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
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