Fair and square: Cake-cutting in two dimensions

Highlights

• The cake-cutting problem is extended by adding 2-dimensional geometric constraints.

• In addition to a fair value, the allotted pieces must have a usable geometric shape.

• Several shapes are examined, focusing on squares and balanced aspect-ratio rectangles.

• Our impossibility results show upper bounds on the attainable fair value per agent.

• Our division procedures give each agent a usable plot worth at least half that value.

Abstract

We consider the classic problem of fairly dividing a heterogeneous good (“cake”) among several agents with different valuations. Classic cake-cutting procedures either allocate each agent a collection of disconnected pieces, or assume that the cake is a one-dimensional interval. In practice, however, the two-dimensional shape of the allotted pieces is important. In particular, when building a house or designing an advertisement in printed or electronic media, squares are more usable than long and narrow rectangles. We thus introduce and study the problem of fair two-dimensional division wherein the allotted pieces must be of some restricted two-dimensional geometric shape(s), particularly squares and fat rectangles. Adding such geometric constraints re-opens most questions and challenges related to cake-cutting. Indeed, even the most elementary fairness criterion–proportionality–can no longer be guaranteed. In this paper we thus examine the level of proportionality that can be guaranteed, providing both impossibility results and constructive division procedures.

Introduction

Fair division of land has been an important issue since the dawn of history. One of the classic fair division procedures, “I cut and you choose”, is already alluded to in the Bible (Genesis 13) as a method for dividing land between two people. The modern study of this problem, commonly termed cake cutting, began in the 1940’s. The first challenge was conceptual — how should “fairness” be defined when the cake is heterogeneous and different people may assign different values to subsets of the cake? Steinhaus (1948) introduced the elementary and most basic fairness requirement, now termed proportionality: each of the $n$ agents should get a piece which he values as worth at least $1/n$ of the value of the entire cake. He also presented a procedure, suggested by Banach and Knaster, for proportionally dividing a cake among an arbitrary number of agents. Since then, many other desirable properties of cake partitions have been studied, including: envy-freeness (e.g.   Weller, 1985, Brams and Taylor, 1996, Su, 1999, Barbanel and Brams, 2004), social welfare maximization (e.g.   Cohler et al., 2011, Bei et al., 2012, Caragiannis et al., 2012) and strategy-proofness (e.g.   Mossel and Tamuz, 2010, Chen et al., 2013, Cole et al., 2013). See the books by Brams and Taylor (1996), Robertson and Webb (1998), Barbanel (2005), Brams (2007) and a recent survey by Procaccia (2015) for more information.

Many economists regard land division as an important application of division procedures (e.g.   Berliant and Raa, 1988, Berliant et al., 1992, Legut et al., 1994, Chambers, 2005, Dall’Aglio and Maccheroni, 2009, Hüsseinov, 2011, Nicolò et al., 2012). Hence, they note the importance of imposing some geometric constraints on the pieces allotted to the agents. The most well-studied constraint is connectivity — each agent should receive a single connected piece. The cake is usually assumed to be the one-dimensional interval [0,1] and the allotted pieces are sub-intervals (e.g.   Stromquist, 1980, Su, 1999, Nicolò and Yu, 2008, Azrieli and Shmaya, 2014). This assumption is usually justified by the reasoning that higher-dimensional settings can always be projected onto one dimension, and hence fairness in one dimension implies fairness in higher dimensions.¹ However, projecting back from the one dimension, the resulting two-dimensional plots are thin rectangular slivers, of little use in most practical applications; it is hard to build a house on a 10×1000 m plot even though its area is a full hectare, and a thin 0.1-inch wide advertisement space would ill-serve most advertises regardless of its height.

We claim that the two-dimensional shape of the allotted piece is of prime importance. Hence, we seek divisions in which the allotted pieces must be of some restricted family of “usable” two-dimensional shapes, e.g. squares or polygons of balanced length/width ratio.

Adding a two-dimensional geometric constraint re-opens most questions and challenges related to cake-cutting. Indeed, even the elementary proportionality criterion can no longer be guaranteed.

Example 1.1

A homogeneous square land-estate has to be divided between two heirs. Each heir wants to use his share for building a house with as large an area as possible, so the utility of each heir equals the area of the largest house that fits in his piece (see Fig. 1). If the houses can be rectangular, then it is possible to give each heir 1/2 of the total utility (a); if the houses must be square, it is possible to give each heir 1/4 of the total utility (b) but impossible to give both heirs more than 1/4 the total utility (c). In particular, when the allotted pieces must be square, a proportional division does not exist.²

This example invokes several questions. What happens when the land-estate is heterogeneous and each agent has a different utility function? Is it always possible to give each agent a 2-by-1 rectangle worth for him at least 1/2 the total value? Is it always possible to give each agent a square worth for him at least 1/4 the total value? Is it even possible to guarantee a positive fraction of the total value? If it is possible, what division procedures can be used? How does the answer change when there are more than two agents? Such questions are the topic of the present paper.

We use the term proportionality to describe the fraction that can be guaranteed to every agent. So when the shape of the pieces is unrestricted, the proportionality is always $1/n$, but when the shape is restricted, the proportionality might be smaller. Naturally, the attainable proportionality depends on both the shape of the cake and the desired shape of the allotted pieces. For every combination of cake shape and piece shape, one can prove impossibility results (for proportionality levels that cannot be guaranteed) and possibility results(for the proportionality that can be guaranteed). While we examined many such combinations, the present paper focuses on several representative scenarios which, in our opinion, demonstrate the richness of the two-dimensional cake-cutting task.

In Example 1.1, the two pieces had to be contained in the square cake. One can think of this situation as dividing a square island surrounded in all directions by sea, or a square land-estate surrounded by 4 walls: no land-plot can overlap the sea or cross a wall.

In practical situations, land-estates often have less than 4 walls. For example, consider a square land-estate that is bounded by sea to the west and north but opens to a desert to the east and south. Allocated land-plots may not flow over the sea shore, but they may flow over the borders to the desert.

Cakes with less than 4 walls can also be considered as unbounded cakes. For example, the above-mentioned land-estate with 2 walls can be considered a quarter-plane. The total value of the cake is assumed to be finite even when the cake is unbounded. When considering unbounded cakes, the pieces are allowed to be “generalized squares” with an infinite side-length. For example, when the cake is a quarter-plane (a square with 2 walls), we allow the pieces to be squares or quarter-planes. When the cake is a half-plane (a square with 1 wall), we also allow the pieces to be half-planes, etc. The terms “square with 2 walls” and “quarter-plane” are used interchangeably throughout the paper.

Intuitively, a piece of cake is usable if its lengths in all dimensions are balanced — it is not too long in one dimension and too short in another dimension. This intuition is captured by the concept of fatness, which we adapt from the computational geometry literature (e.g.   Agarwal et al., 1995, Katz, 1997):

Definition 1.1

A $d$-dimensional object is called $R$-fat, for $R\ge 1$, if it contains a $d$-dimensional cube $c^{-}$ and is contained in a parallel $d$-dimensional cube $c^{+}$, such that the ratio between the side-lengths of the cubes is at most $R$: $\text{len}\left(c^{+}\right)/\text{len}\left(c^{-}\right)\le R$.

A two-dimensional cube is a square. So, for example, a square is 1-fat, a 10-by-20 rectangle is 2-fat, a right-angled isosceles triangle is 2-fat and a circle is $\sqrt{2}$-fat.

Note that $R$ is an upper bound, so if $R_2\ge R_1$, every $R_1$-fat piece is also $R_2$-fat. So a square is also 2-fat, but a 10-by-20 rectangle is not 1-fat.

Our long-term research plan is to study various families of fat shapes. As a first step, we study the simplest fat shape, which is the square (hence the name of the paper). Despite its simplicity, it is still challenging. We also present results for fat rectangles, which are almost identical to the results for squares.

Our results can be broadly summarized as follows.

• Negative results: when the pieces have to be squares or fat rectangles, a proportional division is usually³ not guaranteed to exist. Moreover, there is a constant $A>1$ that depends on the shape of the cake and usable pieces, such that, for some sets of value-functions, it is impossible to give all agents a value of more than $1/(A\cdot n)$.

• Positive results: when the pieces have to be squares or fat rectangles, a constant-factor approximation to a proportional division is usually guaranteed to exist. This means that there is a constant $B>1$ that depends on the shape of the cake and usable pieces, such that, for all sets of value-function, it is possible to give all agents a value of at least $1/(B\cdot n)$.

The constant $A$ in our negative results is at most 2, and the constant $B$ in our positive results is at least 2; this leads us to conjecture that the “real” constant is 2, i.e, a half-proportional division with square pieces always exists, and half-proportionality is the best that can be guaranteed. Currently we can prove this conjecture only in several restricted scenarios, that are presented below.

In the first set of results, the cake is a square bounded in zero or more sides. Table 1 summarizes our negative and positive results:

The Impossibility column shows upper bounds on the attainable proportionality. Each upper bound is proved by showing a specific scenario in which it is impossible to give all agents more than the mentioned fraction of their total value. The upper bound for a square with 4 walls and $n=2$ is $1/\left(2n\right)=1/4$, as was already seen in Example 1.1. The upper bounds for an unbounded plane are valid only when the pieces must be squares parallel to a pre-specified coordinate system, or parallel to each other (as is common in urban planning). The other upper bounds are valid even when the squares are allowed to be non-parallel.

The Possibility column shows our positive results. Each such result is proved constructively by an explicit division procedure that gives each agent at least the mentioned fraction of their total value. The same result means that there exists a different division procedure that guarantees a larger fraction per agent, but this procedure works only when all agents have the same valuations. We do not know whether there exists a division procedure that guarantees this larger fraction for agents with different valuations.

Note that all our impossibility results hold even for agents with the same valuations, and all our division procedures return axes-parallel pieces.

Intuitively, one may think that allowing rectangles instead of just squares should considerably increase the attainable proportionality level. But this is not the case if the pieces need to be fat. As seen in the table, most results for fat rectangles are almost the same as for squares. The only exception is the impossibility result for an unbounded plane, which we have not managed to extend to $R$-fat rectangles.

For $n=2$, the proportionality levels in our possibility results are equal to the impossibility results. For a cake with two or three walls the guaranteed proportionality is equal to the impossibility result for every $n$. This means that in these cases, our procedures are optimal in their worst-case guarantee. For a cake with 4 walls, the guaranteed proportionality for agents with the same value measure is optimal. In the other cases, there is a multiplicative gap of at most 2 between the possibility and the impossibility result.

A secondary consideration in geometric division problems, in addition to value, is the type of cuts used for implementing the division. In some cases, guillotine cuts are preferred. Guillotine cuts are axis-parallel cuts running from one end to the opposite end of an already cut piece. They are considered easier to implement (e.g.  Alvarez-Valdés et al., 2002, Cui et al., 2008, Hifi et al., 2011). In the industry, guillotine cuts are used for cutting stock such as plates of glass. In the context of land division, guillotine cuts may be desired because they may make it easier to build fences between land-plots. Our procedures for a cake with 4 walls find divisions that can be implemented using guillotine cuts. The other procedures use general cuts, and we do not know if it is possible to attain the same value guarantees using guillotine cuts.

While some states in the USA are rectangular (e.g. Colorado or Wyoming), most land-estates have irregular shapes. In such cases, it may be impossible to guarantee any positive proportionality. For example, consider Robinson Crusoe arriving at a circular island. Assume that Robinson’s value measure is such that all value is concentrated in a very thin strip along the shore, as in Fig. 2. The value contained in any single square might be arbitrarily small. Clearly, no division procedure for $n$ agents can guarantee a better fraction of the total value.

Therefore, for arbitrary cakes we use a relative rather than absolute fairness measure. For each agent, we calculate the maximum value that this agent can attain in a square piece if he does not need to share the cake with other agents. We guarantee the agent a certain fraction of this value, rather than a certain fraction of the entire cake value. This fairness criterion is similar to the uniform preference externalities criterion suggested by Moulin (1990). Similar criteria have been recently studied in the context of indivisible item assignment (Budish, 2011, Procaccia and Wang, 2014, Bouveret and Lemaître, 2015).

Table 2 summarizes our bounds on relative proportionality. The impossibility results follow trivially from those for square cakes. The possibility results require new division procedures. They are valid for any cake that is a compact (closed and bounded) subset of the plane. The guarantees are better when the pieces are required to be axis-parallel. This is in accordance with the common practice in urban planning, in which axis-parallel plots are usually preferred.

Most of our division procedures can be presented as sequences of auctions.⁴ The general process is as follows. Initially, each of the $n$ agents receives a ticket with an entitlement to share a certain cake, $C$, in a group of $n$ agents. Then, the divider performs a well-designed sequence of auctions. In each auction, the winning agents exchange their ticket for another ticket with an entitlement to share a smaller cake $C^{\prime }\subset C$ in a smaller group of $n^{\prime }<n$ agents. This goes on until finally each agent holds a private entitlement for a single piece of the cake. Note that there are no monetary payments: the winners ‘pay’ only by giving away their tickets.

We use auctions of two types: mark auction and eval auction.⁵ They are presented briefly below; formal definitions and detailed examples are given in Section  4.

• In a mark auction, each agent bids by marking a piece of cake. All bids must satisfy a given geometric constraint (such as “mark a square at the bottom-left corner”). An agent bidding a piece $X_i$ is interpreted as saying “I am willing to give my ticket in exchange for $X_i$”. The agent bidding the smallest piece is the winner. The winner receives his bid and goes home, while the remaining agents continue to divide the remaining cake.

• In an eval auction, the divider specifies a piece $C^{\prime }\subset C$, and each agent bids by declaring his/her evaluation of $C^{\prime }$. An agent bidding a value $V$ is interpreted as saying “I am willing to give my ticket for sharing $C$ in a group of $n$ agents, in exchange for a ticket for sharing $C^{\prime }$ in a group of up to $f\left(V\right)$ agents”. Here $f:{\mathbb{R}}^{+}\to {\mathbb{Z}}^{+}$ is some weakly-increasing function that depends on the situation (the same function for all agents). The agent or agents bidding the highest values are the winners, since they are willing to share $C^{\prime }$ with the largest number of other agents. The number of winners is determined as the largest value $n^{\prime }$ such that the $n^{\prime }$ highest winners are willing to share $C^{\prime }$ in a group of $n^{\prime }$. These winners go on and divide $C^{\prime }$ among them, while the remaining $n-n^{\prime }$ agents continue to compete on $C\setminus C^{\prime }$.

The geometric constraints are carefully designed in order to guarantee that the final pieces are usable. A key geometric concept here is the cover number — the minimum number of squares required to cover a given region. By making sure that all sub-pieces have a sufficiently small cover-number, we ensure that they can be divided effectively. See Section  4 for details.

For the sake of simplicity, our division procedures are presented as if all agents bid according to their true value functions. However, the guarantees of our procedures are stronger: they are valid for any single agent bidding according to his/her true value function, regardless of what the other agents do. This is the common practice in the cake-cutting world.⁶ On the other hand, our procedures are not dominant-strategy truthful, since an agent who knows the other agents’ valuations may gain from under-bidding, just like in a first-price auction. Designing truthful cake-cutting mechanisms is known to be a difficult problem even with a 1-dimensional cake (Brânzei and Miltersen, 2015).

The remainder of the paper is structured as follows. The introduction section is concluded by reviewing some related research. The model is formally presented in Section  2. Impossibility results are proved in Section  3. Section  4 presents the basic building-blocks for the division procedures: the two auction types and the geometric covering concept. These building-blocks are then used in the division procedures of Section  5. Section  6 discusses several directions for future research.

The most prominent geometric constraint in cake-cutting is one-dimensional: the pieces must be contiguous intervals. Several authors studied a circular cake (Thomson, 2007, Brams et al., 2008, Barbanel et al., 2009), but it is still a one-dimensional circle and the pieces are one-dimensional arcs. Only few cake-cutting papers explicitly consider a two-dimensional cake.

Hill (1983), Beck (1987), Webb (1990) and Berliant et al. (1992) study the problem of dividing a disputed territory between several bordering countries, with the constraint that each country should get a piece that is adjacent to its border.

Iyer and Huhns (2009) describe a procedure that asks each of the $n$ agents to draw $n$ disjoint rectangles on the map of the two-dimensional cake. These rectangles are supposed to represent the desired regions of the agent. The procedure tries to give each agent one of his $n$ desired areas. However, the procedure does not succeed unless each rectangle proposed by an individual intersects at most one other rectangle drawn by any other agent. If even a single rectangle of Alice intersects two rectangles of George (for example), then the procedure fails and no agent receives any piece.

Berliant et al. (1992), Ichiishi and Idzik (1999) and Dall’Aglio and Maccheroni (2009) acknowledge the importance of having nicely-shaped pieces in resolving land disputes. They prove that, if the cake is a simplex in any number of dimensions, then there exists an envy-free and proportional partition of the cake into polytopes. However, this proof is purely existential when the cake has two or more dimensions. Additionally, there are no restrictions on the fatness of the allocated polytopes and apparently these can be arbitrarily thin triangles. Berliant and Dunz (2004) studies the existence of competitive equilibrium with utility functions that may depend on geometric shape; their nonwasteful partitions assumption explicitly excludes fat shapes such as squares. Devulapalli (2014) studies a two-dimensional division problem in which the geometric constraints are connectivity, simple-connectivity and convexity.

In our model (see Section  2), the utility functions depend on geometry, which makes them non-additive. They are not even sub-additive like in the models of Maccheroni and Marinacci (2003), Dall’Aglio and Maccheroni (2005) and Dall’Aglio and Maccheroni (2009).⁷ Previous papers about cake-cutting with non-additive utilities can be roughly divided to two kinds: some (Berliant and Dunz, 2004, Sagara and Vlach, 2005, Hüsseinov and Sagara, 2013) handle general non-additive utilities but provide only pure existence results. Others (Su, 1999, Caragiannis et al., 2011, Mirchandani, 2013) provide constructive division procedures but only for a 1-dimensional cake. Our approach is a middle ground between these extremes. Our utility functions are more general than the 1-dimensional model but less general than the arbitrary utility model; for this class of utility functions, we provide both existence results and constructive division procedures.

Besides fair division problems, geometric methods have been used in many other economics problems,⁸ such as voting (Plott, 1967), trade theory and growth theory (e.g.   Johnson, 1971), tax burdens (Hines et al., 1995), social choice (Cantillon and Rangel, 2002), mechanism design (Goeree and Kushnir, 2011), public good/bad allocation (e.g.   Öztürk et al., 2013, Öztürk et al., 2014, Chatterjee et al., 2016), utility theory (Abe, 2012) and general economics models (Michaelides, 2006).

With square pieces a proportional allocation may not exist, so we have to settle for partial-proportionality. Other goals that justify partial-proportionality are speed of computation (Edmonds and Pruhs, 2006, Edmonds et al., 2008), improving the social welfare (Zivan, 2011, Arzi, 2012) and guaranteeing a minimum-length constraint of a 1-dimensional piece (Caragiannis et al., 2011).