Dynamics in graph games - math.umons.ac.bemath.umons.ac.be/gamenet2019/talks/Hallet Marion.pdf ·...
Transcript of Dynamics in graph games - math.umons.ac.bemath.umons.ac.be/gamenet2019/talks/Hallet Marion.pdf ·...
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