1 The Problem
There is a set of agents who rent a house together. The set of rooms in the house is , with . The total rent is . It is required to assign a price to each room such that the sum of all prices is , and assign each room in to a unique agent in . The agents have different preferences on the rooms. The preferences of an agent are represented by a selection function
— for each price-vector, the agent selects a set of one or more rooms that he/she considers the best rooms given the prices. An allocation in which each agent receives a room from the set of his/her best rooms is called envy-free.
The existence of envy-free allocations has been proved using various techniques and under various assumptions on the agents’ preferences. Gale (1960) made the following assumption:
Quasilinear Tenants assumption: For all and , there is a value representing the value for agent of room . The best rooms of agent in price are the ones maximizing the difference .
Gale proved the existence of envy-free allocations using linear programming duality. Later works provided fast algorithms for calculating an envy-free allocation with quasilinear tenants using various techniques: a compensation procedure(Haake et al., 2002), a market-based mechanism (Abdulkadiroğlu et al., 2004), maximum matching and linear programming (Sung and Vlach, 2004). The latter approach has been later implemented and tested in the popular website spliddit.org (Gal et al., 2017).
Su (1999) (who invented the term “rental harmony”) made a different assumption:
Miserly Tenants assumption: An agent always prefers a free room if one is available. Formally, given a price-vector in which for some , every agent has a best room with .
In addition, he assumed that the selection functions are continuous: if some room is a best room for agent for a convergent sequence of price-vectors, then is a best room for in the limit price-vector.
Miserly tenants, in general, do not satisfy the quasilinear assumption. In particular, their preferences may even depend on the entire price-vector. For example, the miserly tenants assumption allows a tenant to prefer the first room when the price-vector is and prefer the second room when the price-vector is , despite the fact that the prices of both rooms have not changed (Azrieli and Shmaya (2014) describe several situations in which such preferences are reasonable). Thus, attaining rental harmony with miserly tenants requires a different technique, which is described next.
2 Sperner-type Lemmas
Sperner’s lemma and its variants consider an -vertex simplex, which w.l.o.g. is the standard -simplex . For each , we denote by the main vertex of the simplex in which the -th coordinate is and all other coordinates are .
The simplex is triangulated, and each vertex of the triangulation is labeled with a label in . The goal of all Sperner-type lemmas is to identify conditions that guarantee the existence of a fully-labeled sub-simplex — a sub-simplex of the triangulation whose vertices are labeled with different labels.
We say that a labeling satisfies Sperner’s boundary condition if the label on each vertex is some for which . In particular, the label of each main vertex is ; the label of each vertex on the line between and is either or ; and so on. Sperner’s lemma says that every labeling that satisfies Sperner’s boundary condition admits a fully-labeled simplex. An example is shown in Figure 1/Left.
Sperner’s lemma has many variants; of particular interest here is the variant proved by Scarf (1982). We say that a labeling satisfies Scarf’s boundary condition if the label on each vertex is some for which . In particular, when , the label of is either 2 or 3; the label of all vertices on the line between and is 3; and so on. Scarf’s lemma says that every labeling that satisfies Scarf’s boundary condition admits a fully-labeled simplex. An example is shown in Figure 1/Right.
3 Miserly Tenants and Scarf’s Lemma
Su’s proof of rental harmony with miserly tenants has two components. First, he associates each vertex of the triangulation to one of the agents, such that in each sub-simplex, all agents are represented (this is easy to do with the triangulations illustrated in Figure 1). Then, he associates each point in with a price-vector such that, for all ,
Each vertex of the triangulation is labeled with the index of one of the “best rooms” of the agent who owns that vertex.
Each vertex on the boundary of corresponds to a price-vector in which one or more rooms are free. Then, the Miserly Tenants assumption implies that for each agent there is a best room for which . Thus, it is possible to label each vertex of the triangulation such that the labeling satisfies Scarf’s boundary condition. By Scarf’s lemma, there is a fully-labeled simplex. Consider a sequence of finer and finer triangulations. The sequence of fully-labeled simplices has a subsequence that converges to a point. At the limit point, by the continuity of the preferences, each agent has a different best room. Hence there exists an envy-free allocation with the prices determined by the coordinates of the limit point.
Su’s theorem have been extended in various ways:
Azrieli and Shmaya (2014) considered rental harmony with roommates, when each room can accommodate several tenants;
Frick et al. (2017) considered rental harmony with a secretive agent, when only agents are currently available, and they have to determine a price-vector such that, when the -th agent comes back and picks a room he prefers, the other agents can allocate the remaining rooms among them in an envy-free way.
Meunier and Su (2019) considered rental harmony with an extra agent, when agents are currently available, and they have to determine a price-vector such that, when any agent leaves, the remaining agents can allocate the rooms among them in an envy-free way.
Nyman et al. (2017) considered multi-house rental harmony, when there are several -room houses (say, a bedroom house and an office building that are being rented to a set of agents together), and each agent should receive a room in each house.
All these extensions use appropriate generalizations of Scarf’s lemma, and they all make an assumption similar to the Miserly Tenants assumption.
Quasilinear tenants, in general, do not satisfy the miserly tenants assumption. For example, consider a house with total rent and three rooms: a spacious living-room and two basements. Consider a quasilinear agent who values the living-room at and each basement at . If the price-vector is , then the quasilinear agent (quite understandably) strictly prefers the living-room to both basements.111 Su (1999) notes, in the “Comments and Discussion” section, that the Miserly Tenants assumption can be weakened as follows: if a free room is available, then every agent has a best room which is not the most expensive room. The above example shows that quasilinear tenants do not satisfy even this weaker assumption: the quasilinear agent prefers the living room even though it is the most expensive one.
Therefore, to show that the above results are applicable to quasilinear tenants, the next section describes a new proof to the existence of rental harmony with quasilinear tenants — a proof using Sperner’s lemma.
4 Quasilinear Tenants and Sperner’s Lemma
One way to handle quasilinear tenants is to change the interpretation of the points in the -simplex: associate each point in with a price-vector such that, for all ,
Then, the boundary points correspond to price-vectors in which some rooms cost infinity. This means that each price is an element of the (positive) extended real number line , rather than an element of . Note that, since and , all prices are positive and at least one price is finite.
As in Su’s proof, each vertex of the triangulation is labeled with a best room of the vertex owner. A quasilinear agent always prefers a room with a finite price to a room with an infinite price. Hence, the labeling satisfies Sperner’s boundary condition: the label on a boundary vertex is always the index of room for which . By Sperner’s lemma, the labeling admits a fully-labeled sub-simplex. Continuity of preferences is preserved too. Hence there exist an envy-free allocation with some price-vector .
One problem remains: the sum of prices in may be unequal to . However, with quasilinear agents this is easy to solve. First, note that all prices in are finite — otherwise no allocation would have been envy-free. Let . Let be a new price-vector defined by: . For a quasilinear agent, adding a fixed amount to the price of each room does not change the relative preference-ordering between the rooms. Hence, the same allocation is envy-free with price-vector , and the sum of prices is .
Note that adding to all prices might make some prices negative. This means that some tenants are paid to live in their room.222 Negative prices may make sense in some situations. For example, if one of the rooms requires constant maintenance in order to prevent nuisances to the other rooms, then the tenants might agree to pay anyone who will agree to take this room and do the maintenance job. This is inevitable: Brams and Kilgour (2001) have shown that, in some situations with quasilinear agents, there is no envy-free allocation with non-negative prices.
5 Non-linear Tenants
Svensson (1983) and Alkan et al. (1991) generalized the quasilinear tenants model by assuming that each agent has a preference-relation on (room,price) pairs. For each price-vector , the best rooms of agent are the rooms for which the pairs are maximal (by ). They assume that the preference-relation is continuous and monotonic in the price, i.e., whenever . Without further assumptions, an envy-free allocation does not exist. For example, if for all agents and prices , then the agent who receives room always envies the agent who receives room . Therefore they make assumptions whose thrust is that every agent can be convinced to select any room, if its price is sufficiently low relative to the other rooms. The following assumption is similar to the one made by Svensson (1983) before Theorem 1, and by Alkan et al. (1991) at their Theorem 2 proof.
Convincible Tenants assumption: There exists an such that an agent always prefers any room to a room that costs more. Formally, given a price-vector in which , every agent has a best room with .
The Convincible Tenants assumption is more general than the Quasilinear Tenants assumption: quasilinear tenants are always convincible, where . In particular, if each agent assigns a non-negative value to each room, and the sum of all values is , then will do.
Moreover, the Convincible Tenants assumption is more general than the assumptions of Svensson and Alkan et al., since it allows the preferences to depend on the entire price-vector, like in Su’s model (see example at end of Section 1).
An envy-free allocation among convincible tenants always exists.
Associate each point in with a price-vector such that, for all ,
For example, when , . Note that . If for some , then . Moreover, when , there exists a room for which , which implies . Hence . The Convincible Tenants assumption implies that for each agent there is a best room for which . Hence, the labeling satisfies Sperner’s boundary condition and an envy-free allocation exists. ∎
The proof of Section 4 is still useful, as it does not depend on .
The proof of Theorem 1 works with the Miserly Tenants assumption too (each agent prefers a room with a non-positive price to a room with a positive price). It works even with the weaker version of this assumption (whenever there is a room with a non-positive price, all agents prefer a room which is not the most expensive). Assume w.l.o.g. that . If for some , then , while for some room with it holds that . The Miserly Tenants assumption, even in its weaker version, implies that for each agent there is a best room with . Hence, the labeling satisfies Sperner’s boundary condition and an envy-free allocation exists.
Moreover, the prices in this envy-free allocation are guaranteed to be non-negative, since a miserly tenant who gets a room with a positive price always envies a tenant who gets a room with a negative price. Hence, the proof combines advantages of previous proofs: it works both with and without the miserly tenants assumption, and with this assumption it guarantees non-negative prices.
The proof of rental harmony existence using Sperner’s lemma has two advantages over previous proofs.
First, the new proof is arguably simpler than previous ones. Moreover, due to the discrete nature of Sperner’s lemma, and thanks to the beautiful and simple proofs available for it (Su, 1999), it is easy to understand even by students with little background in mathematics.333 This observation is based on my experience teaching fair division to computer programmers.
Second, the new proof is more general, since it extends to all the settings listed in Section 3. This implies the existence of envy-free allocations with roommates, with a secretive agent, with an extra agent, or with multiple houses, when all tenants are convincible. In particular, they hold when all tenants are quasilinear. These existence results, as far as I know, were not known before.444 Rental harmony with roommates under the Quasilinear Tenants assumption was proved directly using agent duplication (Ghodsi et al., 2018). The present note shows that the result holds under the Convincible Tenants assumption too.
These new existence results open up some interesting computational issues. With quasilinear tenants, there are fast algorithms for computing an envy-free allocation (see Section 1). Can these algorithms be extended to handle more advanced settings such as a secretive agent or an extra agent?
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