Deterministic and stochastic cooperation transitions in evolutionary games on networks Nagi Khalil1I. Leyva1 2J.A. Almendral1 2and I. Sendi na-Nadal1 2

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Deterministic and stochastic cooperation transitions in evolutionary games on

networks

Nagi Khalil,1, ∗I. Leyva,1, 2 J.A. Almendral,1, 2 and I. Sendi˜na-Nadal1, 2

1Complex Systems Group & GISC, Universidad Rey Juan Carlos, M´ostoles, 28933 Madrid, Spain

2Center for Biomedical Technology, Universidad Polit´ecnica de Madrid, Pozuelo de Alarc´on, 28223 Madrid, Spain

Although the cooperative dynamics emerging from a network of interacting players has been

exhaustively investigated, it is not yet fully understood when and how network reciprocity drives

cooperation transitions. In this work, we investigate the critical behavior of evolutionary social

dilemmas on structured populations by using the framework of master equations and Monte Carlo

simulations. The developed theory describes the existence of absorbing, quasi-absorbing, and mixed

strategy states and the transition nature, continuous or discontinuous, between the states as the

parameters of the system change. In particular, when the decision-making process is deterministic,

in the limit of zero eﬀective temperature of the Fermi function, we ﬁnd that the copying probabilities

are discontinuous functions of the system’s parameters and of the network degrees sequence. This

may induce abrupt changes in the ﬁnal state for any system size, in excellent agreement with

the Monte Carlo simulation results. Our analysis also reveals the existence of continuous and

discontinuous phase transitions for large systems as the temperature increases, which is explained

in the mean-ﬁeld approximation. Interestingly, for some game parameters, we ﬁnd optimal ”social

temperatures” maximizing/minimizing the cooperation frequency/density.

I. INTRODUCTION

Cooperation and defection are both ubiquitous behav-

iors in natural societies, including indeed those of hu-

mans. While defectors usually receive the highest bene-

ﬁts when acting selﬁshly, cooperators help others in an al-

truistic way at their own cost and, based on the “survival

of the ﬁttest” principle, defection should prevail against

cooperation. Yet Nature provides us with numerous ex-

amples where cooperative interactions among agents (be

either humans, animals, microorganisms, or genes) are at

the origin of more complex and functional systems [1–4].

Understanding the mechanisms driving the evolution

of cooperation within a population is at the core of the

Evolutionary Game Theory [5, 6]. Under this mathemat-

ical framework, social dilemmas are modeled as games

among agents whose strategies are allowed to spread

within the population according to their payoﬀs through

a replicator dynamics [7]. One of the mechanisms known

to favor cooperation is the reciprocity induced by the spa-

tial distribution of the players as shown by Nowak and

May [8]. When interactions are no longer well-mixed and

players are distributed in a spatial/topological structure,

cooperators can cluster together and might survive sur-

rounded by defectors, changing the mean-ﬁeld equilib-

rium panorama of many games [9–12].

Since that seminal work by Nowak and May [8] and

with the rapid development of the complex networks ﬁeld

in the last few decades [13–15], a lot of research has been

focused on the role of the underlying network topology in

the emergence of cooperation, including aspects like net-

work heterogeneity both theoretically [16, 17] and exper-

imentally [18–21], the presence of a layered structure de-

∗nagi.khalil@urjc.es

scribing the diﬀerent types of social relationships [22, 23]

and degree correlations among layers [24], or game re-

ﬁnements by introducing topology dependent payoﬀs [25]

and the inﬂuence of an update rule and connectivity on

the outcome dynamics of structured populations [26, 27].

From a Statistical Physics perspective [28, 29], several

attempts have been made to provide rules that predict

transitions to collective states of cooperation at critical

points, involving the structure connectivity and the game

parameters. For example, Ohtsuki et al. [30] using mean-

ﬁeld and pair approximations derived a simple condition

stating that the ratio of beneﬁt to cost of the altruistic

act has to exceed the mean degree to favor cooperation.

Konno [31], however, suggests that what really matters

is the mean degree of the nearest neighbors. Recently,

Zhuk et al.[32] showed that the unique sequence of de-

grees in a network can be used to predict for which game

parameters major shifts in the level of cooperation can

be expected; this includes phase transitions from absorb-

ing to mixed strategy phases, characterized by agents

switching intermittently between cooperation and defec-

tion. Using ﬁnite-size scaling, Menon et al. [33] investi-

gated the diﬀerent phase transitions between those col-

lective states and found critical exponents dependent on

the connection topology. Phase transitions in evolution-

ary cooperation induced by lattice reciprocity have been

also investigated using the standard Statistical Mechan-

ics of macroscopic systems, showing that the onset of the

phase transition cannot be captured by a purely mean-

ﬁeld approach [34, 35].

Indeed, due to the intrinsic complexity of games on

graphs, their analytical treatment is a challenging task

[36–39]. Here we present a general analytical approach

based on the master and the Fokker-Planck equations

derived for a network of pairwise engaged agents whose

reproductive success, in terms of the replication rate of

their strategy, depends on the payoﬀ obtained during the

arXiv:2210.01710v1 [nlin.AO] 4 Oct 2022

interaction. In this work, the resulting payoﬀ depends on

the actual agents’ strategies through a general matrix of

payoﬀs to account for the full space of two-player social

dilemmas with two strategies, cooperation and defection.

We generalize the results obtained in [32], by examin-

ing in detail the steady and metastable states and the

nature (continuous or discontinuous) of the phase tran-

sitions between absorbing, quasi-absorbing, and mixed

strategy states using the master and Fokker-Planck equa-

tions, for any system size and any eﬀective temperature

describing how often a player makes irrational choices.

The work is organized as follows. In Section II we de-

ﬁne the model (game dynamics, updating rule, and inter-

action network) and introduce the notation used through-

out this work. In Section III we study the most gen-

eral master equation describing the state of the system

and identify some steady-state solutions and discontinu-

ity points depending on the parameters of the system.

The mean-ﬁeld case is thoroughly investigated in Section

IV by means of the corresponding Fokker-Planck equa-

tion, which we solve analytically by artiﬁcially removing

the singularities at the pure absorbing states of the sys-

tem. Monte-Carlo numerical simulations are provided

in Section V to corroborate the analytical predictions of

mean ﬁeld, both for all-to-all interactions and for more

complex interaction networks. Finally, we summarize our

results in the Conclusion section.

II. MODEL DEFINITION

We consider a population of Nagents playing a 2 ×2

game, where each agent can adopt a strategy of coop-

eration (C) or defection (D), that can be changed de-

pending on her performance, her neighbors’ performance,

and some degree of randomness. The population connec-

tivity is structured in a connected and undirected net-

work represented by the adjacency matrix A, such that

Aµ,ν =Aν,µ = 1 if nodes µand νare neighbors, while

Aµ,ν =Aν,µ = 0 otherwise. We denote by Σ the set of all

nodes and by Vσ={ν∈Σ| Aσ,ν = 1}the set of neigh-

bors of a given node σ. The number of elements of Vσis

the degree of σ,kσ=PνAσ,ν . Throughout this work, in

addition to the complete graph (CG) describing all-to-all

interactions, we will consider diﬀerent graph-structured

populations ranging from random regular graphs (RR),

Erd¨os–R´enyi random graphs (ER) [40], to scale-free net-

works (SF) using the Barab´asi-Albert model [41].

As any node σ∈Σ is always occupied by an agent, for

our discussion it is useful to use the Boolean variables cσ

and dσ, indicating if σholds a cooperator or a defector,

respectively. Then, it is readily seen that cσ, dσ∈ {0,1},

cσ+dσ= 1, and cσ·dσ= 0. As a consequence, in order

to specify the state Sof the system at a given time t, we

only need the set S={cσ|σ∈Σ}.

The dynamics, including Monte Carlos simulations,

unfolds in several steps:

(i) First, the network Aand an initial state S0are

selected.

(ii) All agents play the game with their neighbors. The

resulting payoﬀ of a dyadic interaction is given by

the payoﬀ matrix:

C D

CR S

DT P

.(1)

The values R,S,T, and Pclassically represent

the reward for mutual cooperation (R), the sucker’s

payoﬀ (S), the temptation to defect (T), and the

punishment for mutual defection (P). This way,

the payoﬀ gσof an agent at node σdepends on

the parameters of the matrix M, her state, and the

state of her neighbors as

gσ=cσX

ν∈Vσ

(Rcν+Sdν) + dσX

ν∈Vσ

(T cν+P dν).(2)

(iii) After the play, an agent at σand one of her neigh-

bors at νare selected at random. The former copies

the strategy of the latter with a probability

pσ,ν =1

1 + exp −∆gσ,ν

θ,(3)

where θis a non-negative parameter playing the

role of an eﬀective temperature (tuning the proba-

bility of an irrational choice) and

∆gσ,ν =gν−gσ

Tmax(kσ, kν)(4)

is a normalized payoﬀ diﬀerence.

(iv) The time tand the state Sof the system are up-

dated: t→t+t0,S → S0, where t0is an arbitrary

unit of time.

(v) The steps (ii) to (iv) are repeated a desired number

of times.

For zero eﬀective temperature (θ= 0) the copying

mechanism is (almost) deterministic: if ∆gσ,ν >0 then

node σalways copies the strategy of node ν(pσ,ν = 1),

while nothing changes when ∆gσ,ν <0 (pσ,ν = 0).

In the tie case ∆gσ,ν = 0, the copying probability is

pσ,ν =1

2. In this case (θ= 0) and for a very large

and well-mixed population, four diﬀerent categories of

games have been extensively studied as a function of the

parameters R,S,T, and P: Harmony, Snowdrift, Stag

Hunt, and Prisoner’s Dilemma. The Harmony game rep-

resents a category of games satisfying R > S > P and

R > T > P where full cooperation is the only possible

stable outcome in a population [42], while in the Pris-

oner’s Dilemma, T > R > P > S, the evolutionary sta-

ble strategy is a whole population of defectors [43]. The

other two categories represent respectively the classes of

anti-coordination and coordination games. In the Snow-

drift game [44], T > R > S > P , full defection and

cooperation are unstable and the best response is al-

ways doing the opposite of your opponent, giving rise

to a mixed strategy state. In the Stag Hunt game [45],

R > T > P > S, players either always cooperate or

always defect.

In the opposite temperature limit, when θ→ ∞ the

model reduces to the well-known Voter Model [46–48].

In this case, the dynamics is independent of the payoﬀs

(pσ,ν →1

2), and the nodes blindly copy the state of a

randomly chosen neighbor. In our study, we will consider

the eﬀects of small and intermediate values of θon the

ﬁnal state of the system.

III. THEORETICAL DESCRIPTION

A. Master equation

Due to the stochastic character of the dynamics and

the initial state S0, we consider the probability P(S, t)

of ﬁnding the system at state Sat a given time t. The

dynamics is Markovian and completely determined by

the probability rates of the elementary transitions:

•a change of a defector at a node σto a cooperator,

cσ= 0 −→

π+

cσ= 1,(5)

with a rate π+

σ,

•and a change of a cooperator to a defector,

cσ= 1 −→

π−

cσ= 0,(6)

with a rate π−

σ.

Note that the dynamics can be seen as a birth-death

process, hence suitable for being analyzed as in previous

works [12, 49]. Taking into account the steps (ii) and (iii)

of the evolution given in the previous section, the rates

can be written as

π+

σ=dσ

Nkσt0X

ν∈Vσ

cνpσ,ν ,(7)

π−

σ=cσ

Nkσt0X

ν∈Vσ

dνpσ,ν ,(8)

where pσ,ν is provided by Eq. (3).

The probability P(S, t) obeys the following master

equation

∂tP(S, t) = X

σ∈Σ(E+

σ−1)π−

σP(S, t)

+(E−

σ−1)π+

σP(S, t),(9)

where ∂tP(S, t)≡1

t0[P(S, t +t0)− P(S, t)] is the dis-

crete time derivative and the new operator E+

σ(E−

σ) acts

on any function of the state of the system by increasing

(decreasing) the number of cooperators at node σby one.

The master equation can not be solved analytically in

general. Nevertheless, some solutions can be identiﬁed

and analyzed upon changing the parameters of the sys-

tem. In particular, we will be concerned with the values

of the parameters for which there are major changes in

the mean fraction of cooperators hρi, deﬁned in terms of

the probability function P(S, t) as

hρi=1

σ∈Σ

cσP(S, t),(10)

where the sum PSis over all states.

B. Steady, absorbing, and quasi-absorbing states

We assume that, for any initial state, the system al-

ways reaches a steady or metastable state. The steady

states are characterized by a probability function P(S)

verifying

σ∈Σ

(E+

σ−1)π−

σP(S)+(E−

σ−1)π+

σP(S) = 0 (11)

for all states S. The system (11) has an inﬁnite number of

solutions, including the absorbing states for which π−

σ=

π+

σ= 0 for all nodes. It is readily seen that the absorbing

states are, for any value of θ, the consensus states: {cσ=

1}(full cooperation) and {cσ= 0}(full defection).

In the case of positive eﬀective temperature θ > 0,

the probability of copying a neighbor’s strategy is always

positive pσ,ν >0. Hence, for ﬁnite system size N<∞,

there is a nonzero probability for the system to reach and

get trapped into any of the two consensus states starting

from any initial state. As a consequence, the only steady-

state solutions to the master equation when θ > 0 and

N<∞are of the form P(S) = pcPc(S) + pdPd(S),

i.e. linear combinations of the probability functions Pc

(full cooperation) and Pd(full defection) with pcand pd

representing the probabilities of reaching the cooperation

and defection consensus states, respectively.

However, the case of zero temperature (θ= 0) requires

a more careful analysis. Apart from the absorbing states,

we ﬁnd that, depending on the parameters and the net-

work structure, the system can also get trapped to either

a set of what can be called quasi-absorbing states or a

set of mixed strategy states. We show two raster plot ex-

amples in Fig. 1. In the quasi-absorbing states, which

appear mainly but not only when S < 0 and T < 1, see

Fig. 1(a), connected domains of nodes with frozen strat-

egy are separated by a frontier of oscillating nodes. In ad-

dition, the system can also get trapped to a set of mixed

strategy states, in which all nodes change their strategy

along the evolution. These latter states appear for S > 0

and T > 1, as shown in Fig. 1(b). From a dynamical

viewpoint, both the quasi-absorbing and mixed strategy

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DeterministicandstochasticcooperationtransitionsinevolutionarygamesonnetworksNagiKhalil,1,I.Leyva,1,2J.A.Almendral,1,2andI.Sendi~na-Nadal1,21ComplexSystemsGroup&GISC,UniversidadReyJuanCarlos,Mostoles,28933Madrid,Spain2CenterforBiomedicalTechnology,UniversidadPolitecnicadeMadrid,PozuelodeAlarcon,...

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Deterministic and stochastic cooperation transitions in evolutionary games on networks Nagi Khalil1I. Leyva1 2J.A. Almendral1 2and I. Sendi na-Nadal1 2

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