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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

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.

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

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

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Examples: Benaım and Weibull (2001 unpublished)Williams (2002 unpublished)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))

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.

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.

Best response/mutations in a two-strategy coordination game

ε εNx* N(1 – x*)

Best response/mutations in a three-strategy coordination game

εNc εNd

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.

General exit paths in a three-strategy coordination game

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

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.

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

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.

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

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.

Example: A common interest game and its potential function

F(x) =

F1(x)F2(x)F3(x)

1 0 00 2 00 0 3

; f (x) = 12

((x1)2 + 2(x2)2 + 3(x3)2

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).

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.

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

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→∞

Then for all T < ∞ and ε > 0,

limN→∞

supt∈[0,T]

∣∣∣XNt − xt

∣∣∣ < ε = 1.

Figure: Finite horizon deterministic approximation.

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)]+

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).

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.

Example: A common interest game and its potential function

F(x) =

F1(x)F2(x)F3(x)

1 0 00 2 00 0 3

; f (x) = 12

((x1)2 + 2(x2)2 + 3(x3)2

Example: A common interest game and its logit(.2) potential function

F(x) =

1 0 00 2 00 0 3

; f (x) = 5 · 12((x1)2 + 2(x2)2 + 3(x3)2

xi log xi.

F(x) =

1 0 00 2 00 0 3

; f (x) = 35 ·

((x1)2 + 2(x2)2 + 3(x3)2

xi log xi.

Example: A common interest game and its logit(1.5) potential function

F(x) =

1 0 00 2 00 0 3

; f (x) = 23 ·

((x1)2 + 2(x2)2 + 3(x3)2

xi log xi.

(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.

The logit(.2) dynamic for F(x) =(

1 0 00 2 00 0 3

) ( x1x2x3

), plotted atop f η(x) = 5f (x) + h(x).

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)

CLT: limN→∞

P(YN∈ ( 1√

Na, 1√

= 1√

a exp(−y2

2 ) dy.

Now fix d > 0. What is the asymptotic behavior of P(YN ≥ d)?

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.

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

P(YN≥ 3) ≈ exp(−4.5 N)

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)).

P(YN≥ 3) ≈ exp(−1.6137 N).

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.

example: multivariate normal trials

Yi ∼ N(0, I)

H∗(v) = 12 |v|

⇒ P(YN∈ D) ≈ exp

(−N 1

∣∣∣d∗∣∣∣2) 1 2

2 h*(d*) =

P(YN∈ D) ≈ exp(−2N).

Sample path large deviations

Now consider the sequence of Markov chains {{XNt }}∞

Let {xt}t≥0 solve (M) with x0 = limN→∞

xN0 . We have seen that

WLLN: limN→∞

(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.

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

t }t∈[0,T′] ∈ B)≥ − inf

φ∈int(B)c0,T′ (φ);

(ii) lim supN→∞

1N logP

t }t∈[0,T′] ∈ B)≤ − inf

φ∈cl(B)c0,T′ (φ).

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.

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.

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.)

The time-reversed logit(.2) dynamic for F(x) =(

1 0 00 2 00 0 3

) ( x1x2x3

region where large deviations from x 3

are fully determined by “local” considerations

Global behavior over long time spans

We now use these large deviations estimates to study the globalbehavior of the process {XN

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.

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.

XNk+1τ

XNk+1σ

Defining ZNk and ZN

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.

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)

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 .

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 ).

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.)

F(x) =

1 0 00 2 00 0 3

f (x) = 5 · 12((x1)2 + 2(x2)2 + 3(x3)2

xi log xi.

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 .

F(x) =

1 0 00 2 00 0 3

f (x) = 5 · 12((x1)2 + 2(x2)2 + 3(x3)2

xi log xi.

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.

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.

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.

Finding the minimal cost trees (values of −f η shown.)

stable rest points (black) and saddles (green)

undirected edges between stable rest points

Kruskal’s algorithm (step 1)

the minimal spanning tree

the minimal cost x -tree

the minimal cost y -tree

the minimal cost z-tree

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 .

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).

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:

The paper

• Develops a collection of general results.

• Analyzes two settings in detail:(i) two-strategy games under arbitrary revision protocols.

large deviations, reversibility, and equilibrium selection ...whs/research/lepg.pres.pdf ·...

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accounting for entropy class 28.2 objectives qualitatively...

gemeinde waldbrunn ortsteil weisbach bebauungsplan...

reversibility for recoverability