Behavioral Ecology Vol. 12 No. 3: 313-317
© 2001 International Society for Behavioral Ecology
The importance of phenotypic defectors in stabilizing reciprocal altruism
a Department of Biological Sciences, University of Durham, South Road, Durham DH1 3LE, UK b Evolution and Behaviour Research Group, Department of Psychology, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, UK
Address correspondence to T.N. Sherratt. E-mail: t.n.sherratt{at}durham.ac.uk
Received 10 June 2000; revised 5 September 2000; accepted 6 September 2000.
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ABSTRACT |
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At any one time, a population is likely to contain individuals that are either permanently incapable of cooperating or temporarily lack the time, energy, or resources to allow them to act altruistically. These individuals have been called "phenotypic defectors." We show that, rather than prevent cooperation from emerging, these individuals are extremely important to the stability of reciprocal altruism because they prevent the drift toward increasing naivete that is generally associated with highly cooperative environments. By exploring a combination of simulation and analytical models, we demonstrate that both permanent and transient phenotypic defectors readily prevent the intermittent collapses of cooperation that have characterized the majority of evolutionary simulations. The incorporation of this natural class of individuals not only suggests that the widespread "bang-bang" dynamics are a modeling artifact, but also highlights the need to reconsider the types of cooperative strategy that we should expect to see in the natural world.
Key words: cooperation, phenotypic defectors, Prisoner's Dilemma, reciprocal altruism.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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Populations that evolve in the absence of a particular risk factor can show a disastrous naivete when this danger is realized (Quammen, 1996
). In the same
way, unconditional cooperators can drift into a population of retaliatory
cooperative strategies such as tit for tat, generating a population that can
be more readily invaded by cheaters (Bendor and Swistak,
1995,
1997). This succession of
cooperative strategies by increasingly "nicer" but more vulnerable
strategies (their "weak stability";
Bendor and Swistak, 1995) is
considered to represent an important threat to the persistence of cooperative
behavior (Pusey and Packer,
1997). The phenomenon applies not only to the family of
tit-for-tat behaviors (e.g., generous tit for tat, tit for two tats, contrite
tit for tat), but to any strategy that responds to a cooperative act with
cooperation, such as Pavlov (Nowak and
Sigmund, 1993) and firm but fair
(Frean, 1994).
One factor that may prevent more trusting strategies from gradually taking
over by drift is the occasional outbreak of more exploitative forms.
Typically, this event is represented in evolutionary simulations by mutation
(e.g., Nowak and Sigmund,
1992,
1993). So long as mutations
are both frequent and large enough, then more trusting, cooperative strategies
may be regularly selected against. However, even under these conditions,
cooperation is often no more than a transient result. Typically, the
simulations show "bang-bang" characteristics of punctuated change,
whereby at any one time almost all members cooperate, or they almost always
defect (Sigmund, 1997). As
Wahl and Nowak (1999) report,
such oscillatory dynamics are an inherent feature of every model of
cooperation (see Nowak and Sigmund,
1989).
As far as we are aware, there is no evidence that this major property of cooperative models, the "boom-and-bust" dynamics, is a realistic feature of cooperative systems. Therefore, the instability that has been reported may simply reflect a failure of the models to capture an important element of cooperation in the real world. Here we argue that cooperation's toehold need not be so tenuous: stable cooperation is likely if we introduce an additional simple but realistic condition into the traditional models. Furthermore, we propose that the cooperative strategies that evolve under these more stable conditions are likely to behave in different ways from the strategies that have so far been considered to be evolutionarily successful.
The condition that we have introduced is to assume that at any one time
natural populations are likely to contain a number of individuals that are
unable to cooperate, even if they are otherwise predisposed to doing so
(Roberts and Sherratt, 1998).
In an insightful paper, Lotem et al.
(1999) recognized the potential
importance of these individuals to the evolution of cooperation and called
them "phenotypic defectors." Phenotypic defectors will consist of
individuals that are permanently incapable of cooperating, such as the infirm,
but they are also likely to include a much wider class of individuals that
temporarily lack the time, energy, resources, or ability to help others. Lotem
et al. (1999) argued that
permanent phenotypic defectors can help to promote the evolution of
cooperation based on indirect reciprocity. In this paper we extend this work
by demonstrating that both permanent and transient forms of phenotypic
defection can help maintain the stability of cooperative interactions that are
based on direct reciprocal altruism.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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We evaluated the importance of phenotypic defectors in maintaining the long-term stability of altruism using two simple models. Both models assumed that cooperative exchanges between pairs of individuals are asynchronous and perfectly alternating (Frean, 1994
; Nowak and Sigmund,
1994) and that participating individuals simply decide to
cooperate (C) or defect (D) with their partner in a given round
(Axelrod and Hamilton, 1981;
Trivers, 1971). Defection
incurs no cost to the exploiter, but every cooperative act involves a cost,
c, to the altruist and a benefit, b, to the recipient.
Following Frean (1994) and
Nowak and Sigmund (1994), we
assumed that the cooperative strategies adopted by each individual could be
determined primarily by four contingent probabilities,
pcc, pcd, pdc, and
pdd. These probabilities represented the probability of
cooperating with a given individual, dependent on the recent history of
interactions with that partner (e.g., pcd is the
probability of cooperating if you had previously cooperated, and your partner
had subsequently defected). Both of the earlier papers
(Frean, 1994;
Nowak and Sigmund, 1994)
assumed an infinitely long iterated game so that the initial interactions
could be effectively ignored, but in our model we assumed three additional
probabilities (popen, pc,
pd) that allowed us to consider games of finite length
(specifically, R pairwise exchanges between partners). Thus,
popen represented the probability of cooperating with a
stranger, whereas pc and pd represented the
probability of cooperating with an individual when there had been only one
previous cooperative interaction with this individual and it had cooperated or
defected, respectively. As usual, all initial probabilities were set at 0.5
(random), new strategies were introduced through mutation (± 0.01
change to individual contingent probabilities with probability 0.01 per
iteration), and individuals that played strategies that gave them a relatively
high pay-off contributed more offspring to the next generation. Specifically,
we assumed that those individuals whose lifetime pay-off ranked in the top 25%
produced 40% of the offspring in the next generation, the next 25% contributed
30%, the third 25% contributed 20%, and the 25% poorest performers contributed
only 10% of offspring.
Our first model made the standard assumption that individuals are not constrained by the cumulative cooperative costs they incur. To investigate the effects of phenotypic defectors in this system, we simply considered the effects of introducing an additional fixed number of individuals within the population that were permanently incapable of cooperating. In contrast, our second model generated transient phenotypic defectors intrinsically. To do this, we began by simply assuming that each individual had a limited resource capacity (Qmax identical for all individuals) and that the resource level (Qi) of each individual was reduced by g every time they carried out a cooperative act. Crucially, the structure of this model dictated that individuals with Qi < g could not cooperate, even if they were predisposed to doing so. To ensure that individuals that ran out of resources were not permanently unable to cooperate, the resource levels of each individual were replenished with fixed probability prep before each interaction with their partners.
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RESULTS |
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Permanent phenotypic defectors
Figure 1 shows the results of a typical simulation of the basic model in the absence of any phenotypic defectors. In all simulations for b >> c, cooperation readily emerges (Figure 1A), but as in previous work, it is undermined when the retaliatory components of the strategy set (in this case pd, pcd, pdd) become latent and consequently drift. Once pcd drifts toward 1 (high probability of cooperating with an individual that responds to cooperation with defection), then this increased naivete (lack of retaliation) invariably leads to selection for more exploitative strategies (Figure 1B). The end result is the familiar "bang-bang" fluctuations involving all-or-nothing cooperation (Figure 1A).
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Figure 2 shows a simulation of the alternating game conducted under identical conditions to Figure 1, but assumes that there are an additional 10 individuals who do not cooperate under any circumstances. These individuals were replaced each generation and were excluded from the payoff-dependent reproduction algorithm. Clearly, the continued presence of phenotypic defectors prevents drift toward increasing naivete (Figure 2B), which in turn promotes the long-term stability of altruism (Figure 2A). Repeated long-term simulations with 100 potentially cooperative individuals showed that this result was extremely robust to the number of phenotypic defectors that were also present: highly stable cooperation arose when there were at least as many as 120 (the maximum tested) or as few as 2 additional phenotypic individuals in the population at any one time. Similarly, the introduction of a small probability of mistakes each interaction (0.01, 0.02 or 0.05), whereby an individual about to cooperate instead defects (or vice versa), had no discernable influence on the qualitative dynamics: bang-bang oscillations were still seen in the absence of phenotypic defectors, but stable cooperation emerged when they were present.
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To understand the effect of phenotypic defectors in maintaining altruism, consider the long-term sustainability of a cooperative but retaliatory strategy such as tit for tat (pcc = 1, pcd = 0, pdc = 1, pdd = 0). As we have seen, in the absence of phenotypic defectors, less retaliatory cooperators (pcd > 0) can readily drift into such a population, ultimately paving the way for a collapse of cooperation via selection for exploiters. However, if there are any phenotypic defectors present in the population, a reciprocating strategy, once established, will always be at a selective advantage over more naive cooperators (see Table 1). Similarly, it is clear that phenotypic defectors do not significantly undermine the ability of retaliatory cooperative strategies to persist. For instance, if R = 10, b = 4, c = 1, then tit for tat would always gain greater payoff than a defector if there were more than 1.9% tit for tat (i.e., < 98.1% phenotypic defectors) in the population.
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In the periods of the simulations in which altruism did arise in the
absence of phenotypic defectors, the strategies were reminiscent of a strategy
that is considered to perform well in the alternating game firm but fair
(cooperate except when you have been taken advantage of;
pcc = 1, pcd = 0,
pdc = 1, pdd >> 0;
Frean, 1994). However, as
Figure 1B shows, this was not
always the case: pcd often appeared to drift at values
much greater than 0, while pdc and pdd
similarly drifted at values less than 1, due to the latency of retaliatory
strategies. When phenotypic defectors were present, the final competitively
successful strategies consistently emerged with pcc 
1, pcd 0, but in contrast to firm but fair,
pdd consistently remained close to 0, while
pdc was either variable (again, due to latency), or fixed
at 1 (when popen 0 then invariably
pd 1 and pdc 1, such that
individuals can continue to cooperate after an opening defection).
Transient phenotypic defectors
Once again, so long as b >> c, highly stable levels of
cooperation tended to emerge in this system
(Figure 3). Although this was a
robust result, the stability did not arise at extreme values of
Qmax and g. For example, when
Qmax was very large relative to g (e.g.,
Qmax = 1000, g = 1), very few transient
phenotypic defectors were generated, and cooperation underwent the familiar
bang-bang oscillations. Similarly, when Qmax was very
small relative to g (e.g., Qmax = 2, g =
1), very little genuine cooperation took place, and the competitively
successful strategies fluctuated considerably over time.
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Although the genotypes that evolved under less extreme values of
Qmax and g were always such that
pcc 1, the actual proportion of genuinely altruistic
events over all interactions was sometimes relatively small, in part because
of a high proportion of cooperative exchanges that were prevented due to lack
of resources (Figure 3). As
before, the strategies that emerged in the presence of this natural source of
noncooperators were consistently retaliatory (pcd 0)
and were not conciliatory (pdd 0). Similarly,
pdc either evolved to 1 (when popen
0, then pd 1 and pc was
variable so that in order to cooperate, an individual must overlook an initial
defection) or fluctuated due to latency (when popen
1, then pc 1 and pd was
variable).
Of course, one property of this resource-explicit model is that only
cooperators run out of resources. To introduce an uncorrelated form of
resource limitation, we ran alternative sets of simulations in which
individuals were temporarily (one move) unable to cooperate on a purely random
basis (e.g., probability 0.05). Such an assumption is equivalent to
unidirectional mistakes, and indeed this model consistently generated the same
unstable cooperative dynamics as seen earlier for two-way (C 
D and D
C) mistakes.
To understand why some forms of noncooperation can stabilize reciprocal
altruism, but others fail to stabilize it, consider the simpler case of a
population of tit for tat in which unidirectional mistakes are made with fixed
probability. In a noisy environment, the retaliatory ability of tit for tat
means that partners can easily get locked into rounds of mutual defection.
Under these conditions, more forgiving cooperative strategies may be at a
selective advantage (Godfray,
1992; Nowak and Sigmund,
1992). As we have seen from the simulations, such selection for
naivete may ultimately pave the way for a collapse of cooperation. In
contrast, if mistakes were not "one off" but reflected a longer
term inability to provide help, then users of more forgiving strategies will
tend to get repeatedly taken advantage of if they continued to help these
individuals. In this case, retaliation and cautiousness will continue to be
selected for, thereby maintaining the longer term stability of
cooperation.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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Over the past two decades, great efforts have been made to identify the particular types of cooperative behavior that are likely to be stable in the natural world, but there has been much less consideration of the most common feature associated with these predictions: the "bang-bang" dynamics of punctuated change (Nowak and Sigmund, 1989
; Sigmund,
1997). As the intermittent collapse of cooperative behavior is
such a common property of simulation models (e.g., see
Brauchli et al., 1999; Nowak
and Sigmund, 1992,
1993), it is important to ask
whether this boom and bust is a realistic feature of cooperative systems.
Clearly, if such a property turns out to be an artifact, then we cannot be
entirely confident that the right competitively stable cooperative strategies
have been identified. In this paper we have demonstrated that cooperation need not be a transient result. We show that, paradoxically, the long-term stability of cooperation is considerably enhanced if there are individuals present in a population that cannot (or do not) cooperate. Temporary noncooperators tended to induce the same stable dynamics as permanent phenotypic defectors, so long as the failure to cooperate was not one off (a lack of cooperation in one round signaled a disposition to defect in subsequent exchanges). This finding is particularly important because at any one time there are always likely to be individuals present in natural populations who do not have the energy or resources to help others in return for help provided to them: a vampire bat, for instance, may be unable to reciprocate a blood meal if it has not been successful itself.
While Lotem et al. (1999)
recognized the importance of permanent phenotypic defectors in maintaining the
stability of cooperation based on indirect reciprocity, here we have argued
that a similar, but rather more robust, result holds for the potentially much
broader class of altruistic behaviors based on direct reciprocity. As Boyd and
Lorberbaum (1987) report, when
two strategies such as tit for tat and ALLC (the unconditionally cooperative
strategy) interact with each other, their relative fitness depends on their
interactions with other strategies. Boyd and Lorberbaum argue that because
neither strategy can be best against every possible third strategy, no pure
strategy is evolutionarily stable in the repeated Prisoner's Dilemma game. If
phenotypic defectors were prevalent in the population, however, the relative
fitness of these dominant, equivalent strategies would depend primarily on
their interactions with what is effectively a single strategy, a factor that
would considerably increase the potential of evolutionary stability.
Our work was not conducted with the aim of identifying new and
competitively successful forms of cooperative behavior. However, we note that
incorporating phenotypic defectors into models of reciprocal altruism does not
simply prevent a drift toward increasing naivete, but it can also influence
the type of cooperative strategies that emerge. As we have seen from the
simple simulations we have presented, strategies such as firm but fair, which
were formerly considered successful in the alternating Prisoner's Dilemma game
(Frean, 1994), would never
persist in a population that included either transient or permanent phenotypic
defectors because this strategy exhibits the potentially disastrous behavior
of cooperating after a mutual round of defection. So far, the majority of
models of the evolution of cooperation have not considered constraints on the
ability to cooperate, but instead have assumed that individuals can withdraw
cooperative costs from an inexhaustable bank. It is quite possible that in
making this simplifying assumption, we are ignoring one important reason that
it pays to be cautious when cooperating with others, hence overlooking
significant properties of competitively successful cooperative strategies.
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