aa r X i v : . [ m a t h . HO ] A ug Playing Tennis without Envy
Josu´e OrtegaUniversity of GlasgowForthcoming in Mathematics Today
Abstract
A group of friends organize their tennis games by submitting each their availabilityover the weekdays. They want to obtain an assignment such that: each game mustbe a double tennis match, i.e. requires four people, and nobody plays in a day heis unavailable. Can we construct assignments that will always produce efficient, fair,and envy-free outcomes? The answer is no, and extends to any sport that requiresany group size.
In the June 2016 edition of the magazine Mathematics Today, Maher (2016) describedan algorithm to assign tennis double matches among his circle of friends. The algorithmtakes as input the players availability for the weekdays, and maximizes the number oftennis games, subject to three constraints: 1) no agent plays more than once per day, 2)each match has exactly four players, and 3) no agent plays on a day he is not available.The algorithm solves a linear program to maximize the number of games achievedbut the solution is generally not unique. Hence Maher selects among those the ones thatmaximize the number of players that get at least one game. If several assignments remain,then he chooses the ones that maximize the number of players that get at least two games.In case uniqueness is yet not achieved, he selects one solution randomly among those, whichis the one implemented and communicated to each player. Maher writes “ the (membersof the group) appear to trust in the fairness and efficiency of the algorithm ”.It is clear that the final assignment is efficient as it maximizes the number of matches.But is it really fair? In the preferences that appear in Maher’s article, presented in Table1 below for convenience, George StC declares to be available for 4 days - he is the mostflexible player as can play basically any day. However, he only gets one match: the finalassignment appears in brackets in Table 1. Six players (Barry T, Peter W, Colin C, KeithB, Brian F, and Peter K) were all half as flexible as George StC and got twice as gamesas him. George StC may argue the final assignment is treating him unfairly, and probablymost readers would agree with him. Furthermore, everybody except Gordon B has anassignment at least as good as George StC. The property we described is a variant of game-theoretic envy-freeness (see Moulin(1995)). This is, in the assignment presented in Table 1, George StC is envious of Barry This is a actually a simplification of Maher’s problem in which we do not consider individual quotas. able 1: Player’s availability from Prof. Maher’s tennis group.Names Mon Tues Wed Thurs Fri TimesBarry T 0 0 1 (1) 1 (1) 0 2 (2)Tom B 1 (0) 1 (1) 0 1 (1) 0 3 (2)Gordon B 0 0 0 0 1 (0) 1 (0)Peter W 1 (1) 1 (1) 0 0 0 2 (2)Colin C 1 (1) 0 0 1 (1) 0 2 (2)Mike M 0 1 (1) 1 (0) 1 (1) 1 (0) 4 (2)Keith I 0 1 (1) 1 (1) 0 0 2 (1)Alan C 1 (0) 0 0 1 (1) 0 2 (1)John S 0 1 (1) 0 0 0 1 (1)Keith B 1 (1) 0 1 (1) 0 0 2 (2)George StC 1 (0) 1 (0) 1 (0) 1 (1) 0 4 (1)Michael L 0 0 1 (1) 0 0 1 (1)Phil M 0 1 (1) 0 0 0 1 (1)Brian F 1 (1) 1 (1) 0 0 0 2 (2)Peter K 0 1 (1) 0 1 (1) 0 2 (2)Willie McM 0 0 0 1 (1) 0 1 (1)Ken L 0 1 (1) 0 0 0 1 (1)Total 7 10 6 8 2T, who was less flexible but got more games, or envious of Peter W, because George StCprefers his assignment. An algorithm that always produces envy-free assignment has animportant property: players do not want to fictitiously reduce their availability in orderto get more games. Just by one player misreporting his true availability, the assignmentdescribed previously could change dramatically.In this tennis assignment problem, that we describe formally below, players have di-chotomous preferences over the days: either they want to play or they do not. Thesepreferences were first studied by Bogomolnaia and Moulin (2004), and the preferenceshere represent a natural extension of those: agents want to play in as many feasible daysas possible. However, any assignments that gives them a game on a day they are notavailable is considered worse than having no games at all. Hence, players’ preferences canbe captured with a subset of all possible days. The constraints that 4 people are requiredfor a game has been previously imposed by Shubik (1971) over assignments of one dayonly. This note is the natural extension of these two environments.
1. Model
Let A ∗ be a n × m binary matrix containing the preferences of each person i = (1 , . . . , n )about playing on day j = (1 , . . . , m ); the entry a ∗ ik = 1 if person i is available to playon day k and 0 otherwise. A ∗ will be called a tennis problem and represents the players’preferences, who are indifferent about their game partners and just care about the daysin which they play. 2 matrix A ∗ can be reduced to a matrix A by deleting all days when there are notenough people available to create even one match, as in Friday in Table 1. A furtherreduction can be performed by eliminating people that are not available on any remainingdays. The days and players which are eliminated are irrelevant for the type of solutionswe will consider, and hence we will work from now on with the corresponding irreducibletennis problem A . Formally, an irreducible problem A satisfies: ∀ i ∈ { , . . . , n } , m X k =1 a ik ≥ ∀ k ∈ { , . . . , m } , n X i =1 a ik ≥ A is a binary matrix X ( A ), whose elements have the same interpretationas in A , satisfying the following constraints: ∀ i ∈ { , . . . , n } , ∀ k ∈ { , . . . , m } , a ik = 0 = ⇒ x ik = 0 (3) ∀ k ∈ { , . . . , m } , X i x ik ! mod 4 = 0 (4)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 is efficient if there is no assignment X ′ such that P i P k x ′ ik > P i P k x ik .Efficient assignments are exactly those that are Pareto optimal, i.e. those in whichno player can be made better off without hurting another one. The next one propertyconsiders the games received by the individual in the society who is in the worst position,then the ones received by the second worst one, and so on, in the spirit of John Rawls’leximin criterion. Definition 2.
Let G q ( X ) denote the number of rows in X such that P k x ik ≥ q , for anyinteger q : this is the number of players with at least q games. An assignment X is fairerthan another assignment X ′ if there exists an integer q for which G q ( X ) > G q ( X ′ ) andfor any integer q ′ < q , G q ( X ) = G q ( X ′ ). An assignment X is fair if there is no otherassignment which is fairer.This notion of fairness implies an optimality condition: while we can construct assign-ments that are efficient but not fair, every fair assignment is efficient (otherwise anothermatch could be created giving some people more matches, contradicting the fairness prop-erty). Finally we have a variant of envy-freeness. Definition 3.
An assignment X is strongly envy-free if for any two players i, j with P k a ik > P j a jk , we have P k x ik ≥ P k x jk .3trong envy-freeness captures the idea that more flexible people should not be penalizedby the assignment. Strong envy cannot arise from days on which only the envious personis 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’selse schedule, a property which is clearly too hard to satisfy in this case.
2. An Impossibility Result
There are tennis problems that admit no solutions that is strongly envy-free and efficient.Table 2: The impossibility of strong EF and efficient assignments.Names Mon Tues Wed Thur Frid Times a b c d e f g h i j k a , b , c , and d get two games. Then, for whatever way we assign the remaining players to thegames on Wednesday, Thursday, and Friday, one of the agents f to k gets at most onegame while he has an availability of three, so he is envious of any player with availabilitytwo. This shows the aforementioned impossibility.One may think of assignments for other sports. For example, a game of poker thatrequires three players exactly. Or an indoor football match that requires 10 players. Ingeneral, let a q -sport assignment problem be the one that requires q agents per day, withthe tennis assignment problem being its particular case when q = 4. A simple modificationof the example in Table 2 shows that our previous conclusion generalizes for arbitrary q -sport assignment problems (although for q = 2 one needs to add more days). This is Theorem 1.
For any integer q ≥ , there exists q -sport assignment problems which haveno solution that is efficient and strongly envy-free (henceforth no solution that is fair andstrongly envy-free). While we obtained a negative result, we leave many questions unanswered regardinghow 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 by4ale and Shapley (1962), which has led to the improvement of real-life assignments suchas 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 ofproblem, while simple, is always worth considering.
ReferencesBogomolnaia, A. and H. Moulin (2004): “Random Matching Under DichotomousPreferences,”
Econometrica , 72, 257–279.
Gale, D. and L. Shapley (1962): “College Admissions and the Stability of Marriage,”
American Mathematical Monthly , 69, 9–15.
Maher, M. (2016): “A Tennis Assignment Algorithm,”
Mathematics Today , 52, 130–131.
Moulin, H. (1995):
Cooperative Microeconomics: A Game-Theoretic Introduction ,Princeton, NJ: Princeton University Press.
Shubik, M. (1971): “The ”Bridge Game” Economy: An Example of Indivisibilities,”