We study the evolution of cooperation in 2×2 social dilemma games in which players are located on a two-dimensional square lattice. During the evolution, each player modifies her strategy by means of myopic update dynamic to maximize her payoff while composing neighborhoods of different sizes, which are characterized by the corresponding radius, r. An investigation of the sublattice-ordered spatial structure for different values of r reveals that some patterns formed by cooperators and defectors can help the former to survive, even under untoward conditions. In contrast to individuals who resist the invasion of defectors by forming clusters due to network reciprocity, innovators spontaneously organize a socially divisive structure that provides strong support for the evolution of cooperation and advances better social systems.

1.
R.
Axelrod
and
W. D.
Hamilton
, “
The evolution of cooperation
,”
Science
211
,
1390
1396
(
1981
).
2.
E.
Pennisi
, “
How did cooperative behavior evolve?
,”
Science
309
,
93
(
2005
).
3.
J. M.
Smith
,
Evolution and the Theory of Games
(
Cambridge University Press
,
1982
).
4.
J. W.
Weibull
,
Evolutionary Game Theory
(
MIT Press
,
1997
).
5.
J.
Hofbauer
,
K.
Sigmund
et al.,
Evolutionary Games and Population Dynamics
(
Cambridge University Press
,
1998
).
6.
G.
Szabó
and
C.
Tőke
, “
Evolutionary prisoner’s dilemma game on a square lattice
,”
Phys. Rev. E
58
,
69
(
1998
).
7.
G.
Abramson
and
M.
Kuperman
, “
Social games in a social network
,”
Phys. Rev. E
63
,
030901
(
2001
).
8.
F. C.
Santos
and
J. M.
Pacheco
, “
Scale-free networks provide a unifying framework for the emergence of cooperation
,”
Phys. Rev. Lett.
95
,
098104
(
2005
).
9.
C.
Chu
,
C.
Mu
,
J.
Liu
,
C.
Liu
,
S.
Boccaletti
,
L.
Shi
, and
Z.
Wang
, “
Aspiration-based coevolution of node weights promotes cooperation in the spatial prisoner’s dilemma game
,”
New J. Phys.
21
,
063024
(
2019
).
10.
C.
Shen
,
C.
Chu
,
L.
Shi
,
M.
Perc
, and
Z.
Wang
, “
Aspiration-based coevolution of link weight promotes cooperation in the spatial prisoner’s dilemma game
,”
R. Soc. Open Sci.
5
,
180199
(
2018
).
11.
C.
Chu
,
Y.
Zhai
,
C.
Mu
,
D.
Hu
,
T.
Li
, and
L.
Shi
, “
Reputation-based popularity promotes cooperation in the spatial prisoner’s dilemma game
,”
Appl. Math. Comput.
362
,
124493
(
2019
).
12.
C.
Hilbe
,
Š.
Šimsa
,
K.
Chatterjee
, and
M. A.
Nowak
, “
Evolution of cooperation in stochastic games
,”
Nature
559
,
246
249
(
2018
).
13.
C.
Zhu
,
S.
Sun
,
L.
Wang
,
S.
Ding
,
J.
Wang
, and
C.
Xia
, “
Promotion of cooperation due to diversity of players in the spatial public goods game with increasing neighborhood size
,”
Physica A
406
,
145
154
(
2014
).
14.
K.
Donahue
,
O. P.
Hauser
,
M. A.
Nowak
, and
C.
Hilbe
, “
Evolving cooperation in multichannel games
,”
Nat. Commun.
11
,
3885
(
2020
).
15.
C.
Xia
,
Q.
Miao
, and
J.
Zhang
, “
Impact of neighborhood separation on the spatial reciprocity in the prisoner’s dilemma game
,”
Chaos, Solitons Fractals
51
,
22
30
(
2013
).
16.
M.
Diakonova
,
V.
Nicosia
,
V.
Latora
, and
M.
San Miguel
, “
Irreducibility of multilayer network dynamics: The case of the voter model
,”
New J. Phys.
18
,
023010
(
2016
).
17.
K.-K.
Kleineberg
, “
Metric clusters in evolutionary games on scale-free networks
,”
Nat. Commun.
8
,
1
(
2017
).
18.
J.
Wang
,
C.
Xia
,
Y.
Wang
,
S.
Ding
, and
J.
Sun
, “
Spatial prisoner’s dilemma games with increasing size of the interaction neighborhood on regular lattices
,”
Chin. Sci. Bull.
57
,
724
728
(
2012
).
19.
M. A.
Nowak
and
R. M.
May
, “
Evolutionary games and spatial chaos
,”
Nature
359
,
826
829
(
1992
).
20.
D.
Jia
,
T.
Li
,
Y.
Zhao
,
X.
Zhang
, and
Z.
Wang
, “
Empty nodes affect conditional cooperation under reinforcement learning
,”
Appl. Math. Comput.
413
,
126658
(
2022
).
21.
Z.
Song
,
H.
Guo
,
D.
Jia
,
M.
Perc
,
X.
Li
, and
Z.
Wang
, “
Third party interventions mitigate conflicts on interdependent networks
,”
Appl. Math. Comput.
403
,
126178
(
2021
).
22.
D.
Jia
,
X.
Wang
,
Z.
Song
,
I.
Romić
,
X.
Li
,
M.
Jusup
, and
Z.
Wang
, “
Evolutionary dynamics drives role specialization in a community of players
,”
J. R. Soc. Interface
17
,
20200174
(
2020
).
23.
D. J.
Watts
and
S. H.
Strogatz
, “
Collective dynamics of ‘small-world’ networks
,”
Nature
393
,
440
442
(
1998
).
24.
B. J.
Kim
,
A.
Trusina
,
P.
Holme
,
P.
Minnhagen
,
J. S.
Chung
, and
M.
Choi
, “
Dynamic instabilities induced by asymmetric influence: Prisoners’ dilemma game in small-world networks
,”
Phys. Rev. E
66
,
021907
(
2002
).
25.
C. P.
Warren
,
L. M.
Sander
, and
I. M.
Sokolov
, “
Geography in a scale-free network model
,”
Phys. Rev. E
66
,
056105
(
2002
).
26.
S.
Fortunato
,
A.
Flammini
, and
F.
Menczer
, “
Scale-free network growth by ranking
,”
Phys. Rev. Lett.
96
,
218701
(
2006
).
27.
K.
Huang
,
Z.
Wang
, and
M.
Jusup
, “
Incorporating latent constraints to enhance inference of network structure
,”
IEEE Trans. Netw. Sci. Eng.
7
,
466
475
(
2018
).
28.
K.
Huang
,
S.
Li
,
P.
Dai
,
Z.
Wang
, and
Z.
Yu
, “
SDARE: A stacked denoising autoencoder method for game dynamics network structure reconstruction
,”
Neural Netw.
126
,
143
152
(
2020
).
29.
K.
Huang
,
Z.
Xiang
,
W.
Deng
,
C.
Yang
, and
Z.
Wang
, “
False data injection attacks detection in smart grid: A structural sparse matrix separation method
,”
IEEE Trans. Netw. Sci. Eng.
8
,
2545
2558
(
2021
).
30.
W. D.
Hamilton
, “
The genetical evolution of social behaviour. II
,”
J. Theor. Biol.
7
,
17
52
(
1964
).
31.
R. L.
Trivers
, “
The evolution of reciprocal altruism
,”
Q. Rev. Biol.
46
,
35
57
(
1971
).
32.
M. A.
Nowak
, “
Five rules for the evolution of cooperation
,”
Science
314
,
1560
1563
(
2006
).
33.
D. S.
Wilson
, “
Structured demes and the evolution of group-advantageous traits
,”
Am. Nat.
111
,
157
185
(
1977
).
34.
J.
Li
,
Y.
Liu
,
Z.
Wang
, and
H.
Xia
, “
Egoistic punishment outcompetes altruistic punishment in the spatial public goods game
,”
Sci. Rep.
11
,
1
(
2021
).
35.
S.
Podder
,
S.
Righi
, and
F.
Pancotto
, “
Reputation and punishment sustain cooperation in the optional public goods game
,”
Philos. Trans. R. Soc., B
376
,
20200293
(
2021
).
36.
A.
Szolnoki
and
M.
Perc
, “
Coevolution of teaching activity promotes cooperation
,”
New J. Phys.
10
,
043036
(
2008
).
37.
Q.
Jian
,
X.
Li
,
J.
Wang
, and
C.
Xia
, “
Impact of reputation assortment on tag-mediated altruistic behaviors in the spatial lattice
,”
Appl. Math. Comput.
396
,
125928
(
2021
).
38.
X.
Li
,
S.
Sun
, and
C.
Xia
, “
Reputation-based adaptive adjustment of link weight among individuals promotes the cooperation in spatial social dilemmas
,”
Appl. Math. Comput.
361
,
810
820
(
2019
).
39.
Q.
Su
,
A.
Li
, and
L.
Wang
, “
Evolutionary dynamics under interactive diversity
,”
New J. Phys.
19
,
103023
(
2017
).
40.
F. C.
Santos
,
M. D.
Santos
, and
J. M.
Pacheco
, “
Social diversity promotes the emergence of cooperation in public goods games
,”
Nature
454
,
213
216
(
2008
).
41.
X.
Chen
and
L.
Wang
, “
Promotion of cooperation induced by appropriate payoff aspirations in a small-world networked game
,”
Phys. Rev. E
77
,
017103
(
2008
).
42.
M.
Perc
and
Z.
Wang
, “
Heterogeneous aspirations promote cooperation in the prisoner’s dilemma game
,”
PLoS One
5
,
e15117
(
2010
).
43.
J.
Vukov
,
G.
Szabó
, and
A.
Szolnoki
, “
Cooperation in the noisy case: Prisoner’s dilemma game on two types of regular random graphs
,”
Phys. Rev. E
73
,
067103
(
2006
).
44.
U.
Alvarez-Rodriguez
,
F.
Battiston
,
G. F.
de Arruda
,
Y.
Moreno
,
M.
Perc
, and
V.
Latora
, “
Evolutionary dynamics of higher-order interactions in social networks
,”
Nat. Hum. Behav.
5
,
586
595
(
2021
).
45.
M.
Chen
,
L.
Wang
,
S.
Sun
,
J.
Wang
, and
C.
Xia
, “
Evolution of cooperation in the spatial public goods game with adaptive reputation assortment
,”
Phys. Lett. A
380
,
40
47
(
2016
).
46.
Q.
Su
,
A.
Li
,
L.
Wang
, and
H.
Eugene Stanley
, “
Spatial reciprocity in the evolution of cooperation
,”
Proc. R. Soc. B
286
,
20190041
(
2019
).
47.
Q.
Su
,
A.
McAvoy
,
L.
Wang
, and
M. A.
Nowak
, “
Evolutionary dynamics with game transitions
,”
Proc. Natl. Acad. Sci. U.S.A.
116
,
25398
25404
(
2019
).
48.
A.
Matsui
, “
Best response dynamics and socially stable strategies
,”
J. Econ. Theor.
57
,
343
362
(
1992
).
49.
M.
Sysi-Aho
,
J.
Saramäki
,
J.
Kertész
, and
K.
Kaski
, “
Spatial snowdrift game with myopic agents
,”
Eur. Phys. J. B
44
,
129
135
(
2005
).
50.
X.
Chen
and
L.
Wang
, “
Cooperation enhanced by moderate tolerance ranges in myopically selective interactions
,”
Phys. Rev. E
80
,
046109
(
2009
).
51.
C. P.
Roca
,
J. A.
Cuesta
, and
A.
Sánchez
, “
Promotion of cooperation on networks? The myopic best response case
,”
Eur. Phys. J. B
71
,
587
595
(
2009
).
52.
G.
Szabo
,
A.
Szolnoki
,
M.
Varga
, and
L.
Hanusovszky
, “
Ordering in spatial evolutionary games for pairwise collective strategy updates
,”
Phys. Rev. E
82
,
026110
(
2010
).
53.
G.
Szabo
and
A.
Szolnoki
, “
Selfishness, fraternity, and other-regarding preference in spatial evolutionary games
,”
J. Theor. Biol.
299
,
81
87
(
2012
).
54.
G.
Szabo
,
A.
Szolnoki
, and
L.
Czako
, “
Coexistence of fraternity and egoism for spatial social dilemmas
,”
J. Theor. Biol.
317
,
126
132
(
2013
).
55.
A.
Szolnoki
and
M.
Perc
, “
Evolution of extortion in structured populations
,”
Phys. Rev. E
89
,
022804
(
2014
).
56.
G.
Szabó
and
A.
Szolnoki
, “
Congestion phenomena caused by matching pennies in evolutionary games
,”
Phys. Rev. E
91
,
032110
(
2015
).
57.
M. A.
Amaral
and
M. A.
Javarone
, “
Heterogeneous update mechanisms in evolutionary games: Mixing innovative and imitative dynamics
,”
Phys. Rev. E
97
,
042305
(
2018
).
58.
J.
Vukov
,
F. C.
Santos
, and
J. M.
Pacheco
, “
Cognitive strategies take advantage of the cooperative potential of heterogeneous networks
,”
New J. Phys.
14
,
063031
(
2012
).
59.
C.
Hauert
and
M.
Doebeli
, “
Spatial structure often inhibits the evolution of cooperation in the snowdrift game
,”
Nature
428
,
643
646
(
2004
).
60.
M. A.
Amaral
,
M.
Perc
,
L.
Wardil
,
A.
Szolnoki
,
E. J.
da Silva Júnior
, and
J. K.
da Silva
, “
Role-separating ordering in social dilemmas controlled by topological frustration
,”
Phys. Rev. E
95
,
032307
(
2017
).
61.
C. P.
Roca
,
J. A.
Cuesta
, and
A.
Sánchez
, “
Effect of spatial structure on the evolution of cooperation
,”
Phys. Rev. E
80
,
046106
(
2009
).
62.
J.
Tanimoto
, “
Difference of reciprocity effect in two coevolutionary models of presumed two-player and multiplayer games
,”
Phys. Rev. E
87
,
062136
(
2013
).
63.
J.
Tanimoto
, “
Promotion of cooperation by payoff noise in a 2×2 game
,”
Phys. Rev. E
76
,
041130
(
2007
).
64.
G.
Szabó
and
G.
Bunth
, “
Social dilemmas in multistrategy evolutionary potential games
,”
Phys. Rev. E
97
,
012305
(
2018
).
You do not currently have access to this content.