Mixing Time and Stationary Expected Social
Welfare of Logit Dynamics
Francesco Pasquale
Dipartimento di Informatica “R. Capocelli”
Università di Salerno
joint work-in-progress with
Vincenzo Auletta, Diodato Ferraioli, and Giuseppe Persiano
Roma, June 17, 2010
Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Outline
Framework Description
Examples
Research Directions
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Game Theory
...in one slide
G = ([n], S, U)
◮
[n] = {1, . . . , n} players;
◮
S = {S1 , . . . , Sn };
Si = { actions for player i };
◮
U = {u1 , . . . , un };
ui : S1 × · · · × Sn → R utility functions
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Game Theory
...in one slide
G = ([n], S, U)
◮
[n] = {1, . . . , n} players;
◮
S = {S1 , . . . , Sn };
Si = { actions for player i };
◮
U = {u1 , . . . , un };
ui : S1 × · · · × Sn → R utility functions
x = (x1 , . . . , xn ) ∈ S1 × · · · × Sn pure Nash equilibrium if for every
i ∈ [n] and for every y ∈ Si
Framework Description
ui (x−i , y ) 6 ui (x)
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Game Theory
...in one slide
G = ([n], S, U)
◮
[n] = {1, . . . , n} players;
◮
S = {S1 , . . . , Sn };
Si = { actions for player i };
◮
U = {u1 , . . . , un };
ui : S1 × · · · × Sn → R utility functions
x = (x1 , . . . , xn ) ∈ S1 × · · · × Sn pure Nash equilibrium if for every
i ∈ [n] and for every y ∈ Si
ui (x−i , y ) 6 ui (x)
µ = (µ1 , . . . , µn ) ∈ ∆(S1 ) × · · · × ∆(Sn ) mixed Nash equilibrium
if for every i ∈ [n] and for every σ ∈ ∆(Si )
Framework Description
E(µ−i ,σ) [ui ] 6 Eµ [ui ]
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Dynamics
Reaching equilibria
Dynamics: Choose a player, update her strategy, repeat.
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Dynamics
Reaching equilibria
Dynamics: Choose a player, update her strategy, repeat.
Best response dynamics
At her turn, player i chooses the action y ∈ Si that maximizes her
utility
ui (x−i , y ) > ui (x−i , z) for every z ∈ Si
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Dynamics
Reaching equilibria
Dynamics: Choose a player, update her strategy, repeat.
Best response dynamics
At her turn, player i chooses the action y ∈ Si that maximizes her
utility
ui (x−i , y ) > ui (x−i , z) for every z ∈ Si
Questions
◮
Convergence
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Dynamics
Reaching equilibria
Dynamics: Choose a player, update her strategy, repeat.
Best response dynamics
At her turn, player i chooses the action y ∈ Si that maximizes her
utility
ui (x−i , y ) > ui (x−i , z) for every z ∈ Si
Questions
◮
Convergence
If yes then. . .
◮
Speed of convergence.
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Randomized Best Response
Logit Dynamics
Randomized Best Response: Choose for the next round strategy
y with probability proportional to the returned utility.
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Randomized Best Response
Logit Dynamics
Randomized Best Response: Choose for the next round strategy
y with probability proportional to the returned utility.
x ∈ S1 ×· · ·×Sn ,
Framework Description
i ∈ [n],
y ∈ Si
⇒
σi (y | x) ∼ e βui (x−i ,y )
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Randomized Best Response
Logit Dynamics
Randomized Best Response: Choose for the next round strategy
y with probability proportional to the returned utility.
x ∈ S1 ×· · ·×Sn ,
i ∈ [n],
y ∈ Si
⇒
σi (y | x) ∼ e βui (x−i ,y )
e βui (x−i ,y )
βui (x−i ,z)
z∈Si e
σi (y | x) = P
β = “Inverse noise”
Observation
◮
β = 0 players play uniformly at random;
◮
β → ∞ players play best response (u.a.r. over best responses
if more than one)
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Logit Dynamics
Description
Logit dynamics [Blume, GEB’93]
From any profile x, choose a player i ∈ [n] u.a.r and update her
action with probability σi (· | x).
p ∼ e βui (x−i ,y )
x
Framework Description
(x−i , y )
P (x, (x−i , y )) =
1
σi (y | x)
n
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Logit Dynamics
Description
Logit dynamics [Blume, GEB’93]
From any profile x, choose a player i ∈ [n] u.a.r and update her
action with probability σi (· | x).
p ∼ e βui (x−i ,y )
x
(x−i , y )
P (x, (x−i , y )) =
1
σi (y | x)
n
This process defines an ergodic Markov chain
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Markov chains
...in one slide
{Xt : t ∈ N}, (Ω, P).
Framework Description
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Markov chains
...in one slide
{Xt : t ∈ N}, (Ω, P).
◮
Irreducible: for every x, y ∈ Ω, ∃t ∈ N : P t (x, y) > 0;
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Markov chains
...in one slide
{Xt : t ∈ N}, (Ω, P).
◮
◮
Irreducible: for every x, y ∈ Ω, ∃t ∈ N : P t (x, y) > 0;
Aperiodic: for every x, gcd {t > 1 : P t (x, x) > 0} = 1;
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Markov chains
...in one slide
{Xt : t ∈ N}, (Ω, P).
◮
◮
◮
Irreducible: for every x, y ∈ Ω, ∃t ∈ N : P t (x, y) > 0;
Aperiodic: for every x, gcd {t > 1 : P t (x, x) > 0} = 1;
Stationary distribution: π ∈ ∆(Ω), πP = π.
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Markov chains
...in one slide
{Xt : t ∈ N}, (Ω, P).
◮
◮
◮
Irreducible: for every x, y ∈ Ω, ∃t ∈ N : P t (x, y) > 0;
Aperiodic: for every x, gcd {t > 1 : P t (x, x) > 0} = 1;
Stationary distribution: π ∈ ∆(Ω), πP = π.
Irreducible + Aperiodic = Ergodic =⇒
=⇒ π is unique and P t (x, ·) → π
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Markov chains
...in one slide
{Xt : t ∈ N}, (Ω, P).
◮
◮
◮
Irreducible: for every x, y ∈ Ω, ∃t ∈ N : P t (x, y) > 0;
Aperiodic: for every x, gcd {t > 1 : P t (x, x) > 0} = 1;
Stationary distribution: π ∈ ∆(Ω), πP = π.
Irreducible + Aperiodic = Ergodic =⇒
=⇒ π is unique and P t (x, ·) → π
◮
Total variation distance µ, ν ∈ ∆(Ω)
Framework Description
kµ − νk = max |µ(A) − ν(A)| =
A⊆Ω
1X
|µ(x) − ν(x)|
2
x∈Ω
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Markov chains
...in one slide
{Xt : t ∈ N}, (Ω, P).
◮
◮
◮
Irreducible: for every x, y ∈ Ω, ∃t ∈ N : P t (x, y) > 0;
Aperiodic: for every x, gcd {t > 1 : P t (x, x) > 0} = 1;
Stationary distribution: π ∈ ∆(Ω), πP = π.
Irreducible + Aperiodic = Ergodic =⇒
=⇒ π is unique and P t (x, ·) → π
◮
Total variation distance µ, ν ∈ ∆(Ω)
kµ − νk = max |µ(A) − ν(A)| =
A⊆Ω
◮
1X
|µ(x) − ν(x)|
2
x∈Ω
Mixing Time
Framework Description
tmix (ε) = min{t ∈ N : kP t (x, ·) − πk 6 ε for all x ∈ Ω}
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Logit Dynamics
Definition
G = ([n], S, U). Logit dynamics for G is the Markov chain with
state space Ω = S1 × · · · × Sn and transition matrix
Framework Description
n
P(x, y) =
1 X e βui (x−i ,yi )
I
n
Ti (x) {yj =xj for every
j6=i }
i =1
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Logit Dynamics
Definition
G = ([n], S, U). Logit dynamics for G is the Markov chain with
state space Ω = S1 × · · · × Sn and transition matrix
n
P(x, y) =
1 X e βui (x−i ,yi )
I
n
Ti (x) {yj =xj for every
j6=i }
i =1
Logit Dynamics defines an ergodic Markov chain
◮
What is the stationary distribution π?
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Logit Dynamics
Definition
G = ([n], S, U). Logit dynamics for G is the Markov chain with
state space Ω = S1 × · · · × Sn and transition matrix
n
P(x, y) =
1 X e βui (x−i ,yi )
I
n
Ti (x) {yj =xj for every
j6=i }
i =1
Logit Dynamics defines an ergodic Markov chain
◮
What is the stationary distribution π?
◮
What is the stationary expected social welfare Eπ [W ]?
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Logit Dynamics
Definition
G = ([n], S, U). Logit dynamics for G is the Markov chain with
state space Ω = S1 × · · · × Sn and transition matrix
n
P(x, y) =
1 X e βui (x−i ,yi )
I
n
Ti (x) {yj =xj for every
j6=i }
i =1
Logit Dynamics defines an ergodic Markov chain
◮
What is the stationary distribution π?
◮
What is the stationary expected social welfare Eπ [W ]?
◮
How long it takes to get close to the stationary
distribution?
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Logit Dynamics
Some recent related works
[Montanari, Saberi, FOCS’09]:
Hitting time of the best Nash equilibrium in a classical “game”
(Ising model for ferromagnetism. Applications to the spread of
innovations in a network)
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Logit Dynamics
Some recent related works
[Montanari, Saberi, FOCS’09]:
Hitting time of the best Nash equilibrium in a classical “game”
(Ising model for ferromagnetism. Applications to the spread of
innovations in a network)
[Asadpour, Saberi, WINE’09]:
Hitting time of the neighborhood of best Nash equilibria for
Atomic Selfish Routing and Load Balancing.
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Logit Dynamics
Some recent related works
[Montanari, Saberi, FOCS’09]:
Hitting time of the best Nash equilibrium in a classical “game”
(Ising model for ferromagnetism. Applications to the spread of
innovations in a network)
[Asadpour, Saberi, WINE’09]:
Hitting time of the neighborhood of best Nash equilibria for
Atomic Selfish Routing and Load Balancing.
They study hitting time of Nash equilibria. We propose
stationary distribution as equilibrium concept.
Framework Description
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Logit Dynamics
Some recent related works
[Montanari, Saberi, FOCS’09]:
Hitting time of the best Nash equilibrium in a classical “game”
(Ising model for ferromagnetism. Applications to the spread of
innovations in a network)
[Asadpour, Saberi, WINE’09]:
Hitting time of the neighborhood of best Nash equilibria for
Atomic Selfish Routing and Load Balancing.
They study hitting time of Nash equilibria. We propose stationary
distribution as equilibrium concept.
Nash equilibria
Not Unique
Framework Description
Stationary distribution of logit dynamics
Unique
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Logit Dynamics
Some recent related works
[Montanari, Saberi, FOCS’09]:
Hitting time of the best Nash equilibrium in a classical “game”
(Ising model for ferromagnetism. Applications to the spread of
innovations in a network)
[Asadpour, Saberi, WINE’09]:
Hitting time of the neighborhood of best Nash equilibria for
Atomic Selfish Routing and Load Balancing.
They study hitting time of Nash equilibria. We propose stationary
distribution as equilibrium concept.
Nash equilibria
Not Unique
Local
Framework Description
Stationary distribution of logit dynamics
Unique
Global
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Example: Matching Pennies
Examples
◮
If both choose the same
side then Player 1 wins;
◮
If they choose different
sides then Player 2 wins
H
T
H
T
+1, −1 −1, +1
−1, +1 +1, −1
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Example: Matching Pennies
Examples
◮
If both choose the same
side then Player 1 wins;
◮
If they choose different
sides then Player 2 wins
H
T
H
T
+1, −1 −1, +1
−1, +1 +1, −1
No pure Nash, one mixed Nash σ1 = σ2 = (1/2, 1/2)
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Example: Matching Pennies
Examples
◮
If both choose the same
side then Player 1 wins;
◮
If they choose different
sides then Player 2 wins
H
T
H
T
+1, −1 −1, +1
−1, +1 +1, −1
No pure Nash, one mixed Nash σ1 = σ2 = (1/2, 1/2)
Logit dynamics
Example:
σ1 (H | (−, T )) =
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Example: Matching Pennies
Examples
◮
If both choose the same
side then Player 1 wins;
◮
If they choose different
sides then Player 2 wins
H
T
H
T
+1, −1 −1, +1
−1, +1 +1, −1
No pure Nash, one mixed Nash σ1 = σ2 = (1/2, 1/2)
Logit dynamics
Example:
σ1 (H | (−, T )) =
e βu1 (H,T )
e βu1 (H,T ) + e βu1 (T ,T )
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Example: Matching Pennies
Examples
◮
If both choose the same
side then Player 1 wins;
◮
If they choose different
sides then Player 2 wins
H
T
H
T
+1, −1 −1, +1
−1, +1 +1, −1
No pure Nash, one mixed Nash σ1 = σ2 = (1/2, 1/2)
Logit dynamics
Example:
σ1 (H | (−, T )) =
=
e βu1 (H,T )
e βu1 (H,T ) + e βu1 (T ,T )
e −β
e −β + e β
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Example: Matching Pennies
Examples
◮
If both choose the same
side then Player 1 wins;
◮
If they choose different
sides then Player 2 wins
H
T
H
T
+1, −1 −1, +1
−1, +1 +1, −1
No pure Nash, one mixed Nash σ1 = σ2 = (1/2, 1/2)
Logit dynamics
Example:
σ1 (H | (−, T )) =
=
e βu1 (H,T )
e βu1 (H,T ) + e βu1 (T ,T )
1
e −β
=
−β
β
e +e
1 + e 2β
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Matching Pennies
Logit dynamics
Examples


 HH
1
P= 
HT
2

 TH
TT
b=
1
.
1+e −2β
HH
HT
TH
TT
1
b
(1 − b)
0
(1 − b)
b
1
0
0
1
0
(1 − b)
b





b


(1 − b) 
1
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Matching Pennies
Logit dynamics
Examples

HH
HT
TH
TT

1
b
(1 − b)
0
 HH
1

P =  HT (1 − b)
1
0
b
2
b
0
1
(1 − b)
 TH
TT
0
(1-b)
b
1
b=
1
.
1+e −2β








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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Matching Pennies
Logit dynamics
Examples


 HH
1
P= 
HT
2

 TH
TT
b=
◮
HH
HT
TH
TT
1
b
(1 − b)
0
(1 − b)
b
1
0
0
1
0
(1 − b)
b
1
.
1+e −2β
π = 14 (1, 1, 1, 1)





b


(1 − b) 
1
Eπ [W ] = 0
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Matching Pennies
Logit dynamics
Examples


 HH
1
P= 
HT
2

 TH
TT
b=
◮
◮
HH
HT
TH
TT
1
b
(1 − b)
0
(1 − b)
b
1
0
0
1
0
(1 − b)
b
1
.
1+e −2β
π = 14 (1, 1, 1, 1)





b


(1 − b) 
1
Eπ [W ] = 0
kP 3 (x, ·) − πk < 1/2 so
tmix = O(1)
(upper bounded by a constant for every β)
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Logit Dynamics
Potential games
Examples
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)]
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Logit Dynamics
Potential games
Examples
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)]
Logit dynamics for potential games
The stationary distribution is the Gibbs one
π(x) =
e −βΦ(x)
Z
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Logit Dynamics
Potential games
Examples
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)]
Logit dynamics for potential games
The stationary distribution is the Gibbs one
π(x) =
e −βΦ(x)
Z
Observation [Blume’93]
For β → ∞ the stationary distribution π is concentrated over the
global minima of the potential function.
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Logit Dynamics
Potential games
Examples
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)]
Logit dynamics for potential games
The stationary distribution is the Gibbs one
π(x) =
e −βΦ(x)
Z
Observation [Blume’93]
For β → ∞ the stationary distribution π is concentrated over the
global minima of the potential function.
=⇒
opt(W )
→ Price of Stability
Eπ [W ]
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Chicken Game
Description
Examples
◮
If both STOP then none wins;
◮
If both PASS then both lose;
◮
If one PASSes and one STOP,
then who passes win.
S
P
S
P
(0, 0)
(0, 1)
(1, 0) (−1, −1)
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Chicken Game
Description
Examples
◮
If both STOP then none wins;
◮
If both PASS then both lose;
◮
If one PASSes and one STOP,
then who passes win.
S
P
S
P
(0, 0)
(0, 1)
(1, 0) (−1, −1)
Two pure Nash, one mixed Nash.
Potential game: stationary distribution is for free.
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Chicken Game
Description
Examples
◮
If both STOP then none wins;
◮
If both PASS then both lose;
◮
If one PASSes and one STOP,
then who passes win.
S
P
S
P
(0, 0)
(0, 1)
(1, 0) (−1, −1)
Two pure Nash, one mixed Nash.
Potential game: stationary distribution is for free.
Stationary expected social welfare
π(SS) = π(PP) =
π(SP) = π(PS) =
1
2(1+e β )
1
2(1+e −β )
Eπ [W ] =
eβ − 1
eβ + 1
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Chicken Game
Description
Examples
◮
If both STOP then none wins;
◮
If both PASS then both lose;
◮
If one PASSes and one STOP,
then who passes win.
S
P
S
P
(0, 0)
(0, 1)
(1, 0) (−1, −1)
Two pure Nash, one mixed Nash.
Potential game: stationary distribution is for free.
Stationary expected social welfare
π(SS) = π(PP) =
π(SP) = π(PS) =
1
2(1+e β )
1
2(1+e −β )
Eπ [W ] =
eβ − 1
eβ + 1
Observations
◮ Expected social welfare tends to 1 for β → ∞;
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
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Chicken Game
Description
Examples
◮
If both STOP then none wins;
◮
If both PASS then both lose;
◮
If one PASSes and one STOP,
then who passes win.
S
P
S
P
(0, 0)
(0, 1)
(1, 0) (−1, −1)
Two pure Nash, one mixed Nash.
Potential game: stationary distribution is for free.
Stationary expected social welfare
π(SS) = π(PP) =
π(SP) = π(PS) =
1
2(1+e β )
1
2(1+e −β )
Eπ [W ] =
eβ − 1
eβ + 1
Observations
◮ Expected social welfare tends to 1 for β → ∞;
◮ Expected social welfare is fair.
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Chicken Game
Mixing time
Examples
Mixing time is exponential in β
tmix = Θ(e β )
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Chicken Game
Mixing time
Examples
Mixing time is exponential in β
tmix = Θ(e β )
Intuition
◮
The two Nash equilibria have the same stationary probability;
◮
When β is large, it takes a long time to go from one Nash
equilibrium to the other one.
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Chicken Game
Mixing time
Examples
Mixing time is exponential in β
tmix = Θ(e β )
Intuition
◮
The two Nash equilibria have the same stationary probability;
◮
When β is large, it takes a long time to go from one Nash
equilibrium to the other one.
Generalization
Chicken Game is an anti-coordination game. The results extend to
all coordination and anti-coordination games.
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
OR-game
Definition
Examples
A trivial game with a non-trivial mixing time analysis.
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
OR-game
Definition
Examples
A trivial game with a non-trivial mixing time analysis.
Every player has two strategies, say
{0, 1}, and each player pays the OR
of the strategies of all players
(including herself).
ui (x) =
0
−1
if x = 0;
otherwise.
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
OR-game
Definition
Examples
A trivial game with a non-trivial mixing time analysis.
Every player has two strategies, say
{0, 1}, and each player pays the OR
of the strategies of all players
(including herself).
ui (x) =
0
−1
if x = 0;
otherwise.
Almost every profile is Nash equilibrium (except profiles with
exactly one 1)
W (best Nash eq.) = 0;
W (worst Nash eq.) = −n
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
OR-game
Definition
Examples
A trivial game with a non-trivial mixing time analysis.
Every player has two strategies, say
{0, 1}, and each player pays the OR
of the strategies of all players
(including herself).
ui (x) =
0
−1
if x = 0;
otherwise.
Almost every profile is Nash equilibrium (except profiles with
exactly one 1)
W (best Nash eq.) = 0;
W (worst Nash eq.) = −n
Expected social welfare
Eπ [W ] = −
(2n − 1)e −β
·n
1 + (2n − 1)e −β
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
OR-game
Definition
Examples
A trivial game with a non-trivial mixing time analysis.
Every player has two strategies, say
{0, 1}, and each player pays the OR
of the strategies of all players
(including herself).
ui (x) =
0
−1
if x = 0;
otherwise.
Almost every profile is Nash equilibrium (except profiles with
exactly one 1)
W (best Nash eq.) = 0;
W (worst Nash eq.) = −n
Expected social welfare
Eπ [W ] = −
(2n − 1)e −β
·n
1 + (2n − 1)e −β
Eπ [W ] → 0
for β → ∞
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
OR-game
Mixing time
Examples
tmix ≈

O(n log n)



if β < log n



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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
OR-game
Mixing time
Examples
tmix

O(n log n)



Θ(nc )
≈



if β < log n
if β = c log n, c > 1 constant
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
OR-game
Mixing time
Examples
tmix

O(n log n)



Θ(nc )
≈



Θ(2n )
if β < log n
if β = c log n, c > 1 constant
otherwise
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
OR-game
Mixing time
Examples
tmix
Intuition

O(n log n)



Θ(nc )
≈



Θ(2n )
if β < log n
if β = c log n, c > 1 constant
otherwise
◮
When β is large the stationary distribution is concentrated in
profile 0;
◮
From a profile with at least two 1’s, every players choose
u.a.r. over {0, 1}
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
OR-game
Mixing time
Examples
tmix
Intuition

O(n log n)



Θ(nc )
≈



Θ(2n )
if β < log n
if β = c log n, c > 1 constant
otherwise
◮
When β is large the stationary distribution is concentrated in
profile 0;
◮
From a profile with at least two 1’s, every players choose
u.a.r. over {0, 1}
Proof techniques:
◮
Path-coupling for the upper bound;
◮
Bottleneck ratio for the lower bound.
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Further and Future Investigations
◮
Logit dynamics for more interesting (class of) games;
Research Directions
◮
◮
Potential games: Mixing time depends on the shape of the
potential function
(Lipschitz conditions, Number of local minima, Maximum
Slope, . . . );
Stationary expected social welfare vs PoA / PoS for classical
games;
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Further and Future Investigations
◮
Logit dynamics for more interesting (class of) games;
◮
◮
◮
Potential games: Mixing time depends on the shape of the
potential function
(Lipschitz conditions, Number of local minima, Maximum
Slope, . . . );
Stationary expected social welfare vs PoA / PoS for classical
games;
Other randomized best response dynamics
Research Directions
◮
E.g. All players play simultaneously;
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Further and Future Investigations
◮
Logit dynamics for more interesting (class of) games;
◮
◮
◮
Other randomized best response dynamics
◮
◮
Potential games: Mixing time depends on the shape of the
potential function
(Lipschitz conditions, Number of local minima, Maximum
Slope, . . . );
Stationary expected social welfare vs PoA / PoS for classical
games;
E.g. All players play simultaneously;
Connections with other disciplines
Research Directions
◮
◮
Statistical Physics;
Evolutionary biology??
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
Further and Future Investigations
◮
Logit dynamics for more interesting (class of) games;
◮
◮
◮
Other randomized best response dynamics
◮
◮
E.g. All players play simultaneously;
Connections with other disciplines
◮
◮
◮
Potential games: Mixing time depends on the shape of the
potential function
(Lipschitz conditions, Number of local minima, Maximum
Slope, . . . );
Stationary expected social welfare vs PoA / PoS for classical
games;
Statistical Physics;
Evolutionary biology??
What happens when mixing time is exponential?
Research Directions
◮
◮
Chaotic behavior during transient phase?
Polynomially almost stationary distributions?
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
References
Vincenzo Auletta, Diodato Ferraioli, Francesco Pasquale, and Giuseppe
Persiano.
Mixing time and Stationary Expected Social Welfare of Logit Dynamics.
Submitted, 2010 (http://arxiv.org/abs/1002.3474).
Research Directions
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Mixing Time and Stationary Expected Social Welfare of Logit Dynamics
Roma, June 17, 2010
References
Vincenzo Auletta, Diodato Ferraioli, Francesco Pasquale, and Giuseppe
Persiano.
Mixing time and Stationary Expected Social Welfare of Logit Dynamics.
Submitted, 2010 (http://arxiv.org/abs/1002.3474).
Research Directions
Thank you!
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