Playing Tennis
Tennis without
without Envy
Envy∗∗
Playing
© Indigolotos | Dreamstime.com
Josué Ortega
Ortega AMIMA,
AMIMA, University
University of
of Glasgow
Glasgow
Josué
Playing Tennis without Envy*
Josué Ortega AMIMA, University of Glasgow
A
group of
of friends
friends organises
organises their
their tennis
tennis games
games by
by each
each
group
submittingtheir
theiravailability
availabilityover
overthe
theweek.
week. They
Theywant
wantto
to
submitting
obtainan
anassignment
assignmentsuch
suchthat:
that: each
eachgame
gamemust
mustbe
beaadoudouobtain
blestennis
tennismatch,
match,i.e.
i.e.requires
requiresfour
fourpeople,
people,and
andnobody
nobodyplays
playson
onaa
bles
dayhe
heisisunavailable.
unavailable. Can
Canwe
weconstruct
constructassignments
assignmentsthat
thatwill
willalalday
waysproduce
produceefficient,
efficient,fair,
fair,and
andenvy-free
envy-freeoutcomes?
outcomes? The
Theanswer
answer
ways
no,and
andextends
extendsto
toany
anysport
sportthat
thatrequires
requiresany
anygroup
groupsize.
size.
isisno,
In
the
June
2016
edition
of
the
magazine
Mathematics
ToIn the June 2016 edition of the magazine Mathematics Today,
Maher
[1]
described
an
algorithm
to
assign
tennis
doubles
day, Maher [1] described an algorithm to assign tennis doubles
matchesamong
amonghis
hiscircle
circleof
offriends.
friends. The
Thealgorithm
algorithmtakes
takesas
asinput
input
matches
the
players’
availability
for
the
week,
and
maximises
the
number
the players’ availability for the week, and maximises the number
of tennis
tennis games,
games, subject
subject to
to three
three constraints:
constraints: 1)
1) no
no agent
agent plays
plays
of
more
than
once
per
day,
2)
each
match
has
exactly
four
players,
more than once per day, 2) each match has exactly four players,
and3)
3)no
noagent
agentplays
playson
onaaday
dayhe
heisisnot
notavailable.
available.
and
Table1:
1: Players’
Players’availability
availabilityfrom
fromProf.
Prof. Maher’s
Maher’stennis
tennisgroup.
group.
Table
Names
Names
Mon Tues
Tues Wed
Wed Thurs
Thurs Fri
Fri
Mon
Times
Times
BarryTT
Barry
TomBB
Tom
GordonBB
Gordon
PeterW
W
Peter
ColinCC
Colin
MikeM
M
Mike
KeithII
Keith
AlanCC
Alan
JohnSS
John
KeithBB
Keith
GeorgeStC
StC
George
MichaelLL
Michael
PhilM
M
Phil
Brian
Brian FF
PeterKK
Peter
WillieMcM
McM
Willie
Ken
L
Ken L
00
(0)
11(0)
00
(1)
11(1)
(1)
11(1)
00
00
(0)
11(0)
00
(1)
11(1)
(0)
11(0)
00
00
(1)
11(1)
0
0
00
00
00
(1)
11(1)
00
(1)
11(1)
00
(1)
11(1)
(1)
11(1)
00
(1)
11(1)
00
(0)
11(0)
00
(1)
11(1)
1
(1)
1 (1)
1
(1)
1 (1)
00
(1)
11(1)
(1)
11(1)
00
00
00
00
(0)
11(0)
(1)
11(1)
00
00
(1)
11(1)
(0)
11(0)
(1)
11(1)
00
00
00
00
00
(1)
11(1)
(1)
11(1)
00
00
(1)
11(1)
(1)
11(1)
00
(1)
11(1)
00
00
(1)
11(1)
00
00
00
(1)
11(1)
1
(1)
1 (1)
00
00
00
(0)
11(0)
00
00
(0)
11(0)
00
00
00
00
00
00
00
00
00
00
00
(2)
22(2)
(2)
33(2)
(0)
11(0)
(2)
22(2)
(2)
22(2)
(2)
44(2)
(1)
22(1)
(1)
22(1)
(1)
11(1)
(2)
22(2)
(1)
44(1)
(1)
11(1)
(1)
11(1)
2
(2)
2 (2)
2
(2)
2 (2)
1
(1)
1 (1)
(1)
11(1)
Total
Total
77
10
10
66
88
22
Thealgorithm
algorithmsolves
solvesaalinear
linearprogram
programto
tomaximise
maximisethe
thenumnumThe
ber
of
games
achieved
but
the
solution
is
generally
not
unique.
ber of games achieved but the solution is generally not unique.
Hence Maher
Maher selects
selects among
among those
those the
the ones
ones that
that maximise
maximise the
the
Hence
number
of
players
that
get
at
least
one
game.
If
several
assignnumber of players that get at least one game. If several assignmentsremain,
remain,he
hethen
thenchooses
choosesthe
theones
onesthat
thatmaximise
maximisethe
thenumnumments
ber
of
players
that
get
at
least
two
games.
In
case
uniqueness
ber of players that get at least two games. In case uniqueness isis
notyet
yetachieved,
achieved,he
heselects
selectsone
onesolution
solutionrandomly
randomlyamong
amongthose,
those,
not
which
is
the
one
implemented
and
communicated
to
each
player.
which is the one implemented and communicated to each player.
Maher writes:
writes: ‘They
‘They (members
(members of
of the
the group)
group) appear
appear to
to trust
trust in
in
Maher
the
fairness
and
efficiency
of
the
algorithm.’
the fairness and efficiency of the algorithm.’
∗∗Early
EarlyCareer
CareerMathematician
MathematicianCatherine
CatherineRichards
RichardsPrize
Prize2016
2016winning
winningarticle
article
Mathematics TODAY
DECEMBER 2016 288
clearthat
thatthe
thefinal
finalassignment
assignmentisisefficient
efficientas
asititmaximises
maximises
ItItisisclear
the number
number of
of matches.
matches. But
But isis itit really
really fair?
fair? In
In the
the preferences
preferences
the
thatappear
appearin
inMaher’s
Maher’sarticle,
article,presented
presentedin
inTable
Table11below
belowfor
forconconthat
venience,George
GeorgeStC
StCdeclares
declaresto
tobe
beavailable
availablefor
for44days
days––he
heisisthe
the
venience,
most flexible
flexible player
player as
as he
he can
can play
play basically
basically any
any day.
day. However,
However,
most
he only
only gets
gets one
one match:
match: the
the final
final assignment
assignment appears
appears in
in brackbrackhe
ets in
in Table
Table 1.1. Six
Six players
players (Barry
(Barry T,
T, Peter
Peter W,
W, Colin
Colin C,
C, Keith
Keith B,
B,
ets
BrianF,F,and
andPeter
PeterK)
K)were
wereall
allhalf
halfas
asflexible
flexibleas
asGeorge
GeorgeStC
StCand
and
Brian
gottwice
twiceas
asmany
manygames
gamesas
ashim.
him. George
GeorgeStC
StCmay
mayargue
arguethe
thefinal
final
got
assignment isis treating
treating him
him unfairly,
unfairly, and
and probably
probably most
most readers
readers
assignment
wouldagree
agreewith
withhim.
him. Furthermore,
Furthermore,everybody
everybodyexcept
exceptGordon
GordonBB
would
hasan
anassignment
assignmentatatleast
leastas
asgood
goodas
asGeorge
GeorgeStC.
StC.11
has
The property
property we
we described
described isis aa variant
variant of
of game-theoretic
game-theoretic
The
envy-freeness (see
(see [2]).
[2]). This
This is,
is, in
in the
the assignment
assignment presented
presented in
in
envy-freeness
Table1,1,George
GeorgeStC
StCisisenvious
enviousof
ofBarry
BarryT,
T,who
whowas
wasless
lessflexible
flexible
Table
but got
got more
more games.
games. An
An algorithm
algorithm that
that always
always produces
produces envyenvybut
free assignment
assignment has
has an
an important
important property:
property: players
players do
do not
not want
want
free
tofictitiously
fictitiouslyreduce
reducetheir
theiravailability
availabilityin
inorder
orderto
toget
getmore
moregames.
games.
to
Just by
by one
one player
player misreporting
misreporting his
his true
true availability,
availability, the
the assignassignJust
ment
described
previously
could
change
dramatically.
ment described previously could change dramatically.
In this
this tennis
tennis assignment
assignment problem,
problem, which
which we
we describe
describe forforIn
mally
below,
players
have
dichotomous
preferences
over
the
days:
mally below, players have dichotomous preferences over the days:
either they
they want
want to
to play
play or
or they
they do
do not.
not. These
These preferences
preferences were
were
either
first
studied
by
Bogomolnaia
and
Moulin
[3],
and
the
preferences
first studied by Bogomolnaia and Moulin [3], and the preferences
here represent
represent aa natural
natural extension
extension of
of those:
those: agents
agents want
want to
to play
play
here
on
as
many
days
as
possible.
However,
any
assignments
that
give
on as many days as possible. However, any assignments that give
them
a
game
on
a
day
they
are
not
available
is
considered
worse
them a game on a day they are not available is considered worse
thanhaving
havingno
nogames
gamesatatall.
all. Hence,
Hence, players’
players’preferences
preferencescan
canbe
be
than
captured
with
a
subset
of
all
possible
days.
The
constraint
that
captured with a subset of all possible days. The constraint that
fourpeople
peopleare
arerequired
requiredfor
foraagame
gamehas
hasbeen
beenpreviously
previouslyimposed
imposed
four
by
Shubik
[4]
over
assignments
of
one
day
only.
This
note
the
by Shubik [4] over assignments of one day only. This note isisthe
natural
extension
of
these
two
environments.
natural extension of these two environments.
Model
11 Model
Let AA∗∗ be
be aa nn×
×m
m binary
binary matrix
matrix containing
containing the
the preferences
preferences of
of
Let
eachperson
personii=
=(1,
(1,......,,n)
n)about
aboutplaying
playingon
onday
dayjj =
=(1,
(1,......,,m);
m);
each
= 11ififperson
personiiisisavailable
availableto
toplay
playon
onday
daykk and
and00
theentry
entryaa∗ik∗ik =
the
∗∗ will be called a tennis problem and represents the
otherwise.
A
otherwise. A will be called a tennis problem and represents the
players’ preferences,
preferences, who
who are
are indifferent
indifferent about
about their
their game
game partpartplayers’
ners
and
just
care
about
the
days
on
which
they
play.
ners and just care about the days on which they play.
matrix AA∗∗ can
can be
be reduced
reduced to
to aa matrix
matrix AA by
by deleting
deleting all
all
AA matrix
days
when
there
are
not
enough
people
available
to
create
even
days when there are not enough people available to create even
one match,
match, as
as on
on Friday
Friday in
in Table
Table 1.1. AA further
further reduction
reduction can
can be
be
one
performed
by
eliminating
people
who
are
not
available
on
any
performed by eliminating people who are not available on any
remaining days.
days. The
The days
days and
and players
players which
which are
are eliminated
eliminated are
are
remaining
irrelevant
for
the
type
of
solutions
we
will
consider,
and
hence
we
irrelevant for the type of solutions we will consider, and hence we
will
work
from
now
on
with
the
corresponding
irreducible
tennis
will work from now on with the corresponding irreducible tennis
problem A. Formally, an irreducible problem A satisfies:
∀i ∈ {1, . . . , n},
∀k ∈ {1, . . . , m},
m
k=1
n
i=1
aik ≥ 1,
(1)
the aforementioned impossibility, which clearly extends to randomised assignments, i.e. probability distributions over deterministic assignments, like Maher’s original solution.
aik ≥ 4.
(2)
Table 2: The impossibility of strong envy-free and efficient
assignments.
A solution to A is a binary matrix X(A), whose elements
have the same interpretation as in A, satisfying the following constraints:
∀i ∈ {1, . . . , n},∀k ∈ {1, . . . , m}, aik = 0 =⇒ xik = 0, (3)
xik mod 4 = 0. (4)
∀k ∈ {1, . . . , m},
i
There are three types of conditions we look for: efficiency,
fairness, and strong envy-freeness. We look at them in that order.
Definition 1. An assignment
X isefficient
if there is no assignment X such that i k xik > i k xik .
Efficient assignments are exactly those that are Pareto optimal, i.e. those in which no player can be made better off without
hurting another one. The next property considers the games received by the individual in the society who is in the worst position,
then the ones received by the second worst, and so on, in the spirit
of John Rawls’ leximin criterion.
Definition
2. Let Gq (X) denote the number of rows in X such
that k xik ≥ q, for any integer q: this is the number of players
with at least q games. An assignment X is fairer than another
assignment X if there exists an integer q for which Gq (X) >
Gq (X ) and for any integer q < q, Gq (X) = Gq (X ). An assignment X is fair if there is no other assignment which is fairer.
This notion of fairness implies an optimality condition: while
we can construct assignments that are efficient but not fair, every
fair assignment is efficient (otherwise another match could be created giving some people more matches, contradicting the fairness
property). Finally we have a variant of envy-freeness.
Definition 3. An assignment
X is
strongly envy-freeif for any
two
players
i,
j
with
a
>
ik
k
j ajk , we have
k xik ≥
x
.
jk
k
Strong envy-freeness captures the idea that more flexible people should not be penalised by the assignment. Strong envy cannot arise from days on which only the envious person is available
to play, as we are working with the corresponding irreducible tennis problem. It is called strong because standard envy-freeness
means that nobody prefers someone’s else schedule, a property
which is clearly too hard to satisfy in this case without the help
of randomisation.
2 An impossibility result
There are tennis problems which admit no solutions that are
strongly envy-free and efficient.
In the tennis assignment problem in Table 2 there are 11 players. Note that by efficiency, we need to organise five games, one
each day. This implies that players a, b, c, and d get two games.
Then, for whatever way we assign the remaining players to the
games on Wednesday, Thursday, and Friday, one of the agents f
to k gets at most one game while he has an availability of three, so
he is envious of any player with availability of two. This shows
Names
Mon
Tues
Wed
Thur
Fri
Times
a
b
c
d
e
f
g
h
i
j
k
1
1
1
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
1
1
2
2
2
2
3
3
3
3
3
3
3
Total
4
4
7
7
7
One may think of assignments for other sports. For example,
a game of poker that requires exactly three players. Or an indoor
football match that requires 10 players. In general, let a q-sport
assignment problem be one that requires q agents per day, with
the tennis assignment problem when q = 4. A simple modification of the example in Table 2 shows that our previous conclusion
generalises for arbitrary q-sport assignment problems (although
for q = 2 one needs to add more days). This is
Theorem 1. For any integer q ≥ 2, there exist q-sport assignment problems which have no solution that is efficient and
strongly envy-free (hence no solution that is fair and strongly
envy-free).
While we obtained a negative result, we leave many questions unanswered regarding how to construct optimal tennis assignments. We note that this assignment problem, despite being
very simple, is close in spirit to the stable marriage problem proposed by Gale and Shapley [5], which has led to the improvement
of real-life assignments such as those between colleges and students, organs and donors, or junior doctors and hospitals, and for
which Shapley received the Nobel Prize in Economics in 2012.
Hence, this type of problem, while simple, is always worth considering.
Notes
1. This is a actually a simplification of Maher’s problem (where
George StC is available for one game) in which we do not
consider individual quotas.
References
1 Maher, M. (2016) A tennis assignment algorithm, Math. Today,
vol. 52, pp. 130–131.
2 Moulin, H. (1995) Cooperative Microeconomics: A Game-Theoretic
Introduction, Princeton University Press, Princeton, NJ.
3 Bogomolnaia, A. and Moulin, H. (2004) Random matching under dichotomous preferences, Econometrica, vol. 72, pp. 257–279.
4 Shubik, M. (1971) The ‘Bridge Game’ economy: an example of indivisibilities, J. Political Econ., vol. 79, pp. 909–912.
5 Gale, D. and Shapley, L. (1962) College admissions and the stability
of marriage, Am. Math. Mon., vol. 69, pp. 9–15.
Mathematics TODAY
DECEMBER 2016 289