Game Theory.pdf

OPERATIONS RESEARCHGAME THEORY

Ing. Daniel Orlando Rivera Royero

What is Game Theory ?

What is Game Theory?

• Game theory is the study of multi-personal

decision problem, " conflict of interest" .

• It helps to understand the phenomena

observed when agents interact.

• We will only consider non-cooperative

games.

• We will emphasize the effect of the

payments.

Game Theory

Historical Development1713 James

Waldegraveproposed a

minimax solution of the card game

"le Her"1838 Cournot duopoly performs an

analysis based on game theory 1

913 Ernest Zermelo

shows that in every finite zero-sum game with

perfect information

between two people, there is at least one strategy for one player with which you can not

lose

1928 Von Neumann

's Minimax theorem shows 1

944 John von

Neumann and Oskar

Morgenstern published the book " Theory of Games and

Economic Behavior”

1950-1

960 First use of

models of game theory in economics and first checks the predictions of

game theory in experimental economics.

Desarrollo Histórico

19

94 John C. Harsanyi , John

F. Nash and ReinhardSelten received the

Nobel Prize in economics for his

pioneering analysis of equilibria in the theory

of non- cooperative games.

20

07 Leonid Hurwicz , Eric

Maskin and Roger Myerson receive the

Nobel Prize in Economics for having laid the foundations of

mechanism design theory .

Desarrollo Histórico

• http://youtu.be/AKJDlMolZBg

Nash equilibrium

• A set of N Players (2 Players)

• Each player chooses an action 𝑎𝑘 of

𝐴𝑘 possible actions.

• The player k gets a payment

Assumptions

Player 2

A B CP

layer

1

A 51 51.5 52.5

B 50.5 57.5 54

C 52 53 53.5

Game Theory Componets

• The players are rational maximize

profits.

• The players are intelligent They use

their knowledge or experience (they can

deduce and infer).

Assumptions

• Games complete information: Payment

information and action is published.

• Games of complete and imperfect

information : games aren’t sequential.

Game Types

Definition

There is public information of rationality

and each player knows that each player is

rational, and every player knows that each

player knows that each player is rational ...

Infinitum .

Solution methods

Games between 2 people, the strategies of

each of the opponents can be:

• Pure strategies

• Mixed strategies

Solution methods

Average profit per game during many plays.

It corresponds to the minimum value that will

win Player A, always that play intelligently,

no matter how plays the Player B.

V* = Value Play

Value Play

Find how each competitor must combine

their strategies independently as play your

opponent to guarantee himself at least V * .

Solution a Game

1. Each competitor seeks:

• (max(min E(profit))

• (min(max E(losses))

2. Rationalization.

Establish strategies that are rational for the

player.

Solution a Game

Is the point that Player A and Player B always

choose.

This point matches the strategy that maximizes the

minimum profit in Player A, and the strategy that

minimizes the maximum loss in Player B.

NOTE. It is the lowest of the row and the highest

value of the row column.

Saddle point

Player 2

A B CP

layer

1

A 51 51.5 52.5

B 50.5 57.5 54

C 52 53 53.5

Saddle point

• Identify , if any, is theSaddle point.

• If there the Saddlepoint, what would bethe value of the gameafter a number ofmoves ?

Player B

B1 B2

Pla

yer

A A1 -5 4

A2 -4 -8

Player B

B1 B2

Pla

yer

AA1 2 1

A2 -3 -4

A3 -5 -6

Saddle point

• Domain rule for the row: every value of thedominant row must be greater than orequal to the corresponding value in thedominated row.

• Rule domain for columns : each value ofthe dominant column must be less than orequal to the corresponding value of thedominated column.

Rationalization

Find key strategies in

the following situations:

Player B

B1 B2

Pla

yer A

A1 2 6

A2 3 1

Player B

B1 B2

Pla

yer A

A1 -4 -2

A2 -6 -3

Player B

B1 B2 B3 B4

Pla

yer A

A1 -4 -6 2 4

A2 -6 -3 1 2

Rationalization

Steps.

• Find Saddle Point.

• If (Saddle Point Exists){

• Define=> Pure Strategies (Players A & B)

• } else {

• Rationalize

• Define=> Mixed Strategies (Players A & B)

• }

Strategies A % Market A Strategies B % Market B Difference

a1 47 b1 53 -6

a2 54,5 b1 45,5 9

a3 53 b1 47 6

a1 56 b2 44 12

a2 48,5 b2 51,5 -3

a3 47 b3 53 -6

a1 48,5 b3 51,5 -3

a2 56 b3 44 12

a3 54,5 b3 45,5 9

Example MARKET GAME

PLAYER 2

B1 B2 B3

PL

AY

ER

1

A147, 53 56, 44

48.5,

51.5

A2

54.5,

45.5

48.5,

51.556, 44

A353, 47 47, 53

54.5,

45.5

Example

B1 B2 B3

A1 -6 12 -3

A2 9 -3 12

A3 6 -6 9

Example

Are there key strategies ?

Does saddle point ?

B1 B2

A1 -6 12

A2 9 -3

Example

Reduced matrix

Whenever a game has no saddle point,

game theory, each player can assign a

probability distribution for the set of

strategies .

Mixed strategies

Play intelligently is:

Find a combination of strategies indifferent of

how your opponent plays.

Otherwise, your opponent might try to take

advantage of the way you are playing.

Mixed strategies

B1 y B2 1-y

A1 x -6 12

A2 1-x 9 -3

Example

Example

Player A

−6𝑥 + 9 1 − 𝑥 = 12𝑥 − 3 1 − 𝑥𝑥 = 0.4

Player B

−6𝑦 + 12 1 − 𝑦 = 9𝑦 − 3 1 − 𝑦𝑦 = 0.5

Example

B1 0.5 B2 0.5

A1 0.4 -6 12

A2 0.6 9 -3

𝑉 = −6 × 0.4 × 0.5 + 12 × 0.4 × 0.5 + 9 × 0.6 × 0.5 − 3 × 0.6 × 0.5

𝑉 = 3

Question 1

What is the

meaning of Value

Play?

Pregunta 2

What is the saddle

point ?

Question 2

What is a fair game

?

Exercise 1

Exercise 2