Exposé biad game-theory

By:

Ghendir mabrouk nacira

Menaceur khadija

Universitaire Hamma LakhdarDomaine : Mathématique et Informatique

Filière : InformatiqueSpécialité : système distribuée et intelligence artificielle

2014\2015

CONTENTS

Introduction

Agents,a definition

Multiagent Systems,a definition

Game theory,a definition

Game theory in Multiagent Systems

Elements of games.

Basic Concepts of Game Theory

Kinds of Strategies.

Nash Equilibrium.

Types of Games.

Applications of Game Theory.

Conclusion .

References . 2

INTRODUCTION

Game theory is the mathematical analysis of a

conflict of interest to find optimal choices that

will lead to a desired outcome under given

conditions. To put it simply, it's a study of ways

to win in a situation given the conditions of the

situation. While seemingly trivial in name, it is

actually becoming a field of major interest in

fields like economics, sociology, and political

and military sciences, where game theory can

be used to predict more important trends.3

AGENTS, A DEFINITION

An agent is a component that is capable of independent

action on behalf of its user or owner (figuring out what

needs to be done to satisfy design objectives, rather

than constantly being told)

4

MULTIAGENT SYSTEMS, A DEFINITION

A multiagent system is one that consists of a number

of agents, which interact with .one-another

In the most general case, agents will be acting on

behalf of users with different goals and motivations

To successfully interact, they will require the ability to

cooperate, coordinate, and negotiate with each

other, much as people do

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GAME THEORY, A DEFINITION

Developed by Prof. John Von Neumann and

Oscar Morgenstern in 1928 game theory is a field

of knowledge that deals with making decisions

when two or more rational and intelligent opponents

are involved under situations of conflict and

competition.

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GAME THEORY IN MULTI-AGENT SYSTEMS

Game theory is a branch of economics that studies

interactions between self interested agents.

Like decision theory, with which it shares many

concepts, game theory has its roots in the work of von

Neumann and Morgenstern

As its name suggests, the basic concepts of game

theory arose from the study of games such as chess

and checkers. However, it rapidly became clear that

the techniques and results of game theory can equally

be applied to all interactions that occur between self-

interested agents.7

ELEMENTS OF GAMES

The essential elements of a game are:

a. Players: The individuals who make decisions.

b. Rules of the game: Who moves when? What canthey do?

c. Outcomes: What do the various combinations ofactions produce?

d. Payoffs: What are the players’ preferences overthe outcomes?

e. Information: What do players know when theymake decisions?

f. Chance: Probability distribution over chanceevents, if any. 8

BASIC CONCEPTS OF GAME THEORY

1. Game

2. Move

3. Information

4. Strategy

5. Payoff

6. Extensive and Normal Form

7. Equilibria

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1. GAME

A conflict in interest among individuals or groups

(players). There exists a set of rules that define the

terms of exchange of information and pieces, the

conditions under which the game begins, and the

possible legal exchanges in particular conditions.

The entirety of the game is defined by all the moves

to that point, leading to an outcome.

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2. MOVE

The way in which the game progresses between

states through exchange of information and pieces.

Moves are defined by the rules of the game and

can be made in either alternating fashion, occur

simultaneously for all players, or continuously for a

single player until he reaches a certain state or

declines to move further. Moves may be choice or

by chance.

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3. INFORMATION

A state of perfect information is when all

moves are known to all players in a game.

Games without chance elements like chess

are games of perfect information, while

games with chance involved like blackjack

are games of imperfect information.

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4. STRATEGY

A strategy is the set of best choices for a player for

an entire game. It is an overlying plan that cannot

be upset by occurrences in the game itself.

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DIFFERENCE BETWEEN

A Move is a single step

a player can take during

the game.

A strategy is a complete

set of actions, which a

player takes into account

while playing the game

throughout

Move Strategy

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CONT …..

Example

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5.PAYOFF

The payoff or outcome is the state of the

game at it's conclusion. In games such as

chess, payoff is defined as win or a loss. In

other situations the payoff may be material

(i.e. money) or a ranking as in a game with

many players.

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6. EXTENSIVE AND NORMAL FORM

Extensive FormThe extensive form of a game is a complete description of:1. The set of players2. Who moves when and what their

choices are3. What players know when they move4. The players’ payoffs as a function of the

choices that are made.In simple words we also say it is a graphical representation (tree form) of a sequential game. 17

The normal form

The normal form is a matrix representation of

a simultaneous game. For two players, one is the

"row" player, and the other, the "column" player. Each

rows or column represents a strategy and each box

represents the payoffs to each player for every

combination of strategies. Generally, such games are

solved using the concept of a Nash equilibrium. .

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7. EQUILIBRIUM

Equilibrium is fundamentally very complex and

subtle. The goal to is to derive the outcome when

the agents described in a model complete their

process of maximizing behaviour. Determining

when that process is complete, in the short run and

in the long run, is an elusive goal as successive

generations of economists rethink the strategies

that agents might pursue.

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GAME REPRESENTATIONS

Extensive form

player 1

1, 2

3, 4

player 2Up

Down

Left

Right

5, 6

7, 8

player 2

Left

Right

Matrix form

player 1’s

strategy

player 2’s strategy

1, 2Up

Down

Left,

Left

Left,

Right

3, 4

5, 6 7, 8

Right,

Left

Right,

Right

3, 41, 2

5, 6 7, 820

KINDS OF STRATEGIES

I. Pure strategy

II. Mixed Strategy

III. Totally mixed strategy.

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I.PURE STRATEGY

A pure strategy provides a complete

definition of how a player will play a game. In

particular, it determines the move a player

will make for any situation he or she could

face.

A player‘s strategy set is the set of pure

strategies available to that player.

select a single action and play it

Each row or column of a payoff matrix represents

both an action and a pure strategy22

II. MIXED STRATEGY

A strategy consisting of possible movesand a probability distribution (collection ofweights) which corresponds to howfrequently each move is to be played.

A player would only use a mixed strategywhen he is indifferent between several purestrategies, and when keeping the opponentguessing is desirable - that is, when theopponent can benefit from knowing the nextmove.

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III. TOTALLY MIXED STRATEGY.

A mixed strategy in which the player assigns

strictly positive probability to every pure strategy

In a non-cooperative game, a totally mixed strategy

of a player is a mixed strategy giving positive

probability weight to every pure strategy available

to the player.

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NASH EQUILIBRIUM

A Nash equilibrium, named after John Nash, is a

set of strategies, one for each player, such that

no player has incentive to unilaterally change her

action. Players are in equilibrium if a change in

strategies by any one of them would lead that

player to earn less than if she remained with her

current strategy. For games in which players

randomize (mixed strategies), the expected or

average payoff must be at least as large as that

obtainable by any other strategy

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CONT ……..

A strategy profile s = (s1, …, sn) is a Nash

equilibrium if for every i,

si is a best response to S−i , i.e., no agent can do better

by unilaterally changing his/her strategy

Theorem (Nash, 1951): Every game with a finite

number of agents and action profiles has at least

one Nash equilibrium

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TYPES OF GAMES

A. Cooperative /Non-cooperative

B. Perfect Information/Imperfect

Information

C. Zero-sum / Non-zero-sum

D. Simultaneous /Sequential

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A. COOPERATIVE /NON-COOPERATIVE

A cooperative game is one in which players are

able to make enforceable contracts

A non-cooperative game is one in which players

are unable to make enforceable contracts.

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B.PERFECT INFORMATION / IMPERFECT

INFORMATION

A game is said to have perfect Information if

all the moves of the game are known to the

players when they make their move.

Otherwise, the game has imperfect

information.

Example

chess game

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C. ZERO-SUM / NON ZERO SUM

One of the most important classifications .

A game is said to be zero-sum if wealth is neither

created nor destroyed among the players.

Example

a. Rock, Paper, Scissors

A game is said to be non-zero-sum if wealth may be

created or destroyed among the players (i.e. the total

wealth can increase or decrease).

Example

Prisoner's dilemma30

D. SIMULTANEOUS / SEQUENTIAL

A simultaneous game is a game where each player

chooses his action without knowledge of the actions

chosen by other players.

Normal form representations are usually used for

simultaneous games.

Example

Prisoner dilemma .

A sequential game is a game where one player

chooses his action before the others choose theirs.

Importantly, the later players must have some

information of the first's choice, otherwise the difference

in time would have no strategic effect. Extensive

form representations are usually used for sequential

games, since they explicitly illustrate the sequential

aspects of a game.

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APPLICATIONS OF GAME THEORY

Philosophy

Resource Allocation and Networking

Biology

Artificial Intelligence

Economics

Politics

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THE PRISONER’S DILEMMA

Two people are collectively charged with a crime

Held in separate cells

No way of meeting or communicating

They are told that:

if one confesses and the other does not, the confessor will be freed, and the other will be jailed for three years;

if both confess, both will be jailed for two years

if neither confess, both will be jailed for one year

EXEMPLE

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PRISONERS’ DILEMMA GAME

Prisoner 2

Confess

(Defect)

Hold out

(Cooperate)

Prisoner 1

Confess

(Defect)

-8

-8

0

-10

Hold out

(Cooperate)

-10

0

-1

-1

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Prisoner 2

Confess

(Defect)

Hold out

(Cooperate)

Prisoner 1

Confess

(Defect)

-8

-8

0

-10

Hold out

(Cooperate)

-10

0

-1

-1

Whatever Prisoner 2 does, the best that Prisoner 1 can do is Confess

PRISONERS’ DILEMMA GAME

35

Prisoner 2

Confess

(Defect)

Hold out

(Cooperate)

Prisoner 1

Confess

(Defect)

-8

-8

0

-10

Hold out

(Cooperate)

-10

0

-1

-1

Whatever Prisoner 1 does, the best that Prisoner 2 can do is Confess.

PRISONERS’ DILEMMA GAME

36

Prisoner 2

Confess

(Defect)

Hold out

(Cooperate)

Prisoner 1

Confess

(Defect)

-8

-8

0

-10

Hold out

(Cooperate)

-10

0

-1

-1

A strategy is a dominant strategy if it is a

player’s strictly best response to any

strategies the other players might pick.

A dominant strategy equilibrium is a strategy

combination consisting of each players

dominant strategy.

Each player has a dominant

strategy to Confess.

The dominant strategy

equilibrium is

(Confess,Confess)

PRISONERS’ DILEMMA GAME

37

Prisoner 2

Confess

(Defect)

Hold out

(Cooperate)

Prisoner 1

Confess

(Defect)

-8

-8

0

-10

Hold out

(Cooperate)

-10

0

-1

-1

The payoff in the dominant strategy

equilibrium (-8,-8) is worse for both

players than (-1,-1), the payoff in the

case that both players hold out. Thus,

the Prisoners’ Dilemma Game is a

game of social conflict.

Opportunity for multi-agent learning: by

learning during repeated play, the

Pareto optimal solution (-1,-1) can

emerge as a result of learning (also

can arise in evolutionary game theory).

PRISONERS’ DILEMMA GAME

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COCLUSION

By using simple methods of game theory,

we can solve for what would be a

confusing array of outcomes in a real-

world situation. Using game theory as a

tool for financial analysis can be very

helpful in sorting out potentially messy

real-world situations, from mergers to

product releases.

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REFERENCES

Books ;

Game theory: analysis of conflict ,Roger B. Myerson,

Harvard University Press

Game Theory: A Very Short Introduction, K. G.

Binmore- 2008, Oxford University Press.

Links :

http://library.thinkquest.org/26408/math/prisoner.shtml

http://www.gametheory.net

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