Formal Game Theory Stphane Le Roux 1 , Pierre Lescanne 1 and Ren - - PowerPoint PPT Presentation

Motivations Simultaneous Games Sequential Games Summary

Formal Game Theory

Stéphane Le Roux1, Pierre Lescanne1 and René Vestergaard2

1LIP

ENS-Lyon, France

2JAIST

Japan

Workshop Chambéry, Krakow, and Lyon, 2005

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

Outline

1

Motivations

2

Simultaneous Games

3

Sequential Games

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

Game Theory

60 years of history in three dates

1944 J. von Neumann and O. Morgenstern (Theory of Games and Economic Behavior) 1950-1951 J. Nash (Equilibrium points in n-person games, Non-Cooperative Games) 1994 Nash and Nobel Prize of Economy.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

Game Theory

60 years of history in three dates

1944 J. von Neumann and O. Morgenstern (Theory of Games and Economic Behavior) 1950-1951 J. Nash (Equilibrium points in n-person games, Non-Cooperative Games) 1994 Nash and Nobel Prize of Economy.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

Game Theory

60 years of history in three dates

1944 J. von Neumann and O. Morgenstern (Theory of Games and Economic Behavior) 1950-1951 J. Nash (Equilibrium points in n-person games, Non-Cooperative Games) 1994 Nash and Nobel Prize of Economy.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

Applications of Game Theory

“Decision” Making

Economy and Finance: Marketing Strategy and Pricing. Law, Biology, Sociology, etc. Internet: Papadimitriou 2001 (Algorithms, games, and the internet)

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

Applications of Game Theory

“Decision” Making

Economy and Finance: Marketing Strategy and Pricing. Law, Biology, Sociology, etc. Internet: Papadimitriou 2001 (Algorithms, games, and the internet)

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

Applications of Game Theory

“Decision” Making

Economy and Finance: Marketing Strategy and Pricing. Law, Biology, Sociology, etc. Internet: Papadimitriou 2001 (Algorithms, games, and the internet)

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

Formal Methods

100 years of history in four points

1910 - 1930’s: Logic, Type Theory. 1950’s: Induction Principles. 1970’s: Rewriting in Computer Science. 1970 - 1990’s: Automated Proofs.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

Formal Methods

100 years of history in four points

1910 - 1930’s: Logic, Type Theory. 1950’s: Induction Principles. 1970’s: Rewriting in Computer Science. 1970 - 1990’s: Automated Proofs.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

Formal Methods

100 years of history in four points

1910 - 1930’s: Logic, Type Theory. 1950’s: Induction Principles. 1970’s: Rewriting in Computer Science. 1970 - 1990’s: Automated Proofs.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

Formal Methods

100 years of history in four points

1910 - 1930’s: Logic, Type Theory. 1950’s: Induction Principles. 1970’s: Rewriting in Computer Science. 1970 - 1990’s: Automated Proofs.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

What, How, Why, and What to come

Game Theory Formal Methods. Two main reasons:

Many applications. No proper formalization yet.

Improving and Inventing.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

What, How, Why, and What to come

Game Theory Formal Methods. Two main reasons:

Many applications. No proper formalization yet.

Improving and Inventing.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

What, How, Why, and What to come

Game Theory Formal Methods. Two main reasons:

Many applications. No proper formalization yet.

Improving and Inventing.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

What, How, Why, and What to come

Game Theory Formal Methods. Two main reasons:

Many applications. No proper formalization yet.

Improving and Inventing.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Outline

1

Motivations

2

Simultaneous Games Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

3

Sequential Games Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

What is a Simultaneous Game?

Informaly

A system. With some agents. Some options for each agent. Every agent secretly chooses its individual strategy. Then the payoffs are disclosed. This is a one-shot game.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

What is a Simultaneous Game?

Informaly

A system. With some agents. Some options for each agent. Every agent secretly chooses its individual strategy. Then the payoffs are disclosed. This is a one-shot game.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

What is a Simultaneous Game?

Informaly

A system. With some agents. Some options for each agent. Every agent secretly chooses its individual strategy. Then the payoffs are disclosed. This is a one-shot game.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

What is a Simultaneous Game?

Informaly

A system. With some agents. Some options for each agent. Every agent secretly chooses its individual strategy. Then the payoffs are disclosed. This is a one-shot game.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

What is a Simultaneous Game?

Informaly

A system. With some agents. Some options for each agent. Every agent secretly chooses its individual strategy. Then the payoffs are disclosed. This is a one-shot game.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

What is a Simultaneous Game?

Informaly

A system. With some agents. Some options for each agent. Every agent secretly chooses its individual strategy. Then the payoffs are disclosed. This is a one-shot game.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

A Play in a Simultaneous Game

A system B c1 c2 c3 r1 0 1 0 0 1 2 A r2 2 2 1 0 1 1 r3 2 0 2 3 0 1

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

A Play in a Simultaneous Game

With some agents B c1 c2 c3 r1 0 1 0 0 1 2 A r2 2 2 1 0 1 1 r3 2 0 2 3 0 1

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

A Play in a Simultaneous Game

Some options for each of them B c1 c2 c3 r1 0 1 0 0 1 2 A r2 2 2 1 0 1 1 r3 2 0 2 3 0 1

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

A Play in a Simultaneous Game

Every agent secretly chooses its individual strategy. B c1 c2 c3 r1 0 1 0 0 1 2 A r2 2 2 1 0 1 1 r3 2 0 2 3 0 1

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

A Play in a Simultaneous Game

Then the payoffs are disclosed. B c1 c2 c3 r1 0 1 0 0 1 2 A r2 2 2 1 0 1 1 r3 2 0 2 3 0 1 (r1, c3) is a strategy profile.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

A Play in a Simultaneous Game

Then the payoffs are disclosed. B c1 c2 c3 r1 0 1 0 0 1 2 A r2 2 2 1 0 1 1 r3 2 0 2 3 0 1 (r1, c3) is a strategy profile.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

What is a Simultaneous (Pure) Nash Equilibrium?

Informally

A strategy profile, Such that no agent has any incentive to change its own individual strategy.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

What is a Simultaneous (Pure) Nash Equilibrium?

Informally

A strategy profile, Such that no agent has any incentive to change its own individual strategy.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

An example of a Simultaneous Nash Equilibrium

(r2, c1) B c1 c2 c3 r1 0 1 0 0 1 2 A r2 2 2 1 0 1 1 r3 2 0 2 3 0 1

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

An example of a Simultaneous Nash Equilibrium

(r2, c1) B c1 c2 c3 r1 0 1 0 0 1 2 A r2 2 2 1 0 1 1 r3 2 0 2 3 0 1 A checks whether it can increase its payoff or not.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

An example of a Simultaneous Nash Equilibrium

(r2, c1) B c1 c2 c3 r1 0 1 0 0 1 2 A r2 2 2 1 0 1 1 r3 2 0 2 3 0 1 A checks whether it can increase its payoff or not. A has no incentive to change choices.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

An example of a Simultaneous Nash Equilibrium

(r2, c1) B c1 c2 c3 r1 0 1 0 0 1 2 A r2 2 2 1 0 1 1 r3 2 0 2 3 0 1 A checks whether it can increase its payoff or not. A has no incentive to change choices.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

An example of a Simultaneous Nash Equilibrium

(r2, c1) B c1 c2 c3 r1 0 1 0 0 1 2 A r2 2 2 1 0 1 1 r3 2 0 2 3 0 1 B checks whether it can increase its payoff or not. B has no incentive to change choices. Therefore, (r2, c1) is an equilibrium.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

An example of a Simultaneous Nash Equilibrium

(r2, c1) B c1 c2 c3 r1 0 1 0 0 1 2 A r2 2 2 1 0 1 1 r3 2 0 2 3 0 1 B checks whether it can increase its payoff or not. B has no incentive to change choices. Therefore, (r2, c1) is an equilibrium.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

An example of a Simultaneous Nash Equilibrium

(r2, c1) B c1 c2 c3 r1 0 1 0 0 1 2 A r2 2 2 1 0 1 1 r3 2 0 2 3 0 1 B checks whether it can increase its payoff or not. B has no incentive to change choices. Therefore, (r2, c1) is an equilibrium.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Nash Equilibrium may not be “optimal”

From Prisoner’s dilemma

The strategy profile (r2, c2) is the only Nash equilibrium in the game below. B c1 c2 A r1 5 5 0 8 r2 8 0 1 1

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Nash Equilibrium may not be “optimal”

From Prisoner’s dilemma

The strategy profile (r2, c2) is the only Nash equilibrium in the game below. B c1 c2 A r1 5 5 0 8 r2 8 0 1 1

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of Nash Equilibrium

Do Nash Equilibria always exist?

The game below has no Nash equilibrium. B c1 c2 A r1 0 1 1 0 r2 1 0 0 1

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of Nash Equilibrium

Do Nash Equilibria always exist?

The game below has no Nash equilibrium. B c1 c2 A r1 0 1 1 0 r2 1 0 0 1

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of Nash Equilibrium

Do Nash Equilibria always exist?

The game below has no Nash equilibrium. B c1 c2 A r1 0 1 1 0 r2 1 0 0 1 A can increase its payoff.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of Nash Equilibrium

Do Nash Equilibria always exist?

The game below has no Nash equilibrium. B c1 c2 A r1 0 1 1 0 r2 1 0 0 1

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of Nash Equilibrium

Do Nash Equilibria always exist?

The game below has no Nash equilibrium. B c1 c2 A r1 0 1 1 0 r2 1 0 0 1 B can increase its payoff.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of Nash Equilibrium

Do Nash Equilibria always exist?

The game below has no Nash equilibrium. B c1 c2 A r1 0 1 1 0 r2 1 0 0 1

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of Nash Equilibrium

Do Nash Equilibria always exist?

The game below has no Nash equilibrium. B c1 c2 A r1 0 1 1 0 r2 1 0 0 1 Agents cycle: no Nash Equilibrium.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of Nash Equilibrium

Do Nash Equilibria always exist?

The game below has no Nash equilibrium. B c1 c2 A r1 0 1 1 0 r2 1 0 0 1 Agents cycle: no Nash Equilibrium.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of Nash Equilibrium

One should relax the definition of equilibrium in order to guarantee existence. Two proposals: Nash: Introducing probabilities and expected payoffs. Abstract Games: Considering “cycles” as equilibria.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of Nash Equilibrium

One should relax the definition of equilibrium in order to guarantee existence. Two proposals: Nash: Introducing probabilities and expected payoffs. Abstract Games: Considering “cycles” as equilibria.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of Nash Equilibrium

One should relax the definition of equilibrium in order to guarantee existence. Two proposals: Nash: Introducing probabilities and expected payoffs. Abstract Games: Considering “cycles” as equilibria.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Mixed Strategies and Mixed Nash Equilibira

Pure setting: An agent chooses one strategy among some available strategies. Mixed setting: An agent chooses a probability distribution

• ver the set of available strategies.

B 10% 60% 30% 35% 0 1 0 0 1 2 A 20% 2 2 1 0 1 1 45% 2 0 2 3 1 Notions of payoff and equilibrium are easily extended.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Mixed Strategies and Mixed Nash Equilibira

Pure setting: An agent chooses one strategy among some available strategies. Mixed setting: An agent chooses a probability distribution

• ver the set of available strategies.

B 10% 60% 30% 35% 0 1 0 0 1 2 A 20% 2 2 1 0 1 1 45% 2 0 2 3 1 Notions of payoff and equilibrium are easily extended.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Mixed Strategies and Mixed Nash Equilibira

Pure setting: An agent chooses one strategy among some available strategies. Mixed setting: An agent chooses a probability distribution

• ver the set of available strategies.

B 10% 60% 30% 35% 0 1 0 0 1 2 A 20% 2 2 1 0 1 1 45% 2 0 2 3 1 Notions of payoff and equilibrium are easily extended.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Mixed Strategies and Mixed Nash Equilibira

Pure setting: An agent chooses one strategy among some available strategies. Mixed setting: An agent chooses a probability distribution

• ver the set of available strategies.

B 10% 60% 30% 35% 0 1 0 0 1 2 A 20% 2 2 1 0 1 1 45% 2 0 2 3 1 Notions of payoff and equilibrium are easily extended.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of (Mixed) Nash Equilibrium

The game below has no pure Nash equilibrium. B c1 c2 A r1 0 1 1 0 r2 1 0 0 1 But it has a mixed Nash equilibrium: B 50% 50% A 50% 0 1 1 0 50% 1 0 0 1

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of (Mixed) Nash Equilibrium

The game below has no pure Nash equilibrium. B c1 c2 A r1 0 1 1 0 r2 1 0 0 1 But it has a mixed Nash equilibrium: B 50% 50% A 50% 0 1 1 0 50% 1 0 0 1

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of (Mixed) Nash Equilibrium

Guaranteed for finite games (finitely many agents and finitely many available strategies) Brouwer’s fixed point theorem (Hadamard 1910, Brouwer 1912). Non constructive proofs. Constructive proof by Scarf (1967) In Coq?

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of (Mixed) Nash Equilibrium

Guaranteed for finite games (finitely many agents and finitely many available strategies) Brouwer’s fixed point theorem (Hadamard 1910, Brouwer 1912). Non constructive proofs. Constructive proof by Scarf (1967) In Coq?

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of (Mixed) Nash Equilibrium

Guaranteed for finite games (finitely many agents and finitely many available strategies) Brouwer’s fixed point theorem (Hadamard 1910, Brouwer 1912). Non constructive proofs. Constructive proof by Scarf (1967) In Coq?

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of (Mixed) Nash Equilibrium

Guaranteed for finite games (finitely many agents and finitely many available strategies) Brouwer’s fixed point theorem (Hadamard 1910, Brouwer 1912). Non constructive proofs. Constructive proof by Scarf (1967) In Coq?

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

Existence of (Mixed) Nash Equilibrium

Guaranteed for finite games (finitely many agents and finitely many available strategies) Brouwer’s fixed point theorem (Hadamard 1910, Brouwer 1912). Non constructive proofs. Constructive proof by Scarf (1967) In Coq?

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Outline

1

Motivations

2

Simultaneous Games Basic Definitions Simultaneous Nash Equilibrium Properties and Extensions

3

Sequential Games Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a (Finite) Sequential Game?

Informaly

A system, With some agents, Who play in turn, By choosing among finitely many options. The game stops after finitely many steps. And payoffs are disclosed.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a (Finite) Sequential Game?

Informaly

A system, With some agents, Who play in turn, By choosing among finitely many options. The game stops after finitely many steps. And payoffs are disclosed.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a (Finite) Sequential Game?

Informaly

A system, With some agents, Who play in turn, By choosing among finitely many options. The game stops after finitely many steps. And payoffs are disclosed.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a (Finite) Sequential Game?

Informaly

A system, With some agents, Who play in turn, By choosing among finitely many options. The game stops after finitely many steps. And payoffs are disclosed.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a (Finite) Sequential Game?

Informaly

A system, With some agents, Who play in turn, By choosing among finitely many options. The game stops after finitely many steps. And payoffs are disclosed.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a (Finite) Sequential Game?

Informaly

A system, With some agents, Who play in turn, By choosing among finitely many options. The game stops after finitely many steps. And payoffs are disclosed.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a (Finite) Sequential Game?

Semi-formaly

A set of agents, A finite rooted tree, Every internal node is owned by some agent, Leaves are labeled with payoff functions: Agents → Reals

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a (Finite) Sequential Game?

Semi-formaly

A set of agents, A finite rooted tree, Every internal node is owned by some agent, Leaves are labeled with payoff functions: Agents → Reals

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a (Finite) Sequential Game?

Semi-formaly

A set of agents, A finite rooted tree, Every internal node is owned by some agent, Leaves are labeled with payoff functions: Agents → Reals

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a (Finite) Sequential Game?

Semi-formaly

A set of agents, A finite rooted tree, Every internal node is owned by some agent, Leaves are labeled with payoff functions: Agents → Reals

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

A play in a Sequential Game

a

• b
• 1, 2

0, 1 3, 0

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

A play in a Sequential Game

a

• b
• 1, 2

0, 1 3, 0

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

A play in a Sequential Game

b

• 0, 1

3, 0

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

A play in a Sequential Game

b

• 0, 1

3, 0

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

A play in a Sequential Game

3, 0

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a (Binary and Finite) Sequential Game?

Inductive definition

bG ::= bgL Payoffs Payoffs : Agents → Reals | bgN Agents bG bG

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a Sequential Game in COQ?

Parameter Agent : Set. Definition POF := Agent -> Reals. Inductive bG : Set := | bgL : POF -> bG | bgN : Agent -> bG -> bG -> bG.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a Sequential Game in COQ?

Parameter Agent : Set. Definition POF := Agent -> Reals. Inductive bG : Set := | bgL : POF -> bG | bgN : Agent -> bG -> bG -> bG.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a Sequential Strategy Profile?

Graphicaly

a

• b
• b
• po1

po2 po3 po4

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a Sequential Strategy Profile?

Inductive definition

bS ::= bsL Payoffs Choice ::= l | r | bsN Agents Choice bS bS No big changes in Coq.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a Sequential Nash Equilibrium?

Informally

Same intuitive definition as for simultaneous games. A strategy profile, Such that no agent has any incentive to change its own individual strategy,

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

What is a Sequential Nash Equilibrium?

Informally

Same intuitive definition as for simultaneous games. A strategy profile, Such that no agent has any incentive to change its own individual strategy,

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

An example of a Nash Sequential Equilibrium

a

• b
• 1, 2

0, 1 3, 3 If a changes its mind then the associated payoff decreases. If b changes its mind then the associated payoff does not change.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

An example of a Nash Sequential Equilibrium

a

• b
• 1, 2

0, 1 3, 3 If a changes its mind then the associated payoff decreases. If b changes its mind then the associated payoff does not change.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

An example of a Nash Sequential Equilibrium

a

• b
• 1, 2

0, 1 3, 3 If a changes its mind then the associated payoff decreases. If b changes its mind then the associated payoff does not change.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Another Nash Sequential Equilibrium in the same game

a

• b
• 1, 2

0, 1 3, 3 If a changes its mind then the associated payoff decreases. If b changes its mind then the associated payoff decreases.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Another Nash Sequential Equilibrium in the same game

a

• b
• 1, 2

0, 1 3, 3 If a changes its mind then the associated payoff decreases. If b changes its mind then the associated payoff decreases.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Another Nash Sequential Equilibrium in the same game

a

• b
• 1, 2

0, 1 3, 3 If a changes its mind then the associated payoff decreases. If b changes its mind then the associated payoff decreases.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Nash Equilibria may not be “optimal”

The only Nash equilibrium of the game below is not “optimal”. a

• b
• 1, 0

2, 3 0, 4 a

• b
• 1, 0

2, 3 0, 4

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Nash Equilibria may not be “optimal”

The only Nash equilibrium of the game below is not “optimal”. a

• b
• 1, 0

2, 3 0, 4 a

• b
• 1, 0

2, 3 0, 4

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Construction of a “Backward Induction” strategy profile

a

• b
• b
• 3, 0

1, 2 3, 3 0, 4

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Construction of a “Backward Induction” strategy profile

a

• b
• b
• 3, 0

1, 2 3, 3 0, 4

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Construction of a “Backward Induction” strategy profile

a

• b
• b
• 3, 0

1, 2 3, 3 0, 4

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Construction of a “Backward Induction” strategy profile

a

• b
• b
• 3, 0

1, 2 3, 3 0, 4

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Construction of a “Backward Induction” strategy profile

“BI” may not be “optimal”. a

• b
• b
• 3, 0

1, 2 3, 3 0, 4

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Properties of “Backward Induction”

Proved in Coq for binary games

“BI” ⇒ Nash Equilibrium. Guaranteed existence for “BI”.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Properties of “Backward Induction”

Proved in Coq for binary games

“BI” ⇒ Nash Equilibrium. Guaranteed existence for “BI”.

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Deterministic is not enough

A drawback of the “BI” predicate

a

• b
• 1, 0

0, 2 2, 2 What would b do if the play reaches its node?

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Deterministic is not enough

A drawback of the “BI” predicate

a

• b
• 1, 0

0, 2 2, 2 What would b do if the play reaches its node?

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Generalisation and further results

Proved in Coq

Binary games − → variadic games. Deterministic choices − → non-deterministic choices. Payoffs are reals − → arbitray set. Payoff-driven decisions − → substrategy-driven decisions. “BI” predicate − → “BI” computable function. “BI” function, recommendation, and rationality. “BI” is Nash Equilibrium (under reasonnable assumptions).

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Generalisation and further results

Proved in Coq

Binary games − → variadic games. Deterministic choices − → non-deterministic choices. Payoffs are reals − → arbitray set. Payoff-driven decisions − → substrategy-driven decisions. “BI” predicate − → “BI” computable function. “BI” function, recommendation, and rationality. “BI” is Nash Equilibrium (under reasonnable assumptions).

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Generalisation and further results

Proved in Coq

Binary games − → variadic games. Deterministic choices − → non-deterministic choices. Payoffs are reals − → arbitray set. Payoff-driven decisions − → substrategy-driven decisions. “BI” predicate − → “BI” computable function. “BI” function, recommendation, and rationality. “BI” is Nash Equilibrium (under reasonnable assumptions).

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Generalisation and further results

Proved in Coq

Binary games − → variadic games. Deterministic choices − → non-deterministic choices. Payoffs are reals − → arbitray set. Payoff-driven decisions − → substrategy-driven decisions. “BI” predicate − → “BI” computable function. “BI” function, recommendation, and rationality. “BI” is Nash Equilibrium (under reasonnable assumptions).

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Generalisation and further results

Proved in Coq

Binary games − → variadic games. Deterministic choices − → non-deterministic choices. Payoffs are reals − → arbitray set. Payoff-driven decisions − → substrategy-driven decisions. “BI” predicate − → “BI” computable function. “BI” function, recommendation, and rationality. “BI” is Nash Equilibrium (under reasonnable assumptions).

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Generalisation and further results

Proved in Coq

Binary games − → variadic games. Deterministic choices − → non-deterministic choices. Payoffs are reals − → arbitray set. Payoff-driven decisions − → substrategy-driven decisions. “BI” predicate − → “BI” computable function. “BI” function, recommendation, and rationality. “BI” is Nash Equilibrium (under reasonnable assumptions).

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary Basic Definitions Sequential Nash Equilibrium and “Backward Induction” Properties and Extensions

Generalisation and further results

Proved in Coq

Binary games − → variadic games. Deterministic choices − → non-deterministic choices. Payoffs are reals − → arbitray set. Payoff-driven decisions − → substrategy-driven decisions. “BI” predicate − → “BI” computable function. “BI” function, recommendation, and rationality. “BI” is Nash Equilibrium (under reasonnable assumptions).

SLR, PL and RV Formal Game Theory

Motivations Simultaneous Games Sequential Games Summary

Summary

Further formalization and generalisation of Sequential Games. The Formal Methods are suitable to Game Theory. Related work:

New concepts of Abstract Games and Rewriting Equilibrium. Generalisation of Nash Theorem in Abstract Games.

SLR, PL and RV Formal Game Theory