, 1560 (2006);
et al. Martin A. Nowak,
Five Rules for the Evolution of Cooperation
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Five Rules for the Evolution
Martin A. Nowak
Cooperation is needed for evolution to construct new levels of organization. Genomes, cells,
multicellular organisms, social insects, and human society are all based on cooperation. Cooperation
means that selfish replicators forgo some of their reproductive potential to help one another. But
natural selection implies competition and therefore opposes cooperation unless a specific mechanism
is at work. Here I discuss five mechanisms for the evolution of cooperation: kin selection, direct
reciprocity, indirect reciprocity, network reciprocity, and group selection. For each mechanism, a simple
rule is derived that specifies whether natural selection can lead to cooperation.
every cell, and every organism should be de-
signed to promote its own evolutionary success
at the expense of its competitors. Yet we ob-
serve cooperation on many levels of biolog-
ical organization. Genes cooperate in genomes.
Chromosomes cooperate in eukaryotic cells.
Cells cooperate in multicellular organisms. There
are many examples of cooperation among ani-
mals. Humans are the champions of cooperation:
From hunter-gatherer societies to nation-states,
cooperation is the decisive organizing principle
of human society. No other life form on Earth is
engaged in the same complex games of cooper-
ation and defection. The question of how natural
selection can lead to cooperative behavior has
fascinated evolutionary biologists for several
A cooperator is someone who pays a cost,
c, for another individual to receive a benefit,
b. A defector has no cost and does not deal
out benefits. Cost and benefit are measured in
terms of fitness. Reproduction can be genetic
or cultural. In any mixed population, defectors
have a higher average fitness than cooperators
(Fig. 1). Therefore, selection acts to increase
the relative abundance of defectors. After some
time, cooperators vanish from the population.
Remarkably, however, a population of only
cooperators has the highest average fitness,
whereas a population of only defectors has
the lowest. Thus, natural selection constantly
reduces the average fitness of the popula-
tion. Fisher’s fundamental theorem, which
states that average fitness increases under
constant selection, does not apply here be-
cause selection is frequency-dependent: The
fitness of individuals depends on the fre-
quency (= relative abundance) of cooperators in
the population. We see that natural selection in
volution is based on a fierce competition
between individuals and should therefore
reward only selfish behavior. Every gene,
well-mixed populations needs help for establish-
When J. B. S. Haldane remarked, “I will jump
into the river to save two brothers or eight
cousins,” he anticipated what became later known
as Hamilton’s rule (1). This ingenious idea is that
natural selection can favor cooperation if the
donor and the recipient of an altruistic act are
genetic relatives. More precisely, Hamilton’s rule
states that the coefficient of relatedness, r, must
exceed the cost-to-benefit ratio of the altruistic act:
r > c/b (1)
Relatedness is defined as the probability of
sharing a gene. The probability that two brothers
share the same gene by descent is 1/2; the same
probability for cousins is 1/8. Hamilton’s theory
became widely known as “kin selection” or
“inclusive fitness” (2–7). When evaluating the
fitness of the behavior induced by a certain gene,
it is important to include the behavior’s effect on
kin who might carry the same gene. Therefore,
the “extended phenotype” of cooperative behav-
ior is the consequence of “selfish genes” (8, 9).
It is unsatisfactory to have a theory that can ex-
plain cooperation only among relatives. We also
observe cooperation between unrelated indi-
viduals or even between members of different
species. Such considerations led Trivers (10) to
propose another mechanism for the evolution of
cooperation, direct reciprocity. Assume that
there are repeated encounters between the same
two individuals. In every round, each player has
a choice between cooperation and defection. If I
cooperate now, you may cooperate later. Hence,
framework is known as the repeated Prisoner’s
But what is a good strategy for playing this
game? In two computer tournaments, Axelrod
(11) discovered that the “winning strategy”
was the simplest of all, tit-for-tat. This strat-
egy always starts with a cooperation, then it
does whatever the other player has done in the
previous round: a cooperation for a coopera-
tion, a defection for a defection. This simple
concept captured the fascination of all enthu-
siasts of the repeated Prisoner’s Dilemma.
Many empirical and theoretical studies were
inspired by Axelrod’s groundbreaking work
But soon an Achilles heel of the world
champion was revealed: If there are erroneous
moves caused by “trembling hands” or “fuzzy
minds,” then the performance of tit-for-tat de-
clines (15, 16). Tit-for-tat cannot correct mis-
takes, because an accidental defection leads to a
long sequence of retaliation. At first, tit-for-tat
was replaced by generous-tit-for-tat (17), a strat-
egy that cooperates whenever you cooperate,
but sometimes cooperates although you have
defected [with probability 1 − (c/b)]. Natural
selection can promote forgiveness.
Subsequently, tit-for-tat was replaced by
win-stay, lose-shift, which is the even simpler
idea of repeating your previous move when-
ever you are doing well, but changing other-
wise (18). By various measures of success,
win-stay, lose-shift is more robust than either
tit-for-tat or generous-tit-for-tat (15, 18). Tit-
for-tat is an efficient catalyst of cooperation in a
society where nearly everybody is a defector,
but once cooperation is established, win-stay,
lose-shift is better able to maintain it.
Program for Evolutionary Dynamics, Department of Or-
ganismic and Evolutionary Biology, and Department of
Mathematics, Harvard University, Cambridge, MA 02138,
USA. E-mail: email@example.com
Declining average fitness
Fig. 1. Without any mechanism for the evolution of cooperation, natural selection favors defectors. In a
mixed population, defectors, D, have a higher payoff (= fitness) than cooperators, C. Therefore, natural
selection continuously reduces the abundance, i, of cooperators until they are extinct. The average
fitness of the population also declines under natural selection. The total population size is given by N. If
there are i cooperators and N − i defectors, then the fitness of cooperators and defectors, respectively,
is given by fC= [b(i − 1)/(N − 1)] − c and fD= bi/(N − 1). The average fitness of the population is given
by ‾ f = (b − c)i/N.
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The number of possible strategies for the
repeated Prisoner’s Dilemma is unlimited, but
a simple general rule can be shown without
any difficulty. Direct reciprocity can lead to the
evolution of cooperation only if the probability,
w, of another encounter between the same two
individuals exceeds the cost-to-benefit ratio of
the altruistic act:
w > c/b(2)
Direct reciprocity is a powerful mechanism
for the evolution of cooperation, but it leaves
out certain aspects that are particularly impor-
tant for humans. Direct reciprocity relies on
repeated encounters between the same two
individuals, and both individuals must be able
to provide help, which is less costly for the
donor than it is beneficial for the recipient.
But often the interactions among humans are
asymmetric and fleeting. One person is in a
position to help another, but there is no possi-
who are in need. We donate to charities that do
notdonatetous.Directreciprocityislike a barter
economy based on the immediate exchange of
invention of money. The money that fuels the
engines of indirect reciprocity is reputation.
Helping someone establishes a good reputa-
tion, which will be rewarded by others. When
deciding how to act, we take into account the
possible consequences for our reputation. We
feel strongly about events that affect us directly,
but we also take a keen interest in the affairs of
others, as demonstrated by the contents of
In the standard framework of indirect rec-
iprocity, there are randomly chosen pairwise
encounters where the same two individuals
need not meet again. One individual acts as
donor, the other as recipient. The donor can
decide whether or not to cooperate. The inter-
action is observed by a subset of the popu-
lation who might inform others. Reputation
allows evolution of cooperation by indirect
reciprocity (19). Natural selection favors strat-
egies that base the decision to help on the
reputation of the recipient. Theoretical and em-
pirical studies of indirect reciprocity show that
people who are more helpful are more likely to
receive help (20–28).
Although simple forms of indirect reciprocity
can be found in animals (29), only humans seem
to engage in the full complexity of the game.
Indirect reciprocity has substantial cognitive
demands. Not only must we remember our own
interactions, we must also monitor the ever-
changing social network of the group. Language
is needed to gain the information and spread the
gossip associated with indirect reciprocity. Pre-
sumably, selection for indirect reciprocity and
human language has played a decisive role in
the evolution of human intelligence (28). Indirect
reciprocity also leads to the evolution of morality
(30) and social norms (21, 22).
The calculations of indirect reciprocity are
complicated and only a tiny fraction of this uni-
verse has been uncovered, but again a simple
rule has emerged (19). Indirect reciprocity can
only promote cooperation if the probability, q,
of knowing someone’s reputation exceeds the
cost-to-benefit ratio of the altruistic act:
q > c/b(3)
The argument for natural selection of defection
(Fig. 1) is based on a well-mixed population,
where everybody interacts equally likely with
everybody else. This approximation is used by
all standard approaches to evolutionary game
dynamics (31–34). But real populations are not
well mixed. Spatial structures or social net-
works imply that some individuals interact
more often than others. One approach of cap-
turing this effect is evolutionary graph theory
(35), which allows us to study how spatial struc-
ture affects evolutionary and ecological dy-
The individuals of a population occupy the
vertices of a graph. The edges determine who
interacts with whom. Let us consider plain
cooperators and defectors without any strategic
complexity. A cooperator pays a cost, c, for
each neighbor to receive a benefit, b. Defec-
tors have no costs, and their neighbors receive
no benefits. In this setting, cooperators can
prevail by forming network clusters, where
they help each other. The resulting “network
reciprocity” is a generalization of “spatial rec-
Games on graphs are easy to study by com-
puter simulation, but they are difficult to analyze
mathematically because of the enormous num-
ber of possible configurations that can arise.
Nonetheless, a surprisingly simple rule deter-
mines whether network reciprocity can favor
cooperation (41). The benefit-to-cost ratio must
exceed the average number of neighbors, k, per
b/c > k(4)
Selection acts not only on individuals but also
on groups. A group of cooperators might be
more successful than a group of defectors. There
have beenmany theoretical and empirical studies
of group selection, with some controversy, and
recently there has been a renaissance of such
ideas under the heading of “multilevel selection”
A simple model of group selection works as
follows (51). A population is subdivided into
groups. Cooperators help others in their own
group. Defectors do not help. Individuals re-
produce proportional to their payoff. Offspring
are added to the same group. If a group reaches
a certain size, it can split into two. In this case,
another group becomes extinct in order to con-
strain the total population size. Note that only
individuals reproduce, but selection emerges
on two levels. There is competition between
groups because some groups grow faster and
split more often. In particular, pure cooperator
groups grow faster than pure defector groups,
whereas in any mixed group, defectors re-
produce faster than cooperators. Therefore, se-
lection on the lower level (within groups) favors
defectors, whereas selection on the higher level
(between groups) favors cooperators. This model
is based on “group fecundity selection,” which
means that groups of cooperators have a higher
rate of splitting in two. We can also imagine a
model based on “group viability selection,”
Fig. 2. Evolutionary dynamics of cooperators
and defectors. The red and blue arrows indicate
selection favoring defectors and cooperators,
respectively. (A) Without any mechanism for the
evolution of cooperation, defectors dominate. A
mechanism for the evolution of cooperation can
allow cooperators to be the evolutionarily stable
strategy (ESS), risk-dominant (RD), or advanta-
geous (AD) in comparison with defectors. (B)
Cooperators are ESS if they can resist invasion by
defectors. (C) Cooperators are RD if the basin of
attraction of defectors is less than 1/2. (D)
Cooperators are AD if the basin of attraction of
defectors is less than 1/3. In this case, the fixa-
tion probability of a single cooperator in a finite
population of defectors is greater than the in-
verse of the population size (for weak selection).
(E) Some mechanisms allow cooperators to domi-
VOL 314 8 DECEMBER 2006
on January 30, 2007
where groups of cooperators are less likely to go
In the mathematically convenient limit of
weak selection and rare group splitting, we ob-
tain a simple result (51): If n is the maximum
group size and m is the number of groups, then
group selection allows evolution of cooperation,
b/c > 1 + (n/m) (5)
Before proceeding to a comparative analysis of
the five mechanisms, let me introduce some
measures of evolutionary success. Suppose a
game between two strategies, cooperators C and
defectors D, is given by the payoff matrix
The entries denote the payoff for the row
player. Without any mechanism for the evolution
of cooperation, defectors dominate cooperators,
which means a < g and b < d. A mechanism for
the evolution of cooperation can change these
1) If a > g, then cooperation is an evo-
lutionarily stable strategy (ESS). An infinitely
large population of cooperators cannot be in-
vaded by defectors under deterministic selec-
tion dynamics (32).
2) If a + b > g + d, then cooperators are
risk-dominant (RD). If both strategies are
ESS, then the risk-dominant strategy has the
bigger basin of attraction.
3) If a + 2b > g + 2d, then cooperators are
advantageous (AD). This concept is important
for stochastic game dynamics in finite pop-
ulations. Here, the crucial quantity is the fix-
ation probability of a strategy, defined as the
probability that the lineage arising from a
single mutant of that strategy will take over
the entire population consisting of the other
strategy. An AD strategy has a fixation proba-
bility greater than the inverse of the popu-
lation size, 1/N. The condition can also be
expressed as a 1/3 rule: If the fitness of the in-
vading strategy at a frequency of 1/3 is greater
than the fitness of the resident, then the fix-
ation probability of the invader is greater than
1/N. This condition holds in the limit of weak
A mechanism for the evolution of cooper-
ation can ensure that cooperators become
ESS, RD, or AD (Fig. 2). Some mechanisms
even allow cooperators to dominate defectors,
which means a > g and b > d.
We have encountered five mechanisms for the
evolution of cooperation (Fig. 3). Although the
mathematical formalisms underlying the five
mechanisms are very different, at the center of
each theory is a simple rule. I now present a
coherent mathematical framework that allows
the derivation of all five rules. The crucial idea
is that each mechanism can be presented as a
game between two strategies given by a 2 × 2
payoff matrix (Table 1). From this matrix, we
can derive the relevant condition for evolution
For kin selection, I use the approach of
inclusive fitness proposed by Maynard Smith
(31). The relatedness between two players is r.
Therefore, your payoff multiplied by r is added
to mine. A second method, shown in (53), leads
to a different matrix but the same result. For
direct reciprocity, the cooperators use tit-for-tat
while the defectors use “always-defect.” The
expected number of rounds is 1/(1 − w). Two
tit-for-tat players cooperate all the time. Tit-for-
tat versus always-defect cooperates only in the
first move and then defects. For indirect rec-
iprocity, the probability of knowing someone’s
reputation is given by q. A cooperator helps
unless the reputation of the other person in-
dicates a defector. A defector never helps. For
network reciprocity, it can be shown that the
expected frequency of cooperators is described
by a standard replicator equation with a trans-
formed payoff matrix (54). For group selection,
the payoff matrices of the two games—within
Fig. 3. Five mechanisms for the evolution of
cooperation. Kin selection operates when the
donor and the recipient of an altruistic act are
genetic relatives. Direct reciprocity requires re-
peated encounters between the same two individ-
uals. Indirect reciprocity is based on reputation; a
helpful individual is more likely to receive help.
Network reciprocity means that clusters of coop-
erators outcompete defectors. Group selection is
the idea that competition is not only between
individuals but also between groups.
Table 1. Each mechanism can be described by a simple 2 × 2 payoff matrix, which specifies the
interaction between cooperators and defectors. From these matrices we can directly derive the nec-
essary conditions for evolution of cooperation. The parameters c and b denote, respectively, the cost
for the donor and the benefit for the recipient. For network reciprocity, we use the parameter H =
[(b − c)k − 2c]/[(k + 1)(k − 2)]. All conditions can be expressed as the benefit-to-cost ratio
exceeding a critical value. See (53) for further explanations of the underlying calculations.
Cooperation is …
0) 1 (
) 1 (
w…probability of next round
k…number of neighbors
m…number of groups
ESS RD AD
8 DECEMBER 2006 VOL 314
on January 30, 2007
and between groups—can be added up. The
details of all these arguments and their limi-
tations are given in (53).
For kin selection, the calculation shows that
Hamilton’s rule, r > c/b, is the decisive criterion
for all three measures of evolutionary success:
ESS, RD, and AD. Similarly, for network rec-
iprocity and group selection, we obtain the
same condition for all three evaluations, name-
ly b/c > k and b/c > 1 + (n/m), respectively.
The reason is the following: If these con-
ditions hold, then cooperators dominate defec-
tors. For direct and indirect reciprocity, we
find that the ESS conditions lead to w > c/b
and q > c/b, respectively. Slightly more strin-
gent conditions must hold for cooperation to be
RD or AD.
Each of the five possible mechanisms for the
evolution of cooperation—kin selection, direct
reciprocity, indirect reciprocity, network reci-
procity and group selection—can be described
by a characteristic 2 × 2 payoff matrix, from
which we can directly derive the fundamental
rules that specify whether cooperation can
evolve (Table 1). Each rule can be expressed
as the benefit-to-cost ratio of the altruistic act
being greater than some critical value. The
payoff matrices can be imported into standard
frameworks of evolutionary game dynamics.
For example, we can study replicator equations
for games on graphs (54), for group selection,
and for kin selection. This creates interesting
new possibilities for the theory of evolutionary
I have not discussed all potential mechanisms
for the evolution of cooperation. An interest-
ing possibility is offered by “green beard” mod-
els where cooperators recognize each other via
arbitrary labels (56–58). Another way to obtain
cooperation is making the game voluntary rather
than obligatory: If players can choose to cooper-
ate, defect, or not play at all, then some level of
cooperation usually prevails in dynamic oscil-
lations (59). Punishment is an important factor
that can promote cooperative behavior in some
situations (60–64), but it is not a mechanism for
the evolution of cooperation. All evolutionary
models of punishment so far are based on un-
derlying mechanisms such as indirect reciprocity
(65), group selection (66, 67), or network reci-
procity (68). Punishment can enhance the level of
cooperation that is achieved in such models.
Kin selection has led to mathematical the-
ories (based on the Price equation) that are
more general than just analyzing interactions
between genetic relatives (4, 5). The interacting
individuals can have any form of phenotypic
correlation. Therefore, kin selection theory also
anisms for the evolution of cooperation (69, 70).
The two fundamental principles of evolution
are mutation and natural selection. But evolution
is constructive because of cooperation. New
levels of organization evolve when the compet-
ing units on the lower level begin to cooperate.
Cooperation allows specialization and thereby
promotes biological diversity. Cooperation is the
secret behind the open-endedness of the evolu-
tionary process. Perhaps the most remarkable
aspect of evolution is its ability to generate co-
operation in a competitive world. Thus, we
might add “natural cooperation” as a third fun-
damental principle of evolution beside mutation
and natural selection.
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