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J Evol Biol. 2019;00:1–13.
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1
wileyonlinelibrary.com/journal/jeb
1 | INTRODUCTION
Cooperation among nonkin constitutes a puzzle for evolutionary
biologists, and a large body of theoretical models, inspired by
game theory, have been developed to solve it. The most commonly
accepted explanation is that cooperation can be enforced if it trig‐
gers a conditional response on the part of others (West, Griffin,
& Gardner, 2007). Several enforcement mechanisms have been
proposed: direct reciprocity (Axelrod & Hamilton, 1981; Lehmann
& Keller, 2006; Trivers, 1971), indirect reciprocity (Leimar &
Hammerstein, 2001; Nowak & Sigmund, 1998, 2005), punishment
(Bowles & Gintis, 2004; Boyd, Gintis, Bowles, & Richerson, 2003;
Boyd & Richerson, 1992) and partner choice (Bull & Rice, 1991;
Noë & Hammerstein, 1994, 1995; Sachs, Mueller, Wilcox, & Bull,
2004). A growing number of experimental studies support the
idea that among this set of mechanisms, partner choice is likely
to be particularly influential in nature, both in inter‐specific and
in intra‐specific interactions (Bshary & Schäffer, 2002; Fruteau,
Voelkl, Damme, & Noë, 2009; Hammerstein & Noë, 2016; Kiers
et al., 2011; Kiers, Rousseau, West, & Denison, 2003; Schino &
Aureli, 2009; Simms & Taylor, 2002). Besides, partner choice is
Received:8April2019
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Accepted:14June2019
DOI: 10.1111/je b.13508
RESEARCH PAPER
Why cooperation is not running away
Félix Geoffroy1 | Nicolas Baumard2 | Jean‐Baptiste André2
©2019Europea nSociet yForEvolut ionar yBiolog y.JournalofEvolutionaryBiology©2019EuropeanS ociet yForEvoluti onaryBiolog y
FélixGe offroy,Ni colasBa umardan dJean‐B aptist eAndréco ntrib utedequ allytoth is
work.
1Institut des Sciences de l’Évolution, UMR
5554 ‐ CNRS – Université Montpellier,
Montpellier, France
2Instit utJean‐Nicod(CNR S‐EHESS‐
ENS), Département d’Etudes Cognitives,
Ecole Normale Supérieure, PSL Research
University, Paris, France
Correspondence
Félix Geoffroy, Faculté des Sciences de
Montpellier, Place Eugène Bataillon, 34095
Montpellier, France.
Email: felix.geoffroy.fr@gmail.com
Funding information
Agence Nationale de la Recherche, Grant/
Award Number: ANR‐10‐IDEX‐00 01‐02 PSL
and ANR‐10‐LABX‐0087 IEC
Abstract
A growing number of experimental and theoretical studies show the importance of
partner choice as a mechanism to promote the evolution of cooperation, especially
inhumans.Inthispaper,wefocusonthequestionoftheprecisequantitativelevelof
cooperation that should evolve under this mechanism. When individuals compete to
be chosen by others, their level of investment in cooperation evolves towards higher
values, a process called competitive altruism, or runaway cooperation. Using a clas‐
sic adaptive dynamics model, we first show that when the cost of changing partner
is low, this runaway process can lead to a profitless escalation of cooperation. In
the extreme, when partner choice is entirely frictionless, cooperation even increases
up to a level where its cost entirely cancels out its benefit. That is, at evolutionary
equilibrium,individuals gainthesamepayoffthaniftheyhad notcooperatedatall.
Second, importing models from matching theory in economics we, however, show
that when individuals can plastically modulate their choosiness in function of their
own cooperation level, partner choice stops being a runaway competition to outbid
others and becomes a competition to form the most optimal partnerships. In this
case, when the cost of changing partner tends towards zero, partner choice leads
to the evolution of the socially optimum level of cooperation. This last result could
explain the observation that human cooperation seems to be often constrained by
considerations of social efficiency.
KEY WORDS
biological markets, competitive altruism, human cooperation, matching, models, partner
choice
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GEOFFROY Et al.
also believed to play a major role in human cooperation, where
friendships and coalitions are common (Barclay, 2013, 2016;
Baumard, André, & Sperber, 2013) (see also Discussion).
The key idea of partner choice models is that when one happens
to be paired with a defecting partner, one has the option to seek for
another, more cooperative, partner present in the 'biological mar‐
ket' and interact with her instead of the defector. This possibility
allows cooperators to preferentially interact with each other and,
consequently,preventsanyinvasionbyfree‐riders(Eshel&Cavalli‐
Sforza, 1982; Bull & Rice, 1991; Noë & Hammerstein, 1994, 1995;
Ferriere et al., 2002; Bergstrom, 2003; Aktipis, 20 04, 2011; Sachs et
al., 2004; Fu, Hauert, Nowak, & Wang, 20 08; Barclay, 2011).
So far, the primary objective of most partner choice models has
been to explain how some cooperation can exist at all in an evolu‐
tionaryequilibrium.Onthisground,modelshavereachedaclearan‐
swer: partner choice can trigger the evolution of cooperation. In this
paper, however, we are interested in another issue that models gen‐
erally considerwith lessscrutiny:that ofunderstandingthequanti‐
tative level of cooperation that should evolve under partner choice.
Thisanalysis iscrucialbecausethe quantitative levelofcooper‐
ation determines the 'social efficiency', also called the Pareto effi‐
ciency, of interactions. Cooperating too little is inefficient because
individuals miss some opportunities to generate social benefits.
But cooperation, as any investment, is likely to have diminishing re‐
turns (Altmann, 1979; Killingback & Doebeli, 2002; Weigel, 1981).
As a result, there is a 'socially optimal' amount of cooperation, an
intermediate level where the sum of the helper and helpee's pay‐
off is maximized. Cooperating more than this amount is hence also
inefficient, because it increases more the cost of cooperation than
it raises its benefit. In the extreme, there is even a 'wasteful' thresh‐
old beyond which the overall cost of cooperation becomes larger
than its benefit. If two partners cooperate more than this threshold,
the net benefit of their interaction is negative; that is, they are both
worst off than if they had not cooperated at all.
Prima facie, partner choice appears to be a unidirectional pres‐
sure acting on the evolution of cooperation, unlikely to generate
an interme diate equilibr ium. Compet ition to be chose n by others,
called 'competitive altruism' (Hardy & Van Vugt, 2006; Nesse, 2009;
Roberts, 1998), should lead to a runaway of cooperation, as it does
in sexual selection (West‐Eberhard, 1983). In principle, this runaway
should proceed up to the point where the cost of investing into co‐
operation cancels out the benefit of finding a partner (Fisher, 1930;
West‐Eberhard, 1979) that is up to the 'wasteful' threshold where
cooperation becomes fruitless. Is competitive altruism, however,
balanced by opposite forces, leading to an evolutionary stabilization
of cooperation below this threshold? Is this level socially optimal,
or does partner choice lead to the investment into counterproduc‐
tive forms of cooperation to outcompete others as it does in sexual
selection?
In the theoretical literature on partner choice, relatively little
attentionhasbeengiventothesequestions.Firstofall,alargepro‐
portion of models consider cooperation as an all‐or‐nothing decision
andthuscannotstudy its quantitative level (Eshel&Cavalli‐Sforza,
1982; Bergstrom, 2003; Aktipis, 2004, 2011; Fu et al., 2008; Chen,
Fu, & Wang, 2009; Suzuki & Kimura, 2011; Sibly & Curnow, 2012;
CampennìandSchino,2014;Izquierdo,Izquierdo,&Vega‐Redondo,
2014; Chen, Wu, Li, & Wan g, 2016; Wubs, Bshary, & Lehma nn, 2016).
Second , some models con sider cooper ation as a quanti tative trait
but do not entail diminishing returns, and are thus ill‐suited to study
the social efficiency of cooperative interactions (Foster & Kokko,
2006; Nesse, 2009; Sherratt & Roberts, 1998; Song & Feldman,
2013).Third,still other modelsconsider cooperation as aquantita‐
tive trait with diminishing returns, but they only focus on one side
of the problem ‐the evolution of cooperation‐ considering the other
side ‐the strategy employed by individuals to choose their partner‐
as an exogenous parameter (Wilson & Dugatkin, 1997; Ferriere et al.,
2002; Barclay, 2011; Wild & Cojocaru, 2016).
To our knowledge, only one existing model studies the joint evo‐
lution of co operation an d partner cho ice in a quantitat ive setting
with diminishing returns (McNamara, Barta, Fromhage, & Houston,
2008). However, McNamara et al. (2008) make two key assumptions
thatturnoutto haveimport antconsequences:(i)theyassumethat
variability in the amount of cooperation is maintained owing to a
very large genetic mutation rate on this trait, which prevents natural
selection to act efficiently, and (ii) they restrict the set of possible
strategies to choose one's partner in such a way that individuals can
never do so in an optimal manner.
In this paper, we build a model inspired by McNamara et al.
(2008),inwhichaquantitativelevel ofcooperationexpressedby
individualsjointly evolveswitha quantitative levelof choosiness
regarding others' cooperation, while relaxing these two assump‐
tions. First, we observe that competition to be chosen as a partner
leads to a joint rise of both cooperation and choosiness up to a
level that depends on the efficiency of partner choice that is, in
particular, on the cost of changing partner. The more efficient is
partner choice, the higher cooperation is at evolutionary stability.
Moreover, when the cost of changing partner is low, cooperation
can rise beyond its socially optimal level. In fact, in the limit where
partner choice is entirely frictionless (i.e. the cost of changing
partner is zero), cooperation and choosiness rise up to the 'waste‐
ful threshold' where the cost of cooperation entirely cancels out
its benefit. Individuals gain the same payoff than if they had not
cooperated at all. Hence, at first sight, our analyses show that
partner choice generates no systematic trend towards the socially
optimal level of cooperation.
However, we then import tools from the economics literature
and assume that individuals can plastically modulate their choosi‐
ness in function of their own cooperation level. This plasticity allows
every individual to behave optimally on the biological market, which
did not occur in the first model. In this second approach, we show
that assortative matching emerges. That is, more cooperative indi‐
viduals are also choosier and thus interact with more cooperative
partn ers. As a co nsequence of t his assor tment, and p rovided tha t
partner choice is efficient enough, cooperation evolves to the so‐
cially optimal level, where the mutual efficiency of cooperation is
maximized.
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GEOFFROY Et al .
2 | METHODS
2.1 | Partner choice framework
We model par tner choice in an infinite size population using Debove,
André, and Baumard (2015a)'s framework. Solitary individuals ran‐
domly encounter each other in pairs at a fixed rate
𝛽
. In each en‐
counter, the two players decide whether they accept one another
as a partner (see below how this decision is made). If one of the two
individuals (or both) refuses the interaction, the two individuals im‐
mediately split and move back to the solitary pool. If both individuals
accept each other, on the other hand, the interaction takes place and
lasts for an exponentially distributed duration with stopping rate
𝜏
,
after which the two individuals move back to the solitary pool again.
The ratio
𝛽∕𝜏
thus characterizes the 'fluidity' of the biological mar‐
ket. If
𝛽
is high and
𝜏
is low,individualsmeeteachotherfrequently
and interact for a long time. In such an almost frictionless market,
partner choice is almost cost‐free so they should be choosy about
their partner's investment in cooperation. Conversely, if
𝛽∕𝜏
is low,
individuals rarely meet potential partners and interact for a short
time. In such a market, on the contrary, individuals should accept
any partner.
Regarding the encounter rate, here we assume that
𝛽
is a fixed
constant independent of the density of available partners, an as‐
sumption called 'linear search' that captures a situation in which
already paired individuals do not hinder the encounters of soli‐
tary individuals (Diamond & Maskin, 1979). In the Supplementary
Information, however, using simulations we also analyse the model
under the assumption that
𝛽
increases linearly with the propor‐
tion of solitary individuals in the population, an assumption called
'quadratic search'that corresponds to a situation in which already
matched individuals interfere with the encounters of solitary indi‐
viduals(andthatisalsoequivalenttotheclassicmass‐actionkinetics
used in mathematical epidemiology). In the paper, we only describe
the results obtained under linear search. The results obtained under
quadratic search are qualitatively similar (see the Supplementary
Information).
Regarding the nature of the social interaction, we consider a
quantitativeversion of theprisoner'sdilemmain continuous time.
Each individual
i
is genetically characterized by two traits: her coop‐
eration level
xi
and her choosiness
yi
. Cooperation level
xi
represents
thequantitativeamountofef fortthatanindividual
i
is willing to in‐
vest into cooperation. Choosiness
yi
represents the minimal cooper‐
ation level that an individual
i
is willing to accept from a partner; that
is, every potential partner
j
with cooperation
xj≥yi
will be accepted,
whereas every potential partner with
xj<yi
will be rejected. Once
an interaction is accepted by both players, at every instant of the
interaction, each player invests her effort
xi
(see below for the pay‐
off function), and the interaction lasts in expectation for
1∕𝜏
units of
time, where
𝜏
is the stopping rate of the interaction.
When they are solitary, individuals gain a payoff normalized to
zero per unit of time. When involved into an interaction, they gain a
social payoff that depends on both partners' cooperation level. The
cooperative interaction is a continuous prisoner's dilemma: making
an investment brings benefits to the partner but comes at a cost to
the provider. As stated in the introduction, we make the additional
assumption that cooperation has diminishing returns (Altmann,
1979; Killingback & Doebeli, 2002; Weigel, 1981). This induces the
existence of an intermediate level of cooperation at which the sum
of the partners' gains is maximized, the so‐called 'social optimum'.
An individual
i
paired with
j
gains the following social payoff
Π(
x
i,
x
j)
per unit of time:
Hence, the expected payoff of an individual
i
paired with
j
is
where
𝜏
is the stopping rate of the interaction. The socially optimal
level of cooperation is
̂
x=1∕2c
. Beyond this level, the net benefit of
cooperation decreases. Eventually, the interaction becomes entirely
profitless, or even costly, if individuals invest more than the 'waste‐
ful threshold'
x=1∕c
. We allow both cooperation and choosiness to
take any positive real value.
Previous studies demonstrated that the existence of some vari‐
ability among individuals is necessary to stabilize conditional co‐
operation (Ferriere et al., 2002; Foster & Kokko, 2006; McNamara
et al., 2008; McNamara & Leimar, 2010; Song & Feldman, 2013).
Ifevery possible partner is equallycooperative, thenthere is no
needtobechoosywithregardtothequalityofone'spartner,and
choosiness cannot be evolutionarily stable. In order to capture
the effect of variability in the simplest possible way, we assume
that individuals do not perfectly control their investment into
cooperation (as in Song & Feldman, 2013 and André, 2015, for
instance). An individual's actual cooperation level
xi
is a random
variable which follows a truncated‐to‐zero normal distribution
around the individual's gene value x
i
, with standard deviation
𝜎
.
In what follows, we call cooperation level the genetically encoded
cooperation level that individuals aim for, and 'phenotypic cooper‐
ation' the actual level of cooperation that they express after phe‐
notypic noise. For the sake of simplicity, here, we assume that an
individual's cooperation level is randomized at every encounter.
In the Supplementary Information, however, we also consider the
alternative assumption where phenotypic noise occurs only once
at birth (see also Section 3.1).
We are interested in the joint evolution of cooperation, and
choosiness by natural selection. We undertake and compare the
consequences of two distinc t assumptions. In a first approach,
we assume that both cooperation and choosiness are hard‐wired
traits; that is, each individual is characterized by a single level of
cooperation
̄
x
and a single choosiness
y
, both expressed uncondi‐
tionally. In a second approach, we still assume that cooperation
is a hard‐wired trait, but we consider that choosiness is a reac‐
tion norm by which individuals respond to their own phenotypic
cooperation.
Π
(x
i
,x
j
)=x
j
−cx
2
i
x
j−cx
2
i
𝜏
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GEOFFROY Et al.
2.2 | Hard‐wired choosiness
Here, we assume that each individual is genetically characterized
by two traits: his level of cooperation
̄
x
and his choosiness
y
; and
we are interested in the evolution of these two traits by natural se‐
lection. For this, we need to derive the fecundity of a rare mutant
m
playing strategy
(̄
xm,ym)
in a resident population
r
playing strat‐
egy
(̄
xr,yr)
. The mutant fecundity is proportional to her cumulative
lifetime payoff
Gm
. Without loss of generality, we normalize an in‐
dividual's lifetime to unity, such as the cumulative lifetime payoff
Gm
can be written as (see Supporting information for a detailed
analysis of the model):
where
𝛼m
is the mean probability for an encounter between the mu‐
tant and a resident to be mutually accepted and
̄
Πm
is the mutant
mean social payoff (see Table 1 for a list of the parameters of the
model).Thisexpressionissimilartotheclassicalsequentialencoun‐
ter model of optimal diet (Schoener, 1971).
The evolutionary trajectory of both the level of cooperation
̄
x
and the choosiness
y
can be studied from the analysis of their re‐
spective selection gradient:
We could not derive an analytical expression of the evolution‐
arily stable strategy. However, we numerically computed the se‐
lection gradient on each trait, in order to study the evolutionary
trajectories.
2.3 | Plastic choosiness
Because cooperation is subject to phenotypic noise (i.e. one does
not perfectly control one's own level of cooperation), it could make
sense, at least in principle, for individuals to adapt plastically their
degree of choosiness to the actual phenot ypic cooperation that they
happen to express. For instance, it could make sense for those indi‐
viduals who happen to be phenotypically more generous to be also
choosier, and vice versa. In our second model, we aim to explore the
consequences of this possibility.Todoso, weassume thatchoosi‐
ness is not a hard‐wired trait, but a plastic decision that individuals
take in function of their own phenotypic cooperation. An individual's
'choosiness strategy' is thus defined as a reaction norm rather than
a single value.
Our aim in this second model is to study the joint evolution of
cooperation
̄
x
on the one hand, and of the 'choosiness strategy'
y(x)
, defined as the shape of a reaction norm, on the other hand.
One facet of this problem, therefore, consists in seeking for the
equilib rium choosiness s trategy in a situa tion where both one's
own quality (one's phenotypic cooperation level) and the quality
of one's prospective partners vary. Matching theory, a branch of
micro‐economics, provides tools to resolve this problem. Here,
we briefly explain this approach and show how it applies to our
problem.
In a first category of approaches, called matching models, chang‐
ing partner is assumed to be entirely cost‐free (Becker, 1973; Gale &
Shapley, 1962). That is to say, agents have an infinite amount of time
available to find each other. In this setting, theory shows that there
isauniqueequilibriumchoosinessstrategy:anindividualwithphe‐
notypic cooperation
x
should only accept to interact with individuals
with at least the same phenotypic cooperation level
x
; that is, the
equilibriumreactionnorm isthe identityfunction. This equilibrium
strategy leads to a strictly positive assortative matching in which in‐
dividuals are paired with likes.
G
m=
̄
Πm𝛼m𝛽
𝛼
m
𝛽
+
𝜏
⎧
⎪
⎪
⎨
⎪
⎪
⎩
𝜕Gm
𝜕̄
xm�̄
xm=̄
x
r
ym=y
r
𝜕Gm
𝜕ym�̄
xm=̄
x
r
ym=y
r
Parameter Definition
̄
xi
Cooperation level of individual
i
(mean value
before applying noise)
yi
Choosiness of individual
i
𝜎
Standard deviation of the phenotypic cooperation
distribution
βEncounter rate
𝜏
Split rate
Π(xi,xj)
Social payoff of an individual
i
matched with a
partner
j
c
Cost of cooperation
𝛼i
Mean probability for an individual
i
to interact
when she encounters a resident
̄
Πi
Mean social payoff for an individual
i
interacting
with a resident
Gi
Cumulative lifetime payoff of an individual
i
TABLE 1 Parameters of the model
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5
GEOFFROY Et al .
The secon d category of appr oaches, calle d search and matchin g
models, accounts for frictions in the matching process, that is in‐
corporates an explicit cost for changing partner (Chade, Eeckhout,
& Smith, 2017). These models actually correspond exactly to our
own partner choice framework. Individuals randomly encounter
each other at a given rate, and when an individual refuses an inter‐
action, she has to wait for some time before encountering a new
partner.Unfortunately,theequilibriumchoosinessreactionnorm
y∗(x)
cannot be analytically derived in these models. However,
Smith (2006) has shown that a mathematical property of the social
payoff function
Π(
x
i,
x
j)
allows predicting the shape of this reac‐
tion norm. If the social payoff function
Π(
x
i,
x
j)
is strictly log‐super‐
modular, then
y∗(x)
is strictly increasing with
x
. If this is the case,
the more an individual invests into cooperation, the choosier she
should be.Thisequilibriumiscalled aweaklypositiveassortative
matching. Log‐supermodularity is defined as the following:
Π(
x
i,
x
j)
is strictly log‐supermodular only if
Π(
x
i,
x
j)
Π
(
x
k,
x
l)
>Π
(
x
i,
x
l)
Π
(
x
k,
x
j)
for
any investments
xi>xk
and
xj>xl
. Intuitively, in our setting, log‐su‐
permodularity captures the idea that individuals who invest more
into cooperation also benefit relatively more from better partners
(Costinot, 2009). Indeed, by definition, log‐supermodularity im‐
plies that the relative returns to accessing more generous partners
increase in one's level of investment into cooperation:
Π(x
i
,x
j
)
Π
(x
i
,x
l
)>Π
(x
k
,x
j
)
Π(x
k
,x
l)
for
xi>xk
and
xj>xl
. For an exhaustive review on modularity and
matching, see Chade et al. (2017) and Smith (2006).
Matching and search and matching models are, however, only in‐
terestedincharacterizingtheequilibriumchoosinessstrategyofin‐
dividuals, assuming a given, fixed distribution of cooperation levels.
As a result, matching models can offer an insight into the evolution
of choosiness, but not into the joint evolution of choosiness and co‐
operation. To study this joint evolution in the case where choosiness
is a reaction norm, and not a single value, we developed individual‐
based simulations.
2.4 | Individual‐based simulations
In addition to our analytical models, we run individual‐based
simulations coded into Python. We simulate the joint evolution
of cooperation and choosiness in a Wright‐Fisher population of
N
individuals, with the same lifespan
L
and nonoverlapping gen‐
erations. Mutations occur at rate
𝜇
, and mutant genes are drawn
from a normal distribution around the parent's gene value, with
standard deviation
𝜎mut
. Large‐effect mutations are implemented
with probability
𝜇l
. In this case, mutant genes are drawn from a
uniform distribution on the interval
[0,1]
. Large‐effect mutations
donot alter theequilibrium result, and theyallow to speed up
the joint evolution process. We run long enough simulations for
both choosiness and cooperation to stabilize. In contrast to pre‐
vious papers (Foster & Kokko, 2006; McNamara & Leimar, 2010;
Sherratt & Roberts, 1998), here we consider a continuous rather
than discrete trait space, because Sherratt and Roberts (1998)
have shown that too much discretization can produce undesir‐
able consequenceswhen studying a joint evolution process. In
the Supplementary Information, we also present additional simu‐
lations based on a Moran process with overlapping generations,
where the lifespan of individuals is determined by a constant
mortality rate (see also Section 3.1 and McNamara et al., 2008).
We run simulations both under the assumption that choosiness
is hard‐wired and under the assumption that it is a reaction norm.
In the second case, we test two types of reaction norms. First, we
consider polynomial functions, the coefficients of which evolve by
natural selection. Second, we consider step functions with evolving
coefficients coding for the value of choosiness for each interval of
cooperation. In the initial generation, all reaction norms are set to
a constant zero function, so that individuals are never choosy at
initiation.
FIGURE 1 Analytical and numerical results with hard‐wired choosiness. (a) The grey arrows show the vector field of the selection
gradient on both cooperation and choosiness. The red arrows show an evolutionary trajectory from an initial selfish population
(̄
x,y)=(0,0)
uptotheevolutionaryequilibrium.(b)Theredarrowsshowthecorrespondingevolutionofthecumulativelifetimepayoff
G
for a resident
individual. Parameters are
c=1
;
𝜎=0.025
;
𝛽=1
; and
𝜏=0.01
. The socially optimal solution is
̂
x=1∕2
and the interaction becomes profitless if
both individuals invest
x=1
(a) (b)
6
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GEOFFROY Et al.
3 | RESULTS
3.1 | Hard‐wired choosiness
Without variability in cooperation
(𝜎=0)
, there is no selective pres‐
sure to be choosier and, therefore, to be more cooperative. The only
Nash equilibrium is
(̄
x,y)=(0,0)
(see Supporting information for a
demonstration).
When phenotypic cooperation is variable, however, the evolu‐
tionarily stable strategy cannot be formally derived. We therefore
study the joint evolutionary dynamics of cooperation and choosi‐
ness by plotting numerically the selection gradients acting on both
traits. In Figure 1, we show the evolutionary dynamics of coopera‐
tion, choosiness and average payoff, in a case where partner choice
is very effective (high
𝛽∕𝜏
). When starting from an initially selfish
population, cooperation and choosiness jointly rise above zero
(Figure 1). At first, this leads to an increase in the net social payoff
(Figure 1) because cooperation is efficient (i.e. the marginal benefit
of increasing cooperation for the helpee is larger than its marginal
cost for the helper). At some point, however, cooperation reaches
the socially optimal level where the net payoff of individuals is maxi‐
mized. Beyond this level, the marginal cost of increasing cooperation
is larger than the marginal benefit, but the evolutionary runaway
of cooperation and choosiness does not stop. Cooperation keeps
on rising towards higher values, thereby decreasing the net payoff
(Figure 1). Eventually, cooperation and choosiness stabilize when
cooperation is so high, and therefore so inefficient, that its cost
entirely cancels out its benefit (the so‐called 'wasteful threshold').
That is, at ESS, individuals gain the same payoff than if they had not
cooperated at all.
This runaway process, however, only occurs if partner choice
is very efficient. If partner choice has more frictions (low
𝛽∕𝜏
), the
rise of cooperation and choosiness halts at an intermediate level
between 0 and the wasteful threshold. In Figure 2, we plot the
level of cooperation (Figure 2), the level of choosiness (Figure 2)
and the average payoff (Figure 2) reached at evolutionary stability,
in function of the efficiency of partner choice (i.e. in function of
the parameter
𝛽
controlling the fluidity of the social market and
the parameter
𝜎
controlling the extent of phenotypic variability).
As partner choice becomes more efficient, the evolutionarily sta‐
ble cooperation and choosiness monotonously rise from zero up to
the wasteful threshold (Figure 2 and Figure S2). Accordingly, the
net payoff obtained by individuals at evolutionary stability varies
with the efficiency of partner choice in a nonmonotonous way.
Increasing the efficiency of partner choice has first a positive and
then a negative effect on payoff (Figure 2). In the extreme, when
partner choice is frictionless, cooperation and choosiness increase
up to the 'wasteful threshold'
x=1∕c
at which cooperation is
entirely profitless (as shown in Figure 1). Note that in this case,
choosiness is even slightly larger than the 'wasteful threshold' at
equilibriumbecause,due tophenotypicvariability,someindivid‐
uals cooperate beyond
x=1∕c
whichmakesitadaptivetorequest
higher values of cooperation. In fact, when phenotypic variabil‐
ity is too high (large
𝜎
), individuals are so choosy at evolutionary
equilib rium that the equ ilibrium level of co operation is re duced
(Figure 2). These results have been confirmed in individual‐based
simulations (see Supporting information).
The runaway process can be understood intuitively. In any
population, some individuals cooperate more than average, in
FIGURE 2 Analytical results for a range of parameters
withhard‐wiredchoosiness.Equilibriumvaluesareshown
for (a) cooperation, (b) choosiness and (c) cumulative lifetime
payoff as a function of the encounter rate
𝛽
to manipulate the
market fluidity, and for three values of the standard deviation,
𝜎=0.0001; 0.01; and 0.02
respectively for low, medium and high
phenotypic variability. Other parameters are the same as in Figure 1
(a)
(b)
(c)
|
7
GEOFFROY Et al .
particular owing to phenotypic variability. As a result, if part‐
ner choice is sufficiently fluid, it is adaptive to accept only these
hyper‐generous partners. Hence, choosiness increases by natural
selection beyond the average cooperation level. In turn, this fa‐
vours individuals who cooperate more than average; that is, the
mean level of cooperation increases by natural selection, etc. The
extent to which this process goes on depends, however, on the
efficiency of partner choice owing to the existence of a trade‐off
between the cost and benefit of choosiness. The runaway process
stops at the point where the expected benefit of finding a better
partner is not worth the risk of remaining alone.
In our model so far, the cost and benefit of switching partner are
only determined by two parameters (the market fluidity,
𝛽∕𝜏
, and the
amount of phenotypic variability,
𝜎
). Under more realistic biological
assumptions, however, the cost of rejecting a partner should also de‐
pend on other parameters. For instance, one could model mortality
as a stochastic process. The risk of dying while searching for a new
partner would then constitute a supplementary cost of choosiness
(McNamara et al., 2008). In the Supplementary Information, we de‐
velop a model based on a Moran process where individuals are subject
to a constant mortality rate. As expected, ceteris paribus, the run‐
away process results in lower levels of cooperation and choosiness at
evolut io nar yequi li briumwh en themo rta lit yr atei sh ig h.C oo per at io n,
however, still rises beyond the socially optimal level, even up to the
wasteful threshold, if
𝛽
is large and if the mortality rate is not too high.
Also, in our model, so far, we assume that an individual's pheno‐
typic level of cooperation is randomized in every encounter. The
distribution of cooperative types in the solitary population is thus a
fixed and exogenous property. To test the robustness of our results, in
the Supplementary Information, we analyse an alternative case where
the phenotypic level of cooperation of an individual is randomized only
once, at birth. In this case, the distribution of cooperative types in the
solitary population is not an exogenous, fixed, property. More coop‐
erative individuals are less likely to be solitary than average because
they are rapidly accepted as partners (McNamara et al., 2008). Hence,
the population of solitary individuals tends to be biased towards self‐
ish phenotypes. As a result, the cost of being choosy is larger. Yet, in
Supporting information we show that the runaway process still occurs
in this case, including up to the 'wasteful threshold', as long as partner
choice is efficient enough.
Note that Ferriere et al. (2002) and Wild and Cojocaru (2016),
inspired by Barclay (2011), also showed that partner choice could,
under some circumstances, drive the evolution of cooperation up
to a 'wasteful threshold'. However, in both models, the choosiness
strategy was fixed, and not necessarily optimal; it did not evolve
jointly with cooperation. The present results are thus more robust
and general.
3.2 | Plastic choosiness
Here, an individual's choosiness is a reaction norm to her own phe‐
notypic cooperation, and we used search and matching models (see
Section 2.3) to derive the two following predictions regarding the
evolutionarily stable reaction norm:
FIGURE 3 Plastic choosiness at the
equilibrium.Theequilibriumreaction
norms over 30 simulations are shown
in blue, and the corresponding 99%
confident intervals are shown in red with
(a–b) high market fluidity
𝛽=1
, (c–d) low
market fluidity
𝛽=0.01
, (a–c) a polynomial
reaction norm and (b–d) a discrete
reaction norm. The orange dashed
line is the optimal reaction norm for a
frictionless matching market (strong form
of positive assortative matching). The
distribution of phenotypic cooperation at
equilibriumisshowningrey.Parameters
are
c=1
;
𝜎=0.1
;
𝜏=0.01
;
𝜇=0.001
;
𝜎mut =0.05
;
𝜇l=0.05
;
N=300
; and
L=500
(a) (b)
(c) (d)
8
|
GEOFFROY Et al.
1. If the social payoff function is strictly log‐supermodular, an
individual's optimal choosiness is a strictly increasing function
of her own cooperation (weakly positive assortative matching).
2. If the market fluidit y
𝛽∕𝜏
is high, the reaction norm sho uld be close
to
y∗(x)=x∀x
(strictly positive assortative matching).
We first show that our production function
Π
is strictly log‐super‐
modular. Indeed,
Π(xi,xj)
Π
(xk,xl)
>Π
(xi,xl)
Π
(xk,xj)
isequivalentto.
which is true for all
xi>xk
≥
0
and
xj>xl
. Accordingly, search and
matching models show that the optimal choosiness strategy is an
increasing reaction norm; that is, more phenotypically cooperative
individuals should also be choosier, leading to a positive assortative
matching at equilibrium (phenotypically generous individuals are
matched with other generous individuals, and vice versa).
Individual‐based simulations confirm this result. Figure 3 shows
thereactionnorm atevolutionary equilibrium in these simulations:
choosiness is strictly increasing, at least around the levels of pheno‐
typic cooperationthat are actually present at equilibrium.Outside
this range, selection is very weak on the reaction norm, and we ob‐
serve larger confidence intervals. As expected, when the market
tends to be frictionless, the reaction norm becomes very close to
the identity function, that is to a strict positive assortative matching
(Figure 3 and Figure S3 orange dashed line).
Importantly, the evolution of a plastic rather than hard‐wired
choosiness strategy has a key consequence regarding the effi‐
ciency of cooperation at evolutionary equilibrium. In contrast to
the hard‐wired case, when choosiness is plastic, cooperation never
rises above the socially optimal level. As the efficiency of partner
choice (i.e. market fluidity) increases, the level of cooperation at
evolutionary stability increases but, at most, it reaches the socially
optimal level and never more (Figure 4). In particular, when partner
choice is very efficient, cooperation evolves precisely towards the
socially optimal level, that is the level that maximizes the net total
payoff of individuals
(̂
x=1∕2c)
.
This result can also be understood intuitively. In the first model
where choosiness was hard‐wired, it was adaptive to increase one's
cooperation level beyond the population mean because, by doing so,
an individual could s witch from 'being rejected by everyone' to 'being
accepted by everyone'. The runaway process, therefore, proceeded
until cooperation had no benefit at all. In contrast, in the present
model where choosiness is plastic, increasing one's cooperation level
is beneficial because it allows one to access better partners. Hence,
thisisusefulonlyprovidedthebenefitofaccessingahigherquality
partner is larger than the cost of being more cooperative. As a result,
cooperation only rises up to the social optimum, where its net ben‐
efit is maximized.
4 | DISCUSSION
Most theoretical works on the evolution of cooperation by partner
choice aim at explaining how some cooperation can be evolutionarily
stabl e. They do not ai m at unders tanding wh ich specif ic quantit a‐
tive level of cooperation should evolve. In this paper, we have raised
thissecondquestion.Wehave consideredamodelwherecoopera‐
tion has diminishing returns, such that the most efficient level of
cooperation (the level that maximizes social welfare) is intermedi‐
ate. We have investigated whether partner choice can account for
the evolution of an efficient level of cooperation in this case. In this
aim, we have modelled, both numerically and with individual‐based
simulations, the joint evolution of two traits: cooperation, the ef‐
fort invested into helping others, and choosiness, the minimal level
of cooperation that an individual is willing to accept from a partner.
In a first model, we have found that the mechanism of part‐
ner choice entails no systematic force favouring an efficient level
of cooperation. On the contrary, when partner choice is effective
enough, the level of cooperation increases evolutionarily towards
very large values, beyond the socially optimal level. In the extreme,
when partner choice is very effective, cooperation even increases
up to a level where its cost entirely cancels out its benefit. That is,
atevolutionaryequilibrium,individualsgainthesamepayoffthanif
they had not cooperated at all.
To understand intuitively, consider a population with a given
distribution of cooperation levels, with some particularly generous
individuals, some particularly stingy individuals and a given mean
cooperation level. In such a population, provided that the variabil‐
ity of cooperation is sufficiently large and the market sufficiently
fluid, it is always adaptive to accept only partners that are slightly
better than average (McNamara et al., 2008). Hence, natural se‐
lection favours individuals with a choosiness always slightly larger
than the average cooperation level. In turn, this choosiness selects
for mutants whose cooperation level is la rger than the mean, which
(
x
i
−x
k)(
x
j
−x
l)(
x
i
+x
k)
>
0
FIGURE 4 Evolution of cooperation for a polynomial reaction
norm. The average cooperation over 30 simulations is shown
for three values for the encounter rate,
𝛽=0.001; 0.01; and 0.1
respectively for low, medium and high market fluidity. Other
parameters are the same as in Figure 3. The socially optimal
solution is
̂
x=1∕2
and the interaction becomes profitless if both
individuals invest
x=1
|
9
GEOFFROY Et al .
leads to a gradual increase in cooperation. Importantly, this run‐
away process has no particular reason to stop when cooperation
is maximally efficient. Rather, it stops when the cost of searching
for more generous individuals exceeds the benefit of interacting
with them (Figure 2). As long as partner choice is effective (i.e.
the cost of searching is low), it is always worth trying to find a
better‐than‐average partner, irrespective of whether the current
mean level of cooperation is below or beyond the socially optimal
level. Hence, partner choice can prompt individuals to invest into
counterproductive forms of cooperation to outbid others, leading
to an eventually fruitless arms race.
In a second approach, in line with matching models from the eco‐
nomic literature, we have designed a model in which choosiness is
implemented as a reaction norm to the individual's own coopera‐
tion level (see Section 2.3), the shape of which evolves by natural
selection. In this case, both our analytical predictions derived from
search and matching models and our complementary individual‐
based simulations show that the evolutionarily stable reaction norm
is a monotonously increasing function of cooperation (Figure 3). This
implies that more generous individuals are also choosier, leading to a
positive assortative matching: generous individuals tend to interact
with other generous individuals, and vice versa. Furthermore, if the
biological market is fluid enough (i.e. if the cost of changing partner
is low), this positive assortative matching becomes very close to a
perfect matching in which individuals with a given level of coopera‐
tion always interact with other individuals with the exact same level
(Figure 3 and Figure S3).
In this case, and in sharp contrast to the model in which choos‐
iness is a hard‐wired trait, cooperation does not reach the counter‐
productive level where its cost cancels out its benefit when partner
choice is very cheap (Figure 4). More precisely, when the market is
very fluid, the evolutionarily stable cooperation becomes very close
to the social optimum, that is the amount of cooperation that maxi‐
mizes the sum of the partners' payoffs. This can also be understood
intuitively. Because of the strict assortment between cooperative
types, individuals with a given cooperation level interact with other
individuals with the exact same level. Hence, pairs of individuals be‐
come the effective units of selection, like if interactions occurred
among genetic clones (Akçay & Cleve, 2012; Aktipis, 2004; Eshel &
Cavalli‐Sforza, 1982;Wilson & Dugatkin,1997).Consequently,the
socially optimal level of cooperation is favoured.
Hence, the fruitless runaway of cooperation that occurs in a
modelwithhard‐wiredchoosinessisaconsequenceoftheassump‐
tion that individuals cannot optimally adapt their degree of choos‐
iness to local circumstances. If individuals are allowed to behave
optimally, which entails in the present case to adapt plastically their
choosiness to their own generosity, then partner choice looks less
like a competition to outbid others, and more like a competition to
form efficient partnerships with others, which leads to a very differ‐
ent outcome regarding the net benefits of cooperation.
Previous work has shown that assortative matching favours the
evolution of cooperation (Bergstrom, 2003; Eshel & Cavalli‐Sforza,
1982; Hamilton, 1971). For instance, in kin selection, assortment
between relatives drives the evolution of cooperation (Hamilton,
1964; Rousset, 2004). To our knowledge, Wilson and Dugatkin
(1997)firstdiscussedtheconsequencesofassortativematchingfor
the evolution of socially efficient levels of cooperation. Alger and
Weibull (2013, 2016) have studied the evolution of social prefer‐
ences, rather than strategies, under assortative matching. However,
both analyses did not explicitly model a partner choice strategy, let
alone the evolution of this strategy, but merely assumed that assort‐
ment occurs in one way or another. In contrast, here, we have stud‐
ied the joint evolution of choosiness and cooperation, showing how
a positive assortative matching can emerge from a simple partner
choice mechanism.
In another related work, using individual‐based simulations,
McNamara et al. (2008) also observed a form of assortative match‐
ing in the joint evolution of cooperation and choosiness. One of the
main differences with the present approach, however, is that they
assumed that the variability of cooperation is maintained at the ge‐
netic level, via a high mutation rate, rather than at the phenotypic
level. Under this assumption, negative selection on inefficient mu‐
tants(eithertoochoosyortoogenerous)generateslinkagedisequi‐
librium between cooperation and choosiness, resulting in a positive
assortative matching. For this reason, their work is more similar to
our second model where choosiness is plastic than to our first model
where choosiness is hard‐wired. In McNamara et al. (2008)'s simula‐
tions, however, in contrast to our results, cooperation never reaches
the socially optimal level (in the model where they consider a pay‐
off function with diminishing returns). In a complementary analysis
(see Supporting information), we showed that this could be a con‐
sequenceoftheirassumptionthatthegeneticmutationrateisvery
high, which prevents natural selection from fully optimizing social
strategies.
Some scholars have already imported principles from matching
theory into evolutionary biology, especially in the field of sexual
selection.Johnstone,Reynolds,andDeutsch(1996)andBergstrom
and Real (2000) have used matching models, respectively with and
without search frictions, to shed light on mutual mate choice. Both
works focused on the evolution of choosiness with a given, fixed
distributionofindividual'squality.Aswehavepreviouslyshown,the
intensity of assortment may have a dramatic impact on the evolution
of the chosen trait (cooperation, in our case). For instance, further
models could investigate the precise limits of the runaway processes
that occur on weaponry, or on ornamental traits, in sexual selection.
More generally, matching models could be helpful to analyse a large
variety of biological markets (Hammerstein & Noë, 2016; Noë &
Hammerstein, 1994, 1995), including inter‐specific mutualisms, such
as mycorrhizal symbiosis or plant‐rhizobia relationships (Kiers et al.,
2011, 2003; Simms & Taylor, 2002).
As for the human case in particular, several lines of evidence
suggest that partner choice is a likely candidate as a key driving
force in the evolution of cooperation. Numerous experimental stud‐
ies have shown that human beings indeed do choose their social
partners in function of their cooperative reputation (Barclay, 2013,
2016; Barclay & Raihani, 2016; Barclay & Willer, 2007; Baumard et
10
|
GEOFFROY Et al.
al., 2013; E_erson, 2016; Raihani & Smith, 2015; Stovel & Chiang,
2016; Sylwester & Roberts, 2010, 2013; Wu, Balliet, & Lange, 2016).
Anthropological observations show that defection in traditional so‐
cieties is mostly met with a passive abandonment rather than with
more defection in return (see Baumard et al., 2013 for a review).
Also, several theoretical studies have shown that partner choice can
account for the evolution of other important properties of human
cooperation, such as the fact that its benefits are often shared in
proportion to ever yone's respective effort in producing them (André
& Baumard, 2011a, 2011b; Chiang, 2008; Debove, André, et al.,
2015a; Debove Baumard, & André, 2015b, 2017; Takesue, 2017).
Regarding the quantitative level of cooperation, observations
show that humans have precise preferences regarding the amount of
effort that shall be put into helping others. Daily life contains ample
examples of these preferences. For instance, we hold the door for
others in subway stations, but only when they are sufficiently close
to the door already, not when they are very far from it. And this is
truequitegenerally.Asexperimentsinrealsettingsdemonstrate,we
have preferences for specific amounts of cooperation, neither too
little nor too much (L ange & Eggert, 2015; Santamaria & Rosenbaum,
2011). Sometime s this prefere nce is expres sed in a purely q uanti‐
tative manner. At other times, the same preference is expressed in
a more qual itative way, determining the kinds of co operative ac‐
tion that we are willing, or unwilling, to perform. In any case, our
investmentinhelpingisquantitativelybounded.Moreover,thepre‐
cise level of effort we are willing to put in cooperation seems to be
constrained by considerations of social efficiency. Individuals help
one another only when it is mutually advantageous, that is when the
cost of helping is less than the benefit of being helped. Additionally,
recent evolutionary modellings of risk pooling have revealed the
socially optimal nature of helping behaviours (Aktipis et al., 2016;
Aktipis et al., 2011; Cronk, 2007; Hao, Armbruster, Cronk, & Aktipis,
2015; Campenni et al., 2017). They have shown that people's sys‐
tems of mutual help correspond to the most efficient systems of risk
pooling in a volatile environment.
In this paper, we have shown that partner choice can foster the
evolution of such an intermediate and efficient amount of coopera‐
tion, neither too little nor too much. But we have also shown that the
precise evolutionarily stable amount of cooperation should depend
on the fluidity of the biological market and can range from a very
low level of cooperation up to the socially optimal level (Figure 4).
A number of anthropological studies suggest that contemporary
hunter‐gatherer societies exhibit high levels of spatial mobility
(Baumard et al., 2013; Lewis, Vinicius, Strods, Mace, & Migliano,
2014). Therefore, it seems plausible that biological markets were
highly fluid in the social structure that our ancestors experienced.
Our model predicts that in this case, the amount of effort invested
into cooperation should become very close to the social optimum.
Therefore, partner choice can account for the evolution of human
preferences concerning social efficiency.
One could wonder, however, whether other models than partner
choice could account for the evolution of a socially optimal level of
cooperation as well. The most influential model on the evolution of
quantitativecooperationamongnonkinisthecontinuousversionof
the iterated prisoner's dilemma (André, 2015; André & Day, 2007;
Killingback & Doebeli, 2002; Lehmann & Keller, 2006; Roberts &
Sherratt, 1998; Wahl & Nowak, 1999a, 1999b). In this game, André
and Day (2007) have shown that the only evolutionarily stable level
of investment is the one that maximizes the total benefit of the in‐
teraction, that is that natural selection does eventually favour the
socially optimal amount of cooperation (see also Binmore, 1990;
Fundenberg and Maskin 1990; Robson, 1990; Binmore & Samuelson,
1992 in a discrete version of the iterated prisoner's dilemma). Yet, in
this approach, selection for efficient cooperation is only a second‐
order force, that is interactions between rare mutants, which plays
a significant role only because André and Day (2007) assumed the
absence of other first‐order effects. For instance, a slight cognitive
cost of conditional behaviour would have prevented the evolution
of efficient cooperation in their model. In another related study,
Akçay and Cleve (2012) have shown that socially optimal coopera‐
tion is favoured when individuals play a specific class of behavioural
responses to others' cooperative actions. They have also shown that
for a specific case of their model, these behavioural responses can
evolve by natural selection under low levels of relatedness. Here,
we have shown that under the effect of partner choice, efficient
cooperation is favoured by first‐order selective effects even in the
total absence of genetic relatedness. This occurs because, unlike
reciprocity, partner choice is a directional enforcement mechanism.
Whereas reciprocity merely stabilizes any given level of cooperation
(a principle called the folk theorem; see Aumann & Shapley, 1994;
Boyd, 2006), partner choice directionally favours the most efficient
level.
One limit of our model is that we did not introduce an explicit
mechanism for reputation. We simply assumed that in a way or an‐
other, individuals have reliable information regarding the cooper‐
ation level of others, but we did not model the way in which they
obtain this information. Costly signalling theory proposes that some
cooperativebehavioursarecostlysignalsofanindividual'squalit yor
willingness to cooperate (André, 2010; Barclay, 2015; Bird & Power,
2015; Bliege Bird, Ready, & Power, 2018; Gintis, Smith, & Bowles,
2001; Leimar, 1997). Such signals could, in the ory, be far from socially
efficient (Gintis et al., 2001). However, further analyses are needed
to rigorously model signalling in the context of a biological market.
ACKNOWLEDGMENTS
This work was supported by ANR‐10‐LABX‐0087 IEC and
ANR‐10‐IDEX‐0001‐02 PSL. This is contribution 2019‐127 of the
Institut des Sciences de l'Evolution de Montpellier (UMR CNRS
5554). This project analyses benefited from the Montpellier
Bioinformatics Biodiversity platform services. A preprint ver‐
sion of this paper has been reviewed and recommended by Peer
Community In Evolutionary Biology (https ://doi.org/10.24072/
pci.evolb iol.100063).
|
11
GEOFFROY Et al .
CONFLICT OF INTEREST
The authors of this preprint declare that they have no financial con‐
flict of interest with the content of this article.
DATA ACCESSIBILITY
The source code for the simulations is available on the first author's
GitHub repository: https ://github.com/fgeof froy/coope ration_runa‐
way and on the Dryad digital repository: https ://doi.org/10.5061/
dryad.vt66f3m
ORCID
Félix Geoffroy https://orcid.org/0000‐0001‐9800‐4728
Nicolas Baumard https://orcid.org/0000‐0002‐1439‐9150
Jean‐Baptiste André https://orcid.org/0000‐0001‐9069‐447X
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How to cite this article:GeoffroyF,BaumardN,AndréJ‐B.
Why cooperation is not running away. J Evol Biol. 2019;00:1–
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