ARTICLE IN PRESS

Theoretical Population Biology 70 (2006) 486–497
www.elsevier.com/locate/tpb

A mixed strategy model for the emergence and intensiﬁcation of social
learning in a periodically changing natural environment
Joe Yuichiro Wakano, Kenichi Aoki
Department of Biological Sciences, The University of Tokyo, Hongo 7-3-1, Bunkyoku, Tokyo 113-0033, Japan
Received 19 December 2005
Available online 25 April 2006

Abstract
Based on a population genetic model of mixed strategies determined by alleles of small effect, we derive conditions for the evolution ofsocial learning in an inﬁnite-state environment that changes periodically over time. Each mixed strategy is deﬁned by the probabilities
that an organism will commit itself to individual learning, social learning, or innate behavior. We identify the convergent stable strategies(CSS) by a numerical adaptive dynamics method and then check the evolutionary stability (ESS) of these strategies. A strategy that issimultaneously a CSS and an ESS is called an attractive ESS (AESS). For certain parameter sets, a bifurcation diagram shows that thepure individual learning strategy is the unique AESS for short periods of environmental change, a mixed learning strategy is the unique
AESS for intermediate periods, and a mixed learning strategy (with a relatively large social learning component) and the pure innate
strategy are both AESS’s for long periods. This result entails that, once social learning emerges during a transient era of intermediate
environmental periodicity, a subsequent elongation of the period may result in the intensiﬁcation of social learning, rather than a return
to innate behavior.
r 2006 Elsevier Inc. All rights reserved.
Keywords: Alleles of small effect; Evolutionarily stable strategy; Convergent stable strategy; Attractive ESS; Bifurcation diagram; Adaptive dynamics

1. Introduction
The behavior of a biological organism in a particularenvironment—more speciﬁcally, in response to a ﬁtnessrelated aspect of the environment—is a complex process
involving the expression of relevant information possessedby that organism. At the risk of oversimpliﬁcation (Gould
and Marler, 1987), it is possible to classify behaviors interms of the three distinct ways in which this information
can be obtained (Cavalli-Sforza and Feldman, 1983; Boydand Richerson, 1985; Laland et al., 2000; Henrich andMcElreath, 2003; Alvard, 2003). This trichotomous classiﬁcation labels behavior as ‘‘innate,’’ ‘‘socially learned,’’ or‘‘individually learned’’.
A behavior is innate when it entails the direct expressionof information encoded in the genes, which are inheritedfrom the parents via the germ cells. Social learning denotes
the transfer of information between socially interacting
Corresponding author. Fax: +81 3 3818 7547.

E-mail address: kenaoki@biol.s.u-tokyo.ac.jp (K. Aoki).
0040-5809/$ - see front matter r 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.tpb.2006.04.003

organisms, as a result of which the behavior exhibited bythe ‘‘exemplar’’ is adopted by the ‘‘observer’’ (Galef, 1988;
Whiten and Ham, 1992; Heyes, 1993). (We prefer the term
exemplar to ‘‘model’’, which can be confused withmathematical model, and ‘‘demonstrator’’, which implies
active teaching.) The rubric covers teaching, imitation,
local enhancement, and various other psychologicalprocesses. Finally, individual learning refers to learningthat occurs independently of any social inﬂuences. Examples are trial-and-error and insight.
Reviews of the factors contributing to the emergence ofsocial learning emphasize the role played by a temporally
changing environment (Laland et al., 2000; Richerson andBoyd, 2000; Henrich and McElreath, 2003; Alvard, 2003).
These writers agree that individual learning, social learning(from the parental generation), and innate determinationof behavior are favored by natural selection whenenvironmental changes occur at short, intermediate, andlong generation intervals, respectively. Theoretical studies
by Boyd and Richerson (1985, 1988), Rogers (1988), andFeldman et al. (1996) have provided the motivation, andARTICLE IN PRESS
J.Y. Wakano, K. Aoki / Theoretical Population Biology 70 (2006) 486–497

also some support, for this proposal. More recently,
Wakano et al. (2004) and Aoki et al. (2005) rigorously
showed that this consensus view is basically true for both
periodically and randomly changing environments, by
simultaneously comparing individual learners, social learners, and ‘‘innates’’ (organisms behaving innately) when
they are in direct competition with each other.
Wakano et al. (2004) and Aoki et al. (2005) posited
genetic variation at a tri-allelic ‘‘strategy’’ locus, which
determines whether an organism is an individual learner, asocial learner, or an innate. In other words, they assumed
that the three strategies are obligate (also Rogers, 1988;
Feldman et al., 1996). Clearly, a model recognizing the‘‘facultative’’ nature of behavior is more realistic (Boydand Richerson, 1985, 1988; Feldman et al., 1996; Henrichand Boyd, 1998; Kameda and Nakanishi, 2002). In thepresent paper, we propose a new model approximating thisfeature, which is an extension of the mixed strategy modelof Feldman et al. (1996). The mixed strategy model
assumes alleles of small effect instead of major genes.
Since we view evolution as proceeding gradually by the
cumulative substitution of mutations of small effect, ourinterest centers on strategies that are simultaneously
convergent stable (CSS) and evolutionarily stable (ESS).
CSS and ESS are not equivalent concepts (Britton, 2003;
Doebeli et al., 2004): an ESS that is not a CSS cannot bereached by the cumulative substitution of mutations ofsmall effect; a CSS that is not an ESS can be reached, but isinvadable by another strategy. Hence, we restrict ourattention to attractive ESS’s (AESS), which are deﬁned to
be strategies that are simultaneously an ESS and a CSS.
We use a numerical adaptive dynamics method to identify
the CCS (Geritz et al., 1997). Then, we apply analyticallyderived conditions to check that these strategies are also
evolutionarily stable (Maynard Smith, 1982).
Many of our results are consistent with those obtainedby Wakano et al. (2004) and Aoki et al. (2005), which gives
us some conﬁdence that they do indeed apply to thephenomenon being investigated, rather than being dependent on the model. However, some of our predictions arenovel. For example, we ﬁnd that a mixed social andindividual learning strategy, once it emerges during an eraof intermediate environmental stability, may be able toresist invasion by the pure innate strategy, even if an era ofhigh environmental stability should ensue.
2. Description of the model
Feldman et al. (1996) assume an inﬁnite population of
haploid asexual organisms that can use a mixed strategy ofindividual and social learning. We extend their model toinclude innate behavior as an alternative. During development each organism commits itself to one of these threepure strategies. The ‘‘strategy’’ locus, with alleles A and a,
determines the probabilities that behavior will be innate,
individually learned, or socially learned. An organism
carrying an A allele becomes an ‘‘innate’’ (i.e., an organism

487

behaving innately) with probability 1KL, an individual
learner with probability L, and a social learner withprobability K. For allele a, we substitute L þ dL for L andK þ dK for K. Here, dL and dK denote independentchanges of probability (except on the boundaries of theparameter space of K and L as explained below).
The environment changes every l generations. In otherwords, one postchange generation experiences a different
environmental state from the previous generation, and l1
subsequent generations experience the same environmental
state as that postchange generation. Larger values of l imply
more environmental stability. We adopt the inﬁnite environmental state model, which posits that when the environmentchanges it never reverts to an earlier state (Feldman et al.,
1996; Wakano el al., 2004; Aoki et al., 2005).
Behavior may be adaptive (correct) or maladaptive(wrong). Individual learners always achieve a correct
behavior on their own whatever the environmental state,
whereas social learners and innates may get it wrong. A
social learner acquires its behavior by copying a random
organism of the previous generation (oblique transmission,
Cavalli-Sforza and Feldman, 1981). Its behavior will be
adaptive or maladaptive depending on whether the
behavior it acquires is appropriate to its current environmental state. In particular, the inﬁnite environmental statemodel implies that a social learner of the postchange
generation has no suitable exemplars in the prechange
generation from which to acquire a correct behavior (seealso next paragraph).
The information needed for innate behavior is stored
genetically at the ‘‘innate information’’ locus, with twoclasses of alleles M and m. If an innate carries an M allele,
its behavior is adaptive. Carrying an m allele results inmaladaptive behavior. Each of the two classes of alleles
comprises many selectively neutral variants, which arebeing constantly generated by mutation. (However, we donot model the mutation process.) Hence, there may be avariety of adaptive behaviors, which by assumption are
equally adaptive (i.e., have the same ﬁtness), and similarly
a variety of maladaptive behaviors. A small subset
amounting to a fraction r of each class has the special
property of being adaptive when the environment changes.
(Parameter r is analogous to a mutation rate, and we arefollowing standard practice in population genetics when we
assume it to be a constant.) These variants can be regardedas ‘‘preadapted’’ alleles awaiting environmental change.
Nevertheless, an innate carrying a preadapted allele
behaves differently in the prechange and postchange
generations—the behavior of an innate is the result of aninteraction between the allele it carries and the environment it experiences—so that its behavior in the prechange
generation would be maladaptive in the postchange
generation. The innate information locus is not expressed
when the individual or social learning strategy is adopted.
These assumptions may strike the reader as arbitrary, but
in fact are necessary if we are to write down a consistentand analytically tractable recursion model.

ARTICLE IN PRESS
488

J.Y. Wakano, K. Aoki / Theoretical Population Biology 70 (2006) 486–497

Let us illustrate what we mean by a preadapted allele
with two textbook examples from human genetics. An
allele in our haploid asexual organism corresponds, albeit
imperfectly, to a genotype in diploid sexual humans. The
hemoglobin S allele is recessive to the A allele in a malariafree environment, so that the ﬁtnesses of genotypes AA andAS are identical (and greater than the ﬁtness of genotype
SS). If the population is exposed to malaria, however,genotype AS now has a higher ﬁtness than genotype AA
due to overdominance. Hence, in so far as the allele S ispresent at a low frequency before malaria becomes aselective factor, it can be regarded as a preadapted allele.
Our second example is adult lactose absorber, which showsthat the behavior associated with a genetic trait may
depend on the environment in a way that affects ﬁtness. In
the incipient stage of animal domestication, the amount of
milk available to an absorber is likely small and not
sufﬁcient to make a difference in ﬁtness. As domestication
proceeds and milk becomes more abundant, an absorber
who persists in limiting intake is behaving maladaptively
compared to one who increases consumption.
The ﬁtnesses are relative viabilities and are assigned in
the following way. There is a baseline ﬁtness of 1 foradaptive behavior. Maladaptive behavior causes the ﬁtness
to be reduced by s. Social learners bear a direct cost ofdeveloping and maintaining a nervous system supportive oflearning, which is translated into a ﬁtness loss d. Individual
learners suffer a similar—though not necessarily equivalent—direct cost and are also adversely affected by
mistakes made before the mature behavior is realized; thetotal penalty is c. The ﬁtnesses associated with each of thethree strategies when the behavior is correct or wrong aresummarized in Table 1. We assume 0odocosod+so1(Aoki et al., 2005).
There are four genotypes AM, Am, aM, and am, each ofwhich may express either of two phenotypes, i.e., adaptive
or maladaptive behavior. The frequencies of the eightphenogenotypes (genotype–phenotype combinations, Feldman and Cavalli-Sforza, 1984) are listed in Table 2. The life
cycle events are asexual reproduction, learning, and natural
selection in that order. The recursions relating thefrequencies of the eight phenogenotypes in the offspring
generation (with primes) to those in the parental generation(without primes) can be written as follows.

Table 2
Notation for the frequencies of the eight phenogenotypes
Phenogenotype

Frequency

Genotype

Behavior

AM
AM
Am
Am
aM
aM
am
am

Correct
Wrong
Correct
Wrong
Correct
Wrong
Correct
Wrong

u
ū
v
v̄
x
x̄
y
ȳ

First, when the environment changes between generations they are:Vu0 ¼ ðu þ ū þ v þ v̄Þr½1  K  L þ Lð1  cÞ,V ū0 ¼ ðu þ ū þ v þ v̄ÞrKð1  d  sÞ,
Vv0 ¼ ðu þ ū þ v þ v̄Þð1  rÞLð1  cÞ,
V v̄0 ¼ ðu þ ū þ v þ v̄Þð1  rÞ½ð1  K  LÞ
 ð1  sÞ þ Kð1  d  sÞ,Vx0 ¼ ðx þ x̄ þ y þ ȳÞr½1  K  dK
 L  dL þ ðL þ dLÞð1  cÞ,V x̄0 ¼ ðx þ x̄ þ y þ ȳÞrðK þ dKÞð1  d  sÞ,
Vy0 ¼ ðx þ x̄ þ y þ ȳÞð1  rÞðL þ dLÞð1  cÞ,V ȳ0 ¼ ðx þ x̄ þ y þ ȳÞð1  rÞ½ð1  K  dK  L  dLÞ ð1  sÞ þ ðK þ dKÞð1  d  sÞ.

ð1ÞSecond, when the environment remains constant between generations they are:Wu0 ¼ ðu þ ūÞ½1  K  L þ Lð1  cÞþ Kðu þ v þ x þ yÞð1  dÞ,W ū0 ¼ ðu þ ūÞKðū þ v̄ þ x̄ þ ȳÞð1  d  sÞ,
Wv0 ¼ ðv þ v̄Þ½Lð1  cÞ þ Kðu þ v þ x þ yÞð1  dÞ,W v̄0 ¼ ðv þ v̄Þ½ð1  K  LÞð1  sÞþ Kðū þ v̄ þ x̄ þ ȳÞð1  d  sÞ,
Wx ¼ ðx þ x̄Þ½1  K  dK  L  dL þ ðL þ dLÞ0ð1  cÞ þ ðK þ dKÞðu þ v þ x þ yÞð1  dÞ,W x̄ ¼ ðx þ x̄ÞðK þ dKÞðū þ v̄ þ x̄ þ ȳÞð1  d  sÞ,0

Wy0 ¼ ðy þ ȳÞ½ðL þ dLÞð1  cÞ
Table 1Strategy/behavior combinations with associated ﬁtnesses
Strategy/behavior

Fitness

Innate/correct
Innate/wrong
Social learning/correct
Social learning/wrong
Individual learning/correct

1
1s
1d
1ds
1c

0odocosod+so1.

þ ðK þ dKÞðu þ v þ x þ yÞð1  dÞ,W ȳ ¼ ðy þ ȳÞ½ð1  K  dK  L  dLÞð1  sÞ0

þ ðK þ dKÞðū þ v̄ þ x̄ þ ȳÞð1  d  sÞ.

ð2ÞThe quantities V and W in (1) and (2) are the meanﬁtnesses that normalize the equations. Their explicit forms
are not given, but can be obtained by summing the righthand sides. Corresponding to the environmental periodicity of l, one set of iterations entails one application of (1)followed by l1 applications of (2).

ARTICLE IN PRESS
J.Y. Wakano, K. Aoki / Theoretical Population Biology 70 (2006) 486–497

We now explain the derivation of these recursions, which
are a direct extension of (20) and (21) of Feldman et al.
(1996). First, we note that the ﬁrst four lines of (1) and (2)describe changes in the frequencies of genotypes AM andAm, which have the A allele at the strategy locus, while thelast four lines describe changes in the frequencies ofgenotypes aM and am, which have the a allele. Since thederivation of the last four lines is identical to that of theﬁrst four lines except for the substitution of L þ dL for L
and K þ dK for K, the argument will not be repeated.
The recursions with stasis (2) are easier to explain, so wewill deal with them ﬁrst. Since reproduction is asexual, eachoffspring is genetically identical to its parent. Hence thefrequencies of the genotypes AM and Am among newborns
are u þ ū and v þ v̄, respectively. Since these genotypes
have the A allele at the strategy locus, innate behavior,individual learning, and social learning occur with probabilities 1KL, L, and K, respectively. When the innate
strategy is adopted, the behavior of genotype AM is
adaptive since it has an M allele at the innate information
locus (ﬁtness equals 1, ﬁrst line), whereas the behavior of
genotype Am is maladaptive since it has an m allele (ﬁtness
equals 1s, fourth line). With individual learning, behavior
is always adaptive (ﬁtness equals 1c, ﬁrst and third lines).
Socially learned behavior is adaptive if an adaptively
behaving member of the previous generation, whose
frequency is u þ v þ x þ y, is copied (ﬁtness equals 1d,ﬁrst and third lines), and maladaptive if a maladaptively
behaving member, whose frequency is ū þ v̄ þ x̄ þ ȳ, iscopied (ﬁtness equals 1ds, second and fourth lines).
The recursions with environmental change (1) can beobtained by modifying the above argument. Among thenewborns of the postchange generation the frequencies ofgenotypes AM and Am are rðu þ ū þ v þ v̄Þ andð1  rÞðu þ ū þ v þ v̄Þ, respectively. Using the innate strategy, behavior is adaptive for genotype AM (ﬁrst line), butmaladaptive for genotype Am (fourth recursion). All
individual learners behave adaptively (ﬁrst and third lines),
and all social learners behave maladaptively (second andfourth lines).
3. Analysis
Recall that the A and a alleles at the strategy locus
determine two strategies that differ in the probabilities thatbehavior is innate, individually learned, or socially learned.
These differences are expressed in terms of the parameters
dL and dK. Our purpose in Sections 3–5 is to obtain theconditions for local stability of a monomorphism in the‘‘wild type’’ allele A to invasion by the ‘‘mutant’’ allele a. In
particular, if this genetic monomorphism is locally stable
for all combinations of dL and dK, then by deﬁnition it
corresponds to an ESS (Maynard Smith, 1982). We focus
on the case of small dL and dK in this paper.
First, when the population is monomorphic for allele A,
there exists a globally stable equilibrium solution of periodl (Appendix A). Next, as shown in Appendix B, the489

dominant eigenvalue determining the local stability toinvasion by allele a is
l¼a

l 1
Y

gðiÞ þ b

i¼1

l 1
Y

xðiÞ ,

(3a)

i¼1

where
V^ þ ð1  K  LÞð1  rÞsV^
cdL þ ðd þ sÞdK 1,V^ þ ð1  K  LÞð1  rÞs

a¼r

V^  ð1  K  LÞrsb ¼ ð1  rÞV^ðs  cÞdL  ddK
 1þ
,V^  ð1  K  LÞrsgðiÞ ¼xðiÞ ¼ðiÞ
W ðiÞ þ ð1  K  LÞð^vðiÞ þ v̄^ Þs
^ ðiÞ
W
(
)
ðiÞ
cdL þ ½d þ ðūðiÞ þ v̄^ ÞsdK
 1
,
ðiÞ
W ðiÞ þ ð1  K  LÞð^vðiÞ þ v̄^ Þs^ ðiÞ  ð1  K  LÞðu^ ðiÞ þ ū^ ðiÞ Þs
W
^ ðiÞ
W
(
)
ðs  cÞdL  ½d  ðu^ ðiÞ þ v^ðiÞ ÞsdK 1þ
.
^ ðiÞ  ð1  K  LÞðu^ ðiÞ þ ū^ ðiÞ ÞsW

ð3bÞ

Superscript i in (3) denotes the number of generations
that have elapsed since an environmental change and the
hat indicates monomorphic equilibrium. When jljo1, theperiodic solution for the monomorphism in allele A (see(A.1), (A.2) of Appendix A) is locally stable to invasion byallele a.
For small dL and dK we approximate (3) byl  1 þ C L dL þ C K dK,

(4a)

where
(
CL ¼

r½V^ þ ð1  K  LÞð1  rÞs
V^



)
ðiÞ
l1 ^ ðiÞYW þ ð1  K  LÞð^vðiÞ þ v̄^ Þs
^ ðiÞ
W
c

i¼1


 


^
i¼1 W

(
þ



V^ þ ð1  K  LÞð1  rÞs

l1
X

)

c
ðiÞ

ðiÞ

þ ð1  K  LÞðv^ðiÞ þ v̄^ Þs
ð1  rÞ½V^  ð1  K  LÞrs
V^

)
ðiÞ
l 1 ^ ðiÞYW  ð1  K  LÞðu^ ðiÞ þ ū^ Þs
i¼1

^ ðiÞ
W

ARTICLE IN PRESS
J.Y. Wakano, K. Aoki / Theoretical Population Biology 70 (2006) 486–497

490



sc
^
V  ð1  K  LÞrs
l1
X
sc


þ
(
CK ¼

^
i¼1 W

ðiÞ

Hence, C L 40 if and only if)ðiÞ ð1  K  LÞðu^ ðiÞ þ ū^ ÞsðiÞl 1 ^ ðiÞYW þ ð1  K  LÞð^vðiÞ þ v̄^ Þs



þ

ðiÞ

ðiÞ

ðiÞ

Allele A deﬁnes a mixed strategy of individual and social
learning (0oK ¼ 1  Lo1), a pure individual learning
strategy (L ¼ 1), or a pure social learning strategy (K ¼ 1).
It is convenient to rewrite (4) by applying the transformationðiÞþ ð1  K  LÞð^v þ v̄^ Þs
ð1  rÞ½V^  ð1  K  LÞrsV^l 1 ^
Y
W

ðiÞ

i¼1



 


5. Special case of K þ L ¼ 1)

d þ ðū^ þ v̄^ Þs

^
i¼1 W

(



ðiÞ

ðiÞ

 ð1  K  LÞðu^ ðiÞ þ ū^ Þs
^ ðiÞ
W

)
dP ¼

d
V^  ð1  K  LÞrs

l1
X

^
i¼1 W

)

d  ðu^ ðiÞ þ v^ðiÞ Þs
ðiÞ

ðiÞ

 ð1  K  LÞðu^ ðiÞ þ ū^ Þs

.

ð4cÞ

4. Special case of K ¼ L ¼ 0
When allele A deﬁnes a strategy of pure innate behavior,(A.1), (A.2) of Appendix A can be explicitly solved to yield
the periodic solution shown in Appendix C.
If r  ð1  sÞl holds, we have from (4) to leading order
in r:
l1þ

l
½ðs  cÞdL  ddK
1s

(5)

(Appendix D). Since s4c and d40, we observe thatl41 only if dL40. In other words, a small amount ofsocial learning in isolation (dL ¼ 0, dK40) confers
no advantage over the pure innate strategy (sinceld
C K ¼  1so0), so that a successful mutant allele a musthave an individual learning component (dL40).
In fact, we can show that CKo0 is true in general(Appendix E). Hence, a necessary condition for instability
is C L 40. From (4b)
V^

l 1
Y

dK þ dL;2

dQ ¼

dK  dL.
2

In the parameter space of K and L (Fig. 1), dP and dQ
are the components perpendicular and parallel to the line
K þ L ¼ 1, respectively. Clearly dPp0, since the parameter space is a simplex. A mutant allele, a, that introducesa small innate component is characterized by dPo0. On
the other hand, if the probabilities of individual and social
learning are slightly altered without adding an innatecomponent, then dP ¼ 0 and dQa0.
In terms of dP and dQ, the approximate eigenvalue (4a)becomesl  1 þ C P dP þ C Q dQ,

(10a)

where
"

#
ðiÞ
ðiÞ
l1
sþcþd Xsðū^ þ v̄^ Þ þ c þ d
CP ¼  r
þ
^ ðiÞ
V^
W
i¼1
"
#
l1
scd Xsð1 þ u^ ðiÞ þ v^ðiÞ Þ  c  dþ ð1  rÞ
þ
,
^ ðiÞ
V^
W
i¼1
ð10bÞ

CQ ¼ ðiÞðiÞl1scþd Xsðū^ þ v̄^ Þ  c þ d
,

^ ðiÞ
V^
W
i¼1

(10c)

with

^ ðiÞ ¼ V^ þ Kðu^ ðiÞ þ v^ðiÞ Þs.
Wi¼1¼ l½ð1  cÞZðl  1Þ  ZðlÞ,(9)V^ ¼ Lð1  cÞ þ Kð1  d  sÞ,^ ðiÞ  C L ¼ l½rc þ ð1  rÞðs  cÞð1  sÞl1 
W

ð10dÞ

ð6aÞ

where
ZðlÞ ¼ r þ ð1  rÞð1  sÞl .

(8)

there exists a unique integer l m ðX1Þ such that (7) is satisﬁed
for lpl m (Appendix F).
Restated, the pure innate strategy is an ESS if inequality(8) is reversed or if l4l m .

)V^ þ ð1  K  LÞð1  rÞs

l1
X

(7)

Thus, provided

^ ðiÞ
W
d þs

i¼1



 

ð4bÞ

r þ ð1  rÞð1  sÞo1  c,r½V^ þ ð1  K  LÞð1  rÞs
V^



,

ZðlÞ
o1  c.
Zðl  1Þ

(6b)

In Appendix G, the binary sums of the phenogenotypes
ðiÞðiÞ
frequencies, u^ ðiÞ þ v^ðiÞ and ū^ þ v̄^ , are obtained, which
permits explicit evaluation of (10) (also (11) below).

ARTICLE IN PRESS
J.Y. Wakano, K. Aoki / Theoretical Population Biology 70 (2006) 486–497

1

0.8

0.8

0.6

0.6

L

L

1

0.4

0.4

0.2

0.2

0

0
0

0.2

0.4

0.6

0.8

1

K

(a)

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

K

(b)

1

0.8

0.8

0.6

0.6

L

L

1

0.4

0.4

0.2

0.2

0

0
0

(c)

491

0.2

0.4

0.6

0.8

1

K

0

0.2

(d)

0.4

K

Fig. 1. Parameter space of K (probability of social learning) and L (probability of individual learning) is shown, with vector ﬁeld (arrows) indicating themost likely direction that evolution by the cumulative substitution of mutations of small effect will take. Filled circles denote the convergent stablestrategies (CSS), which are also evolutionarily stable (ESS). Other parameters are: for (a)–(c) s ¼ 0:3, c ¼ 0:2, d ¼ 0:19, r ¼ 106 , (a) l ¼ 20, (b) l ¼ 35,(c) l ¼ 70, and for (d) s ¼ 0:3, c ¼ 0:29, d ¼ 0:28, r ¼ 0:05, l ¼ 100.

Eq. (10) shows that a mixed learning strategy (0oK ¼
1  Lo1) can be locally stable against all alternative
strategies with dP ¼ 0 and dQa0 only if C Q ¼ 0, i.e.,
ðiÞ
ðiÞ
l1
scþd Xsðū^ þ v̄^ Þ  c þ d
¼ 0.
þ
^ ðiÞ
V^
W

(11)

i¼1

In addition, we require C P 40 (to ﬁrst order) when dPo0.
Hence, a mixed learning strategy is an ESS (to ﬁrst order)
only if inequality C P 40 and equality (11) are both satisﬁed((11) can be solved numerically for L; Figs. 1 and 2).
Similarly, the pure individual learning strategy (L ¼ 1) islocally stable against alternative strategies with dP ¼ 0 and
dQ40 if CQo0, i.e.,
s
.
(12)
lo
cd

Our assumption 0odocos entails that (12) is satisﬁed
for l sufﬁciently small. If (8) and (12) both hold, the pureindividual learning strategy is an ESS (Appendix H).
Finally, the pure social learning strategy (K ¼ 1) isalways unstable. Conditions (11) and (12) were obtained by
Feldman et al. (1996, p. 227).
6. Numerical work
For general K and L, we investigate the model
numerically. The periodic equilibrium solution for themonomorphism in allele A is uniquely determined by
iterating (A.1) and (A.2) of Appendix A. Moreover, local
stability to invasion by allele a can be resolved byevaluating the eigenvalue (4). Let us represent the wild
type strategy by a point (K, L) in the parameter space of K

ARTICLE IN PRESS
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492

1

0.8

L*

0.6

K*=1-L*

0.4

0.2
K*=0

0
10

20

30

40

50

60

70

80

90

100

l
Fig. 2. A bifurcation diagram of AESS’s. Attractive evolutionarily stableprobability of individual learning, L*, is plotted against the period ofsenvironmental change, l. For lo cd, the pure individual learning strategys(K  ¼ 0, L ¼ 1) is the unique AESS (crosses). For cdolpl m , a mixed
learning strategy (0oL*o1, K  ¼ 1  L ) is the unique AESS (triangles).
For l4l m , the pure innate strategy (K  ¼ 0, L ¼ 0) coexists as an AESS
(circles) with a mixed learning strategy. Hence, the model exhibits‘‘bistability’’ when l4l m . Parameters are: s ¼ 0:3, c ¼ 0:2, d ¼ 0:19,
s
r ¼ 106 . Critical values of the period are cd¼ 30, l m ¼ 37, l M ¼ 61.

and L (Fig. 1) and the mutant strategy by a neighboring
point (K þ dK, L þ dL). If (K, L) is an interior point, then(K þ dK, L þ dL) may occupy any one of the four
quadrants centered on (K, L). For a boundary point(K, L), there are restrictions on (dK, dL), for example as
noted in Section 5 and Appendix H. With each point (K, L)
there is associated a vector of coefﬁcients (CK, CL) deﬁnedby (4b) and (4c), which in conjunction with the vector ofdifferences (dK, dL) determines whether the mutant
strategy will invade.
The wild type strategy is locally stable to invasion by any
mutant strategy for which C L dL þ C K dK (the innerproduct of the two vectors) is negative. This will be thecase when the angle between (dK, dL) and (CK, CL) exceeds
901. On the other hand, the wild type strategy will
be invaded by any mutant strategy for which
C L dL þ C K dK40. This inequality will hold if the angle
between (dK, dL) and (CK, CL) is less than 901. Hence, if weview evolution as proceeding gradually by the cumulative
substitution of mutations of small effect (not necessarily at
the same genetic locus), the vector (CK, CL) serves as arough indicator of the direction in parameter space thatevolution will take.
In Fig. 1 we have drawn ‘‘vector ﬁelds’’ obtained byplotting the vectors (CK, CL) at various interior andboundary points of the parameter space. Thus, an orbit inthis vector ﬁeld represents an evolutionary path. A CSS is
obtained numerically as the stable equilibrium to which the
dynamical system
K_ ¼ C K ; L_ ¼ C L(13)converges in the simplex 0pKp1, 0pLp1, and
0pK+Lp1 (Fig. 1). The dots over K and L in (13) signify
differentiation by a time variable, and of course the CSS
may depend on the initial conditions. The method
described here is similar to the adaptive dynamics
approach (e.g., Geritz et al., 1997), where the gradient of
the ﬁtness landscape is (CK, CL). We ﬁrst solve (13)numerically to locate the CSS’s. Then we check that theseequilibria are also ESS’s. As shown in Sections 4 and 5,
tests of evolutionary stability are available in the special
cases K ¼ L ¼ 0 and K þ L ¼ 1. (Appendix I illustrates
the difference between CSS and ESS.)The numerical work suggests that there are no interior
CSS’s, which would require C K ¼ C L ¼ 0. Neither do we
ﬁnd CSS’s on the L-axis (K ¼ 0, 0oLo1) or the K-axis(L ¼ 0, 0oKp1). Thus, a strategy with both an innate anda learning (whether individual or social) component cannot
be a CSS. The CSS’s are the pure individual learning
strategy, a mixed learning strategy, and the pure innate
strategy. In fact, as we see below, each CSS is also an ESS,
hence an AESS.
Fig. 1a illustrates the case where the pure individual
learning strategy is the unique CSS. Fig. 1b shows that amixed learning strategy can be the unique CSS. In Fig. 1cthere are two CSS’s, the pure innate strategy and a mixed
learning strategy. Although not shown, the pure innate
CSS may coexist with the pure individual learning CSS.
Finally, the pure innate strategy can be the sole CSS as
shown in Fig. 1d. Each pure individual learning CSS andpure innate CSS is also an ESS, as can be veriﬁed byapplication of (7), (8), and (12). In addition, numerical
work using (10b) and (11) suggests that each mixed
learning CSS is an ESS; since (11) is necessary but not
sufﬁcient, local stability against mutant strategies represented by neighboring points of the parameter space waschecked by iterating (1) and (2).
Fig. 2 is a ‘‘bifurcation diagram’’ illustrating thedependence of the AESS’s on parameter l (the environmental periodicity). Inequality (8) holds in this ﬁgure(other parameter are: s ¼ 0:3, c ¼ 0:2, d ¼ 0:19, r ¼ 106 ),whence the pure individual learning strategy is an ESS forslo cd(i.e., when (12) is satisﬁed), and the pure innate
strategy is an ESS for l4l m (i.e., when (7) is reversed).
Numerical work using (10b) and (11) suggests that a mixedslearning ESS exists for each l4 cd. All points indicated bycrosses, triangles, or circles have been shown numericallyto be CSS’s. Hence, there are apparently two AESS’s
when l4l m .
On the basis of extensive numerical work, we haveidentiﬁed two other bifurcation patterns when l is varied.
First, if inequality (8) is reversed (because the penalty
suffered by individual learners, c, is almost as large as thecost of maladaptive behavior, s), the pure innate strategy
appears to be the only ESS. In this case, learning cannotsevolve whatever the value of l. Second, if l m o cd(e.g., forparameter values s ¼ 0:4, c ¼ 0:39, d ¼ 0:38, r ¼ 106 ), thepure individual learning ESS coexists with the pure innate

ARTICLE IN PRESS
J.Y. Wakano, K. Aoki / Theoretical Population Biology 70 (2006) 486–497
s
ESS for l m olo cd, whereas a mixed learning ESS and thespure innate ESS coexist for l4 cd. All ESS’s are also
CSS’s, hence AESS’s.
It is also possible to plot bifurcation diagrams against
other parameters. Let us brieﬂy note the effect ofparameter r. The pure innate strategy is an ESS if CLo0in (6a), or equivalentlyr4ðs  cÞð1  sÞl1c þ ðs  cÞð1  sÞl1

.

(14)

In words, the frequency of preadapted alleles at the
innate information locus must be sufﬁciently high. From(11) we see that the probability of individual learning, L, ata mixed learning strategy is independent of r. Evolutionary
stability, however, requires that C P 40 in (10b). A
necessary but not sufﬁcient condition for this inequality
to hold is that r be sufﬁciently small. Numerical workshows that an ESS is also a CSS, hence an AESS. Thus, apossible bifurcation pattern comprises two horizontal
straight lines, one representing the pure innate strategy,
which is an AESS when (14) is satisﬁed, and the other amixed learning strategy, which is an AESS for sufﬁciently
small values of r.
7. Discussion
We studied the evolution of social learning in a model
where small genotypic differences occur in the probabilities
that an organism will commit itself to individual learning,social learning, or innate determination of behavior.
Analytical and numerical methods were employed to
identify the evolutionarily stable pure or mixed strategies(ESS) that are also CSS. The possible AESS are the pure
innate strategy (K ¼ 0, L ¼ 0), a mixed learning strategy(0oLo1, K ¼ 1  L), and the pure individual learning
strategy (K ¼ 0, L ¼ 1).
It is perhaps surprising that no AESSs are found withboth an innate and a learning component. This result
apparently contradicts the observation that some strategies
currently used by humans, for example in language
acquisition, involve learning subject to innate constraints(Chomsky, 1975; Bickerton, 1983). We can think of at least
three reasons for the discrepancy. First, the present model
may be too simple—we have assumed that each organism
commits itself during development to one of three purestrategies, whereas actual examples of a mixed strategy
often involve the simultaneous application of two or morepure strategies by the same organism—which is the most
likely explanation. Second, a current strategy may not be
an AESS, since the waiting time for the appropriate
mutations may be prohibitively long. Third, the fault may
lie with our reliance on numerical method alone to locate
the CSS’s—no numerical survey can claim to be complete.
According to the bifurcation diagram of Fig. 2, the pureindividual learning strategy is the unique
AESS
if thesenvironmental periodicity is short lo cd, a mixed

493

learning strategy is the unique
AESS if changes occur with solpl m , and two AESSs—aintermediate frequency cdmixed learning strategy (with a relatively large social
learning component, K) and the pure innate
strategy—
coexist if the environment is highly stable l4l m . Except
for the continued presence of a mixed learning AESS
beyond the threshold l m , with implications discussed
below, the dependence on the environmental periodicity
is essentially the same as that noted by Wakano et al.
(2004) and Aoki et al. (2005).
Let us assume the bifurcation diagram of Fig. 2. Then, toreiterate, the pure innate strategy is an AESS if theenvironmental periodicity l4l m . On the other hand, if theenvironment is sufﬁciently changeable so that lpl m (i.e.,
when (7) holds), the pure innate strategy is invadable, but
only by a mutant strategy with an individual learningcomponent that is large relative to the social learningcomponent (0pdKodL, see Section 4). Hence, the initial
evolution of learning from an ancestral state of innate
behavior likely involves a stage in which learning, on therare occasions that it occurs, is entirely or predominantly
individual. Nevertheless, by the cumulative substitution of
mutations of small effect, the relative reliance on socialslearning will subsequently increase if cdolpl m (depending on the parameter set, this interval may not be as narrow
as Fig. 2 would suggest). Ultimately, a mixed learning
strategy with a non-negligible social learning component
will evolve, as this is the unique AESS (Fig. 1b) in this case.
Still assuming the same bifurcation diagram, supposethat a population at this mixed learning AESS (Fig. 1b)
now experiences a more stable environment (l4l m ). Two
AESS’s, the pure innate strategy and a mixed learning
strategy with a relatively larger social learning component(Fig. 1c), replace the previous unique AESS (Fig. 1b). This
population will not necessarily evolve back to the pure
innate condition (K ¼ 0, L ¼ 0,). Rather, the vector ﬁeld ofFig. 1c suggests it is more likely to evolve along the edge
K þ L ¼ 1 toward the new AESS with the greater relianceon social learning (a larger value of K). Hence, the model
exhibits ‘‘irreversibility’’. Once social learning emerges
during a temporary era of intermediate environmental
stability, a subsequent elongation of the environmental
periodicity toward increased stability may result in theintensiﬁcation of social learning, rather than a return toinnate behavior.
Richerson and Boyd (2005) note a possible connection
between the evolution of social learning in humans and theincreased variability of the world’s climate during the
Pleistocene (beginning about 2 million years ago). On theother hand, the climate has been very stable according to
the ice core data for the last 10,000 years, or about 500
generations. (Nevertheless, even in a stable climate, range
expansion and migration may have caused drastic andrapid changes in the environment governing the evolutionof early humans.) The irreversibility of our model may
account for the observation that our social learning ability
has not been lost.

ARTICLE IN PRESS
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494

In stark contrast, the Wakano et al. (2004) major genemodel predicts a sharp threshold, l M , above and below
which obligate innate behavior and a polymorphism of
obligate individual and social learners, respectively, areglobally stable. Here, l M is deﬁned as the largest integerthat satisﬁes the inequalityZðlÞoð1  cÞ

l

(15)

where Z(l) is given by (6b). Incidentally, l m ol M (Appendix
F), which entails more stringent conditions for the invadability of the pure innate strategy in the present model.
We have begged the question of the accuracy of social
learning, by assuming that the behavior of the exemplar isfaithfully reproduced in the observer. However, a more
fundamental analysis would require us to show that social
learning does indeed evolve toward greater precision.
Given that it does, when the reliability of social learning
has attained a level sufﬁcient for the development of acumulative culture, we would expect that culture to affectthe environment, in particular render it more, or less,stable. Clearly, such ‘‘cultural niche construction’’ (Laland
et al., 2000; Odling-Smee et al., 2003; Ihara and Feldman,2004) could have far reaching implications. (We say more,or less, because it is not obvious what the long-term, as
opposed to the immediate effects, of an innovation will be.
For example, the control of ﬁre and the making of clothing
greatly modiﬁed the hominid niche, but these innovations
also provided protection against temperature ﬂuctuations.)
To fully understand the nature of human social learning, itis necessary to take into account the fact that a signiﬁcantpart of our environment is, and has been, ‘‘man-made’’.
Note our use of the adjective ‘‘natural’’ in the title of thepresent paper to emphasize the limitations of the research
described here.
Acknowledgments

whereV^ ¼ ð1  K  LÞ½r þ ð1  rÞð1  sÞþ Lð1  cÞ þ Kð1  d  sÞ,

ðA:1bÞ

and from (2) thatði1Þ
Þ½1  K  L þ Lð1  cÞu^ ðiÞ ¼ ðu^ ði1Þ þ ū^^ ði1Þ ,þ Kðu^ ði1Þ þ v^ði1Þ Þð1  dÞ=W
ðiÞ
ði1Þ
ði1Þ
ði1Þ
ū^ ¼ ðu^ ði1Þ þ ū^ÞKðū^þ v̄^Þ^ ði1Þ , ð1  d  sÞ=Wði1Þv^ðiÞ ¼ ð^vði1Þ þ v̄^Þ½Lð1  cÞ^þ Kðu^ ði1Þ þ v^ði1Þ Þð1  dÞ=WðiÞv̄^ ¼ ð^vði1Þ þ v̄^þ Kðū^

ði1Þ

ði1Þ

ði1Þ

,

Þ½ð1  K  LÞð1  sÞ

þ v̄^

ði1Þ

^ ði1Þ ,Þð1  d  sÞ=W

ðA:2aÞ

where^ ði1Þ ¼ ð1  K  LÞ½u^ ði1Þ þ ū^ ði1ÞWði1Þþ ð^vði1Þ þ v̄^Þð1  sÞþ Lð1  cÞ þ K½ðu^ ði1Þ þ v^ði1Þ Þð1  dÞþ ðū^

ði1Þ

þ v̄^

ði1Þ

Þð1  d  sÞðA:2bÞ

for 2pipl. Recursions (A.2) can be solved iteratively with
(A.1) as the initial conditions to yield a periodic solution.
Global stability in the genetically monomorphic subspaceis assured, since the phenogenotype frequencies in the
postchange generation assume ﬁxed values (see formulae
ð1Þð1Þ
for u^ ð1Þ ; ū^ ; v^ð1Þ ; v̄^ in (A.1)).
Appendix B
Take xðiÞ þ x̄ðiÞ and yðiÞ þ ȳðiÞ as the small variables. Letx þ x̄ð0Þ and yð0Þ þ ȳð0Þ be the initial deviations from theð0Þð0Þ
equilibrium point ðu^ ð0Þ ; ū^ ; v^ð0Þ ; v̄^ Þ, where the superscript
0 denotes the prechange generation. Then, after one
application of (1) followed by l1 applications of (2), we
have
!
!
!
!
l 1 gðiÞYa axðlÞ þ x̄ðlÞxð0Þ þ x̄ð0Þ
0
¼

,
b b
yðlÞ þ ȳðlÞyð0Þ þ ȳð0Þ0 xðiÞ
i¼1
ð0Þ

We thank Yasuo Ihara, Wataru Nakahashi, and theanonymous referees for comments on a previous draft.
This research was supported in part by the 21st Century
Center of Excellence Program of the Department of
Biological Sciences and the Department of Biochemistry
and Biophysics, University of Tokyo.
Appendix A
Using superscripts to denote the number of generations
that have elapsed after an environmental change, we
observe from (1) thatu^ ð1Þ ¼ r½1  K  L þ Lð1  cÞ=V^ ,where the elements of the coefﬁcient matrices are (3b).
Clearly, the dominant eigenvalue is (3a).
Appendix C
Substitute K ¼ L ¼ 0 in (A.1) and (A.2). Then,ð1Þū^ ¼ rKð1  d  sÞ=V^ ,v^ð1Þ ¼ ð1  rÞLð1  cÞ=V^ ,

ðiÞ

ð1Þ
v̄^ ¼ ð1  rÞ½ð1  K  LÞð1  sÞ þ Kð1  d  sÞ=V^ ,ðA:1aÞū^ ¼ v^ðiÞ ¼ 0 for 1pipl,(C.1a)

V^ ¼ r þ ð1  rÞð1  sÞ,(C.1b)^ ðiÞ ¼ u^ ðiÞ þ v̄^ ðiÞ ð1  sÞW

for 1pipl  1.

(C.1c)ARTICLE IN PRESS
J.Y. Wakano, K. Aoki / Theoretical Population Biology 70 (2006) 486–497

Next, taking the ratio of the ﬁrst and fourth lines of(A.2a) yields the linear recursion

Appendix E
The terms multiplied by d or d þ s on the right-hand sideof (4c) are negative. Hence,u^ ðiÞ u^ ði1Þ 1,¼ ði1Þ
ðiÞ
1s
v̄^
v̄^

(
CK o

which can be solved with the initial conditionru^ ð1Þ
.
¼
ð1Þ
ð1  rÞð1  sÞv̄^r½V^ þ ð1  K  LÞð1  rÞs
V^



ðiÞ
l 1 ^ ðiÞYW þ ð1  K  LÞð^vðiÞ þ v̄^ Þs

(



(


i
r
1
u^ ðiÞ
¼,
ðiÞ
1r 1sv̄^þor equivalently



ðiÞ

l1
X

^
i¼1 W

ðiÞ

ðiÞ
þ ð1  K  LÞð^vðiÞ þ v̄^ Þs

ð1  rÞ½V^  ð1  K  LÞrsV^ðiÞl 1 ^ ðiÞYW  ð1  K  LÞðu^ ðiÞ þ ū^ Þs

(

(C.2)



)

ðiÞ

ðū^ þ v̄^ Þs

l1
X

^
i¼1 W

)

^ ðiÞ
W

i¼1

r
.
r þ ð1  rÞð1  sÞi

)

^ ðiÞ
W

i¼1

Thus

u^ ðiÞ ¼495)

ðu^ ðiÞ þ v^ðiÞ Þs
ðiÞ

ðiÞ

 ð1  K  LÞðu^ ðiÞ þ ū^ Þs.

Substituting (C.2) in (C.1) gives
ðiÞ

iþ1

^ ðiÞ ¼ r þ ð1  rÞð1  sÞWr þ ð1  rÞð1  sÞi

for 1pipl  1.

(C.3)

Appendix D

Since K þ L ¼ 0, and Appendix C informs us that ū^ ¼ðiÞ^ ðiÞ ¼ u^ ðiÞ þ v̄^ ðiÞ ð1  sÞ,v^ ¼ 0, V^ ¼ r þ ð1  rÞð1  sÞ, and W
this simpliﬁes to
V^

l 1
1Y
1
.
^
^ ðiÞ
V
W

(D.1a)

i¼1

Similarly from (4c)
( "

l1
X
ðiÞ
v̄^
r ld þ s 1 þ

CK ¼ 
" ld  s

l1

l1
l1
X
X
u^ ðiÞ sðiÞ.
v̄^ s þ ð1  rÞð1  sÞl
1s
i¼1
i¼1

Thus, substituting the explicit values of u^ ðiÞ and v̄^Appendix C, we obtainV^!#þ ð1  rÞð1  sÞ

^ ðiÞ  C K o  rWi¼1

Substituting K ¼ L ¼ 0 and (C.1) in (4b) gives
C L ¼ ½rlc þ ð1  rÞlðs  cÞð1  sÞl1 

l 1
Y

l 1
Y

^ ðiÞ  C K orð1  rÞs
W

i¼1

l1
X
ð1  sÞl1  ð1  sÞii¼1

r þ ð1  rÞð1  sÞi

ðiÞ

from

o0.

i¼1

l1
X

#)
u^

ðiÞ

i¼1

l 1
1Y
1
.
^
^ ðiÞ
V
W

ðD:1bÞ

i¼1

Using (C.1b) and (C.3), we achieve the simpliﬁcation
l 1
1Y
1
1
¼
.
ðiÞ
^
^
V
r þ ð1  rÞð1  sÞlW

Appendix FZðlÞThe ratio Zðl1Þ
, where Z(l) is deﬁned in (6b), is monotone
increasing in l. Moreover, Zð1Þ
Zð0Þ ¼ r þ ð1  rÞð1  sÞ andZðlÞ
lim Zðl1Þ ¼ 1. Hence, provided (8) holds, there exists a
l!1

(D.2)

unique root, y, of the equationi¼1

We now use the assumption r  ð1  sÞl to obtain the
zeroth order approximation in the small quantity r. First,
1^ ðiÞ in(D.2) reduces to ð1sÞl . Second, (C.2) shows that u(D.1b) are ﬁrst order in r, and hence can be neglected.
Thus,
CL ¼

lðs  cÞ;1s

as required.

CK ¼ 

ld
1s

ZðyÞ
¼1c
Zðy  1ÞNext, we show that (7) is more stringent than (15).
Suppose lpl m . ThenZðlÞ ¼Zð1Þ Zð2ÞZðlÞ  oð1  cÞl ,Zð0Þ Zð1ÞZðl  1Þsince Zð0Þ ¼ 1. Hence, lpl m implies lpl M .

ARTICLE IN PRESS
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496

Appendix G
Setting K þ L ¼ 1 in (A.1a) and taking the ratio of thebinary sums of appropriate lines yields
Lð1  cÞu^ ð1Þ þ v^ð1Þ
.
¼
ð1Þ
ð1Þ
Kð1  d  sÞū^ þ v̄^Next, doing the same with (A.2a) gives
Lð1  cÞ þ Kðu^ ði1Þ þ v^ði1Þ Þð1  dÞu^ ðiÞ þ v^ðiÞ
¼
ðiÞ
ðiÞ
ði1Þ
ði1Þ
Kðū^
þ v̄^
Þð1  d  sÞū^ þ v̄^¼

Lð1  cÞ þ Kð1  dÞ u^ ði1Þ þ v^ði1Þ
ði1Þ
ði1Þ
Kð1  d  sÞ
ū^
þ v̄^
Lð1  cÞ.
þ
Kð1  d  sÞLð1cÞ
Hence, setting a ¼ Lð1cÞþKð1dÞKð1dsÞ and b ¼ Kð1dsÞ, we obtain

1  aiu^ ðiÞ þ v^ðiÞ
,
¼
b
ðiÞ
ðiÞ
1a
ū^ þ v̄^or equivalentlyu^ ðiÞ þ v^ðiÞ ¼

bð1  ai Þ.
1  a þ bð1  ai Þ

Appendix H
Any alternative to the pure individual learning strategy
has dPp0 and dQ40, where consistency requires
dQodPp0. When L ¼ 1,
CP ¼

½2lð1  rÞ  1s  lðc þ dÞ,1cs  c þ d  ðl  1Þðc  dÞ,1cwhere (12) entails CQo0. If CP40, then clearly lo1, andwe are done. If CPo0, the constraint dQodPp0
implies CPdPoCPdQ, so that lo1(CPCQ)dQ.
But CPCQ ¼ 2l[(1r)sc]/(1c). Hence, lo1 whenco(1r)s, or equivalently (8).

CQ ¼ 

Appendix I
The eigenvalue (3) is a function of the wild type strategy(K, L) and the mutant strategy (K þ dK; L þ dL). As apurely hypothetical example, supposel ¼ 1 þ a½ðK  K  ÞdK þ ðL  L ÞdL þ b½ðdKÞ2 þ ðdLÞ2 ,
where (K*, L*) is an interior point, and a, b are constants.
Then, C K ¼ aðK  K  Þ and C L ¼ aðL  L Þ. Clearly,(K*, L*) is an ESS if bo0, i.e., if the quadratic form in
dK and dL is negative deﬁnite. However, if a40, then all
vectors originating at neighboring points (K  þ dK,

L þ dL) are directed outward from (K*, L*). Hence,(K*, L*) is not a CSS. Conversely, if ao0 but b40, then
(K*, L*) is a CSS but not an ESS, since all vectors
originating at neighboring points (K  þ dK, L þ dL) aredirected toward (K*, L*), but the eigenvalue at (K*, L*) is
greater than one.
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