# Population dynamics and games of variable size

Matheus<EMAIL_ADDRESS>Center for Mathematics and
Applications (NOVA Math), NOVA FCT, Universidade NOVA de Lisboa, Quinta da
Torre, 2829-516, Caparica, Portugal  Fabio A. C. C. Chalub222Corresponding
author<EMAIL_ADDRESS>Center for Mathematics and Applications (NOVA Math),
NOVA FCT, Universidade NOVA de Lisboa, Quinta da Torre, 2829-516, Caparica,
Portugal  Department of Mathematics, NOVA FCT, Universidade NOVA de Lisboa,
Quinta da Torre, 2829-516, Caparica, Portugal

###### Abstract

This work introduces the concept of Variable Size Game Theory (VSGT), in which
the number of players in a game is a strategic decision made by the players
themselves. We start by discussing the main examples in game theory:
dominance, coexistence, and coordination. We show that the same set of pay-
offs can result in coordination-like or coexistence-like games depending on
the strategic decision of each player type. We also solve an inverse problem
to find a $d$-player game that reproduces the same fixation pattern of the
VSGT. In the sequel, we consider a game involving prosocial and antisocial
players, i.e., individuals who tend to play with large groups and small
groups, respectively. In this game, a certain task should be performed, that
will benefit one of the participants at the expense of the other players. We
show that individuals able to gather large groups to perform the task may
prevail, even if this task is costly, providing a possible scenario for the
evolution of eusociality. The next example shows that different strategies
regarding game size may lead to spontaneous separation of different types, a
possible scenario for speciation without physical separation (sympatric
speciation). In the last example, we generalize to three types of populations
from the previous analysis and study compartmental epidemic models: in
particular, we recast the SIRS model into the VSGT framework: Susceptibles
play 2-player games, while Infectious and Removed play a 1-player game. The
SIRS epidemic model is then obtained as the replicator equation of the VSGT.
We finish with possible applications of VSGT to be addressed in the future.

keywords: Game theory $|$Fixation $|$Evolution of eusociality $|$Speciation
$|$Epidemic models.

##  1 Introduction

Game theory is the part of mathematics that models strategic behavior (von
Neumann and Morgenstern, 2004). A symmetric $d$-player game is defined by $d$
individuals (_players_), that have to choose among $n$ _strategies_
$\mathbf{e}_{i}$, $i=1,\dots,n$, and as a result each one receive a certain
_pay-off_. A common assumption when modeling economic behavior is so-called
_rationality_ : each player acts to maximize his or her pay-off function. A
set of strategies in which no rational player has incentives to deviate from
his or her strategy is called a _Nash equilibrium_. It is possible to prove
that, if players are allowed to choose their strategies according to certain
probability distributions, there is always at least one Nash equilibrium in
every game defined as above. More general formulations of games, in particular
with applications to modeling economic behavior, can be found in (Gintis,
2009) and references therein.

In biology, it is customary to replace the rationality assumption with a given
dynamics. In particular, a population of individuals of $n$ different types
where each type adopts one of the strategies $\mathbf{e}_{i}$, $i=1,\dots,n$
is considered. The fraction of individuals of type $i$ at time $t\geq 0$ is
given by $x_{i}(t)\in[0,1]$, and the _state_ of the population is given by
$\mathbf{x}(t)=(x_{1}(t),\dots,x_{n}(t))\in
S^{n-1}:=\\{\mathbf{z}\in\mathbb{R}^{n}_{+},\sum_{i=1}^{n}z_{i}=1\\}$, the
$n-1$-dimensional _simplex_ , for all $t\geq 0$. Pay-off functions, which are
frequently identified as _fitness_ in the biological literature, are smooth
functions $\psi_{i}:S^{i-1}\to\mathbb{R}$, $i=1,\dots,n$; more precisely, we
say that, at state $\mathbf{x}$, $i$-type individuals have fitness
$\psi_{i}(\mathbf{x})$. The state of the population evolves according to a
certain given dynamics, from which the most popular is the _replicator
dynamics_ , introduced in Taylor and Jonker (1978):
$x_{i}^{\prime}=x_{i}\left(\psi_{i}(\mathbf{x})-\overline{\psi}(\mathbf{x})\right)$,
where $\overline{\psi}(\mathbf{x}):=\sum_{i=1}^{n}x_{i}\psi_{i}(\mathbf{x})$
is the average pay-off; cf. (Hofbauer and Sigmund, 1998) and references
therein.

One common limitation of the previous description is that the number of
players $d$ in a given game is assumed fixed. However, the number of
individuals we interact with at any given period of time in our lives is
variable, with several factors impacting this number, e.g., race, gender,
employment status, and family situation (Zhaoyang et al., 2018). One important
determinant of the number of social interactions in a typical day is the time
left in life (Carstensen et al., 1999).

To a certain extent, the number of interactions we have in our daily lives is
a consequence of our present and past choices; that will depend, among other
factors, on the challenges we face in our lives, in particular at work. If we
have to perform complex tasks, it is necessary to be part of a team, and the
size of the team will, in general, be related to the complexity of the task.
In this sense, we characterize the present work as a continuation and a
generalization of previous works where the number of players in a game is not
fixed _a priori_. We refer, in particular, to (Souza et al., 2009; Archetti,
2009), which studied the effect of the size of the groups on the outcome of
given games. In (Izquierdo et al., 2014; Kurokawa, 2019), an individual opt-
out strategy was introduced, having as a pratical consequence the variation of
the size of the game. Symmetric games with variable number of players in
neural networks also appear in (Gatchel, 2021). If the possible interactions
in a population are modeled using a graph, games played by each individual
might be variable as in (Broom and Rychtář, 2012; Erovenko et al., 2019).
Furthermore, the individual contact network may evolve according to the
outcome of the game, and, therefore, the number of interactions in each game
varies with time (Taylor and Nowak, 2006; Skyrms and Pemantle, 2000; Pacheco
et al., 2006).

The objective of this work is to go beyond these previous works and to
consider that the definition of the number of players that will participate in
a given game is a player’s decision. In _variable size game theory_ (VSGT),
there is always a first step in which an individual is chosen from the
population — the so-called _focal player_ —, and he or she will decide the
number of players that will participate in the game. Only after that initial
step will the group of players be selected and the game be played.

The outline of the paper is as follows: we introduce the basic notation in
Sec. 2. In Sec. 3, we present several applications. In particular, in Subsec.
3.1, we discuss the traditional structure of game theory: dominance,
coordination, and coexistence, stressing the fact that a single change in a
player’s strategy, without changing the game’s pay-off, may change game types.
We also find the game with a fixed number of players that reproduces the
fixation pattern of the given VSGT. In Subsec 3.2, we propose a simple model
for the evolution of eusociality, introducing a game with prosocial (i.e.,
large $d$) and antisocial (i.e., small $d$) players, while in Subsec. 3.3 we
show that a player strategy may provide a reason for the spontaneous
separation of types, without a clear physical barrier, providing one possible
scenario for speciation. In the last example, in Subsec. 3.4, we show how to
derive classical differential equations from compartimental models using VSGT.
Finally, in Sec. 4, we discuss our findings, possible future work, and
applications.

##  2 Definitions

Let us consider a fixed-size population, composed of two different types, say,
$\mathbb{A}$ and $\mathbb{B}$. We define a family of $d$-player symetric
games, where $1\leq d\leq d_{\mathrm{max}}<\infty$, $d\in\mathbb{N}$. For any
value of $d$, $\mathbf{a}^{(d)},\mathbf{b}^{(d)}\in\mathbb{R}^{d}$, where
$a_{k}^{(d)}$ ($b_{k}^{(d)}$) indicates the pay-off of an $\mathbb{A}$-type
($\mathbb{B}$-type, respectively), in a $d$-player game, where $k$ individuals
are of type $\mathbb{A}$. If there are $i$ individuals of type $\mathbb{A}$ in
a population of $N$ individuals, pay-offs are defined by, respectively
(Gokhale and Traulsen, 2010; Lessard, 2011),

$\displaystyle\psi_{\mathbb{A}}^{(d)}(i)$
$\displaystyle=\sum_{k=0}^{d-1}\frac{\binom{i-1}{k}\binom{N-i}{d-1-k}}{\binom{N-1}{d-1}}a_{k}^{(d)}\
,$ (1) $\displaystyle\psi_{\mathbb{B}}^{(d)}(i)$
$\displaystyle=\sum_{k=0}^{d-1}\frac{\binom{i}{k}\binom{N-i-1}{d-1-k}}{\binom{N-1}{d-1}}b_{k}^{(d)}\
.$ (2)

In this case, we say that the population is at state $i$. We assume that the
$d$ players were drafted from the population with equal probability. Note that
$\psi_{\mathbb{A},\mathbb{B}}^{(d)}$ are polynomials of degree $d-1$.

The first step in the proposed model is to select an individual in the
population, with equal probability, to act as the _focal player_. The focal
player will decide the number of players that will participate in the game,
according to the probability distributions

$\bm{\lambda}_{\mathbb{A}}:=\left(\lambda_{\mathbb{A}}^{(d)}\right)_{d\leq
d_{\max}}\quad\text{or}\quad\bm{\lambda}_{\mathbb{B}}:=\left(\lambda_{\mathbb{B}}^{(d)}\right)_{d\leq
d_{\max}}\ ,$

if the focal player is of type $\mathbb{A}$ or type $\mathbb{B}$,
respectively. The probability distribution is such that
$\lambda^{(d)}_{\mathbb{A},\mathbb{B}}\geq 0$, and
$\sum_{d=1}^{d_{\max}}\lambda^{(d)}_{\mathbb{A},\mathbb{B}}=1$.

The pay-off of the focal player is given by
$\Psi_{\mathbb{A},\mathbb{B}}(i):=\sum_{d=1}^{d_{\max}}\lambda_{\mathbb{A},\mathbb{B}}^{(d)}\psi_{\mathbb{A},\mathbb{B}}^{(d)}(i)$,
for type $\mathbb{A}$ and type $\mathbb{B}$ individuals, respectively. For
non-focal players, pay-offs are given by
$\mu\sum_{d=1}^{d_{\max}}\lambda_{\mathbb{X}}^{(d)}\psi_{\mathbb{A},\mathbb{B}}^{(d)}(i)$,
where $\mathbb{X}=\mathbb{A}$ or $\mathbb{B}$ indicates the focal player. The
parameter $\mu\in[0,1]$ introduces an asymmetry between focal and non-focal
players that is natural in the model, given a possibly reduced pay-off to non-
focal players. In this work, we will consider only the case in which each
strategist defines precisely the number of players to be involved in the game,
i.e. $\lambda_{\mathbb{A}}^{(d)}=\delta_{d,d_{\mathbb{A}}}$ and
$\lambda_{\mathbb{B}}^{(d)}=\delta_{d,d_{\mathbb{B}}}$, with
$d_{\mathbb{A}},d_{\mathbb{B}}\leq d_{{\max}}$, and $\delta_{\cdot,\cdot,}$ is
the Kronecker delta. In the examples studied, we will consider only $\mu=0$,
which means that the focal player acts as group leader, dictator, or similar
and $\mu=1$, in which case the game is symmetric.

######  Remark 1.

Note that if $\bm{\lambda}_{\mathbb{A}}=\bm{\lambda}_{\mathbb{B}}$, the payoff
of a player will not depend on the strategy of the focal type. As all players
have the same probability of being focal, all pay-offs may be replaced by
equivalent functions, and the parameter $\mu$ may be ignored. If the above
equality does not hold, there are no simple equivalent pay-off functions, as
it is necessary to consider that the probability that a given individual will
be selected to participate in a game as a non-focal player will depend on the
strategy of both players and the population state.

From now on, we will use also the expression _absolute fitness_ or simply
_fitness_ , to refer to the average pay-off of the individuals of a given
type, once all individuals in the population have had the opportunity to be
the focal player. In this sense, the pay-off is a game property, while fitness
is a state property. The relative fitness $\Theta(i)$ is the ratio between the
fitness of type $\mathbb{A}$ and $\mathbb{B}$ at state $i$. When $\mu=0$, this
equals

$\Theta(i)=\frac{\Psi_{\mathbb{A}}(i)}{\Psi_{\mathbb{B}}(i)}\ .$ (3)

We define the _fitness potential_

$V\left(\frac{i}{N}\right)=-\sum_{j=0}^{i}\log\Theta(j)\ ,$ (4)

which provides simple heuristics to analyze the game dynamics, cf. (Chalub and
Souza, 2018, 2016). In this work, we will only refer to the fitness potential
if $\mu=0$.

######  Remark 2.

Assume $\mu=0$. Let us define $x=i/N$, and assume the _weak selection
principle_

$\psi_{\mathbb{A},\mathbb{B}}^{(d)}(x)=1+\frac{1}{N}\varphi_{\mathbb{A},\mathbb{B}}^{(d)}(x)+o(N^{-2})\
,$ (5)

for large values of $N$, uniformly on $x$. We use the “small o” notation to
indicate the asymptotic behavior of relevant variables in the limit
$N\to\infty$. We define
$\Phi_{\mathbb{A},\mathbb{B}}(x):=\sum_{d=1}^{d_{\max}}\lambda_{\mathbb{A},\mathbb{B}}^{(d)}\varphi_{\mathbb{A},\mathbb{B}}^{(d)}(x)$.
Therefore,

$\displaystyle V(x)$
$\displaystyle=-\sum_{j=0}^{i}\log\frac{\Psi_{\mathbb{A}}(x)}{\Psi_{\mathbb{B}}(x)}$
$\displaystyle=-\sum_{j=0}^{i}\log\frac{1+\frac{1}{N}\Phi_{\mathbb{A}}(x)+o(N^{-2})}{1+\frac{1}{N}\Phi_{\mathbb{B}}(x)+o(N^{-2})}$
$\displaystyle=-\sum_{j=0}^{i}\left\\{\frac{1}{N}\left(\Phi_{\mathbb{A}}(x)-\Phi_{\mathbb{B}}(x)\right)+o(N^{-2})\right\\}$
$\displaystyle=-\int_{0}^{x}\left(\Phi_{\mathbb{A}}(x)-\Phi_{\mathbb{B}}(x)\right)\mathrm{d}x+o(N^{-1})\
.$

As discussed in (Chalub and Souza, 2014, 2018), the fixation probability
$\phi(x)$ is the stationary solution of the adjoint Kimura equation with
appropriate boundary conditions; namely

$\phi(x)=\frac{\int_{0}^{x}\mathrm{e}^{\frac{2}{\kappa}V(y)}\mathrm{d}y}{\int_{0}^{1}\mathrm{e}^{\frac{2}{\kappa}V(y)}\mathrm{d}y}\
,$

where $\kappa$ is the selection strength (inverse population size). This
result is valid in the limit $N\to\infty$ for fitnesses/pay-off functions
given by Eq. (5). In this case, type $\mathbb{A}$ individuals will be the most
likely to fixate if the boundary $x=1$ can be reached with less (fitness
potential) energy than the boundary $x=0$; otherwise, type $\mathbb{B}$ will
be the most likely to fixate. If both boundaries require a comparable amount
of energy to be reached, there will be a metastable coexistence state and
fixation will take longer. For further details, see (Chalub and Souza, 2018).
In this work, except when otherwise stated, we will depart from the weak
selection regime, cf. (Chalub and Souza, 2016).

A differential equation assumes at least implicitly an infinite population; in
part of the examples described in the present work, we will consider finite
populations, which are compatible with infinite ones if we assume the weak
selection principle as described above.

Throughout this work, we will always consider the Wright-Fisher (WF) model for
finite populations (Fisher, 1922; Wright, 1931). The WF model is characterized
by a $(N+1)\times(N+1)$ stochastic transition matrix
$\mathbf{M}=\left(M_{ij}\right)_{i,j=0,\dots.N}$, where $N$ is the population
size and $M_{ij}=\binom{N}{i}\mathsf{s}_{j}^{j}(1-\mathsf{s}_{j})^{N-i}$ is
the transition probability from state $j$ to $i$. The type selection
probability
$\bm{\mathsf{s}}:=(\mathsf{s}_{0},\mathsf{s}_{1},\dots,\mathsf{s}_{N})$ was
introduced in (Chalub and Souza, 2017) and is related to the fitness by

$\mathsf{s}_{i}=\frac{i\Psi_{\mathbb{A}}(i)}{i\Psi_{\mathbb{A}}(i)+(N-i)\Psi_{\mathbb{B}}(i)}\
.$

We assume no mutation in the model, i.e., $s_{0}=0$, $s_{N}=1$.

The fixation probability vector $\mathbf{F}=(F_{0},\dots,F_{N})$ is obtained
considering the Wright-Fisher process. Let $F_{i}$ be the probability that
type $\mathbb{A}$ is fixated in the population, starting from a presence of
$i=0,\dots,N$ type $\mathbb{A}$ individuals, i.e., the fixation probability
from initial condition $i$ is equal to to the sum over all $j$ of the
probabilities that, in the first step, the state changes from $i$ to $j$, and
then it fixates from $j$, $F_{i}=\sum_{j}F_{j}M_{ji}$. Then, $\mathbf{F}$ is
the only solution of the eigenvalue problem
$\mathbf{F}^{\dagger}\mathbf{M}=\mathbf{F}^{\dagger}$, $F_{0}=1-F_{N}=0$.
Namely, we define
$\widetilde{\mathbf{M}}=\left(M_{ij}\right)_{i,j=1,\dots,N-1}$,
$\mathbf{b}=\left(M_{Ni}\right)_{i=1,\dots,N-1}$,
$\widetilde{\mathbf{F}}=\left(F_{i}\right)_{i=1,\dots,N-1}$ and then
$\widetilde{\mathbf{F}}^{\dagger}=\mathbf{b}^{\dagger}\left(\mathbf{I}_{N-1}-\widetilde{\mathbf{M}}\right)^{-1}$,
where $\mathbf{I}_{N}$ is the $N\times N$ identity matrix. In the above
formulas, vectors are column-vectors and $(\cdot)^{\dagger}$ denotes their
transpose. See (Chalub and Souza, 2017) for further details.

######  Remark 3.

Other finite population models (e.g., the Moran process (Moran, 1962)) could
be used as well. In this work, we will only use the WF process for two main
reasons. The first one is that the reproductive and the Darwinian fitnesses
are compatible for all choices of parameters. A second reason is that the
inverse problem (to be discussed in Subsec. 3.1) can be applied to any
fixation function — it is important to note that in $d$-player game theory,
for $d>2$, the fixation probability is not necessarily increasing in. For
further details, see (Chalub and Souza, 2017, 2019).

##  3 Examples

###  3.1 Dominance, coexistence, and coordination

The replicator equation for a two-type population is given by

$x^{\prime}=x(1-x)\Delta\psi(x)\ ,$

where $\Delta\psi:=\psi^{\mathbb{A}}-\psi^{\mathbb{B}}$ is the fitness
differences between the types; if $\Delta\psi(x)$ does not change sign, we say
that $\mathbb{A}$ dominates $\mathbb{B}$ (if the difference is positive) and
that $\mathbb{B}$ dominates $\mathbb{A}$ otherwise. Note that
$x(t)\stackrel{{\scriptstyle t\to\infty}}{{\longrightarrow}}1$ in the first
case and $x(t)\stackrel{{\scriptstyle t\to\infty}}{{\longrightarrow}}0$
otherwise for all non-trivial initial conditions. If the same fitness
difference is used to model the Wright-Fisher process, the fixation
probability will be close to 1 in the first case and close to 0 in the second,
except if the initial condition is near the boundaries. In the following
discussion, we will ignore cases in which $x(0)$ is close to 0 or 1. We also
assume that $\Delta\psi$ is smooth.

If $\Delta\psi(x)>0$ for $x<x_{*}\in(0,1)$ and $\Delta\psi(x)<0$ for
$x>x_{*}$, then $x(t)\stackrel{{\scriptstyle
t\to\infty}}{{\longrightarrow}}x_{*}$, and the fixation probability of the
Wright-Fisher dynamics is characterized by a plateau, apart from the
boundaries. This is characteristic of a loss of memory of the initial
condition, as initially, the system converges to an inner equilibrium at
$x_{*}$. In the finite population case, the system will fixate or will be
extinct with comparable probabilities. In the replicator dynamics, $x_{*}$ is
an inner stable equilibrium. This is the coexistence case.

On the other hand, if $\Delta\psi(x)<0$ for $x<x_{*}\in(0,1)$ and
$\Delta\psi(x)>0$ for $x>x_{*}$, then $x(t)\stackrel{{\scriptstyle
t\to\infty}}{{\longrightarrow}}0$ or $x(t)\stackrel{{\scriptstyle
t\to\infty}}{{\longrightarrow}}1$ according to the case $x(0)<x_{*}$,
$x(0)>x_{*}$, respectively. The fixation probability is close to 0 for
$x<x_{*}$ and close to 1 for $x>x_{*}$, with a sharp discontinuity around
$x_{*}$. This is the coordination case.

For a $2$-player game, this exhausts all possibilities as $\Delta\psi$ is an
affine function. The analysis is simple if we study the potential defined in
the weak selection approximation $V$, which is a quadratic function. An inner
maximum (minimum) indicates an unstable (stable, respect.) equilibrium in the
replicator equation, and a sharp discontinuity (plateau, respect.) in the
fixation probability of the Wright-Fisher process. For $d$-player games,
$\Delta\psi$ is a polynomial of degree $d-1$ and the situation is more
complex, with sharp discontinuities and plateaux coexisting in the same
fixation function (as inner stable and unstable equilibria will be present in
the associated replicator dynamics), cf. (Chalub and Souza, 2016, 2018).

In the discussion above, it is clear that the fixation pattern will depend on
the function $\Delta\psi$ and, therefore, only on the pay-offs of the game.
However, the situation is different in VSGT, as we have families of games,
indexed by $d$ with possibly different classifications. Therefore, without
changing the family of pay-offs, but changing the game size preferences of the
players, we may change from one pattern to a completely different one.

On the other hand, in a previous work of one of the authors, it was proved
that any fixation pattern can be arbitrarily approximated by the Wright-Fisher
dynamics, with pay-offs given by a certain $d$-player game, if $d$ is large
enough. To obtain the minimum $d$ that approximates a given fixation pattern
within an acceptable approximation, it is necessary to solve an inverse
problem, cf. (Chalub and Souza, 2019). It is clear that the equivalent fixed-
size game will be strongly dependent not only on the family of pay-offs, but
also on the individual strategies with respect to game sizes.

More precisely, in this subsection, we consider a population of two types of
individuals which evolves according to a variable $d$-player game, in which
$d_{\max}=4$. We assume that only the focal player receives the pay-off, i.e.,
$\mu=0$. The relative fitness is given by Eq. (3). The fixation probability is
obtained by solving the associated Wright-Fisher process, as explained in the
introduction, and in the sequel we consider the inverse problem, discussed in
(Chalub and Souza, 2019), to obtain a symmetric game with a fixed number of
players $d=d_{\mathrm{fp}}$ that reproduces with accuracy the fixation
probability; in this case, we say that the game of fixed size is _equivalent_
to the game of variable size proposed. In Fig. 1 we explore the three
traditional cases studied in game theory, dominance, coexistence, and
coordination (Hofbauer and Sigmund, 1998). In the dominance case, we
considered $d$ and $k$ independent pay-offs, such that
$b_{k}^{(d)}>a_{k}^{(d)}$, and $d_{\mathbb{B}}>d_{\mathbb{A}}=1$. In the long
run, the population will fixate in the $\mathbb{B}$ type, but as the pay-offs
are independent of the values of $k$ and $d$, the fixation probability can be
obtained from an equivalent 1-player game, i.e., the constant fitness case.

We also show that we can move from the typical fixation of a coexistence-like
game to a coordination-like game without changing the payoff. In the former
case, there is an essentially constant plateau for the fixation probability,
for initial presence $x$ not too close to 0 (extinction) nor 1 (fixation). The
plateau indicates the existence of an inner equilibrium of the replicator
equation. In the latter case, there is a jump from 0 to 1 in the fixation
probability at $0\ll x_{*}\ll 1$. We showed that we could move from one case
to the other changing only the strategic decision of type $\mathbb{A}$ and
type $\mathbb{B}$ individuals with respect to the choice of the number of
players in each given game.

Figure 1: (a) For any $d$-player game, the pay-off of $\mathbb{B}$ individuals
is larger, $a_{k}^{(d)}=1$, $b_{k}^{(d)}=1.02$ for all $k$ and $d\leq
d_{\max}=4$. We assume that type $\mathbb{A}$ tries to avoid the game, while
$\mathbb{B}$ individuals want games as large as possible, i.e.,
$d_{\mathbb{A}}=1$, $d_{\mathbb{B}}=4$. The equivalent fixed-player game is
presented in subfigure (b), with $d_{\mathrm{fp}}=1$ and pay-offs given by
$\mathbf{a}^{(1)}_{\mathrm{fp}}=(93.07)$,
$\mathbf{b}^{(1)}_{\mathrm{fp}}=(94.94)$. Note that all pay-off may be
multiplied by a constant without changing the game’s result. In this case,
$\mathbf{b}^{(1)}_{\mathrm{fp}}=1.02\times\mathbf{a}^{(1)}_{\mathrm{fp}}$. In
subfigures (c) and (d), we consider the direct and inverse problem,
respectively, for
$\mathbf{a}=\left(\mathbf{a}^{(1)},\dots,\mathbf{a}^{(4)}\right)=\left((1),(6,2),(1,1,1),(1,2,3,2)\right)$
and
$\mathbf{b}=\left(\mathbf{b}^{(1)},\dots,\mathbf{b}^{(4)}\right)=\left((1),(4,1),(1,1,1),(5,3,4,2)\right)$,
i.e., $\mathbb{A}$ dominates the 2-player game, $\mathbb{B}$ dominates the
4-player game and the others are neutral. We assume $d_{\mathbb{A}}=2$ and
$d_{\mathbb{B}}=4$. The pay-offs are chosen such that every time an
$\mathbb{A}$ or $\mathbb{B}$ player is selected as focal, his or her fitness
will increase more than it will decrease in case his or her opponent is
selected. The resulting game is of coexistence type and the equivalent fixed-
player game requires at least 6 players in each game with
$\mathbf{a}^{(6)}_{\mathrm{fp}}=(0.0160,-0.0425,-0.159,5.38,-0.982,0.905)$ and
$\mathbf{b}^{(6)}_{\mathrm{fp}}=(0.0138,-0.0322,-0.221,5.37,-0.839,0.883)$. In
subfigures (e) and (f), we consider the same pay-off as (c) and (d), but
assume that $d_{\mathbb{A}}=4$ and $d_{\mathbb{B}}=2$, i.e., each player
chooses (in a non-rational way) a game in which he or she is at a
disadvantage. The resulting game is equivalent to a coordination type game,
with $d_{\mathrm{fp}}=5$ and
$\mathbf{a}^{(5)}_{\mathrm{fp}}=(1.03,0.849,-0.335,-0.194,0.501)$,
$\mathbf{b}^{(5)}_{\mathrm{fp}}=(4.16,-0.367,0.0973,-0.331,0.531)$. In the
last two cases we also plot the fixation probability of the game with higher
accuracy for suboptimal $d_{\mathrm{fp}}$, with consistent colors in
subfigures (d) and (f). In the three figures above, fixation probabilities are
represented by continuous black lines, while fitness potentials are given by
continuous blue lines. In the three pictures below, fixation probabilities are
also denoted by continuous black lines, but they are superimposed on the
fixation of the equivalent $d$-player game.

######  Remark 4.

Note that the definition of the potential $V$ in Eq. (4) requires the value of
$\Theta(0)$ and $\Theta(N)$. Therefore, the values of $\Psi^{(\mathbb{A})}(0)$
and $\Psi^{(\mathbb{B})}(N)$, and, consequently the definition of the pay-off
functions of a given type, when that given type is absent of the population,
i.e., $\psi^{(d)}_{\mathbb{A}}(0)$ and $\psi^{(d)}_{\mathbb{B}}(N)$, must be
defined for all values of $d$ such that $\lambda^{(d)}_{\mathbb{A}}\neq 0$, or
$\lambda^{(d)}_{\mathbb{B}}\neq 0$, respectively. On the other hand,

$\binom{-1}{k}=\frac{1}{k!}\prod_{i=0}^{k-1}(-1-i)=\frac{(-1)^{k}}{k!}\prod_{i=1}^{k}i=(-1)^{k}$

is the only natural extension of the binomial function, a fact explicitly used
to define $\psi^{(d)}_{\mathbb{A}}(0)$ and $\psi^{(d)}_{\mathbb{B}}(N)$ from
Eqs. (1) and (2), and, therefore to correctly define $V$ from Eqs. (3) and
(4).

###  3.2 Evolution of eusociality

According to Maynard Smith and Szathmáry (1997), a group of individuals is
_eusocial_ if it contains sterile workers that help their parents; typical
examples are insects, but this is also the case of the naked mole rat, a
mammal. However, in (Nowak et al., 2010), the necessity to consider
relatedness between individuals in the same group was questioned, challenging
the foundations of the inclusive fitness theory. This idea generated a large
discussion in the specialized literature, cf., e.g. (Ferriere and Michod,
2011; Herre and Wcislo, 2011) and the answer in (Nowak and Allen, 2015). Here
we explore a possible explanation for the evolution of eusocial behavior
within the framework of VSGT, and therefore, relatedness among individuals is
not required.

We define the eusocial behavior as a given individual in a population working
for the benefit of a different, not necessarily genetically related,
individual. In the proposed model, we consider two different types of players,
that decide to work or not to the benefit of the focal player: type
$\mathbb{A}$ players have the cognitive capacity to gather larger groups of
individuals and collaborate in the pursuit of complex tasks; on the other
hand, type $\mathbb{B}$ individuals are unable to gather larger groups and
even if he or she joins a group large enough to fulfill the task, will be
unable to collaborate or is not willing to do so. The result of the task will
benefit only the focal player; considering successive rounds of the game,
every individual will have the opportunity to be the focal player and get the
lion’s share of the completed task.

More precisely, we consider that $k_{\min}$ eusocial (type $\mathbb{A}$)
individuals are required to fulfill the task, which gives an increase in $G>0$
in the absolute fitness of the focal player, which also pays a cost $c>0$ if
he or she is of type $\mathbb{A}$, and pays nothing otherwise. For all the
other players, the pay-off is identically zero, independently of the fact that
the task has been completed or not. More precisely,
$0<a_{k}^{(d)}=b_{k}^{(d)}=\varepsilon$, for $k<k_{\min}$ (where $\varepsilon$
prevents the existence of zero entries in the transition matrix and is given
to all individuals in the population regardless of their participation or
strategy). If $k\geq k_{\min}$, pay-off’s are given by
$a_{k}^{(d)}=\varepsilon+G-c$ and $b_{k}^{(d)}=\varepsilon+G$. We assume
$\mu=0$, and therefore only the focal player receives the pay-off. The pay-
offs of other players are identically zero. In particular, participation in
the pursuit of the task is evolutionarily neutral to non-focal individuals,
stressing that the model puts all focus on the strategic definition of the
group size.

If there is only a small number of type $\mathbb{A}$ individuals, evolution
will be locally neutral, as no group will have the required number of eusocial
individuals. If we assume $d_{\mathbb{A}}>k_{\min}>d_{\mathbb{B}}$, the same
will happen whenever a type $\mathbb{B}$ individual is selected as a focal
individual. If the number of type $\mathbb{A}$ individuals is sufficiently
large when compared to $k_{\min}$, the presence of type $\mathbb{A}$
individuals may increase randomly such that it is possible to fulfill the
task.

If $d_{\mathbb{A}}>k_{\min}>d_{\mathbb{B}}$, and the initial fraction of type
$\mathbb{A}$ individuals is smaller than a certain critical value, the result
of evolution is uncertain; after that critical value, the long-term dominance
by eusocial individuals is almost certain. This follows from the fact that
$\mathbb{B}$-type individuals are unable to gather sufficiently large groups
to perform the task. When $d_{\mathbb{B}}=d_{\mathbb{A}}$, type $\mathbb{B}$
has similar skills as type $\mathbb{A}$ individuals in forming larger groups
and, therefore, will dominate in the long run. The eusocial behavior will not
prevail, and the same is true if $d_{\mathbb{B}}>d_{\mathbb{A}}$. If
$k_{\min}<d_{\mathbb{B}}<d_{\mathbb{A}}$, type $\mathbb{B}$ individuals are
able to gather large enough groups, but the presence of type $\mathbb{A}$
individuals in these groups (i.e., that the group will have a sufficient
amount of prosocial individuals to perform the task) is not guaranteed.
Therefore, this is a transition region where the result will present a strong
dependence on the value of $G$; a larger value of $G$ will favor prosocial
behavior. See Fig. 2.

Figure 2: Consider the model for the evolution of eusociality, with gain
$G=5,10,20$ (marked with different colors), cost $c=1$, $\varepsilon=1$,
$\mu=0$, minimum size group $k_{\min}=8$, and $d_{\mathbb{A}}=14$. The
fixation probability is obtained from the Wright-Fisher process with
population $N=400$ (continuous lines, scale to the left). Fitness potentials
are indicated by dashed lines of the same color, with values to the right of
the graph. We consider three cases, $6=d_{\mathbb{B}}<k_{\min}$ (left), where
prosocial individuals will prevail if their initial presence is large enough;
$14=d_{\mathbb{B}}=d_{\mathbb{A}}$ (right), where prosocial individuals will
be eliminated by natural evolution (except if their initial presence is close
to 1, in which case they can prevail due to stochasticity); and
$k_{\min}<11=d_{\mathbb{B}}<d_{\mathbb{A}}$, a transition regime, where the
viability of the prosocial individual is uncertain. In the last case, natural
selection will lead to a coexistence regime, associated with a local minimum
of the potential $x_{*}\in(0,1)$ and the final state will be defined by
stochasticity, cf. the discussion at (Chalub and Souza, 2016, 2018).

###  3.3 Spontaneous geographic separation

Following ideas from Nagylaki (1992), we use the Island Model to study
spontaneous separation, according to the game size that each type plays. In
the proposed model, contrary to what may be suggested by the terminology, we
discuss a possible geographic separation, but without assuming any physical
barrier between different environments, i.e., there is no spatial segregation
among types. We consider a model of a small island, with population
$N_{\mathrm{i}}$, close to a large continent with population
$N_{\mathrm{c}}\gg N_{\mathrm{i}}$. These numbers also correspond to the
carrying capacities of both environments, and the population will be
continuously adjusted such that after each interaction the real population
size is equal to the carrying capacity. We refer to the island and the
continent as _patches_ , and the name only reflects the difference in
population size; in particular we allow free migration between both patches,
depending on the individual’s strategy.

The population consists of two types of individuals, $\mathbb{A}$ and
$\mathbb{B}$, that are similar with respect to all characteristics, except
regarding game size preferences $d_{\mathbb{A}}$ and $d_{\mathbb{B}}$. Type
$\mathbb{A}$ likes gathering large groups, while $\mathbb{B}$ prefer small
groups, i.e., $d_{\mathbb{A}}>d_{\mathbb{B}}$. This is similar to the model
developed in Subsec. 3.2, but in this case, there is no task to be performed.
Both types are initially identically distributed (50%-50%) in both patches.
Each type feels comfortable in an environment if they meet in the game
partners with a certain amount of players of identical type $k_{\min}$, namely
$a^{(d)}_{k}=1$ for $k\geq k_{\min}$, and $0$ otherwise, and $b^{(d)}_{k}=1$
for $k<d-k_{\min}$ and 0 otherwise. To stress that all the conclusions follow
solely from the game size that each type plays, $k_{\min}$ will be the same
independently of the focal player. The pay-off is converted, uniquely to the
focal player (i.e., $\mu=0$), into the willingness to stay in the same patch
(if the fitness is $1$) or to migrate to a different one with probability
$p\in(0,1]$, if the fitness is 0.

After each individual in the population has played as a focal player, we
update the population in both patches, and, consequently, the population size
in each patch will be different from the carrying capacity. If it exceeds it,
the exceeding number is removed with an equal probability for all individuals
in the given patch. If the population is below the carrying capacity, then
individuals are randomly selected to reproduce, with an equal probability for
all individuals in the given patch, until the carrying capacity is reached.

The idea of considering migration as a result of a given game pay-off appears
in (Broom and Rychtář, 2012; Erovenko et al., 2019), where the authors
consider the spatial structure of individuals in a given population modeled
through a graph, allowing games played between groups of different size and
analyzing the influence of different graph topologies. In this subsection, we
use VSGT to model spontaneous type separation according to the strategy. This
idea is natural, as the typical structure of the game is of coordination type,
i.e., each type has a high fitness if his or her type predominates among game
players. In fact, in our model, this is not exactly true, as $k_{\min}$ can be
small when compared to $d_{\mathbb{B}}$. The more interesting phenomenon is
not the separation, but the fact that individuals who prefer to play in large
groups (i.e., type $\mathbb{A}$) individuals concentrate in the continent,
with its larger populations, while type $\mathbb{B}$ individuals move to the
island, mimicking migration from rural (_island_) to urban (_continent_) areas
and vice-versa. The separation will be faster depending on the absolute value
of $d_{\mathbb{A}}-d_{\mathbb{B}}$ and on $p$. In the limit case in which
$d_{\mathbb{A}}=d_{\mathbb{B}}$, evolution is purely random, stressing the
fact that the spontaneous separation is due uniquely to the strategic
difference in game size, see Fig. 3.

Finally, we would like to stress that the above model resembles the concept of
_sympatric speciation_. Speciation occurs when a single species splits into
two different species. There are several evolutionary mechanisms possibly
involved in the speciation process. The most relevant one is _allopatric
speciation_ when a geographical (physical) barrier provides segregation of the
parent species into two groups with small or non-existent gene flow among
them. We note that in our model there is no physical barrier between the
island and the continent (despite the choice of words), and every individual
is free to migrate but decides not to do so if he or she is in an environment
of similar individuals. If, in the long run, we identify the types with newly
formed species (due to accumulated mutation in possibly different loci), then
we have an example of _sympatric speciation_ , cf. (Gavrilets, 2004; Mayr,
2001).

Figure 3: We consider three sets of simulations of the continent-island model.
In the first row, we consider $d_{\mathbb{A}}=d_{\mathbb{B}}=5,15,25$ (left,
middle, right, respectively), with $k_{\min}=4,12,20$, always smaller than
$d_{\mathbb{B}}$. Evolution is “quasi-neutral” and, after 100-time steps,
there is no clear spatial separation. Note that, in each one-time step, all
individuals in both patches have the opportunity to migrate, according to the
result of the game. For $d_{\mathbb{A}}>d_{\mathbb{B}}$ (second and third
rows), the separation is clear, with the continent dominated by
type-$\mathbb{A}$ individuals and the island by type-$\mathbb{B}$ individuals.
Values for $d_{\mathbb{A}}$, $d_{\mathbb{B}}$ and $k_{\min}$ are indicated in
the figures. The size of the population of the island is $N_{\mathrm{i}}=50$
and the continent is $N_{\mathrm{c}}=1000$, initial conditions are 25 and 500
individuals of each type in each patch, and the migrations probability is
given by $p=0.2$. Error bars (shadows) are calculated after 100 simulations.

###  3.4 Compartimental models

In this section, we show how to derive the SIRS epidemic model as a VSPG game.
This requires a level of generalization of the theory discussed before, as in
this case each individual has more than two possible strategies, each one
corresponding to the possible compartments in the model.

We will assume a symmetric game, i.e., with $\mu=1$. Consider a population of
$N$ individuals, of three types, Suscpetible, Infectious, and Recovered. The
number of individuals in each class will be given by $S$, $I$, and $R$,
respectively. Let $N:=S+I+R$ be the total population.

We compute payoffs according to the structure of the SIRS model, namely
$S+I\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}I+I$,
$I\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}R$,
$R\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}S$. Note that in our
parlance, the first transition will be represented by a two-player game, while
the last two as one-player games. In this model, all participants receive the
payoffs and not only the focal player.

Consider a given S individual. If he or she was selected as the focal player,
then a two-player game is played and his or her payoff will be changed only if
the second player is of type I. His or her payoff will also be changed if the
focal player is R, as R will _donate_ part of his or her payoff to the entire
pool of S players (transition
$R\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}S$). Namely

$\Psi_{\textbf{S}}(S,I,R)=-\beta\frac{I}{N}+\alpha\frac{R}{S}\ .$ (6)

Now, consider an I individual. If he or she is the focal player, the pay-off
decreases by a constant, representing the transition
$I\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}R$ (one-player game). The
payoff of an I individual is also changed if the focal player was of S type
and he or she was selected as the second player in the two-player game, what
happens with probability $1/N$. Therefore

$\Psi_{\textbf{I}}(S,I,R)=\beta\frac{S}{N}-\gamma\ .$ (7)

Finally, the pay-off of an R individual decreases if it is focal and increases
if I is focal, i.e.,

$\Psi_{\textbf{R}}(S,I,R)=\gamma\frac{I}{R}-\alpha.$ (8)

See also Fig. 4 for further information in the how Eqs. (6)–(8) were obtained.

It is clear that the average fitness is zero, i.e.,

$\displaystyle\overline{\Psi}(S,I,R)$
$\displaystyle\quad:=S\Psi_{\textbf{S}}(S,I,R)+I\Psi_{\textbf{I}}(S,I,R)+R\Psi_{\textbf{R}}(S,I,R)$
$\displaystyle\quad=0\ .$

The replicator equation is exactly the SIRS model:

$\displaystyle S^{\prime}$
$\displaystyle=S\left(\Psi_{S}(S,I,R)-\overline{\Psi}(S,I,R)\right)=-\frac{\beta
SI}{N}+\alpha R\ ,$ $\displaystyle I^{\prime}$
$\displaystyle=I\left(\Psi_{I}(S,I,R)-\overline{\Psi}(S,I,R)\right)=\frac{\beta
SI}{N}-\gamma I\ ,$ $\displaystyle R^{\prime}$
$\displaystyle=R\left(\Psi_{R}(S,I,R)-\overline{\Psi}(S,I,R)\right)=\gamma
I-\alpha R\ .$

Figure 4: Consider a population of $N=10$ individuals of one of the three
types: S, I, and R. The three figures above show the fitness loss from the
focal individual to all individuals in the population, while the row below
shows the fitness gain of an individual from all the other individuals in a
given game. If the individual selected as the focal player is a type I
individual, its fitness is decreased by $\gamma$, distributed among all R
individuals (a). The fitness of an R individual will be increased by
$\gamma/R$ times $I$, the number of $I$ individuals in the population (d). If
the focal type is of type R, its fitness is decreased by $\alpha$ (b), and the
fitness of type S is increased by $\alpha/S$ for each R present in the
population (d). Finally, if the focal individual is of type S, it is necessary
to select a second individual. If the second individual is of type I (what
happens with probability $I/N$), then the S individual loses a fitness $\beta$
and nothing happens otherwise (c). On the other hand, an I individual receives
a fitness $\beta$ from an S individual _if he or she was selected as the
second player_ , what happens with probability $1/N$ (f). All non-represented
arrows mean a fitness transfer of 0; cf. Eqs (6), (7), and (8).

##  4 Discussion

The main novelty of the present work is to consider that games inside a
population may vary in the number of participants as a consequence of a
player’s strategy.

We started by showing that the traditional examples of game theory can be
recast into the framework of VSGT, and it is not necessary to change pay-offs
to move from coexistence- to coordination-type games and vice-versa. A change
in the prosociality (i.e., the typical number of participants in a game, when
we are allowed to choose the game size) of the players involved in the game
can have dramatic consequences in the long run.

As a second example, we showed that this framework is compatible with the
fixation of a strategy in which individuals work for the benefit of a single
one. The focal player is defined at random, which seems artificial if we think
of a population of individuals, but natural when we consider that the evolving
unit is the gene. The loosely defined _eusocial gene_ will be present in all
individuals of a given type, but cannot know (abusing the language) which of
its copies will manifest at the leader.

As a third example, in Subsec. 3.3, we showed how a difference in the size of
the game each type plays, along with the preference to be with similar
individuals, may provide spontaneous geographical separation between types,
even in the absence of a rigid barrier between different environment. This
makes prosocial individuals migrate to the larger environment, while
antisocial individuals move toward regions with smaller populations, in a
phenomenon that resembles sympatric speciation.

As a last example, we also showed how to reformulate an epidemic model as a
game in which transmission is seen as a two-player game and the other
transitions, that do not depend on interactions in the populations, are
considered as a one-player game. Each individual may play a two-player game if
the focal player is of the Susceptible type and selects this individual –
possibly against his or her will – to be part of the game.

In this work, we do not consider certain classical topics in game theory, for
example, proofs of existence and classification of Nash equilibria, or the
study of evolutionarily stable strategies. The proof of the existence of Nash
equilibria follows without change from the one used in games of fixed size, as
they are based only on the continuity of the pay-off function and the
convexity and compactness of the domain of possible strategies (i.e.,
$S^{n-1}$ is a convex, bounded, and closed subset of $\mathbb{R}^{n}$). Note
that in this case we probably will have to resort to mixed strategy games,
defined, in this case, as non-trivial probability distributions
$\bm{\lambda}$. The details will be developed in a follow-up work, where we
also plan to study Evolutionarily Stable Strategies. One interesting question
is the relation between these traditional concepts in game theory and the
stability of the disease-free and endemic equilibria in epidemic models.

Many applications of these ideas can be expected in the near future. The most
obvious application is in economic modeling, where the focal player may
deliberately define the size of the game. Consider, for example, a
hierarchical structure, as a company (or an academic department), in which all
employees have to deal with one boss, but the boss has to deal with dozens, or
more, employees, that can be or not be structured in smaller groups. Our
framework may be used for the design of an optimal hierarchical structure.
Another idea is to study how strategies vary over time, modeling the
discussion present in (Carstensen et al., 1999), in which when the number of
expected future interactions decrease — as a consequence of aging, for example
— more priority is given to interactions with affective value, i.e., with
smaller $d$ but larger pay-offs. Note that the strategy-dependence on future
expected interaction is a well-established concept in evolutionary game
theory, following the seminal work by Axelrod and Hamilton (1981); see also
(Axelrod, 1984).

In biology, one idea is to study haplodiplontic cycles, in which, e.g., a gene
manifested in a male will play a 1-player game, and the same gene in a female
will play a 2-player game. The study of species with alternations of
generations is another possible application of the ideas developed here. We
also envisage an application in the study of parent-offspring conflict, in
particular, the evolution of genomic imprinting, as the number of players
changes during the development (for example comparing the number of players
during gestational time and after that period). Another application is the
study of the coexistence between one large individual of a given species with
a cohort of individuals in a mutualistic or parasitic interaction.

## Acknowledgement

FACCC and MH thanks C. Gokhale for suggesting applying the ideas from (Chalub
and Souza, 2019) to a non-fixed number of players. This idea is at the origin
of the present work. MH and FACCC are funded by national funds through the FCT
– Fundação para a Ciência e a Tecnologia, I.P., under the scope of the
projects UIDB/00297/2020 (https://doi.org/10.54499/UIDB/00297/2020) and
UIDP/00297/2020 (https://doi.org/10.54499/UIDP/00297/2020) (Center for
Mathematics and Applications). FACCC also acknowledges the support of the
project _Mathematical Modelling of Multi-scale Control Systems: applications
to human diseases_ 2022.03091.PTDC (https://doi.org/10.54499/2022.03091.PTDC),
supported by national funds (OE), through FCT/MCTES. (CoSysM3). MH and FACCC
contributed equally to the development of the work’s ideas, computational
codes, data analysis, discussions, and writing of the final version of the
manuscript. We also thank both referees for many suggestions that helped to
improve this work, and Renata Ramalho (Universidade NOVA de Lisboa) for
helping in the language editing of the manuscript.

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