We thus see that A should always play A2 and B always B3 and the value of the game

is 4 as before.



-5 10 20 8

5 -10 -10 6

5 -20 -20 7

B1 B2 B3 B4

A1

A2

A3

Strategy B3 is dominated by the

Strategies B1 and B2 and so can be

eliminated

B4 is inferior to B1 so is deleted., we get:



-5 10

5 -10

5 -20

B1 B2A1

A2

Eliminating the strategy A3 we get the

reduced payoff matrix:

For player A, strategy A3 is dominated by

the strategy A2.


A3


B1 B2A1

A2

The game has no saddle point as

was the case originally.


-5 10

5 -10


Used for (2 a n) or (m a 2) games, i.e.,

for two-person zero-sum games where

at least one player has only 2 strategies

It is assumed that the player with two

strategies, chooses a mixture of both

the strategies with some fixed but

unknown probabilities, to be calculated

Mixed Strategies:

Graphical Method


a11 a12 . a1n

a21 a22 . a2n

B1 B2 . Bn

A1

A2

player A selects two strategies A1 and A2

with probabilities p1 and p2 respectively

Graphical Method: 2 x n Game

Probability

p1

p2

player Bplayer A

Probability p1 p2 .. pn


Such that 1,2 0 and 1 + 2 = 1


Expected Payoff for Player A

Bs pure Strategies As Expected Payoff

B1 + B2 + B3 + . .

. .

. .

Bn +


Now plot probabilities on a- axis and

Expected payoff on Y- axis

Choose the lower envelope if initial

payoff was for A, and

player A has two strategies (as here)



Graphical: Illustration 1

2 4 3 8 4

5 6 3 7 8

10 5 9 8 7

4 2 8 4 3

B1 B2 B3 B4 B5

A1

A2

A3

A4

A1 and A4 are dominated by A3



5 6 3 7 8

10 5 9 8 7

B1 B2 B3 B4 B5

A2

A3

B1 is dominated by B3,so B1 gets deleted



6 3 7 8

5 9 8 7

B2 B3 B4 B5

A2

A3

Probability

p1

p2

Assume that player A selects Strategies

A2 and A3 with probabilities p1 and p2respectively


Such that 1,2 0 and 1 + 2 = 1

Graphical : Illustration 1



B2 + B3 + B4 + B5 +


M

As Expected payoff, A1

probability

B2

9

8

7

6

5

4

3

2

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1(0, 0)


9

8

7

6

5

4

3

2

1

p1=1, p2=0 p1=0, p2=1

(0, 0)

As Exapected payoff, A2B3B4

B5

L

N


Out of three points L, M and N; M

represents Maximin, so M is the

optimal mixed strategy

Solve payoff equations for B2 and B3

+ = + , and

+ =



+ ( ) = + ( )

= + = +

= + = +

= =

There for =

and =

and =



Illustration 1: Calculation for B

6 3

5 9

B2 B3

A2

A3

probability q2 q3


Such that 2, 3 0 and 2 + 3 = 1


Expected Payoff for Player B

As pure Strategies Bs Expected Payoff

A2 + A3 +


+ ( ) = + ( )

= + = +

= + = +

= =

There for =

and =

and =



Simplex Method

a11 a12 a1n

a21 a22 ........ a2n

.

. aij

am1 am2 . amn

A1

A2

.

.

Am

B1 B2 Bn


Str

ate

gie

s o

f p

layer

AProbability

1

2

Probability q1 q2 qn


= =

= , , , . . ,

Simplex Method

,

:


Simplex Method

a11 p1+ a21 p2+ ... + am1 pm V

a12 p1 + a22 p2 + ... + am2 pm V

:

a1n p1 + a2n p2 + ... + amn pm V

p1 + p2 + ... + pm = 1

pi 0 (Non-negativity constraints) = 1,2, 3, . . ,

, ,


Simplex MethodDivide all constraints by V

a111

+ a21

2

+ ... + am1

1

a121

+ a22

2

+ ... + am2

1

:

a1n1

+ a2n

2

+ ... + amn

1

1

+ 2

+ ... +

= 1

0 = 1,2, 3, . . ,


Simplex Method

Let

= so that

a11 1+ a21 2+ ... + am1 1

a12 1+ a22 2+ ... + am2 1

:

a1n 1+ a2n 2+ ... + amn 1

1+2+ ... += 1

0 = 1,2, 3, . . ,


Simplex Method

=

, there for,

=

= 1+2+ ... + ,

a11 1+ a21 2+ ... + am1 1

a12 1+ a22 2+ ... + am2 1

:

a1n 1+ a2n 2+ ... + amn 1

0 = 1,2, 3, . . ,


Simplex Method

a11 a12 a1n

a21 a22 ........ a2n

.

. aij

am1 am2 . amn

A1

A2

.

.

Am

B1 B2 Bn


Str

ate

gie

s o

f p

layer

AProbability

1

2

Probability q1 q2 qn


= =

= , , , . . ,

Simplex Method

,

:


Simplex Method

a11 q1+ a12 q2+ ... + a1n qn V

a21 q1 + a22 q2 + ... + a2n qn V

:

am1 q1 + am1 q2 + ... + amn qn V

q1 + q2 + ... + qn = 1

qj 0 (Non-negativity constraints) = 1,2, 3, . . ,

, ,



a111

+ a12

2

+ ... + a1n

1

a211

+ a22

2

+ ... + a2n

1

:

am11

+ am2

2

+ ... + amn

1

1

+ 2

+ ... +

= 1

0 = 1,2, 3, . . ,


Simplex Method

Let

= so that

a11 1+ a12 2+ ... + a1n 1

a21 1+ a22 2+ ... + a1n 1

:

am1 1+ am2 2+ ... + amn 1

1+2+ ... += 1

0 = 1,2, 3, . . ,


Simplex Method

=

, there for,

=

= 1+2+ ... + ,

a11 1+ a12 2+ ... + a1n 1

a21 1+ a22 2+ ... + a2n 1

:

am1 1+ am2 2+ ... + amn 1

0 = 1,2, 3, . . ,


Solve for ys as there are only slack variables

X values will be calculated from dual

values of slack in the primal for y

For x values, take the absolute value of

(cj-zj), i.e., | cj -zj| against slack variables

For x1 take | cj -zj| against s1, for x2 take

| cj -zj| against s2 and like wise

Simplex Method


Note: As

= 0 and

= 0

For this, value must be non-negative, i. e., 0 ,there for all aij 0

For this, add {| largest negative aij | +1}to all aij . If value of the game now is .

Original value of the game ( ) now is

= { | | +1}

Simplex Method


1 -1 3

3 5 -3

6 2 -2

B1 B2 B3

A1

A2

A3

MaxCol 6 5 3 5

Row min

-1

-3

-2

minimax

maximin

Simplex Method: Illustration


Clearly there is no saddle point

Value of the game follows

Here V can be negative, so add

{| largest negative aij | +1} i.e., {| -3| +1}= 4 to all aij



Modified Payoff Matrix

Probability 1 2 3

5 4 7

7 9 1

10 6 2

B1 B2 B3

A1

A2

A3

Probability

1

2

3



F ; ,, and + + =

Simplex Method : Illustration



B1 + + B2 + + B3 + +


Simplex Method

5 p1+7 p2+ 10 p3 V

3 p1+9 p2+ 6 p3 V

7 p1 + p2 + 2 p3 V

p1 + p2 + p3 = 1

p1, p2 , p3 0

, ,



5

+ 7

+ 10 1

3

+ 9

+ 6 1

7

+

+ 2 1

+

+

=

,

, 0


Simplex Method

Let

= ,

= and

= so that

5 + 7 + 10 1

3 + 9 + 6 1

7 + + 2 1

++ =

,, 0


Simplex Method

=

, there for,

=

= ++ ,

5 + 7 + 10 1

3 + 9 + 6 1

7 + + 2 1

,, 0


F ; ,, and + + =

Simplex Method : Illustration

Expected Payoff for Player B

As pure Strategies Bs Expected Payoff

A1 + + A2 + + A3 + +


Simplex Method

5 q1 + 3 q2 + 7 q3 V

7 q1 + 9 p2 + q3 V

10 q1 + 6 q2 + 2 q3 V

q1 + q2 + q3 = 1

q1, q2 , q3 0

, ,



5

+ 3

+ 7 1

7

+ 9

+ 1

10

+ 6

+ 2 1

+

+

=

,

, 0


Simplex Method

Let

= ,

= and

= so that

5 + 3 + 7 1

7 + 9 + 1

1 + 6 + 2 1

++ =

,, 0


Simplex Method

=

, there for,

=

= ++ ,

5 + 3 + 7 1

7 + 9 + 1

1 + 6 + 2 1

,, 0



The Standard form of LPP with constraints is:

= 1+ 2 + + 01 + 02 + 03

Subject to

5 1+ 3 2+ 7 3+ 1 1+ 0 2+ 0 3 = 1

7 1+ 9 2+ 3+ 0 1+ 1 2+ 0 3 = 1

10 1+ 6 2+ 2 3+ 0 1+ 0 2+ 1 3 = 1

1 0, 2 0, 3 0, 1 0, 2 0, 3 0


Final Tableau with optimal solution:

Cj 1 1 1 0 0 0

CB

Basic

Variable

(B)

Basic

Soln(YB)y1 y2 y3 s1 s2 s3

1 y3 1/10 2/5 0 1 3/20 -1/10 0

1 y2 1/10 11/15 1 0 -1/60 7/60 0

0 s3 1/5 24/5 0 0 -1/5 -3/5 1

17/15 0 0 2/15 1/15 0

-2/15 0 0 -2/15 -1/15 0

Zj

(Net Evaluation)Cj - Zj


Zq= 1/5


Optimal strategy for B:

= , =

=

=

There for =

=

= there for = = =

= there for = =

=

= there for = =

=



As =

=

Original value of the game ( ) now is

= { | | +1}

= { } =



Optimal strategy for A:

=

, =

=

= there for = =

=

= there for = =

=

= there for = = =

= (

,

, ), = ,

,

, = 1



End of Unit - I

Mst541 Unit i

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