Theoretical and empirical research highlights the role of punishment in promoting collaborative efforts1, 2, 3, 4, 5.
However, both the emergence and the stability of costly punishment are
problematic issues. It is not clear how punishers can invade a society
of defectors by social learning or natural selection, or how
second-order free-riders (who contribute to the joint effort but not to
the sanctions) can be prevented from drifting into a coercion-based
regime and subverting cooperation. Here we compare the prevailing model
of peer-punishment6, 7, 8
with pool-punishment, which consists in committing resources, before
the collaborative effort, to prepare sanctions against free-riders.
Pool-punishment facilitates the sanctioning of second-order free-riders,
because these are exposed even if everyone contributes to the common
good. In the absence of such second-order punishment, peer-punishers do
better than pool-punishers; but with second-order punishment, the
situation is reversed. Efficiency is traded for stability. Neither
other-regarding tendencies or preferences for reciprocity and equity,
nor group selection or prescriptions from higher authorities, are
necessary for the emergence and stability of rudimentary forms of
sanctioning institutions regulating common pool resources and enforcing
collaborative efforts.
Many economic experiments on ‘public goods games’ (PGGs) have
shown that a substantial fraction of players are willing to incur costs
to impose fines on exploiters, that is, those who do not contribute to
the joint effort1, 2, 3, 4, 5, 6, 7, 8.
As a consequence, the threat of punishment looms credibly enough to
increase the average level of pro-social contributions. However, the
sanctioning system is itself a public good. Thus, punishers are often
seen as altruistic, because others benefit from their costly efforts9, 10, 11, 12, 13.
Conversely, those who refrain from punishing exploiters are
‘second-order free-riders’. Among self-interested agents, second-order
free-riding should spread and ultimately cause the collapse of
cooperation.
A solution is to punish second-order free-riders also14.
But such ‘second-order punishment’ risks being subverted by third-order
free-riders in turn, leading to infinite regress. Moreover, if everyone
contributes to the public good, second-order free-riders will not be
spotted. Their number can grow through neutral drift, ultimately
allowing defectors to invade with impunity. We show how a simple
mechanism can overcome this problem.
There exist a variety of
sanctioning systems. Most experiments on public goods with punishment
have considered peer-punishment: after the PGG, individuals can impose
fines on exploiters, at a cost to themselves. Interestingly, the first
experiment on public goods with punishment15
considered a different mechanism: players decide whether to contribute
to a ‘punishment pool’ before contributing to the public goods. This can
be viewed as a first step towards an institutionalized mechanism for
punishing exploiters, and compared with the self-financed contract
enforcement games in Governing the Commons16. It is like paying towards a police force, whereas peer-punishers take law enforcement into their own hands.
Peer-
and pool-punishment are both expensive ways to impose negative
incentives on free-riders. In many economic experiments, the increase in
cooperation is more than matched by the costs of punishment, and an
overall reduction of total pay-off is observed8, 9.
Because the costs of pool-punishment arise even when there are no
exploiters to be punished, it seems even more socially expensive than
peer-punishment. However, the issue of second-order punishment favours
pool-punishment. If everyone contributes to the public good, then
peer-punishers are not distinguishable from second-order free-riders. By
contrast, pool-punishers declare themselves beforehand. We may expect
that pool-punishment leads more easily to a second-order punishment
regime and, hence, to more stability.
Because sanctioning
institutions, as known from social history, usually forbid individuals
to take the law into their own hands, it is also worthwhile to
investigate the competition between peer- and pool-punishment. A model
based on evolutionary game theory shows that both peer- and
pool-punishment can emerge, if participation in the joint effort is
optional rather than compulsory. Pool-punishment requires second-order
punishment, whereas peer-punishment is little affected by it. Both
sanctioning mechanisms can evolve if players simply imitate whatever
yields the highest pay-off. If peer-punishers compete with
pool-punishers, all depends on second-order punishment. Without it, the
population is dominated by peer-punishers. With it, pool-punishers take
over, although the average income is thereby reduced.
A
‘punishment fund’ can be viewed as a rudimentary institution to uphold
the common interest. Many small-scale societies use this principle, for
instance by hiring an enforcer. In Governing the Commons16,
several examples of self-financed contract enforcement are described.
They concern the provisioning and the appropriation of common resources,
for instance high mountain meadows (the ‘commons’), irrigation systems
or inshore fisheries. Our model shows that individuals can spontaneously
adopt a self-governing institution to monitor contributions and
sanction free-riders. It needs no top-down prescriptions from higher
authorities, nor great feats of planning: trial and error, and the
imitation of successful examples, can lead to a social contract among
individuals guided by self-interest.
To model a PGG, we assume that if N≥2 individuals participate in the interaction, each can decide whether to contribute a fixed amount, c>0, to the common pool. This amount will be multiplied by a factor of r>1 and then divided among the N−1 other players. If all contribute, they obtain (r−1)c
each. Because contributors do not benefit from their own contribution,
self-interested players ought to contribute nothing. If all do this,
their pay-off will be zero. This reveals a social dilemma.
Pool-punishers not only contribute c to the PGG, but also, beforehand, an amount, G, to a punishment pool. Free-riders will be fined an amount, BNv, proportional to the number, Nv,
of pool-punishers. In the case of second-order punishment, second-order
free-riders will be fined the same amount. Peer-punishers contribute c to the PGG, and after the game impose a fine, β, on each free-rider in their group, at a cost γ. If Nw peer-punishers are in the group, each defector pays a total fine βNw. In case of second-order punishment, second-order defectors are treated just like defectors.
Let us assume that the game is not compulsory11, 17. Some players may abstain from the joint enterprise. They can do something else instead, and earn a pay-off, σ,
independent of what others are doing. If only one player is willing to
engage in the joint effort, there will be no PGG and the solitary
would-be participant also earns σ.
Let M denote the population size; X the number of players who participate in the PGG and contribute, but do not punish; Y the number of defectors, who participate but contribute neither to the PGG nor to the sanctions; Z the number of non-participants; V the number of pool-punishers; and W the number of peer-punishers. Random samples of N
individuals are faced with the opportunity of a joint enterprise.
Social learning leads to preferential copying of successful strategies.
We obtain their long-run frequencies by numerical simulations (compare
with Figs 1, 2 and 3). In a limiting case, we obtain analytic results (Supplementary Information) that we now describe.
Let us first neglect peer-punishment, and assume that the pay-off, σ, for non-participants lies between zero (obtained if all free-ride) and (r−1)c−G (obtained if all contribute to the PGG and the punishment pool). The inequality
highlights that participating in the joint enterprise is a
venture that succeeds if most participants contribute and fails if most
do not.
In the absence of second-order punishment, the long-run frequencies in the (X, Y, Z, V)
subpopulations are (2, 2, 2, 1)/7 and little cooperation is achieved.
With second-order punishment, the corresponding long-run frequencies are
(0, 0, 0, 1).The population is dominated by pool-punishers enforcing
cooperation. If the game is compulsory (that is, Z = 0), the population consists of free-riders only.
Alternatively, if we neglect pool-punishment, and assume that
the long-run frequencies in the (X, Y, Z, W) subpopulations are (2, 2, 2, M+2)/(M+8)
and punishers prevail, with or without second-order punishment. Again,
if the game is compulsory, only free-riders survive in the long run.
In
the competition between peer- and pool-punishers without second-order
punishment, peer-punishers win. The long-run frequencies in the (X, Y, Z, V, W) subpopulations are (6, 6, 4, 1, 3M+6)/(3M+23). With second-order punishment, pool-punishers win, and the corresponding frequencies are (0, 0, 0, 1, 0).
Repression of free-riding is a basic theme for several major transitions in evolution18, and can lead to evolutionarily stable strategies allocating part of the contribution towards suppressing competition19. In human societies, sanctions are ubiquitous4, 16, 20, 21.
Peer-punishment emerges more easily than pool-punishment, because it
requires no second-order punishment, and inequality (2) is weaker than
inequality (1). But with second-order sanctions, pool-punishment
out-competes peer-punishment, despite being socially expensive. Both
types of punishment only emerge, in our model, if players can opt out of
the joint enterprise. This restricts the range of applications22, 23. However, there is considerable evidence that cooperation can increase, if participation is voluntary rather than compulsory24, 25, 26 (see Supplementary Information for an intuitive explanation).
Many
early experiments on public goods with punishment terminated after six
or ten rounds, and although punishment usually increased the propensity
to cooperate, the overall income was often less than without punishment2, 8, 9. But if the number of rounds is sufficiently large, cooperation becomes common3. As long as players avoid antisocial punishment of contributors5
(a feature not included in our model), peer-punishment becomes cost
free. Pool-punishment entails fixed costs and thus is less efficient.
However, peer-punishment is ill-suited for second-order punishment, as
has also been observed empirically27.
Pool-punishment is more conducive to second-order punishment. A
sanctioning institution should view anyone not contributing to its
upkeep as a defector and resort to second-order punishment. Adding
second-order punishment may add to the cost of sanctioning, but as long
as inequality (1) holds, the results are unaffected.
Experimental PGGs allowing players to opt, from round to round, between treatments with or without peer-punishment28, or to vote on whether to forbid antisocial punishment29,
suggest intermediary stages towards pool-punishment. Further steps
towards endogenous institution formation are analysed in refs 23, 30. We considered players motivated entirely by self-interest, and did not assume preferences for reciprocity or equity21.
This obviously does not mean that such preferences do not exist. Their
emergence may actually have been favoured by the prevalence of
sanctioning institutions over thousands of years.
We left out many
important issues, such as quorum-sensing and signalling, reputation and
opportunism, repeated interactions and graduated punishment, and did
not specify how pool-punishment is actually set up. Our model is
minimalistic, but allows proof of principle. Origins of institutions are
notoriously difficult to trace, but we have shown that they can emerge
spontaneously among self-interested individuals.
We apply evolutionary game theory to populations of fixed size, M, and variable composition, X, Y, Z, V and W
(the numbers of players using the five strategies for the optional PGG
with peer- or pool-punishment). We compute the pay-offs obtained by
players using these strategies. The pay-off differences define the
probabilities that the strategies are copied through social learning, as
a function of a parameter, s≥0, measuring ‘imitation strength’. Together with an ‘exploration rate’, μ≥0,
which specifies the propensity to switch randomly to another strategy,
this defines a stochastic process describing the evolution of the
frequencies X, Y, Z, V and W. We
compute their stationary distributions (which correspond to the relative
frequencies in the long run) both numerically and, in a limiting case,
analytically, and check these values by individual-based simulations.
This allows us to compare the evolution of any subset of the five
strategies under social learning. For further details, see Supplementary Information.
Lehmann, L., Rousset, F., Roze, D. & Keller, L.Strong reciprocity or strong ferocity? A population genetic view of the evolution of altruistic punishment. Am. Nat.170, 21–36 (2007)
Hauert, C., Traulsen, A., Nowak, M. A. & Brandt, H. H. &. Sigmund, K. Via freedom to coercion: the emergence of costly punishment. Science316, 1905–1907 (2007)
Ertan, A., Page, T. & Putterman, L.Who to punish? Individual decisions and majority rule in mitigating the free rider problem. Eur. Econ. Rev.53, 495–511 (2009)
Faculty of Mathematics, University of Vienna, A-1090 Vienna, Austria
Karl Sigmund
International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria
Karl Sigmund
WU (Vienna University of Economics and Business), A-1090 Vienna, Austria
Hannelore De Silva
Max Planck Institute for Evolutionary Biology, 24306 Ploen, Germany
Arne Traulsen
Department of Mathematics, University of British Columbia, Vancouver, British ColumbiaV6T 1Z2, Canada
Christoph Hauert
Contributions
All authors were involved in the design and analysis of the model.
H.D.S. and C.H. ran the simulations, A.T. and C.H. did the numerical
analysis, and K.S. wrote the paper.
Competing financial interests
The authors declare no competing financial interests.
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Cooperation
in evolutionary games can be stabilized by the punishment of
non-cooperators, at a cost to those doing the punishing. It can take
various forms, including peer punishment, where individuals punish
free-riders after the event, and pool punishment, where a fund for
sanctioning is set up beforehand. The former can be summed up as 'taking
the law into your own hands', the latter as having police to do the
job. Using a computational model, Sigmund et al. show that pool
punishment has the edge over peer punishment in dealing with
second-order free-riders — those who cooperate in the main game but
refuse to contribute to punishment. The model shows that individuals can
spontaneously adopt a self-governing institution to monitor
contributions and sanction free-riders. It needs no top-down
prescriptions or planning. Trial and error, and the imitation of
successful examples, suffice to produce a social contract between
individuals guided by self-interest.
Cooperation in evolutionary games can be
stabilized by the punishment of non-cooperators, at a cost to those
doing the punishing. It can take various forms, including peer
punishment, where individual…
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