<s>
A	O
highest-averages	O
method	O
,	O
also	O
called	O
a	O
divisor	B-Algorithm
method	I-Algorithm
,	O
is	O
a	O
class	O
of	O
methods	O
for	O
allocating	O
seats	O
in	O
a	O
parliament	O
among	O
agents	O
such	O
as	O
political	O
parties	O
or	O
federal	O
states	O
.	O
</s>
<s>
A	O
divisor	B-Algorithm
method	I-Algorithm
is	O
an	O
iterative	B-Algorithm
method	I-Algorithm
:	O
at	O
each	O
iteration	O
,	O
the	O
number	O
of	O
votes	O
of	O
each	O
party	O
is	O
divided	O
by	O
its	O
divisor	O
,	O
which	O
is	O
a	O
function	O
of	O
the	O
number	O
of	O
seats	O
(	O
initially	O
0	O
)	O
currently	O
allocated	O
to	O
that	O
party	O
.	O
</s>
<s>
The	O
inputs	O
to	O
a	O
divisor	B-Algorithm
method	I-Algorithm
are	O
the	O
number	O
of	O
seats	O
to	O
allocate	O
,	O
denoted	O
by	O
h	O
,	O
and	O
the	O
vector	O
of	O
parties	O
 '	O
entitlements	O
,	O
where	O
the	O
entitlement	O
of	O
party	O
is	O
denoted	O
by	O
(	O
a	O
number	O
between	O
0	O
and	O
1	O
determining	O
the	O
fraction	O
of	O
seats	O
to	O
which	O
is	O
entitled	O
)	O
.	O
</s>
<s>
A	O
divisor	B-Algorithm
method	I-Algorithm
is	O
parametrized	O
by	O
a	O
function	O
,	O
mapping	O
each	O
integer	O
to	O
a	O
real	O
number	O
(	O
usually	O
in	O
the	O
range	O
)	O
.	O
</s>
<s>
An	O
equivalent	O
definition	O
directly	O
gives	O
the	O
outcome	O
of	O
the	O
divisor	B-Algorithm
method	I-Algorithm
as	O
follows	O
.	O
</s>
<s>
Rounding	O
down	O
is	O
equivalent	O
to	O
using	O
the	O
D'Hondt	B-Algorithm
method	I-Algorithm
,	O
while	O
rounding	O
to	O
the	O
nearest	O
whole	O
number	O
is	O
equivalent	O
to	O
the	O
Sainte-Laguë	B-Architecture
method	I-Architecture
.	O
</s>
<s>
The	O
tables	O
used	O
in	O
the	O
D'Hondt	O
or	O
Sainte-Laguë	B-Architecture
methods	I-Architecture
can	O
then	O
be	O
viewed	O
as	O
calculating	O
the	O
highest	O
quotient	O
possible	O
to	O
round	O
off	O
to	O
a	O
given	O
number	O
of	O
seats	O
.	O
</s>
<s>
Formally	O
,	O
given	O
the	O
vector	O
of	O
entitlements	O
and	O
house-size	O
,	O
a	O
divisor	B-Algorithm
method	I-Algorithm
can	O
be	O
defined	O
as:where	O
the	O
method	O
of	O
rounding	O
is	O
defined	O
by	O
the	O
divisor	O
function	O
d	O
.	O
</s>
<s>
Every	O
divisor	B-Algorithm
method	I-Algorithm
can	O
be	O
defined	O
using	O
a	O
min-max	O
inequality	O
:	O
a	O
is	O
an	O
allocation	O
for	O
the	O
divisor	B-Algorithm
method	I-Algorithm
with	O
divisor	O
d	O
,	O
if-and-only-if	O
.	O
</s>
<s>
The	O
Imperiali	O
highest	B-Algorithm
average	I-Algorithm
method	I-Algorithm
(	O
not	O
to	O
be	O
confused	O
with	O
the	O
Imperiali	O
quota	O
which	O
is	O
a	O
Largest	B-Algorithm
remainder	I-Algorithm
method	I-Algorithm
)	O
has	O
divisors	O
1	O
,	O
1.5	O
,	O
2	O
,	O
2.5	O
,	O
3	O
,	O
3.5	O
etc.	O
,	O
or	O
equivalently	O
2	O
,	O
3	O
,	O
4	O
,	O
5	O
,	O
etc.	O
,	O
corresponding	O
to	O
a	O
divisor	O
function	O
.	O
</s>
<s>
It	O
is	O
called	O
the	O
D'Hondt	B-Algorithm
formula	I-Algorithm
.	O
</s>
<s>
The	O
Webster/Sainte	B-Architecture
-Laguë	I-Architecture
method	I-Architecture
divides	O
the	O
number	O
of	O
votes	O
for	O
each	O
party	O
by	O
the	O
odd	O
numbers	O
(	O
1	O
,	O
3	O
,	O
5	O
,	O
7	O
etc	O
.	O
</s>
<s>
This	O
system	O
is	O
more	O
favourable	O
to	O
smaller	O
parties	O
than	O
Hondt	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
and	O
thus	O
it	O
encourages	O
splits	O
.	O
</s>
<s>
The	O
Webster/Sainte	B-Architecture
-Laguë	I-Architecture
method	I-Architecture
is	O
sometimes	O
modified	O
by	O
increasing	O
the	O
first	O
divisor	O
from	O
1	O
to	O
e.g.	O
</s>
<s>
In	O
the	O
Huntington	B-Algorithm
–	I-Algorithm
Hill	I-Algorithm
method	I-Algorithm
,	O
the	O
divisor	O
function	O
is	O
,	O
which	O
makes	O
sense	O
only	O
if	O
every	O
party	O
is	O
guaranteed	O
at	O
least	O
one	O
seat	O
:	O
this	O
effect	O
can	O
be	O
achieved	O
by	O
disqualifying	O
parties	O
receiving	O
fewer	O
votes	O
than	O
a	O
specified	O
number	O
of	O
votes	O
,	O
be	O
it	O
a	O
percentage	O
threshold	O
or	O
a	O
quota	O
such	O
as	O
Hare	O
,	O
Droop	O
,	O
or	O
Imperiali	O
.	O
</s>
<s>
He	O
perceived	O
Jefferson	B-Algorithm
's	I-Algorithm
method	I-Algorithm
to	O
allocate	O
too	O
few	O
seats	O
to	O
smaller	O
states	O
.	O
</s>
<s>
It	O
can	O
be	O
described	O
as	O
the	O
inverse	O
of	O
Jefferson	B-Algorithm
's	I-Algorithm
method	I-Algorithm
;	O
it	O
awards	O
a	O
seat	O
to	O
the	O
party	O
that	O
has	O
the	O
most	O
votes	O
per	O
seat	O
before	O
the	O
seat	O
is	O
added	O
.	O
</s>
<s>
Like	O
the	O
Huntington-Hill	B-Algorithm
method	I-Algorithm
,	O
this	O
results	O
in	O
a	O
value	O
of	O
0	O
for	O
the	O
first	O
seats	O
to	O
be	O
appointed	O
for	O
each	O
party	O
,	O
resulting	O
in	O
an	O
average	O
of	O
∞	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
Hondt	B-Algorithm
's	I-Algorithm
method	I-Algorithm
,	O
in	O
the	O
row	O
,	O
the	O
numbers	O
are	O
just	O
the	O
parties	O
 '	O
votes	O
(	O
divided	O
by	O
)	O
.	O
</s>
<s>
For	O
the	O
Saint-Lague	B-Architecture
method	I-Architecture
,	O
in	O
the	O
row	O
,	O
the	O
numbers	O
are	O
the	O
votes	O
divided	O
by	O
3	O
(	O
the	O
second	O
element	O
in	O
the	O
sequence	O
of	O
divisors	O
)	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
As	O
can	O
be	O
seen	O
in	O
the	O
example	O
,	O
D'Hondt	O
,	O
Sainte-Laguë	B-Architecture
and	O
Huntington-Hill	O
allow	O
different	O
strategies	O
by	O
parties	O
looking	O
to	O
maximize	O
their	O
seat	O
allocation	O
.	O
</s>
<s>
D'Hondt	O
and	O
Huntington	O
–	O
Hill	O
can	O
favor	O
the	O
merging	O
of	O
parties	O
,	O
while	O
Sainte-Laguë	B-Architecture
can	O
favor	O
splitting	O
parties	O
(	O
modified	O
Saint-Laguë	O
reduces	O
the	O
splitting	O
advantage	O
)	O
.	O
</s>
<s>
In	O
these	O
examples	O
,	O
under	O
D'Hondt	O
and	O
Huntington	O
–	O
Hill	O
the	O
Yellows	O
and	O
Greens	O
combined	O
would	O
gain	O
an	O
additional	O
seat	O
if	O
they	O
merged	O
,	O
while	O
under	O
Sainte-Laguë	B-Architecture
the	O
Yellows	O
would	O
gain	O
if	O
they	O
split	O
into	O
six	O
lists	O
with	O
about	O
7,833	O
votes	O
each	O
.	O
</s>
<s>
All	O
divisor	B-Algorithm
methods	I-Algorithm
satisfy	O
the	O
basic	O
properties	O
of	O
anonymity	O
,	O
balance	O
,	O
concordance	O
,	O
exactness	O
and	O
completeness	O
.	O
</s>
<s>
All	O
the	O
divisor	B-Algorithm
methods	I-Algorithm
satisfy	O
house	O
monotonicity	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
divisor	B-Algorithm
methods	I-Algorithm
avoid	O
the	O
Alabama	O
paradox	O
.	O
</s>
<s>
Moreover	O
,	O
all	O
divisor	B-Algorithm
methods	I-Algorithm
satisfy	O
pairwise	O
population	O
monotonicity	O
.	O
</s>
<s>
Moreover	O
,	O
the	O
divisor	B-Algorithm
methods	I-Algorithm
are	O
provably	O
the	O
only	O
methods	O
satisfying	O
this	O
form	O
of	O
monotonicity	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
divisor	B-Algorithm
methods	I-Algorithm
are	O
the	O
only	O
ones	O
avoiding	O
the	O
population	O
paradox	O
.	O
</s>
<s>
On	O
the	O
negative	O
side	O
,	O
divisor	B-Algorithm
methods	I-Algorithm
might	O
violate	O
the	O
quota	O
rule	O
:	O
they	O
might	O
give	O
some	O
agents	O
less	O
than	O
their	O
lower	O
quota	O
(	O
quota	O
rounded	O
down	O
)	O
or	O
more	O
than	O
their	O
upper	O
quota	O
(	O
quota	O
rounded	O
up	O
)	O
.	O
</s>
<s>
This	O
can	O
be	O
fixed	O
by	O
using	O
quota-capped	O
divisor	B-Algorithm
methods	I-Algorithm
(	O
see	O
below	O
)	O
.	O
</s>
<s>
Simulation	O
experiments	O
show	O
that	O
different	O
divisor	B-Algorithm
methods	I-Algorithm
have	O
greatly	O
different	O
probabilities	O
of	O
violating	O
quota	O
(	O
when	O
the	O
number	O
of	O
votes	O
is	O
selected	O
by	O
an	O
exponential	O
distribution	O
)	O
:	O
</s>
<s>
The	O
probability	O
for	O
Webster/Sainte	O
-Laguë	O
is	O
the	O
smallest	O
-	O
only	O
0.16	O
%	O
.	O
</s>
<s>
A	O
divisor	B-Algorithm
method	I-Algorithm
is	O
called	O
stationary	O
if	O
its	O
divisor	O
is	O
of	O
the	O
form	O
for	O
some	O
real	O
number	O
.	O
</s>
<s>
A	O
quota-capped	O
divisor	B-Algorithm
method	I-Algorithm
is	O
an	O
apportionment	O
method	O
in	O
which	O
the	O
next	O
seat	O
is	O
allocated	O
only	O
to	O
a	O
party	O
from	O
a	O
set	O
of	O
eligible	O
parties	O
.	O
</s>
<s>
The	O
Balinsky-Young	O
quota	O
method	O
is	O
the	O
quota-capped	O
variant	O
of	O
the	O
D'Hondt	B-Algorithm
method	I-Algorithm
(	O
also	O
called	O
:	O
Quota-Jefferson	O
)	O
.	O
</s>
<s>
Every	O
quota-capped	O
divisor	B-Algorithm
method	I-Algorithm
satisfies	O
house-monotonicity	O
.	O
</s>
<s>
If	O
eligibility	O
is	O
based	O
on	O
quotas	O
as	O
above	O
,	O
then	O
the	O
quota-capped	O
divisor	B-Algorithm
method	I-Algorithm
satisfies	O
upper	O
quota	O
by	O
definition	O
,	O
and	O
it	O
can	O
be	O
proved	O
that	O
it	O
satisfies	O
lower	O
quota	O
as	O
well	O
.	O
</s>
<s>
However	O
,	O
quota-capped	O
divisor	B-Algorithm
methods	I-Algorithm
might	O
violate	O
a	O
property	O
of	O
population	O
monotonicity	O
:	O
it	O
is	O
possible	O
that	O
some	O
party	O
i	O
wins	O
more	O
votes	O
,	O
while	O
all	O
other	O
parties	O
win	O
the	O
same	O
number	O
of	O
votes	O
,	O
but	O
party	O
i	O
loses	O
a	O
seat	O
.	O
</s>
<s>
There	O
are	O
similar	O
examples	O
for	O
all	O
quota-capped	O
divisor	B-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
A	O
rank-index	O
methods	O
is	O
a	O
generalization	O
of	O
a	O
divisor	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Divisor	B-Algorithm
methods	I-Algorithm
are	O
a	O
special	O
case	O
of	O
rank-index	O
methods	O
:	O
a	O
divisor	B-Algorithm
method	I-Algorithm
with	O
divisor	O
function	O
is	O
equivalent	O
to	O
a	O
rank-index	O
method	O
with	O
rank-index	O
function	O
.	O
</s>
<s>
This	O
immediately	O
follows	O
from	O
the	O
iterative	B-Algorithm
procedure	I-Algorithm
.	O
</s>
