Game Theory

• Theory of games started in 20 th century, developed by John Von Neumann and Morgenstern

In many practical problems, it is required to take decision in a situation where there are two or more opposite parties with conflicting interests and the action of one depends upon the action which the opponent takes.

• A great variety of competitive situations are commonly seen every day life

• e.g in military battles, political campaigns marketing campaigns

Competitive games

Following properties1- There are finite no of competitors2- Each player has available to him a list of finite no of

possible courses of actions.3 - A play is said to be played when each of the players

chooses a single course of action from the list.Here it is assumed that the choices are made simultaneously so that no player knows his opponents choice until he has decided his own course of action.

4 - Every play determines an outcome.

Two person Zero sum game

A game with only two players in which the gains of one player are the losses of another player is called two person zero sum game

Pay off matrix

• In a two person zero sum game the resulting gain can be represented in the form of matrix called pay off matrix.

• Suppose player A has m courses of action and B has n courses of action.

• Pay off can be represented by m by n matrix• Rows are of courses of action available for A• columns are of courses of action available for B

• In A’s payoff matrix aij represents the payment to A when A

chooses action “i “and B chooses action” j”• B’s pay off matrix will be negative of A’s pay off matrix

+1 -1

-1 +1

B

A

1 finger 2 finger

1 finger

2 fingers

A’s pay off matrix

Strategy

• The strategy of a player is the predetermined rule

by which • a player decides his course of action from his

own list of courses of action during the game

Pure / mixed strategies

A Pure strategy is a decision in advance of all plays, always to choose a particular course of action

A mixed strategy is a decision in advance of all plays,to choose a course of action for each play in accordance with some particular probability distribution.

Opponents are kept guessing as to which course of action is to be selected by the other on any particular occasion.

Pure strategy is a special case of mixed strategy.

Solution of a game

By solving a game we mean to find the best strategies for both the players and the value of the game.

Value of the game Is the maximum guaranteed gain to player A (A maximixing player) if both the players use their best strategies.

It is generally denoted by “v” and is unique.

Fair game : if the value of the game is zero.

Maximin and minimax criterion of optimality

It states that if a player lists his worst possible outcomes of all his potential strategies then he will choose that strategy which corresponds to the best of these worst outcomes.

• Maxmin for A is given by max {min aij } = apq I j

Minimax fo B min {max aij } = ars

j I

a pq ≤ a r s

Maximin for A ≤ minimax for B

If v is the value of the game

Maxmin for A ≤ v ≤ minimax for B

Saddle pint

• if minimax = maximin = value of the game then game is called a game with saddle point.

Def: A saddle point of a pay off matrix is that position in the matrix where the maximum of row min’s coincides with the minimum of column ‘s maxima.

The cell entry at that saddle point is called the value of the game. In a game with saddle point the players use pure

strategies i.e they choose the same course of action through out the game

Method for detecting a saddle point

Find the minimum value in each row and write it in row minima

Find the maximum value in each column and put it in column maxima.

Select the largest element in row minima and enclose it in circle and select the lowest element in column max and encloses it in rectangle.

Find the element which is same in the circle and rectangle and mark the position of such element in matrix..

It is the saddle point which represents the value of the game.

9 3 1 8 06 5 4 6 72 4 3 3 85 6 2 2 1

Player BPlayer A

Row minima0

421

9 6 4 8 8

4

4

4

MAXIMIN = MINMAX = VLUE OF THE GAME = 4

GAME IS NOT FAIR

Col max

Best strategy for player A is second action

while for player B best action is third one

Rules of dominance

Rule 1

if all the elements in a row (say I th)of payoff matrix are less than or equal to the corresponding elements of other row(say j th) then player A will never choose the I th strategy or in other words I th strategy is dominated by the j th strategy.

Rule 2

• If all the elements in a column(say r th) of payoff matrix are greater than or equal t o the corresponding elements of other column (say s th) then the player B will never choose the r th strategy or in other words the r th strategy is dominated by the s th strategy

Rule 3

• A pure strategy may be dominated if it is inferior to an average of two or more other pure strategies.

-1 -2 8

7 5 -1

6 0 12

No saddle point exists,

For A pure strategy I is dominated by strategy III.also for player B pure strategy I is dominated by strategy II.

Solution of 2 x n or m x 2 games without saddle point(graphical method)

• Either of the players has only two undominated pure strategies available.

• By using graphical method it is aimed to reduce a game to the order of 2 x 2 by identifying and eliminating the dominated strategies and then solve it by analytical method.

• Consider the following 2 x n payoff matrix game without saddle point

• Player A has two strategies A1 and A2 with probabilities p1 and p2 resp such that

• p1 + p2 =1,

A11 A12 - - a1nA21 A22 - - a2n

Player BB1 B2 Bn probPlayer A

A1

A2

P1p2

• For each of the pure strategies available to player B, expected payoff for player A would be as follows:

Player B’s pure strategy

Player A’s expected payoff

B1

B2--

Bn

a11 p1 + a21 p2

a12 p1 +a22 p2

a1n p1 +a2np2

According to the maximum criterion for mixed strategy games player A should select the value of probabilities p1,p2 to maximize his minimum expected payoff.This can be done by plotting st lines representing player A’s expected payoffs. The highest point of the lower boundary of these lines(lower envelope) will give maximum expected payoff and the optimum values of probabilities p1 and p2. Now the two strategies of player B corresponding to those lines which pass through the maximum point can be determined

• The m x 2 games are also treated in the same way except that the upper boundary of the st lines corresponding to B’s expected payoff will give the maximum expected payoff to player B and the lowest point on this boundary (upper envelope) will then give the minimum expected payoff and the optimum values of prob q1 and q2.

Solve the game graphically with the following payoff matrix

B1 B2 B3 B4

A1 8 5 -7 9

A2 -6 6 4 -2

When B chooses B1 expected payoff for A shall be8 p1 + (-6)(1-p1) or 14 p1 -6

Similarly expected payoff functions in respect to B2,B3 and B4 can be derived as 6-p1; 4- 11p1 ; 11 p1 -2 resp.We can represent these graphically plotting each payoff as a function of p1

• Lines are marked B1,B2,B3 and B4 and they represent the respective strategies. For each value of p1 the height of the lines at that point denotes the payoff of each of B’s strategy against (p1,1-p1) for A.A is concerned with his least payoff when he plays a particular strategy, which is represented by the lowest of the four lines at that point and wishes to choose p1 in order to maximize this minimum payoff.

• This is at K where the lower envelope lowest of the lines at his point is the highest. this point lies at the intersection of the lines representing B1 and B2 strategies.distance

• KL = -0.4 represents the game value V• Alternatively the game can be written as a 2

by 2 game as follows with strategies A1 and A2 for A and B1 and B3 for B

PLA AYer

B1 B3

A1

A2

8 -7

-6 4

Player B

Using algebraic method opt strategy for A and B are (2/5, 3/5) and (11/25, 0, 14/25,0)

Theorem

• For any zero sum two person game where the optimum strategies are not pure and for player A payoff matrix is A, the optimal strategies are (x1,x2) and (y1,y2) given by

•

)2112()2211(

21122211

2111

1222

2

1

1211

2122

2

1

aaaa

aaaav

isAtogametheofvaluetheand

aa

aa

y

y

and

aa

aa

x

x

If (x1,x2) and(y1,y2) are the mixed strategies for players A and B resp then

x1 +x2 = 1y1 + y2 = 1

x1≥ 0, x2 ≥0, y1 ≥ 0 , y2 ≥ 0

Expected gain to A when B uses strategy 1a11 x1 + a21x2Expected gain to A when B uses strategy IIa12 x1 + a22x2Similarly expected loss for B when A uses strategy Ia11y1 +a12y2Similarly expected loss for B when A uses strategy IIa21 y1+ a22 y2

• If v is the value of the game then

• since A expects to get at least v• a11 x1 + a21x2 ≥ v• a12 x1 + a22x2 ≥ v• Also B expects atmost v• a11y1 +a12y2≤v• a21 y1+ a22 y2 ≤v

a11 x1 + a21x2=va12 x1 + a22x2 =va11y1 +a12y2=va21 y1+ a22 y2 =v

(a11 – a12 ) x1 = (a22-a21) x2

2111

1222

2

1

1211

2122

2

1

aa

aa

y

y

similarlyaa

aa

x

x

Using the eq x1 +x2 =1

)2112()2211(

12112

)2112()2211(

21221

aaaa

aax

aaaa

aax

similarly

)2112()2211(

21112

)2112()2211(

12221

aaaa

aay

aaaa

aay

Value of the game

)2112()2211(

21122211

aaaa

aaaav

By using the dominance properties we always try to reduce the size of payoff matrix to 2 x 2.

In case the payoff matrix reduces to size 2 x n or m x 2 then graphical method is used

Q Solve the game whose payoff matrix is

-1 -2 8

7 5 -1

6 0 12

No saddle point exists,

For A pure strategy I is dominated by strategy III.also for player B pure strategy I is dominated by strategy II.

5 -1

0 12

II III

II

III

)2112()2211(

12112

)2112()2211(

21221

aaaa

aax

aaaa

aax

X1 = 2/3

X2 = 1/3

)2112()2211(

21112

)2112()2211(

12221

aaaa

aay

aaaa

aay

Y1 = 13/18

Y2 =5/18

)2112()2211(

21122211

aaaa

aaaav

Value of the game is 10/3

Thus optimal strategy for A is (0,2/3 1/3)AndFor B (0,13/18,5/18)Value of the game for A is 10/3

Q: solve the game whose pay off matrix is

I Ii Iii iv

I 1 3 -3 7

II 2 5 4 -6