# Mean-field interactions in evolutionary spatial games

Dmitriy Antonov1 Evgeni Burovski1 Lev Shchur1,2 1 HSE University, 101000
Moscow, Russia  2 Landau Institute for Theoretical Physics, 142432
Chernogolovka, Russia

###### Abstract

We introduce a mean-field term to an evolutionary spatial game model. Namely,
we consider the game of Nowak and May, based on the Prisoner’s dilemma, and
augment the game rules by a self-consistent mean-field term. This way, an
agent operates based on local information from its neighbors and non-local
information via the mean-field coupling. We simulate the model and construct
the steady-state phase diagram, which shows significant new features due to
the mean-field term: while for the game of Nowak and May, steady states are
characterized by a constant mean density of cooperators, the mean-field game
contains steady states with a continuous dependence of the density on the
payoff parameter. Moreover, the mean-field term changes the nature of
transitions from discontinuous jumps in the steady-state density to jumps in
the first derivative. The main effects are observed for stationary steady
states, which are parametrically close to chaotic states: the mean-field
coupling drives such stationary states into spatial chaos. Our approach can be
readily generalized to a broad class of spatial evolutionary games with
deterministic and stochastic decision rules.

Introduction— The mean-field approximation, initially devised in 1907 in the
context of (ferro)magnetic properties of materials Weiss1907 , is since
playing a truly central role in a variety of branches of physics, ranging from
superconductivity Sordi2012  to ferromagnetism of disordered metals
Andreev2004 ; from ultracold quantum gases Aikawa2012  to spin glasses
Gross1985 , to name just a few. The main idea—that a behavior of a collection
of a macroscopic number of strongly interacting components can be reduced to a
single-particle problem in a self-consistent external potential—proves
fruitful for multiple research fields outside of traditional physics: for
instance, describing spatially extended networks Bonamassa2019 , droplet
traffic Engl2005 , traffic flow Schreckenberg1995 , and social dilemmas
Lin2019 .

In this Letter, we demonstrate how mean-field ideas can be applied to a system
that does not have an explicit representation in the language of statistical
mechanics. Instead, it is a deterministic dynamical system with a steady-
state, i.e., a long-lived state with a well-defined “thermodynamic limit” of
the infinite system size. We start with the completely deterministic
evolutionary game model of Nowak and May Nowak1992  and add a novel
ingredient, the coupling to the instantaneous mean density of cooperators. The
modification does not change the deterministic nature of the process in
contrast with the previous research (see, f.e., Perc2017 ; Perc2021 ;
Javarone2016 ; Javarone2016_memory ; Javarone2018 ) in which the stochastic
nature of modifications obviously opens the possibility to using methods of
statistical physics. Once again, our model is deterministic (specifically,
there is no temperature, no fluctuations, and no random forces). The
introduction of the mean-field interaction leads to a nontrivial diagram of
steady states and opens new directions for possible applications of structured
multi-agent systems. Our approach can be generalized to any deterministic or
stochastic spatial evolutionary game.

Studying the dynamics of ensembles of agents is a common theme in biology,
economics, and social sciences. Collective behavior emerges through repeated
pairwise interactions between individuals, where each agent acts to maximize
its fitness. The evolutionary game theory approach model elementary
interactions as game-theoretic contests between agents having varying
strategies Smith1982 . For well-mixed populations—in physics language, this
corresponds to a mean-field approximation—the macroscopic description proceeds
via the master equation for the time evolution of populations of strategies,
the so-called replicator equation Weibull1995 ; Hofbauer2008 .

One limitation of the replicator equation approach is that it does not allow
for _structured_ populations, where the interaction range is restricted to a
local neighborhood. Such populations are known to exhibit long-time behavior
beyond standard replicator dynamics Hauert2005 . Various local arrangements
have been considered: following the pioneering work Nowak1992 , noisy dynamics
were studied on regular, decorated, and frustrated two-dimensional lattices
Szolnoki2005 ; Hauert2005 ; Szolnoki2017  and coevolving random networks
Szolnoki2009 . Deterministic imitation dynamics have been investigated on
regular square Vainstein2014 ; Kolotev2018  and triangular lattices
Burovski2019 , on a simple cubic lattice 3D Moskalenko2020 , and diluted 2D
lattices Vainstein2001 ; Vainstein2014 .

In the Letter, we construct an evolutionary game model which features both
kinds of effects: purely local interactions with nearest neighbors, _and_ a
self-consistent, mean-field-type coupling to the order parameter. We start
with the classic game of Nowak and May (NM) Nowak1992 ; Nowak1993 , where the
order parameter can be chosen as the mean density of cooperators
$\left<f_{c}\right>$ in the steady-state regime of evolution. The mean-field
term is the coupling to the instantaneous density of cooperators, $f_{c}(t)$,
which fluctuates due to the game dynamics and has a well-defined mean value in
the steady-state regime. The addition of the MF term to the decision-making
rule is quite realistic. It can be interpreted as the influence of the
“averaged strategy of society” or a mass-media influence that reflects and
drives society’s opinions. In the limit of vanishing mean-field (MF) term, the
phase diagram reproduces known results.

We note that our approach is very different from what is known as mean-field
games in mathematical economics LasryLions2007 : the latter is a class of
stochastic differential games (which, in turn, can be mapped onto the
nonlinear Schroedinger equation Swiecicki2016 ). In our work, we only consider
discrete deterministic games based on the Prisoner’s dilemma.

The Prisoner’s dilemma.— Consider two agents, $\alpha$ and $\beta$, which
interact via the rules of the Prisoner’s dilemma game. An agent is
characterized by a _strategy_ , which takes one of two values: cooperate,
$\mathcal{C}$, or defect, $\mathcal{D}$. In an “interaction”, agents receive
payoffs which depend on their strategies Axelrod-1981 . Encoding strategies of
an agent by length-two vectors, $\vec{s}$, such that $\mathcal{C}{=}(1,0)^{T}$
and $\mathcal{D}{=}(0,1)^{T}$, the payoff of an agent $\alpha$ is
$P_{\alpha}{=}\vec{s}_{\alpha}^{\,T}\widehat{H}\vec{s}_{\beta}$, and the
payoff of the agent $\beta$ is
$P_{\beta}{=}\vec{s}_{\beta}^{\,T}\widehat{H}\vec{s}_{\alpha}$, where
$\widehat{H}$ is the payoff matrix, and $\vec{s}_{\alpha}$ and
$\vec{s}_{\beta}$ are the states of agents $\alpha$ and $\beta$, respectively.
Following Ref. Nowak1992 , we take

$\widehat{H}=\begin{pmatrix}1&0\\\ b&0\end{pmatrix}\;.$ (1)

Essentially, Eq. (1) means that in a $\mathcal{C}$-$\mathcal{C}$ interaction,
both agents receive a unit payoff (we set it to unity to fix the payoff
scale), and in a $\mathcal{D}$-$\mathcal{C}$ interaction, the $\mathcal{D}$
receives a payoff of $b$. We set all other payoffs to zero so that a single
parameter, $b$, governs the game. In general, one may vary the two additional
parameters, which correspond to the payoffs in the $\mathcal{D}$-$\mathcal{D}$
and $\mathcal{C}$-$\mathcal{D}$ interactions. We keep both these parameters
equal zero for simplicity and to allow a direct comparison with Ref. Nowak1992
.

The rules of the Nowak-May spatial game Nowak1992 ; Nowak1993 . — Consider a
population of $L^{2}$ players arranged at the vertices of a regular $L\times
L$ square lattice with periodic boundary conditions. The game is played at
discrete time steps, $t{=}0,1,2,\ldots,T$. The transition of the system from
time $t$ to the next time step $t{+}1$ consists of the update of strategies of
all players in parallel (we call this a game round). At time step $t$, an
agent located at the lattice site x “interacts” with its neighbors and
receives a total payoff,

$P_{\mathbf{x}}(t)=\sum_{\mathbf{y}\in\mathrm{n.n.}}\vec{s}_{\mathbf{x}}^{\,T}\widehat{H}\vec{s}_{\mathbf{y}}\;,$
(2)

where $\vec{s}_{\mathbf{x}}{\equiv}\vec{s}_{\mathbf{x}}(t)$ is the strategy of
an agent $\mathbf{x}$ at a time step $t$, and the summation runs over the
eight neighbors of the agent $\mathbf{x}$ (similar to the chess king
moves)111Refs. Nowak1992 ; Kolotev2018  also considered a modification of Eq.
(2) to include a self-interaction term,
$\vec{s}_{\mathbf{x}}^{\,T}\widehat{H}\vec{s}_{\mathbf{x}}$. This modification
does not lead to any qualitative differences. We therefore follow Ref.
Nowak1993  and do not include self-interaction.. After all, payoffs are
calculated, each player changes its state,
$\vec{s}_{\mathbf{x}}(t){\to}\vec{s}_{\mathbf{x}}(t{+}1)$, to accept the
strategy of their neighbor with the largest payoff,

$\max\\{P_{\mathbf{x}}(t),\max_{\mathbf{y}\in\mathrm{n.n.}}\\{P_{\mathbf{y}}(t)\\}\\}\;.$
(3)

This concludes the game round, and the game repeats with the new strategies of
players.

To fully specify the time dynamics of the game, we need to define the initial
state at $t=0$. We follow Ref. Nowak1992  and start the evolution with a
random initial state where each player is $\mathcal{C}$ with a probability
$f_{0}$ and a $\mathcal{D}$ with a probability $1-f_{0}$. In the numerical
simulations, we average over several realizations of initial conditions at
fixed $f_{0}$.

Note that the initial conditions provide the only source of randomness in the
game: the dynamics is fully deterministic given the initial state of all
strategies at $t{=}0$. The strategy of a player $\mathbf{x}$ at the next time
step, $\vec{s}_{\mathbf{x}}($t+1$)$, depends on the current state of itself,
$\vec{s}_{\mathbf{x}}(t)$, its eight neighbors, and their neighbors—i.e., on
states of 25 players in total. In the cellular automata language, the spatial
evolutionary game of Nowak-May (NM game) is a cellular automaton with the
single-player’s transition matrix containing $2^{25}$ rules.

The steady-state of the spatial evolutionary game.— The game starts with some
initial distribution of strategies at $t{=}0$ and proceeds repeatedly. After a
certain transition time, the system reaches a steady-state, with the nature of
the steady-state strongly dependent on the value of the game parameter $b$
Nowak1992 . Following Ref. Nowak1992 , we characterize the instantaneous state
of the game by the density of cooperators, $f_{c}(t)$, and the steady-state is
characterized by the mean density of cooperators $\left<f_{c}\right>$:

$\displaystyle\left<f_{c}\right>$ $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\frac{1}{T-\tau}\sum_{t=\tau}^{T}{f_{c}(t)}\;,$
(4) $\displaystyle f_{c}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{L^{2}}\sum_{\mathbf{x}}\delta\left(\vec{s}_{\mathbf{x}}(t)=\mathcal{C}\right)\;.$
(5)

The summation in Eq. (5) runs over the states of all agents at time $t$, and
$\delta\left(\vec{s}_{\mathbf{x}}(t){=}\mathcal{C}\right)$ is an indicator
variable taking the value of 1 if $\vec{s}_{\mathbf{x}}(t){=}C$ and zero
otherwise. In other words, Eq. (5) is nothing but the number of cooperators
divided by the total number of agents. In Eq. (4), $T$ is the total number of
game rounds, and $\tau$ is the burn-in time such that
$|\left<f_{c}\right>{-}f_{c}(t)|{\ll}\left<f_{c}\right>$ for $t{>}\tau$.

The mean-field modification.— In this Letter, we include the mean-field term
as follows: instead of Eq (2) we consider payoffs of the form

$\displaystyle
P_{\mathbf{x}}(t)=\sum_{\mathbf{y}\in\mathrm{n.n.}}\vec{s}_{\mathbf{x}}^{\,T}\widehat{H}\vec{s}_{\mathbf{y}}+\lambda\,\vec{s}_{\mathbf{x}}^{\,T}\widehat{H}_{\mathrm{mf}}\vec{s}_{\mathbf{x}}\;,$
(6) where $\displaystyle\widehat{H}_{\textrm{mf}}=\begin{pmatrix}f_{c}(t)&0\\\
0&b\,f_{c}(t)\\\ \end{pmatrix}\;.$ (7)

In other words, the interaction of a player with its neighbors is identical to
the the NM game, Eq. (2). Additionally, we introduce coupling to the
instantaneous density of cooperators, $f_{c}(t)$. This way, transition rules
become non-local in space due to the dependence on the density of cooperators
at the current time step $f_{c}(t)$. Here, we stress that $f_{c}(t)$ is the
instantaneous density so that the game is memory-less.

In Eqs. (6)-(7), $\lambda$ is the relative strength of the MF coupling. The
limit $\lambda{\to}0$ is simply the NM game, Eq. (2). In this work, we only
consider $\lambda{=}1$, so that $b$ is the only parameter that governs the
game dynamics.

The transition from time step $t$ to $t{+}1$ is identical to the NM game:
After all payoffs are calculated, each agent changes its strategy according to
Eq. (3), and the game repeats with the new strategies of players.

Dynamical regimes.— The discrete structure of the payoffs in the NM game
(1)-(2) leads to a very specific dependence of the game dynamics on the payoff
parameter $b$. Comparing the payoffs of $\mathcal{C}$ and $\mathcal{D}$ in the
neighborhood of an agent, one finds for the NM game Nowak1993 ; Kolotev2018
that the dynamics of the game changes at the values of $b{=}m/n$ with
$1{\leqslant}m,n{\leqslant}8$. Here $m$ and $n$ are integers and are related
to the numbers of $\mathcal{C}$ in a local neighborhood. The steady-state
density of cooperators, $\langle f_{c}\rangle$ takes discontinuities at these
special values of $b$, as shown in Fig. 1(a) with the open circles.

For the mean-field game (6)-(7) (MF game), a similar comparison of the payoffs
of neighboring agents shows that an agent changes state depending on the ratio
of $m+f_{c}(t)$ and $n+f_{c}(t)$ — which self-consistently depends on the
density of cooperators at time $t$. Since in the steady state we expect
$f_{c}(t){\approx}\langle f_{c}\rangle$, we conjecture that the steady state
density is related to the payoff parameter via

$b=\frac{m+\left<f_{c}\right>}{n+\left<f_{c}\right>}\;,$ (8)

where $m$ and $n$ are integers and $1{\leqslant}m,n{\leqslant 8}$. Note that
Eq. (8) cannot be directly interpreted as giving the boundaries of dynamical
regimes. To clarify the status of Eq. (8) we compare its predictions to direct
numerical simulations.

Numerical simulations.— We simulate the dynamics of the game on the square
lattice with linear size $L{=}400$. We use periodic boundary conditions to
minimize finite-size effects. Additional simulations with system size up to
$L{=}800$ demonstrate no significant finite-size effects. Therefore we believe
our simulations correspond to the thermodynamic limit of the model 222At
smaller values of $L$ some $b$ realizations of initial conditions contain
long-lived artifacts, which disappear as the field size is increased. . For
all values of $b$, we simulate 100 runs with random initial distributions of
$\mathcal{C}$ and $\mathcal{D}$ using the initial density of cooperator states
$f_{0}$. Each configuration converges to a particular steady-state, and we
compute averages for both time fluctuations in the steady-state and
realizations of initial conditions. The steady-state behavior is largely
insensitive to the precise value of $f_{0}$, and we fix $f_{0}{=}0.9$ in most
simulation runs 333At lower values of $f_{0}$ there are initial states where
all $\mathcal{C}$ are isolated at $t{=}0$ and thus immediately disappear at
$t{=}1$, resulting in $\left<f_{c}\right>{=}0$. When these initial states are
discarded, the steady-state observables are independent on $f_{0}$., which is
the value used in Refs Nowak1993 ; Kolotev2018 . The number of game rounds is
typically a few thousand for the burn-in time towards the steady-state,
$\tau$, cf Eq. (4), which is larger than the typical relaxation time for the
initial state. After burn-in, we collect statistics for a few thousand further
game rounds, $T{-}\tau$.

Figure 1: (a) Steady-state mean density of cooperators $\left<f_{c}\right>$
for the NM game (open blue circles) and MF game (full red squares) as a
function of the payoff parameter $b$. Errorbars are shown for MF game at all
points and are typically less than symbol size. We also show hyperbolas, Eq.
(8) for $m,n{=}(8,5)$ (green dashed line) and $m,n=(5,3)$ (cyan dash-dotted
line). In these simulations, $f_{0}{=}0.9$, $L=400$, $\tau{=}10^{4}$,
$T{=}1.6{\cdot}10^{4}$. Inset shows the zoom-in of the region
$1.51{<}b{<}1.66$.
(b) Steady-state persistence, $P(\tau,\infty)$, for the NM game (open blue
circles) and the MF game (red squares). We also show $P(0,\infty)$ for the MF
game (yellow crosses). See text for discussion.

Figure 1(a) shows the dependence of the steady state mean density of
cooperators, $\langle f_{c}\rangle$, on the payoff parameter $b$ for both NM
and MF games. The values of $\langle f_{c}\rangle$ are essentially identical
outside of the interval $3/2{<}b{<}5/3$. In this interval, however, the NM and
MF games are strikingly different. For the MF game, the steady state density,
$\langle f_{c}\rangle$, varies continuously with $b$ — the values of $\langle
f_{c}\rangle$ closely follow the hyperbolas, Eq. (8), with $(m,n){=}(8,5)$ and
$(5,3)$. This behavior is to be contrasted with the NM game, where $\langle
f_{c}\rangle$ only changes via discontinuous jumps at $b{=}3/2$, $8/5$ and
$5/3$.

Another remarkable feature of the MF game, Fig. 1(a), is the flat segment
between the $(m,n){=}(8,5)$ and $(5,3)$ states at $\langle
f_{c}\rangle{\approx}0.3$—which corresponds to the value of $\langle
f_{c}\rangle$ in the chaotic regime of NM game. The switch between two regimes
is reminiscent of the first-order phase transitions, driven by the competition
between two stable states. In our case, it is the switch between two competing
states—with $(m,n){=}(8,5)$ and $(5,3)$—driven by the mean-field coupling.

The mean-field coupling, Eqs. (6)– (7), changes the nature of the transitions
between steady state regimes: for the NM game the transitions are
discontinuous; for the MF game, the density $\langle f_{c}\rangle$ is
continuous, but its derivative w.r.t. $b$ jumps. The locations of transitions,
$b_{*}$, are defined by the crossings of constant values $f=f_{*}$ and
hyperbolas, Eq. (8). In the vicinity of a transition, expanding Eq. (8) for
small $|b-b_{*}|$, we find

$f-f_{*}=A_{(m,n)}|b-b_{*}|^{\beta}\;,$ (9)

where the amplitudes $A_{m,n}\propto 1/(m-n)$ and the pseudo-critical exponent
$\beta=1$. Table 1 lists the numeric values of amplitudes for transitions
shown in Fig. 1(a).

$(m,n)$ | $f_{*}$ | $b_{*}$ | $A_{(m,n)}$ |
---|---|---|---|---
(8, 5) | 0.74 | 1.522 | -10.992 | $b>b_{*}$
(8, 5) | 0.3 | 1.566 | 9.362 | $b<b_{*}$
(5, 3) | 0.3 | 1.606 | -5.444 | $b>b_{*}$
(5, 3) | 0.23 | 1.618 | 5.232 | $b<b_{*}$
Table 1: Amplitudes for the transitions between steady states in Fig. 1(a).
See text for discussion.

Spatial chaos.— A salient feature of the NM game is the appearance of the
chaotic regime: for $8/5{<}b{<}5/3$ all agents change their strategy with
typical time scales of several game rounds while maintaining the mean density
constant, $\langle f_{c}\rangle{\approx}0.3$ Nowak1993 . (For the NM game with
self-interactions, the chaotic regime occurs for $9/5{<}b{<}2$ with a slightly
larger density, $\langle f_{c}\rangle{\approx}12\ln{2}{-}8{\approx}0.318$
Nowak1992 ).

The standard way of probing the chaotic behavior is via the so-called
persistence Derrida1994 ; Stauffer1994 ; Majumdar1999  $P(t_{1},t_{2})$, which
in the present context is nothing but the fraction of players who never change
their strategy between time steps $t_{1}$ and $t_{2}$. Note that we are
primarily interested in the _asymptotic steady state persistence_ ,
$P(t_{1},t_{2}{\to}\infty)$. To this end, we compute $P(\tau,T)$ with $\tau$
and $T$ having the same meaning as in Eq. (4). Averaging this value over the
realizations of initial conditions, we obtain an estimate of the asymptotic
value, $P(0,\infty)$. We stress that $P(\tau,\infty)$ is qualitatively
different from $P(0,\infty)$: The latter is sensitive to the overlap between
the steady state and the initial distribution of strategies at $t{=}0$.

Fig. 1(b) compares the asymptotic persistence for the NM and MF games. Two
kinds of regimes are seen for both games: those with $P(\tau,\infty){\neq}0$ —
where the steady-state features finite clusters (or “cliques”) of agents who
cooperatively maintain a fixed strategy, and chaotic regimes where
$P(\tau,\infty){=}0$ — where everyone changes their strategy at least once.

The dependence of $P(\tau,\infty)$ for the NM and MF games is very similar,
and the main difference is observed for $3/2{<}b{<}5/3$: The NM game displays
chaotic steady states for $8/5{<}b{<}5/3$ as expected; for the MF game the
chaotic regime is shifted to $1.53{\lesssim}b{\lesssim}1.6$, which is exactly
the range where the mean density of cooperators, $\langle f_{c}\rangle$,
follows Eq. (8).

In other words, the effect of the mean-field coupling, Eqs. (6)-(7), is
drastic for steady states of the NM game, which are stable but is
parametrically close in $b$ to chaotic states: the MF coupling drives such
stationary states into spatial chaos.

We also show in Fig. 1(b) the behavior of $P(0,\infty)$ for the MF game. For
$b{<}1$, about 40% of agents keep their initial strategy. Upon increasing $b$,
$P(0,\infty)$ jumps to below ${\sim}20$% at $b{=}1$ and then further decreases
before dropping off to zero at the onset of the chaotic regime. Note that
$P(0,\infty)$ keeps being essentially zero for $b{>}1.63$ where
$P(\tau,\infty)$ is non-zero. This means that while the finite fraction of
agents stabilize, the steady-state of an agent is not related to the initial
state at $t{=}0$, which is forgotten over the first ${<}\tau$ time steps..

Figure 2: Typical snapshots of the steady-state configurations of players for
the MF game. Here we show $100\times 100$ part of the field with $L{=}400$,
and values of $b{=}1.2$(a), $b{=}1.6$(b), $b{=}1.62$(c) and $b{=}1.63$(d).
Blue (light grey) color corresponds to the cooperator state $\mathcal{C}$, red
(dark grey) color is the defector state $\mathcal{D}$, green color is a
$\mathcal{C}$ which was a $\mathcal{D}$ at the previous time step, and yellow
is a $\mathcal{D}$ which was a $\mathcal{C}$.

Game field configurations.— Fig. 2 shows representative snapshots of steady
states of the game field of the MF game depending on the payoff parameter $b$.
For $b{<}1.53$ typical configurations feature web-like structures of
$\mathcal{D}$ players. These webs are random, i.e., precise positions of the
branches vary depending on an initial state of the field(see Fig. 2(a)). In
the steady state, these webs remain mostly static, with possible “blinking”
players at the interfaces. The widths of the branches of defectors increases
with increasing the payoff parameter $b$, and at around $b{\approx}1.53$—which
is the onset of the chaotic regime— the $\mathcal{D}$ webs melt into
collections of smaller clusters of $\mathcal{C}$ and $\mathcal{D}$, which are
no longer static and grow, shrink and collide instead.

At $b{>}1.53$ field is dominated with clusters of $\mathcal{C}$ and
$\mathcal{D}$ of varying shapes and sizes (Fig. 2(b)). Upon further increase
of the payoff parameter, for $1.6{<}b{<}1.62$ the field is dominated by
square-shaped clusters of $\mathcal{C}$(see Fig. 2(c)). Isolated clusters grow
in all directions, and disintegrate upon colliding with neighboring clusters.
For $b{>}1.63$ the fields feature static collections of small square-shaped
clusters of $\mathcal{C}$ embedded into the $\mathcal{D}$ background(see Fig.
2(d)).

The rectangular shapes of spatial patterns that emerge in Fig. 2 can be
readily understood from the synchronous evolution rules and the payoff
structure of the NM game. Consider an interface between regions of
$\mathcal{C}$ and $\mathcal{D}$ agents. The interface can be either straight
(i.e., oriented along the square lattice directions), or “angled” (e.g.,
oriented along the diagonal of the lattice). For the NM game ($\lambda=0$ in
Eq. (7)), a square cluster of $\mathcal{C}$ is stable for $5/3<b<8/3$; however
the “angled” interface is only stable for $7/3<b<8/3$. For $b<7/3$, the angled
interface evolves into a straight one in several time steps—and the straight
interface is stable. Introducing the mean-field term, Eq. (7) changes switch-
over ratios, $m/n$, into density-dependent hyperbolas, Eq. (8).

It is instructive to examine the distribution of the sizes of clusters of
$\mathcal{C}$ and $\mathcal{D}$ in different steady-state regimes. Figure 3
shows the exponential decay in the distribution of the cluster sizes in
chaotic regimes, $b{=}1.55$ and $1.59$, while it is slower than exponential
for $b{=}1.2$, the regime with the long memory of initial states (see Fig.
1(b)).

Asymptotically, the fractal dimension of the clusters as well as of the
interfaces between clusters of cooperators $\mathcal{C}$ and defectors
$\mathcal{D}$ tends to the dimension of the plane, $d{=}2$, which also
coincides with the scaling dimension of masses of clusters Kolotev2018 . These
interfaces form a special type of random fractals, quite different from those
at the second-order or first-order phase transitions Bunde1996 .

Figure 3: Distribution of the cluster sizes (i.e. the number of agents in a
cluster) of $\mathcal{D}$ (red) and $\mathcal{C}$ (blue) in the steady states
for $L{=}200$, and values of $b{=}1.2$ (a), $b{=}1.55$ (b), $b{=}1.59$ (c) and
$b{=}1.63$ (d). We only show clusters of up to 250 agents.

Discussion.— We propose a way of introducing self-consistent mean-field-like
couplings into evolutionary games with spatially structured populations. We
perform direct numerical simulations of a simplest yet non-trivial model—where
decision rules include non-local information due to the mean-field
coupling—and find a variety of steady states: chaotic states and self-
organized stationary states where a finite fraction of agents form stable
clusters (or “cliques”). These states are not critical because cluster sizes
are distributed exponentially.

We find that the mean-field coupling induces qualitative changes for
stationary steady states, which are parametrically close to chaotic states.

In this work, we only consider a restricted version of the Prisoner’s Dilemma,
Eq. (1). Our approach however, can be readily generalized to a wide class of
games, including PD games with non-zero $\mathcal{C}$-$\mathcal{D}$ and
$\mathcal{D}$-$\mathcal{D}$ payoffs and versions of the Public Goods game.

The introduction of the mean-field type couplings opens several intriguing
avenues for possible future work. It would be interesting to clarify the
effect of the strength of the MF coupling, $\lambda{\neq}1$. Non-uniform,
site-dependent couplings (e.g., $\lambda_{\mathbf{x}}$ drawn from some random
distribution) would reflect natural variations between individuals. The role
of the local connectivity of the population needs to be clarified: this
includes both higher-dimensional regular lattices and disordered and
hierarchical random lattices and graphs. One may introduce an external field.
It would be interesting to see the interplay between the mean-field couplings
and noisy decision rules (e.g., where the decision rules contain pseudo-
temperature Hauert2005 ; Szolnoki2005 ; Javarone2016 ).

###### Acknowledgements.

L.S. is supported within the framework of State Assignment of Russian Ministry
of Science and Higher Education. E.B. and A.D. acknowledge support within the
Project Teams framework of MIEM HSE. We thank J.J. Arenzon and A. Szolnoki for
valuable suggestions. Simulations were carried out in part through
computational resources of HPC facilities at HSE University.

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