Existing studies have revealed a paradoxical phenomenon in public goods games, wherein destructive agents, harming both cooperators and defectors, can unexpectedly bolster cooperation. Building upon this intriguing premise, our paper introduces a novel concept: constructive agents, which confer additional benefits to both cooperators and defectors. We investigate the impact of these agents on cooperation dynamics within the framework of public goods games. Employing replicator dynamics, we find that unlike destructive agents, the mere presence of constructive agents does not significantly alter the defective equilibrium. However, when the benefits from constructive agents are outweighed by the damage inflicted by destructive agents, the addition of constructive agents does not affect the ability of destructive agents to sustain cooperation. In this scenario, cooperators can be maintained through a cyclic dominance between cooperators, defectors, and destructive agents, with constructive agents adding complexity but not fundamentally changing the equilibrium. Conversely, if the benefits from constructive agents surpass the harm caused by destructive agents, the presence of constructive agents nullifies the ability of destructive agents to foster cooperation. Our results highlight the nuanced role of constructive agents in cooperation dynamics, emphasizing the necessity of carefully assessing incentive balances when encouraging cooperation.

1.
R.
Axelrod
and
W. D.
Hamilton
, “
The evolution of cooperation
,”
Science
211
,
1390
1396
(
1981
).
2.
R. L.
Riolo
,
M. D.
Cohen
, and
R.
Axelrod
, “
Evolution of cooperation without reciprocity
,”
Nature
414
,
441
443
(
2001
).
3.
M.
Archetti
and
I.
Scheuring
, “
Game theory of public goods in one-shot social dilemmas without assortment
,”
J. Theor. Biol.
299
,
9
20
(
2012
).
4.
J. W.
Weibull
,
Evolutionary Game Theory
(
MIT Press
,
Cambridge
,
1997
).
5.
M. A.
Nowak
and
R.
Highfield
,
SuperCooperators: Altruism, Evolution, and Why We Need Each Other to Succeed
(
Free Press
,
New York
,
2011
).
6.
D. G.
Rand
and
M. A.
Nowak
, “
Human cooperation
,”
Trends Cogn. Sci.
17
,
413
425
(
2013
).
7.
M. A.
Nowak
, “
Five rules for the evolution of cooperation
,”
Science
314
,
1560
1563
(
2006
).
8.
L.
Schmid
,
K.
Chatterjee
,
C.
Hilbe
, and
M. A.
Nowak
, “
A unified framework of direct and indirect reciprocity
,”
Nat. Hum. Behav.
5
,
1292
1302
(
2021
).
9.
O.
Leimar
and
P.
Hammerstein
, “
Evolution of cooperation through indirect reciprocity
,”
Proc. R. Soc. Lond. Ser. B
268
,
745
753
(
2001
).
10.
C.
Xia
,
J.
Wang
,
M.
Perc
, and
Z.
Wang
, “
Reputation and reciprocity
,”
Phys. Life Rev.
46
,
8
45
(
2023
).
11.
J.
Wang
and
C.
Xia
, “
Reputation evaluation and its impact on the human cooperation—A recent survey
,”
Europhys. Lett.
141
,
21001
(
2023
).
12.
H.
Guo
,
Z.
Wang
,
J.
Xing
,
P.
Tao
, and
Y.
Shi
, “Cooperation and coordination in heterogeneous populations with interaction diversity,” in Proceedings of the 23rd International Conference on Autonomous Agents and Multiagent Systems (AAMAS, 2024, Auckland, New Zealand, May 6 – 10, 2024, IFAAMAS), pp. 752–760.
13.
S.
Ren
,
Z.
Cui
,
R.
Song
,
Z.
Wang
, and
S.
Hu
, “Emergence of social norms in large language model-based agent societies,” arXiv:2403.08251 (2024).
14.
Y.
Xu
,
J.
Wang
,
C.
Xia
, and
Z.
Wang
, “
Higher-order temporal interactions promote the cooperation in the multiplayer snowdrift game
,”
Sci. China Inf. Sci.
66
,
222208
(
2023
).
15.
Y.
Xu
,
J.
Wang
,
J.
Chen
,
D.
Zhao
,
M.
Özer
,
C.
Xia
, and
M.
Perc
, “
Reinforcement learning and collective cooperation on higher-order networks
,”
Knowl.-Based Syst.
301
,
112326
(
2024
).
16.
E.
Fehr
and
S.
Gächter
, “
Cooperation and punishment in public goods experiments
,”
Am. Econ. Rev.
90
,
980
994
(
2000
).
17.
D.
Balliet
,
L. B.
Mulder
, and
P. A.
Van Lange
, “
Reward, punishment, and cooperation: A meta-analysis
,”
Psychol. Bull.
137
,
594
(
2011
).
18.
A.
Szolnoki
and
M.
Perc
, “
Reward and cooperation in the spatial public goods game
,”
Europhys. Lett.
92
,
38003
(
2010
).
19.
X.
Li
,
M.
Jusup
,
Z.
Wang
,
H.
Li
,
L.
Shi
,
B.
Podobnik
,
H. E.
Stanley
,
S.
Havlin
, and
S.
Boccaletti
, “
Punishment diminishes the benefits of network reciprocity in social dilemma experiments
,”
Proc. Natl. Acad. Sci. U.S.A.
115
,
30
35
(
2018
).
20.
A.
Szolnoki
and
M.
Perc
, “
Second-order free-riding on antisocial punishment restores the effectiveness of prosocial punishment
,”
Phys. Rev. X
7
,
041027
(
2017
).
21.
J.
Andreoni
,
W.
Harbaugh
, and
L.
Vesterlund
, “
The carrot or the stick: Rewards, punishments, and cooperation
,”
Am. Econ. Rev.
93
,
893
902
(
2003
).
22.
X.
Chen
,
T.
Sasaki
,
Å.
Brännström
, and
U.
Dieckmann
, “
First carrot, then stick: How the adaptive hybridization of incentives promotes cooperation
,”
J. R. Soc. Interface
12
,
20140935
(
2015
).
23.
C.
Hauert
,
S.
De Monte
,
J.
Hofbauer
, and
K.
Sigmund
, “
Volunteering as red queen mechanism for cooperation in public goods games
,”
Science
296
,
1129
1132
(
2002
).
24.
G.
Szabó
and
C.
Hauert
, “
Phase transitions and volunteering in spatial public goods games
,”
Phys. Rev. Lett.
89
,
118101
(
2002
).
25.
G.
Szabó
and
C.
Hauert
, “
Evolutionary prisoner’s dilemma games with voluntary participation
,”
Phys. Rev. E
66
,
062903
(
2002
).
26.
C.
Shen
,
M.
Jusup
,
L.
Shi
,
Z.
Wang
,
M.
Perc
, and
P.
Holme
, “
Exit rights open complex pathways to cooperation
,”
J. R. Soc. Interface
18
,
20200777
(
2021
).
27.
S.
Li
,
Z.
He
,
D.
Jia
,
C.
Shen
,
L.
Shi
, and
J.
Tanimoto
, “
Granting leaders priority exit options promotes and jeopardizes cooperation in social dilemmas
,”
Neurocomputing
583
,
127566
(
2024
).
28.
S.
Bogaert
,
C.
Boone
, and
C.
Declerck
, “
Social value orientation and cooperation in social dilemmas: A review and conceptual model
,”
Br. J. Soc. Psychol.
47
,
453
480
(
2008
).
29.
R. O.
Murphy
,
K. A.
Ackermann
, and
M. J. J.
Handgraaf
, “
Measuring social value orientation
,”
Judgm. Decis. Mak.
6
,
771
781
(
2011
).
30.
A.
Arenas
,
J.
Camacho
,
J. A.
Cuesta
, and
R. J.
Requejo
, “
The joker effect: Cooperation driven by destructive agents
,”
J. Theor. Biol.
279
,
113
119
(
2011
).
31.
R. J.
Requejo
,
J.
Camacho
,
J. A.
Cuesta
, and
A.
Arenas
, “
Stability and robustness analysis of cooperation cycles driven by destructive agents in finite populations
,”
Phys. Rev. E
86
,
026105
(
2012
).
32.
K.
Khatun
,
C.
Shen
,
L.
Shi
, and
J.
Tanimoto
, “Stability of pairwise social dilemma games: Destructive agents, constructive agents, and their joint effects,”
Phys. Rev. E
110
,
034307
(
2024
).
33.
A.
Szolnoki
and
X.
Chen
, “
Strategy dependent learning activity in cyclic dominant systems
,”
Chaos, Solitons Fractals
138
,
109935
(
2020
).
34.
H.-W.
Lee
,
C.
Cleveland
, and
A.
Szolnoki
, “
Mercenary punishment in structured populations
,”
Appl. Math. Comput.
417
,
126797
(
2022
).
35.
H.-W.
Lee
,
C.
Cleveland
, and
A.
Szolnoki
, “
Supporting punishment via taxation in a structured population
,”
Chaos, Solitons Fractals
178
,
114385
(
2024
).
36.
L.
Liu
,
Z.
Xiao
,
X.
Chen
, and
A.
Szolnoki
, “
Early exclusion leads to cyclical cooperation in repeated group interactions
,”
J. R. Soc. Interface
19
,
20210755
(
2022
).
37.
A.
Szolnoki
and
M.
Perc
, “
Competition of tolerant strategies in the spatial public goods game
,”
New J. Phys.
18
,
083021
(
2016
).
38.
H.
Gintis
,
S.
Bowles
,
R.
Boyd
, and
E.
Fehr
, “
Explaining altruistic behavior in humans
,”
Evol. Hum. Behav.
24
,
153
172
(
2003
).
39.
J.-K.
Choi
and
T.
Ahn
, “
Strategic reward and altruistic punishment support cooperation in a public goods game experiment
,”
J. Econ. Psychol.
35
,
17
30
(
2013
).
40.
A.
Gneezy
and
D. M. T.
Fessler
, “
Conflict, sticks and carrots: War increases prosocial punishments and rewards
,”
Proc. R. Soc. B
279
,
219
223
(
2012
).
41.
F. C.
Santos
,
F. L.
Pinheiro
,
T.
Lenaerts
, and
J. M.
Pacheco
, “
The role of diversity in the evolution of cooperation
,”
J. Theor. Biol.
299
,
88
96
(
2012
).
42.
M.
Perc
and
A.
Szolnoki
, “
Social diversity and promotion of cooperation in the spatial prisoner’s dilemma game
,”
Phys. Rev. E
77
,
011904
(
2008
).
43.
J.
Maynard Smith
,
Evolution and the Theory of Games
(
Cambridge University Press
,
Cambridge
,
1982
).
44.
M.
Archetti
, “
Definetti: A mathematica program to analyze the replicator dynamics of 3-strategy collective interactions
,”
SoftwareX
11
,
100415
(
2020
).
45.
H.-W.
Lee
,
C.
Cleveland
, and
A.
Szolnoki
, “
Suppressing defection by increasing temptation: The impact of smart cooperators on a social dilemma situation
,”
Appl. Math. Comput.
479
,
128864
(
2024
).
You do not currently have access to this content.