z
i,τ(T) z
ρ(i),τ-1(T)
z
i,τ(T) z
ρ(i),τ-1(T)
Figure 1
0 0.2 0.4 0.6 0.8 1
0 5 10 15 20 25 30 35 40 COS
q=1-10-6 (N=106) q=1-10-4 (N=104) q=1-10-2 (N=102)
(a) x*
0 0.2 0.4 0.6 0.8 1
0 5 10 15 20 25 30 35 40 COS
q=1-10
-6 (N=10
6) q=1-10
-4 (N=10
4) q=1-10
-2 (N=102 )
(b) v*
0 5 10 15 20 25 30 35 40 1
102 104 106 108
10-2
(c) z*(T)
COS
q=1-10-6 (N=106) q=1-10-4 (N=104) q=1-10-2 (N=102)
~
Figure 2
0 0.2 0.4 0.6 0.8 1
0 5 10 15 20
Generation [x10
5]
0 50 100 150 200 250 300
0 5 10 15 20
(a)
(b)
x v
z(T)
Figure 3
0 0.2 0.4 0.6 0.8 1
0 5 10 15 20 25 30 35 40 q=1.0
q=0.99 q=0.9 Theory (q=1.0)
0 0.2 0.4 0.6 0.8 1
0 5 10 15 20 25 30 35 40 q=1.0
q=0.99 q=0.9 Theory (q=1.0)
0 20 40 60 80 100 120 140
0 5 10 15 20 25 30 35 40 q=1.0
q=0.99 q=0.9
(a)
(b)
(c)
x
v
z(T)
Theory (q=1.0)
Figure 4
0 0.2 0.4 0.6 0.8 1
0 200 400 600 800 1000 1200
0 5 10 15 20
Mutant (CO
S) fitness
Resident (ESS) fitness
Probability of mutant survival
Generation
F it ne ss
P roba bi li ty
Figure 5
N Population size
q Vertical transmission rate
T Lifetime
Efficiency of social learning
Efficiency of individual learning
,
vi The fraction of the lifetime invested in learning by individual (i,)
,
xi
The fraction of the learning time invested in individual learning by individual (i,)
)
, (t
zi The z-value of individual (i,) at within-generation time t.
)
~(T
z The equilibrium mature z-value in a genetically monomorphic population
,
wi The fitness of individual (i,)
w~ The equilibrium fitness in a genetically monomorphic population v, x, ~z(T) The COS values of vi, , xi,, and ~z(T), respectively.
*
v , x*, ~z*(T) The ESS values of vi,, xi,, and ~z(T), respectively.
v, x, z(T) The population averages of vi,, xi,, and ~z(T), respectively.
Table 1: Notation Tables
Supporting information
A Paradox of Cumulative Culture
Yutaka Kobayashi
a,b, Joe Y. Wakano
c, Hisashi Ohtsuki
dMarch 12, 2015
aResearch Center for Social Design Engineering, Kochi University of Technology, Kochi 782-8502, Japan, bDepartment of Management, Kochi
University of Technology, Kochi 782-8502, Japan, cMeiji Institute for Advanced Study of Mathematical Sciences, Nakano, Tokyo 164-8525, Japan, dThe Graduate University for Advanced Studies, Shonan Village,
Hayama, Kanagawa 240-0193, Japan
Appendix A: Derivation of the COS
1
To derive the COS, let us assume that the population is monomorphic for
2
a learning strategy (x, v). Solving eq. (1) in the main text with respect to
3
1
zi,τ(t) under the assumption that zi,τ(0) = 0 and (xi,τ, vi,τ) = (x, v), we have
4
zi,τ(t) =zρτ(i),τ−1(T)(1−e−βt). (11)
It follows that thez-value at the end of the social learning stage (t=v(1−x))
5
is given by
6
zi,τ(v(1−x)) =zρτ(i),τ−1(T)(1−e−βv(1−x)). (12) Further, from eq. (2) in the main text, the value of zi,τ(t) at the end of the
7
individual-learning stage (t=v) is given by
8
zi,τ(v) = zi,τ(v(1−x)) +vx
= zρτ(i),τ−1(T)(1−e−βv(1−x)) +vx. (13)
Noting that zi,τ(v) = zi,τ(T), we have
9
zi,τ(T) = zρτ(i),τ−1(T)(1−e−βv(1−x)) +vx. (14)
This equation gives the between-generation dynamics of zi,τ(T). From
10
eq. (14), the equilibrium value of zi,τ(T), denoted by ˜z(T), is given by
11
˜
z(T) = lim
τ→∞zi,τ(T) = vxeβv(1−x). (15)
2
The equilibrium fitness function, denoted by ˜w, is therefore given by
12
˜
w= lim
τ→∞wi,τ = lim
τ→∞zi,τ(T)·(1−v)
= v(1−v)xeβv(1−x). (16)
The COS is the strategy (x, v) which maximizes eq. (16). It is easily shown
13
that the strategy (x◦, v◦) given by eq. (6) in the main text maximizes eq.
14
(16) and hence gives the COS.
15
Appendix B: Derivation of the ESS in an
infi-16
nite population
17
We define an evolutionarily stable learning strategy in an infinite population
18
as a learning strategy that is resistant against invasion by rare mutants with
19
any slightly deviated strategy. We will derive eq. (7) in the main text, which
20
an ESS must satisfy.
21
Let (x, v) and (x′, v′) denote the resident and mutant strategies,
respec-22
tively. We assume that the resident population is at cultural equilibrium, so
23
that all residents have thez-value given by eq. (15) at the end of the learning
24
stage. In order to derive the ESS, we classify individuals as follows. Residents
25
are class 0. The mutants who socially learned from residents are class 1. The
26
mutants who socially learned from class-1 individuals are class 2. Class-j
27
individuals are defined recursively. Note that offspring of class-j mutants fall
28
3
back to class 1 when their cultural role models are residents (oblique social
29
learning). In this case, cultural accumulation over j generations by mutants
30
is reset.
31
From eq. (14), the mature z-value of an individual (i, τ) in class j ≥ 1
32
satisfies
33
zi,τ(T) =zρτ(i),τ−1(T)(1−e−βv′(1−x′)) +v′x′. (17) Note that the above equation recursively applies, so thatzρτ(i),τ−1(T) is given
34
as a function of zρτ−1(ρτ(i)),τ−2(T), which is in turn given as a function of
35
zρτ−2(ρτ−1(ρτ(i))),τ−3(T), and so on. Given that individual (i, τ) belongs to
36
class j, individual (ρτ−(j−1)(ρτ−(j−2)(. . .(ρτ−1(ρτ(i))). . .)), τ −j) belongs to
37
class 0 and is hence a resident. Noting this and eq.(15), eq. (17) can be
38
solved to yield
39
zi,τ(T) =v′x′eβv′(1−x′)+rCτ(i)(vxeβv(1−x)−v′x′eβv′(1−x′)), (18)
where Cτ(i) denotes the class of individual (i, τ) and
40
r= 1−e−βv′(1−x′). (19)
Note that eq. (18) does not depend on iand τ but only on the classCτ(i) of
41
individual (i,τ). This implies that the fitness of an individual also depends
42
4
only on its class. Therefore, we let w′j denote the fitness of class-j mutants:
43
w′Cτ(i) :=zi,τ(T)(1−v′), (Cτ(i)≥1) (20)
It is easily confirmed that mutants have the same fitness as residents
irre-44
spective of classes (i.e. wj′ = ˜w = v(1−v)xeβx(1−v) for arbitrary j ≥ 1) if
45
they adopt the same strategy as residents ((x′, v′) = (x, v)).
46
Letpj,τ denote the frequency of class-j mutants (j ≥1) in the population
47
in generationτ. Since mutants are rare, we may assume that a mutant’s role
48
model is a mutant only when vertical transmission occurs. The offspring of
49
a class-j mutant hence belong to class-(j+ 1) and class-1 with probabilities
50
q and 1−q, respectively. Further, because of rarity of mutants, the average
51
fitness of the population is approximated by the residents’ fitness ˜wgiven by
52
eq. (16). From these arguments, it holds that
53
p1,τ+1 =
∞
∑
j=1
(1−q)wj′
˜
wpj,τ, (21)
54
pj+1,τ+1 =qw′j
˜
wpj,τ, (22)
where j ≥1.
55
Note that the above equation is formally equivalent to the standard model
56
of age structure. Substitutingpj,τ+1 =λpj,τ into eqs. (21) and (22) and
rear-57
ranging the resulting equations, it is easily shown that the leading eigenvalue
58
λ, i.e. the asymptotic growth rate of mutants, should satisfy the following
59
5
(Euler-Lotka) characteristic equation:
60
1 =
∞
∑
i=0
(1−q)qiλ−i−1
i+1
∏
j=1
wj′
˜
w. (23)
Note that, when mutants have the same fitness as residents (i.e. wj = ˜w
61
for all j’s), λ = 1 is the only solution of eq. (23). This implies that the
62
frequency of mutants remains constant when they adopt the same strategy
63
as residents.
64
Differentiating eq. (23) with respect to a mutant strategic variable y′
65
(y′ ∈ {x′, v′}) yields
66
0 =
∞
∑
i=0
(1−q)qi(−i−1)λ−i−2∂λ
∂y′
i+1
∏
j=1
wj′
˜ w +
∞
∑
i=0
(1−q)qiλ−i−1
i+1
∑
k=1
wk′−1∂w′k
∂y′
i+1
∏
j=1
w′j
˜
w. (24)
Substitutingx′ =x,v′ =v,w′j = ˜w, and λ= 1 into eq. (24) and rearranging
67
the resulting equation yield
68
˜ w ∂λ
∂y′
x′
=x,v′=v
= ∂w′
∂y′
x′
=x,v′=v
, (25)
where
69
w′ =
∞
∑
i=1
(1−q)qi−1wi′. (26) If the stationary growth rate of mutants is larger than one, mutants can
70
invade. Therefore, for the resident strategy (x, v) to be evolutionarily stable,
71
6
λ must be maximized at (x′, v′) = (x, v) as a function of the mutant strategy
72
(x′, v′). However, this and eq. (25) together imply that w′ is maximized at
73
(x′, v′) = (x, v). Thus, for our ESS analysis we may treatw′ like the mutant
74
invasion fitness.
75
In fact, w′ can be interpreted as the asymptotic average of the mutant
76
invasion fitness, as follows. Note that the leading eigenvector of the system
77
(21-22) is given by (1, q, q2, . . . , qi−1, . . .). This means that the fraction of
78
class i among mutants asymptotically approaches (1−q)qi−1 when selection
79
is absent ((x′, v′) = (x, v)). Thus, when selection is sufficiently weak, the
80
average fitness of mutants is asymptotically given by∑∞i=1(1−q)qi−1wi′ =w′.
81
Using eq. (18), (26) and (20), we find that
82
w′ = (1−v′)v′x′eβv′(1−x′) +(1−v′)r(1−q)
1−rq (vxeβv(1−x)−v′x′eβv′(1−x′)). (27)
For (x, v) to be the ESS, w′ as a function of (x′, v′) must be maximized at
83
(x′, v′) = (x, v). Thus, the ESS (x∗, v∗) satisfies
84
∂w′
∂x′
x′
=x=x∗,v′=v=v∗
= 0, (28)
85
∂w′
∂v′
x′=x=x∗,v′=v=v∗
= 0. (29)
It is easily shown that these equations reduce to eqs. (7a) and (7b) in the
86
main text. Finally, substituting eq. (7a) in the main text into eq. (15) yields
87
7
eq. (7c).
88
Appendix C: Derivation of the ESS in a finite
89
population
90
Here we derive the ESS in a finite population assuming pure vertical
trans-91
mission (q = 1) (eq. (9) in the main text). More specifically, we show that
92
the ESS for a finite population of size N under q = 1 is identical with the
93
ESS for an inifinite population under q = 1−1/N. Thus, in terms of the
94
ESS, decreasing the population size from ∞ to N under q = 1 has exactly
95
the same effect as decreasing q by 1/N in an infinite population.
96
To compute the ESS under q = 1, we need the fixation probability of a
97
mutant strategy that is initially expressed by a single individual. For this
98
purpose, we apply the method introduced by Rousset (2004) below.
99
Imagine that a mutant strategy (x′, v′) is expressed by a single individual
100
in the population of the resident strategy (x, v). For convenience sake, let us
101
reuse the classification of individuals introduced in Appendix B. Then, the
102
initial single mutant is obviously of class 1 because there is no mutant in the
103
previous generation. Sinceq = 1 (pure vertical transmission), any mutant in
104
any generation τ inherits culture from its own parent, which is a mutant in
105
generationτ−1. This implies that all mutants in generationτ belong to class
106
τ (Cτ(i) = τ for any mutant (i, τ)), given that the mutant was introduced
107
in generation 1. Therefore, all mutants in generation τ have equal fitnesses
108
8
given by wτ′ in eq. (20). It is important that the mutant fitness is not a
109
stochastic variable but is determined by the number of generations passed
110
since introduction of the initial mutant. By virtue of this property, we can
111
treat this process as a Wright-Fisher process in which the selection coefficient
112
depends deterministically on time (see below).
113
LetPτ denote the frequency of mutants in generationτ. Since all mutants
114
in generation τ belong to class τ, it holds that Pτ = ∑jpj,τ = pτ,τ in
Ap-115
pendix B’s notation. Note that we assume a Wright-Fisher-type update for
116
the genetic state of the population and also culture is transmitted between
117
adjacent generations; thus, Pτ obeys a time-inhomogeneous Markov process
118
with the initial state P1 = 1/N. Obviously, this stochastic process has only
119
two absorbing states: Pτ = 1 (fixation) and Pτ = 0 (extinction). Let π
120
denote the fixation probability of the mutant strategy. Then, the expected
121
frequency of mutants in the infinitely distant future should be given by
122
τlim→∞E[Pτ] = 1·π+ 0·(1−π) =π, (30)
where E[·] denotes expectation. Below we use this relationship to compute
123
π.
124
Note that we can write
125
Pτ =P1+ ∆P1+ ∆P2 +. . .+ ∆Pτ−1, (31)
where ∆Pτ = Pτ+1−Pτ denotes the frequency change between generations
126
9
τ and τ+ 1 and is a stochastic variable itself. Substituting eq. (31) into eq.
127
(30) yields
128
π = E[P1+
∞
∑
τ=1
∆Pτ]
= 1
N +
∞
∑
τ=1
E[∆Pτ], (32)
where we used E[P1] = P1 = 1/N. From the standard theory of population
129
genetics, the frequency change ∆Pτ is given by
130
∆Pτ = w′τ−w˜
˜
w+Pτ(wτ′ −w)˜ Pτ(1−Pτ), (33)
where ˜w is the equilibrium fitness of residents given by eq. (16). Let us
131
define the selection coefficient sτ as
132
sτ = w′τ −w˜
˜
w . (34)
Substituting (34) into eq. (33) yields
133
∆Pτ = sτ
1 +Pτsτ
Pτ(1−Pτ)≈sτPτ(1−Pτ), (35)
where the approximation holds for small sτ.
134
Substituting eq. (35) into eq. (32) yields
135
π ≈ 1 N +
∞
∑
t=1
sτE[Pτ(1−pτ)]. (36)
10
Note that the expectationE[Pτ(1−Pτ)] in the above equation is itself affected
136
by selection coefficients of up to generation τ −1 (i.e., s1, s2, s3, . . . , sτ−1).
137
However, Rousset (2004) has shown that the expectation E[·] can be
approx-138
imately replaced by the expectation under neutrality (i.e. s0 = s1 = . . . =
139
st = . . . = 0) provided selection is sufficiently weak. We denote the
expec-140
tation under neutrality by E◦[·] following Rousset (2004). Thus, it holds
141
that
142
π ≈ 1 N +
∞
∑
t=1
sτE◦[Pτ(1−Pτ)]. (37) Note that E◦[2Pτ(1−Pτ)] can be interpreted as the probability that two
143
individuals drawn at random with replacement from generation τ have
dif-144
ferent genotypes under selective neutrality. Such two individuals can have
145
different genotypes only if their ancestral lineages trace back to generation 1
146
without coalescing and, in addition, only one of them hits the initial mutant.
147
From the standard coalescent theory this probability is given by
148
E◦[2Pτ(1−Pτ)] =
(
1− 1 N
)τ−1
·2P1(1−P1)
= 2 1 N
(
1− 1 N
)τ
, (38)
where we used P1 = 1/N.
149
Substituting eqs. (34) and (38) into eq. (37) yields
150
π ≈ 1
N + 1 N
∞
∑
τ=1
(w′τ
˜ w −1
)(
1− 1 N
)τ
11
= 1 N +
(
1− 1 N
)(
w′
˜ w −1
)
, (39)
where
151
w′ =
∞
∑
τ=1
wτ′ 1 N
(
1− 1 N
)τ−1
. (40)
Remember that for a finite population we define an ESS as the strategy
152
that never allows a mutant strategy expressed by a single individual to have
153
a fixation probability higher than 1/N (i.e. the fixation probability of the
154
ESS itself). This implies that for our ESS analysis we can treat w′ like the
155
mutant invasion fitness in the standard ESS analysis in an infinite-population
156
model. Note that eq. (40) is formally identical with eq. (26) except that qis
157
replaced by 1−1/N. This implies that the ESS for a finite population under
158
pure vertical transmission (q = 1) is equivalent with the ESS for an infinite
159
population with q= 1−1/N.
160
Appendix D: Probabilistic engagement in
so-161
cial and individual learning
162
In the main text, we assumed that social and individual learning occur in
163
separate stages of life. In this Appendix, we instead assume that each
in-164
dividual engages in individual and social learning with probabilities x and
165
1−x, respectively, at any moment in the learning stage and derive eq. (14)
166
under some additional assumptions. Thus, the results of the present paper
167
12
all apply to this modified model.
168
Suppose that zi,τ(t) represents the amount of knowledge that the
indi-169
vidual (i, τ) acquires by time t through individual and social learning. Let
170
zi,τ,IL(t) and zi,τ,SL(t) denote the amounts of knowledge acquired through
171
individual and social learning, respectively, by time t. In addition, assume
172
that the knowledge acquired through individual learning never overlaps with
173
that acquired through social learning. This implies that any piece of
knowl-174
edge produced by an individual through individual learning is always new
175
to the role model of the focal individual as well as the focal individual
it-176
self. Then, the total amount of knowledge individual (i, τ) bears is given by
177
zi,τ(t) = zi,τ,SL(t) +zi,τ,IL(t).
178
Note that each individual engages in social learning with probability 1−x
179
at any moment in the learning stage. This implies thatzi,τ,SL(t) grows in the
180
learning stage as follows:
181
d
dtzi,τ,SL(t) =β(1−x)(zρτ(i),τ−1(T)−zi,τ,SL(t)). (0≤t≤v) (41)
Likewise, zi,τ,IL(t) follows
182
d
dtzi,τ,IL(t) = αx=x. (0≤t≤v) (42)
Integrating both equations yield
183
zi,τ,SL(v) =zρτ(i),τ−1(T)(1−e−βv(1−x)). (43)
13