GAME THEORY

GAME THEORY 2

Game Theory _ Introduction

• Game theory is a study of how to mathematically determine the best strategy for given conditions in order to optimize the outcome

• Finding acceptable, if not optimal, strategies in conflict situations.

• Abstraction of real complex situation• Game theory is highly mathematical• Game theory assumes all human interactions can

be understood and navigated by presumptions.

GAME THEORY 3

Why Game Theory?

• All intelligent beings make decisions all the time.

• AI needs to perform these tasks as a result.

• Helps us to analyze situations more rationally and

formulate an acceptable alternative with respect to

circumstance.

GAME THEORY 4

Factors in Game Theory

• Number of players – If the game involves two person it is called two person game else it is called n-person game.

• Sum of gains & Losses – If the gains of one player is equal to the losses of the other player then it is called zero-sum game else it is called non-zero sum game.

• Strategy – The strategy of the player is all possible actions that he will take for every payoff.

GAME THEORY 5

Game Theory

• Zero Sum Game The sum of the payoffs remains constant during the

course of the game. Two sides in conflict Being well informed always helps a player

• Non Zero Sum game The sum of payoffs is not constant during the course of

game play. Players may co-operate or compete Being well informed may harm a player.

GAME THEORY 6

Strategy of Game Theory

• Pure Strategy It is the decision rule which is always used by the

player to select the particular strategy (course of action). Thus each player knows in advance all strategies out of which he selects only one particular strategy.

• Mixed Strategy Courses of action that are to be selected on a particular

occasion with some fixed probability are called mixed strategies are called mixed strategy

GAME THEORY 7

Two Person Zero Sum Game

• Payoff Matrix Payoff is a quantitative measure of satisfaction a player gets at the

end of play. It can be market share, profit, etc. Gain of one person is loss of other person. Thus it is sufficient to construct payoff table for one player only. Each player has available to him a finite no of possible strategies. Player attempts to maximise his gains while player attempts to

minimise losses. Decisions are made simultaneously and known to each other. Both players know each other’s payoff’s.

GAME THEORY 8

General Payoff MatrixPlayer A Strategy

Player B strategy

  B1 B2 . Bn

A1 a11 a12 . a1n

A2 a21 a22 . a2n

. . . . .

. . . . .

. . . . .

Am am1 am2 . amn

GAME THEORY 9

Minimax & Maximin Principle

• Optimal Pure Strategy (Minimax Criterion) To locate the optimal pure strategy for the row player,

first circle the row minima: the smallest payoff's) in each row. Then select the largest row minimum. (If there are two or more largest row minima, choose either one.)

To locate the optimal pure strategy for the column player, first box the column maxima: the largest payoff in each column. Then select the smallest column maximum. (If there are two or more smallest column maxima, choose either one.)

GAME THEORY 10

Saddle Point

• Saddle Points; Strictly Determined Games A saddle point is an entry that is simultaneously a row

minimum and a column maximum. If a game has one or more saddle points, it is strictly determined.

All saddle points will have the same payoff value, called the value of the game. A fair game has a value of zero; otherwise it is unfair, or biased.

Choosing the row and column through any saddle point gives optimal strategies for both players under the minimax criterion.

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Example (2 person sum game)

Union Strategies

Company Strategies  

I II III IV Row Min

I 20 15 12 35 12

II 25 14 8 10 8

III 40 2 10 5 2

IV 5 4 11 0 0

Column Max 40 15 12 35  

Maximin = Minimax = Value of Game =12

GAME THEORY 12

Mixed Game Strategy

• If pure strategy is both applicable then we have to apply mixed strategy.

• Both players must determine mixed strategy to optimize their payoff’s.

• Mixed strategy is evolved using probability.• The expected payoff to a player in a game with arbitrary

payoff matrix [aij] of order mxn is defined as

• Where pi are probabilities of player A and qj are probabilities of player B

• Can be solved using Algebraic, Matrix, Graphical or LP methods

m

i

n

j

jiji qapqpE1 1

),(

GAME THEORY 13

Rules of Dominance

• Reduction by Dominance Check whether there is any row in the (remaining)

matrix that is dominated by another row (this means that it is ≤ some other row). If there is one, delete it.

Check whether there is any column in the (remaining) matrix that is dominated by another column (this means that it is ≥ some other column). If there is one, delete it.

Repeat steps 1 and 2 in any order until there are no dominated rows or columns

GAME THEORY 14

Algebraic Method

Probability of A player

Player A Strategy

Player B strategy

  B1 B2 . Bn

p1 A1 a11 a12 . a1n

p2 A2 a21 a22 . a2n

. . . . . .

. . . . . .

. . . . . .

. Am am1 am2 . amn

pn Probability of B player

q1 q2 qn

GAME THEORY 15

Algebraic Method

1..q

....

.

.....

....

1..

....

.

.....

....

321

2211

2222212

1212111

321

2211

2222212

1212111

qnqqwhere

Vqaqaqa

Vqaqaqa

Vqaqaqa

ppppwhere

Vpapapa

Vpapapa

Vpapapa

nmnnn

nn

nn

m

mmnnn

mm

mm

GAME THEORY 16

Algebraic Method

• The equations shown earlier need to be solved.• To do this we first convert the equations as

equality.• Solve to arrive at the p’s and q’s

GAME THEORY 17

Algebraic Method

Player APlayer B

 

  q1 q2

p1 a11 a12

p2 a21 a2212

21122211

12221

12

21122211

21221

1

)(

1

)(

qq

aaaa

aaq

pp

aaaa

aap

GAME THEORY 18

Example

Player A

Player B

B1 B2 B3 B4

A1 3 2 4 0

A2 3 4 2 4

A3 4 2 4 0

A4 0 4 0 8

The example does not have a saddle point. We will apply rules of dominance

For player A first row is dominated by third row, hence delete first row

In the second matrix, column B1 is dominated by column B3

Player A

Player B

B1 B2 B3 B4

A2 3 4 2 4

A3 4 2 4 0

A4 0 4 0 8

GAME THEORY 19

Example

B2 B3 B4

A2 4 2 4

A3 2 4 0

A4 4 0 8

Now we cannot find any dominant strategies, however the average payoff of B3 & B4 is greater than B2 and hence we may delete column B2

B3 B4

A2 2 4

A3 4 0

A4 0 8

Similarly the average payoff of rows A3 & A4 is better than A2 and hence we can eliminate row A2

GAME THEORY 20

Example

B3 B4

A3 4 0

A4 0 8

The problem has been reduced to 2x2 matrix which can be solved using algebraic methods.

GAME THEORY 21

Graphical Method

• Approach to find solutions for 2 x n or m x 2 games.

• Let player A have two strategies A1 & A2, and B have n strategies B1, B2,…Bn.

• For B1 strategy the expected gain for player will be a11p1+a21p2.

• Similarly for each strategy of B we will have one equation in p1 & p2.

• Draw the straight lines using these equations.

GAME THEORY 22

Graphical Method

• The vertical axes will have the strategy of player A and the horizontal axes will have the probability of achievement.

• The highest point on the lower boundary of these lines will give maximum expected payoff among the minimum expected payoff’s and the optimum value of probability p1 & p2.

• The m x 2 game is alo treated similarly except that the upper boundary of the straight lines corresponding to B’s expected payoff will give minimum expected payoff.

GAME THEORY 23

Example

Player A

Player B

B1 B2 B3 B4

A1 2 2 3 -2

A2 4 3 2 6

A1 A2

87 76 65 54 43 32 21 10 0

-1 -1-2 -2-3 -3

p1=4/9

B4

GAME THEORY 24

LP Method