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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.
References
[1] F. Bassetti and G. Toscani. Explicit equilibria in a kinetic model of gambling. Phys. Rev. E,
81(6):066115/1–7, 2010.
[2] P. Binde. Why people gamble: a model with five motivational dimensions. Int. Gambl. Stud.,
13(1):81–97, 2013.
[3] A. V. Bobylev and ˚
A. Windfall. Kinetic modeling of economic games with large number of
participants. Kinet. Relat. Models, 4(1):169–185, 2011.
26
[4] G. Dimarco and L. Pareschi. Numerical methods for kinetic equations. Acta Numerica,
23:369–520, 2014.
[5] G. Dimarco and G. Toscani. Kinetic modeling of alcohol consumption. Preprint:
arXiv:1902.08198, 2019.
[6] M. H. Ernst and R. Brito. Scaling solutions of inelastic Boltzmann equations with over-
populated high energy tails. J. Statist. Phys., 109(3):407–432, 2002.
[7] G. Furioli, A. Pulvirenti, E. Terraneo, and G. Toscani. Fokker-Planck equations in the
modeling of socio-economic phenomena. Math. Models Methods Appl. Sci., 27(1):115–158,
2017.
[8] S. Gualandi and G. Toscani. Call center service times are lognormal: A Fokker-Planck
description. Math. Models Methods Appl. Sci., 28(8):1513–1527, 2018.
[9] S. Gualandi and G. Toscani. Pareto tails in socio-economic phenomena: a kinetic description.
Economics, 12(2018-31):1–17, 2018.
[10] S. Gualandi and G. Toscani. Human behavior and lognormal distribution. Math. Models
Methods Appl. Sci., 2019. doi:10.1142/S0218202519400049.
[11] J. Jonsson, I. Munck, R. Volberg, and P. Carlbring. GamTest: Psychometric evaluation and
the role of emotions in an online self-test for gambling behavior. J. Gambl. Stud., 33(2):505–
523, 2017.
[12] D. Kahneman and A. Tversky. Prospect theory: An analysis of decision under risk. Econo-
metrica, 47(2):263–91, 1979.
[13] S. Kristiansen, M. C. Trabjerg, and G. Reith. Learning to gamble: early gambling experiences
among young people in Denmark. J. Youth Stud., 18(2):133–150, 2015.
[14] S. Lederman. Alcool, alcoolisme, alcolisation. Presses Universitaire de France, Paris, 1956.
[15] I. Lund. The population mean and the proportion of frequent gamblers: Is the theory of total
consumption valid for gambling? J. Gambl. Stud., 24(2):247–256, 2008.
[16] L. Pareschi and G. Toscani. Interacting Multiagent Systems: Kinetic equations and Monte
Carlo methods. Oxford University Press, 2013.
[17] L. Pareschi and G. Toscani. Wealth distribution and collective knowledge: a Boltzmann
approach. Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 372(2028):20130396/1–15, 2014.
[18] L. Pareschi and M. Zanella. Structure preserving schemes for mean-field equations of col-
lective behavior. In C. Klingenberg and M. Westdickenberg, editors, Theory, Numerics and
Applications of Hyperbolic Problems II, HYP 2016., volume 237 of Springer Proceedings in
Mathematics & Statistics, pages 405–421, 2018.
[19] L. Pareschi and M. Zanella. Structure preserving schemes for nonlinear Fokker-Planck equa-
tions and applications. J. Sci. Comput., 74(3):1575–1600, 2018.
[20] F. Slanina. Inelastically scattering particles and wealth distribution in an open economy.
Phys. Rev. E, 69(4):046102/1–7, 2004.
[21] G. Toscani. Kinetic and mean field description of Gibrat’s law. Phys. A, 461(1):802–811,
2016.
[22] G. Toscani, C. Brugna, and S. Demichelis. Kinetic models for the trading of goods. J. Stat.
Phys., 151(3-4):549–566, 2013.
27
[23] G. Toscani, A. Tosin, and M. Zanella. Opinion modeling on social media and marketing
aspects. Phys. Rev. E, 98(2):022315/1–15, 2018.
[24] C. Villani. Contribution `a l’´etude math´ematique des ´equations de Boltzmann et de Landau
en th´eorie cin´etique des gaz et des plasmas. PhD thesis, Paris 9, 1998.
[25] X. Wang and M. Pleimling. Behavior analysis of virtual-item gambling. Phys. Rev. E,
98(1):012126/1–12, 2018.
28
