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Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Large Deviations, Reversibility,and Equilibrium Selection
under Evolutionary Game Dynamics
Michel Benaım, Universite de Neuchatel
William H. Sandholm, University of Wisconsin
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Population Games, Revision Protocols,and Evolutionary Dynamics
Population games model strategic interactions among large numbersof small, anonymous agents.
To model the dynamics of behavior, we use revision protocols todescribe how myopic individual agents decide whether and whento switch strategies.
A population game F, a revision protocol σ, and a population size Ndefine a Markov process {XN
t } on the set of population states.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Deterministic and stochastic evolutionary dynamics: overview
There have been two main approaches to studying the process {XNt }.
1. If the population is large, then a suitable law of large numbersshows that over finite time spans, the evolution of aggregatebehavior is well approximated by solutions to an ODE, the meandynamic.
The behavior of these differential equations is the focus ofdeterministic evolutionary game theory.
Examples:• replicator dynamic (Taylor and Jonker (1978))
• best response dynamic (Gilboa and Matsui (1991))
• logit dynamic (Fudenberg and Levine (1998))
• BNN dynamic (Brown and von Neumann (1950))
• Smith dynamic (Smith (1984))
• many others
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Deterministic and stochastic evolutionary dynamics: overview
3. To study infinite horizon behavior, one instead looks at thestationary distributions of the original stochastic process.
The behavior of these stationary distributions is the focus ofstochastic evolutionary game theory.
When certain limits are taken (in the noise level ε, in the populationsize N), this approach can provide a unique prediction of verylong run behavior: the stochastically stable state.
(More discussion to follow.)
Examples:• best response w. mutations (Kandori et al. (1993), Young (1993))
• logit choice (Blume (1993))
• noisy imitation (Binmore and Samuelson (1997))
• others
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Deterministic and stochastic evolutionary dynamics: overview
2. For a middle way between the finite and infinite horizon analyses,one can consider large deviations analysis.
This describes the occasional excursions of the stochastic processagainst the flow of the mean dynamic.
Large deviations analysis provides a general foundation forstochastic stability analysis, and is of interest in its own right.
Examples: Benaım and Weibull (2001 unpublished)Williams (2002 unpublished)
(Related work in macro: Cho, Williams, and Sargent (2002)Williams (2009))
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Stationary Distributions and Stochastic Stability
In games with multiple equilibria, predictions generally depend onthe initial behavior of the system.
But suppose that
(i) if we are interested in very long time spans, and
(ii) there is noise is agents’ decisions.
Then we can base predictions on the unique stationary distributionµN of the Markov process {XN
t }, which describes the infinitehorizon behavior of the process.
When the noise level is small or the population size large, thisdistribution often concentrates its mass around a single state,which is said to be stochastically stable.
Intuition: Rare but unequally unlikely transitions between equilibria.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
The standard stochastic stability model: best responses and mutations(Kandori, Mailath, and Rob (1993), Young (1993))
Each agent occasionally receives opportunities to switch strategies.
At such moments, the agent plays a best response with probability1 − ε, chooses a strategy at random with probability ε.
Key assumption of the BRM model: the probability of any givensuboptimal choice is independent of the payoff consequences
⇒ Analysis of equilibrium selection via mutation counting:
To compute the probability of a transition between equilibria, countthe number of mutations required.
Then use graph-theoretic methods to determine the asymptotics ofthe stationary distribution.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Best response/mutations in a two-strategy coordination game
0 1
ε εNx* N(1 – x*)
x*
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Best response/mutations in a three-strategy coordination game
e*
e* e*
1
2 3
εNa
εNb
εNc εNd
εNe
εNf
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Mutations and mutation counting are analytically convenient.
But in many settings, one expects mistake probabilities to depend onpayoff consequences.
ex.: logit choice protocol (Blume (1993))
Lj(π) =exp(η−1πj)∑
k∈S exp(η−1πk)(η > 0 is the noise level)
Once mistake probabilities depend on payoffs, evaluatingthe probabilities and paths of transitions between basins ofattraction becomes much more difficult.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
General exit paths in a three-strategy coordination game
e*
e* e*
1
2 3
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Large deviations analysis of the stochastic evolutionary process
To describe transitions between basins of attraction, we usetechniques from large deviations theory to describe the behaviorof the process in the large population limit.
The analysis proceeds in three steps:
1. Obtain deterministic approximation via the mean dynamic (M).
2. Evaluate probabilities of excursions away from stable rest points of (M).
3. Combine (1) and (2) to understand global behavior:
• probabilities of transitions between rest points of (M)
• expected times until exit from general domains
• asymptotics of the stationary distribution & stochastic stability
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
The paper:
• Develops a collection of general results, building onDupuis (1986) and Freidlin and Wentzell (1998).
• Analyzes two settings in detail:
(i) two-strategy games under arbitrary revision protocols.
(ii) potential games under the logit choice protocol.
In both of these settings,
(a) large deviations can be described analytically,allowing a full description of local behavior.
(b) the process {XNt } is “reversible in the large”,
allowing a full description of global behavior.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
The Model
I. Population games
For simplicity, we consider games played by a single unit-masspopulation (e.g., in traffic networks, one origin/destination pair).
S = {1, . . . ,n} strategiesX = {x ∈ Rn
+ :∑
i∈S xi = 1} population states/mixed strategiesFi : X→ R payoffs to strategy iF : X→ Rn payoffs to all strategies
State x ∈ X is a Nash equilibrium if
xi > 0⇒[Fi(x) ≥ Fj(x) for all j ∈ S
]
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Example: Matching in (symmetric two-player) normal form games
A ∈ Rn×n payoff matrixAij = e′i Aej payoff for playing i ∈ S against j ∈ SFi(x) = e′i Ax total payoff for playing i against x ∈ XF(x) = Ax payoffs to all strategies
When A is symmetric (Aij = Aji), it is called a common interest game.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Example: Congestion games (Beckmann, McGuire, and Winsten (1956))
Home and Work are connected by paths i ∈ S consisting of links λ ∈ Λ.
The payoff to choosing path i is−(the delay on path i) = −(the sum of the delays on links in path i)
Fi(x) = −∑λ∈Λi
cλ(uλ(x)) payoff to path ixi mass of players choosing path iuλ(x) =
∑i: λ∈Λi
xi utilization of link λcλ(uλ) (increasing) cost of delay on link λ
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Potential games (Monderer and Shapley (1996), Sandholm (2001))
F is a potential game if it admits a potential function f : Rn→ R:
∇f (x) = F(x) for all x ∈ X (i.e., ∂ f∂xi
(x) = Fi(x)).
Example: Matching in common interest games
F(x) = Ax, A symmetric
f (x) = 12 x′Ax = 1
2 × average payoffs
Example: Congestion games
Fi(x) = −∑λ∈Λi
cλ(uλ(x))
f (x) = −∑λ∈Λi
∫ uλ(x)
0 cλ(v) dv , −∑λ∈Λi
uλ(x) cλ(uλ(x)) = average payoffs
Theorem: Solutions of deterministic dynamics satisfying (NS) and (PC)ascend the potential function and converge to Nash equilibria.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Example: A common interest game and its potential function
1
2 3
F(x) =
F1(x)F2(x)F3(x)
=
1 0 00 2 00 0 3
x1
x2
x3
; f (x) = 12
((x1)2 + 2(x2)2 + 3(x3)2
).
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
II. Revision protocols
Agents make decisions by applying a revision protocol.
σ : Rn× X→ Rn×n
+
σij(π, x) is the conditional switch rate from strategy i to strategy j
Example: The logit choice protocol (Blume (1993))
σij(π, x) = Lj(π) =exp(η−1πj)∑
k∈S exp(η−1πk).
Interpretation:
In each period of length 1N , a randomly chosen player receives
a revision opportunity.
If he is an i player, he switches to strategy j with probability σij(F(x), x).
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
III. The Markov chain
A population game F, a revision protocol σ, and a population size Ndefine a Markov chain {XN
t } on the finite state space
X N = X ∩ 1N Zn = {x ∈ X : Nx ∈ Zn
} .
The evolution of {XNt } is described recursively by
XNt+1/N = XN
t + 1Nζx,
where the random vector ζx represents a normalized incrementof {XN
t } from state x:
P(ζx = ej − ei) =
xiσij(x) if i , j,∑k∈S
xkσkk(x) if i = j.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Finite horizon deterministic approximation
How does the process {XNt } behave over time interval [0,T]
when N is large?
The mean dynamic generated by ρ and F is defined bythe expected increments of the Markov process:
(M) x = V(x) ≡ Eζx.
Expressed componentwise:
xi =∑j∈S
xjσji(F(x), x) − xi
= inflow into i − outflow from i
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Over finite time spans, the process is well-approximated by solutionstrajectories of the mean dynamic.
Theorem (Benaım and Weibull (2003)):
Let {{XNt }}∞
N=N0be a sequence of stochastic evolutionary processes.
Let {xt}t≥0 be the solution to the mean dynamic x = V(x) from x0 = limN→∞
xN0 .
Then for all T < ∞ and ε > 0,
limN→∞
P
supt∈[0,T]
∣∣∣XNt − xt
∣∣∣ < ε = 1.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
x0
x1
x2
x3
0X
1X
2X
3X
=
Figure: Finite horizon deterministic approximation.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Revision protocols and ev. dynamics: five fundamental examples
Revision protocol Evolutionary dynamic Name
ρij = xj[πj − πi]+ xi = xi(Fi(x) − F(x)
)replicator
ρi· ∈ argmaxy∈X
y′π(x) x ∈ argmaxy∈X
y′F(x) − x best response
ρij =exp(η−1πj)∑
k∈S exp(η−1πk)xi =
exp(η−1Fi(x))∑k∈S exp(η−1Fk(x))
− xi logit
ρij = [πj]+ xi = m [Fi(x)]+ − xi∑j∈S
[Fj(x)]+ BNN
ρij = [πj − πi]+
xi =∑j∈S
xj[Fi(x) − Fj(x)]+
−xi∑j∈S
[Fj(x) − Fi(x)]+
Smith
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Evolution under the logit choice protocol in potential games
Let σ be the logit(η) choice protocol:
σij(π, x) =exp(η−1πj)∑
k∈S exp(η−1πk).
The mean dynamic generated by this protocol is the logit dynamic:
(L) xi =exp(η−1Fi(x))∑
k∈S exp(η−1Fk(x))− xi.
The rest points of the logit dynamic are called logit equilibria.
Let F : X→ Rn be a potential game with potential function f : X→ R:
F(x) = ∇f (x).
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
The logit potential function
The logit(η) potential function is defined by
f η(x) = η−1f (x) + h(x),
where h(x) = −∑i∈S
xi log xi is the entropy function.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Example: A common interest game and its potential function
1
2 3
F(x) =
F1(x)F2(x)F3(x)
=
1 0 00 2 00 0 3
x1
x2
x3
; f (x) = 12
((x1)2 + 2(x2)2 + 3(x3)2
).
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Example: A common interest game and its logit(.2) potential function
F(x) =
1 0 00 2 00 0 3
x1
x2
x3
; f (x) = 5 · 12((x1)2 + 2(x2)2 + 3(x3)2
)−
∑i
xi log xi.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Example: A common interest game and its logit(.6) potential function
F(x) =
1 0 00 2 00 0 3
x1
x2
x3
; f (x) = 35 ·
12
((x1)2 + 2(x2)2 + 3(x3)2
)−
∑i
xi log xi.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Example: A common interest game and its logit(1.5) potential function
F(x) =
1 0 00 2 00 0 3
x1
x2
x3
; f (x) = 23 ·
12
((x1)2 + 2(x2)2 + 3(x3)2
)−
∑i
xi log xi.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
(L) xi =exp(η−1Fi(x))∑
k∈S exp(η−1Fk(x))− xi.
f η(x) = η−1f (x) −∑i∈S
xi log xi. logit(η) potential function
Observation:
The logit(η) equilibria (≡ rest points of (L)) are the critical points of f η.
Theorem (Hofbauer and Sandholm (2007)):
f η is a strict Lyapunov function for the logit dynamic (L).
Thus, all solutions of (L) converge to logit equilibria.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
1
2 3
The logit(.2) dynamic for F(x) =(
1 0 00 2 00 0 3
) ( x1x2x3
), plotted atop f η(x) = 5f (x) + h(x).
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Large deviations: a brief introduction
Let {Yi}Ni=1 be a sequence of i.i.d., mean 0, variance 1 random variables.
Let YN = 1N
∑Ni=1 Yi be their sample mean.
Two basic results about the distribution of YN:
WLLN: limN→∞
P(YN ∈ (−ε, ε)
)= 1.
CLT: limN→∞
P(YN∈ ( 1√
Na, 1√
Nb))
= 1√
2π
∫ b
a exp(−y2
2 ) dy.
Now fix d > 0. What is the asymptotic behavior of P(YN ≥ d)?
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Cramer’s Theorem:
There is a convex function h∗ : R→ R∗+ with h∗(0) = 0 such that
− limN→∞
1N logP(YN
≥ d) = h∗(d).
In words: h∗(d) is the exponential rate of decay of P(YN ≥ d).
P(YN ≥ d) ≈ exp(−N h∗(d)).
The function h∗, the Cramer transform of Yi, is defined as follows:
Let h(u) = logE[exp(uYi)] be the log m.g.f. of Yi.
Then h∗(v) = supu
uv − h(u) is the Legendre transform of h.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
example: normal trials
Suppose Yi ∼ N(0, 1).
h(u) = logE[exp(uYi)] = 12 u2
h∗(v) = supu uv − h(u) = 12 v2
⇒ P(YN ≥ d) ≈ exp(−N · 12 d2)
4 2 2 4
2
4
6
8
d
h*(d)
0 6
P(YN≥ 3) ≈ exp(−4.5 N)
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
example: demeaned exponential trials
Suppose Yi ∼ “exp(1) − 1”.
h(u) = logE[exp(uYi)] = −u−log(1−u)
h∗(v) = supu uv − h(u) = v − log(v + 1)
⇒ P(YN ≥ d) ≈ exp(−N·(d+log(d+1)).
0 2
2
4 6
4
6
d
h*(d)
P(YN≥ 3) ≈ exp(−1.6137 N).
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Large deviations in Rn
Let {Yi}Ni=1 be a sequence of i.i.d., mean 0 random vectors
whose log m.g.f. h(u) = logE[exp(u′Yi)] exists.
Let YN = 1N
∑Ni=1 Yi be their sample mean.
Cramer’s Theorem in Rn:There is a convex function h∗ : Rn
→ R∗+ with h∗(0) = 0 such that
− limN→∞
1N logP(YN
∈ D) = mind∈D
h∗(d)
for all nice sets D ⊂ Rn, where again h∗(v) = supu u′v − h(u).
Thus the exponential rate of decay of P(YN∈ D) is H∗(d∗),
where d∗ is the “most likely” outcome in D.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
example: multivariate normal trials
Yi ∼ N(0, I)
H∗(v) = 12 |v|
2.
⇒ P(YN∈ D) ≈ exp
(−N 1
2
∣∣∣d∗∣∣∣2) 1 2
1 2
2 h*(d*) =
D d*
4 8
P(YN∈ D) ≈ exp(−2N).
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Sample path large deviations
Now consider the sequence of Markov chains {{XNt }}∞
N=N0.
Let {xt}t≥0 solve (M) with x0 = limN→∞
xN0 . We have seen that
WLLN: limN→∞
P
(maxt∈[0,T]:
∣∣∣XNt − xt
∣∣∣ < ε) = 1.
analogy: realizations of YN ≈ sample paths of {XNt }
We now consider large deviations of {XNt } from solutions of (L).
To measure the “unlikelihood” of path φ , we add up the“unlikelihoods” of each “step” along the path.
To assess the probability of moving from state x to state y,we must find the “most likely” path from x to y.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Large deviations analysis
Recall: ζx is the normalized increment of {XNt } from x.
x = Eζx is the mean dynamic.
The Cramer transform of ζx is
H∗(x, v) = supu∈TX
(u′v − logE exp(u′ζx)
).
The path cost function adds up the unlikelihoods the steps on path φ:
cT,T′ (φ) =
∫ T′
TH∗(φt, φt) dt
Theorem: Large deviations principle (Dupuis (1986))
For any Borel set B ⊆ Cx[0,T′],
(i) lim infN→∞
1N logP
({XN
t }t∈[0,T′] ∈ B)≥ − inf
φ∈int(B)c0,T′ (φ);
(ii) lim supN→∞
1N logP
({XN
t }t∈[0,T′] ∈ B)≤ − inf
φ∈cl(B)c0,T′ (φ).
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Exit from basins of attraction
The transition cost function describes the unlikelihoodof the most likely path from x to y:
C(x, y) = inf{cT,T′ (φ) : φT = x, φT′ = y, and [T,T′] ⊂ (−∞,∞)
}Lemma: If there is a solution of (M) leading from x to y, then it is an
extremal yielding C(x, y) = 0.
Theorem (Exit from basins of attraction):
Let x be an asymptotically stable rest point of (M).
Let y = argminz∈bd(basin(x ))
C(x , z). If play begins near x , then for large enough N :
(i) Escape from basin(x ) occurs near y with probability close to 1.
(ii) The expected escape time is of order exp(N C(x , y )).
(iii) The realized escape time is of order exp(N C(x , y )) with prob. close to 1.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Computing C(x, y) means solving a multivariatecalculus of variations problem.
There are two cases in which we can solve this problem analytically:
(i) two-strategy games under arbitrary revision protocols.
(ii) potential games under the logit choice protocol.
Example: logit choice in potential games
Proposition: For any path φ ∈ C[T,T′] from x to y, we havecT,T′ (φ) ≥ f η(x) − f η(y). Thus C(x, y) ≥ f η(x) − f η(y).
Proposition: If there is a solution of (L) leading from y to x, then itstime-reversed version is an extremal yielding C(x, y) = f η(x) − f η(y).
(Proofs via the Hamilton-Jacobi equation and the canonical Euler equation.)
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
1
2 3
The time-reversed logit(.2) dynamic for F(x) =(
1 0 00 2 00 0 3
) ( x1x2x3
).
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
1
2 3
region where large deviations from x 3
are fully determined by “local” considerations
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Global behavior over long time spans
We now use these large deviations estimates to study the globalbehavior of the process {XN
t }:
• probabilities of transitions between rest points of (M)
• expected times until exit from general domains
• asymptotics of the stationary distribution & stochastic stability
Our results for the general model are adapted from Freidlin andWentzell’s (1998) analysis of ODEs with Brownian perturbations.
Maintained assumption:
(F) The mean dynamic (M) has finitely many ω-limit points.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
The auxiliary Markov chain
We begin by considering an auxiliary Markov chain {ZNk }.
It records the position of {XNt } at times τk when it is near a rest point
after having been away from all rest points.
XNkτ
XNk+1τ
XNk+1σ
Defining ZNk and ZN
k+1.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Theorem: Let x and y be rest points of (M)
Suppose that n ≥ 3 (or that n = 2 and that x and y are adjacent).
Let PN denote the transition law of {ZNk }.
With appropriate choices/order of quantifiers, for large enough N,
−1N log PN(x ,Bε(y )) ≈ C(x , y ).
This result describes the probabilities of all transitions from a stablerest point, not just the one to the “closest” saddle point.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Asymptotics of the stationary distribution
Let R∗ be the set of asymptotically stable states of (M)
A directed tree with tx root x ∈ R∗ (an x -tree) is a directed graph withno outgoing edges from x , exactly one outgoing edge from eachy , x , and a unique path from each y , x to x .
For x ∈ R∗, I(x ) is the minimal cost of any x -tree, where the cost of atree is the sum of the transition costs of its edges:
I(x ) = mintx ∈Tx
C (tx ), where C (tx ) =∑
(y ,z)∈tx
C(y , z).
For other states y ∈ X − R∗,
I(y) = minx ∈R∗
(I(x ) + C(x , y)
).
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Shift the graph of I vertically so that its minimum value is 0.
∆I(x) = I(x) −miny∈R∗
I(y ).
Theorem: With appropriate choices/order of quantifiers, for large enough N,
1N logµN(Bε(x)) ≈ −∆I(x).
Thus, if x ∈ R∗ is the unique minimizer of I, then µN⇒ δx .
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Mean exit times from general domains
Let R∗D ⊂ R∗ be such that D is connected, where
D = int(⋃
x ∈R∗D cl(basin(x ))).
What is the expected time EτN required to exit D from some x ∈ R∗D?
Let ID(∂D) = mint∈T∂D
∑(y ,z)∈t
CD(y , z).
JD(x ) = min
∑
(y ,z)∈gCD(y , z) : g ∈ G∂D⊕x ∪
⋃y∈R∗D−{x }
G x y∂D⊕y
.Theorem: If play begins near x ∈ R∗D, the expected time to exit D satisfies
limN→∞
1N logEτN = ID(∂D) − JD(x ).
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Global behavior: the logit/potential case
Asymptotics of the stationary distribution
A natural conjecture
µN(x) ≈ exp(N ∆f η(x)), where ∆f η(x) = f η(x) −maxy∈X
f η(y).
(Compare Blume (1997) on logit choice in normal form potential games.)
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Example: A common interest game and its logit(.2) potential function
F(x) =
1 0 00 2 00 0 3
x1
x2
x3
f (x) = 5 · 12((x1)2 + 2(x2)2 + 3(x3)2
)−
∑i
xi log xi.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Intuition: passage through saddles and reversibility in the large
Focus on the set of stable rest points R∗.
For x , y ∈ R∗, define the saddle point sx ,y as follows:
Bx ,y = cl(basin(x )) ∩ cl(basin(y ));
sx ,y = argmaxz∈Bx ,y
f η(z).
Evidently, sx ,y = sy ,x .
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Example: A common interest game and its logit(.2) potential function
F(x) =
1 0 00 2 00 0 3
x1
x2
x3
f (x) = 5 · 12((x1)2 + 2(x2)2 + 3(x3)2
)−
∑i
xi log xi.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Intuition: passage through saddles and reversibility in the large
Recall that a Markov chain is reversible if
µ(x ) P(x , y ) = µ(y ) P(y , x ) for all x , y .
Summing over x shows that µ is a stationary distribution.
In the logit/potential model, since sx ,y = sy ,x , we can verify that
µN(x ) PN(x , y ) = exp(N ∆f η(x )) exp(−N C(x , y )))
= exp(N ∆f η(x )) exp(−N (f η(x )) − f η(sx ,y ))
= exp(N ∆f η(y )) exp(−N (f η(y )) − f η(sx ,y ))
= exp(N ∆f η(y )) exp(−N C(y , x )))
= µN(y ) PN(y , x ).
This intuition does not lead directly to a proof.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Optimal z-trees via minimal spanning trees
Focus on the stable rest points R∗.
Suppose it is known that if an optimal x -tree tx contains an edgefrom y to z , then y and z are adjacent, and C(y , z) = f η(y ) − f η(sy ,z ).(This cannot be shown in isolation.)
Then C (tx ) =∑
(y ,z)∈tx
C(y , z)
=∑
(y ,z)∈tx
(f η(y ) − f η(sy ,z )
)=
∑y∈R∗
f η(y ) − f η(x ) +∑
(y ,z)∈tx
(−f η(sy ,z )
).
Thus the sole determinants of C (tx ) are
(i) the identity of the root x ;
(ii) the saddles through which the edges pass.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
C (tx ) =∑y∈R∗
f η(y ) − f η(x ) +∑
(y ,z)∈tx
(−f η(sy ,z )
)The sole determinants of C (tx ) are
(i) the identity of the root x ;
(ii) the saddles through which the edges pass.
Because of this, to find all of the minimum cost x -trees, we need onlyfind a single minimum cost spanning tree problem.
In other words: rather than many optimal directed graphs, we findjust one optimal undirected graph.
We accomplish the latter using Kruskal’s algorithm:always add the lowest cost edge that does not introduce a cycle.
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Finding the minimal cost trees (values of −f η shown.)
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stable rest points (black) and saddles (green)
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Finding the minimal cost trees (values of −f η shown.)
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undirected edges between stable rest points
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Finding the minimal cost trees (values of −f η shown.)
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Kruskal’s algorithm (step 1)
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Kruskal’s algorithm (step 2)
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Kruskal’s algorithm (step 3)
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Kruskal’s algorithm (step 4)
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Kruskal’s algorithm (step 5)
the minimal spanning tree
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x
the minimal cost x -tree
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Finding the minimal cost trees (values of −f η shown.)
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y
the minimal cost y -tree
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z
the minimal cost z-tree
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Let t ∗ be the minimal cost spanning tree. Then
I(x ) = C (tx ) =∑y∈R∗
f η(y ) − f η(x ) +∑
(y ,z)∈t ∗
(−f η(sy ,z )
)∆I(x ) = I(x ) −min
y∈R∗I(y ) = −f η(x ) + max
y∈R∗f η(y ) = −∆f η(x )
Further arguments show that ∆I(x ) = −∆f η(x ) for x < R∗.
Theorem: With appropriate choices/order of quantifiers, for large enough N,
1N logµN(Bε(x)) ≈ ∆f η(x).
Thus, if x ∈ R∗ is the unique maximizer of f η, then µN⇒ δx .
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Worst-case expected exit times
Let R∗D ⊂ R∗ and D = int(⋃
x ∈R∗Dcl(basin(x ))
).
What is the worst-case expected time to exit D?
Define E(D) = maxx ∈X∗D
miny∈X∗−X∗D
minχ∈Cxy [0,1]
(f η(x ) − min
t∈[0,1]f η(χ(t))
).
In words: For each x ∈ R∗D, find the smallest “drop” in the value of f η
on any path connecting x to ∂D.
Then maximize this minimum drop over x ∈ R∗D.
Theorem: Let τD be the time at which {XNt } hits ∂D. Then
maxx∈D
1N logExτ
N = E(D).
Longwinded introduction The model Mean dynamics Large deviations 1 Large deviations 2 Global behavior 1 Global behavior 2 End
Summary
The large deviations analysis of {XNt } proceeds in three steps:
1. Obtain deterministic approximation via the mean dynamic (M).
2. Evaluate probabilities of excursions away from stable rest points of (M).
3. Combine (1) and (2) to understand global behavior:
• probabilities of transitions between rest points of (M)
• expected times until exit from general domains
• asymptotics of the stationary distribution & stochastic stability
The paper
• Develops a collection of general results.
• Analyzes two settings in detail:(i) two-strategy games under arbitrary revision protocols.
(ii) potential games under the logit choice protocol.