logit dynamics with concurrent updates
TRANSCRIPT
Vincenzo Auletta (Univ. of Salerno)Diodato Ferraioli (Univ. of Salerno)Francesco Pasquale (Univ. of Rome)Giuseppe Persiano (Univ. of Salerno)
Logit Dynamicswith
Concurrent UpdatesPaolo Penna (LIAFA, Univ. Paris Diderot)
joint work with
Aussois, Displexity Workshop, March 2015
Complex Systems
Lots of simpleparticles
Complex Systems
Lots of simpleparticles
Ferromagnetism
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“like” to agree with the neighbors
Coordination Games
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Coordination Games
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Best Response
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Coordination Games
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Best Response
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Coordination Games
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Best Response
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Coordination Games
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Best Response
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Coordination Games
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Best Response
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Coordination Games
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Best Response
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Temperature (or Noise)!
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or
low temperature high temperature
Temperature (or Noise)!
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or
low temperature high temperature
Bounded rationality
Temperature (or Noise)!
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or
low temperature high temperature
Noisy best response
Strategies with higher payoff chosen withhigher probability
€10
€1
Bounded rationality
prob ∝ e10/temperature
prob ∝ e1/temperature
Coordination Games
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Coordination Games
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Coordination Games
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Best Response
Coordination Games
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Best Response
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Noisy Best Response
Coordination Games
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Best Response
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Noisy Best Response
Logit Dynamics (Blume’93):• Pick a player at random• This players updates strategy according to noisy
best-response
Coordination Games
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1 0
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Best Response
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Noisy Best Response
Where are we after 1000 steps?
Logit Dynamics (Blume’93):• Pick a player at random• This players updates strategy according to noisy
best-response
Coordination Games
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Best Response
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Noisy Best Response
Most likely here
Where are we after 1000 steps?
Logit Dynamics (Blume’93):• Pick a player at random• This players updates strategy according to noisy
best-response
Coordination Games
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1 0
0 2
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1 0
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Best Response
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1 0
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Noisy Best Response
Most likely here
Where are we after 1000 steps?
Logit Dynamics (Blume’93):• Pick a player at random• This players updates strategy according to noisy
best-response
with prob ∝ ev/temperaturein state v
Logit DynamicsLogit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Logit DynamicsLogit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Ferromagnetism (Ising model): Spontaneous symmetrybreaking, transition phases
Logit DynamicsLogit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Ferromagnetism (Ising model): Spontaneous symmetrybreaking, transition phases
Equilibrium selection in games
Logit DynamicsLogit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Ferromagnetism (Ising model): Spontaneous symmetrybreaking, transition phases
Equilibrium selection in games
Diffusion of technologies/innovations
Logit DynamicsLogit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Ferromagnetism (Ising model): Spontaneous symmetrybreaking, transition phases
Equilibrium selection in games
Diffusion of technologies/innovations
Efficiency
Logit DynamicsLogit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Ferromagnetism (Ising model): Spontaneous symmetrybreaking, transition phases
Equilibrium selection in games
Diffusion of technologies/innovations
Efficiency
Logit DynamicsLogit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Ferromagnetism (Ising model): Spontaneous symmetrybreaking, transition phases
Equilibrium selection in games
Diffusion of technologies/innovations
Efficiency
natural alternatives
Logit DynamicsLogit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Ferromagnetism (Ising model): Spontaneous symmetrybreaking, transition phases
Equilibrium selection in games
Diffusion of technologies/innovations
Efficiency
convenient for the analysis natural alternatives
Logit DynamicsLogit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Ferromagnetism (Ising model): Spontaneous symmetrybreaking, transition phases
Equilibrium selection in games
Diffusion of technologies/innovations
Efficiency
two players
?
Logit DynamicsLogit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Ferromagnetism (Ising model): Spontaneous symmetrybreaking, transition phases
Equilibrium selection in games
Diffusion of technologies/innovations
Efficiency
different probabilities
?
Logit DynamicsLogit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Ferromagnetism (Ising model): Spontaneous symmetrybreaking, transition phases
Equilibrium selection in games
Diffusion of technologies/innovations
Efficiency?
independent players
Logit DynamicsLogit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Ferromagnetism (Ising model): Spontaneous symmetrybreaking, transition phases
Equilibrium selection in games
Diffusion of technologies/innovations
Efficiency?
independent players
Known: Equilibria at nearly zero temperature for manynatural player selections (Alos-Ferrer – Netzer’10).
Logit DynamicsLogit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Ferromagnetism (Ising model): Spontaneous symmetrybreaking, transition phases
Equilibrium selection in games
Diffusion of technologies/innovations
Efficiency
Known: Equilibria at nearly zero temperature for manynatural player selections (Alos-Ferrer – Netzer’10).
What depends on the scheduler and what on the players?
Our Work
Consider the extreme case of concurrent updates.
Our Work
Consider the extreme case of concurrent updates.
Logit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Select all players
All-Logit Dynamics
Our Work
Consider the extreme case of concurrent updates.
Equilibria at every temperature for a very natural class of games
Logit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Select all players
All-Logit Dynamics
Our Work
Consider the extreme case of concurrent updates.
Equilibria at every temperature for a very natural class of games• Reversibility equivalent to local interaction games
Logit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Select all players
All-Logit Dynamics
Our Work
Consider the extreme case of concurrent updates.
Equilibria at every temperature for a very natural class of games• Reversibility equivalent to local interaction games
Differences between one-logit and all-logit:
Logit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Select all players
All-Logit Dynamics
Our Work
Consider the extreme case of concurrent updates.
Equilibria at every temperature for a very natural class of games• Reversibility equivalent to local interaction games
Differences between one-logit and all-logit:• “Observable” quantities
Logit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Select all players
All-Logit Dynamics
Our Work
Consider the extreme case of concurrent updates.
Equilibria at every temperature for a very natural class of games• Reversibility equivalent to local interaction games
Differences between one-logit and all-logit:• “Observable” quantities
Efficiency (mixing time)
Logit Dynamics (Blume’93):• Pick one player uniformly at random• This player updates strategy according to noisy
best-response
Select all players
All-Logit Dynamics
Characterization
When is the analysis doable?
Local InteractionGame
Game
Reversibility of all-logit
Characterization
When is the analysis doable?
Local InteractionGame
Game
GG′
Reversibility of all-logit potentialgames
Characterization
When is the analysis doable?
Local InteractionGame
Game
GG′
Reversibility of all-logit potentialgames
Directed Potential
Local Interaction Games
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Local Interaction Games
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Local Interaction Games
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Local Interaction Games
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Diffusion of new technology/innovation (...,Montanari-Saberi’10,...)
Local Interaction Games
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Local Interaction Games
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Local Interaction Games
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Local Interaction Games
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Local Interaction Games
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Reversibility
yxπ(x)P (x, y) = π(y)P (y, x)
π is the stationary distribution
P t(x, y)→ π(y) as t→∞
If there is π satisfying
Directed Potential
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1/4 1/4
1/41/4
Game Stationarydistribution
Directed Potential
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1/4 1/4
1/41/4
Game Stationarydistribution
Directed Potential
DirPot(x, y) =∑i Pot(yi, x−i)− (n− 2)Pot(x)
π(x) ∝∑y e
βDirPot(x,y)
+− ++
−− −+
0 1
01
β = 1/temperature
Directed Potential
DirPot(x, y) =∑i Pot(yi, x−i)− (n− 2)Pot(x)
π(x) ∝∑y e
βDirPot(x,y)
+− ++
−− −+
0 1
01
β = 1/temperature
Directed Potential
DirPot(x, y) =∑i Pot(yi, x−i)− (n− 2)Pot(x)
π(x) ∝∑y e
βDirPot(x,y)
+− ++
−− −+
0 1
01
1
β = 1/temperature
Directed Potential
DirPot(x, y) =∑i Pot(yi, x−i)− (n− 2)Pot(x)
π(x) ∝∑y e
βDirPot(x,y)
+− ++
−− −+
0 1
01 0
β = 1/temperature
Directed Potential
DirPot(x, y) =∑i Pot(yi, x−i)− (n− 2)Pot(x)
π(x) ∝∑y e
βDirPot(x,y)
+− ++
−− −+
0 1
012
β = 1/temperature
Directed Potential
DirPot(x, y) =∑i Pot(yi, x−i)− (n− 2)Pot(x)
π(x) ∝∑y e
βDirPot(x,y)
+− ++
−− −+
0 1
01
2
π(−−) ∝ 1 + 2eβ + e2β
1
01
β = 1/temperature
Directed Potential
DirPot(x, y) =∑i Pot(yi, x−i)− (n− 2)Pot(x)
π(x) ∝∑y e
βDirPot(x,y)
+− ++
−− −+
0 1
011 2
1
π(+−) = 1 + 2eβ + e2βπ(−−) = 1 + 2eβ + e2β
1/4 1/4
1/41/4
β = 1/temperatureStationarydistribution
Observables
Different equilibria...
...but you may observe the same thing!
e.g. magnetization
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− +1 0
0 1
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− +1 0
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Observables
Different equilibria...
...but you may observe the same thing!
e.g. magnetization
one-logit−
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− +1 0
0 1
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− +1 0
0
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Observables
Different equilibria...
...but you may observe the same thing!
e.g. magnetization
all-logit−
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− +1 0
0 1
−
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− +1 0
0
-+
Observables
Different equilibria...
...but you may observe the same thing!
e.g. magnetization
all-logit−
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− +1 0
0 1
−
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− +1 0
0
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+ -# #-
Observables
Different equilibria...
...but you may observe the same thing!
e.g. magnetization
-+
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-+
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+ -# #-
Observables
Different equilibria...
...but you may observe the same thing!
e.g. magnetization
-+
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+ -# #-For every game and every bipartite graph.
Open Questions
Different players selections...reversibility, observables?
Open Questions
Different players selections...reversibility, observables?• Two-logit?
Open Questions
Different players selections...reversibility, observables?• Two-logit?When the stationary of all-logit has similar propertiesas that of one-logit?
Open Questions
Different players selections...reversibility, observables?• Two-logit?When the stationary of all-logit has similar propertiesas that of one-logit?• Entropy maximization?
Open Questions
Different players selections...reversibility, observables?• Two-logit?When the stationary of all-logit has similar propertiesas that of one-logit?• Entropy maximization?• Transition phase?
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Mercie!