Defector-accelerated cooperativeness and punishment in public goods games with mutations.

Abstract

We study the evolution of cooperation in spatial public goods games with four competing strategies: cooperators, defectors, punishing cooperators, and punishing defectors. To explore the robustness of the cooperation-promoting effect of costly punishment, besides the usual strategy adoption dynamics we also apply strategy mutations. As expected, frequent mutations create kind of well-mixed conditions, which support the spreading of defectors. However, when the mutation rate is small, the final stationary state does not significantly differ from the state of the mutation-free model, independently of the values of the punishment fine and cost. Nevertheless, the mutation rate affects the relaxation dynamics. Rare mutations can largely accelerate the spreading of costly punishment. This is due to the fact that the presence of defectors breaks the balance of power between both cooperative strategies, which leads to a different kind of dynamics.

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Defector-accelerated cooperativeness and punishment in public goods games with mutations
Dirk Helbing,1,2,3Attila Szolnoki,4Matjaˇ z Perc,5Gy¨ orgy Szab´ o4
1ETH Zurich, CLU E1, Clausiusstr. 50, 8092 Zurich, Switzerland
2Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
3Collegium Budapest - Institute for Advanced Study, Szenth´ aroms´ ag u. 2, 1014 Budapest, Hungary
4Research Institute for Technical Physics and Materials Science, P.O. Box 49, H-1525 Budapest, Hungary
5Faculty of Natural Sciences and Mathematics, University of Maribor, Koroˇ ska cesta 160, SI-2000 Maribor, Slovenia
We study the evolution of cooperation in spatial public goods games with four competing strategies: cooper-
ators, defectors, punishing cooperators, and punishing defectors. To explore the robustness of the cooperation-
promoting effect of costly punishment, besides the usual strategy adoption dynamics we also apply strategy
mutations. As expected, frequent mutations create kind of well-mixed conditions, which support the spreading
of defectors. However, when the mutation rate is small, the final stationary state does not significantly differ
from the state of the mutation-free model, independently of the values of the punishment fine and cost. Never-
theless, the mutation rate affects the relaxation dynamics. Rare mutations can largely accelerate the spreading
of costly punishment. This is due to the fact that the presence of defectors breaks the balance of power between
both cooperative strategies, which leads to a different kind of dynamics.
PACS numbers: 02.50.Le, 87.10.Hk, 87.23.Ge
Evolutionary game theory formalizes the dynamics of pop-
ulations of interacting individuals, considering the success
(payoff)oftheirinteractions. Whilethis approachhasapplica-
tions in biology [1–5], economics [6], and the social sciences
[7], it has attracted a great deal of interest among physicists
as well due to the relevance of methods from non-linear dy-
namics [8], statistical physics [9], cellular automata [10], and
many-particle simulations [11].
One of the grand scientific challenges in this field con-
cerns the question, how the outcome of interactions in so-
cial dilemma situations can be improved. In social dilemmas
such as the public goods game, the collective well-being de-
pends on the cooperationof individuals, which howeveris un-
likely, as selfish behaviorcan generate higherpersonalprofits.
It has been proposed that reputation and costly punishment
can fight free-riding (defection) and promote cooperation in
public goods situations [12]. It is puzzling, however, why
people would make punishment efforts, as this reduces their
payoffs comparedto others who do not punish (“second-order
free-riders”). In fact, punishing strategies disappear in public
goods games, when the interactions between individuals are
well mixed, creating again a tragedy of the commons. How-
ever,whenindividualshavespatialneighborhoodinteractions,
free-ridersmay be eliminated(both, conventionaland second-
order ones) [13, 14]. This is due to the fact that the different
cooperative strategies form clusters and segregate from each
other. Inthis way, punishingcooperatorsavoid to be exploited
by second-order free-riders (non-punishing cooperators) and
can efficiently fight against defectors. Adding strategy muta-
tions, however, endangers homogeneous clusters of individu-
als pursuingthe same strategy,as they support the intrusionof
competing strategies (“enemies”). For example, if defectors
manage to enter a cooperative cluster, it can quickly erode.
As a consequence, one would expect that mutations under-
mine the spreading of punishing strategies, thereby restoring
the “second-order free-rider problem” and the “tragedy of the
commons”.
Therefore,thispaperinvestigatestheimpactofstrategymu-
tations on the evolution of cooperation in the spatial public
goods game with punishing strategies. As interaction graph,
we assume a square lattice.
meansoftwoadditionalstrategiesbesidescooperators(C)and
defectors (D). These two strategies are punishing cooperators
(PC) and punishing defectors (PD), both of which impose a
fine on defectors at a personal cost. The public goods game
is iteratively played on a fully occupied square lattice of size
L × L with periodic boundary conditions, where each player
x holds a strategy sx ∈ {C, D, PC, PD}. Initially, the four
strategies are equally and uniformly distributed over the L2
lattice sites. Each player x is a member of G = 5 groups
consisting of 5 individuals each. Each of these groups corre-
sponds to a Neumann neighborhoodof the focal individual or
one of the direct neighbors.
In each iteration, an individualx plays a public goods game
in all groups it belongs to. Cooperative individuals (playing C
or PC) make a contribution of 1, while non-cooperative indi-
viduals (D or PD) contribute nothing. Afterwards, the sum of
all contributionsin each groupis multiplied with the “synergy
factor” r, and the resulting amount is equally shared between
all of its members, irrespective of their contribution.
Let P∗
ceives in all of the G groups it participates in. Then P∗
responds to the overall payoff of individual x in the absence
of punishment. This payoff is modified by punishment fines
and punishment costs as follows: If sx = D or PD, player
x is punished with a fine f in such a way that the remaining
payoff is P′
x= P∗
the groups containing player x. πpis given by the number of
punishers (PC or PD) in each group (not considering player
x), divided by G − 1. Moreover, if sx = PC or PD, player
x invests a punishment cost c such that the finally remaining
payoff is Px= P′
all the groups containing player x. πdis given by the number
of defecting individuals around player x in each group (D or
PD), divided by G − 1. The division by G − 1 serves to scale
for the group size G.
Punishment is introduced by
xdenote the sum of the shares that individual x re-
xcor-
x−?fπp. Herein, the sum runs over all
x−?cπd. Herein, the sum runs again over

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The strategies are updated according to the following
Monte Carlo procedure: In each elementary step, a player
x and one of its neighbors y is randomly chosen.
both individuals, the payoffs Px and Py are determined
as described above. It is assumed that individual y imi-
tates the strategy sxof individual x with probability W =
{1 + exp[(Py− Px)/K]}−1, where K denotes the uncer-
tainty of strategy adoptions [15]. Here, we use the value
K = 0.5. During one full iteration (Monte Carlo step MCS),
the strategy of each player may be copied once on average.
Following the work of Traulsen et. al. [16], mutation is in-
troduced as a separate process. Accordingly, a player changes
his or her strategy randomly (independently of the neighbor-
hood)with a probabilityµ, while the abovedescribed strategy
adoption process is executed with probability 1 − µ. In other
words, in the limit µ → 1, the game-specificstrategyadoption
is completely ignored.
Initially, each player follows a strategy at random. For
all combinations of cost and fine parameters, the simulations
were performed for systems with L ≥ 400. Values greater
than 400 (up to 1600) were chosen in the vicinity of the phase
boundaries. This served to avoid that small strategy clusters
would disappear by accident (by chance). The fractions ρs
of individuals using the strategies s were determined after the
transient time (up to 106iterations, dependingon the speed of
convergence).
In the absenceof punishmentand mutation, cooperatorsdie
out at r = 3.74, as can be concluded from Fig. 2 of Ref.
[17]. For lower synergyfactors, defectors dominate, while for
higher values of r, cooperators can survive, or even spread all
over the system (if r > 5.45). Taking these values as a refer-
ence, Fig. 1 shows a representative phase diagram of the spa-
tial public goods game with punishment. As the punishment
fine f is increased, it can be observed that (for intermediate
values of the punishment cost c) the system goes from a pure
D phase over a mixed D+PC phase to a pure PC phase. If
the cost of punishment is high (c > 0.51), the mixed D+PC
phase disappears completely, and the system directly changes
from a pure D to a pure PC phase via a discontinuous phase
transition. In the other extreme, if the cost of punishment is
low (c < 0.013), we have an additional area characterized
by a coexistence of PC and PD (see inset of Fig. 1). Quite
surprisingly, the second-order free-rider strategy C is not sus-
tainable for r = 3.5. Only if r is increased, the pure D phase
becomes a mixed D+C phase, which is the only phase where
non-punishingcooperators can survive [13]. For lower values
of r, the mixed D+PC phase vanishes altogether, thus leaving
the pure D and the pure PC phases as the only sustainable so-
lutions, with a discontinuous transition between both phases
when a critical c(f) line is crossed [13].
The problem of second-order free riders results from the
fact that pure cooperators bear no punishment cost, while re-
ceiving the same share of the public good as punishing coop-
erators (given the spatial strategy configuration is the same).
This is, why non-punishing cooperators (“second-order free-
riders”) crowd out punishing one under well-mixed condi-
tions. However, the resulting tragedy of the commons is natu-
rally resolved in structured populations [13, 14]. There, the
For
0.00

0.01

0.02

0.03
0

0.2
0.4
0.6
DD+PC PC
PD+PC
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3
fine
0.4 0.5 0.6
cost
DD+ PC PC
FIG. 1: (color online) Phase diagram of the spatial public goods
game with cooperators (C), defectors (D), and two punishing strate-
gies, PC and PD, in the absence of mutations (µ = 0) for a synergy
factor of r = 3.5 (after [13]). The inset magnifies the part of the
phase diagram for small cost values, where the two punishing strate-
gies PC and PD can coexist. Dashed lines indicate a first-order, solid
lines a continuous phase transition.
victory of the punishing cooperators is not based on a di-
rect competition between the C and PC strategies, but rather
on their different success in encounters with defectors. Due
to the fixed, finite neighborhood, both the PC and C strate-
gies form homogeneous clusters on the spatial grid and are
exploited by defectors. If the fine is sufficiently large, pun-
ishing cooperatorscan overcomedefectors, while cooperators
can not. (Remember that r > 3.74 is needed for cooperators
to be sustainable in the presence of defectors.) Thus, punish-
ing cooperators can spread when competing with defectors,
while non-punishing cooperators are crowded out by them.
As a consequence, second-order free-riders (cooperators) dis-
appear, while punishing cooperators take over.
In the following, we investigate how robust this mechanism
basedontheclusteringandsegregationis withrespectto strat-
egy mutations. We proceed similarly as in Ref. [16], but for a
spatial setting and consideringpunishingdefectorsrather than
loners. For each phase displayed in Fig. 1, we find the follow-
ing typical behavior: Small mutation rates do not significantly
change the strategy distribution as compared to the mutation-
less case. However, for µ ≈ 10−2or higher, the fraction of
defectors increases quickly to values close to 1, as mutations
generate kind of well-mixed conditions, then. Finally, in the
limit µ ≈ 1, mutations dominate the dynamics, leading to
an equidistribution of strategies (i.e. the fraction of defectors
drops again). Figure 2 shows a typical example for the PC
phase, where the dominance of punishing cooperators is sus-
tained until approximately µ ≈ 10−3.
Naturally, the value of the mutation rate, beyond which
defectors can efficiently spread, is highly dependent on the
f/c-ratio. Increasing the fine f can reduce the impact of mu-
tations, because this strengthens punishing cooperators com-
pared to defectors. Nevertheless, sufficiently high value of
µ eventually promote the spreading of defectors through the
creation of a kind of well-mixed state. As emphasized before,
a successful spatial clustering and segregation of strategies is

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0.0
0.2
0.4
0.6
0.8
1.0
10 -6
10 -5
10 -4
10 -3
10 -2
10 -1
10 0
fractions
mutation rate
FIG. 2: (color online) Fractions of all four strategies in dependence
of the mutation rate µ for c = 0.6 and f = 0.6, which for µ = 0
lies in the PC phase. Filled green circles represent punishing coop-
erators (PC), open blue circles cooperators (C), filled yellow squares
punishing defectors (PD), and open red squares defectors (D).
a precondition for the spreading of cooperative behavior and
punishment in the public goods game.
While the introduction of moderate mutations does not sig-
nificantly affect the final outcome of the competition between
strategies,thisdoesnotnecessarilyapplytothedynamics,par-
ticularly when both cooperativestrategies (PC and C) become
equivalent after the extinction of defecting strategies (D and
PD). When non-punishing compete with punishing coopera-
tors, a slow logarithmic coarsening (in the absence of surface
tension) takes place, which is equivalent to the dynamics of
thevotermodel. Despitetheslowdynamics,thefixationtothe
absorbingPC phaseis relativelyfast, because,aftertheextinc-
tion of defectors, the fraction of punishing cooperatorsis high
comparedtothefractionofcooperators. Thisis adirectconse-
quence of the greater success of punishing cooperators in the
competition with defectors during the early stages of strategy
competition,when the punishmentcost is largeenough. How-
ever, if the fractions of punishing and non-punishing cooper-
ators were about the same and no defecting strategies were
present, it would require exceptionally long to reach any of
the absorbing states (C-only or PC-only). Such a scenario is
illustratedin Fig.3(solidgreenline),wheretheinitial fraction
of punishing cooperatorsis assumed to be 0.6 and the fraction
of cooperators is assumed to be 0.4.
Remarkably, even if a tiny mutation rate is introduced,
which occasionally creates defectors and punishing defectors,
this generates an enormous advantage of punishing coopera-
tors over cooperators in the battle with defecting strategies.
The presence of defecting strategies destroys the equivalence
of cooperators and punishing cooperators and breaks the bal-
ance of power in favor of punishing cooperators. This results
in a striking acceleration of the spreading of punishing coop-
erators, as depicted by the dashed green line in Fig. 3. It is
notable that a mutation rate as tiny as µ = 0.000001 evokes
such an enormous change in the system dynamics.
The mutation-induced acceleration of the coarsening pro-
cess relies on the same effect that creates the dominance of
0.5
0.1

0.6

0.7

0.8

0.9

1.0

ρ PC
0.0

10 2
10 3
10 4
10 5
10 6
ρ D
time [MCS]
(a)
(b)
FIG. 3: (color online) Time evolution of the fraction ρPCof punish-
ing cooperators for a small mutation rate of µ = 10−6(dashed green
linein panel (a)), and in the absence of defecting strategiesand muta-
tions (solid green line). In both cases, the initial state was assumed to
consist of 60% punishing cooperators and 40% cooperators. Panel
(b) shows the time evolution of the fraction ρD of defectors in the
presence of small strategy mutations (µ = 10−6). Curves are aver-
ages over 100 independent runs for a grid of size 1600 × 1600.
punishing cooperators over cooperators. It gives rise to a
D+PC phase and, solves the second-order free-rider problem
dueto the disappearanceof non-punishingcooperators. When
defectors occur in the vicinity of cooperators, they can spread
efficiently because of the low value of r. Defectors, however,
cannot succeed against punishing cooperators, if the fine is
sufficiently high. Consequently,punishingcooperatorsspread
at the cost of defectors,while these crowdout cooperators. As
a consequence, a quick victory of punishing cooperators over
cooperators requires an interaction between three strategies:
C, PC and D. The snapshots of Fig. 4 demonstrate the coars-
ening process impressively for the cases with mutation (bot-
tomrow)andwithout(toprow). Startingwith identicalspatial
distributions of cooperators and punishing cooperators, it can
be observed that the fractions of the two strategies remain al-
most the same, when no mutations take place. However, in
the presence of a small rate of strategy mutations (µ = 10−6),
defectors can temporarily spread in the population at the ex-
pense of cooperators. This, in turn, provides conditions for
a fast spreading of punishing cooperators at the expense of
defectors. This dynamics replaces the slow logarithmiccoars-
ening in the absence of defectors. Notably, once punishing
cooperators take over the majority of the spatial grid, a sig-
nificant fraction of defectors can no longer exist. This can be
seen on the bottom of Fig. 4 as well as in panel (b) of Fig. 3,
which demonstrates the temporary uprise of defectors just be-
fore punishing cooperators prevail in the system.
In summary, we have studied the evolution of coopera-
tion in public goods games with mutation and punishment,
where punishing cooperators and punishing defectors were

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FIG. 4: (color online) Typical snapshots of the simulation grid after
0, 900, 1,200, and 10,000 iterations during the coarsening process
with strategy mutations (bottom) and without (top). The initial con-
figurations are identical, and the parameter values agree with those
of Fig. 3. All panels show a 100×100 part of the 1600×1600 grid.
Red (grey) sites correspond to defectors, green (light) ones to pun-
ishing cooperators, and blue (dark) ones to cooperators. Punishing
defectors cannot survive. It can be clearly seen that the presence of
defectors due to strategy mutations largely accelerates the spreading
of punishing cooperators.
taken into account besides conventional cooperators and de-
fectors. Considering structured populations naturally solves
the second-order free-rider problem by spatially separating
the interaction of cooperators with defectors and of defec-
tors with punishing cooperators. Since punishing cooperators
are able to outperform defectors at sufficiently large punish-
ment fines and defectors are superior to cooperators, punish-
ing cooperators are the winners of the strategy competition in
space. This mechanism is robust to modest mutation rates,
while large mutation rates create kind of well-mixed interac-
tions, which promote a spreading of defectors and, thereby, a
tragedy of the commons. Naturally, in the limit µ → 1 mu-
tations become so strong that they create a game-independent
random strategy distribution.
Despite the robustness of the final outcome to moderate
mutation rates, we could demonstrate that even tiny mutation
rates can have an enormous impact on the evolutionary dy-
namics, particularlywhenpunishingcooperatorswould other-
wise competewith cooperatorsonly. Whenno otherstrategies
are present, punishing and non-punishing cooperators receive
the same payoffs, which leads to a slow logarithmic coarsen-
ingas in the votermodel. Theoccurrenceofdefectorsthrough
strategy mutations breaks the balance of power between the
two cooperative strategies. This can dramatically accelerate
the spreading of punishing cooperators. Note that, in many
systems, mutations lead to a different outcome. In the model
studiedhere, however,mutationshavean effectlike a catalyst:
theyspeedup a processwhile the outcomeof thesystem is not
affected.
We acknowledgepartial financial supportby the Future and
Emerging Technologies programme FP7-COSI-ICT of the
European Commission through the project QLectives (grant
no.: 231200) and by the ETH Competence Center “Coping
with Crises in Complex Socio-Economic Systems” (CCSS)
through ETH Research Grant CH1-01 08-2 (D.H.), by the
Hungarian National Research Fund (grant K-73449 to A.S.
and G.S.), the Bolyai Research Grant (to A.S.), the Slove-
nian Research Agency (grant Z1-2032-2547 to M.P.), and
the Slovene-Hungarian bilateral incentive (grant BI-HU/09-
10-001 to A.S., M.P. and G.S.).
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