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By:
Ghendir mabrouk nacira
Menaceur khadija
Universitaire Hamma Lakhdar
Domaine : Mathématique et Informatique
Filière : Informatique
Spécialité : système distribuée et intelligence artificielle
20142015
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
5
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.
6
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 can
they do?
c. Outcomes: What do the various combinations of
actions produce?
d. Payoffs: What are the players’ preferences over
the outcomes?
e. Information: What do players know when they
make decisions?
f. Chance: Probability distribution over chance
events, 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
9
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.
10
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.
11
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.
12
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.
13
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
14
CONT …..
Example
15
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.
16
6. EXTENSIVE AND NORMAL FORM
Extensive Form
The extensive form of a game is a complete
description of:
1. The set of players
2. Who moves when and what their
choices are
3. What players know when they move
4. 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. .
18
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.
19
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, 8
20
KINDS OF STRATEGIES
I. Pure strategy
II. Mixed Strategy
III. Totally mixed strategy.
21
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 strategy
22
II. MIXED STRATEGY
 A strategy consisting of possible moves
and a probability distribution (collection of
weights) which corresponds to how
frequently each move is to be played.
 A player would only use a mixed strategy
when he is indifferent between several pure
strategies, and when keeping the opponent
guessing is desirable - that is, when the
opponent can benefit from knowing the next
move. 23
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.
24
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
25
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
26
TYPES OF GAMES
A. Cooperative /Non-cooperative
B. Perfect Information/Imperfect
Information
C. Zero-sum / Non-zero-sum
D. Simultaneous /Sequential
27
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.
28
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
29
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 dilemma
30
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.
31
APPLICATIONS OF GAME THEORY
 Philosophy
 Resource Allocation and Networking
 Biology
 Artificial Intelligence
 Economics
 Politics
32
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
33
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
34
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
38
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.
39
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
40
41

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Exposé biad game-theory

  • 1. By: Ghendir mabrouk nacira Menaceur khadija Universitaire Hamma Lakhdar Domaine : Mathématique et Informatique Filière : Informatique Spécialité : système distribuée et intelligence artificielle 20142015
  • 2. 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
  • 3. 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
  • 4. 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
  • 5. 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 5
  • 6. 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. 6
  • 7. 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
  • 8. 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 can they do? c. Outcomes: What do the various combinations of actions produce? d. Payoffs: What are the players’ preferences over the outcomes? e. Information: What do players know when they make decisions? f. Chance: Probability distribution over chance events, if any. 8
  • 9. BASIC CONCEPTS OF GAME THEORY 1. Game 2. Move 3. Information 4. Strategy 5. Payoff 6. Extensive and Normal Form 7. Equilibria 9
  • 10. 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. 10
  • 11. 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. 11
  • 12. 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. 12
  • 13. 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. 13
  • 14. 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 14
  • 16. 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. 16
  • 17. 6. EXTENSIVE AND NORMAL FORM Extensive Form The extensive form of a game is a complete description of: 1. The set of players 2. Who moves when and what their choices are 3. What players know when they move 4. 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
  • 18. 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. . 18
  • 19. 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. 19
  • 20. 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, 8 20
  • 21. KINDS OF STRATEGIES I. Pure strategy II. Mixed Strategy III. Totally mixed strategy. 21
  • 22. 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 strategy 22
  • 23. II. MIXED STRATEGY  A strategy consisting of possible moves and a probability distribution (collection of weights) which corresponds to how frequently each move is to be played.  A player would only use a mixed strategy when he is indifferent between several pure strategies, and when keeping the opponent guessing is desirable - that is, when the opponent can benefit from knowing the next move. 23
  • 24. 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. 24
  • 25. 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 25
  • 26. 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 26
  • 27. TYPES OF GAMES A. Cooperative /Non-cooperative B. Perfect Information/Imperfect Information C. Zero-sum / Non-zero-sum D. Simultaneous /Sequential 27
  • 28. 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. 28
  • 29. 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 29
  • 30. 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 dilemma 30
  • 31. 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. 31
  • 32. APPLICATIONS OF GAME THEORY  Philosophy  Resource Allocation and Networking  Biology  Artificial Intelligence  Economics  Politics 32
  • 33. 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 33
  • 34. 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 34
  • 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 2 does, the best that Prisoner 1 can do is Confess PRISONERS’ DILEMMA GAME 35
  • 36. 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
  • 37. 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
  • 38. 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 38
  • 39. 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. 39
  • 40. 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 40
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