Asymptotic behaviors of a kinetic approach to the collective dynamics of a rock-paper-scissors binary game Hugo Martin

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Asymptotic behaviors of a kinetic approach to the collective

dynamics of a rock-paper-scissors binary game

Hugo Martin ∗

October 7, 2022

Abstract

This article studies the kinetic dynamics of the rock-paper-scissors binary game in a

measure setting given by a non local and non linear integrodiﬀerential equation. After proving

the wellposedness of the equation, we provide a precise description of the asymptotic behavior

in large time. To do so we adopt a duality approach, which is well suited both as a ﬁrst step

to construct a measure solution by mean of semigroups and to obtain an explicit expression

of the asymptotic measure. Even thought the equation is non linear, this measure depends

linearly on the initial condition. This result is completed by a decay in total variation norm,

which happens to be subgeometric due to the nonlinearity of the equation. This relies on an

unusual use of a conﬁning condition that is needed to apply a Harris-type theorem, taken

from a recent paper [2] that also provides a way to compute explicitly the constants involved

in the aforementioned decay in norm.

Keywords: kinetic equations, binary games, measure solutions, nonlinearity, long-time behav-

ior, explicit limit, subgeometric convergence rate.

MSC 2010: Primary: 45K05, 35B40, 35R06; Secondary: 91A05

1 Introduction

In a recent article [6], Pouradier Duteil and Salvarani introduced a kinetic equation to describe

a large population of agents that interact by mean of random encounters and wealth exchange.

Transcient pairs are formed with probability η, then if both of the agents are rich enough (in

a sense to be clariﬁed below), they play a game of rock-paper-scissors, and based on the result

possibly exchange money. If the outcome of the game is a draw, then the transcient pair is

unmade without change in the players’ wealth. Otherwise, the winner receives a ﬁxed quantity

h > 0from the other player. The payoﬀ matrix of player 1 for this zero-sum rock-paper-scissors

game is given by

R P S

R0h−h

P−h0h

S h −h0

Standard results from game theory state that the optimal strategy is in this case a mixed strategy

and a Nash equilibrium, that is selecting at random and uniformly one of the three moves [9].

∗Université Rennes 1 &Institut d’Agro, Rennes, France. Email: hugo.martin1@polytechnique.edu

arXiv:2210.02781v1 [math.AP] 6 Oct 2022

The population of players, structured in wealth, is described by a distribution function u=u(t, y)

deﬁned on R2

+. Given a subset A⊂R+, the integral

u(t, y)dy

represents the number of individuals whose wealth belongs to A. Denoting uin the initial wealth

distribution, this model reads











∂tu(t, y) = η

3Z∞

u(t, y0)dy0u(t, y −h)1[2h,∞)(y) + u(t, y +h)1[0,∞)(y)−2u(t, y)1[h,∞)(y)

u(0, y) = uin(y),∀y>0.

(1)

The intensity of the exchange between players is proportional to the integral

Z∞

u(t, y0)dy0

which makes this equation non linear. This term comes from the rule that forbids debts, implying

that players involved in a transcient pair only play the game if they own at least h.

The authors of the aforementionned article provided (among other results) wellposedness of this

equation in a L1setting as well as its behavior as hvanishes under the diﬀusive scalling t←t/h2.

The large time asymptotics when hremains ﬁxed was yet to be investigated: so is the purpose of

the present paper. Our goal is to provide a precise asymptotic behavior to this equation, drawing

inspiration from the methodology developped in [7] for a critical case of the growth-fragentation

equation. In this article, the authors worked in a measure framework and adopted a combination

of semigroup and duality approach. They obtained both an uniform exponential decay using the

results from [8] and a formula for the invariant probability measure, that was explicit in term

of direct and adjoint eigenvector of the growth-fragmentation equation (see [4] for a rich survey

as well as very general assumptions ensuring the existence of such functions for this equation).

This equation is non local yet linear, unlike the one studied in the present article which is both

non local and non linear. Such feature can arguably be considered as the hallmark of the lost of

exponential relaxation toward a stationary solution, that is indeed veriﬁed in the present case, see

Theorem 1 below. Taking advantage of the particular expression of Equation (1), an appropriate

time rescaling enables to use results from the recent paper [2] from Cañizo and Mischler.

The remaining of the paper is organized as follows. In the next section, we introduce the

framework that is required to solve Equation (1) in the sense of measures and state our main

result. Our methodoloy relies on a duality approach, so Section 3 is devoted to the study of

the adjoint equation. In Section 4, we build on previous results to construct a measure solution

to Equation (1). Section 5 contains a precise description of the asymptotic behavior of the

solution. These results are illustrated by numerical simulations in Setion 6. Finally in Section 7,

we propose some possible continuations of this paper.

2 Preliminaries and the main result

We start by recalling brieﬂy the notions from measure theory that we need to establish our

results. For a more complete introduction on this ﬁeld, we refer the reader to [12] in which the

focus is on the total variation norm, and the recent book [5] for a rich exposition on measure

solutions to PDEs, in particular using the topology of the ﬂat norm or dual bounded Lipschitz

norm.

We endow R+= [0,+∞)with its standard topology and the associated Borel σ−algebra B(R+).

Throughout the paper, we shall consider discrete subsets of R+and unions of such sets, so for a

subset Ω⊂R+, we denote by M(Ω) the space of real-valued Radon measures with Hahn-Jordan

decomposition µ=µ+−µ−on Ωsuch that

kµk:= ZΩ

d|µ|<∞

where |µ|=µ++µ−is the total variation measure of µ. To obtain the desired asymptotic results,

we shall work in spaces of weighted measures. For a measurable (weight) function V: Ω →[1,∞),

we denote by MV(Ω) the subspace of ﬁnite signed measures µon Ωsuch that

kµkV=ZΩ

Vd|µ|<∞,

and simply k·kwhenever V≡1.

Now we denote BV(Ω) the space of Borel functions f: Ω →R+such that

kfkBV(Ω) := sup

y∈Ω

|f(y)|

V(y)<∞.

If a function f∈ BV(Ω) is also continuous, we denote f∈ CV(Ω). In the case V≡1, we simply

write k·k∞for this norm, and more generally omit the index V. For every µ∈ MV(Ω), one can

deﬁne a linear form on BV(Ω) through the duality bracket

f7→ hµ, fi:= ZΩ

fdµ.

With a slight abuse of notation, for a measurable set A, we will write µ(A)instead of hµ, 1Ai.

The norm k·kVcan be expressed as

kµkV= sup

kfkBV(Ω)61hµ, fi.

Now we deﬁne a weaker norm on the space of measures. First, for a function fcontinuous on Ω,

we deﬁne

|f|Lip := sup

y6=z

|f(y)−f(z)|

|y−z|.

Then we can deﬁne the Lipschitz bounded norm

kfkBL(V):= kfkBV(Ω) +|f|Lip.

The dual bounded Lipschitz norm is thus deﬁned as

kµkBL∗(V):= sup ZΩ

fdµ:f∈ C(Ω),kfkBL(V)61

with C(Ω) denoting the set of continuous functions on Ω. Since the supremum is taken on a

smaller set, it is clear that for any measure µ, one has kµkBL∗(V)6kµkV. In particular, one has

kδy−δzk= 2 if y6=zbut kδy−δzkBL∗(1) = min(1,|y−z|), so (M(Ω),k·kBL∗)enjoys better

topological properties that (M(Ω),k·kT V ).

It remains to deﬁne a notion of measure solutions to Equation(1). We choose an equation of the

"mild" type, in the sense that it relies on an integration in time. Let us give our motivation for

this choice. Assume that u(t, y)∈ C1(R+; L1(R+)) satisﬁes (1) in the classical sense. Integrating

this equation multiplied by f∈ B(R+), we obtain

Z∞

f(y)u(t, y)dy=Z∞

f(y)uin(y)dy

+Zt

0Z∞

3Z∞

u(z, s)dz[f(y+h) + f(y−h)−2f(y)] 1[h,∞)(y)u(s, y)dyds.

In order to state the equivalent of this equation for a general signed measure, we deﬁne the

operator Aon B(R+)by

Af:y7→ [f(y+h) + f(y−h)−2f(y)] 1[h,∞)(y).

Deﬁnition 1. A family (µt)t>0⊂ M(R+)with initial condition µin is called a measure solution

to Equation (1) if for all f∈ B(R+)one has

hµt, fi=hµin, fi+Zt

0Dµs,η

3µs([h, ∞))AfEds. (2)

We introduce now another slight abuse of notation. For a measurable set A⊂R+and a

positive real number a, we denote

A+a:= {y∈R+:∃x∈A, y =x+a}.

Finally, we are ready to state the main result of this paper, which is about the wellposedness of

Equation (1) in the measure setting, as well as the asymptotic bahaviour of the solution, when

the exchange parameter hremains ﬁxed.

Theorem 1. For any initial condition µin ∈ M(R+), there exists a unique measure solution

(µt)t>0to Equation (1) in the sense of Deﬁnition 1, a projection operator Phdeﬁned on M(R+)

and constants C, λ > 0independant of ηand µin([h, ∞)) such that

∀t>0,

µt−µinPh

6C

1 + ηµin([h,∞))

3tλ

µin −µinPh

.(3)

The constants Cand λcan be computed explicitely, and the measure µinPhis given explicitely

in terms of the initial condition µin and the exchange parameter h:

•supp µinPh⊂[0, h)

•for all measurable set A⊂[0, h)

µinPh(A) =

∞

k=0

µin (A+kh).

Remark. A particular feature of Equation (1) is the conservation of the population and total

wealth. Indeed, a formal integration against the measures dyand ydyover [0,∞)leads to the

balance laws d

dtZ∞

u(t, y)dy=d

dtZ∞

yu(t, y)dy= 0.

In the language of measures, this translates by µt(R+) = µin(R+)and hµt, Idi=hµin, Idiwith

Id the identity function, provided µin as a ﬁnite ﬁrst moment. If (µt)t>0is a measure solution,

then the ﬁrst equality is satisﬁed by deﬁnition, since 1R+lies in B(R+). If in addition hµin, Idi

is ﬁnite, one can deﬁne the formula (2) on test functions f∈ B1+Id(R+), i.e. functions that are

Borel and satisfy

sup

y∈Ω

|f(y)|

1 + y<∞.

The function Id lies in this set and satisﬁes Af= 0. Thus the equality hµt, Idi=hµin, Idiis

satisﬁed, meaning that the total wealth is conserved at any ﬁnite time t.

Now, we make a statement about the behavior of the ‘asymptotic in time’ measure µinPh

when hvanishes.

Theorem 2. Denoting P0the linear operator acting on measures deﬁned by µP:= µ(R+)δ0,

one has

~Ph− P0~BL∗6h

with ~·~BL∗the operator norm on (M(R+),k·kBL∗)deﬁned by

~P~BL∗:= sup

kµkBL∗61

kµPkBL∗

kµkBL∗.

3 Dual equation and right semigroup

This section is devoted to the study of the wellposedness of a family of equations that are related

to the adjoint equation. Assume (µt)t>0is the unique solution to (2). Then, the (backward)

adjoint equation is given by

∂

∂tf(t, s, y) = η

3µt([h, ∞))Af(t, s, y)

=η

3µt([h, ∞)) [f(t, s, y +h) + f(t, s, y −h)−2f(t, s, y)] 1[h,∞)(y),(4)

with (t, s, y)∈R3

+and s6t, supplemented with the terminal condition f(t, t, y) = f0(y). The

second argument sis included to account for the inhomogeneity in the time evolution, since the

solution of (4) depends on the values of t7→ µt([h, ∞)). Due to the indicator function 1[h,∞),

a solution fto Equation (4) satisﬁes f(t, s, y) = f0(y)for all y∈[0, h)and 06s6t. On the

interval [h, ∞), we solve a mild version of this equation, that is obviously more complicated than

on [0, h). All in all, we search for a function fthat satisﬁes

f(t, s, y) =f0(y)e−2η

31[h,∞)(y)Rt

sµσ([h,∞))dσ

+η

31[h,∞)(y)Zt

µσ([h, ∞))e−2η

3Rt

σµτ([h,∞))dτ[f(σ, s, y +h) + f(σ, s, y −h)]dσ

(5)

for all y > 0. In this form, this equation depends on the solution (µt)t>0of the direct problem, so

we introduce related equations by replacing µ•([h, ∞)) by a generic non negative and continuous

bfunction. The equation we study is then

f(t, s, y) =f0(y)e−2η

31[h,∞)(y)Rt

sb(σ)dσ

+η

31[h,∞)(y)Zt

b(σ)e−2η

3Rt

σb(τ)dτ[f(σ, s, y +h) + f(σ, s, y −h)]dσ. (6)

In this section, we will solve Equation (6) for a ﬁxed function b. The wellposedness of this

problem, as well as useful properties, are collected in the next proposition. First, we introduce

some notations. For ˜

Ω⊂R2

+and Ω⊂R+, we denote C˜

Ω,B(Ω)the set of functions fthat

are deﬁned and continuous on ˜

Ωsuch that for all (t, s)∈˜

Ω, the function f(t, s, ·)lies in B(Ω).

Similarly, we deﬁne a subset of the previous one, denoted C1˜

Ω,B(Ω)made of the functions

such that ∂tfand ∂sflie in C˜

Ω,B(Ω). For x∈[0, h)we deﬁne

Cx:= {x+kh, k ∈N}.

Since players can only gain or loose hafter each game, they shall remain in the same ‘class of

wealth’ at all time, depending on their initial wealth only. Such classes are precisely these sets

Cxfor x∈[0, h). Since any f∈ B(R+)can be written as

f=X

x∈[0,h)

f|Cx

it is enough to prove properties on B(Ω ×Cx), which we do for the next result.

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Asymptoticbehaviorsofakineticapproachtothecollectivedynamicsofarock-paper-scissorsbinarygameHugoMartin*October7,2022AbstractThisarticlestudiesthekineticdynamicsoftherock-paper-scissorsbinarygameinameasuresettinggivenbyanonlocalandnonlinearintegrodierentialequation.Afterprovingthewellposednessofthee...

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Asymptotic behaviors of a kinetic approach to the collective dynamics of a rock-paper-scissors binary game Hugo Martin

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