# Jackpot statistics, a physicist’s approach

István Gere  Physics Department, Babeş-Bolyai University, Cluj-Napoca, 400347,
Romania<EMAIL_ADDRESS>Szabolcs Kelemen  Physics Department,
Babeş-Bolyai University, Cluj-Napoca, 400347, Romania  Zoltán Néda  Physics
Department, Babeş-Bolyai University, Cluj-Napoca, 400347, Romania  and Tamás
S. Biró  HUN-REN Wigner Research Centre for Physics, H-1525 Budapest, P.O.Box
49, Hungary  Complexity Science Hub, Vienna, Austria

###### Abstract

At a first glance lottery is a form of gambling, a game in which the chances
of winning is extremely small. But upon a deeper look, considering that the
Jackpot prize of lotteries is a result of the active participation of millions
of players, we come to the conclusion that the interaction of the simple rules
with the high number of players creates an emergent complex system. Such a
system is characterized by its time-series that presents some interesting
properties. Given the inherent stochastic nature of this game, it can be
described within a mean-field type approach, such as the one implemented in
the Local Growth and Global Reset (LGGR) model. We argue that the Jackpot
time-series behaves ergodic for six lotteries with diverse formats and player
pools. Specifying this consideration in the framework of the LGGR model, we
model the lotteries with growth rates confirmed by the time-series. The reset
rate is deduced mathematically and confirmed by data. Given these parameters
we calculate the probability density of the Jackpot prizes, that fits well the
experimentally observed ones. We propose to use a single $w$ parameter, as the
product of the player pools found under the jurisdiction of the lottery and
the chance that a single lottery ticket wins.

## Introduction

Making decisions based on chance has always been a practical solution applied
by people in numerous territories of life. Such a practice in the ancient
times was the basis of divination methods, such as the I Ching [1], but they
also came handy to modern physicists in the form of Monte-Carlo simulations.
It may be used to settle disputes, to assign land or work [2]. In this sense,
lotteries constitute a type of gambling where the fundamental mechanism
involves the random choce of a winner from a pool of players. In this paper we
look at pseudo-active lotteries, where players buy a chance to win a Jackpot
prize by selecting a series of numbers on a ticket. When no winner occurs, the
Jackpot prize (the prize pool) grows incrementally, with a given portion of
the weekly sales being added to it. Winners are selected through a periodic
random number draw, which is scheduled at regular intervals, weekly or bi-
weekly in general. If a ticket (or tickets) happens to contain the identical
numbers as those drawn, the winner will receive the prize pool (in case of
multiple winners they share the prize), whereas in the absence of a winner,
the prize pool persists in growing through each successive draw until a winner
is finally declared [3] . One of the most popular lotteries is the ’6/49’
Lotto, in which participants choose six numbers from a range of 1 to 49,
explaining its ’6/49’ designation [4]. Depending on the specific lottery
rules, participants who match fewer numbers than what is required to win the
Jackpot (but still more than a minimum threshold) may be eligible for smaller
prizes. These smaller prizes can be funded from either a portion of the
Jackpot prize pool or from a dedicated fraction of the weekly sales. Jackpot
lotteries, such as the Powerball in the United States, guarantee a weekly
increase of the prize value until someone wins [5, 6].

Throughout history and in contemporary times, lotteries have typically been
administered by states, providing a consistent source of revenue for state
budgets, with usually only a fraction (typically $50\%$) of the sales
contributing to the prize pool [6, 7]. The remaining portion of the sales is
allocated to cover the administrative expenses of the lottery, which are then
legally taxed. Additionally, when a winner is chosen, the Jackpot prize is
also subject to taxation [2]. The extremely slim chances of winning the
lottery (in the case of $6/49$ lotto there are $C(49,6)=13983816$ distinct
combinations for selecting $6$ numbers from $49$, meaning that the chance of
winning with a single ticket is roughly 1 to 14 millions)[4, 3] allows the
jackpot to grow after consecutive plays, attracting even more players [8].
This high Jackpot, coupled with the extremely small chances to win, fascinates
people, making lotteries a popular and addictive game [8].

To provide an example, the North American Association of State and Provincial
Lotteries reported that United States Lotteries achieved total sales of
$\$98,519,173,817$ in the fiscal year of 2021 [9]. Lotteries, as we can
observe, can be approached via their mathematical aspects [3] (combinatorics,
chance of winning etc.), psychological aspects (why do people play) [4],
economical aspects (optimizing their rules in favour of state revenue) [10],
and from statistical physics viewpoint, as we argue in this article.

Approaching lotteries as complex systems, by focusing on the emergent dynamics
resulted from the combination of a set of rules and the decisions of millions
of players, and analysing them using the methodology of statistical physics,
lends novelty and an intriguing nature to the present study.

Figure 1 presents a part of the Jackpot value time-series for the Powerball
lotto created from freely available Jackpot dataset, collected by players
[11]. Such time series of the Jackpot values summarize the dynamics emerging
from the actions of a huge number of players. The time evolution of the
Jackpot presented in Figure 1 consists of qualitatively two different types of
regions. First, there are relatively longer regions dominated by gradual
growth of the Jackpot value, due to the fonds accumulated by the sold tickets
when no winner tickets were sold (green segments of the time series). The
regions characterised my a monotonic growth of the Jackpot are augmented by
intervening and prompt ressetting events of the Jackpot to a default value,
when the Jackpot is won (red deshed sections of the time series). The
processes controlling lottery jackpot values are identical to those serving as
the foundation for the LGGR (Local Growth and Global Reset) mean field type
dynamical model, built on a master equation, which, akin to the previously
described, comprises alternating probabilistic growth and reset processes over
time [12]. The LGGR approach has recently been applied successfully to
demonstrate the underlying dynamics of a decent number of phenomena, ranging
from the field of economics through the field of social sciences to biology
[13, 14, 15, 16].

This paper is organised as follows: in Section Jackpot value time-series, and
ergodicity we present the time-series of the Jackpot prize for six different
lotteries, and argue for the ergodicity of these data along the chosen time
periods. In Section The LGGR approach the LGGR model is presented, along with
the growth and reset rates that are consistent with the dynamics of lotteries.
This section contains the theorethical explanations that lead to the general
form of the stationary probability density function that fits the experimental
results. Section Discussions and validation of the elaborated model is devoted
to further discussions on the chosen rates, also from the perspective of the
gathered data. In the last section (Conclusion we briefly summarize the main
results of this work.

Figure 1: Time series of the Jackpot prize, for the Powerball from 03.2017 to
07.2019. [11]

## Jackpot value time-series, and ergodicity

To understand and mathematically model the Jackpot prize dynamics, we analysed
time-series data from various lotteries, each with distinct rule sets and
player populations [6, 17, 7, 18, 19, 20]. In Table 1 we present the main
statistical properties of the lotteries investigated in this study for the
time periods considered. The reason for focusing only on limited time windows
of lottery histories, instead of studying the available time-series entirely,
is that the rules and the territory in which a given lottery is played change
from time to time. Thus, we analyze only time periods with unchanging
conditions. Besides the general information about the lotteries we also list
for each of the six lotteries the values given by the product of the Jackpot
winning probability by a single ticket ($P_{w}$) with the total population of
the geographic area where the given lottery is played ($Pop$). The expected
number of wins is therefore $w=P_{w}\times Pop$. Although, this single
parameter is not capable of entirely characterizing the lottery by itself, we
assume that it conveys valuable information about the lottery. We further
elaborate on the importance of this parameter $w$ in the discussion section.

The data and the rules of the US lotteries (Powerball, Mega Millions and Texas
lotto) were scraped from the Lottery Report web-portal [6, 21, 22, 5, 23, 24,
7, 25, 26, 27, 18], where citizens collect data regarding the lotteries. In
the case of the other lotteries we used the following sources: the time series
for the Uk lotto $6/49$ was obtained from the BeatLottery web-portal [20] that
is a portal similar to Lottery Report for lottery enthusiasts. The Canada 6/49
lotto data was scraped from the web-portal National-Lottery [19]. The time
series of the Euromillions lottery were obtained from the Loterieplus web-
portal [28], while the rules from the official Euromillions webpage [17]. In
the sections A of the combined Figure 2 we present the time-series of the six
lotteries collected in Table 1.

As we have anticipated in the previous section, we plan to apply the LGGR
framework for modelling the Jackpot dynamics that is designed to describe
ergodic Markovian processes. The ergodic property implies, however, that in
such systems the time average of a characteristic quantity is equivalent to
the ensemble average of the same quantity. Accordingly, the stationary
probability density over the system’s possible states $\rho(x)$ is calculated
as an ensemble average from numerous replicas of the single-element-system,
and is equivalent with the probability density of the states’ occurences,
reflected in the time fractions spent by a single element of the system in the
respective states (assuming a sufficiently long existence of the system) [29].
In the context of lotteries, the intricate system consists of a single agent
(the Jackpot prize) whose development is probabilistically influenced by the
lottery rules and the players via their ticket purchases. Thus, before
applying the LGGR model, it is necessary to test the ergodic nature of the
Jackpot time-series.

To establish the ergodicity of the lottery time-series, we underpin our
argument by demonstrating their stationary characteristics, observed through
the mean Jackpot value’s convergence across an extended time window $\langle
x(t)\rangle_{t}=c$. The convergence of the mean Jackpot values in time, for
each studied lottery games, are shown in sections B of Figure 2. Furthermore,
we confirm their aperiodic nature by revealing the absence of a sharp
autocorrelation ($ACF(s)=Corr(x(t),x(t+s))$). As it is shown in sections C of
Figure 2, for relatively long time fractions (after $4$ to $6$ draws) the
autocorrelation tends to a negligible value. For much longer time periods it
practically tends to $0$ ($lim_{s\to\infty}ACF(s)=0$). In general, these two
features are the necessary criteria for a Markov chain to be considered
ergodic [30]. A similar argumentation was applied to demonstrate the ergodic
nature of hydrology data in [31]. These measures relate to the stability of
the statistical moments of the time-series, denoting stationarity. In most
applications, however, stationary processes without proving their aperiodic
nature are considered to be ergodic [30]. In the present study, being these
two statistical properties (the mean and the autocorrelation) of the Jackpot
time-series proven, we consider the here studied lotteries to be ergodic.

Lottery | Studied time-frame | Format | $P_{w}$ | Pop. | w
---|---|---|---|---|---
Powerball [21, 22] | 2015-08-05 to 2020-02-15 |  $5/69+1/26$ [6] |  $\approx 1/(292\times 10^{6})$ [6] |  $\approx 292.2\times 10^{6}$ [6] | $\approx 1$
Megamillions [23, 24] | 2005-06-22 to 2010-01-31 |  $5/56+1/46$ [7] |  $\approx 1/(258\times 10^{6})$ [7] |  $\approx 152.4\times 10^{6}$ [24] | $\approx 0.58$
Euromillions [28] | 2012-01-12 to 2016-09-01 |  $5/50+2/9$ [17] |  $\approx 1/(116\times 10^{6})$ [17] |  $\approx 272\times 10^{6}$ [32] | $\approx 2.34$
Canada lotto 6/49 [19] | 2007-03-03 to 2019-02-27 | $6/49$ | $\approx 1/(13\times 10^{6})$ |  $\approx 35.2\times 10^{6}$ [32] | $\approx 2.7$
UK lotto 6/49 [20] | 1995-01-07 to 2023-09-09 | $6/49$ | $\approx 1/(13\times 10^{6})$ |  $\approx 62.5\times 10^{6}$ [32] | $\approx 4.8$
Texas lotto [25, 26, 27] | 1994-07-14 to 2000-07-18 |  $6/50$ [18] | $\approx 1/(17\times 10^{6})$ |  $\approx 18.9\times 10^{6}$ [33] | $\approx 1.12$
Table 1: The lotteries studied in this article, along the time frame where the
rules are consistent. We present the format of the lottery, the chance of
winning with a single ticket, the possible maximal player population and the
lottery parameter $w$. Figure 2: The time series of the lotteries studied
(general information provided in Table 1). The rows represent the following:
(A.) The time series plot of the lotteries, on the time range where the rules
and the player pools are not changing, (B.) the mean of the time series
$\langle x(t)\rangle_{t}$ along with the convergence value, (C.) the
autocorrelation function of the time series $ACF(s)$ until $s$ days along with
the confidence interval with $p\leq 0.05$

## The LGGR approach

In order to describe the statistics of Jackpot values, we utilize the LGGR
(Local Growth and Global Reset) model [12]. This model accounts for a process
with a unidirectional probabilistic growth and an augmented reset, the same
processes that are driving the dynamics of the Jackpot value. Instead of using
an ensemble of elements that all evolve according to such a dynamics, here we
consider a long-term evolution of the Jackpot value, where the dynamics
consist of multiple regions with probabilistic growth concluded by a reset
process, as described in the introduction.

We assume that we have a Markov process characterised by a stationary
stochastic matrix. The stationarity has been proven in the previous Section
Jackpot value time-series, and ergodicity, devoted to data analyses.

Given the stationarity of the Jackpot time-series in the previous section, we
consider it ergodic for the remaining part of this study. We assume that the
Markovian process is characterized by a time independent but possibly state
dependent stochastic transition matrix.

The successive values of the Jackpot values, $x(t)$, constitute the states of
the considered dynamical system. The distribution of the $x$ values at a time
moment $t$ is described by the $\rho(x,t)$ density function. Following the
general considerations for the LGGR master-type equations [34, 12], the time
evolution of this distribution function writes as:

$\frac{\partial\rho(x,t)}{\partial t}=-\frac{\partial}{\partial
x}\left[\mu(x)\rho(x,t)\right]-\gamma(x)\rho(x,t)+\langle\gamma(x)\rangle(t)\delta(x).$
(1)

Here $\mu(x)$ accounts for the state dependent local growth rate (probability
per unit time to growth from state $x$ to a state between $x$ and $x+dx$, for
$dx=1$). The reset is governed by the $\gamma(x)$ reset rate: by the
probability that the system resets from state $x$ to state $0$ in a unit time
interval. The last term of the evolution equation is the re-feeding at $x=0$
imposed by the Dirac delta function $\delta(x)$. This term serves to preserve
the normalization of $\rho(x,t)$. The mean value of the reset rate
($\langle\gamma\rangle$) is given as:

$\langle\gamma(x)\rangle(t)=\int_{\\{x\\}}\limits\\!\gamma(x)\rho(x,t)\,dx$
(2)

The model is complete, it allows for a mathematical solution once the $\mu(x)$
and $\gamma(x)$ state-dependent rates are given. The $\rho_{s}(x)$ stationary
distribution is readily obtained by imposing $\partial\rho(x,t)/\partial t=0$:

$\rho_{s}(x)\>=\>\frac{C}{\mu(x)}\,e^{-\int_{\\{x\\}}\limits\\!\frac{\gamma(u)}{\mu(u)}\,du},$
(3)

with $C$ being a normalization constant. Convergence to the stationary state
$\rho_{s}(x)$ is proven for quite general conditions [35].

In order to apply this model for the monitored Jackpot amounts, we need to
obtain reset $\gamma(x)$ and growth $\mu(x)$ rates realistically. The increase
per unit time (one week) in the Jackpot value is determined by the number of
sold tickets. As the Jackpot value is higher, the number of sold tickets also
rises, so one deals with a multiplicative growth. The simple
$\mu(x)=a(1+b\,x)$ linear form as a function of the $x$ Jackpot value seems
therefore to be a reasonable approximation. The reset probability per unit
time also depends on the number of sold tickets, which in turn depends on the
increase in the Jackpot values. Taking into account that some of the sold
tickets might not introduce new numbers in the lottery, for determining the
$\gamma(y)$ reset probability for $y$ number of sold tickets one should take
into account the number of differently completed lottery tickets (tickets with
different numbers). The probability of a reset (winning the jackpot) depends
solely on the number of differently completed lottery tickets. We thus write
up a simple equation

$\gamma(y+dy)=\gamma(y)+dy\,\alpha(1-\gamma(y)),$ (4)

which states that by increasing the number of sold ticket with an amount of
$dy$ the increase in the win-probability to reset the Jackpot depends on the
$dy$ input amount and on the $1-\gamma(y)$ probability that each of them are
not in the previous set of $y$. Equation (4) leads to the differential
equation:

$\frac{d\gamma(y)}{dy}=\alpha\left(1-\gamma(y)\right).$ (5)

Considering the imposed boundary condition $\gamma(0)=1-\beta$, we get the
solution:

$\gamma(y)=(1-\beta\,e^{-\alpha\,y})$ (6)

Assuming that $y\propto\mu(x)$, we get the general form in which one should
consider the growth and reset rates:

$\mu(x)=a(1+b\,x),$ (7)

and

$\gamma(x)=\left(1-\beta e^{-\kappa(1+b\,x)}\right),$ (8)

with $\kappa\propto a\,\alpha$. Having these rates, it is straightforward to
determine the $\rho_{s}(x)$ stationary distribution. According to equation (3)
we obtain:

$\rho_{s}(x)=C\,(1+b\,x)^{-\lambda-1}e^{\nu\,\text{Ei}\left(-\kappa(1+b\,x)\right)}$
(9)

In the above form $\lambda=1/(a\,b)$ ,$\nu=\beta\lambda$, $\text{Ei}(x)$
denotes the exponential integral function

$\text{Ei}(x)=\int_{-x}^{\infty}\limits\\!\frac{e^{-t}}{t}\,dt,$ (10)

and $C$ is a normalization constant, ensuring that
$\int_{0}^{\infty}\rho_{s}(x)dx=1$.

For the obtained results some comments are now in order:

  1. 1.

For $\beta>1$ the reset rate given by equation (8) can be negative. Indeed if

$x<\frac{1}{b}-\frac{1}{\kappa\,b}\ln(\beta),$ (11)

the reset rate is negative. For an $x\geq 0$ value this negative reset rate
means that after the reset process the growth dynamics starts with a nonzero
$x$ jackpot value. The probability rate for such an event is $-\gamma(x)$. As
we will see later in the real word data, this is the case for many lottery
systems.

  2. 2.

The asymptotic behavior ($x\rightarrow\infty$) of the $\rho_{s}(x)$
probability density (equation (9)) is governed by the $(1+b\,x)^{-\lambda-1}$
Tsallis-Pareto term, since $\lim_{x\rightarrow-\infty}\text{Ei}(x)=0$, and
therefore
$\lim_{x\rightarrow\infty}e^{\nu\,\text{Ei}\left(-\kappa(1+b\,x)\right)}=1$.
This means that the tail of the distribution of the Jackpot values has to
scale with the exponent $-\lambda-1$. The actual form of the power-law like
tail is determined solely by the $a$ and $b$ parameters governing the
multiplicative growth.

  3. 3.

The $C$ normalization constant cannot be written in an analytic form, and has
to be computed numerically after fixing the model parameters: $a$, $b$,
$\kappa$ and $\beta$.

## Discussions and validation of the elaborated model

After having introduced the LGGR-model and the stationary jackpot time-series
for different lotteries, we argue for the applicability of our model. In order
to make the understanding of the available Jackpot data more intuitive and
unified, we rescale each time series to the time average of the corresponding
Jackpot value, $z=\frac{x}{\langle x\rangle_{t}}$.

First, as a proof of concept, we verify whether the probability distribution
function, obtained as the stationary solution of the LGGR model (Equation 9)
assuming preferencial growth rate for the Jackpot values (Equation 7) fits the
experimental probability distributions. In Figure 3. the experimental
distributions of the relative Jackpot values, $\rho_{s}(z)$, are presented,
along with the probability density function given by Equation 9. In cases of
the Canadian and the UK lotteries, the probability distribution’s shape
differs considerably from the Jackpot distributions of the other four
lotteries. This qualitative distinction can be attributed to the distinct rule
system. Among others, the dissimilarity originates primarily from the
different reset processes characteristic of these lotteries. While the Euro
millions lottery and the lotteries played in the US are characterised by
conventional, total reset processes to a minimal Jackpot value [6, 21, 22, 5,
23, 24, 7, 25, 26, 27, 18] for the Canadian and UK lotteries a so-called
"smart reset" process is typical [19, 20]. In the cases when a total reset
applies, the value of the $\beta$ parameter is $1$, leaving us with only three
free parameters. The smart reset rate however, ($\beta\neq 1$) implies that
after the Jackpot is won, its value can be reset to values higher than the
minimum Jackpot value. In the case of such lotteries, the smaller prizes,
awarded for players matching fewer numbers than the maximum possible lottery
numbers, are also paid from the Jackpot found [19, 20].

Alongside the difference in reset processes, the relatively large value of the
$w$ measure, defined in the introduction (see Table 1), also suggests a
deviation from the other lotteries. This indicates that for these lotteries
the total number of potential players is relatively high compared to the
probability of winning the Jackpot, resulting in a notably accelerated Jackpot
dynamics.

The fitting parameters were chosen based on visual inspection, as the
objective here is not to provide a perfect fit for the experimentally obtained
probability density functions. This would be impractical due to the limited
amount of experimental data available and the fact that the rules of lotteries
change from time to time, while our modelling focuses on relatively short
periods with unchanging conditions. Rather, our objective is to construct a
statistical mean-field model capable of providing a consistent explanation for
the dynamics governing the stationary evolution of Jackpot values.

Figure 3: The stationary probability distribution of the Jackpot prizes
$\rho_{s}(\frac{x}{\langle x\rangle_{t}})$ as a function of the mean rescaled
Jackpot $\frac{x}{\langle x\rangle_{t}}$, for the six studied lotteries.

### Preferencial growth rate and consistency of the $\mathbf{a}$, $\mathbf{b}$
parameters

The assumed form of the growth rate given in Equation 7 contains two
parameters that influence the form of the probability density function.
Considering the relation, $\lambda=1/(a\cdot b)$, between the $a$, $b$ and
$\lambda$ parameters, the value of the parameter $a$ can be deduced from the
values obtained from fitting the probability distributions of the relative
Jackpot values on Figure 3.

We comment here also on the meaning of the two parameters of the growth kernel
function from the point of view of the lottery. The parameters $a$ and $b$ are
clearly characterising the player pool. Parameter $a$ is linked to the size of
the active player society, representing the proportion of sales contributed by
the subset of players who participate in the lottery regardless of the
Jackpot’s value. If we consider the form given by Equation 7 the value of $a$
can be given as the extrapolated average sales after the lottery is won,
$\mu(0)=a$. The $b$-parameter on the other hand addresses the player
attraction success, as in the linear growth rate (Equation 7) the term $a\cdot
b$ defines the slope according to which the sales increases with the Jackpot
value. As the Jackpot grows, sales also increase because the alluring prize
encourages the purchase of multiple tickets.

As mentioned in Section Jackpot value time-series, and ergodicity the time-
series contain all the information needed to elaborate the form of the growth
and reset rates. Thus, we determined the form of the growth and reset rates
based on the experimental data and compare against the assumed kernel
functions (Equations 7 and 8) with the chosen values of the fitting parameters
in Figure 3.

The growth rate was determined experimentally by calculating the average
relative increase of the Jackpot values in between consecutive draws, when no
winners were announced. Then, the calculated growth rate is represented in the
function of the relative Jackpot values averaged within consecutive relative
jackpot value intervals.

$\mu(\langle\frac{x}{\langle x\rangle_{t}}\rangle_{z})=\langle\frac{1}{\langle
x\rangle_{t}}\cdot\frac{\Delta x}{\Delta t}\rangle_{z},$ (12)

where $\langle\rangle_{z}$ denotes the averaging within bins defined based on
the relative Jackpot value $z=\frac{x}{\langle x\rangle_{t}}$.

In Figure 4 the experimentally obtained growth rates, defined in Equation 12,
are represented. On each figure we indicated the uncertainty of each data
points’ positions using the standard deviation of the data in the
corresponding bins along both axis. We split the graphs along the $x$ axis
into two parts, based on the amount of data, $N_{data}$, that fell into the
respective regions. By changing the background color of the figures, we
indicate the reliability of the data within the regions, green meaning
reliable, while red meaning unreliable. In each case, we fit parameters to the
data only from the green region of the graphs.

The theoretical fitting curves shown in the subfigures of Figure 4 are given
by Equation 7, while for the values of the parameters $a$ and $b$ we
considered the ones used for fitting the probability distribution functions in
Figure 3. The alignment between the assumed preferential growth rates
determined using the fitting parameters in Figure 3 and the experimentally
measured growth rates strongly supports the idea of linearly increasing growth
rates in the context of lotteries. Additionally, this finding validates the
suitability and internal consistency of the proposed model.

Figure 4: The experimental growth rate $\mu(x/\langle x\rangle_{t})$ presented
as the function of $x/\langle x\rangle_{t}$, along with the theoretical fit
and the parameters for the studied lotteries. The experimental data averages
are presented along with the error bars for the averages (standard deviation
in the ensamble) in the bins for both axes. We also present the number of
datapoints, $N_{data}$, used for calculating the averages in order to
demonstrate that the red regions in the plots are not reliable for statistical
inference.

### Comments on the form of the assumed and experimental reset rates

In the Section The LGGR approach we stated that there is a connection between
the two rates that leads to the reset rate in the form of Equation 8.
Fortunately, this connection between the rates resolves the difficulty of
determining the realistic form of the reset rate, as it is much more
challenging to establish experimentally due to the infrequency of resetting
compared to growth. Moreover, the growth-reset relation introduces another two
free model parameters, $\alpha$ (or $\kappa=a\cdot\alpha$) and $\beta$.

The parameter $\beta$ dictates the character of the reset process. When
$\beta=1$, it results in a total reset, but if $\beta>e^{\kappa(1+b\cdot
x_{c})}$, then the reset becomes negative for $x<x_{c}$, evolving into a smart
reset rate. The negative reset rate indicates an incoming process into the
system at the state characterised by the value $x$. The parameter $\alpha$
represents the reciprocal of the unit price of the lottery tickets, ensuring
that in Equation 8, the number of sold tickets appears in the exponent.

Due to the less frequent nature of resetting, especially in the case of
lotteries characterised by total reset and a small $w$ measure, there have not
been enough resetting events occurring during the studied time periods to
construct and verify the shape of the reset rate.

On the other hand, in the case of the Canadian and UK lotteries, due to their
accelerated dynamics (explained by a relatively large $w$) and the presence of
a smart reset allowing the Jackpot to be reset to values greater than the
minimum Jackpot value more frequently, we have a sufficient number of
resetting records available to construct the reset rate from real world data.
Another advantageous circumstance is that these two lotteries offer the
longest datasets with unchanging rulesets.

Because the connection between the reset rate and the growth rate (Equation 6)
inherent to the design of lottery games, it holds true for other types of
lotteries as well. As a result, we can confidently assume that the
experimentally determined shape of the reset rate (provided we have sufficient
data) could be matched with the theoretically predicted kernel function for
the reset rate for other lotteries, too.

Experimentally the reset rate can be defined as a function of the scaled
Jacpot value $x/\langle x\rangle_{t}$. It is computed by subtracting the total
number of events when the Jackpot value is reset from $x/\langle
x\rangle_{t}$, from the total number of events when the Jackpot is reset to
the same value from a greater relative value. The result is then divided by
the total number of realizations when the relative value of the Jackpot is
$x/\langle x\rangle_{t}$.

$\gamma(\frac{x}{\langle x\rangle_{t}})=\frac{N_{win}(\frac{x}{\langle
x\rangle_{t}})-N_{ret}(\frac{x}{\langle x\rangle_{t}})}{N(\frac{x}{\langle
x\rangle_{t}})}$ (13)

On Figure 5 the experimentally determined reset-rates are shown for the
Canadian and UK lotteries, as a function the mean rescaled Jackpot $x/\langle
x\rangle_{t}$. Once more it’s important to highlight that the last few bins in
these plots have a small number of data points, making this part of the plot
less reliable.

As we did with the growth rate, we fit the experimental reset rates using the
reset rate form incorporated in the model’s design to demonstrate the
consistency of the model’s general assumptions regarding the reset and the
chosen parameter values.

The theoretical fit, displayed alongside the experimental data, is governed by
Equation 8, using the same parameters employed for fitting the probability
distribution function of the relative Jackpot values in Figure 3 and retaining
only one free parameter, $\alpha$. Although the matching of the curves is
modest, it is still a remarkable result given the poor quantity of processed
data.

Figure 5: The experimental reset rate calculated by Equation 13, fitted with
the theoretical reset-rate, for the Canada lottery and UK lottery. We used the
parameters obtained for fitting the probability density functions.

## Conclusion

The value of the Jackpot in lotteries is constantly followed by millions of
people, yet there are very few of them who think of it as a complex system.
Although, the format of the lotteries can differ, the resulting dynamics of
the combined system of players and the ruleset defined by the lottery
associations is very similar. The simple rules, combined with the behaviour of
the players, result in an emergent, non deterministic evolution of the Jackpot
prize. The dynamics of the Jackpot is governed by a persistent growth process,
that is the result of the ticket sales and a probabilistic reset rate, induced
by the rare winnings. This stochastic dynamics serves as a classic example of
phenomena that can be effectively described using the LGGR model. In case of
the lottery, hovewer, instead of having a system with multiple elements that
are acting at the same conditions (considering a mean field type model), there
is only a single component behaving according to the same rules. The dynamics
of this single element is entirely characterized by the Jackpot time-series.

In order to be handled as an ergodic Markovian process, we considered such
fractions of the entire historical data, for that the rules and the size of
the player pool were stationary (not changing the rules and maintaining the
$w$ value). For such time periods the lottery can be considered ergodic as we
have demonstrated it by proving their stationarity and aperiodic nature
(Figure 2). The design of the LGGR model required the establishment of both
the growth and reset rates. As we have demonstrated in the previous sections,
it is an inherent property of the systems featuring the connection
$\gamma(y)=(1-\beta\,e^{-\alpha\cdot\mu(y)})$ between the growth and reset
rates. Being aware of this, it is sufficient to identify the form of the
growth rate only.

For choosing a linear growth rate, our intuition was tested and proved to be
correct by real world data (see Figure 4). The relation stated in Equation 6
between growth and reset rates was also validated by the experimentally
computed reset rate, cf. Figure 5. By using these rates, both proven to be
realistic by experimental data (Figure 4 and Figure 5), in the stationary
limit. The LGGR model produced a probability distribution function (Equation
9) that with the right parameters fits in a totally consistent manner, the
observed distributions of the Jackpot values in all six lotteries we studied.
A notable feature of this model is its ability to depict two types of lottery
dynamics: one defined by a total reset (Euromillions, Powerball, Megamillions,
Texas lottery) and the other by an incomplete reset (UK Lotto and Canadian
Lotto) process.

Another observation is worth noting here, related to the straightforward $w$
measure, which despite its apparent simplicity can effectively capture some
key properties of lotteries.

As shown in Table 1, the $w$ value is consistently maintained at an optimal
low value ($0.5<w<3$), determining the ideal speed of the dynamics, and
keeping the players interested.

## Acknowledgements

This work was supported by the project “A better understanding of socio-
economic systems using quantitative methods from Physics” funded by the
European Union – NextgenerationEU and the Romanian Government, under National
Recovery and Resilience Plan for Romania, contract no 760034/23.05.2023, code
PNRR-C9-I8-CF255/29.11.2022, through the Romanian Ministry of Research,
Innovation and Digitalization, within Component 9, Investment I8. The work of
Sz.K. is also supported by the Collegium Talentum Program of Hungary.

## Author contributions statement

T.S.B and Z.N. have conceptualized and created the LGGR model. Its application
to lottery Jackpot time series was suggested by I.G. Data curation and
analysis was carried out by I.G. and Sz.K., supervised by Z.N. Visualization
by I.G. and Sz.K., The manuscript has been written by Sz.K. and I.G. and Z.N.,
edited and corrected by T.S.B. All authors contributed to discussing the
results and form of presentation.

## Competing interests

The authors declare no competing interests.

## Data availability

The data used in this study are available at the refernced internet links. The
data curated by us used in this study can be found in the following online
repository: https://doi.org/10.6084/m9.figshare.24310669.

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