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Logit Dynamics with Concurrent Updates for Local Interaction Games - - PowerPoint PPT Presentation

Logit Dynamics with Concurrent Updates for Local Interaction Games Francesco Pasquale Sapienza Universit` a di Roma joint work with Vincenzo Auletta, Diodato Ferraioli, Paolo Penna, and Giuseppe Persiano ESA: European Symposium on


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Logit Dynamics with Concurrent Updates for Local Interaction Games

Francesco Pasquale

“Sapienza” Universit` a di Roma

joint work with Vincenzo Auletta, Diodato Ferraioli, Paolo Penna, and Giuseppe Persiano ESA: European Symposium on Algorithms Sophia Antipolis, September 2013

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Motivating example Write down a number between 1 and 100. Your number should be as close as possible to half of the average

  • f all numbers we write.

Motivating example 2/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Motivating example

The standard game-theoretic way

◮ Numbers are at most 100, so the average will be at most 100,

and half of the average will be at most 50

◮ I will not write a number larger than 50

Motivating example 3/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Motivating example

The standard game-theoretic way

◮ Numbers are at most 100, so the average will be at most 100,

and half of the average will be at most 50

◮ I will not write a number larger than 50 ◮ If none writes a number larger than 50, then the average will

be at most 50, and half of the average will be at most 25

◮ I will not write a number larger than 25

Motivating example 3/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Motivating example

The standard game-theoretic way

◮ Numbers are at most 100, so the average will be at most 100,

and half of the average will be at most 50

◮ I will not write a number larger than 50 ◮ If none writes a number larger than 50, then the average will

be at most 50, and half of the average will be at most 25

◮ I will not write a number larger than 25 ◮ If none writes a number larger than 25,. . . ◮ . . . ◮ Prediction: Everyone writes 1!

Motivating example 3/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Motivating example

The standard game-theoretic way

◮ Numbers are at most 100, so the average will be at most 100,

and half of the average will be at most 50

◮ I will not write a number larger than 50 ◮ If none writes a number larger than 50, then the average will

be at most 50, and half of the average will be at most 25

◮ I will not write a number larger than 25 ◮ If none writes a number larger than 25,. . . ◮ . . . ◮ Prediction: Everyone writes 1!

Do you believe that prediction?

Motivating example 3/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Motivating example

A previous experiment

STOC poster session at FCRC’11 Half of the average

12.2

Motivating example 4/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Motivating example

A previous experiment

STOC poster session at FCRC’11 Half of the average

12.2

Standard game theoretic assumption Rationality common knowledge This is too strong assumption in several cases

◮ Limited knowledge ◮ Limited computational power ◮ Limited rationality

Motivating example 4/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Randomized best-response

Nash equilibria = Steady states of best-response dynamics

Idea

Relaxation of best-response dynamics

Logit dynamics and stationary distribution 5/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Randomized best-response

Nash equilibria = Steady states of best-response dynamics

Idea

Relaxation of best-response dynamics Best-response

Logit dynamics and stationary distribution 5/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Randomized best-response

Nash equilibria = Steady states of best-response dynamics

Idea

Relaxation of best-response dynamics Best-response

Logit dynamics and stationary distribution 5/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Randomized best-response

Nash equilibria = Steady states of best-response dynamics

Idea

Relaxation of best-response dynamics Best-response Randomized best-response

Logit dynamics and stationary distribution 5/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Randomized best-response

Nash equilibria = Steady states of best-response dynamics

Idea

Relaxation of best-response dynamics Best-response Randomized best-response

Logit dynamics and stationary distribution 5/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Randomized best-response

Nash equilibria = Steady states of best-response dynamics

Idea

Relaxation of best-response dynamics Best-response Randomized best-response

Logit Choice Function [McFadden, 1974]

From profile x = (x1, . . . , xn) player i chooses strategy y with probability proportional to eβui(x−i,y).

Logit dynamics and stationary distribution 5/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Randomized best-response

Logit choice function

pi(y | x) ∼ eβui(x−i,y)

Logit dynamics and stationary distribution 6/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Randomized best-response

Logit choice function

pi(y | x) ∼ eβui(x−i,y)

β = “Rationality level”

◮ β = 0 players play uniformly at random ◮ β → ∞ players best-respond

Logit dynamics and stationary distribution 6/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Randomized best-response

Logit choice function

pi(y | x) ∼ eβui(x−i,y)

β = “Rationality level”

◮ β = 0 players play uniformly at random ◮ β → ∞ players best-respond

Logit dynamics [Blume, GEB’93]

◮ Revision process: choose one player u.a.r. ◮ Update rule: logit choice function

Logit dynamics and stationary distribution 6/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Previous works on logit dynamics

◮ Economics

[Blume, GEB’93]: Equilibrium selection when β → ∞ [Al´

  • s-Ferrer and Netzer, GEB’10]:

Characterization of stochastically stable states

◮ Computer Science

[Montanari and Saberi, FOCS’09]: Hitting time of the best Nash equilibrium [Asadpour, Saberi, WINE’09]: Hitting time of the neighborhood of best Nash equilibria for Atomic Selfish Routing and Load Balancing.

◮ Statistical Mechanics

Logit dynamics vs Glauber dynamics

Logit dynamics and stationary distribution 7/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Our previous questions

  • 1. What is the equilibrium notion for this dynamics? Does

it always exists? Is it unique?

Logit dynamics and stationary distribution 8/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Our previous questions

  • 1. What is the equilibrium notion for this dynamics? Does it

always exists? Is it unique? Stationary distribution of logit dynamics always exists and it is unique. [Auletta et al, SAGT’10]

Logit dynamics and stationary distribution 8/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Our previous questions

  • 1. What is the equilibrium notion for this dynamics? Does it

always exists? Is it unique? Stationary distribution of logit dynamics always exists and it is unique. [Auletta et al, SAGT’10]

  • 2. Starting from an arbitrary initial configuration, how long

does it take to reach equilibrium?

Logit dynamics and stationary distribution 8/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Our previous questions

  • 1. What is the equilibrium notion for this dynamics? Does it

always exists? Is it unique? Stationary distribution of logit dynamics always exists and it is unique. [Auletta et al, SAGT’10]

  • 2. Starting from an arbitrary initial configuration, how long does

it take to reach equilibrium? Analysis of mixing time of logit dynamics for some classes of games [Auletta et al, SPAA’11]

Logit dynamics and stationary distribution 8/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Our previous questions

  • 1. What is the equilibrium notion for this dynamics? Does it

always exists? Is it unique? Stationary distribution of logit dynamics always exists and it is unique. [Auletta et al, SAGT’10]

  • 2. Starting from an arbitrary initial configuration, how long does

it take to reach equilibrium? Analysis of mixing time of logit dynamics for some classes of games [Auletta et al, SPAA’11]

  • 3. When the time to reach equilibrium is long, can we say

something about what happens in the meanwhile?

Logit dynamics and stationary distribution 8/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Our previous questions

  • 1. What is the equilibrium notion for this dynamics? Does it

always exists? Is it unique? Stationary distribution of logit dynamics always exists and it is unique. [Auletta et al, SAGT’10]

  • 2. Starting from an arbitrary initial configuration, how long does

it take to reach equilibrium? Analysis of mixing time of logit dynamics for some classes of games [Auletta et al, SPAA’11]

  • 3. When the time to reach equilibrium is long, can we say

something about what happens in the meanwhile? Metastability of logit dynamics [Auletta et al, SODA’12]

Logit dynamics and stationary distribution 8/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

In this paper

Logit choice function (pi(y | x) ∼ eβui(x−i,y)) Revision process (pick one single player at random)

All-logit reversibility 9/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

In this paper

Logit choice function (pi(y | x) ∼ eβui(x−i,y)) Revision process (pick one single player at random)

What happens when all players play simultaneously?

◮ All-logit ergodic (unique stationary distribution and

convergence)

All-logit reversibility 9/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

In this paper

Logit choice function (pi(y | x) ∼ eβui(x−i,y)) Revision process (pick one single player at random)

What happens when all players play simultaneously?

◮ All-logit ergodic (unique stationary distribution and

convergence)

◮ How do stationary distribution for all-logit differ from

stationary distribution for one-logit?

All-logit reversibility 9/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

In this paper

Logit choice function (pi(y | x) ∼ eβui(x−i,y)) Revision process (pick one single player at random)

What happens when all players play simultaneously?

◮ All-logit ergodic (unique stationary distribution and

convergence)

◮ How do stationary distribution for all-logit differ from

stationary distribution for one-logit?

◮ Are there any meaningful invariant quantities (that are the

same for the one-logit and the all-logit)?

All-logit reversibility 9/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Stationary distribution

Reversibility

What is the stationary distribution for the all-logit dynamics?

All-logit reversibility 10/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Stationary distribution

Reversibility

What is the stationary distribution for the all-logit dynamics?

Stationary distribution is easy for reversible chains. Reversibility: π(x)P(x, y) = π(y)P(y, x)

All-logit reversibility 10/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Stationary distribution

Reversibility

What is the stationary distribution for the all-logit dynamics?

Stationary distribution is easy for reversible chains. Reversibility: π(x)P(x, y) = π(y)P(y, x)

Kolmogorov criterion for reversibility

P is reversible if and only if for every cycle (x0, x1, . . . , xk, x0) P(x0, x1)P(x1, x2) · · · P(xk, x0) = P(x0, xk)P(xk, xk−1) · · · P(x1, x0)

x0 x1 x2 xk

All-logit reversibility 10/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

All-logit dynamics

Potential games and local interaction games

One-logit reversibility Theorem (Blume, GEB’93)

One-logit for game G is reversible if and only if G is a potential game.

All-logit reversibility 11/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

All-logit dynamics

Potential games and local interaction games

One-logit reversibility Theorem (Blume, GEB’93)

One-logit for game G is reversible if and only if G is a potential game.

All-logit reversibility

???

All-logit reversibility 11/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

All-logit dynamics

Potential games and local interaction games

One-logit reversibility Theorem (Blume, GEB’93)

One-logit for game G is reversible if and only if G is a potential game.

All-logit reversibility Theorem

All-logit for game G is reversible if and only is G is a local interaction game.

All-logit reversibility 11/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Local interaction games

Potential Games

G = ([n], S, U). Φ : S1 × · · · × Sn → R exact potential if for every profile x, for every player i, and for every action y ui(x−i, y) − ui(x) = − [Φ(x−i, y) − Φ(x)]

Local interaction games 12/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Local interaction games

Potential Games

G = ([n], S, U). Φ : S1 × · · · × Sn → R exact potential if for every profile x, for every player i, and for every action y ui(x−i, y) − ui(x) = − [Φ(x−i, y) − Φ(x)]

Local interaction games

◮ Players are nodes of a

graph

Local interaction games 12/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Local interaction games

Potential Games

G = ([n], S, U). Φ : S1 × · · · × Sn → R exact potential if for every profile x, for every player i, and for every action y ui(x−i, y) − ui(x) = − [Φ(x−i, y) − Φ(x)]

Local interaction games

◮ Players are nodes of a

graph

◮ Edges are two-player

potential games

Local interaction games 12/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Local interaction games

Potential Games

G = ([n], S, U). Φ : S1 × · · · × Sn → R exact potential if for every profile x, for every player i, and for every action y ui(x−i, y) − ui(x) = − [Φ(x−i, y) − Φ(x)]

Local interaction games

◮ Players are nodes of a

graph

◮ Edges are two-player

potential games

Observation

A local interaction game is a potential game.

Local interaction games 12/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

All-logit dynamics and local interaction games

Idea of proof

Theorem

Logit dynamics for game G is reversible if and only if G is a local interaction game. Idea of proof.

Local interaction games 13/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

All-logit dynamics and local interaction games

Idea of proof

Theorem

Logit dynamics for game G is reversible if and only if G is a local interaction game. Idea of proof.

  • 1. All logit for G reversible implies G potential game

[It follows from Monderer and Shapley characterization of potential games]

Local interaction games 13/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

All-logit dynamics and local interaction games

Idea of proof

Theorem

Logit dynamics for game G is reversible if and only if G is a local interaction game. Idea of proof.

  • 1. All logit for G reversible implies G potential game

[It follows from Monderer and Shapley characterization of potential games]

  • 2. All-logit for a potential game G reversible if and only if

for every pair of profiles x, y K(x, y) = K(y, x) (1) where K(x, y) = n

i=1 Φ(x−i, yi) − (n − 2)Φ(x)

[It follows from the Kolmogorov criterion for reversibility applied to the all-logit for a potential game]

Local interaction games 13/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

All-logit dynamics and local interaction games

Stationary

  • 3. Show that K(x, y) = K(y, x) if and only if G is a local

interaction game [A potential function satisfies K(x, y) = K(y, x) if and only if it is a sum of 2-player potential functions]

Local interaction games 14/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

All-logit dynamics and local interaction games

Stationary

  • 3. Show that K(x, y) = K(y, x) if and only if G is a local

interaction game [A potential function satisfies K(x, y) = K(y, x) if and only if it is a sum of 2-player potential functions]

Stationary distribution

G local interaction game πall(x) ∼

  • y∈S

e−βK(x,y) K(x, y) = n

i=1 Φ(x−i, yi) − (n − 2)Φ(x)

Local interaction games 14/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

All-logit dynamics and local interaction games

Stationary

  • 3. Show that K(x, y) = K(y, x) if and only if G is a local

interaction game [A potential function satisfies K(x, y) = K(y, x) if and only if it is a sum of 2-player potential functions]

Stationary distribution

G local interaction game πall(x) ∼

  • y∈S

e−βK(x,y) K(x, y) = n

i=1 Φ(x−i, yi) − (n − 2)Φ(x)

For the one-logit it is πone(x) ∼ e−βΦ(x)

Local interaction games 14/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Observables

Example

F : {strategy profiles} → R

Question

◮ Local interaction game G ◮ πone, πall stationary distributions of one-logit and all-logit

Is there any meaningful observable F such that Eπone [F] = Eπall [F].

Observables 15/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Observables

Example

F : {strategy profiles} → R

Question

◮ Local interaction game G ◮ πone, πall stationary distributions of one-logit and all-logit

Is there any meaningful observable F such that Eπone [F] = Eπall [F].

Example (Ising model)

±1

◮ Φ(x) = − {i,j}∈E xixj

(Energy)

◮ F(x) = n i=1 xi

(Magnetization)

Observables 15/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Observables

Example

F : {strategy profiles} → R

Question

◮ Local interaction game G ◮ πone, πall stationary distributions of one-logit and all-logit

Is there any meaningful observable F such that Eπone [F] = Eπall [F].

Example (Ising model)

±1

◮ Φ(x) = − {i,j}∈E xixj

(Energy)

◮ F(x) = n i=1 xi

(Magnetization) Eπone [F] = Eπall [F]

Observables 15/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Observables

Decompositions

Local interaction game G, potential Φ, set of strategy profiles S

Decomposition

A permutation σ = (σ1, σ2) of S × S such that for every pair of profiles

◮ σ1(x, y) = σ2(y, x) ◮ K(x, y) = Φ(σ1(x, y)) + Φ(σ2(x, y))

Observables 16/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Observables

Decompositions

Local interaction game G, potential Φ, set of strategy profiles S

Decomposition

A permutation σ = (σ1, σ2) of S × S such that for every pair of profiles

◮ σ1(x, y) = σ2(y, x) ◮ K(x, y) = Φ(σ1(x, y)) + Φ(σ2(x, y))

Lemma

G local interaction game on a bipartite graph then G admits a decomposition.

Observables 16/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Observables

Decompositions

Decomposable observables

Observable F decomposable if decomposition σ exists such that for all x, y F(x) + F(y) = F(σ1(x, y)) + F(σ2(x, y))

Observables 17/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Observables

Decompositions

Decomposable observables

Observable F decomposable if decomposition σ exists such that for all x, y F(x) + F(y) = F(σ1(x, y)) + F(σ2(x, y))

Theorem

If F is a decomposable observable then Eπone [F] = Eπall [F]

Observables 17/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Conclusions and future works

  • 1. Logit choice function as a model for players with limited

rationality

  • 2. Stationary distribution depends on the revision process

Research directions 18/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Conclusions and future works

  • 1. Logit choice function as a model for players with limited

rationality

  • 2. Stationary distribution depends on the revision process
  • 3. All-logit reversible if and only if local interaction game

Research directions 18/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Conclusions and future works

  • 1. Logit choice function as a model for players with limited

rationality

  • 2. Stationary distribution depends on the revision process
  • 3. All-logit reversible if and only if local interaction game
  • 4. Sufficient conditions for invariant observables

Research directions 18/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Conclusions and future works

  • 1. Logit choice function as a model for players with limited

rationality

  • 2. Stationary distribution depends on the revision process
  • 3. All-logit reversible if and only if local interaction game
  • 4. Sufficient conditions for invariant observables

Open problems

◮ Game theoretic interpretation of K(x, y)

Research directions 18/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Conclusions and future works

  • 1. Logit choice function as a model for players with limited

rationality

  • 2. Stationary distribution depends on the revision process
  • 3. All-logit reversible if and only if local interaction game
  • 4. Sufficient conditions for invariant observables

Open problems

◮ Game theoretic interpretation of K(x, y) ◮ Mixing time and metastabilty of all-logit for local

interaction games

Research directions 18/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Conclusions and future works

  • 1. Logit choice function as a model for players with limited

rationality

  • 2. Stationary distribution depends on the revision process
  • 3. All-logit reversible if and only if local interaction game
  • 4. Sufficient conditions for invariant observables

Open problems

◮ Game theoretic interpretation of K(x, y) ◮ Mixing time and metastabilty of all-logit for local interaction

games

◮ Other invariant observables

Research directions 18/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Conclusions and future works

  • 1. Logit choice function as a model for players with limited

rationality

  • 2. Stationary distribution depends on the revision process
  • 3. All-logit reversible if and only if local interaction game
  • 4. Sufficient conditions for invariant observables

Open problems

◮ Game theoretic interpretation of K(x, y) ◮ Mixing time and metastabilty of all-logit for local interaction

games

◮ Other invariant observables ◮ Other revision processes: Players selected according to

some distribution

Research directions 18/ 19

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Logit Dynamics with Concurrent Updates for Local Interaction Games ESA - Sophia Antipolis, September 2013

Thank you!

Research directions 19/ 19