Dynamics in graph games

Marion Hallet

14 March 2019Theory and Algorithms in Graph and Stochastic Games

General context Definitions Particular games Dynamics Results Computer Networking

1 General context

2 Definitions

3 Particular kind of gamesSequential gamesReachability games

4 Dynamics

5 ResultsSequential gamesReachability games

6 Computer Networking

Hallet Marion Dynamics in graph games March 2019 2 / 25

General context Definitions Particular games Dynamics Results Computer Networking

1 General context

2 Definitions

3 Particular kind of gamesSequential gamesReachability games

4 Dynamics

5 ResultsSequential gamesReachability games

6 Computer Networking

Hallet Marion Dynamics in graph games March 2019 3 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Reactive systems

Reactive system Environment Specificationinteraction

Coffee machine Customer Give coffee, give change,does not explode, ...

Plane’s autopilot Weather Arrive safe

Whatever theenvironment does

Possible solutions :

Testing

Model-checking

Hallet Marion Dynamics in graph games March 2019 4 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Reactive systems

Reactive system Environment Specificationinteraction

Coffee machine Customer Give coffee, give change,does not explode, ...

Plane’s autopilot Weather Arrive safe

Whatever theenvironment does

Possible solutions :

Testing

Model-checking

Hallet Marion Dynamics in graph games March 2019 4 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Reactive systems

Reactive system Environment Specificationinteraction

Coffee machine Customer Give coffee, give change,does not explode, ...

Plane’s autopilot Weather Arrive safe

Whatever theenvironment does

Possible solutions :

Testing

Model-checking

Hallet Marion Dynamics in graph games March 2019 4 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Reactive systems

Reactive system Environment Specificationinteraction

Coffee machine Customer Give coffee, give change,does not explode, ...

Plane’s autopilot Weather Arrive safe

Whatever theenvironment does

Possible solutions :

Testing

Model-checking

Hallet Marion Dynamics in graph games March 2019 4 / 25

General context Definitions Particular games Dynamics Results Computer Networking

What is model-checking ?

Real systemplane,...

Specificationarrive safely,...|=?

Abstract modelautomaton,...

Logic formulaFO, LTL,...|=?

Algorithm

YES/NO

Hallet Marion Dynamics in graph games March 2019 5 / 25

General context Definitions Particular games Dynamics Results Computer Networking

What is model-checking ?

Real systemplane,...

Specificationarrive safely,...|=?

Abstract modelautomaton,...

Logic formulaFO, LTL,...|=?

Algorithm

YES/NO

Hallet Marion Dynamics in graph games March 2019 5 / 25

General context Definitions Particular games Dynamics Results Computer Networking

What is model-checking ?

Real systemplane,...

Specificationarrive safely,...|=?

Abstract modelautomaton,...

Logic formulaFO, LTL,...|=?

Algorithm

YES/NO

Hallet Marion Dynamics in graph games March 2019 5 / 25

General context Definitions Particular games Dynamics Results Computer Networking

From model-checking to (algorithmic) game theory

Environment weather,...

Real systemSplaneS,...

Quant. Spec.energy cons.,...

Equilibrium?

Abstract modelgame

Payoffpayoff functions

Equilibrium?

Algorithm

NO/YES + An equilibrium (as simple as possible)

Hallet Marion Dynamics in graph games March 2019 6 / 25

General context Definitions Particular games Dynamics Results Computer Networking

From model-checking to (algorithmic) game theory

Environment weather,...

Real systemSplaneS,...

Quant. Spec.energy cons.,...

Equilibrium?

Abstract modelgame

Payoffpayoff functions

Equilibrium?

Algorithm

NO/YES + An equilibrium (as simple as possible)

Hallet Marion Dynamics in graph games March 2019 6 / 25

General context Definitions Particular games Dynamics Results Computer Networking

From model-checking to (algorithmic) game theory

Environment weather,...

Real systemSplaneS,...

Quant. Spec.energy cons.,...

Equilibrium?

Abstract modelgame

Payoffpayoff functions

Equilibrium?

Algorithm

NO/YES + An equilibrium (as simple as possible)

Hallet Marion Dynamics in graph games March 2019 6 / 25

General context Definitions Particular games Dynamics Results Computer Networking

From model-checking to (algorithmic) game theory

Environment weather,...

Real systemSplaneS,...

Quant. Spec.energy cons.,...

Equilibrium?

Abstract modelgame

Payoffpayoff functions

Equilibrium?

Algorithm

NO/YES + An equilibrium (as simple as possible)

Hallet Marion Dynamics in graph games March 2019 6 / 25

General context Definitions Particular games Dynamics Results Computer Networking

1 General context

2 Definitions

3 Particular kind of gamesSequential gamesReachability games

4 Dynamics

5 ResultsSequential gamesReachability games

6 Computer Networking

Hallet Marion Dynamics in graph games March 2019 7 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profileWe only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profileWe only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profileWe only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profileWe only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1

2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play

: Infinite path

Strategy profileWe only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play

: Infinite path

Strategy profileWe only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1

2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play

: Infinite path

Strategy profileWe only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play

: Infinite path

Strategy profileWe only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play

: Infinite path

Strategy profileWe only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play

: Infinite path

Strategy profileWe only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profileWe only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profile

We only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1

2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profile

We only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profile

We only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profile

We only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1

2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profile

We only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profile

We only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profile

We only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

1

1

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profile

We only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profile

We only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profile

We only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

1

1

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profile

We only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite graph games

1 2

1

1 2

1

1 2

11

Nodes

Edges

Players

Initialized or not

Play : Infinite path

Strategy profileWe only deal with positionalstrategies (pure andmemoryless)

Hallet Marion Dynamics in graph games March 2019 8 / 25

General context Definitions Particular games Dynamics Results Computer Networking

1 General context

2 Definitions

3 Particular kind of gamesSequential gamesReachability games

4 Dynamics

5 ResultsSequential gamesReachability games

6 Computer Networking

Hallet Marion Dynamics in graph games March 2019 9 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite sequential game or game played on tree

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

• n players• Finite play• Quantitatif• Selfish players who want maximise their payoff

Hallet Marion Dynamics in graph games March 2019 10 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite sequential game or game played on tree

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

1

• n players• Finite play• Quantitatif• Selfish players who want maximise their payoff

Hallet Marion Dynamics in graph games March 2019 10 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite sequential game or game played on tree

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

• n players• Finite play• Quantitatif• Selfish players who want maximise their payoff

Hallet Marion Dynamics in graph games March 2019 10 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite sequential game or game played on tree

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

2

• n players• Finite play• Quantitatif• Selfish players who want maximise their payoff

Hallet Marion Dynamics in graph games March 2019 10 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite sequential game or game played on tree

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

• n players• Finite play• Quantitatif• Selfish players who want maximise their payoff

Hallet Marion Dynamics in graph games March 2019 10 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite sequential game or game played on tree

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

(7, 1)

• n players• Finite play• Quantitatif• Selfish players who want maximise their payoff

Hallet Marion Dynamics in graph games March 2019 10 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite sequential game or game played on tree

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

• n players• Finite play• Quantitatif• Selfish players who want maximise their payoff

Hallet Marion Dynamics in graph games March 2019 10 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite sequential game or game played on tree

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

• n players• Finite play• Quantitatif• Selfish players who want maximise their payoff

Hallet Marion Dynamics in graph games March 2019 10 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite sequential game or game played on tree

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

• n players• Finite play• Quantitatif• Selfish players who want maximise their payoff

Hallet Marion Dynamics in graph games March 2019 10 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite sequential game or game played on tree

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

• n players• Finite play• Quantitatif• Selfish players who want maximise their payoff

Hallet Marion Dynamics in graph games March 2019 10 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite sequential game or game played on tree

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

• n players• Finite play• Quantitatif• Selfish players who want maximise their payoff

Hallet Marion Dynamics in graph games March 2019 10 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Finite sequential game or game played on tree

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

−→ We will study stable strategy profiles

Hallet Marion Dynamics in graph games March 2019 10 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Nash Equilibrium (NE)

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

(7, 1)

A strategy profile is a Nash Equilibrium (NE) if none of the players has aprofitable deviation as long as the other players don’t change their strategy.

Hallet Marion Dynamics in graph games March 2019 11 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Nash Equilibrium (NE)

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)(5, 2) (4, 0)

(7, 1)

A strategy profile is a Nash Equilibrium (NE) if none of the players has aprofitable deviation as long as the other players don’t change their strategy.

Hallet Marion Dynamics in graph games March 2019 11 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Nash Equilibrium (NE)

1

2

1

(4, 2) (3, 0)(3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

(7, 1)

A strategy profile is a Nash Equilibrium (NE) if none of the players has aprofitable deviation as long as the other players don’t change their strategy.

Hallet Marion Dynamics in graph games March 2019 11 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Other notions of Equilibrium

Subgame Perfect Equilibrium

A strategy profile is a Subgame Perfect Equilibrium (SPE) if this is aNash Equilibrium in every subgame.

Strong Nash Equilibrium

A strategy profile is a Strong Nash Equilibrium (SNE) if no coalition ofplayers has a profitable deviation.

Hallet Marion Dynamics in graph games March 2019 12 / 25

General context Definitions Particular games Dynamics Results Computer Networking

1 General context

2 Definitions

3 Particular kind of gamesSequential gamesReachability games

4 Dynamics

5 ResultsSequential gamesReachability games

6 Computer Networking

Hallet Marion Dynamics in graph games March 2019 13 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Particular Reachability Game

1 2

1

p(10) < p(120)

Not initialized

Every player has the sametarget→ not owned by a player anddeadlock

Players have preferences overthe paths→ Ex:shortest path

Hallet Marion Dynamics in graph games March 2019 14 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Particular Reachability Game

1 2

p(10) < p(120)

Not initialized

Every player has the sametarget→ not owned by a player anddeadlock

Players have preferences overthe paths→ Ex:shortest path

Hallet Marion Dynamics in graph games March 2019 14 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Particular Reachability Game

1 2

p(10) < p(120) Not initialized

Every player has the sametarget→ not owned by a player anddeadlock

Players have preferences overthe paths

→ Ex:shortest path

Hallet Marion Dynamics in graph games March 2019 14 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Particular Reachability Game

1 2

p(10) < p(120) Not initialized

Every player has the sametarget→ not owned by a player anddeadlock

Players have preferences overthe paths→ Ex:shortest path

Hallet Marion Dynamics in graph games March 2019 14 / 25

General context Definitions Particular games Dynamics Results Computer Networking

1 General context

2 Definitions

3 Particular kind of gamesSequential gamesReachability games

4 Dynamics

5 ResultsSequential gamesReachability games

6 Computer Networking

Hallet Marion Dynamics in graph games March 2019 15 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Static approach

• The game is played only once.

• Before playing, players decide which strategy they will play.

• If they decide to play a Nash Equilibrium, none of the player hasinterest to change his strategy.

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

Hallet Marion Dynamics in graph games March 2019 16 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Static approach

• The game is played only once.

• Before playing, players decide which strategy they will play.

• If they decide to play a Nash Equilibrium, none of the player hasinterest to change his strategy.

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

Hallet Marion Dynamics in graph games March 2019 16 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Static approach

• The game is played only once.

• Before playing, players decide which strategy they will play.

• If they decide to play a Nash Equilibrium, none of the player hasinterest to change his strategy.

1

2

1

(4, 2) (3, 0)

(7, 1)

2

(8, 1) 1

(5, 2) (4, 0)

Hallet Marion Dynamics in graph games March 2019 16 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Dynamic approach

If we discover a new game

• Find immediately a good strategy is concretely impossible.

• If we play several plays, we will improve our strategy.

• With enough different plays, will we eventually stabilize?

• If so, will this strategy be a “good” strategy?

→ Learning in games (e.g. fictitious play);

→ Strategy improvement (e.g. in parity games);

→ Evolutionary game theory (continuous time).

Hallet Marion Dynamics in graph games March 2019 17 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Dynamic approach

If we discover a new game

• Find immediately a good strategy is concretely impossible.

• If we play several plays, we will improve our strategy.

• With enough different plays, will we eventually stabilize?

• If so, will this strategy be a “good” strategy?

→ Learning in games (e.g. fictitious play);

→ Strategy improvement (e.g. in parity games);

→ Evolutionary game theory (continuous time).

Hallet Marion Dynamics in graph games March 2019 17 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Dynamic approach

If we discover a new game

• Find immediately a good strategy is concretely impossible.

• If we play several plays, we will improve our strategy.

• With enough different plays, will we eventually stabilize?

• If so, will this strategy be a “good” strategy?

→ Learning in games (e.g. fictitious play);

→ Strategy improvement (e.g. in parity games);

→ Evolutionary game theory (continuous time).

Hallet Marion Dynamics in graph games March 2019 17 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Dynamic approach

If we discover a new game

• Find immediately a good strategy is concretely impossible.

• If we play several plays, we will improve our strategy.

• With enough different plays, will we eventually stabilize?

• If so, will this strategy be a “good” strategy?

→ Learning in games (e.g. fictitious play);

→ Strategy improvement (e.g. in parity games);

→ Evolutionary game theory (continuous time).

Hallet Marion Dynamics in graph games March 2019 17 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Dynamic approach

If we discover a new game

• Find immediately a good strategy is concretely impossible.

• If we play several plays, we will improve our strategy.

• With enough different plays, will we eventually stabilize?

• If so, will this strategy be a “good” strategy?

→ Learning in games (e.g. fictitious play);

→ Strategy improvement (e.g. in parity games);

→ Evolutionary game theory (continuous time).

Hallet Marion Dynamics in graph games March 2019 17 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Dynamics

Allowing players to reconsider their strategy and update it into a betterone, observing some conditions.

1

(1, 0) 2

(0, 2) (2, 1)

1

(1, 0) 2

(0, 2) (2, 1)

Questions

• What does better one means? What are the conditions over theupdates?

• Does the dynamics always terminates?

• If so, what are the terminal profiles?

Hallet Marion Dynamics in graph games March 2019 18 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Dynamics

Allowing players to reconsider their strategy and update it into a betterone, observing some conditions.

1

(1, 0) 2

(0, 2) (2, 1)

1

(1, 0) 2

(0, 2) (2, 1)

Questions

• What does better one means? What are the conditions over theupdates?

• Does the dynamics always terminates?

• If so, what are the terminal profiles?

Hallet Marion Dynamics in graph games March 2019 18 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Dynamics

Allowing players to reconsider their strategy and update it into a betterone, observing some conditions.

1

(1, 0) 2

(0, 2) (2, 1)

1

(1, 0) 2

(0, 2) (2, 1)

Questions

• What does better one means? What are the conditions over theupdates?

• Does the dynamics always terminates?

• If so, what are the terminal profiles?

Hallet Marion Dynamics in graph games March 2019 18 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Dynamics

Allowing players to reconsider their strategy and update it into a betterone, observing some conditions.

1

(1, 0) 2

(0, 2) (2, 1)

1

(1, 0) 2

(0, 2) (2, 1)

Questions

• What does better one means? What are the conditions over theupdates?

• Does the dynamics always terminates?

• If so, what are the terminal profiles?

Hallet Marion Dynamics in graph games March 2019 18 / 25

General context Definitions Particular games Dynamics Results Computer Networking

1 General context

2 Definitions

3 Particular kind of gamesSequential gamesReachability games

4 Dynamics

5 ResultsSequential gamesReachability games

6 Computer Networking

Hallet Marion Dynamics in graph games March 2019 19 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Sequential games

I SI A L 1P Games Termination Final Profiles

× × • • • × not appl.• X • • • acyclic prefs X ⊇ SPEsX • X • • acyclic prefs X ⊇ NEsX × × × • × not appl.

X × × X ×

swo prefs prefs can be layered (s.)

= SNEsswo prefs prefs out of pattern (n.)slo prefs prefs out of pattern (n. & s.)

swo prefs, 2 player prefs out of pattern (n. & s.)X × × X X acyclic prefs X = NEs

I = Improvement

SI = Subgame Improvement

A = Atomicity

L = Lazyness (along the play induced by the updated strategy)

1P = One player

Hallet Marion Dynamics in graph games March 2019 20 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Reachability games

Lazy Subgametwo players n players two players n players

Termination NE Termination NE Termination SPE Termination SPE

Qualitative reachability X X ??? ??? × X × ???

Quantitative reachability ??? ??? × ??? × × × ×Mean payoff ??? ??? × × × × × ×

Hallet Marion Dynamics in graph games March 2019 21 / 25

General context Definitions Particular games Dynamics Results Computer Networking

1 General context

2 Definitions

3 Particular kind of gamesSequential gamesReachability games

4 Dynamics

5 ResultsSequential gamesReachability games

6 Computer Networking

Hallet Marion Dynamics in graph games March 2019 22 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Computer Networking

1 2

p(10) < p(120) p(20) < p(210) Send a message to a specificdestination = A uniquetarget

Only care about pathstarting at this node = Notinitialized

Has preference over thepaths

Hallet Marion Dynamics in graph games March 2019 23 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Computer Networking

1 2

p(10) < p(120) p(20) < p(210) Send a message to a specificdestination = A uniquetarget

Only care about pathstarting at this node = Notinitialized

Has preference over thepaths

Hallet Marion Dynamics in graph games March 2019 23 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Computer Networking

1 2

p(10) < p(120) p(20) < p(210) Send a message to a specificdestination = A uniquetarget

Only care about pathstarting at this node = Notinitialized

Has preference over thepaths

Hallet Marion Dynamics in graph games March 2019 23 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Computer Networking

1 2

p(10) < p(120) p(20) < p(210)

Send a message to a specificdestination = A uniquetarget

Only care about pathstarting at this node = Notinitialized

Has preference over thepaths

Hallet Marion Dynamics in graph games March 2019 23 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Computer Networking

1 2

p(10) < p(120) p(20) < p(210)

Send a message to a specificdestination = A uniquetarget

Only care about pathstarting at this node = Notinitialized

Has preference over thepaths

Hallet Marion Dynamics in graph games March 2019 23 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Computer Networking

1 2

p(10) < p(120) p(20) < p(210) Send a message to a specificdestination = A uniquetarget

Only care about pathstarting at this node = Notinitialized

Has preference over thepaths

Hallet Marion Dynamics in graph games March 2019 23 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Border Gateway Protocol (BGP) ≡ Concurrent dynamics

1 2

p(10) < p(120) p(20) < p(210)

Hallet Marion Dynamics in graph games March 2019 24 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Border Gateway Protocol (BGP) ≡ Concurrent dynamics

1 2

p(10) < p(120) p(20) < p(210)

Hallet Marion Dynamics in graph games March 2019 24 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Border Gateway Protocol (BGP) ≡ Concurrent dynamics

1 2

p(10) < p(120) p(20) < p(210)

Hallet Marion Dynamics in graph games March 2019 24 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Border Gateway Protocol (BGP) ≡ Concurrent dynamics

1 2

p(10) < p(120) p(20) < p(210)

Hallet Marion Dynamics in graph games March 2019 24 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Border Gateway Protocol (BGP) ≡ Concurrent dynamics

1 2

p(10) < p(120) p(20) < p(210)

Hallet Marion Dynamics in graph games March 2019 24 / 25

General context Definitions Particular games Dynamics Results Computer Networking

Results

On the Stability of Interdomain Routing LUCA CITTADINI, GIUSEPPE DI BATTISTA, and MASSIMO RIMONDINI

Hallet Marion Dynamics in graph games March 2019 25 / 25