<s>
A	O
strongly-proportional	B-Algorithm
division	I-Algorithm
(	O
sometimes	O
called	O
super-proportional	B-Algorithm
division	I-Algorithm
)	O
is	O
a	O
kind	O
of	O
a	O
fair	O
division	O
.	O
</s>
<s>
Formally	O
,	O
in	O
a	O
strongly-proportional	B-Algorithm
division	I-Algorithm
of	O
a	O
resource	O
C	O
among	O
n	O
partners	O
,	O
each	O
partner	O
i	O
,	O
with	O
value	O
measure	O
Vi	O
,	O
receives	O
a	O
share	O
Xi	O
such	O
that.Obviously	O
,	O
a	O
strongly-proportional	B-Algorithm
division	I-Algorithm
does	O
not	O
exist	O
when	O
all	O
partners	O
have	O
the	O
same	O
value	O
measure	O
.	O
</s>
<s>
In	O
1948	O
,	O
Hugo	O
Steinhaus	O
conjectured	O
the	O
existence	O
of	O
a	O
super-proportional	B-Algorithm
division	I-Algorithm
of	O
a	O
cake:It	O
may	O
be	O
stated	O
incidentally	O
that	O
if	O
there	O
are	O
two	O
(	O
or	O
more	O
)	O
partners	O
with	O
different	O
estimations	O
,	O
there	O
exists	O
a	O
division	O
giving	O
to	O
everybody	O
more	O
than	O
his	O
due	O
part	O
(	O
Knaster	O
)	O
;	O
this	O
fact	O
disproves	O
the	O
common	O
opinion	O
that	O
differences	O
estimations	O
make	O
fair	O
division	O
difficult.In	O
1961	O
,	O
Dubins	O
and	O
Spanier	O
proved	O
that	O
the	O
necessary	O
condition	O
for	O
existence	O
is	O
also	O
sufficient	O
.	O
</s>
<s>
That	O
is	O
,	O
whenever	O
the	O
partners	O
 '	O
valuations	O
are	O
additive	O
and	O
non-atomic	O
,	O
and	O
there	O
are	O
at	O
least	O
two	O
partners	O
whose	O
value	O
function	O
is	O
even	O
slightly	O
different	O
,	O
then	O
there	O
is	O
a	O
super-proportional	B-Algorithm
division	I-Algorithm
in	O
which	O
all	O
partners	O
receive	O
more	O
than	O
1/n	O
.	O
</s>
<s>
In	O
1986	O
,	O
Douglas	O
R	O
.	O
Woodall	O
published	O
the	O
first	O
protocol	O
for	O
finding	O
a	O
super-proportional	B-Algorithm
division	I-Algorithm
.	O
</s>
<s>
The	O
extension	O
of	O
this	O
protocol	O
to	O
n	O
partners	O
is	O
based	O
on	O
Fink	B-Algorithm
's	I-Algorithm
"	I-Algorithm
Lone	I-Algorithm
Chooser	I-Algorithm
"	I-Algorithm
protocol	I-Algorithm
.	O
</s>
<s>
Suppose	O
we	O
already	O
have	O
a	O
strongly-proportional	B-Algorithm
division	I-Algorithm
to	O
i-1	O
partners	O
(	O
for	O
i≥3	O
)	O
.	O
</s>
<s>
Julius	O
Barbanel	O
extended	O
Woodall	O
's	O
algorithm	O
to	O
agents	O
with	O
different	B-Algorithm
entitlements	I-Algorithm
,	O
including	O
irrational	O
entitlements	O
.	O
</s>
<s>
Janko	O
and	O
Joo	O
presented	O
a	O
simpler	O
algorithm	O
for	O
agents	O
with	O
different	B-Algorithm
entitlements	I-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
they	O
showed	O
how	O
to	O
reduce	O
a	O
problem	O
of	O
strongly-proportional	B-Algorithm
division	I-Algorithm
(	O
with	O
equal	O
or	O
different	B-Algorithm
entitlements	I-Algorithm
)	O
into	O
two	O
problems	O
of	O
proportional	B-Algorithm
division	I-Algorithm
with	I-Algorithm
different	I-Algorithm
entitlements	I-Algorithm
:	O
</s>
