gains in evolutionary dynamics€¦ · set up j population games gains in evolutionary dynamics...

Gains in evolutionary dynamics

unifying rational framework for dynamic stability

Dai ZUSAI

Hitotsubashi University (Econ Dept.)Economic Theory Workshop October 27, 2016

1 Introduction

Motivation and literature Review

Outline of this paper

2 Set up

Population games

Evolutionary dynamics

3 Properties

Basic properties

Gains

4 Equilibrium stability

Stability theorems

Extensions

Gains in evolutionary dynamics | Zusai | 2

Abstract

We investigate gains from strategy revisions in deterministic evolutionary dynamics.

• To clarify the gain from revision, we propose a framework to reconstruct an evolutionarydynamic from optimal decision with stochastic (possibly restricted) available action setand switching cost. Many of major non-imitative dynamics can be constructed in thisframework.

• We formally define net gains from revisions and obtain several general properties ofthe gain function,which leads to Nash stability of contractive games—generalization of concave potentialgames—and local asymptotic stability of a regular evolutionary stable state.

• The unifying framework allows us to apply the Nash stability to mixture ofheterogeneous populations, whether heterogeneity is observable or unobservable orwhether heterogeneity is in payoffs or in revision protocols.

• This extends the known positive results on evolutionary implementation of socialoptimum through Pigouvian pricing to the presence of heterogeneity and non-aggregatepayoff perturbations.

• While the analysis here is confined to general strategic-form games, we finally discussthat the idea of reconstructing evolutionary dynamics from optimization with switchingcosts and focusing on net revision gains for stability is promising for further applicationsto more complex situations.

Keywords: evolutionary dynamics, equilibrium stability, Lyapunov function,contractive/negative definite/ stable games, passivity, evolutionary implementation

Introduction Gains in evolutionary dynamics | Zusai | 3

1 Introduction

Motivation and literature Review

Outline of this paper

2 Set up

3 Properties


Introduction | Motivation Gains in evolutionary dynamics | Zusai | 4


In (deterministic) evolutionary game theory, westudy various evolutionary dynamics• These differential equations define different

dynamicsof the social state (distribution of actions)in large population (continuum of agents)in continuous time.

• We investigate off-equilibrium processes in whichagents occasionally switch their actions inresponse to the change in the social state and thepayoffs.

• The variety reflects a variety in economic agents’decision rules: optimization, imitation, etc.

• In the study of deterministic dynamics, weexamine if a well-known equilibrium concept (e.g.Nash, ESS) can be justified by dynamic stabilityunder these dynamics.


In search for Lyapunov...

Different evolutionary dynamics may behave differently;in particular, they might yield different results on equilibrium stability.

Theorem Lyapunov stability theorem Zusai (2015)

Let A be a closed subset of a compact space X and A′ be a neighborhood of A. Supposethat two continuous functions W : X → R and W : X → R satisfy (i) W(x) ≥ 0 andW(x) ≤ 0 for all x ∈ X and (ii) W−1(0) = W−1(0) = A. In addition, assume that W isLipschitz continuous in x ∈ X with Lipschitz constant K ∈ (0, ∞). If any Caratheodorysolution {xt} starting from A′ satisfies

W(xt) ≤ W(xt) for almost all t ∈ [0, ∞),then A is asymptotically stable and A′ is its basin of attraction.

• I call W the Lyapunov function and W the decaying rate function.• This is a general version for a differential inclusion (set-value dynamic to allow multiple transition

vectors: e.g. best response dynamic).• If it is a differential equation, it is sufficient to verify that[

W(x) ≥ 0 for all x ∈ A′]

and[W(x) = 0 ⇔ x ∈ A

].

No need to specify the decaying rate function W.

In general, a Lyapunov function may vary with gamesand it also depends on specification of the dynamic.


Nash stability of potential games

In potential games, stability of Nash eqm is commonly guaranteed for major dynamicsthanks to positive correlation.

Definition Potential game Sandholm (2001)

A population game F : ∆A → RA is called a (full) potential game if ∃ a C1-functionf : RA → R s.t.

∂ f∂xa

(x) = Fa(x)− F(x) for all a ∈ A, i.e.,[

d fdx

(x)]T

= F(x)− F(x)1.

F: payoff function, A = {1, ·, A}: action set, ∆A: simplex over A, x: action distributionxa: mass of action-a players, Fa(x) payoff from a, F(x) = x · F(x): average payoff.

• Equivalent to externality symmetry: ∂Fa/∂xb(x) = ∂Fb/∂xa(x) for alla, b ∈ A, x ∈ ∆A.

• e.g. Congestion games with payoff=−traveling time; Binary action games

Major evolutionary dynamics (BRD, pairwise payoff comparison, etc.) satisfypositive correlation:

z · F(x) ≥ 0 ∀z ∈ V(x), = 0 ⇔ x ∈ NE(F).Then, local max of f in ∆A ⇒ a Nash eqm, and

f = (d f /dx)x = F(x)x ≥ 0 by positive correlation.Potential game: immune to small changes in payoffs that break externality symmetry.


Nash stability of contractive gameContractive/stable game: a generalization of a concave potential game

no scalar-valued summary measure of the state (action distribution) of the game.

Definition Contractive game Hofbauer & Sandholm (2009, JET)

A population game F is a contractive game if(y− x) · (F(y)− F(x)) ≤ 0 for all x, y ∈ ∆A.

or (if F is C1) equivalent to z · dFdx

(x)z ≤ 0 for any z ∈ RA.

• e.g. Congestion games with payoff heterogeneity; Two-player zero sum games;War of attrition

• (regular) ESS: local version (see later).HS verified Nash stability under major dynamics by finding a Lyapunov function for each.


δ-passivity

Is there any general condition to guarantee Nash stability of contractive games??© General economic intuition behind this commonality of Nash stability over different dynamics.?© Robustness to perturbation/mis-identification of dynamics (individuals’ decision rules).

Definition δ-passivity Micheal J. Fox & Jeff Shamma (2013, Games)

Evolutionary dynamic x = V(x, π) is δ-passive, if there exists a storage function L s.t.∂L∂x

∆x +∂L∂π

∆π ≤ ∆x · ∆π for any ∆π (with ∆x = V(x, π)).

If we allow differential inclusion, a decay function W should be added to the RHS.If the game F is contractive and the dynamic V is δ-passive, then we have

L =∂L∂x

x +∂L∂π

π ≤ x · π = x · dFdx

(x)x ≤ 0(

with π =dFdx

(x)x)

.

Taking L as a Lyapunov function, we can confirm global stability of L−1(0).

δ-passivity points out exactly the property that should be satisfied for evolutionarydynamics, separately from games, in order to guarantee Nash stability of contractivegames.

But, it looks purely mathematical. (Passivity is originally a concept in electric engineering.)?© What behavioral/economical aspects of those well-behaved dynamics result in δ-passivity??© Isn’t there any general principle to find a storage function?

Introduction | Outline of this paper Gains in evolutionary dynamics | Zusai | 9

Gains from revision

Basic idea: Track the gain from revision, i.e.,Gross gain=[payoff from new action]−[payoff from old action].

• At Nash eqm, “gain”= 0 for all agents.• If not at Nash eqm, “gain”> 0 for some agents.

However, except BRD, agents do not choose an action to achieve the greatest payoff.e.g. Pairwise comparison dynamics (Smith dynamic):switch to a sampled action as long as it is better than the current action

So, agents may leave some of possible gains not wholly exploited;Wouldn’t it be possible that the gain may not steadily decrease over timebefore reaching eqm or may not vanish to zero in some games?

Indeed so, if it is not a potential game.

We need to rationalize such irrationality and refine the notion of “gains from revision.”

Introduction | Outline of this paper Gains in evolutionary dynamics | Zusai | 10

Net gains from revision

To rationalize the irrationality, I reconstruct an evolutionary dynamicby optimization protocolwith randomly limited available action set and stochastic switching costs.

That is, a revising agent is supposedto have a random draw of the available set and the switching costand then to choose{

the best among the available actions if the gross gain >switching costkeeping the current action if the gross gain <switching cost.

Then, the net gain is defined asNet gain=[payoff from new action]−[payoff from old action]−[switching cost].

The aggregate gain function works generally as a scalar-valued function to measure howfar the current state is from an eqm. We use it to prove δ-passivity of such a dynamic andNash stability in contractive games; this extends to local stability of a (regular) ESS.

Generality of our idea allows to make it robust to any mixture of such dynamics (whetherheterogeneity is observable or not).

Set up Gains in evolutionary dynamics | Zusai | 11

1 Introduction

2 Set up

Population games


3 Properties


Set up | Population games Gains in evolutionary dynamics | Zusai | 12

Population game

For now, start from a single-population game.• A continuum of agents; total mass=1• Each agent chooses an action from A =: {1, . . . , A}.• x ∈ ∆A: the action distribution (the social state). ∆A := {x ∈ RA | ∑a∈A xa = 1}.• F : ∆A → RA: payoff function. Fa(x) ∈ R: payoff of action a in state x.

Assumption Assumption on the game(A-F1) F is continuously differential.

Set up | Evolutionary dynamics Gains in evolutionary dynamics | Zusai | 13

Evolutionary dynamics and revision protocol

Standard formulation of an evolutionary dynamic Sandholm 2010

• Each agent occasionally receives an opportunity to revise the actionRevision opportunities follow a Poisson process; the arrival rate is fixed at 1.

• An agent uses a certain revision protocol to choose a new action, depending on thecurrent payoff vector π∈ RA and the social state x ∈ ∆A.

• Evolutionary dynamic x ∈ V(x) is obtained as the aggregation of such individualrevision processes in large population.

e.g. BRD: optimization revision protocol: pick an action that yields the greatest payoffe.g. Smith: randomly pick another action b with equal probability

and switch to it (from a) with prob proportional to the payoff difference, as longas b is better than a.


Reconstruction of evolutionary dynamics from optimization

To rationalize deviation from exact optimization, I propose to reconstruct evolutionarydynamics from optimization protocol with switching cost and restricted action set.

General reconstruction by optimization-based revision protocol• Upon receipt of a revision opportunity, a revising agent compares payoffs from the

current action a and from other available actions.• The available action set A′a is a subset of A \ {a}.• If the revising agent chooses to switch an action, he needs to pay switching cost q.

The switching cost is common to all the available actions.• Given payoff vector π ∈ RA, the revising agent chooses an action to optimize his net

payoff within the available set:max{πa, max

b∈A′aπb − q}

N.B. In other words, the current action a is status quo in the decision of new action and q is the bias toprefer the status quo.

Allowing the randomness• A′a is drawn from from probability distribution PAa over the power set of A \ {a}.• q is drawn from probability distribution PQ over R.• Assume that these random draws are independent of each other, the agent’s current

action and the social state and also their probability distributions are invariant over time.


Dynamics and Examples

Va: change in the action distribution (mixed str) of agents who have been taking a so far:Va[π] := ∑

A′a⊂A\{a}PAa[A′a]Q(π∗[A′a]− πa)(∆A(b∗[π; A′a])− ea),

where b∗ [π; A′a] = argmaxa∈A′aπa and ∆A(A′) = {x ∈ ∆A | xa > 0⇒ a ∈ A′}.

V : the dynamic of the social state, induced from the constrained optimization protocol:x ∈ V [π] := ∑

a∈AxaVa[π].

When the payoff function is specified as F, we denote V [F(x)] by VF(x).• Standard BRD: always switch to the current optimal action

⇐ PAa(A \ {a}) = 1 for any a ∈ A and Q(q) = 1 for all q > 0 while Q(0) = 0.• Tempered BRD: switch to the current optimal action with prob Q(π∗ − πa)

⇐ PAa(A \ {a}) = 1 for any a ∈ A and this Q as c.d.f. of PQ.• Symmetric pairwise payoff comparison: randomly pick another action b and switch to

it with prob Q([πb − πa]+) (Smith: Q(π) = π).⇐ PAa({b}) = 1/(A− 1) for all b ∈ A \ {a} and this Q.

• Smoothed BRD (e.g. logit dyn) can be seen as aggregate BRD under payoff heterogeneity.• Excess payoff target dynamics (e.g. Brown-von Neumann-Nash dyn) do not fall into this framework

as it is. But, the protocol can be interpreted as a constrained optimization protocol by taking themixed strategy x (random assignment based on the current action distribution) as the status quoand charging the cost on switching from this status quo. Then, the gain function satisfies theproperties that we will see.


Assumptions

Assumption Assumptions on the revision protocol(A-Q1) Nonnegativity of q:

Q(0) = PQ((−∞, 0)) = 0, i.e., q ≥ 0 almost surely.(A-Q2) Infinitesimally small positive q: ∀q > 0,

Q(q) = PQ((−∞, q)) > 0.(A-A1) Ex ante availability of all actions: ∀a ∈ A, b ∈ A \ {a},

PAa( {A′a | b ∈ A′a}︸︷︷︸b is switchable from a

) > 0

(A-A2) Symmetry of available action set distribution: ∀a, b ∈ A, A0 ⊂ A \ {a, b},PAa({A′a | A′a ∩ A0 6= ∅}︸︷︷︸

∃an action in A0

switchable from a

) = PAb({A′b | A′b ∩ A0 6= ∅}︸︷︷︸∃an action in A0

switchable from b

).

(A-A3) Invariance of A′a to the social state:PAa is independent of x.

The first mentioned classes of unobservatory dynamics (BRD, pairwise payoff comparison)satisfy these assumptions.The second (perturbed BRD, excess payoff) need some modifications.Observatory dynamics (imitative dynamics like replicator) do not meet A3.

Properties Gains in evolutionary dynamics | Zusai | 17

1 Introduction

2 Set up

3 Properties

Basic properties

Gains


Properties | Basic properties Gains in evolutionary dynamics | Zusai | 18

BR stationarity and positive correlation

Let b∗[π] = argmaxa∈A πa, the set of optimal actions given payoff vector π.

Theorem Best response/Nash stationarityAssume Assumptions Q1, Q2 and A1. Then,

i) x ∈ ∆A(b∗[π]) =⇒ V(x)[π] = {0}; ii) x /∈ ∆A(b∗[π]) =⇒ 0 /∈ V(x)[π].If F is C1, these imply Nash stationarity: VF(x) = {0} ⇐⇒ x ∈ NE(F).

Theorem Positive correlation and Nash stability of potential gamesAssume Assumptions Q1, Q2 and A1. Then,

i) ∆x ·π ≥ 0 for any ∆x ∈ V(x)[π];

ii)[

∆x ·π = 0 with some ∆x ∈ V(x)[π]]

⇔ x ∈ ∆A(b∗[π]).If F is a potential game, these imply that i) NE(F) is globally attracting,and ii) each local maximizer of f is Lyapunov stable.

• Nash stability of potential games is immediate from positive correlation:

f =d fdx· x = F(x) · x ≥ 0.

• Note that ∆x ·π is the aggregate gross gain:Let ∆x ∈ ∑a∈A xa(ya − ea) where ya is the distribution of new actions forthose who have been taking action a before the revision. Then, we have

∆x ·π = ∑a∈A xa(ya ·π − πa).

Properties | Gains Gains in evolutionary dynamics | Zusai | 19

Individual expected net gains

The expected (first-order) net gain for an action-a playerga∗[π] := ∑

A′a⊂A\{a}PAa[A′a]EQ[π∗[A′a]− πa − q︸︷︷︸

net gain of switchfrom a

to the best in A′a

]+.

Theorem Properties of first-order net gain functionAssume Assumptions Q1, Q2 and A1.(g0) i) g·∗[·] ≥ 0. ii) ga∗[π] = 0 if and only if a ∈ b∗(π).(g1) Further assume Assumption A2. Consider arbitrary two actions a, b;

i) πa ≤ πb ⇒ ga∗[π] ≥ gb∗[π]; ii) πa < πb ⇒ ga∗[π] > gb∗[π].(g2) For each a ∈ A, function ga∗ : RA → R is differentiable almost everywhere in RA.If it is differentiable at π ∈ RA,dga∗dπ

[π]∆π = ∑A′a⊂A\{a}

PAa[A′a]Q(π∗[A′a]−πa)(∆π∗[A′a]−∆πa) for any ∆π ∈ RA

(g0) g·∗ surely represents the gain from revision:[Net gain] ≥ 0 always; [Net gain]=0⇔ the current action is optimal.

(g1) Ranking of actions by g·∗reversed! payoff ranking by π·

(g2) Differentiability of g·∗ and derivative formula (better to see the next slide).


Individual expected net second-order gainsNote that (g1) implies that any of the best actions in available action set A′a yield

the smallest gain among actions in this set:b ∈ b∗[π] =⇒ gb∗[π] = min

c∈A′agc∗[π] =: g∗∗[π; A′a].

The expected second-order net gain for an action-a player is defined asha∗[π] := ∑

A′a⊂A\{a}PAa[A′a]Q(π∗[A′a]− πa)︸︷︷︸

Prob of switchfrom a to the best in A′a

(g∗∗[π; A′a]− ga∗[π])︸︷︷︸the change in gain

by the switch

.

Let g∗[π] = (ga∗[π])a∈A be the vector that collects the expected gains of all the actions,and similarly h∗[π] = (ha∗[π])a∈A.

Theorem Properties of second-order net gain functionAssume that g∗ satisfies g1.(gh) For any a ∈ A and π ∈ RA,

ha∗[π] = za · g∗[π] for any za ∈ Va[π].(h) i) h·∗[·] ≤ 0. ii) ha∗[π] = 0 if and only if a ∈ b∗(π).

(gh) ha∗=expected change in gain g·∗ by a revision for an agent who currently takes a(h) Gain g·∗ can only decrease after a revision;

Gain ga∗ remains unchanged⇔ action a is optimal given payoff vector π.


Aggregate gains and δ-passivityAggregate first and second order gains:

G(x)[π] = x · g∗[π] and H(x)[π] = x · h∗[π].

Theorem Properties of aggregate gain functions

Assume that g∗ and h∗ satisfy properties g0, g2, h and gh.Then, their aggregates G and H satisfy the followings.(G) Property g0 implies i) G ≥ 0 and, ii) G(x, π) = 0 ⇔ x ∈ ∆A(b∗[π]).(H) Property h implies i) H ≤ 0 and, ii) H(x, π) = 0 ⇔ x ∈ ∆A(b∗[π]).(GH-0) Property gh implies∀x ∈ ∆A, π ∈ RA H(x)[π] = ∆x · g∗[π] for any ∆x ∈ V(x)[π].

(GH-1) i) Property g2 implies

∀x ∈ ∆A, π ∈ RA ∂G∂π

(x, π)∆π = ∆x ·∆π for any ∆x ∈ V(x)[π], ∆π ∈ RA.

ii) Further assume Assumption (A-A3). Then, properties gh and g2 imply

∀x ∈ ∆A, π ∈ RA ∂G∂x

(x, π)∆x = H(x, π) for any ∆x ∈ V(x)[π].

These properties of G, H imply δ-passivity:

G =∂G∂x

(x, π)x +∂G∂π

(x, π)π = H(x, π)︸︷︷︸≤0

+x · π ≤ x · π.

Equilibrium stability Gains in evolutionary dynamics | Zusai | 22

1 Introduction

2 Set up

3 Properties


Stability theorems

Extensions

Equilibrium stability | Stability theorems Gains in evolutionary dynamics | Zusai | 23

Nash stability of contractive games

Further, with π = (dF/dx)x,

G =∂G∂x

(x, π)x +∂G∂π

(x, π)π = H(x, π)︸︷︷︸≤0

+x · π ≤ x · π = x · dFdx

x︸︷︷︸≤0⇑

F:contractive

≤ 0.

Use G for W and H for W in the Lyapunov stability theorem.

Theorem Nash stability in contractive gamesConsider a contractive game with continuously differentiable payoff function F.Assume an evolutionary dynamic that generate the total expected first/second-order gainfunctions G, H satisfying properties G, H, and GH-0,1.Then, NE(F) is asymptotically stable under the evolutionary dynamic.


Logical relation

Note that these are not equivalent relations.Even if a dynamic does not fit into this reformulation or satisfy the assumptions, it mayretain δ-passivity and Nash stability of contractive games; e.g. Excess payoff targetdynamic.


Local stability of regular ESSEvolutionary stability: robustly optimal to small mutation

Any sufficiently small mutation that was equally optimal at the equilibriumworse off the mutants themselves compared to the incumbents.

Definition Regular ESSx∗ is a regular (Taylor) evolutionary stable state if it satisfies both of the following twoconditions:

i) x∗ is a quasi-strict equilibrium:Fb(x∗) = F∗(x∗) > Fa(x∗) whenever x∗b > 0 = x∗a ;

ii) DF(x∗) is negative definite with respect to RAS0:

z · DF(x∗)z < 0 for all z ∈ RAS0 \ {0}.

Here S is the set of actions used in x∗, i.e., S := {b ∈ A | x∗b > 0}; and, let

RAS0 :=

{z ∈ RA | [zb = 0 for any b /∈ S] and ∑

a∈Aza = 0

}= {z ∈ RA

0 | zb = 0 if x∗b = 0}.

Theorem Local stability of a regular ESSConsider a game with a regular ESS x∗ and an evolutionary dynamic with AssumptionsQ1, Q2, and A1. Then, x∗ is (locally) asymptotically stable.

We modify the Lyapunov function: G∗(x) = G(x) + ∑a/∈S xa.

Equilibrium stability | Extensions Gains in evolutionary dynamics | Zusai | 26

Robustness of contractiveness

(Fox & Shamma) The following modifications to payoffs π keep the payoff dynamicanti-passive and thus retains stability of Nash equilibria:• Smoothed payoff modification: a revising agent’s decision is based on an exponentially

weighted moving average of past payoffs. That is, given the payoff stream {πt}t≥0 anddiscount rate λ > 0, the perceived payoff vector at time t is

e−λtπ0 +∫ t

0e−λ(t−τ)πτdτ.

• Anticipatory payoff modification: there is a predictor function πe of near-future payoffssuch as πe = λ(π −πe). With weight k on the predictor, a revising agent’s decision isbased on the perceived payoff such as π + kπe. So, if the actual payoff for an action isgreater than the predicted payoff, an agent becomes more favor of that action.


Extension to multi-population games

Extending to a multi-population game• Finitely many populations P

Each population p consists of a continuous mass mp of agents. ∑p∈P mp = 1.• Each population p may have

. a different action set Ap;

. a different payoff function Fp : X P → RAp; and/or

. a different revision protocol.(X P = ×p∈Pmp∆A

p: the set of the profiles of action distributions.)

Theorem Aggregability of δ-passivitySuppose that, for each population p ∈ P , the aggregate first and second order gainfunctions Gp, HP : ∆A

p ×RAp → R satisfy the properties G, H, and GH-0,1.Then, the total of them GP , HP : X P ×RAP → R given by

GP (xP , πP ) := ∑p∈P

Gp(xp, πp), and HP (xP , πP ) := ∑p∈P

Hp(xp, πp)

also satisfy the properties G, H, and GH-0,1.


Unobservable heterogeneity

Unobservable heterogeneity with additive separability in payoffs

• Aggregate game: Payoff Fp depends only on aggregate state x := ∑q∈P xq ∈ ∆A.

• Additive separability: each population’s payoff function Fp : X P → RA consists ofcommon payoff function F0 : ∆A → RA and constant payoff perturbation θp ∈ RA:

Fp(xP ) = F0( ∑q∈P

xq) + θp.

Theorem Robustness of contractiveness to unobservable payoff heterogeneity

Suppose that the common payoff function F0 : ∆A → RA is contractive.⇒ the extended payoff function FP = (Fp)p∈P : X P → RAP

is also contractive.

Corollary Robustness of Nash stability to heterogeneityNash stability of a contractive game is robust to heterogeneity in payoffs and revisionprotocols, as long as• Payoff heterogeneity is additively separable, and• Heterogeneity in revision protocols keeps the properties G and H.


Evolutionary implementation

Social optimum in an aggregate game• The aggregate game with additively separable payoff heterogeneity:

Fp(xP ) = F0(∑q∈P xq) + θp.

Here, the base game F0 can be anything.• The social optimum: maximum of the total payoff

FP (xP ) = ∑q∈P

xq · Fq(xP ) = x · F0(x) + ∑q∈P

xq · θq.

Pigouvian pricing

• Adding a monetary payment T : ∆A → RA to the payoff: Fp(x) + T(x) for each p.• Dynamic Pigouvian pricing:

Ta(x) := x · ∂F0

∂xa(x) = ∑

b∈Axb

∂F0b

∂xa(x).

• FP + T is a potential game with potential FP .

Evolutionary implementation (Sandholm, ’03,’05 RES)(A-F1) Assume that the common payoff function F0 is C1

and its Hessian is negative definite on the tangent space of ∆A (negative externality).⇒ The total payoff function FP is strictly concave.⇒ The social optimum is globally stable under any dynamic that satisfies Nash

stationarity and positive correlation.


Evolutionary implementation: perturbation

Perturbation to the game

• Fpδ (xP ) = Fp(xP ) + δFp(xP ) with δ ∈ R.

• Imagine that the central planner does not know the perturbation δFp andthus designs the Pigouvian pricing scheme based on FP .

• Then the game is no longer a potential game.

Robustness of evolutionary implementation(A-F2) Assume that FP is C1 and there exists an upper bound on |∂FP/∂xP |.⇒ With strict concavity of F0 (A-F1), this implies that

the perturbed game Fpδ is still a contractive game as long as δ is sufficiently small.

⇒ Further, the total payoff maximization is approximately achieved at the limit stateunder any evolutionary dynamics that guarantee Nash stability in contractive games.

Theorem Robustness of evolutionary implementationConsider the perturbed game as above that satisfies (A-F1) and (A-F2).For any ε > 0 there is δ(ε) > 0 such that

|δ| < δ(ε) =⇒ | limt→∞

FPδ (xPδ,t)− maxxP∈X P

FPδ (xP )| < ε,

where {xPδ,t}t≥0 is the trajectory starting from an arbitrary initial state xPδ,0 ∈ XP in the

population game FPδ under an evolutionary dynamic that satisfies all the assumptions.

Concluding remarks Gains in evolutionary dynamics | Zusai | 31

Concluding remarks

Summary• We reconstruct evolutionary dynamics from optimization-based protocol• A variety of dynamics can be retrieved by random restriction to available actions and

random switching costs.• Then, we can quantitatively identify the net gain from revision.• While the aggregate gross gain serves as a Lyapunov function for Nash stability of

potential games,the aggregate net gain works generally for Nash stability of contractive games.

• With small modification, it extends to local stability of a regular ESS.• Generality of these stability results allow us to mix different dynamics and to add

perturbation; they are robust to heteterogeneity.• With Nash stability of contractive games, these robust stability results imply robustness

of evolutionary implementation of social optimum by dynamic Pigouvian pricing.

Extension• I believe that this idea is generally applicable to a various (possibly more complex)

dynamics even if the assumptions do not hold or the dynamic may not fit exactly into theframework just as presented here.

• e.g. Sawa & Zusai (2016) consider the “multitasking BRD” in which an agent engagessimultaneously in multiple games but can change action in only one of them. The mBRDdoes not exactly fit into our framework. But, applying the same idea, we define the gainfunction and use it to prove Nash stability.

gains in evolutionary dynamics€¦ · set up j population games gains in evolutionary dynamics...

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