7. game theory

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7

Games Theory


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Game Theory

An activity between two or more persons involving activitiesby each person according to a set of rules , at the end of

each person receives some benefit or satisfaction or suffers

loss (negative benefits)

The set of rules defines the Game

Going through the set of rules once by the participantsdefines a Play.

Application:

Labour unions striking against company management

Players in chess game Army generals engaged in fighting the enemy

Firm striving for larger share of market in duopolistic market

condition. etc


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Game Theory

The Mathematical analysis of competitive problems is

fundamentaly based upon the minimax (maximin) criteria of

J. Von Neumann (called father of game theory)

This criteria implies the assumption of rationality from whichit is argued that each player will act so as to maximize its

minimum gain or minimize its maximum loss.


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Characteristics

Chance of strategy- if in a game, activities are determined byskills, it is game of strategy, if by chance, it is called as

Game of Chance

Number of persons- a game is called n person game , if the

number of player is n

Number of activities- finite or infinite Number of alternatives(Choices) available to each person-

finite or infinite.

Information to the players about the past activities of other

players- completely available, partly available or not

available at all Payoff : quantitative measure of satisfaction a person gets at

the end of play.


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Game Theory- Basic Definitions

Competitive Game- game with a competitive situation;

Properties:

there are finite number(n) of players such than n> 2 . In case

n=2 it is called 2 person game and n>2 it is called n person

game. Each player has a finite number of possible activities.

A play is said to occur when each player chooses one of his

activities. Choices are made simultaneously.

Every combination of activities determines an outcome.


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Two-Person Non- zero sum game

Zero sum game ;if the player makes payment to each other ,

i.e. the loss of one is gain of others and nothing comes from

outside , the competition is zero sum.

A game which is not zero sum is called Non zero-sum game. Most of the competitive games are non zero sum games.

Eg. Poker ( a certain part of the pot is removed from the

house before the final payoff)


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Two-Person zero-sum (Rectangular) game

A game with only two players (Say, A and B).

If the losses of one player are equivalent to the gains of the

other, so that the sum of their net gains is zero.

Usually represented by a pay-off matrix in rectangular form.




Strategy- A rule of decision making of a player in advance of all the

plays by which it decides the activities to be adopted.

Pure Strategy: if a player knows exactly what the other

player is going to do, deterministic situation is obtained andobjective function is to maximize the gain. Pure strategy is a

decision rule always to elect a particular course of action.

Mixed strategy: a player is guessing as to which activity is to

be selected by the other on any particular occasion, aprobabilistic situation is obtained and objective function is to

maximize the gains. A mixed strategy a selection among

pure strategies with fixed probabilities.




Pay-off Matrix-

Row designations for each matrix are activities available to

player A

Column designations for each matrix are activities available

to player B Cell entry Vijis the payment to player Aiand Bj

With a zero sum two person game the cell entry in the player

Bs pay-off matrix will be negative of the corresponding cell

entry Vijin the player As pay off matrix so that the sum of

pay-off matrixes for player A and B is ultimately Zero.



Minimax Criterion and Optimalstrategy

The minimax criterion of optimality states that if a player liststhe worst possible outcomes of all his potential strategies , it

will choose that strategy to be most suitable for itself which

corresponds to the best of these worst outcomes. Such a

strategy is called an Optimal strategy.

Example: (Two Person Zero-sum Game) with respect to

player A

B

A -3 -2 6

2 0 2

5 -2 -4



Procedure To Determine SaddlePoint

Step 1: Select the minimum element in each row and

enclose it in a square () Step 2: Select the maximum element in each column and

enclose it in a circle () Step 3: Find out the element which is enclosed by the square

as well as the circle. Such element is the value of the game

and that position is called as the saddle point.



Minimax (Maximin)Criterion andOptimal strategy

Example: (Two Person Zero-sum Game)

BA -3 -2 6

2 0 2

5 -2 -4

Saddle point

Row Minimum

-3

Column maximum 5 0 6

-4

0

Minimax value

MaximinValue



Saddle Point, Optimal Strategies,and Value of Game

A saddle point of a Payoff matrix is the position of such an

element in the payoff matrix which is minimum in its rowand maximum in column.

Optimal Strategies- If the payoff matrix has a saddle pointthen the players (A and B) are said to have rth and sthoptimal strategies respectively.

Value of Game- the payoff at the saddle point is calledValue of Game and its obviously equal to maximin andminimax of the game.

The game is said to be Fair game if maximin= minimax = 0 The game is strictly determined if maximin= minimax = v


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Games with Saddle Point

To obtain a solution of a rectangular game, it is feasibleto find out;

The best strategy for player A The best strategy for player B The value of Game

The value of Game to the player A is the element at thesaddle point, and the value to the player B will be itsnegative.


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Games without Saddle Point

The game has no optimal strategies , The games without saddle point are not strictly

determined i.e. maximin minimax v The players use mixed strategy. Mixed strategy represents a combination of two or more

strategies that are selected one at a time, according topre-determined probabilities.

In employing a mixed strategy, a player decide to mixhis choices among several alternatives in a certainratio.


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A Game with No Saddle Point

AsStrategy

Bs Strategy Row Minimab1 b2 b3

a1 8 5 14 5

a2 22 -6 8 -6

a3 7 9 12 7

Column Maxima 22 9 14

Maximin Strategy = a3, Payoff = 7 Minimax Strategy = b2, Payoff = 9 There is no Saddle Point

Minimax Maximin


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Games with no Saddle Point If it is a 2 2 game

If A plays a1 with probabilityxand a2 with probability 1-x; and B plays b1with probability yand b2 with probability 1-y, then

a22a21x= --------------------------------

(a11 + a22) (a21 + a12)

a22a12y = -------------------------------- and

(a11 + a22) (a21 + a12)

(a11 a22) (a21 a12)v= ----------------------------------

(a

11 +a

22) (a

21 +a

12)

Player B

b1 b2

P

layerA

a1 a11 a12

a21 a22a2


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Arithmetic method (Oddments) 2 2 Game

Find the difference of two numbers in column I and put itunder the column II , neglecting the negative sign if occurs.

Find the difference of two numbers in column II and put itunder the column I , neglecting the negative sign if occurs.

Repeat the above two steps for the two rows also.

The values thus obtained are called oddments. These are thefrequencies with which the players must use their courses ofaction in their optimum strategies.


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Arithmetic method (Oddments) 2 2 Game

8 -3

-3 1

H T

H

T

4

11

4 11

4/(11+4 )= 4/15

11/(11+4 )= 11/15

4/15 11/15

Taking Bs oddments:Playing H , v= Rs. (4x8 + 11x-3)/(11+4) = -1/5Playing T , v= Rs. (4x-3 + 11x1)/(11+4) = -1/5Taking As oddments:

Playing H , v= Rs. (4x8 + 11x-3)/(4+11) = -1/5Playing T , v= Rs.( 4x-3 + 11x1)/(4+11)= -1/5Optimum strategy for A is (4/15, 11/15)Optimum strategy for B is (4/15, 11/15)Value of game for A ,v= -1/5,Value of game for B ,v= 1/5,

B

A


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Dominance Property OfReducing The Size Of The Matrix

Inferior and Superior Strategies: If a>b then for player Athe strategy corresponding to B is said to be inferior tothe strategy corresponding to A, and equivalently, thestrategy corresponding to A is said to be superior to thestrategy corresponding to B.

If A>B then A is said to Dominate .

The superior strategies are said to dominate the inferiorones

Thus a player would not like to use strategies which aredominated by others


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Dominance Property OfReducing The Size Of The Matrix

Principle of Dominance :If one pure strategy of a player is better orsuperior than another one(irrespective to thestrategy emplayed by his opponent), then the

inferior strategy may be simply ignored byassigning a zero probability while searching foroptimal strategies.


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Dominance Property OfReducing The Size Of The Matrix We can sometimes reduce the size of a games pay off matrix by

eliminating a course of action which is so inferior to another asnever to be used. Such a course of action is said to be dominatedby the other. The concept of dominance is especially useful for theevaluation of two-person zero-sum games where a saddle pointdoes not exist.

General rule1. If all the elements of a row, say kth, are less than or equal to the

corresponding elements of any other row, say rth, then kth row isdominated by the rth row.

2. If all the elements of a column, say k th are greater than or equal tothe corresponding elements of any other column, say rth, then kthcolumn is dominated by rth column.

3. Omit dominated rows or columns.

4. If some linear combination of some rows dominates ith row, then ithrow will be deleted. Similar argument follows for columns.

START


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START

Write the payoff matrix

Is there a purestrategy solution

Is it a 2Xn ormX2 Game?

Use dominance rule to reducethe matrix as far as possible

Is this reduced toa 2X2 Game?

Is it a 2X2Game?

Solve by using analyticalformula mixed strategy

Use Graphical method toreduce the problem to a2X2 game

Formulate and solveas a linearprogramming problem

STOP

Obtain the solutions

Yes

No

No

Yes

Yes

YesNo

No

Alternatively

2 2 G ith


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2 n or an m 2 Game with noSaddle Point

Plot expected pay-off of each strategy on a graph Locate the highest point in the lower envelop (in case of a 2n

game) and the lowest point in the upper envelop (in case of

an m2 game) Consider the pair of lines whose intersection yields thehighest/lowest point and use the strategies represented by it

This reduces the game to a 22 game and it is solvedaccordingly


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Solution to a m n Game with noSaddle Point

Attempt to reduce the order of the problem by applyingdominance rule

If it can be reduced to a 22 game, solve it accordingly

If it can not be reduced to a 22 game, solve it as an LPP

7. game theory

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