In a small town, there is a public basketball court in which ten players (two teams of five players each) can play at a time. On a certain Sunday, 30 players want to play on the court, and they have different preferences regarding the time of day at which they prefer to play. The court manager therefore needs to partition the players into three groups of ten players each, and divide the time of day into three contiguous intervals—one interval per group—so that each group of ten can play in its designated time slot. To avoid complaints, the manager would like the partition to be envy-free: every player should believe that the time slot in which he or she plays is at least as good as each of the time slots given to the other two groups.
In this note, we show that no matter what the players’ preferences on the time of day are, there always exists a partition of the players together with an envy-free division of the time into contiguous time slots.
2 Cake Cutting.
Before we describe our setting formally, let us take a step back and ask a simpler question: Instead of a group activity like basketball, what if we are trying to divide the time for an individual activity, say running on a treadmill? In this case, there is no need to partition the players, since only one player will use the treadmill at a time. The challenge thus lies in dividing the time and assigning the resulting time slots to the players in such a way that no player envies another.
It turns out that this simpler setting falls precisely under the well-studied framework of cake cutting.111Several excellent surveys on the subject are available, e.g., [3, 11]. The cake serves as a metaphor for resources such as time or land, and the aim is to divide the cake between players in a fair manner. Formally, there are players, and the cake is represented by an interval of length . Each (contiguous) -partition of the cake can be defined by an -tuple with , where denotes the length of the th piece from the left. Given a partition, we say that a player prefers a certain piece if the player thinks there is no better piece in the partition; note that this preference can depend on the entire partition. Each player prefers at least one piece in any given partition, and may prefer more than one piece in case of ties.
Hungry players. Players never prefer an empty piece.
Closed preference sets. Any piece that is preferred for a convergent sequence of partitions is also preferred at the limiting partition.
A seminal result in cake cutting is that, as long as these two assumptions are satisfied, an envy-free division of the cake always exists. It was proved some decades ago in this Monthly by Stromquist  via topological arguments and Su  using Sperner’s lemma, and generalizes earlier results by Dubins and Spanier , also in this Monthly, and Woodall .
Let be any positive integer. For hungry players with closed preference sets, there is always a partition of the cake into contiguous pieces such that each player prefers a different piece.
Theorem 1 is an existence result, so the reader may naturally wonder how one could compute a desired solution. When there are two players, a contiguous envy-free division can be found using the so-called “cut-and-choose” protocol: the first player cuts the cake into two pieces so that she prefers both of them (this is always possible due to our two assumptions), and the second player chooses a piece that she prefers. For any number of players, however, the problem becomes surprisingly difficult—for example, with three players, Stromquist  showed that no finite algorithm can always compute such a division. We refer the reader to  and the references therein for more details.
3 Our Result.
Let us now return to our basketball court example and introduce a generalization of the cake-cutting model that captures it. Again, there are players who have preferences over the cake. However, unlike in canonical cake cutting, we are also given positive integers whose sum is , and the players should be divided into groups with group containing players. The cake is then partitioned into pieces, and group is assigned the th piece from the left. Our example thus corresponds to the case where , , and . In general, the group sizes may be different—for example, if only players come to the court, the manager may decide to split them into two groups of ten and one group of eight.
A first idea that comes to mind for proving the existence of an envy-free allocation is to apply Theorem 1 to the individual players, and then group them according to their time slots. Specifically, given an envy-free allocation to the individuals such that player gets piece in the -partition , we construct groups , , and so on. We then take the -partition where , and allocate the th piece from the left to group . Unfortunately, this idea does not work—the resulting allocation might not be envy-free. Indeed, going back to our basketball example, it is possible that player 7 prefers piece 7 to all other 29 pieces in the partition , but does not prefer the union of pieces 1, 2, , 10 to the union of pieces 11, 12, , 20 in the partition .
Nevertheless, we establish below that an envy-free allocation is guaranteed to exist in this general setting.
Let be any positive integers, and let be positive integers such that . For hungry players with closed preference sets, there is always a partition of the cake into contiguous pieces, along with a division of the players into groups with group containing players, such that each player in group prefers the th piece from the left in the partition.
If the partition of the players into groups is fixed in advance (in contrast to our setting), then an envy-free allocation is no longer guaranteed to exist. For example, suppose there are two groups, each of which contains a player who prefers to play in the morning (specifically, only prefers the piece containing the interval if such a piece exists in the partition) and a player who prefers to play in the evening (specifically, only prefers the piece containing the interval if such a piece exists in the partition). Regardless of how the interval is partitioned and assigned to the two groups, it is not hard to see that at least one player will be envious.
In light of this negative result, relaxations have been proposed for the fixed-group setting, for instance allowing more than one connected piece per group  or ensuring fairness only for a certain fraction of the players in each group [2, 13]. Both of these solutions are far from ideal in our basketball example. Hence, the possibility of constructing ad hoc groups plays a crucial role in ensuring that each group receives a contiguous piece and all players are nonenvious.222Envy-freeness with variable groups as in our model has recently been studied for different problems: allocating indivisible items between two groups , and dividing rooms and rent [1, 7]. For fixed groups, fair division has been investigated using a probabilistic approach  and with respect to other fairness notions ; we refer to  for an overview.
Proof of Theorem 2.
We first introduce some notation. For every positive integer , denote by the set . For each , denote by the sum of sizes of the first groups, i.e., , and let . Let . Note that and .
We are given hungry players. The preferences of each player are represented by a demand function , which assigns to each -partition a nonempty subset of representing the indices of the pieces that player prefers in that partition.
For each player , we construct a new demand function , which assigns to each -partition a nonempty subset of . The function is constructed as follows.
Given an -partition , construct an -partition by uniting the leftmost intervals into a single interval, the next intervals into another single interval, and so on. Formally, for each , let .
Find — the preferred piece(s) of player in the -partition .
For every , let . That is, contains the indices of the interval(s) in that are longest among the subintervals of interval in .
Going back to our basketball example, suppose that , so in partition player prefers to play in the middle time slot. Suppose that in partition the longest among the pieces are pieces and . Then .
Next, we show that satisfies the two assumptions that are necessary in order to apply Theorem 1. Since the function represents a hungry player, does not contain any empty piece. Hence, , too, does not contain any empty piece, so also represents a hungry player.
We now claim that satisfies the closed preference sets assumption. To see this, consider a sequence of -partitions such that for all partitions in the sequence, contains some fixed . Suppose that the sequence converges to a partition . To establish the claim, it suffices to show that .
For each , let be the -partition derived from by uniting adjacent intervals as in the construction of . The sequence converges to some -partition . The assumption that implies that for all :
, where is the unique index in for which ;
for all pieces .
From (1) and since satisfies the closed preference sets assumption, we have . In addition, it follows from (2) that for all . The construction of now implies that , as claimed.
We have shown that the preferences represented by satisfy the two requirements of Theorem 1. Therefore, there exists an -partition in which each player “prefers” a different piece according to . Let be a bijection such that player prefers piece . In particular, we have .
Finally, let be the -partition derived from by uniting adjacent intervals as in the construction of . For each , allocate the th piece in to group . For every player , since , the construction of implies that , that is, player prefers piece in . Hence the constructed -allocation is envy-free for the original players. ∎
4 Concluding Remarks.
Theorem 2 allows the players’ preferences to depend on the group size. For instance, suppose , , and . Then, given a partition, each player may decide whether he or she prefers to play in the earlier time slot in a group of eight, or in the later time slot in a group of ten.333A related problem is the group activity selection problem, where each player also has a preference over certain activities and the number of players involved in each activity . However, in our setting a player’s preference depends in addition on the “resource” to which his or her activity is allocated. The resulting allocation respects these preferences as long as they satisfy the closed preference sets assumption and the hungry players assumption—in particular, if the earlier piece is empty, then every player must prefer the later piece, and vice versa, regardless of the group sizes.
However, the theorem does not allow the players’ preferences to depend on the identity of the other players assigned to the same group. In fact, with this extension, the existence guarantee ceases to hold. For instance, if there is a popular player, say Alice, such that all other players prefer to play with Alice than not, then the players who are assigned to a different group than hers will necessarily be envious.
For groups, the result of Stromquist  implies that no finite protocol can find a contiguous envy-free division even in the simplest case in which each group consists of a single player. However, for groups (and players), a simple protocol finds a contiguous envy-free division, as long as the players’ preferences satisfy a third condition (in addition to “hungry players” and “closed preference sets”):
Monotone preferences. Let and be partitions such that for some , the th piece of is contained in the th piece of , while for every such that , the th piece of is contained in the th piece of . Then, if a player prefers the th piece of , he or she also prefers the th piece of .
Note that Stromquist’s impossibility result for three groups is valid even with monotone preferences. The protocol for two groups works as follows:
Let each player mark a point so that the player prefers both and . Such a point must exist by the hungry players and closed preference sets assumptions.
Order the marks from left to right, breaking ties arbitrarily.
Cut the cake at some point between the th mark and the st mark from the left.
Assign the piece to the players who made the leftmost marks, and the piece to the remaining players. The monotone preferences assumption guarantees that all agents prefer their allocated piece.
The problem of computing an envy-free allocation without the monotone preferences assumption remains for future work.
-  Azrieli, Y., Shmaya, E. (2014). Rental harmony with roommates. J. Econom. Theory. 153: 128–137.
-  Bade, S., Segal-Halevi, E. (2018). Fair and efficient division among families. Available at: arxiv.org/abs/1811.06684.
-  Brams, S. J., Taylor, A. D. (1996). Fair Division: From Cake-Cutting to Dispute Resolution. Cambridge, UK: Cambridge University Press.
-  Darmann, A., Elkind, E., Kurz, S., Lang, J., Schauer, J., Woeginger, G. (2012). Group activity selection problem. In: Goldberg, P. W., Guo, M., eds. Proceedings of the 8th International Workshop on Internet and Network Economics. Berlin-Heidelberg: Springer, pp. 156–169.
-  Dubins, L. E., Spanier, E. H. (1961). How to cut a cake fairly. Amer. Math. Monthly. 68(1): 1–17.
-  Goldberg, P. W., Hollender, A., Suksompong, W. (2019). Contiguous cake cutting: Hardness results and approximation algorithms. Available at: arxiv.org/abs/1911.05416.
Ghodsi, M., Latifian, M., Mohammadi, A., Moradian, S., Seddighin, M.
Rent division among groups.
In: Kim, D., Uma, R. N., Zelikovsky, A., eds.
Proceedings of the 12th International Conference on Combinatorial Optimization and Applications. Cham: Springer, pp. 577–591.
Kyropoulou, M., Suksompong, W., Voudouris, A. A. (2019). Almost envy-freeness in group resource allocation. In: Kraus, S., ed.
Proceedings of the 28th International Joint Conference on Artificial Intelligence
. Macao: International Joint Conferences on Artificial Intelligence, pp. 400–406.
-  Manurangsi, P., Suksompong, W. (2017). Asymptotic existence of fair divisions for groups. Math. Soc. Sci. 89: 100–108.
-  Meunier, F., Su, F. E. (2019). Multilabeled versions of Sperner’s and Fan’s lemmas and applications. SIAM J. Appl. Algebra Geom. 3(3): 391–411.
-  Robertson, J., Webb, W. (1998). Cake-Cutting Algorithms: Be Fair if You Can. Boca Raton, FL: Peters/CRC Press.
-  Segal-Halevi, E., Nitzan, S. (2019). Fair cake-cutting among families. Soc. Choice Welf. 53(4): 709–740.
-  Segal-Halevi, E., Suksompong, W. (2019). Democratic fair allocation of indivisible goods. Artificial Intelligence. 277: 103167.
-  Suksompong, W. (2018). Approximate maximin shares for groups of agents. Math. Soc. Sci. 92: 40–47.
-  Suksompong, W. (2018). Resource allocation and decision making for groups. PhD dissertation. Stanford University, Stanford, USA.
-  Stromquist, W. (1980). How to cut a cake fairly. Amer. Math. Monthly. 87(8): 640–644.
-  Stromquist, W. (2008). Envy-free cake divisions cannot be found by finite protocols. Electron. J. Comb. 15: R#11.
-  Su, F. E. (1999). Rental harmony: Sperner’s lemma in fair division. Amer. Math. Monthly. 106(10): 936–942.
-  Woodall, D. R. (1980). Dividing a cake fairly. J. Math. Anal. Appl. 78(1): 233–247.