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Second-Order Free-Riding on Antisocial Punishment Restores the Effectiveness of Prosocial Punishment

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Abstract

Economic experiments have shown that punishment can increase public goods game contributions over time. However, the effectiveness of punishment is challenged by second-order freeriding and antisocial punishment. The latter implies that non-cooperators punish cooperators, while the former implies unwillingness to shoulder the cost of punishment. Here we extend the theory of cooperation in the spatial public goods game by considering four competing strategies, which are traditional cooperators and defectors, as well as cooperators who punish defectors and defectors who punish cooperators. We show that if the synergistic effects are high enough to sustain cooperation based on network reciprocity alone, antisocial punishment does not deter public cooperation. Conversely, if synergistic effects are low and punishment is actively needed to sustain cooperation, antisocial punishment does act detrimental, but only if the cost-to-fine ratio is low. If the costs are relatively high, cooperation again dominates as a result of spatial pattern formation. Counterintuitively, defectors who do not punish cooperators, and are thus effectively second-order freeriding on antisocial punishment, form an active layer around punishing cooperators, which protects them against defectors that punish cooperators. A stable three-strategy phase that is sustained by the spontaneous emergence of cyclic dominance is also possible via the same route. The microscopic mechanism behind the reported evolutionary outcomes can be explained by the comparison of invasion rates that determine the stability of subsystem solutions. Our results reveal an unlikely evolutionary escape from adverse effects of antisocial punishment, and they provide a rationale for why second-order freeriding is not always an impediment to the evolutionary stability of punishment.
Second-order freeriding on antisocial punishment restores the effectiveness of prosocial punishment
Attila Szolnoki1, and Matjaˇ
z Perc2, 3, 4,
1Institute of Technical Physics and Materials Science, Centre for Energy Research,
Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
2Faculty of Natural Sciences and Mathematics, University of Maribor, Koroˇ
ska cesta 160, SI-2000 Maribor, Slovenia
3CAMTP – Center for Applied Mathematics and Theoretical Physics,
University of Maribor, Mladinska 3, SI-2000 Maribor, Slovenia
4Complexity Science Hub, Josefst¨
adterstraße 39, A-1080 Vienna, Austria
Economic experiments have shown that punishment can increase public goods game contributions over time.
However, the effectiveness of punishment is challenged by second-order freeriding and antisocial punishment.
The latter implies that non-cooperators punish cooperators, while the former implies unwillingness to shoul-
der the cost of punishment. Here we extend the theory of cooperation in the spatial public goods game by
considering four competing strategies, which are traditional cooperators and defectors, as well as cooperators
who punish defectors and defectors who punish cooperators. We show that if the synergistic effects are high
enough to sustain cooperation based on network reciprocity alone, antisocial punishment does not deter public
cooperation. Conversely, if synergistic effects are low and punishment is actively needed to sustain cooperation,
antisocial punishment does act detrimental, but only if the cost-to-fine ratio is low. If the costs are relatively
high, cooperation again dominates as a result of spatial pattern formation. Counterintuitively, defectors who
do not punish cooperators, and are thus effectively second-order freeriding on antisocial punishment, form an
active layer around punishing cooperators, which protects them against defectors that punish cooperators. A
stable three-strategy phase that is sustained by the spontaneous emergence of cyclic dominance is also possible
via the same route. The microscopic mechanism behind the reported evolutionary outcomes can be explained
by the comparison of invasion rates that determine the stability of subsystem solutions. Our results reveal an
unlikely evolutionary escape from adverse effects of antisocial punishment, and they provide a rationale for why
second-order freeriding is not always an impediment to the evolutionary stability of punishment.
Keywords: human cooperation, antisocial punishment, pattern formation, evolutionary game theory, cyclic dominance
I. INTRODUCTION
Cooperation is widespread in human societies [1–7]. Like
no other species, we champion personal sacrifice for the com-
mon good [8, 9]. Not only are people willing to incur costs to
help unrelated others, economic experiments have shown that
many are also willing to incur costs to punish those that do not
cooperate [10–16]. Unfortunately, cooperation is jeopardized
by selfish incentives to freeride on the selfless contributions
of others. What is more, individuals that abstain from pun-
ishing such freeriders are often called second-order freerid-
ers for their failure to bear the additional costs of punishment
[17, 18]. Several evolutionary models have been developed to
study the effects of punishment on cooperation [19–27], and it
has been pointed out that second-order freeriding is amongst
the biggest impediments to the evolutionary stability of pun-
ishment [28–31].
In addition to second-order freeriding, the effectiveness of
punishment is challenged by antisocial punishment. The fact
that non-cooperators sometimes punish cooperators has been
observed experimentally in different human societies [32–37],
and it has been shown theoretically that this antisocial punish-
ment can prevent the successful coevolution of punishment
and cooperation [38, 39]. In fact, if antisocial punishment is
an option, prosocial punishment may no longer increase coop-
eration, deteriorating instead to a self-interested tool for pro-
szolnoki.attila@energia.mta.hu
matjaz.perc@uni-mb.si
tecting oneself against potential competitors [40]. While the
punishment of freeriders is aimed at increasing cooperation,
antisocial punishment can be a form of retaliation for punish-
ment received in repeated games [32, 41], or is simply aimed
at cooperators without a retaliatory motive [36, 37].
Given the potential drawbacks associated with punishment
related to second-order freeriding and antisocial punishment,
it has been rightfully pointed out that the maintenance of co-
operation may be better achievable through less destructive
means. In particular, rewards may be as effective as pun-
ishment and lead to higher total earnings without potential
damage to reputation or fear from retaliation [42, 43]. Al-
though many evolutionary models confirm the effectiveness of
positive incentives for the promotion of cooperation [44–52],
in this case too the challenges associated with second-order
freeriding and antisocial rewarding persist [53, 54].
Here we use methods of statistical physics to show how the
two long-standing problems – namely second-order freerid-
ing and antisocial punishment – cancel each other out in an
unlikely and counterintuitive evolutionary outcome, and in
doing so restore the effectiveness of prosocial punishment
to promote cooperation. We extend the theory of coopera-
tion by considering the spatial public goods game with non-
punishing cooperators and defectors, as well as with coopera-
tors who punish defectors and defectors who punish coopera-
tors. As we will show in detail, spatial pattern formation leads
to unconditional defectors forming an active layer around
punishing cooperators, which protects them against defectors
that punish cooperators. This is a new evolutionary escape
from adverse effects of antisocial punishment, which in turn
arXiv:1710.04636v1 [physics.soc-ph] 12 Oct 2017
2
also reveals unexpected benefits stemming from second-order
freeriding.
In what follows, we first present the spatial public goods
game with prosocial and antisocial punishment, and then pro-
ceed with the results and a discussions of their implications for
the successful coevolution of cooperation and punishment.
II. PUBLIC GOODS GAME WITH PROSOCIAL AND
ANTISOCIAL PUNISHMENT
The traditional version of the public goods game is simple
and intuitive, and it captures the essence of the puzzle that
is human cooperation [55, 56]. In a group of players, each
one can decide whether to cooperate (C) or defect (D). Co-
operators contribute a fixed amount (equal to 1without loss
of generality) to the common pool, while defectors contribute
nothing. The sum of all contributions is multiplied by a multi-
plication factor r > 1, which takes into account synergistic ef-
fects of cooperation, and the resulting amount of public goods
is divided equally amongst all group members irrespective of
their strategies. Defection thus yields highest short-term in-
dividual payoffs, while cooperation is best for the group as a
whole.
Here we extend this game by introducing two additional
strategies, namely cooperators that punish defectors (PC), and
defectors that punish cooperators (PD). The former represent
prosocial punishment, while the later represent antisocial pun-
ishment. Technically, PCplayers punish Dand PDplayers,
while PDplayers punish Cand PCplayers. In a ggroup of
size Gthe resulting payoffs are
Πg
D= ΠP GG βNPC
G1,(1)
Πg
C= ΠP GG βNPD
G11,(2)
Πg
PD= ΠP GG βNPC
G1γNC+NPC
G1,(3)
Πg
PC= ΠP GG βNPD
G1γND+NPD
G11,(4)
where
Πg
P GG =r(NC+NPC)
G(5)
and NC,ND,NPCand NPDare, respectively, the number of
non-punishing cooperators, non-punishing defectors, punish-
ing cooperators and punishing defectors in the ggroup. In
addition to the multiplication factor rwe have two additional
parameters, which are βas the maximal fine imposed on a
player if all other players within the group punish her, and γas
the maximal cost of punishment that can apply. Importantly,
the values of both βand γare kept the same for prosocial and
antisocial punishment so as to not give either a default evolu-
tionary advantage or disadvantage.
This public goods game is staged on a square lattice with
periodic boundary conditions where L2players are arranged
into overlapping groups of size G= 5 such that everyone
is connected to its G1nearest neighbors. Accordingly,
each player belongs to g= 1, . . . , G different groups, each
of size G. Notably, the square lattice is the simplest of net-
works that takes into account the fact that the interactions
among us are inherently structured rather than random. By
using the square lattice, we continue a long-standing tradition
that begun with the work of Nowak and May [57], and which
has since emerged as a default setup to reveal all evolution-
ary outcomes that are feasible within a particular version of
the public goods game [56]. We should note, however, that
our observations are robust and do not restricted to this inter-
action topology. The only crucial criteria are players should
have limited and stable connections with others, which allow
network reciprocity to work.
Monte Carlo simulations are carried out as follows. Initially
each player on site xis designated either as a non-punishing
cooperator, non-punishing defector, punishing cooperator or
punishing defector with equal probability. The following el-
ementary steps are then iterated repeatedly until a stationary
solution is obtained, i.e., until the average fractions of strate-
gies on the square lattice become time-independent. During
an elementary step a randomly selected player xplays the
public goods game in all the Ggroups where she is mem-
ber, whereby her overall payoff Πsxis thus the sum of all the
payoffs Πg
sxacquired in each individual group, as described
above in Eqs. (1-5). Next, a randomly selected neighbor of
player xacquires her payoff Πsyin the same way. Lastly,
player yimitates the strategy of player xwith a probability
given by the Fermi function
Γ(sxsy)=1/{1 + exp[(ΠsyΠsx)/K],(6)
where Kquantifies the uncertainty by strategy adoptions [58],
implying that better performing players are readily adopted,
although it is not impossible to adopt the strategy of a player
performing worse. In the K0limit, player yimitates the
strategy of player xif and only if Πsx>Πsy. Conversely,
in the K limit, payoffs seize to matter and strategies
change as per flip of a coin. Between these two extremes play-
ers with a higher payoff will be readily imitated, although the
strategy of under-performing players may also be occasion-
ally adopted, for example due to errors in the decision mak-
ing, imperfect information, and external influences that may
adversely affect the evaluation of an opponent. Without loss
of generality we use K= 0.5, in agreement with previous
research that showed this to be a fully representative value
[58–60]. Repeating all described elementary steps L2times
constitutes one full Monte Carlo step (MCS), thus giving a
chance to every player to change its strategy once on average.
We note that imitation is a fundamental process by means of
which humans change their strategies [61–64]. The applica-
tion of imitation-based strategy updating based on the Fermi
function is thus appropriate and justified, although, as we will
show, our results are robust to changes in the details that de-
termine the microscopic dynamics of the studied public goods
game. In terms of the application of the square lattice, we em-
phasize that, despite its simplicity, it fully captures the most
relevant aspect of human interactions – namely the fact that
nobody interacts randomly with everybody else, not even in
3
small groups, and that our interaction range is thus inherently
limited. Applications of more complex interaction topologies
are of course possible, but this does not affect our results. This
is because our key argument is based on the limited number
of interactions a players has, but it does not in any way rely
on the specific properties of the square lattice topology.
The average fractions of all four strategies on the square lat-
tice are determined in the stationary state after a sufficiently
long relaxation time. Depending on the proximity to phase
transition points and the typical size of emerging spatial pat-
terns, the linear system size was varied from L= 400 to 6000,
and the relaxation time was varied from 104to 106MCS to
ensure that the statistical error is comparable with the size of
symbols in the figures. We emphasize that the usage of a suf-
ficiently large system size is a decisive factor that allows us to
identify the correct evolutionary stable solutions. Using a too
small system size may easily prevent this, for example if the
linear size of the lattice is comparable to or smaller than the
typical size of the emerging spatial patterns.
III. RESULTS
Before presenting the main results, we briefly summarize
the evolutionary outcomes in a well-mixed population. In the
absence of a limited interaction range the behavior is largely
trivial and resembles that reported before for the traditional
two-strategy public goods game [29, 55]. In particular, if rex-
ceeds the group size Gthen both cooperative strategies domi-
nate while all defectors die out. Conversely, below this thresh-
old both defector strategies dominate while all cooperators die
out. This behavior is also in agreement with the well-mixed
results published in [39]. In short, all the non-trivial evolu-
tionary solutions reported here in the continuation are due to
the consideration of a structured population and remain com-
pletely hidden if a well-mixed population is assumed.
In what follows, we focus on two representative values of
the multiplication factor that cover two relevantly different
public goods game scenarios. First, we use r= 3.8, where
the spatial selection allows cooperators to survive even in the
absence of punishment – this is the well-known manifestation
of network reciprocity, where the limited interactions among
players allow cooperators to organize themselves into com-
pact clusters, which confers them competitive payoffs in com-
parison to defectors [65]. Subsequently, we also use a suffi-
ciently small r= 3.0value, where cooperators can no longer
survive solely due to network reciprocity and thus require ad-
ditional support [58]. For both values of rwe determine the
stationary fractions of strategies in dependence on the punish-
ment fine βand the punishment cost γ, and we pinpoint the
location and type of phase transitions from the Monte Carlo
simulation data.
In Fig. 1, we show the full βγphase diagram, as ob-
tained for r= 3.8. As discussed above, we refer to this
as the strong network reciprocity region. Presented results
reveal that antisocial punishment is hardly viable, with PD
players surviving only in a tiny region of the βγparameter
space. Conversely, as the fine βincreases, punishing coopera-
0.0
0.2
0.4
0.6
0.8
1.0
0 0.2 0.4 0.6 0.8 1
cost, γ
fine, β
D+C PC
D+PC
PD+PC
FIG. 1. Full βγphase diagram of the spatial public goods game
with prosocial and antisocial punishment, as obtained for r= 3.8.
Solid lines denote continuous phase transitions while dashed lines
denoted discontinuous phase transitions. Two representative cross-
sections of this phase diagram are presented in Fig. 2.
tors subvert non-punishing cooperators, first via a discontinu-
ous phase transition from the two-strategy D+Cphase to the
two-strategy D+PCphase, and subsequently via a continuous
phase transition to the absorbing PCphase. The discontinu-
ous phase transition is due to indirect territorial competition,
which emerges between Cand PCplayers competing against
defectors [29], while the continuous phase transition is due to
an increasing effectiveness of punishment that stems from the
larger fines.
The two representative cross-sections of the phase diagram
in Fig. 2 provide a more quantitative insight into the nature
of these phase transitions. In both cases the application of
small punishment fines yields a punishment-free state, where
traditional cooperators and defectors coexist due to network
reciprocity. If the cost of punishment is considerable, as in
panel (a), the D+Cphase suddenly gives way to the D+PC
phase at a critical value of the punishment fine βc= 0.229,
and by increasing βfurther, a defector-free state is reached
at βc= 0.361. This succession of phase transitions remains
the same if the cost of punishment is tiny, shown in panel (b),
apart from a narrow intermediate region of β, where antisocial
punishment replaces non-punishing defectors via a discontin-
uous D+PCPD+PCphase transition at βc= 0.284.
Interestingly, this phase transition is qualitatively identical to
the preceding D+CD+PCphase transition – in both
cases non-punishing strategies are subverted by their pun-
ishing counterparts on the grounds of increasing punishment
fines. It can also be observed that the emergence of a stable
PD+PCphase involves a slight decay of the fraction of PC
players, although they quickly recover to full dominance as
the punishment fine is increased further.
To sum up, in the strong network reciprocity region anti-
social punishment has a negligible impact on the evolution of
cooperation. In a small region of the βγparameter space an-
tisocial punishers can outperform non-punishing defectors to
form a stable coexistence with prosocial punishers. But apart
4
0.0
0.2
0.4
0.6
0.8
1.0
0 0.1 0.2 0.3 0.4
fractions
D
C
P
D
P
C
(a)
0.0
0.2
0.4
0.6
0.8
1.0
0 0.1 0.2 0.3 0.4
fractions
fine, β
D
C
P
D
P
C
(b)
FIG. 2. Two representative cross-sections of the phase diagram de-
picted in Fig. 1, as obtained for the punishment cost γ= 0.4(top)
and γ= 0.02 (bottom). Depicted are stationary fractions of the four
competing strategies in dependence on the punishment fine β.
from this, and despite the fact that both forms of punishment
are implemented equally effective (the values of both βand
γare kept the same for prosocial and antisocial punishment),
antisocial punishment fails and is evolutionary uncompetitive.
An exciting question now is what if the network reciprocity
alone is not strong enough to support the coexistence of coop-
erators and defectors? Although previous research has shown
that prosocial punishment can be effective if the imposed fines
are sufficiently high [66, 67], this results was obtained in the
absence of antisocial punishment. However, if cooperators
can also be punished the situation changes significantly. A
subsystem analysis of the public goods game entailing only
punishing cooperators and punishing defectors actually re-
veals that at r= 3.0cooperation is unable to survive regard-
less of the values of βand γ, and regardless of the fact that
punishing cooperators also benefit from network reciprocity.
Quite remarkably, the evolutionary outcome of the full 4-
strategy public goods game can be very different. The full
βγphase diagram presented in Fig. 3 reveals that punish-
ing cooperators can actually dominate completely in a sizable
region of the parameter plane. Of course, if the cost of pun-
ishment is too high in relation to the imposed fines, defectors
dominate. More precisely, Cand PCplayers die out fast, with
only Dand PDplayers remaining. In the absence of the two
0.0
0.2
0.4
0.6
0.8
1.0
0 0.2 0.4 0.6 0.8 1
cost, γ
fine, β
D (PD )
PC
PD
D+PD+PC
D+PC
FIG. 3. Full βγphase diagram of the spatial public goods game
with prosocial and antisocial punishment, as obtained for r= 3.0.
Solid lines denote continuous phase transitions while dashed lines
denoted discontinuous phase transitions. Representative spatial evo-
lutions of the four competing strategies are presented in Figs. 4-6,
while two representative cross-sections of this phase diagram are pre-
sented in Fig. 7.
cooperative strategies the relation between Dand PDplayers
is neutral since the latter do not need to bear the punishment
cost. This yields a logarithmically slow coarsening without
surface tension, as in the voter model [68, 69]. In this case the
probability to reach either the absorbing Dor the absorbing
PDphase depends on the fraction of these two strategies [70]
when all cooperators die out, which is typically higher for D
and hence the D(PD)notation in Fig. 3.
Returning to the absorbing PCphase, since network reci-
procity at r= 3.0is weak, an additional mechanism must be
at work that allows the dominance of cooperation despite the
low multiplication factor and despite antisocial punishment.
This mechanism is illustrated in Fig. 4, where we show a rep-
resentative spatial evolution of the four competing strategies
from a random initial state for parameter values that yield
the absorbing PCphase. It is important to emphasize that
a sufficiently large square lattice must be used, since other-
wise the evolutionary process can quickly lead to a mislead-
ing outcome, i.e., to a solution that is not stable in the large
population size limit, or to a solution that is highly sensitive
on the initial fraction of strategies, as reported in [39]. This
is, however, just a finite-size effect because the evolutionary
stable solution can spread in the whole population if it has
a chance to emerge somewhere locally. Due to the random
initial state the number of non-punishing and punishing co-
operators starts dropping fast because support from network
reciprocity is lacking, both because the value of ris low, and
even more so because compact clusters are not yet formed
so early in the process. During this stage, if the population
would be small, an accidental extinction of PCplayers would
be very likely. Indeed, even with L= 800 they manage to
just barely survive, as indicated in panel (c) by a white circle.
At this point the temporary winners appear to be Dand PD
players, which in the absence of cooperators are neutral, and
5
FIG. 4. Representative spatial evolution of the four competing strate-
gies. Depicted are snapshots of the square lattice, as obtained for
β= 0.8,γ= 0.36, and r= 3. Non-punishing (punishing) coopera-
tors are depicted light blue (dark blue), while non-punishing (punish-
ing) defectors are depicted light red (dark red). From a random initial
state (a) both cooperative strategies start vanishing quickly (b). The
only chance for cooperation to survive is if a lucky PCseed starts
growing in the sea of Dplayers, as indicated in panel (c) by a white
circle. It turns out, however, that PCplayers, surrounded by a thin
active layer of Dplayers, can rise to complete dominance over time,
as shown in panels (d-f) (the final state, where only PCremain after
6000 MCS, is not shown). The linear size of the lattice is L= 800.
hence perform a logarithmically slow coarsening [69].
However, the unlikely evolutionary twist is yet to come and
reveals itself in panels (d-f). Since PCplayers are weaker than
PDplayers, the only chance for the former to survive is if they
form a compact cluster inside a Ddomain (Ccan not survive
either way because r= 3.0is too small). Although one might
suspect that this “hanging by a thread”-like survival of PC
players is merely temporary because the superior PDplayers
will eventually invade their cluster, this does in fact never hap-
pen. On the contrary, punishing cooperators eventually rise to
complete dominance (the final state is not shown in Fig. 4).
Crucial for the understanding of this counterintuitive evo-
lutionary outcome is the realization that punishing defectors
suffer from second-order freeriding of non-punishing defec-
tors as soon as they both meet in the vicinity of punishing
cooperators. More precisely, when Dplayers meets with PD
players in the vicinity of PCplayers, then PDplayers have
to bear the additional cost of punishment while Dplayers are
of course free from this burden. The same argument is tradi-
tionally put forward when it is time to explain why punishing
cooperators are uncompetitive next to non-punishing cooper-
ators near defectors, and why in fact punishment is evolution-
ary unstable. When antisocial punishment is present, how-
ever, this very same reasoning helps punishing cooperators to
beat defectors that punish them. As a result, Dplayers start
invading PDdomains, but in parallel PCplayers also invade
Dplayers from the other side of the interface. The thin ac-
tive layer of Dplayers thus acts as a protection, shielding PC
players from a direct invasion of PD. As can be observed in
panels (d-f), the shield is not passive, but expands permanently
FIG. 5. An illustration of how second-order freeriding on antisocial
punishment restores the effectiveness of prosocial punishment. A
small domain of PCplayers (dark blue), surrounded by a thin layer
of Dplayers (light red) is inserted into the sea of PDplayers (dark
red) in the bottom left corner of the lattice (a). Similarly, a sizable
domain of PCplayers, but without the protective Dlayer, is inserted
into the sea of PDplayers in upper right corner of the lattice (a).
While the large PCdomain without the protective layer shrinks over
time, the small PCdomain with the protective Dlayer grows (b-f).
The absorbing PCphase is reached after 2000 MCS (not shown).
The linear size of the square lattice in this case is L= 200. Other
parameters are β= 0.8, γ = 0.3, and r= 3.
because Dplayers become successful when meeting PDplay-
ers close to cooperators. (This process will be quantified via
an effective invasion rate in the following section.) At the end,
when PDplayers die out, the Dshield falls victim to the inva-
sion of PCplayers, which thus rise to complete dominance. A
direct illustration of this mechanism is shown in Fig. 5, where
a prepared initial state was used for clarity. The comparison
of the evolution of a large but lonely PCdomain, and a tiny
but D-protected seed of the same strategy illustrates nicely
that the previously described “activated layer” mechanism is
effective to overcome the danger of simultaneous presence of
antisocial punishment.
We show the above-described pattern formation in the an-
imation provided in [71]. At this point, we also emphasize
that the key mechanism that is responsible for the recovery
of prosocial punishment is not restricted to the application
of imitation-based strategy updating and is in fact robust to
changes in the microscopic dynamics. For example, if we
apply the so-called “score-dependent viability” strategy up-
dating [39, 72], the trajectory of evolution remains the same
[73]. The only visible difference is that, in the latter case, the
interfaces that separate the competing domains are rugged and
strongly fluctuating, which in turn decelerates the evolution-
ary dynamics and prolongs the time needed to arrive at the
same final outcome.
As we have pointed out, snapshots in Fig. 4 illustrate clearly
that the size of the lattice plays a decisive role in reaching
the correct evolutionary outcome from a random initial state.
For some parameter values that bring the population closer
to a phase transition point or because of the large fluctua-
6
abcdef
FIG. 6. Representative spatial evolution of the four competing strategies from a prepared initial state towards the three-strategy D+PD+PC
phase that is sustained by cyclic dominance. Note that blue and red colors dominate cyclically over the course of 16000 MCS from left to
right. The colors used are the same as in Figs. 4 and 5. Depicted are snapshots of the square lattice, as obtained for β= 0.52,γ= 0.065, and
r= 3. Since a prepared initial state is used, a small square lattice with linear size L= 100 can be used for demonstration.
tions of strategy abundance during the pattern formation even
L= 6000 (linear system size) can turn out to be too small.
In such cases a prepared initial state, as depicted in panel (a)
of Fig. 6, consisting of sizable patches of the four competing
strategies, can help to determine the correct composition of
the stationary solution. We have used this approach to deter-
mine the stability of the three-strategy D+PD+PCphase,
which according to the phase diagram in Fig. 3, also forms an
important part of the solution. Figure 6 illustrates that such
a solution can be observed even if using a very small lattice
size, if only suitable initial conditions are used. The alter-
nating oscillations of read and blue indicate that this three-
strategy phase is sustained by cyclic dominance. Indeed, due
to using a different set of βand γvalues from those used in
Fig. 4, here PDplayers beat PCplayers because of the low
value of r,PCplayers beat Dplayers because of prosocial
punishment, and Dplayers beat PDplayers near cooperators
because of second-order freeriding. However, the balance of
these invasions is such that neither strategy dies out, and hence
the three-strategy D+PD+PCphase is stable.
The two representative cross-sections of the phase dia-
gram in Fig. 7 show that the average fraction of competing
strategies changes similarly as in the canonical rock-paper-
scissors model [74–77]. The stability of the three-strategy
D+PD+PCphase hinges strongly on the continuous, al-
beit oscillating and sometimes nearly vanishing, presence of
all three strategies. As the inset in the bottom panel of Fig. 7
shows, the average fraction of Dplayers can be extremely low,
and therefore this three-strategy phase can be stable only if the
size of the lattice is large enough. We have used L= 5400
to produce results presented in Fig. 7. If the size of the lattice
would be smaller, a strategy could easily die out due to ran-
dom fluctuations, in which case the evolution would terminate
into a single-strategy phase.
The invasion rates within the three-strategy D+PD+PC
phase, which exists under conditions that are more favorable
for the survival of the PDstrategy – specifically, if the cost of
punishment γis lower – hence stabilizing the three-strategy
phase, can be measured directly by monitoring the fractions
of strategies when the evolution is initialized from straight do-
main interfaces [78, 79]. While the meaning of w1and w2(see
the diagram inserted in Fig. 8) is clear from the payoff differ-
ences, the determination of w3requires further clarification.
Namely, if only PDand Dplayers would be present along the
interface, then we would of course measure a net zero invasion
rate because the two strategies are neutral in the absence of co-
operators. However, since we are interested in their relation
when PCplayers are present too, we use parallel interfaces
of PCand PDplayers, separated by thin (width of 5lattice
sites) layers of Dplayers. In this way, although PCand PD
0.0
0.2
0.4
0.6
0.8
1.0
0.36 0.4 0.44 0.48 0.52 0.56 0.6
fractions
fine, β
D
C
P
D
P
C
(a)
0.0
0.2
0.4
0.6
0.8
1.0
0.09 0.1 0.11 0.12 0.13
fractions
cost, γ
D
C
P
D
P
C
(b)
0
0.004
0.008
0.09
0.11
0.13
γ
FIG. 7. Two representative cross-sections of the phase diagram de-
picted in Fig. 7, as obtained for the punishment cost γ= 0.04 (top)
and the punishment fine β= 0.9(bottom). Depicted are stationary
fractions of the four competing strategies in dependence on the other
punishment parameter from the one used for the cross-section. The
inset in the bottom panel shows just how tiny the fraction of non-
punishing defectors in the stationary state can be.
7
-0.4
-0.2
0.0
0.2
0.4
0 0.1 0.2 0.3
w
cost, γ
PC
PD
D
0.0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3
invasion rates, w
cost, γ
PDD
PC
w1w2
w3
w2
w1
w3
FIG. 8. Invasion rates wibetween three competing strategies (see in-
serted diagram) in dependence on the punishment cost γ, as obtained
for β= 0.8and r= 3.0. We note that Dinvades PDonly in the
vicinity of PCplayers, since otherwise the two defecting strategies
are neutral. This fact is indicated by a dashed w3arrow. For de-
tails how this specific invasion rate is measured we refer to the main
text. The inset shows the difference wbetween different pairs of
invasion rates, which explains how and why the fraction of strategies
changes as a result of the increasing punishment cost (see main text
for details).
players do not interact directly, the setup properly describes
the movement of Dlayer that is followed by punishing coop-
erators. This “effective” invasion, which emerges only in the
presence of the third party, is highlighted by a dashed arrow
in the legend of Fig. 8.
The decay of w2in the main panel of Fig. 8 highlights that
the PCDinvasions are relevant and in fact occur rather
frequently, but also that their intensity deteriorates as the cost
of punishment increases. Similarly, the PDPCinvasions
are also a recurring phenomenon based on the positive value
of the corresponding invasion rate w1, which indicates that
PDplayers would dominate PCplayers during a direct com-
petition as a consequence of the small value of r. Neverthe-
less, this invasion rate also decays slightly as γincreases be-
cause the costs associated with the main public goods game
come to play second fiddle to the costs of punishment that both
these strategies should bear. Lastly, the w3(γ)function is also
always positive, because even a small cost evokes the second-
order freeriding effect (in this case of course associated with
avoiding the costs of antisocial punishment), such that in the
presence of PCplayers Dplayers can invade PDplayers. Ac-
cordingly, as the value of γincreases, so does the w3invasion
rate, as shown in Fig. 8, which illustrates directly that second-
order freeriding on antisocial punishment is responsible for
the evolutionary success of prosocial punishment, specifically
for the survival or even for the complete dominance of the PC
strategy.
The differences between these three invasion rates, depicted
in the inset of Fig. 8, allow us to understand how the rela-
tive abundance of strategies changes as a result. For example,
w3w2quantifies how the fraction of non-punishing defec-
tors changes when we vary the cost of punishment. An in-
crease in the value of γwill support strategy D, but the actual
beneficiary will be her predator, which is strategy PC. This
seemingly paradoxical response of the population to the in-
crease of the punishment cost is a well-known consequence
of cyclic dominance, i.e., when directly supporting a partic-
ular species will actually support her predator [80]. This in
turn explains why PCplayers rise to full dominance when we
increase the value of γ, as well as why PDplayers dominate
when we decrease it. Namely, decreasing the punishment cost
supports the PCstrategy, which is the prey of punishing de-
fectors.
IV. DISCUSSION
We have shown how second-order freeriding on antiso-
cial punishment restores the effectiveness of prosocial pun-
ishment, thus providing an unlikely and counterintuitive evo-
lutionary escape from adverse effects of antisocial punish-
ment. When the synergistic effects of cooperation are low to
the point of network reciprocity failing to sustain it, coopera-
tors that punish defectors can still rise to dominance because
non-punishing defectors enable their evolutionary success by
capitalizing on second-order freeriding and eliminating anti-
social punishers as a result. If conditions for punishment are
somewhat more lenient, we have shown that a three-strategy
phase consisting of non-punishing defectors, punishing coop-
erators, and punishing defectors becomes stable. The relations
within this phase, and its termination to an absorbing punish-
ing cooperator or an absorbing punishing defector phase, can
be fully understood in terms of invasion rates along straight
interfaces that separate different strategy domains. We have
demonstrated that these results are robust to changes in the
microscopic dynamics, and we have emphasized that the only
important property of the interaction structure is the limited
interaction range rather than its topological details. Indeed,
the mechanism relies solely on spatial pattern formation, and
is the first stand-alone remedy against adverse effects of an-
tisocial punishment, not relying on any additional strategic
complexity or other assumptions limiting its general validity.
Paradoxically, it turns out that antisocial punishment is vul-
nerable to the same second-order freeriding that is tradition-
ally held responsible for preventing evolutionary stability of
prosocial punishment.
We emphasize that these phenomena can not be observed
in well-mixed populations. Furthermore, a reliable study of
competing subsystem solutions requires a careful finite-size
analysis of the spatial system. Additionally, the usage of ran-
dom initial conditions may be misleading, especially if using
a small system size, because it does not necessarily allow for
all possible subsystem solutions to emerge (before they could
compete with one another). This difficulties can be overcome
by using suitable prepared initial states, which allow the evo-
lutionary stable subsystem solution to form before competi-
tion between them unfolds. Since pattern formation and in-
vasions of propagating fronts are general features of multi-
strategy complex systems, such an analysis is a must when
determining the consequences of spatiality.
8
Our research also reveals that under conditions that favor
cooperation, for example when the multiplication factor of
the public goods game is sufficiently high for the spatial se-
lection alone to sustain cooperation, antisocial punishment is
overall uncompetitive. In fact, even though we have used a
fully symmetrical implementation of prosocial and antisocial
punishment throughout our paper, antisocial punishers could
survive only in very small regions of the parameter space. We
may thus conclude that under such conditions cooperators that
punish defectors should not be afraid of retaliatory antisocial
punishment by defectors.
In comparison to previous findings concerning the sym-
metrical implementation of prosocial and antisocial rewarding
[54], we find that with punishment there is no lower bound
on the multiplication factor that would be impossible to com-
pensate with a sufficiently effective punishment system. But
there is one with rewarding, i.e., below a critical value of the
multiplication factor full defection is unavoidable, and this re-
gardless of just how efficient the rewarding system might be.
In case of punishment, ever lower values of the multiplication
factor simply require ever higher fines at a given cost for co-
operation to be sustained. Quite remarkably, the very same
process puts a noose around antisocial punishers, which are
defeated by second-order freeriding in their own ranks.
Although social preference models of economic decision-
making predict that antisocial punishment should not occur
[81, 82], and despite the fact that antisocial punishment is also
inconsistent with rational self-interest and the hypothesis that
punishment facilitates cooperation, it is nevertheless remark-
ably common across human societies [32–37]. In the light
of this fact, it was important to extend the theory of coopera-
tion in the spatial public goods game with the option that non-
cooperators can punish cooperators. Rather unexpectedly, the
detrimental effects of such antisocial punishment on the co-
evolution of punishment and cooperation turned out to be mi-
nor simply by taking into account the fact that the interactions
among humans are inherently structured, entailing a limited
number of frequently used links, rather than being random or
well-mixed.
ACKNOWLEDGMENTS
This research was supported by the Hungarian National Re-
search Fund (Grant K-120785) and the Slovenian Research
Agency (Grants J1-7009 and P5-0027).
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... Besides, inspired by real observations, sociologists and physicists also provide solutions from a sociological point of view such as reward [10], punishment [11] and exclusion [12]. In this regard, they have successfully instigated the transition from the common unique absorbing state to coexistence and cyclic dominance, opening up opportunities for altruism, including but not limited to cooperation, in situations where their survival has traditionally been considered impossible [13,14]. Further, being well aware of the fact that real networks are often dynamic rather than static [15], some embark on this issue by resorting to the so-called coevolution dynamics [16]. ...
... The ability to solidly maintain a higher level of cooperation, particularly with a more hostile condition, once again advertises the improve-ment generated by the synergy of different social norms. Excitingly, there exists the rare oscillation of behavior that commonly arises in a three-strategy game [13,14], thus supplementing the evidence of strategic oscillation for a two-strategy game. Other parameters: L = 50, ρ = 0.5, and μ = 0.01 Fig. 6 Collapse (persistence) of cooperation in the absence (presence) of conformity under noisy conditions. ...
... As a consequence, cooperation flourishes and withers alternatively as time goes by, unable to steadily dominate the population at all. Notably, such an apparently non-trivial burst pattern has been rarely traced in a two-strategy game before, whereas in a three-strategy game, similar cyclic dominance has been frequently reported [11,13,14]. To probe the story behind this phenomenon, a more detailed examination is carried out in Fig. 7. ...
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... Understanding the emergence and maintenance of cooperation has long been a challenge in evolutionary biology and ecology. Past decades have seen many powerful mechanisms proposed to explain the evolution of cooperation, such as direct reciprocity [1,2], indirect reciprocity [3][4][5][6], spatial reciprocity [7][8][9][10][11] and costly punishment (also called altruistic punishment) [12][13][14][15][16][17][18][19]. Among them, both mechanisms of direct and indirect reciprocity indicate the principle of personal responsibility that once taking defection, one would incur others' retaliations, whether in the repeated interactions with the same opponent repeatedly (direct reciprocity) or with different opponents (indirect reciprocity). ...
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