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Multiple-interaction kinetic modelling of a virtual-item

gambling economy

Giuseppe Toscani∗Andrea Tosin†Mattia Zanella‡

Abstract

In recent years, there has been a proliferation of online gambling sites, which made

gambling more accessible with a consequent rise in related problems, such as addiction. Hence,

the analysis of the gambling behaviour at both the individual and the aggregate levels has

become the object of several investigations. In this paper, resorting to classical methods of

the kinetic theory, we describe the behaviour of a multi-agent system of gamblers participat-

ing in lottery-type games on a virtual-item gambling market. The comparison with previous,

often empirical, results highlights the ability of the kinetic approach to explain how the simple

microscopic rules of a gambling-type game produce complex collective trends, which might

be diﬃcult to interpret precisely by looking only at the available data.

Keywords: Multiple-collision Boltzmann-type equation, linearised kinetic models, Fokker-

Planck equation, lognormal distribution, gamma and inverse gamma distributions.

Mathematics Subject Classiﬁcation: 35Q20, 35Q84, 82B21, 91D10.

1 Introduction

Gambling is usually perceived as a complex multi-dimensional activity fostered by several diﬀerent

motivations [2]. Due to the rapid technological developments, in the last decade the possibility

of online gambling has enormously increased [13], leading to the simultaneous rise of related

behavioural problems. As remarked in [11], structural characteristics of online gambling, such as

the speed and the availability, led to conclude that online gambling has a high potential risk of

addiction.

A non-secondary aspect of the impressive increase in online gambling sites is related to economic

interests. Indeed, the expansion of the video-gaming industry has resulted in the formation of a

new market, in which gamblers are the actors, that has reached a level of billions of dollars. The

continuous expansion of this market depends on many well-established reasons, which include its

easy accessibility, low entry barriers and immediate outcome.

As documented in [25], mathematical modelling of these relatively new phenomena attracted

the interest of current research, with the aim of understanding the aggregate behaviour of a system

of gamblers. In [25], the behaviour of online gamblers has been studied by methods of statistical

physics. In particular, the analysis has been focused on a popular type of virtual-item gambling,

the jackpot, i.e. a lottery-type game which occupies a big portion of the gambling market on

the web. As pointed out in [25], to be able to model the complex online gambling behaviour at

both the individual and the aggregate levels is quickly becoming a pressing need for adolescent

gambling prevention and eventually for virtual gambling regulation.

∗Department of Mathematics “F. Casorati”, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy

(giuseppe.toscani@unipv.it)

†Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24,

10129 Torino, Italy (andrea.tosin@polito.it)

‡Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24,

10129 Torino, Italy (mattia.zanella@polito.it)

1

The gambling datasets used in [25] have been extracted from the publicly available history

page of a gambling site. The huge number of gambling rounds, and the time period (more than

seven months) taken into account, allowed for a consistent ﬁtting. The analysis of the dataset

has been essentially split in two main parts. A ﬁrst part deals with the behavioural distribution

of the gambler activities. Here, the main result concerns the cumulative distribution function

of the number of rounds played by individual gamblers, which was found to be best ﬁtted by a

lognormal distribution. A second part of the analysis in [25] is concerned instead with the study

of the distribution function of the winnings and of the related correlations. As it happens in

many socio-economic phenomena involving multi-agent systems [16], the best ﬁtting curve for the

winnings has been found to be a power-law-type distribution with cut-oﬀ. While the possible

reasons leading to the formation of a lognormal distribution for the number of rounds played by

the gamblers has been left largely unexplored, the formation of a power law distribution for the

incomes has been explained in [25] by resorting to three diﬀerent random walk models. As clearly

outlined by the authors, their aim was to gain insights into the ingredients necessary to obtain

from these models results with qualitative properties similar to those of the data derived from the

gambling logs.

The huge number of gamblers and the well-deﬁned rules of the game allow us to treat the

system of gamblers as a particular multi-agent economic system, in which the agents invest (risk)

part of their personal wealth to obtain a marked improvement of their economic conditions. Unlike

classical models of the trading activity [16], in this gambling economy particular attention needs

to be paid to the behavioural reasons pushing people to gamble even in presence of high risks.

By looking at the jackpot game from this perspective, and resorting to the classical modelling of

multi-agent systems via kinetic equations of Boltzmann and Fokker-Planck type [16], we will be

able to obtain a detailed interpretation of the datasets collected in [25].

This approach has proved to be powerful in many situations, ranging from the formation of

knowledge in a modern society [9,17] to the spreading of the popularity of online content [23]

or the description of the reasons behind the formation of a lognormal proﬁle in various human

activities characterised by their skewness [8].

Our forthcoming analysis will be split in two parts. In a ﬁrst part, we will discuss the kinetic

modelling of the jackpot gambling and we will study, in particular, the distribution in time of

the tickets played and won by the gamblers. Our modelling approach is largely inspired by the

similarities of the jackpot game with the so-called winner takes it all game described in detail

in [16]. Nevertheless, the high number of gamblers taking part to the game, the presence of a

percentage cut on the winnings operated by the site, and the continuous reﬁlling of tickets to play,

introduce essential diﬀerences.

In a second part, we will deal with the behavioural aspects linked to the online gambling.

This is a phenomenon that may be fruitfully described by resorting to a skewed distribution and

that, consequently, may be modelled along the lines of the recent papers [8,10]. The behavioural

aspects of the gambling and their relationships with other economically relevant phenomena have

been discussed in a number of papers, cf. e.g. [15] and the references therein. Also, the emergence

of the skewed lognormal distribution was noticed before. The novelty of the present approach is

that we enlighten the principal behavioural aspects at the basis of a reasonable kinetic description.

Going back to the kinetic description of the jackpot game, it is interesting to remark that some

related problems have been studied before. The presence of the site cut, which can be regarded

as a sort of dissipation, suggests that the time evolution of the distribution function of the tickets

played and won by the gamblers may be described in a way similar to other well-known dissipative

kinetic models, such as e.g. that of the Maxwell-type granular gas studied by Ernst and Brito [6] or

that of the Pareto tail formation in self-similar solutions of an economy undergoing recession [20].

However, essential diﬀerences remain. Unlike the situations described in [6,20], where the loss

of the energy or of the mean value, respectively, was artiﬁcially restored by a suitable scaling of

the variables, in the present case the percentage cut on each wager, leading to an exponential

loss of the mean value of the winnings, is reﬁlled randomly because of the persistent activity of

the gamblers even in the presence of losses. A second diﬀerence concerns the necessity to take

into account a high number of participants in the jackpot game. In [25], it is conjectured that

2

the shape of the steady state distribution emerging from the game rules does not change as the

number of participants increases. Consequently, all models studied there were limited to describe

the evolution of winnings in a game with a very small number of gamblers. Here, we adopt instead

a diﬀerent strategy, inspired by the model introduced by Bobylev and Windfall [3]. In that paper,

it is shown that the kinetic description of an economy with transactions among a huge number of

trading agents can be suitably linearised, leading to a simpler description. Hence, following [3],

we will consistently simplify the jackpot game description by introducing a suitable linearisation

of the problem, which makes various explicit computations possible.

Out of the detailed kinetic description of the online jackpot game, and unlike the analysis

proposed in [25], we conclude that the game mechanism does not actually give rise to a power-

law-type steady distribution of the tickets played and won by the gamblers. The formation of such

a fat tail may, however, be obtained by resorting to a diﬀerent linearisation of the game, which,

while apparently close to the actual non-linear version, may be shown numerically to produce

quite diﬀerent trends.

In more detail, the paper is organised as follows. In Section 2, we introduce the microscopic

model of the jackpot game with Ngamblers and its non-linear Boltzmann-type kinetic description

with multiple-interactions (Section 2.1). Next, in the limit of Nlarge, we replace the N-interaction

dynamics with a sort of mean ﬁeld individual interaction, which gives rise to a linear Boltzmann-

type model (Section 2.2). We study the large time trend of the linear model by means of a

Fokker-Planck asymptotic analysis, which shows that no fat tails are produced at the equilibrium

(Section 2.3). Finally, by resorting to a diﬀerent linearisation of the multiple-interaction model

based on the preservation of the ﬁrst two statistical moments of the distribution function, we

produce an alternative kinetic model, whose equilibrium distribution has indeed a power-law-type

fat tail (Section 2.4). Nevertheless, we argue that such a new linear model does not provide a

description of the gambling dynamics completely equivalent to the original multiple-interaction

model and, hence, that it does not describes exactly the original jackpot game. In Section 3, we

discuss a model for the distribution of the tickets which the gamblers purchase to participate in

successive rounds of the jackpot game. This study complements the previous one on the gambling

dynamics, as it provides the basis to model the reﬁlling of tickets mentioned above. In Section 4,

we illustrate the evolution of the real game predicted by the multiple-interaction kinetic model and

that of the various linearised models by means of several numerical experiments, which conﬁrm

the theoretical ﬁndings of the previous sections. Finally, in Section 5, we summarise the main

results of the work.

2 Kinetic models of jackpot games

2.1 Maxwell-type models

The jackpot game we are going to study is very simple to describe. At given intervals of time,

which may last from a few seconds to several minutes, the site opens a new round of the game

that the gamblers may attend. The gamblers participate in the game by placing a bet with a

certain number of lottery tickets purchased with one or several skins deposited to the gambling

site. There is only one winning ticket in each round of the game. The winning ticket is drawn

when the total number of skins deposited as wagers in that round exceeds a certain threshold. The

draw is based on a uniformly distributed random number with a range equal to the total number

of tickets purchased in that round. The gambler who holds the winning ticket wins all the wagers,

i.e. the deposited skins in that round, after a site cut (percentage cut) has been subtracted.

As usual in the kinetic description, we assume that the gamblers are indistinguishable [16].

This means that, at any time t≥0, the state of a gambler is completely characterised by their

wealth, expressed by the number x≥0 of owned tickets. Consequently, the microscopic state of

the gamblers is fully characterised by the density, or distribution function, f=f(x, t).

3

The precise meaning of the density fis the following. Given a subdomain D⊆R+, the integral

ZD

f(x, t)dx

represents the number of individuals possessing a number x∈Dof tickets at time t≥0. We

assume that the density function is normalised to one, i.e.

ZR+

f(x, t)dx = 1,

so that fmay be understood as a probability density.

The time evolution of the density fis due to the fact that rounds are programmed at regular

time intervals and gamblers continuously upgrade their number of tickets xat each new round.

In analogy with the classical kinetic theory of rareﬁed gases, we refer to a single upgrade of the

quantity xas an interaction.

The game has evident similarities with the winner takes it all game described in detail in [16,

Chapter 5]. The main diﬀerences are the presence of a high number of participants and of the

site cut. Indeed, while the microscopic interactions in the winner takes it all game are pointwise

conservative, any round of the online jackpot game leads to a loss of the value returned to the

gamblers.

Let us consider a number Nof gamblers, with N1, who participate in a sequence of rounds.

At the initial time, the gamblers (indexed by k= 1, . . . , N) buy certain numbers xk=xk(0) of

tickets, with the intention to play for a while. While it is clear that actually xk∈N+, in order to

avoid inessential diﬃculties, and without loss of generality, we will consider xk∈R+. Moreover,

we may ﬁx a unitary price for the tickets, so as to identify straightforwardly the number of tickets

with the amount of money owned by the gamblers. We assume that each gambler participates in

a round by using only a small fraction of their tickets, say αkxk, where 0 < 1 while the αk’s

may be either constant or random coeﬃcients. In the simplest case, i.e. αk= 1 for all k, the total

number of tickets played by the gamblers in a single round is PN

k=1 xk.

At ﬁxed time intervals of length ∆t > 0, a ticket is chosen randomly. The owner of that ticket

wins an amount of money corresponding to the value of the total number of tickets played in that

round, minus a certain ﬁxed cut operated by the site. Let us denote by xk(t−1) the number of

tickets possessed by the kth gambler right before the next round. If δ > 0 denotes the percentage

cut operated by the site, after the new round the quantities xk(t−1) update to

xk(t) = (1 −)xk(t−1) + (1 −δ)

N

X

j=1

xj(t−1)I(A(t−1) −k), k = 1,2, . . . , N. (1)

In (1), A(t−1) ∈ {1, . . . , N }is a discrete random variable giving the index of the winner in the

forthcoming round. Since the winner is chosen by extracting uniformly one of the played tickets,

the random variable A(t−1) may be characterised by the following law:

P(A(t−1) = k) = xk(t−1)

N

P

j=1

xj(t−1)

, k = 1,2, . . . , N. (2)

Furthermore, in (1) the function I(n), for n∈Z, is deﬁned by

I(0) = 1, I(n) = 0 ∀n6= 0.

Because of the ﬁxed cut operated by the site, the total number of tickets, viz. the amount of

money, in the hands of the gamblers diminishes at each round, so that, in the long run, the gamblers

remain without tickets to play. On the other hand, as noticed in the recent analysis [25], the data

published by the jackpot site certify that this never happens. One may easily identify at least two

4

explanations. First, gamblers with high losses are continuously replaced by new gamblers entering

the game. Second, in presence of repeated losses the gamblers continuously reﬁll the amount of

money available to their wagers by drawing on their personal reserves of wealth. Notice that we

may easily identify the new gamblers entering the game with those leaving it by simply assuming

that the number Nof gamblers remains constant in time. Taking this non-secondary aspect into

account, we modify the upgrade rule (1) as follows:

xk(t) = (1 −)xk(t−1) + βYk(t−1) + (1 −δ)

N

X

j=1

xj(t−1)I(A(t−1) −k),(3)

k= 1, . . . , N . In (3), β≥0 is a ﬁxed constant, which identiﬁes the rate of reﬁlling of the tickets.

Moreover, the Yk’s are non-negative, independent and identically distributed random variables

giving the number of reﬁlled tickets. In agreement with [25], and as explained in full details in

Section 3, one can reasonably assume that the random variables Ykare lognormally distributed.

The upgrade rules (2), (3) lead straightforwardly to a Boltzmann-type kinetic model describing

the time evolution of the density f(x, t) of a population of gamblers who play an N-player jackpot

game, independently and repeatedly, according to the interaction

x0

k= (1 −)xk+βYk+(1 −δ)

N

X

j=1

xjI(A−k), k = 1,2, . . . , N, (4)

where A∈ {1, . . . , N }is a discrete random variable with law

P(A=k) = xk

N

P

j=1

xj

, k = 1, . . . , N.

In (4), the quantity xkrepresents the number of tickets, hence the amount of money, put into the

game by the kth gambler, while the quantity x0

kis the new number of tickets owned by the kth

gambler after the draw of the winning ticket.

Starting from the microscopic interaction (4), the study of the time evolution of the distribution

function fmay be obtained by resorting to kinetic collision-like models [16]. Speciﬁcally, the

evolution of any observable quantity ϕ, i.e. any quantity which may be expressed as a function of

the microscopic state x, is given by the Boltzmann-type equation

d

dt ZR+

ϕ(x)f(x, t)dx =1

τN ZRN

+

N

X

k=1hϕ(x0

k)−ϕ(xk)i

N

Y

j=1

f(xj, t)dx1··· dxN,(5)

where τdenotes a relaxation time and h·i is the average with respect to the distributions of the

random variables Yk,Acontained in (4). Note that the interaction term on the right-hand side

of (5) takes into account the whole set of gamblers, and consequently it depends on the N-product

of the density functions f(xj, t), j= 1, . . . , N . Thus, the evolution of fobeys a highly non-linear

Boltzmann-type equation.

Remark 1.In the classical kinetic theory of rareﬁed gases, the binary collision integral depends

on a non-constant collision kernel, which selects the collisions according to the relative velocities

of the colliding particles. Conversely, the interaction integral in (5) has a constant kernel, chosen

equal to 1 without loss of generality. This corresponds, in the jargon of the classical kinetic theory,

to consider Maxwellian interactions. Remarkably, in the case of the jackpot game, this assumption

corresponds perfectly to the description of the game under investigation, since one may realistically

assume that the numbers of tickets played by diﬀerent gamblers are uncorrelated.

Choosing ϕ(x) = 1 in (5) yields

d

dt ZR+

f(x, t)dx = 0,

5

meaning that the total mass of the system is conserved in time. It is worth pointing out that, as

a matter of fact, this is the only conserved quantity in (5).

In order to better understand the time evolution of f, as well as the role of the site cut, we begin

by considering the situation in which the gamblers do not reﬁll their tickets, which corresponds to

letting β= 0. In this case, the interactions (4) being linear in the xk’s, we can compute explicitly

the evolution in time of the mean number of tickets

m(t) := ZR+

xf(x, t)dx

owned by the gamblers. Indeed, since

*N

X

k=1

x0

k+= (1 −)

N

X

k=1

xk+(1 −δ)

N

X

j=1

xj

N

X

k=1

P(A=k) = (1 −δ)

N

X

k=1

xk,(6)

choosing ϕ(x) = xin (5) we obtain

dm

dt =−δ

τm. (7)

As expected, the presence of a percentage cut δ > 0 in the jackpot game leads to an exponential

decay to zero of the mean number of tickets at a rate proportional to δ

τ.

As far as higher order moments of the distribution function fare concerned, analytic results

may be obtained at the cost of more complicated computations, due to the non-linearity of the

Boltzmann-type equation (5). This unpleasant fact is evident by computing, e.g. the second order

moment, i.e. the energy of the system, which amounts to choosing ϕ(x) = x2in (5). In this case,

we have:

*N

X

k=1

(x0

k)2+=(1 −)2+ 2(1 −)(1 −δ)N

X

k=1

x2

k+2(1 −δ)2 N

X

k=1

xk!2

.(8)

Notice that the term PN

k=1 xk2, once integrated against the N-product of the distribution func-

tions, produces a dependence on both the second moment and the square of the ﬁrst moment,

whose decay law has been established in (7).

It is now clear that, while giving a precise picture of the evolution of the jackpot game, the

highly non-linear Boltzmann-type equation (5) may essentially be treated only numerically.

2.2 A linearised model

A considerable simpliﬁcation occurs in presence of a large number Nof participants in the game.

In this situation, at any time t > 0 we have

N

X

k=1

xk=N·1

N

N

X

k=1

xk≈Nm(t).(9)

In practice, if Nis large enough we may approximate the empirical mean number of tickets

1

NPN

k=1 xkof the gamblers participating in a round of the game with the theoretical mean number

of tickets mowned by the entire population of potential gamblers. Hence, still considering for the

moment the case β= 0, the interaction (4) may be restated as

x0

k= (1 −)xk+N (1 −δ)m(t)I(˜

A−k), k = 1,2, . . . , N, (10)

where ˜

A∈ {1, . . . , N }is the discrete random variable with (approximate) law

P(˜

A=k)≈xk

Nm(t), k = 1, . . . , N .

6

Remark 2.Owing to the approximation (9), the usual properties P(˜

A=k)≤1 and PN

k=1 P(˜

A=

k) = 1 may be fulﬁlled, in general, only in a mild sense, which however becomes tighter and tighter

as Ngrows. We refrain from investigating precisely the proper order of magnitude of N, because,

as we will see in a moment, we will be mostly interested in the asymptotic regime N→ ∞.

Before proceeding further, we observe that in the recent paper [3] the linearisation resulting

from considering a large number of gamblers has been proposed in an economic context. The same

type of approximation has also been used in [22] to linearise a Boltzmann-type equation describing

the exchange of goods according to micro-economy principles.

The main consequence of the new interaction rule (10) is that the each post-interaction number

of tickets x0

kdepends linearly only on the pre-interaction number xkand on the (theoretical) mean

number of tickets m(t). Plugging (10) into (5) leads then to a linear Boltzmann-type equation.

In particular, the time evolution of the observable quantities ϕ=ϕ(x) is now given by

d

dt ZR+

ϕ(x)f(x, t)dx =1

τZR+hϕ(x0)−ϕ(x)if(x, t)dx, (11)

where

x0= (1 −)x+N (1 −δ)m(t)I(¯

A−1) (12)

and the random variable ¯

A∈ {0,1}is such that

P(¯

A= 1) = x

Nm(t).(13)

In practice, since it is no longer necessary to label the single gamblers participating in a round

of the jackpot game, we use ¯

Asimply to decide whether the randomly chosen gambler xwins

(¯

A= 1) or not ( ¯

A= 0) in that round.

Equation (11) allows for a simpliﬁed and explicit computation of the statistical moments of

the distribution function f. In particular, it gives the right evolution of the ﬁrst moment like

in (7). We remark, however, that the simpliﬁed interaction rules (12)-(13) have two main weak

points. First, since the mean value m(t) follows the decay given by (7), thus it is in particular

non-constant in time, the interaction (12) features an explicit dependence on time. Second, if

is ﬁxed independently of N, the number N of tickets played in a single game tends to blow as

Nincreases. At that point, the kinetic model does not represent the target jackpot game any

more. Therefore, while maintaining the fundamental linear characteristics, which make the model

amenable to analytical investigations, it is essential to combine the large number of gamblers in

each round with a simultaneously small value of . Indeed, it is realistic to assume that the product

N, which characterises the percent number of tickets played in each game, remains ﬁnite for every

N1 and 1. We express this assumption by letting ∼κN−1, where κ > 0 is a constant,

so that

lim

N→∞

N =κ. (14)

Remark 3.Notice that the rate of decay of the mean value min the linear model (11), which,

as already observed, equals the one of the non-linear model given by (7), is bounded away from

zero for any value of if and only if τ∼. Therefore, in order to maintain the correct decay

of the mean value for any value of ,Nin the linearised model, we will assume, without loss of

generality, τ=.

We are now ready to re-include in the dynamics also the reﬁlling of money operated by the

gamblers drawing on their personal reserves of wealth. Assuming a very large number N1

of gamblers together with (14) and taking also Remark 3into account, the jackpot game with

reﬁlling is well described by the linear kinetic equation

d

dt ZR+

ϕ(x)f(x, t)dx =1

ZR2

+hϕ(x0)−ϕ(x)if(x, t)Φ(y)dx dy, (15)

7

where

x0= (1 −)x+βY +κ(1 −δ)m(t)I(¯

A−1),(16)

with ¯

A∈ {0,1}and, recalling (13),

P(¯

A= 1) = x

κm(t).

In (15), we denoted by Φ : R+→R+the probability density function of the random variable Y

describing the reﬁlling or money operated by the gamblers. Motivated by the discussion contained

in the next Section 3, we assume that Φ is a lognormal probability density function. This agrees

with the behaviour of the gamblers observed in [25] and ensures that the moments of Yare all

ﬁnite. In particular:

M:= ZR+

yΦ(y)dy < +∞.(17)

Taking ϕ(x) = xin (15), we obtain that the mean number of tickets owned by the gamblers

obeys now the equation dm

dt =−δm +βM,

whence

m(t) = m0e−δt +βM

δ1−e−δt(18)

with m0:= m(0) ≥0. Remarkably, mdoes not depend on . Moreover, in presence of reﬁlling, m

is uniformly bounded in time from above and from below:

min m0,βM

δ≤m(t)≤max m0,βM

δ.

Note that, for β, M > 0, the mean number of tickets mno longer decays to zero but tends

asymptotically to the value β M

δ.

Choosing now ϕ(x) = e−iξx, where ξ∈Rand iis the imaginary unit, we obtain the Fourier-

transformed version of the kinetic equation (15):

∂tˆ

f(ξ, t) = 1

ZR2

+De−iξx0−e−iξx Ef(x, t)Φ(y)dx dy, (19)

where, as usual, ˆ

fdenotes the Fourier transform of the distribution function f:

ˆ

f(ξ, t) := ZR+

f(x, t)e−iξx dx.

Taking (13) into account, the right-hand side of (19) can be written as the sum of two contributions:

A(ξ, t) = 1

ZR+e−iβξy −1"ZR+

e−iξ[(1−)x+κ(1−δ)m(t)] x

κm(t)f(x, t)dx

+ZR+

e−i(1−)ξx 1−x

κm(t)f(x, t)dx#Φ(y)dy,

and

B(ξ, t) = 1

ZR+e−iξ[(1−)x+κ(1−δ)m(t)] −e−iξx x

κm(t)f(x, t)dx

+1

ZR+e−i(1−)ξx −e−iξx 1−x

κm(t)f(x, t)dx.

8

In the limit →0+, viz. N→ ∞, we obtain

lim

→0+A(ξ, t) = −iβM ξ ˆ

f(ξ, t)

lim

→0+B(ξ, t) = i

κm(t)e−iκm(t)(1−δ)ξ−1−ξ∂ξˆ

f(ξ, t),

which shows that, for N1 and in the regime (14), the non-linear kinetic model (5) with the

scaling τ=(cf. Remark 3) is well approximated by the Fourier-transformed linear equation

∂tˆ

f=i

κm(t)e−iκm(t)(1−δ)ξ−1−ξ∂ξˆ

f−iβM ξ ˆ

f. (20)

This equation may be used to compute recursively the time evolution of the statistical moments

of f, upon recalling the relationship

mn(t) := ZR+

xnf(x, t)dx =in∂n

ξˆ

f(0, t), n ∈N,(21)

and to check their possible blow up indicating the formation of fat tails in f.

2.3 Explicit steady states and boundedness of moments

To gain further information on (20) in the physical variable x, let us consider at ﬁrst the case in

which the constant κis small, say κ1. Expanding the exponential function appearing in (20)

in Taylor series up to the second order, we obtain

i

κm(t)e−iκm(t)(1−δ)ξ−1−ξ∂ξˆ

f≈−δξ −iκm(t)

2(1 −δ)2ξ2∂ξˆ

f. (22)

Within this approximation, we can go back from (20) to the physical variable xby the inverse

Fourier transform. In particular, we get

∂tf(x, t) = κ(1 −δ)2m(t)

2∂2

x(xf(x, t)) + ∂x(δx −βM )f(x, t),(23)

which is a Fokker-Planck-type equation with variable diﬀusion coeﬃcient. Notice that the mean

value of the solution to (23) coincides with (18). In particular, if m0=βM

δthen the mean value

remains constant in time:

m(t)≡βM

δ∀t > 0.

In this simple case, (23) has a stationary solution, say f∞=f∞(x), which is easily found by

solving the diﬀerential equation

κ(1 −δ)2

2·βM

δ∂x(xf∞)+(δx −βM)f∞= 0

and which turns out to be a gamma probability density function:

f∞(x) = 2δ2

κ(1−δ)2βM 2δ

κ(1−δ)2

Γ2δ

κ(1−δ)2x2δ

κ(1−δ)2−1e−2δ2

κ(1−δ)2βM x.(24)

Since f∞has moments bounded of any order, we conclude that no fat tail is produced in this case.

In the general case, i.e. without invoking the approximation (22), we may check that the same

qualitative asymptotic trend emerges by resorting to the following argument. Let us deﬁne

D(ξ, t) := i

κm(t)e−iκm(t)(1−δ)ξ−1−(1 −δ)ξ,

9

so that (20) may be rewritten as

∂tˆ

f=D(ξ, t)∂ξˆ

f−δξ∂ξˆ

f−iβM ξ ˆ

f. (25)

The function D(ξ, t) satisﬁes

D(0, t) = ∂ξD(0, t)=0,

while, for n≥2,

∂n

ξD(0, t)=(iκm(t))n−1(1 −δ)n,

and further, owing to the Leibniz rule,

∂n

ξD(ξ, t)∂ξˆ

f(ξ, t)ξ=0 =

n

X

k=2 n

k∂k

ξD(0, t)∂n−k+1

ξˆ

f(0, t).(26)

Notice that the highest order derivative of ˆ

fappearing on the right-hand side of (26) is of order

n−1. Therefore, taking the nth ξ-derivative of (25) and computing in ξ= 0, while recalling (21),

yields, for n≥2, dmn

dt =−nδmn+E(m1, . . . , mn−1),(27)

where Eis a term containing only moments of order at most equal to n−1. The exact expression

of Emay be obtained from (21)-(26) but, in any case, (27) shows recursively that the statistical

moments of fof any order are uniformly bounded in time if they are bounded at the initial time.

Therefore, we conclude that fat tails do not form also in the general case described by (20).

Remark 4.The uniform boundedness of all moments of fhas been actually proved only for

the linearised kinetic model (11)-(12) in the limit regime →0+, viz. N→ ∞. Nevertheless,

the result so obtained suggests that also the “real” kinetic model, described by the highly non-

linear Boltzmann-type equation (5), may behave in the same way. This is in contrast with the

conclusions drawn in [25], where, resorting to some simpliﬁed models, the authors justify the

formation of power law tails in the distribution of the gambler winnings.

It is noticeable that equation (20), obtained in the limit of a very large number of gamblers

participating in a round of the jackpot game, maintains all the essential features of the game. In

particular, it preserves the fact that there is only one winner in each round. This imposes a strong

correlation between the winnings of the gamblers, which clearly remains also in the limit. These

characteristics are very close to those of the situation described in [1], where explicit steady states

for a model of a pure gambling between two players are found. Speciﬁcally, if in each round there

is exactly one winner and one loser then it is proved that the steady state possesses all moments

bounded. Conversely, if both gamblers may simultaneously win or lose in a round then power law

tails appear at equilibrium.

2.4 Are power law tails correct?

As brieﬂy outlined in Remark 4, the solution to the linearised kinetic model of the jackpot game

does not possess fat tails. In order to investigate the possible reasons behind the fat tails apparently

observed in [25], in the following we introduce an alternative linear kinetic model of the jackpot

game, still derived from the microscopic interaction (4), whose equilibrium density exhibits indeed

power-law-type fat tails. This new model may be obtained by resorting to a diﬀerent linearisation

of (5). Nevertheless, as observed via numerical experiments in the next Section 4, such a linearised

equation, while apparently very close to the original non-linear model, produces a quite diﬀerent

large-time trend compared to the one described by (20).

Let us ﬁx β= 0 in (4) and let us assume, without loss of generality, that the extracted winner

is the gambler k= 1. Then:

x0

1= (1 −)x1+(1 −δ)

N

X

j=1

xj

x0

k= (1 −)xk, k = 2,3, . . . , N,

10

which implies (cf. also (8)):

N

X

k=1

(x0

k)2= (1 −)2

N

X

k=1

x2

k+ 2(1 −)(1 −δ)x1

N

X

k=1

xk+2(1 −δ)2 N

X

k=1

xk!2

.

Taking into account the expression (6) of the mean value, we obtain

N

N

P

k=1

(x0

k)2

N

P

k=1

x0

k2=(1 −)2

(1 −δ)2

N

N

P

k=1

x2

k

N

P

k=1

xk2+N2(1 −δ)2+ 2N (1 −)(1 −δ)x1

N

P

k=1

xk

≈

N

N

P

k=1

x2

k

N

P

k=1

xk2

for N1 large and, consequently, 1 small. Indeed,

x1

N

P

k=1

xk

=

1

Nx1

1

N

N

P

k=1

xk

≈x1

Nm(t)

N→∞

−−−−→ 0.

In other words, for a large number Nof gamblers and a correspondingly small percentage of

tickets played in a single game, the relationship (14) implies that the quantity

χ:=

N

N

P

k=1

x2

k

N

P

k=1

xk2(28)

may be regarded approximately as a collision invariant of the interaction (4). Since

N

X

k=1

xk!2

≤N

N

X

k=1

x2

k,

it follows that χ≥1. Note that this result does not depend on the choice of the winner in each

round of the jackpot game.

Using (14) and (28) in (8), in this asymptotic approximation we obtain:

*N

X

k=1

(x0

k)2+=(1 −)2+ 2(1 −)(1 −δ)N

X

k=1

x2

k+(1 −δ)2κ

χ

N

X

k=1

x2

k

=(1 −δ)2+(1 −δ)2κ

χ− N

X

k=1

x2

k,(29)

whence, choosing ϕ(x) = x2in (5),

dm2

dt =(1 −δ)2κ

χ−2−2(1 −2δ)m2.(30)

This equation shows that the ratio κ/χ is of paramount importance to classify the large-time trend

of the energy of the distribution f, hence also of fitself. Indeed, the sign of the coeﬃcient

c(κ, χ, ) := (1 −δ)2κ

χ−2−2(1 −2δ)

11

determines if fconverges asymptotically in time to a Dirac delta centred in x= 0 (when

c(κ, χ, )<0) or if it spreads on the whole positive real line (when c(κ, χ, )>0).

This discussion suggests a consistent way to eliminate the time dependence in the interac-

tion (12), while preserving the main macroscopic properties of the jackpot game, such as the right

time evolutions of the mean, cf. (7), and of the energy, cf. (30). Speciﬁcally, we proceed as follows.

For all observable quantities ϕ=ϕ(x), we consider the linear kinetic model (11) with the following

linear interaction rule:

x0= (1 −δ)x+√xη,(31)

where > 0. In (31), ηis a discrete random variable taking only the two values −√(1 −δ),

M/√with probabilities

Pη=−√(1 −δ)= 1 −p,Pη=M

√=p,

where p∈[0,1] and M>0 are two constants to be properly ﬁxed.

We interpret the rule (31), together with the prescribed values of η, as follows: a gambler,

who enters the game with a number of tickets (viz. an amount of money) equal to x, may either

win a jackpot equal to (M−δ)xwith probability por lose the amount x put into the game

with probability 1 −p.

In particular, we determine pby imposing hηi= 0, which guarantees that (31) reproduces the

correct evolution of the mean provided by (12) (indeed, in such a case we have hx0i= (1 −δ)x).

We ﬁnd then

p=(1 −δ)

M+(1 −δ).

Using this, we discover hη2

i=M(1 −δ), whence

h(x0)2i=(1 −δ)2+(1 −δ)Mx2.(32)

A comparison between formulas (29) and (32) allows us to conclude that the choice

M= (1 −δ)κ

χ−

further implies a time evolution of the energy identical to (30). Notice that the positivity of M

is guaranteed by choosing 1 small enough.

After deriving the linearised model for β= 0, we may re-include the reﬁlling of tickets/money

in the interaction rule:

x0= (1 −δ)x+βY +√xη,(33)

where, as stated in Section 2.2, the random variable Y∈R+is described by a prescribed lognormal

probability density function Φ : R+→R+.

Within this approximation of the dynamics, the evolution of the distribution function g=

g(x, t) of the tickets (viz. the money) played and won by a large number of gamblers participating

in the jackpot game is then described by the linear kinetic equation (cf. also (15)):

d

dt ZR+

ϕ(x)g(x, t)dx =1

τZR2

+hϕ(x0)−ϕ(x)ig(x, t)Φ(y)dx dy (34)

with x0given by (33).

2.4.1 Fokker-Planck description of the jackpot game

The linear kinetic equation (34) describes the evolution of the distribution function due to inter-

actions of type (33). As discussed in Section 2.2, for large values of the number Nof gamblers

participating in a round, and therefore, in view of (14), a small value of , the interaction (33)

12

produces a small variation in the number of tickets owned by a gambler. We say then that,

in such a regime, the interaction (33) is quasi-invariant or grazing. Consequently, a ﬁnite (i.e.,

non-inﬁnitesimal) evolution of the distribution function gmay be observed only if each gambler

participates in a huge number of interactions (33) during a ﬁxed period of time. This is achieved

by means of the scaling τ∼like in Section 2.2, cf. Remark 3.

In this scaling, the kinetic model (34) is shown to approach its continuous counterpart given

by a Fokker-Planck-type equation [7,16,24]. In the present case, (34) is well approximated by

the following weak form of a new linear Fokker-Planck equation with variable coeﬃcients:

d

dt ZR+

ϕ(x)g(x, t)dx =ZR+−ϕ0(x)(δx −βM) + ˜σ

2ϕ00(x)x2g(x, t)dx, (35)

where Mis the mean reﬁlling of tickets, cf. (17), and where we have deﬁned

˜σ:= lim

→0+M= (1 −δ)κ

χ.

Then, provided the boundary terms produced by the integration by parts vanish, (35) may be

recast in strong form as

∂tg(x, t) = ˜σ

2∂2

x(x2g(x, t)) + ∂x((δx −βM)g(x, t)).(36)

This equation describes the evolution of the distribution function gof the number of tickets x∈R+

played by the gamblers at time t > 0 in the limit of the grazing interactions. The advantage of

this equation over (34) is that its unique steady state g∞with unitary mass may be explicitly

computed:

g∞(x) = 2βM

˜σ1+2δ

˜σ

Γ1 + 2δ

˜σ·e−2βM

˜σx

x2+ 2δ

˜σ

.(37)

We observe that this is an inverse gamma probability density function with parameters linked to

the details of the microscopic interaction (33).

Remark 5.A comparison between (23) and the Fokker-Planck equation (36) shows that, while the

drift term is the same, the coeﬃcient of the diﬀusion term is proportional to xin (23) and to x2

in (36). This diﬀerence determines, in the latter case, the formation of fat tails, which is consistent

with the claim made in [25]. Nevertheless, as brieﬂy explained before, the approach based on the

interaction (33) leading to (36) in the quasi-invariant regime does not actually describe exactly

the jackpot game. Indeed, it admits that all gamblers may win simultaneously, although with a

very small probability.

2.4.2 The case β= 0

Further explicit computations on the Fokker-Planck equation (36) may be done in the case β= 0,

which corresponds to the situation in which gamblers enter the game with a certain number

of tickets, viz. amount of money, and use only those tickets, viz. money, to play. Then, the

distribution function g=g(x, t) solves the equation

∂tg(x, t) = ˜σ

2∂2

x(x2g(x, t)) + δ∂x(xg(x, t)).(38)

Setting

˜g(x, t) := e−δt g(e−δt x, t),

which is easily checked to be in turn a distribution function with unitary mass at each time t > 0,

we see that ˜gsolves the diﬀusion equation

∂t˜g(x, t) = ˜σ

2∂2

x(x2˜g(x, t)) (39)

13

with the same initial datum as that prescribed to (38), because ˜g(x, 0) = g(x, 0).

The unique solution to (39) corresponding to an initial datum g0(x) is given by the expression:

˜g(x, t) = ZR+

1

zg0x

zLt(z)dz, (40)

where

Lt(x) := 1

√2π˜σtx exp −(log x+˜σ

2t)2

2˜σt !

is a lognormal probability density. Indeed, (39) possesses a unique source-type solution given by

a lognormal density with unit mean, which at time t= 0 coincides with a Dirac delta centred in

x= 1, cf. [21].

Both the mass and the mean of (40) are conserved in time, while initially bounded moments

of order n≥2 grow exponentially at rate n(n−1). Moreover, (40) can be shown to converge in

time to Lt(x) in various norms, see [21].

Starting from (40), we easily obtain that the unique solution to the original Fokker-Planck

equation (38) is given by

g(x, t) = ZR+

1

zg0x

z˜

Lt(z)dz, (41)

where

˜

Lt(x) = 1

√2π˜σtx exp −log x+ (δ+˜σ

2)t2

2˜σt !.(42)

Notice that, as expected, the mean value of the lognormal density (42) decays exponentially in

time: ZR+

x˜

Lt(x)dx =e−δt.

Consequently, if Xt∼g(x, t) is a stochastic process with probability density equal to the solution

of (38), the mean of Xtdecays exponentially to zero at the same rate and

hXti=ZR+

xg(x, t)dx =e−δt hX0i.

Taking advantage of the representation formula (41), we can easily compute also higher order

moments of the solution. In particular, the variance of Xtis equal to

hX2

ti−hXti2=hX2

0ie(˜σ−2δ)t− hX0i2e−2δt.

From here, we see that the large time trend of the variance depends on the sign of the quantity

˜σ−2δ. If ˜σ < 2δ, the variance converges exponentially to zero, thus all gamblers tend, in the long

run, to lose all their tickets (viz. money). Conversely, if ˜σ > 2δ, the variance blows up for large

times. This situation is analogous to the winner takes it all behaviour [16], where the asymptotic

steady state is a Dirac delta centred in zero but at any ﬁnite time a small decreasing number of

gamblers possesses a huge number of tickets, suﬃcient to sustain the growth of the variance.

3 Agent behaviour on gambling

A non-secondary aspect of the online gambling is related to the behavioural trends of the gamblers.

The data analysis in [25] focuses, in particular, on two characteristics of the gambling activity:

ﬁrst, the waiting time, deﬁned as the time, measured in seconds, between successive bets by the

same gambler; second, the number of rounds played by individual gamblers. The study of this

second aspect may shed light on the reasons behind a high gambling frequency and therefore also

on possible addiction problems caused by gambling.

14

The ﬁtting of the number of rounds played by individual gamblers during the period covered by

gambling logs allowed the authors of [25] to conclude that the number of rounds is well described

by a lognormal distribution. This result is in agreement with other studies, cf. e.g. [15] and

references therein, where the mean gambling frequency is put in close relation with the alcohol

consumption. Starting from the pioneering contribution [14], it has long been acknowledged that

there exists a positive correlation between the level of alcohol consumption in a population and

the proportion of heavy drinkers in the society. This relationship is known under several names,

such as the total consumption model or the single distribution theory. Previous research has also

found that its validity is not limited to the alcohol consumption but extends to diﬀerent human

phenomena.

In some recent papers [8,10], we introduced a kinetic description of a number of human beha-

vioural phenomena, which recently has been applied also to the study of alcohol consumption [5].

The modelling assumptions in [5] allowed us to classify the alcohol consumption distribution as a

generalised gamma probability density, which includes the lognormal distribution as a particular

case. Recalling that, as discussed above, alcohol consumption shows a lot of similarities with the

gambling activity and taking inspiration from [5,10], we may explain exhaustively two main phe-

nomena linked to the gambler behaviour. On one hand, the distribution of the number of tickets

which individual gamblers play (including the reﬁlling) in a single round of the jackpot game.

On the other hand, the distribution of the number of rounds played by individual gamblers in

time. Concerning this second aspect, a ﬁtting of empirical data is presented and analysed in [25].

Conversely, no mention is made therein about the ﬁrst aspect. For this reason, in the following

we will be mainly interested in the problem of the distribution of the number of tickets used by

gamblers in each round, which provides the law of the Yk’s appearing in (3), (4) and of Yappearing

in (16). From the discussion about it, it will be possible to draw conclusions also on the problem

of the number of rounds played by individual gamblers, since both problems are actually subject

to identical microscopic rules, cf. Remark 8below.

3.1 Kinetic modelling and value functions

The evolution of the number density of tickets which the gamblers purchase to participate in

successive rounds of the jackpot game may be still treated resorting to the principles of statistical

mechanics. Speciﬁcally, one can think of the population of gamblers as a multi-agent system:

each gambler undergoes a sequence of microscopic interactions, through which s/he updates the

personal number of tickets. In order to keep the connection with the classical kinetic theory of

rareﬁed gases, these interactions obey suitable and universal rules, which, in the absence of well-

deﬁned physical laws, are designed so as to take into account at best some of the psychological

aspects related to gambling.

Due to the nature of the game, the players know that there is a high probability to lose and

a small one to win. For this reason, they are usually prepared to participate in a sequence of

rounds, hoping to win in at least one of them. The involvement in the game pushes the gamblers

to participate in successive rounds by purchasing an increasing number of tickets, so as to increase

the probability to win. On the other hand, the attempt to safeguard the personal wealth suggests

them to ﬁx an a priori upper bound to the number of tickets purchased. These two aspects, clearly

in conﬂict, are characteristic of a typical human behaviour, which has been recently modelled in

similar situations [5,8,10]. There, the microscopic interactions have been built taking inspiration

from the pioneering analysis by Kahneman and Twersky [12] about decisional processes under

risk.

In the present case, the aforementioned safeguarding tendency may be modelled by assuming

that the gamblers have in mind an ideal number ¯w > 0 of tickets to buy in each round and,

simultaneously, a threshold ¯wL>¯w, which they had better not exceed in order to avoid a (highly

probable) excessive loss of money. Hence, the natural tendency of the gamblers to increase their

number of tickets w > 0 bought for the forthcoming rounds has to be coupled with the limit value

¯wL, which it would be wise not to exceed. Following [8,10], we may realise a gambler update via

15

01−∆s11 + ∆s

−µ

Ψ(1 −∆s)

0

Ψ(1 + ∆s)

s

Figure 1: The function Ψ given in (44).

the following rule:

w0=w−Ψw

¯wLw+wη. (43)

In (43), w,w0denote the numbers of tickets played in the last round and in the forthcoming one,

respectively. The function Ψ plays the role of the so-called value function in the prospect theory

by Kahneman and Twersky [12]. Speciﬁcally, it determines the update of the number of tickets in

askewed way, so as to reproduce the behavioural aspects discussed above. Analogously to [8], we

let

Ψ(s) := µsα−1

sα+ 1, s ≥0,(44)

where µ, α ∈(0,1) are suitable constants characterising the agent behaviour. In particular, µ

denotes the maximum variation in the number of tickets allowed in a single interaction (43), indeed

|Ψ(s)| ≤ µ∀s≥0.(45)

Hence, a small value of µdescribes gamblers who buy a regular number of tickets in each round.

The function Ψ given in (44) maintains most of the physical properties required to the value

function in the prospect theory [12] and is particularly suited to the present situation. In the

microscopic interaction (43), the minus sign in front of Ψ is related to the fact that the desire to

increase the probability to win pushes a gambler to increase the number wof purchased tickets

when w < ¯wL. At the same time, the tendency to safeguard the personal wealth induces the

gambler to reduce the number of purchased tickets when w > ¯wL. Moreover, the function Ψ is

such that

−Ψ(1 −∆s)>Ψ(1 + ∆s),∀∆s∈(0,1),

cf. Figure 1. This inequality means that, if two gamblers are at the same distance from the

limit value ¯wLfrom below and from above, respectively, the gambler starting from below will

move closer to the optimal value ¯wLthan the gambler starting from above. In other words, it is

typically easier for a gambler to allow her/himself to buy more tickets, when the optimal threshold

has not been exceeded, than to limit her/himself, when the optimal threshold has already been

exceeded.

Finally, in order to take into account a certain amount of human unpredictability in buying

tickets in a new round, it is reasonable to assume that the new number of tickets may be aﬀected

16

by random ﬂuctuations, expressed by the term wη in (43). Speciﬁcally, ηis a centred random

variable

hηi= 0,hη2i=λ > 0,

meaning that the random ﬂuctuations are negligible on average. Moreover, to be consistent with

the necessary non-negativity of w0, we assume that η > −1 + µ, i.e. that the support of ηis

bounded from the left.

Remark 6.The behaviour modelled by (43), which in principle concerns only the losers, may

actually be applied also to the unique winner. Indeed, if the winner remains into the game, the

pleasure to play will be dominant, so that it is reasonable to imagine that the future behaviour

will not depend too much on the number of tickets gained in the last round.

Remark 7.As discussed in [8], the function (44) may be modiﬁed to better match the phenomenon

under consideration. For example, in order to diﬀerentiate the rates of growth and of decrease of

Ψ and to stress the diﬃculty of the gamblers to act against such a skewed trend, one may consider

the following modiﬁed value function:

Ψ(s) = µsα−1

νsα+ 1 , s ≥0,

with ν > 1, so that the bounds (45) modify to

−µ≤Ψ(s)≤µ

ν< µ.

In this case, the possibility to go against the natural tendency is slowed down. Also, as discussed

in [5], the shape of the value function (44) can be generalised so as to better take into account

possible addiction eﬀects. The general class of value functions considered there is given by

Ψ(s) = µe(sδ−1)/δ −1

e(sδ−1)/δ + 1, s ≥0,(46)

where δ∈(0,1] is a constant. This choice leads to diﬀerent skewed steady states, in the form of

generalised gamma densities.

Remark 8.The discussion set forth applies also to the modelling of the number of rounds played

by individual gamblers in a ﬁxed period of time, which has been considered in [25]. In particular,

we may assume that the gamblers establish a priori to play for a limited number of times, in order

to spend only a certain total amount of money. But then, as it happens in the single game, it is

more diﬃcult to stop than to continue. This can be well described by the rule (43) and by the

value function (44), where now wrepresents the number of rounds played in the time period.

Let now h=h(w, t) be the distribution function of the number of tickets purchased by a

gambler in a certain round of the jackpot game. As anticipated at the beginning of this section,

its time evolution may be obtained by resorting to kinetic collision-like models [16] based on (43).

In particular, since the interaction (43) depends only on the behaviour of a single gambler, hobeys

a linear Boltzmann-type equation of the form

d

dt ZR+

ϕ(w)h(w, t)dw =1

τZR+hϕ(w0)−ϕ(w)ih(w, t)dw, (47)

cf. (11), where the constant τ > 0 measures the interaction frequency and ϕis any observable

quantity.

Since the elementary interaction (43) is non-linear with respect to w, the only conserved

quantity in (47) is obtained from ϕ(w) = 1:

d

dt ZR+

h(w, t)dw = 0,

17

which implies that the solution to (47) remains a probability density at all times t > 0 if it is so at

the initial time t= 0. The evolution of higher order moments is diﬃcult to compute explicitly. As

a representative example, let us take ϕ(w) = w, which provides the evolution of the mean number

of tickets purchased by the gamblers over time:

m(t) := ZR+

wh(w, t)dw.

Since

hw0−wi=µwα−¯wα

L

wα+ ¯wα

L

w,

we obtain dm

dt =µ

τZR+

wα−¯wα

L

wα+ ¯wα

L

wh(w, t)dw. (48)

This equation is not explicitly solvable. However, in view of (45), mremains bounded at any time

t > 0 provided it is so initially, with the explicit upper bound, cf. [10],

m(t)≤m0eµ

τt,

where m0:= m(0). From (48) it is however not possible to deduce whether the time variation of

mis or is not monotone.

Taking now ϕ(w) = w2in (47) and considering that

h(w0)2−w2i=Ψ2w

¯wL−2Ψ w

¯wL+λw2≤(3µ+λ)w2

because of (45) together with 0 <µ<1, we see that the boundedness of the energy at the initial

time implies that of the energy at any subsequent time t > 0, with the explicit upper bound

m2(t)≤m2,0e3µ+λ

τt,

where m2,0:= m2(0).

3.2 Fokker-Planck description and equilibria

The linear kinetic equation (47) is valid for every choice of the parameters α,µand λ, which

characterise the microscopic interaction (43). In real situations, however, a single interaction,

namely a participation in a new round of the jackpot game, does not induce a marked change in

the value of w. This situation is close to that discussed in Section 2.4.1, where we called these

interactions grazing collisions [16,24].

Similarly to Section 2.4.1, we may easily take such a smallness into account by scaling the

microscopic parameters in (43), (47) as

α→α, λ →λ, τ =, (49)

where 0 < 1. A thorough discussion of these scaling assumptions may be found in [7,8]. In

particular, here we mention that the rationale behind the coupled scaling of the parameters α,λ

and of the frequency of the interactions τis the following: since the scaled interactions are grazing,

and consequently produce a very small change in w, a ﬁnite (i.e. non-inﬁnitesimal) variation of the

distribution function gmay be observed only if each gambler participates in a very large number

of interactions within a ﬁxed period of time.

As already observed in Section 2.4.1, when grazing interactions dominate, the kinetic model (47)

is well approximated by a Fokker-Planck type equation [16,24]. Exhaustive details on such an

approximation in the kinetic theory of socio-economic systems may be found in [7]. In short, the

mathematical idea is the following: if ϕis suﬃciently smooth and w0≈wbecause interactions are

18

grazing, one may expand ϕ(w0) in Taylor series about w. Plugging such an expansion into (47)

with the value function (44) and taking the scaling (49) into account one obtains:

d

dt ZR+

ϕ(w)h(w, t)dw =ZR+−αµ

2ϕ0(w)wlog w

¯wL

+λ

2ϕ00(w)w2h(w, t)dw +1

R(w, t),

where Ris a remainder such that 1

R→0 as →0+, cf. [7]. Therefore, under the scaling (49),

the kinetic equation (47) is well approximated by the equation

d

dt ZR+

ϕ(w)h(w, t)dw =ZR+−αµ

2ϕ0(w)wlog w

¯wL

+λ

2ϕ00(w)w2h(w, t)dw.

This equation may be recognised as the weak form of the following Fokker-Planck equation with

variable coeﬃcients:

∂th(w, t) = λ

2∂2

w(w2h(w, t)) + αµ

2∂wwlog w

¯wL

h(w, t),(50)

upon assuming that the boundary terms produced by the integration by parts vanish. Like in

Section 2.4.1, the Fokker-Planck description (50) is advantageous over the original Boltzmann-

type equation (47) because it allows for an explicit computation of the steady state distribution

function, say h∞=h∞(w). The latter solves the following ﬁrst order ordinary diﬀerential equation:

λ

2

d

dw (w2h∞(w)) + αµ

2wlog w

¯wL

h∞(w)=0,

whose unique solution with unitary mass is

h∞(w) = 1

√2πσw exp −(log w−θ)2

2σ!,(51)

where

σ:= λ

αµ, θ := log ¯wL−σ.

Therefore, in very good agreement with the observations made in [25], the equilibrium distribution

function predicted by the microscopic rule (43) with the value function (44) in the grazing interac-

tion regime is a lognormal probability density, whose mean and variance are easily computed from

the known formulas for lognormal distributions:

m∞:= ¯wLe−σ

2,Var(g∞) := ¯w2

L(1 −e−σ).

In particular, these quantities are fractions of ¯wL, ¯w2

L, respectively, depending only on the ratio

σ=λ

αµ between the variance λof the random ﬂuctuation ηand the portion αµ of the maximum

rate µof variation in the number of tickets purchased by a gambler in a single round. If

σ > 2 log ¯wL

¯w

then the asymptotic mean m∞is lower than the ﬁxed ideal number ¯wof tickets to be purchased

in each round. This identiﬁes a population of gamblers capable of not being too deeply involved

in the jackpot game.

Figure 2shows that the asymptotic proﬁle (51) describes excellently the large time distribution

of the Boltzmann-type equation (47) in the quasi-invariant regime (i.e., small in (49)). The

solution to (47) has been obtained numerically via a standard Monte Carlo method.

Remark 9.As shown in [5], using the value function (46) in place of (44) yields a skewed steady

state distribution in the class of the generalised gamma densities. Such densities share most of the

properties of the lognormal density and, as it happens in the problem of the alcohol consumption,

might provide a better correspondence with the empirically observed proﬁles if they are used to

ﬁt the number of tickets purchased by the gamblers. In any case, the main aspect of the steady

state, namely its rapid decay at inﬁnity due to a slim tail, remains unchanged.

19

Figure 2: Comparison of (51) with the numerically computed large time solution (at the computa-

tional time T= 10) to the Boltzmann-type equation (47). The value function is (44) with µ= 0.5,

α= 1. Moreover, the binary interaction is (43) with λ=1

10 and wL=eλ

2µ. We considered the

quasi-invariant scaling (49) with = 10−1(left panel) and = 10−2(right panel).

4 Numerical tests

In this section, we provide numerical insights into the various models discussed before, resorting

to direct Monte Carlo methods for collisional kinetic equations and to the recent structure pre-

serving methods for Fokker-Planck equations. For a comprehensive presentation of these numerical

methods, the interested reader is addressed to [4,16,18,19].

We begin by integrating the multiple-interaction Boltzmann-type model (5), so as to assess

its equivalence with the linearised model (15) in the case N1 with N =κ > 0, as predicted

theoretically in Section 2.2. Next, we also test numerically the consistency of the Fokker-Planck

equation (23) with the linearised Boltzmann-type equation (15). Subsequently, we investigate the

kinetic model with fat tails discussed in Section 2.4.1. In particular, we evaluate numerically some

discrepancies that it presents with the other models.

4.1 Test 1: the multiple-interaction Boltzmann-type model and its lin-

earised version

The multiple-interaction Boltzmann-type equation (5) can be fruitfully written in strong form, to

put in evidence the gain and loss parts of the integral operator:

∂tf(x, t) = 1

*ZRN−1 1

J

N

Y

k=1

f(0

xk, t)−

N

Y

k=1

f(xk, t)!dx2. . . dxN+

=1

Q+(f, . . . , f )(x, t)−1

f(x, t),(52)

where Q+is the gain operator:

Q+(f, . . . , f )(x, t) := 1

*ZRN−1

N

Y

k=1

1

Jf(0

xk, t)dx2. . . dxN+

and Jis the Jacobian of the transformation (4) from the pre-interaction variables {0

xk}N

k=1 to the

post-interaction variables {xk}N

k=1

We discretise (52) in time through a forward scheme on the mesh tn:= n∆t, ∆t > 0. With

the notation fn(x) := f(x, tn), we obtain the following semi-discrete formulation:

fn+1(x) = 1−∆t

fn(x) + ∆t

Q+(fn, . . . , f n)(x).

20

0 1 2 3 4 5

0

0.1

0.2

0.3

0.4

0.5

(a) t= 0.1

0 1 2 3 4 5

0

0.2

0.4

0.6

(b) t= 1

0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

(c) t= 2

Figure 3: Test 1 – Without reﬁlling. Evolution of the multiple-interaction Boltzmann-type

model with either N= 5 gamblers (empty circular markers) or N= 100 gamblers (triangular

markers) and of its linearised version (ﬁlled circular markers) in the time interval [0,2] for δ= 0.2,

β= 0. We considered κ= 0.1.

By choosing ∆t=, the loss part disappears and at each time step only the gain operator Q+

needs to be computed.

We recall that the multiple-interaction microscopic dynamics are given by (4). In particular,

motivated by the results of Section 3, we choose the Yk’s as independent and identically distributed

random variables with lognormal probability density:

Φ(y) = 1

√4πy exp −(log y+ 1)2

2!.(53)

A comparison with (51) shows that this corresponds to σ= 2 and ¯wL=e, so that M=hYki= 1

for all k.

Parallelly, we consider the linearised Boltzmann-type equation (15), which we have shown to

be formally equivalent to the multiple-interaction model for a large number of gamblers N. The

semi-discrete in time formulation of the linearised model reads

fn+1(x) = 1−∆t

fn(x) + ∆t

*ZR+

1

Jfn(0

x)dx+,

where now the microscopic dynamics are given by (16) with κ=N and Y∼Φ(y) like before,

cf. (53).

In both cases, we solve the interaction dynamics by a Monte Carlo scheme, considering a

random sample of 106particles with initial uniform distribution in the interval [0,2], thus f0(x) :=

f(x, 0) = 1

21[0,2](x), where 1denotes the characteristic function.

In Figure 3, we compare the evolutions of the two models in the time interval t∈[0,2] for

δ= 0.2, β= 0 in (4), (16), cf. also (10), i.e., in particular, with no reﬁlling. In Figure 4, we

perform the same test in the larger time interval t∈[0,25] for δ=β= 0.2, i.e. by including also

the reﬁlling. In both cases, we clearly see that, if Nis suﬃciently large, the linearised model is

able to catch the multiple-interaction dynamics at each time, whereas diﬀerences can be observed

if Nis relatively small.

Moreover, in the linearised model, we know that the mean number of tickets owned by the

gamblers during the jackpot game is given by (18). In Figure 5, we show instead the time evolution

of the mean of the solution to the multiple-interaction Boltzmann-type model for several choices

of the reﬁlling parameter β. We observe a good agreement with the theoretical results and, in

particular, we see that the mean value tends indeed asymptotically to βM

δ, as expected.

4.2 Test 2. Fokker-Planck approximation for large N

In the case , κ 1, the interactions (16) are quasi-invariant, hence the linearised Boltzmann-type

model (15) is well described by the Fokker-Planck equation (23). In the case of a constant mean

21

012345

0

0.1

0.2

0.3

0.4

0.5

0.6

(a) t= 1

012345

0

0.2

0.4

0.6

0.8

1

1.2

(b) t= 5

012345

0

0.5

1

1.5

2

(c) t= 25

Figure 4: Test 1 – With reﬁlling. Evolution of the multiple-interaction Boltzmann-type model

with either N= 5 gamblers (empty circular markers) or N= 100 gamblers (triangular markers)

and of its linearised version (ﬁlled circular markers) in the time interval [0,25] for δ=β= 0.2

(lognormal reﬁlling sampled from (53)). We considered κ= 0.1.

0 5 10 15 20 25

0

0.5

1

1.5

Figure 5: Test 1 – Evolution of the mean. Evolution of the mean number of tickets m(t) in

the time interval [0,25] for δ= 0.2, κ= 0.1 and several choices of β.

value m(t)≡m0=βM

δof the number of tickets owned by the gamblers, the steady distribution

is the gamma probability density function (24). In this section, we compare numerically the large

time distributions produced by either the multiple-interaction Boltzmann-type model (5) or the

linearised Boltzmann-type model (15) with (24).

Like before, we consider a uniform initial distribution f0(x) in the interval [0,2] and moreover a

random variable Ylognormally distributed according to (53), thus in particular with mean M= 1.

We also set β=δ= 0.2 in the microscopic interactions (4), (16), so that the mean value of the

ticket distribution is always m0=βM

δ= 1, consistently with the Fokker-Planck regime in which

we are able to compute explicitly the steady distribution (24).

In Figure 6(a), we compare the large time distribution of the multiple-interaction Boltzmann-

type model for an increasing number of gamblers participating in each round of the jackpot game

(N= 102,N= 103, respectively) with the asymptotic gamma probability density (24) computed

from the Fokker-Planck equation. We clearly see that, for Nlarge enough, the Fokker-Planck

steady solution provides a good approximation of the equilibrium distribution of the real multiple-

interaction model. In Figure 6(b), we show the log-log plot of the same distributions, which allows

us to appreciate that, in particular, the Fokker-Planck solution reproduces correctly the tail of

the equilibrium distribution of the multiple-interaction model, thereby conﬁrming that no fat tails

have to be expected in the distribution of the tickets owned by the gamblers.

In Figure 6(c), we compare instead the large time distribution of the linearised Boltzmann-type

22

0 1 2 3 4 5

0

0.5

1

1.5

(a)

100101

10-2

100

(b)

0 1 2 3 4 5

0

1

2

3

(c)

100

10-2

100

(d)

Figure 6: Test 2. Top row: (a) Comparison of the steady distribution of the multiple-interaction

Boltzmann-type model (5) with the Fokker-Planck asymptotic distribution (24) (solid line) for

N= 102(empty circular markers), N= 103(ﬁlled circular markers) and ﬁxed κ= 0.1. (b)

Log-log plot of (a). Bottom row: (c) Comparison of the steady distribution of the linearised

Boltzmann-type model (15) with the Fokker-Planck asymptotic distribution (24) (solid line) for

κ= 0.1 (empty circular markers) and κ= 0.01 (ﬁlled circular markers). (d) Log-log plot of (c).

model with the asymptotic gamma probability density (24) for decreasing values of κ(κ= 0.1,

κ= 0.01, respectively). In Figure 6(d), we show the log-log plot of the same distributions to

stress, in particular, the goodness of the approximation of the tail provided by (24).

4.3 Test 3. The fat tail case

In Section 2.4.1, we derived the alternative linear Boltzmann-type model (33)-(34), which preserves

some of the main macroscopic properties of the original multiple-interaction model (4)-(5). In

particular, it accounts for the right evolution of the ﬁrst and second moment of the distribution

function.

We grounded such a derivation on the consideration that, for Nlarge and small, the quantity

χdeﬁned in (28) may be treated approximately as a collision invariant of the N-gambler dynamics.

In Figure 7(a), we test numerically this assumption by taking N= 102and some values of the

scaling parameter decreasing from 10−2to 10−4. In particular, since χdepends actually on the

evolving microscopic states x1, . . . , xNof the agents, we plot the time evolution of χfor t∈[0,25].

23

0 5 10 15 20 25

0

0.5

1

1.5

2

2.5

(a)

100

10-200

100

(b)

Figure 7: Test 3. (a) Estimate of the approximate collision invariant χ, cf. (28). (b) Log-log plot

of the distributions (24), (37).

0 1 2 3 4 5

0

0.2

0.4

0.6

(a) t= 1

0 1 2 3 4 5

0

0.5

1

1.5

(b) t= 5

0 1 2 3 4 5

0

1

2

3

(c) t= 25

Figure 8: Test 3. Comparison between the time evolutions of the Boltzmann-type model (33), (34)

(circular markers) and of its Fokker-Planck approximation (36) (starred markers) in the quasi-

invariant regime. The following parameters have been used: β=δ= 0.2, M= 1, κ= 10−2.

The value of the approximate collision invariant χis estimated from the multiple-interaction

Boltzmann-type model like in Figure 7.

Such a time evolution is computed with the Monte Carlo method described in Section 4.1, starting

from an initial sample of S= 106particles. Therefore, we get N= 102sub-samples of S/N = 104

particles, each of which produces a Monte Carlo estimate of the time trend of χ. Out of these

samples, we compute ﬁnally the average time trend of χ, namely each of the curves plotted in

Figure 7(a). Consistently with our theoretical ﬁndings, we observe that, for small enough, χ

may be actually regarded as a collision invariant.

In the quasi-invariant limit, the solution to the linear Boltzmann-type model (33)-(34) has

been shown to approach that of the Fokker-Planck equation (36). Its explicitly computable steady

state is the inverse gamma probability density (37), which, unlike the equilibrium distribution (24)

approximating the trend of the multiple-interaction model for large N, exhibits a fat tail. In

Figure 7(b), we show the log-log plot of the distributions (24), (37), which stresses the diﬀerence

in their tails.

In order to check the consistency of the Fokker-Planck regime described, in the quasi-invariant

limit, by (36) with the Boltzmann-type model (33), (34), in Figure 8we show the time evolution

of the distribution function gcomputed with both models for t∈[0,25], starting from an initial

uniform distribution for x∈[0,2]. In both cases, we treat χas a collision invariant of the N-

gambler model. Thus, we ﬁrst computed the value of χfrom model (4), (5) (with N= 104), then

24

0 20 40 60 80 100

10-2

100

Figure 9: Test 3. Relative L1-error (54) between the solutions to the Fokker-Planck equations (23)

and (36). The numerical solution of both models is obtained by means of semi-implicit SP methods

over the computational domain [0,10] in the xvariable, with ∆t= ∆x= 10/Nxand Nx= 401

nodes.

we used it in the binary rules (33), where χdetermines the values that ηcan take, and in the

diﬀusion coeﬃcient ˜σof (36). From Figure 8, we see that the two models remain close to each

other at every time and approach the same steady distribution for large times, as expected.

Finally, we quantify the distance between the solution fto the Fokker-Planck equation (23),

which reproduces the large time trend of the multiple-interaction Boltzmann-type model (4), (5),

cf. the previous Test 2 (Section 4.2), and the solution gto Fokker-Planck equation (36), which

describes instead the large time trend of the linear diﬀusive Boltzmann-type model (33), (34). We

consider, in particular, the following relative L1-error

Eκ(t) := ZR+

|g(x, t)−f(x, t)|

f(x, t)dx, (54)

for several values of the constant κ, cf. (14), which appears as a coeﬃcient in both Fokker-Planck

equations. In particular, we consider κ= 10−1,10−2,10−3and we take f(x, 0) = g(x, 0) =

1

21[0,2](x) as initial (uniform) distribution. By means of semi-implicit SP methods, we guarantee

the positivity and the large time accuracy of the numerical solution to both models. The interested

reader is referred to [19] for further details on this numerical technique). From Figure 9, we see that

Eκdecreases with κ, although its order of magnitude remains non-negligible. Hence, the diﬀusive

model with fat tails may approach, in a sense, the non-diﬀusive one with slim tails, but visible

diﬀerences remain between them as a consequence of the fact that the diﬀusive model describes

a jackpot game which is not completely equivalent to the real one caught by the non-diﬀusive

model.

5 Conclusions

In this paper, we introduced and discussed kinetic models of online jackpot games, i.e. lottery-type

games which occupy a big portion of the web gambling market. Unlike the classical kinetic theory

of rareﬁed gases, where binary collisions are dominant, in this case the game is characterised by

simultaneous interactions among a large number N1 of gamblers, which leads to a highly

non-linear Boltzmann-type equation for the evolution of the density of the gambler’s winnings.

When participating in repeated rounds of the jackpot game, the gamblers continuously reﬁll the

number of tickets available to play and, at the same time, their winnings undergo a percentage cut

operated by the site which administers the game. Hence, through the study of the evolution of

the mean number of tickets and of its variance, one realises that the solution of the model should

25

approach in time a non-trivial steady state describing the equilibrium distribution of the gambler’s

winnings.

In the limit N→ ∞, we showed that the multiple-interaction kinetic model can be suitably

linearised, so as to get access to analytical information about the large time trend of its solution.

We proposed two diﬀerent linearisations, which, while apparently both consistent with the original

non-linear model, exhibit marked diﬀerences for large times. The solution to the linear model

presented in Section 2.2 converges towards a steady state with all moments bounded. In some

cases, such a steady state can be written explicit in the form of a gamma probability density

function. Conversely, the solution to the linear model considered in Section 2.4 converges towards

a steady state in the form of an inverse gamma probability density function, hence with Pareto-

type fat tails. We explained the diﬀerent trend of the second model as a consequence of a too

strong loss of correlation among the gamblers, which is instead present in the original non-linear

multiple-interaction model and also in its linear approximation proposed in Section 2.2. Numerical

results showed indeed that the solution to this linear model is in perfect agreement with that to

the full non-linear kinetic model.

The main conclusion which can be drawn from the present analysis is that the wealth economy

of a multi-agent system in which the trading activity relies on the rules of the jackpot game does

not lead to a stationary distribution exhibiting Pareto-type fat tails, as it happens instead in a real

economy. Unlike the real trading economy, where the small richest part of the population owns

a relevant percentage of the total wealth, in the economy of the jackpot game the class of rich

people is still very small but it does not own a consistent percentage of the total wealth (measured

in terms of tickets played and won in time). In other words, it is exceptional to become rich by

just playing the jackpot game and, in such a case, it is further exceptional to become very rich.

A non-secondary conclusion of the present analysis is that the rules of the jackpot game imply

a strong correlation among the gamblers participating in the game. Indeed, in each round of the

game there is just one gambler who wins, while all the other gamblers lose. Any approximation

of the full non-linear model needs to take into account this aspect. This is clearly in contrast

with a real trading economy, where the agents may instead take advantage simultaneously of their

trading activity.

Acknowledgements

This research was partially supported by the Italian Ministry of Education, University and Re-

search (MIUR) through the “Dipartimenti di Eccellenza” Programme (2018-2022) – Department

of Mathematics “F. Casorati”, University of Pavia and Department of Mathematical Sciences “G.

L. Lagrange”, Politecnico di Torino (CUP: E11G18000350001) and through the PRIN 2017 project

(No. 2017KKJP4X) “Innovative numerical methods for evolutionary partial diﬀerential equations

and applications”.

This work is also part of the activities of the Starting Grant “Attracting Excellent Professors”

funded by “Compagnia di San Paolo” (Torino) and promoted by Politecnico di Torino.

All the authors are members of GNFM (Gruppo Nazionale per la Fisica Matematica) of INdAM

(Istituto Nazionale di Alta Matematica), Italy.

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