Game theory

Brief History of Game Theory

• 1913 - E. Zermelo provided the first theorem of

game theory asserts that chess is strictly determined

• 1928 - John von Neumann proved the minimax

theorem

• 1944 - John von Neumann / Oskar Morgenstern’s

wrote "Theory of Games and Economic Behavior”

• 1950-1953, John Nash describes Nash equilibrium

• 1972 - John Maynard Smith wrote “Game Theory

and The Evolution of Fighting”

What is Game Theory?

Game theory is a study of how to mathematically

determine the best strategy for given conditions

in order to optimize the outcome.

Basic Terms or Terminology

• Player: The competitors in the game are known as

players. A player may be individual or group of

individuals, or an organization.

• Strategy: A strategy for a player is defined as a set

of rules or alternative courses of action that he

should adopt. A strategy may be of two type: Pure

Strategy and Mixed Strategy.

• Pure Strategy:- If the players select the same

strategy each time, then it is referred to as pure-

strategy. In this case each player knows exactly

Basic Terms or Terminology…

what the other player is going to do, the objective of

the players is to maximize gains or minimize losses.

• Mixed Strategy:-When the players use a

combination of strategies and each player always

kept guessing as to which course of action is to be

selected by the other player at a particular occasion

then this is known as mixed strategy. Thus there is a

probabilistic situation and objective of the player is

to maximize expected gains or to minimize expected

losses.

Basic Terms or Terminology…• Optimum strategy: A course of action or play which

puts the player in the most preferred position,

irrespective of the strategy of his competitor, is called

an optimum strategy.

• Payoff: Each combination of a course of action and

an event is associated with a payoff, which measures

the net benefit to the decision maker the accrues

(accumulate) from a given combination of decision

alternatives and events. ( conditional profit values or

conditional economic consequences )

Basic Terms or Terminology…

• Value of the Game: It is the expected payoff of

play when all the players of the game follow their

optimum strategies. The game is called fair if the

value of the game is zero and unfair, if it is non-

zero.

• Payoff Matrix: When the players select their

particular strategies, the payoffs (gains or losses)

can be represented in the form of a matrix called the

payoff matrix.

Two Person Zero-Sum game

• When there are there are two competitors playing a game, it

is called a two-person game. In case the number of

competitors exceeds two, say n, then the game is termed as

a n-person game.

• Zero-sum means that the algebraic sum of gains and losses

of all the players is zero. Zero-sum game with two players

are called two-person zero sum games. In it the loss (gain)

of one player is exactly equal to the gain (loss) of the other.

• If the sum of losses or gains is not equal to zero, then the

game is of non-zero sum game.

Two Person Zero-Sum game…

• Let player A has m strategies A1, A2, ...,Am and

players B has n strategies B1,B2,…,Bm.

• Assumptions:

Each player has his choices from amongst the pure

strategies.

Player A is always the gainer and player B is

always the loser. (i.e. all payoffs assumed in terms

of player A). Let aij be the payoff which player A

gains from player B if player A chooses strategy Ai

and B chooses Bj. Then the payoff matrix of A is:

mnm3m2m1

3n333231

2n232221

1n131211

n321

a ..... a a a

a ..... a a a

a ..... a a a

a ..... a a a

A Player

B B B B

BPlayer

The payoff matrix for players B is -aij

Two Person Zero-Sum game…

• Example:- Consider a two person coin tossing

game. Each player tosses an unbiased coin

simultaneously. Player B pays Rs. 7 to A, if (H,H)

occurs and Rs. 4, if (T,T) occurs; otherwise player A

pays Rs. 3 to B.

(This two-person game is zero-sum game)

Pure strategies for players are H and T.

The maximin-minimax Principle

• For player A, minimum value in each row

represents the least gain (payoff) to him if he

chooses his particular strategy. Then the player A

selects the strategy that maximizes his minimum

gains. This choice of player A is called the maximin

principle and corresponding gain is called maximin

value of the game.

• On the other hand, player B likes to minimize his

losses. The maximum value of each column

represents the maximum loss to him if he chooses

The maximin-minimax Principle…his particular strategy. These are written in the

matrix by column maxima. He will then select the

strategy that minimize his maximum losses. This

choice of player B is called the minimax principle,

and the corresponding loss is the minimax value of

the game.

• If the maximin value equals the minimax value, then

the game is said to have a saddle point

(equilibrium) and the corresponding strategies are

called optimum strategies. The amount of payoff at an

equilibrium point is known as the value of the game.

The maximin-minimax Principle…Example: Consider a two-person zero sum game

with a 3X2 payoff matrix for player A.

4 6 A

6 8 A A Player

2 9 A

B B

BPlayer

3

2

1

21

Pure strategies for player A (SA) = A1, A2, A3

Pure strategies for player B (SB) = B1, B2

The maximin-minimax Principle…Example:…….

Suppose player A starts the game ( he knows fully

that whatever strategy he adopts, B will select that

particular counter strategy which will minimize the

payoff to A)

If A selects A1 then B will reply by selecting B2, as

this corresponds to the minimum payoff to A in in

row-1 corresponding to A1.

Similarly if A chooses A2, he may gain 8 or 6

depending upon B’s strategy.

Games without saddle points —Mixed Strategy Games

• It is not necessary that there is always a saddle point

exist.

• In such cases to solve games, both the players must

determine an optimal mixture of strategies to find a

saddle point.

• The optimal strategy mixture for each player may be determined by assigning to each strategy its probability of being chosen. The strategies so determined are called mixed strategies because they are probabilistic combination of available choices of strategies

Mixed Strategy Games… • The value of game obtained by the use of mixed

strategies represents which least player A can expect

to win and the least which player B can lose. The

expected payoff to a player in a game with arbitrary

payoff matrix (aij)of order m*n is defined as:

m

1i

n

1jjiji qapq)E(p,

where p and q denote the mixed strategies for players A and B respectively

Maximin-Minimax Xriterion• Consider a m*n game without any saddle point, i.e.

strategies are mixed. Let p1, p2, ….,pm be the

probabilities with which player A will play his

moves A1, A2, ….,Am respectively; and let q1, q2,

…., qn be the probabilities with which player A will

play his moves B1, B2, …., Bn respectively.

• Also, p1+p2+ ….+pm=1 and q1+ q2+ ….+qn =1

• Pi >=0 and qj >=0

• The expected payoff function for player A is :

m

1i

n

1jjiji qapq)E(p,

• For player A (maximin-minimax criterion)

strategyjth his uses Bwhen

A, gain to expected th denotes a p min. Here,

a p ,.......,a p ,a p min. max.

a p min. max. q)E(p, min. max.v

ij

m

1ii

j

in

m

1iii2

m

1iii1

m

1ii

jp

ij

m

1ii

jpqp

• For player B (maximin-minimax criterion)

strategy.ith his usesA

whenB, of loss expected thedenotes a q max. Here,

a q,.......,a q,a q max. min.

a q max. min. q)E(p, max.min.

ij

n

1jj

i

jm,

n

1jjj2,

n

1jjj1,

n

1jj

iq

ij

n

1jj

iqpq

v

The relationship holds good in general and when pi and qj

correspond to the optimal strategies the relation holds in ‘equality’

sense and the expected value for both the players becomes equal to the

optimum expected value of the game.

vv

• For any 2x2 two-person zero-sum game without any saddle

point having the payoff matrix for player

• the optimum mixed strategies

• and determined by

a a A A Player

a a A

B B

BPlayer

22212

12111

21

q q

B B S and

p p

A A S

21

21

B

21

21

A

1211

2122

2

1

1211

2122

2

1

aa

aa

q

q ,

aa

aa

p

p

)a(aaa

aaaa v

bygiven isA togame

theof valueThe 1.qq and 1pp where

21122211

12212211

212 1

Example-1

• For the game with the following payoff matrix,

determine the optimum strategies and the value of

the game:

1qq,1pp

;q q

B B S and

p p

A A S

4 3

1 5A

B

2121

21

21

B

21

21

A

17/5 vis game theof value theand

2/5 3/5

B BS,

4/5 1/5

A AS

:are B andA for strategy optimum Hence

17/5. down to )q,E(p keep

can B 3/5,q because 17/5, than more of sure bet can'A

17/5.atleast isn expectatio his that ensuresA 1/5,p If

5/17)5/3)(5/1(5

435p

)q)(1p4(1)q(11.p)qp3(1q5p q)E(p,

thenfunction, payoff expected thedenotes q)E(p, If

21

A

21

A

11

1

1

11

1111

11111111

qp

qpq

Example-2

• Consider a modified form of matching biased coins

game problem. The matching player is paid Rs. 8 if

the two coins turn both heads and Rs. 1 if the coins

turns both tails. The non-matching player is paid Rs.

3 when the two coins do not match. Given the

choice of being the matching or non-matching

player, which one would you choose and what

would be your strategy?

• The payoff matrix for the matching player is given by

1 3- T Palyer Matching

3- 8 H

T H

player matching-Non

The payoff matrix does not posses any saddle point.

The players will used mixed strategies. Optimum

mixed strategies for matching player and non-

matching player are:

15

4

)33(18

)3(1q

)a(aaa

aaq

15

4

)33(18

)3(1p

)a(aaa

aap

121122211

12221

121122211

21221

The expected value of the game is :

15

1

)33(118

)3)(3(1*8

)a(aaa

aaaav

21122211

21122211

Graphic Solution of 2xn and mx2 games

• Consider the following 2xn game:

2n232221

1n131211

n321

a ..... a a a

a ..... a a aA Player

B B B B

BPlayer

• It is assumed that the game does not have a saddle point.

• Let the optimum mixed strategy for A be given by

1pp, p p

A A S 21

21

21

A

• The expected payoff of A when he plays SA against

B’s pure strategies B1, B2,…Bn is given by:

22n11n1nn

222112122

221111111

papa)(pE B

.

.

.

papa)(pE B

papa)(pE B

E(p) Payoff Expected sA' Strategies Pure sB'

• According to maxmin criterion for mixed strategy

games, player A should select the value of p1 & p2

so as to maximize his min. expected payoffs.

• This may be done by plotting the expected payoff

lines Ej(p1)=(a1j-a2j) p1 + a2j (j=1,2,…,n)

• The highest point on the lower envelope of these

lines will give max. of the min. expected payoffs to

player to player A as also the max. value of pi.

• The two lines passing through maximin point

identify the two critical moves of B, which

combined with two of A, yield a 2x2 matrix that can

• This may be done by plotting the expected payoff

lines Ej(p1)=(a1j-a2j) p1 + a2j (j=1,2,…,n)

• The highest point on the lower envelope of these

lines will give max. of the min. expected payoffs to

player to player A as also the max. value of pi.

• The two lines passing through maximin point

identify the two critical moves of B, which

combined with two of A, yield a 2x2 matrix that can

be used to determine the optimum strategies of the

two players, for the original game, using the results

of mixed strategies (without saddle point).

• Similarly, in mx2 games, the upper envelope of the straight

lines corresponding to B’s expected payoffs will give

expected payoff to player B and the lowest point on this

then gives the minimum expected payoff (minimax value)

and the optimum value of q1.

• EXAMPLE:

Solve the following game graphically:

2 3 0 1 A A Palyer

2- 0 1 2 A

B B B B

BPlayer

2

1

4321

• Since maximin.# minimax. (no saddle point), so players

have to choose mixed strategies, Let the player A plays the

mixed strategy SA against B

1pp ;p p

A A S 21

21

21

A

Then A’s expected payoffs against B’s pure moves are given by

2p4)(pE B

3p3)(pE B

p)(pE B

1p)(pE B

E(p) Payoff Expected sA' Strategies Pure sB'

1144

1133

1122

1111

• These expected payoff equations are plotted as

functions of p1 as shown in figure which shows the

payoffs of each column represented as points on two

vertical axis 1 and 2, unit distance apart.

• Thus line B1 joins the first payoff element 2 in the

first column represented by +2 on the axis 2, and the

second payoff element 1 in the first column

represented by +1 on axis 1.

• Similarly, lines B2, B3, b4 joins the corresponding

representation of payoff elements in the 2nd, 3rd and

4th columns.

• Since the player A wishes to maximize his

minimum expected payoff we consider the highest

point of intersection H on the lover envelope of the

A’s expected payoff equations. This point H

represents the maximin expected value of the game

for A.

• The lines B2 & B4, passing through H, define the

two relevant moves B2 & B4 that alone B needs to

play.

• The solution of 2x4 game, boils down to the simpler

game with 2x2 payoff matrix:

2 0 A A Palyer

2- 1 A

B B

BPlayer

2

1

42

5

1q-1q ,

5

4

)2(21

)2(2q

)a(aaa

aaq

5

3p-1p ,

5

2

)2(21

02p

)a(aaa

aap

thenB, andA for strategy optimum thebe

q q

B B S and

p p

A A S if Now

24 221122211

12222

12 121122211

21221

42

21

B

21

21

A

5

2

2)(021

2)x0(1x2

)a(aaa

aaaa v

is game theof valueexpected The

1/5 0 4/5 0

B B B B S and

3/5 2/5

A A S

are B andA for strategies optimum theNow,

21122211

21122211

4321

B

21

A

Dominance Property

• It is observer that one of the pure strategies of either

player is always inferior to at least one of the

remaining ones.

• The superior strategies are said to dominate the

inferior ones.

• Clearly, a player would have no incentive to use

inferior strategies which are dominated by the

superior ones. In such cases of dominance, we can

reduce the size of the payoff matrix by deleting

those strategies which are dominated by the others.

Dominance Property…

• Thus if each element in one row, say kth of the

payoff matrix is less than or equal to the

corresponding elements in some other row, say rth,

then player A will never choose kth strategy.

• The value of the game and the non-zero choice of

probabilities remain unchanged even after the

deletion of kth row from the payoff matrix. In such

a case the kth strategy is said to be dominated by the

rth one.

• General rules for dominance are:

Dominance Property…

• 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 is dominated by rth row.

• If all the elements of a column, say kth are greater

then or equal to the corresponding elements of any

other column, say, rth, then kth column is

dominated by the rth column.

• Dominated rows or columns may be deleted to

reduce the size of the matrix, as the optimal

strategies will remain unaffected.

Dominance Property…

The modified dominance property: The dominance

property is not always based on the superiority of

pure strategies only. A given strategy can also be

said to be dominated if it is inferior to an average of

two or more other pure strategies only. More

generally, if some convex linear combination of

some rows dominates the ith row, then ith row will

be deleted. Similar arguments follow for columns.

Example

• Two firms are competing for business under the

condition so that one firm’s gain is another firm’s

loss. Firm A’s payoff matrix is

10 14 16 Adv.)(High A

15 12 13 Adv.) (MediumA A Firm

2- 05 10 Adv.) (NoA

Adv.)(High B Adv.) (Medium B Avd.) (No B

B Firm