Module 5- Game Theory

7/28/2019 Module 5- Game Theory

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GAME THEORY

Competitive situation is called a game.

Game represents conflict between two or more parties.

Game is defined as an activity between two or more persons involvingactivities by each person according to a set of rules ,at the end of which eachperson receives some benefit or satisfaction or suffers loss.

Game theory is concerned with the study of decision making in situationswhere two or more opponents are involved under conditions of competitionand conflicting interests

Main objective in the theory of games is to determine the rules of rationalbehavior in the game situations, in which outcomes are dependent on theactions of the interdependent players.

Each players definitely has number of possible outcomes with different valuesto them


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All competitive situations having the following properties aretermed as game:

1. There are a finite number of competitors or players.

2. For each of the competitors, there are a number ofpossible courses of action or strategies.

3. The interest of each other is conflicting in nature.

4. A play occurs when each player chooses one of hiscourses of action.

5. Every combination of course of action determined an

outcome, which results in a gain to each player. (Loss isconsidered as negative gain)


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Number of players

2Players Two-person game

More than two players n person game

Zero Sum game -If in a game the gains to one player are exactly equal to

the losses to another player, then the sum of the gains and losses equals to

zero

Strategy - It is the list of all possible actions (moves or course of action)

that he will take for every outcome that might arise.

Optimal strategy - The particular strategy by which a player

optimizes his gains or losses without knowing the competitors

strategies is called optimal strategy.

Value of a game - The expected outcome per play when players

follow their best or optimal strategy.


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Two types of strategies

1. Pure strategy:

It is decision rate, which is always used by the player to select the

particular course of action.

If a player knows exactly what the other player is going to do, a

deterministic situation is obtained and Objective of the players is tomaximize gains

A pure strategy is usually represented by a number with which the

course of action is associated

2. Mixed strategy:

When both the players are guessing as to which course of action

is to be selected on a particular occasion ,a probabilistic situation is

obtained and the objective function is to maximize the expected gain

The mixed strategy is a selection among pure strategies with fixed

probabilities


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Two person zero sum game :A game which has only two players, where a gain of one

player is the loss of the other, so that the resultant is zero

(rectangular game).

Pay offis the outcome of playing the game.

A pay off matrix is a table showing the amounts received by

the player named at the left hand side after all possible plays of

the game. The player named at the top of the table makes the

payment.

If the value of the game is zero, then the game is

called as Fair game ,otherwise the game is strictly

determinable


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If a player A has m-courses of action and player B has n-

courses, then a pay off matrix is constructed.

1. Row designations for each matrix are the courses of action

available to A.2. Column designations for each matrix are the courses of

action available to B.

3. With a two person zero sum game, the cell entries in Bs

payoff matrix will be the negative of the correspondingentries in As pay off matrix and the matrices will appear as

follows.


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As pay off matrix

Player B

B1 B2 B3 Bn

A1

a11

a12

a13

a1n

A2 a21 a22 a23 a2n

A3

a31

a32

a33

a3n

Player A . . . . . .

. . . . .

Am am1 am2 am3 amn


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Bs pay off matrix

Player B

B1 B2 B3 Bn

A1 -a11 -a12 -a13 -a1n

A2 -a21 -a22 -a23 -a2nA

3-a

31-a

32-a

33 -a

3n

Player A . . . . . .

. . . . .

Am -am1 -am2 -am3 -amn


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Pure strategies (MINIMAX & MAXMIN

Principles) Games with saddle point:

1. For player A minimum value in each row represents the least gain if A

chooses a particular strategy. These are written in the matrix by row

minima. Then A selects the strategy that gives largest gain among the row

minimum values. The choice of player A is maximin principle and thecorresponding gain is maxmin value of the game.

2. For player B, who is assumed to be a loser, the maximum value in each

column represents the maximum loss if B chooses a particular strategy.

They are column maxima. B has to select a strategy that minimizes the

maximum loss. This choice of player b is called the minimax principle and

the corresponding loss is minimax value of the game.


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Algebraic method

Player B

a11 a12

Player A a21 a22

(A plays strategy a1 with probability p1 and plays strategy a2 with probability 1-p1) a22 - a21

P1 = a11 + a22 - (a12 + a21)

P2 = 1P1;a22 - a12

q1 =

a11 + a22 - (a12 + a21)

q2

= 1q1;

a11 a22 - a12 a21

V = a11 + a22 - (a12 + a21)

A game is said to be fair game, if minimax = maximin value and both equal tozero.

Said to be strictly determinable if minimax = maximin and both equals to value

of the game.


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Rules to determine saddle point:

1. Select the minimum (lowest) element in each row of the payoff matrix

and write them under row minima heading. Then select a largest element

among these elements and enclose it in a rectangle.

2. Select the maximum (largest) element in each column of the payoff

matrix and write them under column maxima heading. Then select a

lowest element among these elements and enclose it in a circle.

3. Find out the element (s), which is same in the circle as well as rectangle,and mark the position of such element (s) in the matrix. This element

represents the value of the game and is the saddle point.


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Rules of Dominance

For player B, if each element in a column say Cr, is greater than or equal tothe corresponding element in another column say Cs, then the column Cr is

dominated by the column Cs. So delete column Cr from the pay off matrix.

i.e., player B will lose more by choosing strategy Cr than by choosing strategy

for column Cs.

2. For player A, 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 dominated by

row Rs. So delete row Rr from the pay off matrix.

i.e., player A will never use the strategy corresponding to Rr because he

will gain less for choosing such strategy.

3.Dominance need not be based on the superiority of pure strategies only. A

given strategy can be dominated if it is inferior to an average of two or more

other strategies.

Pay off matrix is a profit matrix for player A and loss matrix for player B.


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Arithmetic method

Find the difference of two nos in column I, andput it under the column II, neglect the negativesign it occurs

Find the diff of two nos in column II,put it underthe column I, neglect the negative sign

Repeat the same for rows also.

The values are called as oddments. These are thefrequencies with which the players must use theircourses of action in their optimum strategies

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Module 5- Game Theory