ZUSAI, Hetero Evol Dyn

JEA

Micro-based construction of evolutionary dynamics
Game (Payoff function)
Payoff vector

Strategy dist

Str composition
Evolutionary dynamics∈Revision protocol→→ ,Individual agent’sswitching rate

Individual agent’s decisionof switching the choice→

Deterministic mean dynamic,

⋅,

⋅→ ,,

→⋅,

Aggregation of individual
agents’ switches→→

An agent occasionally reconsiders the choice,when it receives a “revision opportunity,”
which arrives randomly.
(to make the dynamic differentiable with respect to infinitesimalchange in time.)

At that opportunity, the agent
finds the candidate of a new strategy,
decides whether to switch to it or not.

∈and→At each moment of time,for each strategy, we countthe agents who switches to it
and those who switches from it.

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ZUSAI, Hetero Evol Dyn

Evolutionary dynamics as a differential equation
An evolutionary dynamic is mathematically a differential equation such aswhere state variable is strategy distribution
Homogeneous with finite strategy space
State variable:, finite dimension.
has infinite/continuous dimension
Continuous type space with finite :Zusai 2023.
State variable:
whereis strategy distribution in agents of type ,( is a density function of ).
Or equivalently,: joint probability measure on• The same rev protocol can be used.
• General extension for various dynamics
all at once.

a differential equation on infinite dim.
Homogenous & cont’s action space :Oechessler & Riedel, Cheung, Lahkar.
State variable:,strategy distribution over(a density function of ).
Or equivalently,: probability measure on• Rev protocol itself must be newly definedin its math formulation.
• Piecewise extension for each dynamic.

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ZUSAI, Hetero Evol Dyn

Main and most fundamental results: Well-definedness of a hetero evoldynamic
Lipschitz continuous rev protocols
Switching rate : L-cont in payoffs→→∃•

E.g. Most of evol dynamics

Exact optimization rev protocol• Switching rate to action a becomes 0
when a becomes not optimal• While it is optimal, switching rate changes
continuously (or unchanged) with payoffs.
•E.g. BRD, tempered BRD

Assumption 1. Lipschitz continuity of payoff function
Assumption 2. Uniform boundedness of switching rate (with the domain
restricted to the range of payoffs actually taken under ).
Assumption 3. Continuous change in best responses: type distribution on

Mass of types whose BRswitch from s to s’.
Theorem 1. (Well-definedness) Under these assumptions,
there always exists a unique solution path from any initial state.

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Eqm stationarity in general
Best Response stationarity. Rev protocol→

satisfies BR stationarity if→∈𝒮∈𝒮

Average change in from 𝟎given payoff vecfor the agentfor anyand.

Known fact. Under any homog pop game
BR stationarity implies eqm stationarity:,Theorem 2. Under any hetero pop game ,BR stationarity implies eqm stationarity:

Side work: existence of an eqm
Theorem 3. : Assmp 1 (L-cont), equicont, and nearly bounded over types.
is equicontiuous:for ∀, almost everys.t.

.

An eqm exists.

is nearly bounded over types:s.t.
∈𝒮

ZUSAI, Hetero Evol Dyn

Eqm stability in potential games
satisfies Positive correlation, i.e.,𝟎

is a potential game:
function s.t.

Theorem 4. Under a hetero potential game ,PC implies eqm stability.
• The set of eq’ia is globally stable.
scalar-valued • A local max of is locally stable.

Homogenous setting

Heterogenous setting

for eachWhat’s a derivative??
Finite dimension

Infinite dimension→Fréchet differentiation

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ZUSAI, Hetero Evol Dyn

Adding heterogeneity
or: a base game in homogeneous setting.
•Additively separable anonymous Incomplete information gamegame (ASAG)∈Type

is an idiosyncratic payoff vector

Type is a signal.
Posterior belief
is an aggregate strategy distribution.

whereStructured population game(continuous version of Wu & Zusai, Nature Human Behav., ‘19)

Typeis a location.

Theorem 5. Base

Distance between locations: a potential game ⇒ Modified : also a potential game.

Cor. For each eqm in the base game, its local stability is robust under these modifications.
e.g. Dynamic Pigouvian pricing to asymptotically implement the social optimum(evolutionary implementation, by Bill Sandholm, ‘02, ‘05):
extendable to continuous heterogeneity.

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