Abstract
We study a model of social learning on rooted regular trees. An agent is stationed at each vertex of
double struck upper T Subscript m
T
m
$\mathbb{T}_{m}$
, the rooted tree in which each vertex has precisely
m
children, and at any time step
t element of double struck upper N 0
t
∈
N
0
$t \in \mathbb{N}_{0}$
, the agent is allowed to select one of two available technologies:
B
and
R
. Let the technology chosen by the agent at vertex
v
of
double struck upper T Subscript m
T
m
$\mathbb{T}_{m}$
, at time step
t
, be
upper C Subscript t Baseline left parenthesis v right parenthesis
C
t
(
v
)
$C_{t}(v)$
. We begin with the independent and identically distributed (i.i.d.) collection
StartSet upper C 0 left parenthesis v right parenthesis colon v element of double struck upper T Subscript m Baseline EndSet
{
C
0
(
v
)
:
v
∈
T
m
}
$\{C_{0}(v)\,:\, v \in \mathbb{T}_{m}\}$
, where
upper C 0 left parenthesis v right parenthesis equals upper B
C
0
(
v
)
=
B
$C_{0}(v)=B$
with probability
pi 0
π
0
$\pi_{0}$
. During the epoch
t
, the agent at vertex
v
performs an experiment that results in success with probability
p Subscript upper B
p
B
$p_{B}$
if
upper C Subscript t Baseline left parenthesis v right parenthesis equals upper B
C
t
(
v
)
=
B
$C_{t}(v)=B$
, and with probability
p Subscript upper R
p
R
$p_{R}$
if
upper C Subscript t Baseline left parenthesis v right parenthesis equals upper R
C
t
(
v
)
=
R
$C_{t}(v)=R$
. If the children of
v
are denoted
v 1 comma ellipsis comma v Subscript m Baseline
v
1
,
…
,
v
m
$v_{1}, \ldots, v_{m}$
, the agent at
v
updates their technology to
upper C Subscript t plus 1 Baseline left parenthesis v right parenthesis equals upper B
C
t
+
1
(
v
)
=
B
$C_{t+1}(v)=B$
if the number of successes among all
v Subscript i
v
i
$v_{i}$
(where
i element of StartSet 1 comma 2 comma ellipsis comma m EndSet
i
∈
{
1
,
2
,
…
,
m
}
$i \in \{1,2,\ldots,m\}$
) with
upper C Subscript t Baseline left parenthesis v Subscript i Baseline right parenthesis equals upper B
C
t
(
v
i
)
=
B
$C_{t}(v_{i})=B$
exceeds, strictly, the number of successes among all
v Subscript j
v
j
$v_{j}$
(where
j element of StartSet 1 comma 2 comma ellipsis comma m EndSet
j
∈
{
1
,
2
,
…
,
m
}
$j \in \{1,2,\ldots,m\}$
) with
upper C Subscript t Baseline left parenthesis v Subscript j Baseline right parenthesis equals upper R
C
t
(
v
j
)
=
R
$C_{t}(v_{j})=R$
. If these two numbers are equal then the agent at
v
sets
upper C Subscript t plus 1 Baseline left parenthesis v right parenthesis equals upper B
C
t
+
1
(
v
)
=
B
$C_{t+1}(v)=B$
with probability
1 divided by 2
1
/
2
$1/2$
. In all other cases,
upper C Subscript t plus 1 Baseline left parenthesis v right parenthesis equals upper R
C
t
+
1
(
v
)
=
R
$C_{t+1}(v)=R$
. We show that
StartSet upper C Subscript t Baseline left parenthesis v right parenthesis colon v element of double struck upper T Subscript m Baseline EndSet
{
C
t
(
v
)
:
v
∈
T
m
}
$\{C_{t}(v)\,:\, v \in \mathbb{T}_{m}\}$
is i.i.d. as well, with
upper C Subscript t Baseline left parenthesis v right parenthesis equals upper B
C
t
(
v
)
=
B
$C_{t}(v)=B$
with probability
pi Subscript t
π
t
$\pi_{t}$
, where the sequence
left brace pi Subscript t Baseline right brace Subscript t element of double struck upper N 0
{
π
t
}
t
∈
N
0
$\{\pi_{t}\}_{t \in \mathbb{N}_{0}}$
converges to a fixed point
pi
π
$\pi$
, in [0, 1], of a function
g Subscript m
g
m
$g_{m}$
. We show that for
m greater than or slanted equals 3
m
⩾
3
$m \geqslant 3$
, there exists a
p left parenthesis m right parenthesis element of left parenthesis 0 comma 1 right parenthesis
p
(
m
)
∈
(
0
,
1
)
$p(m) \in (0,1)$
such that
g Subscript m
g
m
$g_{m}$
has the unique fixed point
1 divided by 2
1
/
2
$1/2$
when
p less than or slanted equals p left parenthesis m right parenthesis
p
⩽
p
(
m
)
$p \leqslant p(m)$
, and three distinct fixed points, of the form
alpha
α
$\alpha$
,
1 divided by 2
1
/
2
$1/2$
, and
1 minus alpha
1
−
α
$1-\alpha$
, for some
alpha element of left bracket 0 comma 1 divided by 2 right parenthesis
α
∈
[
0
,
1
/
2
)
$\alpha \in [0,1/2)$
when
p greater than p left parenthesis m right parenthesis
p
>
p
(
m
)
$p > p(m)$
. When
m equals 3
m
=
3
$m=3$
,
p Subscript upper B Baseline equals 1
p
B
=
1
$p_{B}=1$
, and
p Subscript upper R Baseline element of left bracket 0 comma 1 right parenthesis
p
R
∈
[
0
,
1
)
$p_{R} \in [0,1)$
, we show that the function
g 3
g
3
$g_{3}$
(i) has a unique fixed point, 1, when
p Subscript upper R Baseline less than StartRoot 3 EndRoot minus 1
p
R
<
3
−
1
$p_{R} < \sqrt{3}-1$
, (ii) has two distinct fixed points, one of which is 1, when
p Subscript upper R Baseline equals StartRoot 3 EndRoot minus 1
p
R
=
3
−
1
$p_{R} = \sqrt{3}-1$
, and (iii) has three distinct fixed points, one of which is 1, when
p Subscript upper R Baseline greater than StartRoot 3 EndRoot minus 1
p
R
>
3
−
1
$p_{R} > \sqrt{3}-1$
. When
g Subscript m
g
m
$g_{m}$
has multiple fixed points, we also specify which of these fixed points
pi
π
$\pi$
equals, depending on
pi 0
π
0
$\pi_{0}$
. Finally, for
m equals 2
m
=
2
$m=2$
, we describe the behaviour of
g 2
g
2
$g_{2}$
for
all
values of
p Subscript upper B
p
B
$p_{B}$
and
p Subscript upper R
p
R
$p_{R}$
.