<s>
Maximin	B-Algorithm
share	I-Algorithm
(	O
MMS	O
)	O
is	O
a	O
criterion	O
of	O
fair	O
item	O
allocation	O
.	O
</s>
<s>
Given	O
a	O
set	O
of	O
items	O
with	O
different	O
values	O
,	O
the	O
1-out-of-n	O
maximin-share	B-Algorithm
is	O
the	O
maximum	O
value	O
that	O
can	O
be	O
gained	O
by	O
partitioning	O
the	O
items	O
into	O
parts	O
and	O
taking	O
the	O
part	O
with	O
the	O
minimum	O
value	O
.	O
</s>
<s>
An	O
allocation	O
of	O
items	O
among	O
agents	O
with	O
different	O
valuations	O
is	O
called	O
MMS-fair	O
if	O
each	O
agent	O
gets	O
a	O
bundle	O
that	O
is	O
at	O
least	O
as	O
good	O
as	O
his/her	O
1-out-of-n	O
maximin-share	B-Algorithm
.	O
</s>
<s>
The	O
set	O
attaining	O
this	O
maximin	O
value	O
is	O
called	O
the	O
"	O
1-out-of-3	O
maximin-share	B-Algorithm
"	O
-	O
it	O
is	O
the	O
best	O
subset	O
of	O
items	O
that	O
can	O
be	O
constructed	O
by	O
partitioning	O
the	O
original	O
set	O
into	O
parts	O
and	O
taking	O
the	O
least	O
valuable	O
part	O
.	O
</s>
<s>
The	O
1-out-of-n	O
maximin-share	B-Algorithm
of	O
from	O
is	O
defined	O
as:Here	O
,	O
the	O
maximum	O
is	O
over	O
all	O
partitions	O
of	O
into	O
disjoint	O
subsets	O
,	O
and	O
the	O
minimum	O
is	O
over	O
all	O
subsets	O
in	O
the	O
partition	O
.	O
</s>
<s>
Their	O
algorithm	O
is	O
based	O
on	O
"	O
ordering	O
"	O
the	O
instance	O
(	O
i.e.	O
,	O
reducing	O
the	O
instance	O
to	O
one	O
in	O
which	O
all	O
agents	O
agree	O
on	O
the	O
ranking	O
of	O
goods	O
)	O
,	O
and	O
then	O
running	O
the	O
envy-graph	B-Algorithm
procedure	I-Algorithm
starting	O
at	O
the	O
most	O
valuable	O
good	O
.	O
</s>
<s>
A	O
simple	O
algorithm	O
for	O
1/10	O
-fraction	O
MMS	O
for	O
the	O
more	O
challenging	O
case	O
of	O
submodular	B-Algorithm
valuations	I-Algorithm
-	O
based	O
on	O
round-robin	B-Algorithm
item	I-Algorithm
allocation	I-Algorithm
.	O
</s>
<s>
For	O
submodular	B-Algorithm
valuations	I-Algorithm
:	O
a	O
polynomial-time	O
algorithm	O
for	O
1/3	O
-fraction	O
MMS-fairness	O
,	O
and	O
an	O
upper	O
bound	O
of	O
3/4	O
-fraction	O
.	O
</s>
<s>
Budish	O
showed	O
that	O
the	O
Approximate	B-Algorithm
Competitive	I-Algorithm
Equilibrium	I-Algorithm
from	I-Algorithm
Equal	I-Algorithm
Incomes	I-Algorithm
always	O
guarantees	O
the	O
1-of-( )	O
MMS	O
,	O
However	O
,	O
this	O
allocation	O
may	O
have	O
excess	O
supply	O
,	O
and	O
-	O
more	O
importantly	O
-	O
excess	O
demand	O
:	O
the	O
sum	O
of	O
the	O
bundles	O
allocated	O
to	O
all	O
agents	O
might	O
be	O
slightly	O
larger	O
than	O
the	O
set	O
of	O
all	O
items	O
.	O
</s>
<s>
Barman	O
and	O
Biswas	O
present	O
an	O
algorithm	O
reducing	O
the	O
problem	O
to	O
a	O
problem	O
with	O
no	O
constraints	O
but	O
with	O
submodular	B-Algorithm
valuations	I-Algorithm
,	O
and	O
then	O
use	O
the	O
algorithm	O
of	O
to	O
attain	O
1/3	O
-fraction	O
MMS-fairness	O
.	O
</s>
<s>
Algorithms	O
for	O
finding	O
the	O
optimal	O
MMS	O
approximation	O
of	O
a	O
given	O
instance	O
,	O
based	O
on	O
algorithms	O
for	O
multiway	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
.	O
</s>
<s>
The	O
algorithm	O
can	O
be	O
seen	O
as	O
a	O
generalization	O
of	O
the	O
LPT	B-Algorithm
algorithm	I-Algorithm
for	O
identical-machines	B-Algorithm
scheduling	I-Algorithm
.	O
</s>
<s>
Their	O
algorithm	O
can	O
be	O
seen	O
as	O
a	O
generalization	O
of	O
the	O
Multifit	B-Algorithm
algorithm	I-Algorithm
for	O
identical-machines	B-Algorithm
scheduling	I-Algorithm
.	O
</s>
<s>
The	O
above	O
scaling	O
requires	O
to	O
compute	O
the	O
MMS	O
of	O
each	O
agent	O
,	O
which	O
is	O
an	O
NP-hard	O
problem	O
(	O
multiway	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
)	O
.	O
</s>
<s>
Construct	O
a	O
picking	B-Algorithm
sequence	I-Algorithm
in	O
which	O
the	O
agent	O
who	O
received	O
in	O
picks	O
first	O
,	O
the	O
agent	O
who	O
received	O
in	O
picks	O
second	O
,	O
etc	O
.	O
</s>
<s>
This	O
is	O
an	O
NP-hard	O
optimization	O
problem	O
,	O
but	O
it	O
has	O
several	O
approximation	O
algorithms	O
;	O
see	O
Multiway	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
and	O
Identical-machines	B-Algorithm
scheduling	I-Algorithm
.	O
</s>
<s>
Both	O
problems	O
can	O
be	O
approximated	O
by	O
a	O
PTAS	B-Algorithm
,	O
assuming	O
that	O
the	O
number	O
of	O
agents	O
is	O
fixed	O
.	O
</s>
<s>
When	O
agents	O
 '	O
valuations	O
are	O
binary	O
,	O
or	O
additive	O
and	O
based	O
on	O
Borda	O
score	O
,	O
maximin	B-Algorithm
share	I-Algorithm
allocations	O
can	O
always	O
be	O
found	O
efficiently	O
.	O
</s>
<s>
When	O
their	O
valuations	O
are	O
nonadditive	O
,	O
there	O
are	O
instances	O
in	O
which	O
an	O
r-fraction	O
MMS	O
allocation	O
does	O
not	O
exist	O
for	O
any	O
positive	O
r	O
.	O
However	O
,	O
for	O
a	O
class	O
of	O
symmetric	O
submodular	B-Algorithm
utilities	O
,	O
there	O
exists	O
a	O
tight	O
1/2	O
-fraction	O
MMS	O
allocation	O
,	O
and	O
it	O
can	O
be	O
approximated	O
to	O
within	O
a	O
factor	O
of	O
1/4	O
.	O
</s>
<s>
EF1	O
allocations	O
can	O
be	O
found	O
,	O
for	O
example	O
,	O
by	O
round-robin	B-Algorithm
item	I-Algorithm
allocation	I-Algorithm
or	O
by	O
the	O
envy-graph	B-Algorithm
procedure	I-Algorithm
.	O
</s>
<s>
The	O
following	O
maximin-share	B-Algorithm
approximations	O
are	O
implied	O
by	O
PROP*( 	O
n-1	O
)	O
,	O
hence	O
also	O
by	O
EF1	O
:	O
</s>
<s>
An	O
allocation	O
is	O
called	O
pairwise-maximin-share-fair	O
(	O
PMMS-fair	O
)	O
if	O
,	O
for	O
every	O
two	O
agents	O
i	O
and	O
j	O
,	O
agent	O
i	O
receives	O
at	O
least	O
his	O
1-out-of-2	O
maximin-share	B-Algorithm
restricted	O
to	O
the	O
items	O
received	O
by	O
i	O
and	O
j	O
.	O
</s>
<s>
An	O
allocation	O
is	O
called	O
groupwise-maximin-share-fair	O
(	O
GMMS-fair	O
)	O
if	O
,	O
for	O
every	O
subgroup	O
of	O
agents	O
of	O
size	O
k	O
,	O
each	O
member	O
of	O
the	O
subgroup	O
receives	O
his/her	O
1-out-of-k	O
maximin-share	B-Algorithm
restricted	O
to	O
the	O
items	O
received	O
by	O
this	O
subgroup	O
.	O
</s>
<s>
Farhadi	O
,	O
Ghodsi	O
,	O
Hajiaghayi	O
,	O
Lahaie	O
,	O
Pennock	O
,	O
Seddighin	O
and	O
Seddigin	O
introduce	O
the	O
Weighted	O
Maximin	B-Algorithm
Share	I-Algorithm
(	O
WMMS	O
)	O
,	O
defined	O
by:Intuitively	O
,	O
the	O
optimal	O
WMMS	O
is	O
attained	O
by	O
a	O
partition	O
in	O
which	O
the	O
value	O
of	O
part	O
j	O
is	O
proportional	O
to	O
the	O
entitlement	O
of	O
agent	O
j	O
.	O
</s>
<s>
A	O
1/n	O
-fraction	O
WMMS-fair	O
allocation	O
always	O
exists	O
and	O
can	O
be	O
found	O
by	O
round-robin	B-Algorithm
item	I-Algorithm
allocation	I-Algorithm
.	O
</s>
<s>
Now	O
,	O
the	O
Ordinal	O
Maximin	B-Algorithm
Share	I-Algorithm
(	O
OMMS	O
)	O
is	O
defined	O
by:For	O
example	O
,	O
if	O
the	O
entitlement	O
of	O
agent	O
i	O
is	O
any	O
real	O
number	O
at	O
least	O
as	O
large	O
as	O
2/3	O
,	O
then	O
he	O
is	O
entitled	O
to	O
at	O
least	O
the	O
2-out-of-3	O
MMS	O
of	O
C	O
.	O
Note	O
that	O
,	O
although	O
there	O
are	O
infinitely	O
many	O
pairs	O
satisfying	O
with	O
,	O
only	O
finitely-many	O
of	O
them	O
are	O
not	O
redundant	O
(	O
not	O
implied	O
by	O
others	O
)	O
,	O
so	O
it	O
is	O
possible	O
to	O
compute	O
the	O
OMMS	O
in	O
finite	O
time	O
.	O
</s>
<s>
They	O
define	O
it	O
in	O
two	O
equivalent	O
ways	O
;	O
one	O
of	O
them	O
is	O
clearly	O
a	O
strengthening	O
of	O
the	O
maximin	B-Algorithm
share	I-Algorithm
.	O
</s>
