Cutting a Cake Is Not Always a "Piece of Cake": A Closer Look at the Foundations of Cake-Cutting Through the Lens of Measure Theory

11/09/2021
by   Peter Kern, et al.
University of Düsseldorf
TU Dresden
0

Cake-cutting is a playful name for the fair division of a heterogeneous, divisible good among agents, a well-studied problem at the intersection of mathematics, economics, and artificial intelligence. The cake-cutting literature is rich and edifying. However, different model assumptions are made in its many papers, in particular regarding the set of allowed pieces of cake that are to be distributed among the agents and regarding the agents' valuation functions by which they measure these pieces. We survey the commonly used definitions in the cake-cutting literature, highlight their strengths and weaknesses, and make some recommendations on what definitions could be most reasonably used when looking through the lens of measure theory.

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1 Introduction

Since the groundbreaking work of ste:j:fair-division, cake-cutting is a metaphor for the so-called fair division problem for a divisible, heterogeneous good, which addresses the problem to split a contested quantity (a “cake”) in a fair way among several parties ; each party may have its own idea about the value of the different parts of the cake. A traditional way of fair division between two parties and would be to let divide the cake into two pieces (depending on their own valuation) while has the right to choose one of the pieces, the so-called cut & choose protocol. There are other possibilities for two parties as well as extensions to more than two parties [<]see, e.g.,>[for an overview]pro:b:handbook-comsoc-cake-cutting-algorithms,lin-rot:b:economics-and-computation-cake-cutting. Yet, while the basic rules of the game are pretty clear, the assumptions on the actual cutting process are often treated in a gentlemanlike manner. If the whole cake is represented by an interval, say , many authors think of the pieces as “intervals,” without specifying whether the intervals are open , half-open , or closed , and how to treat the – possibly twice counted – end points, i.e., vs. ; this is, of course, not an issue if a one-point set like has zero value for all parties. However, this simple example shows that a formal mathematical approach to cake-cutting needs to address questions like:

  • Are (open, closed, half-open) intervals the only possible pieces of cake?

  • Do we allow for finitely many or infinitely many cuts (a “cut” being the split of any subset of at a single point)?

  • Which properties should a valuation function have, and how does it interact with the family of admissible pieces of cake?

For some cases, there is an obvious answer: If we use only finitely many cuts, finite unions of intervals of the form  – where the angular braces indicate either open or closed ends – is all we can get; and if, in addition, any single point has zero value, we do not have to care about the open or closed ends anymore. We will see in Section 2.1 below that this rather implicit assumption brings us in a much more potent framework that can effectively deal with a countably infinite number of cuts.

As soon as we allow for countably infinitely many cuts, things change dramatically, as the following example shows.

Example 1.1 (Cantor dust; Cantor’s ternary set).

Start with the complete cake as a single piece, i.e., . Now, cut out the middle third of to obtain the intermediate piece comprising two closed intervals. Next, cut out the middle third of both remaining pieces in to obtain a union of four closed intervals , see Figure 1. If this procedure is repeated on and on, we will remove countably many open intervals, and the remainder set is . The set is the Cantor (ternary) set [<]see, e.g.,>[§ 2.5]schi:b:counterexamples, and one can show that this is a closed set, which has more than countably many points, does not contain any interval, and is dense in itself, i.e., each of its points is a limit point of a sequence inside . In the usual measuring scale, the original cake had length , and the recursively removed pieces have total length

so that has zero “length,” but it still contains more than countably many points.

The same construction principle, removing at each stage identical open middle intervals, each having length for some , , leads to the Cantor set , which is, again, closed, uncountable, and does not contain any interval. If, say, , the removed intervals have total length and the remaining Cantor dust has “length” . This is not quite expected.

While it is intuitive that the removed intervals should have a certain length, it feels unnatural to speak of the “length” of a dust-like set as . In fact, we are dealing here with (one-dimensional) Lebesgue measure, which is the mathematically formal extension of the familiar notion of “length.”

An alternative, slightly more formal way of illustrating the Cantor dust is given in the appendix as Example A.1.

This example shows that, as soon as we allow for countably many cuts, there can appear sets which may not be written as a countable union of intervals; moreover, although these sets consist of limit points only, they may have strictly positive length.

Figure 1: Step-wise pieces to be cut for a Cantor-like piece of cake.

An important feature of this example is the fact that we extend the family of intervals to a family of subsets which contains (i) finite unions, (ii) countable intersections, and (iii) complements of its members, leading to fairly complicated subsets as, e.g., . Moreover, when calculating the length of all removed intervals, we tacitly assumed

  • the (finite) additivity of length: the length of two disjoint sets is the sum of their lengths

  • the countable or -additivity which plays the role of a continuity property: the length of a countable union is the limit of the length of the union of the first sets as .

As it will turn out, these are two far-reaching assumptions on the interplay of the valuation function (here: length) with its domain; we will see how this relates to the desirable property that we can cut off pieces of arbitrary length , , from the cake (allowing for any valuation values of the cut-off pieces).

Commonly, in cake-cutting theory [<]see, e.g.,>bra-tay:b:fair-division-from-cake-cutting-to-dispute-resolution,pro:b:handbook-comsoc-cake-cutting-algorithms,lin-rot:b:economics-and-computation-cake-cutting a (piece-wise constant) valuation function , where is some family of subsets of the cake, is represented as shown in Figure 2: The cake is split horizontally into multiple pieces and the number of vertically stacked boxes per piece describes the piece’s valuation from some agent’s perspective. For example, the valuation function in Figure 2 evaluates the piece with .

1

2

3

4

5

6
Figure 2: Common representation for a valuation function in cake-cutting.

Having this example in mind, one is tempted to assume that can always be taken as the power set .

The following classical example from measure theory shows that there cannot exist a valuation function that assigns to intervals their natural length , and which is additive, -additive (in the sense explained above), and able to assign a value to every set . Things are different if we do not require -additivity [<]see the discussion in>[§ 7.31]schi:b:counterexamples.

Example 1.2 (vit:c:vitali-sets; <see also, e.g.,>schi:b:measures, and schi:b:counterexamples).

Let be the standard cake, and assume that the valuation function is -additive (see Definition 2.2), assigning to any interval its natural length. This means, in particular, that is invariant under translations and evaluates the complete cake with . Let us define the relation as follows: We say that two real numbers satisfy the relation if, and only if, , i.e., their difference is rational. The relation is an equivalence relation and the corresponding equivalence classes lead to a disjoint partitioning of . By the axiom of choice, there is a set which contains exactly one representative of every equivalence class . A set like is called a Vitali set. Clearly, and the sets , , are a disjoint partition of ; thus . By assumption, is -additive and assigns to each the same value (translation invariance). Hence, we end up with the contradiction

Thus cannot have the power set of the cake as its domain.

The above examples highlight some of the problems when evaluating sets. A Cantor-like piece can only be evaluated if the valuation function is not too simplistic. On the other hand, a Vitali set cannot be evaluated at all if we request too many properties of a valuation function, i.e., the domain consisting of all possible pieces of cake is, in general, too large.

Across the research field of cake-cutting [<]see, e.g., the textbooks by>[and the book chapters by pro:b:handbook-comsoc-cake-cutting-algorithms,lin-rot:b:economics-and-computation-cake-cutting]bra-tay:b:fair-division-from-cake-cutting-to-dispute-resolution,rob-web:b:cake-cutting-algorithms-be-fair-if-you-can, there exist several different assumptions on the underlying model. Our goal is to review thoroughly and comprehensively all the different models that are currently applied in the literature. Furthermore, we study the relationships between these models and formulate some related results. It turns out that some of these models are problematic and should not be used as they are formulated. We highlight these models’ problems and provide specific examples showing why they are problematic. Our overall goal is to determine a model, which is as simple as possible, yet powerful enough to cope with these problems and still compatible with many of the currently used models.

Frequently, authors proposing cake-cutting protocols abstain from making formal assumptions or from formalizing their model in detail. For example, [p. 553]bra-tay-zwi:c:moving-knife write:

“Many feel that the informality adds to the subject’s simplicity and charm, and we would concur. But charm and simplicity are not the only factors determining the direction in which mathematics moves or should move. Our analysis in this paper raises several issues that may only admit a resolution via some negative results. While such results may not require complete formalization of what is permissible, they do appear to require partial versions. We will refer to such partial limitations as theses.”

It would thus be desirable to have some common consensus on which models are useful for any given purpose, and which are not. If we allow only a fixed number of cuts, splitting the cake into a finite number of pieces of the type , a naive approach is always possible: The valuation should be additive and its domain contains unions of finitely many intervals. If, on the other hand, there are potentially infinitely many cuts – e.g., if the players play a game resulting in an a priori not fixed number of rounds – the limiting case cannot any longer be treated by a finitely additive valuation and a domain containing only finite unions, see Example 1.1.

We propose to use ideas from measure theory, which provides the right toolbox to tackle the issues described above. We will see that, at least for the cake , even the naive approach plus the requirement that we can split every piece by a single cut into any proportion (in fact, a slightly weaker requirement will do, cf. Definition 2.2 ), automatically leads to the measure-theoretic point of view. That is to say that in many natural situations the naive standpoint is “practically safe” since its obvious shortcomings are automatically “fixed by (measure) theory”, if one uses the correct formulation.

2 The Rules of the Game

Throughout this paper, denotes a standard cake, and the power set are all possible pieces of cake from a set-theoretic point of view. We define as the set of all admissible pieces of , i.e., those pieces which (a) can be allocated to some players via a cake-cutting protocol, and (b) can be evaluated by the players using their valuation functions. Some results remain true in a more general setting with more general cakes ; to highlight this, we will speak of an “abstract” cake . For example, an abstract cake might be contained in an -dimensional unit cube: .

2.1 Dividing a Cake with Finitely Many Cuts

We start by formulating requirements for regarding the admissible pieces of cake. Obviously, we want to be able to allocate the complete cake as well as an empty piece to a player and therefore, and must hold. If is already allocated to some player, i.e., , then we want to be able to give the remainder of the cake to another player; so for all , we demand that the complement of , denoted by , is in . Furthermore, we want to be able to cut and combine pieces of cake; so for all , we require . Note that and , so our previously formulated requirements also allow us to allocate the intersection of a finite number of pieces of cake and to evaluate the difference of two pieces of cake.

Definition 2.1.

Let be an arbitrary abstract cake. A family is called an algebra over if and for all it holds that and .

It is worth noting that only by the formulation of intuitive requirements with respect to the set of all admissible pieces of cake, we ended up with a well-studied, structured concept from measure theory: an algebra. For instance, if , then and are algebras – in fact these are the largest possible and the smallest possible algebras over . Another useful algebra is the family of all unions of finitely many intervals in  – and it is easy to check that is the smallest algebra containing all closed (or all open or all half-open) intervals from . While it is obvious that is useless for our purpose, as then only two possible pieces can be allocated, the complete cake and an empty piece, we might – at the other extreme – also take as the set for the admissible pieces of . However, when choosing , we must also ensure that meaningful valuation functions can exist for this set, and Example 1.2 shows that for a rather natural valuation function – geometric length – is too big.

Let us list the common requirements for the players’ valuation functions. A valuation function shall assign to any admissible piece of cake some nonnegative real number, i.e., . In order to normalize the players’ valuations and keep them comparable, we demand that and hold. Hence, we can further limit the valuation function’s range to , i.e., we have . The next definition lists further requirements for a valuation function.

Definition 2.2.

Let be an abstract cake and the algebra of admissible pieces. A valuation function is a function , which is normalized, i.e., and .111Note that the normalization requirement immediately implies that (as a measure, cf. Remark 2.3) is finite. Moreover, is called

  1. monotone if for with , one has ;

  2. additive or finitely additive if for all such that , one has ;

  3. -additive or countably additive if for any sequence of pieces in such that () and , one has ;

  4. divisible if for every and for every real number , , there exists some with such that .

Clearly, implies  – take , , and for  – and is equivalent to the so-called strong additivity, defined as : Just observe that , i.e., counts towards both and but only once in , hence the correction . Finally, (strong) additivity implies monotonicity.

The assumption that is an algebra makes sure that we can indeed perform all of the above manipulations with sets without ever leaving . Note, however, that and impose further assumptions on the structure of .

Remark 2.3.

Let be a(n abstract) cake and an algebra over . Any additive valuation is also a finitely additive measure with total mass [<]see, e.g.,>[Chapter 4]schi:b:measures.

Requirement not only demands more from but also from . Specifically, entails that any which does not contain a nonempty and strictly smaller piece of cake – this is an atom, i.e., an undivisible piece of cake – must have zero valuation.

Definition 2.4.

Let be an algebra over an abstract cake and be a finitely additive valuation. A set is an atom if and every , , satisfies with or .

Clearly, a valuation which enjoys property cannot have atoms. If and contains all intervals of type , then all singletons are in , and they are the only possible atoms. In this case, entails that does not charge single points: for all . This is the proof of the following lemma.

Lemma 2.5.

Let be a cake and an algebra of admissible sets. Every additive valuation function that satisfies is atom-free. In particular, if contains all intervals, then for all .

Quite often, we require valuation functions to satisfy continuity, a property that is crucial for so-called moving-knife cake-cutting protocols to work.

Definition 2.6.

Let be a finitely additive valuation function on the algebra of finite unions of intervals from .

  1. The function , , is the distribution function of the valuation .

  2. The valuation is said to be continuous if is continuous.

Since is additive, is positive, monotonically increasing, and bounded by . Note that a continuous valuation function on cannot have atoms, as

The continuity of can also be cast in the following way: For all and with satisfying and , and for every , there exists some such that . This explains the close connection between continuity and divisibility of . In fact, assuming divisibility of , it can be shown that the distribution function is necessarily continuous. The following proof of this statement is inspired by [sch-sto:t:continuity-assumptions-in-cake-cutting, Example 3.4].

Lemma 2.7.

Let be an additive valuation for the standard cake , where denotes the family of admissible pieces. If is divisible, then the distribution function is a continuous function with .

Proof.

We have seen in Lemma 2.5 that a divisible additive valuation has no atoms, so . Since

is monotone and bounded, the one-sided limits

and exist for all and .

Assume that is not continuous. Then there exists some such that or . If , then there exists some such that . Set and observe that . Pick an arbitrary which is contained in . Since is a finite union of intervals, differs from its closure by at most finitely many points; as for any , we have .

We distinguish between two cases: If is not an isolated point, then . If or if is an isolated point, then we have due to that

Hence, it is not possible to select a piece of cake with and , which contradicts divisibility.

If , a similar argument applies. ∎

Conversely, if the distribution function of a finitely additive valuation defined on is continuous with , then it is easy to see that is divisible. Hence we get:

Corollary 2.8.

A finitely additive valuation on is divisible if, and only if, its distribution function is continuous with . This is also equivalent to being atom-free.

We will see in the next section that every finitely additive, divisible valuation can be extended to become and identified with a unique -additive measure that is defined on a the Borel -algebra ; this is the smallest family of sets that contains all intervals and that is stable under complements and countable unions of its members. This enables us to evaluate sets in that are not finite unions of intervals, such as the Cantor set in Example 1.1.

2.2 Measure Theory: The Art of Dividing a Cake by Countably Many Cuts

Up to now we have only allowed finitely many cuts when dividing the cake. But we may easily come into the situation where the number of cuts is not limited; not all protocols in the cake-cutting literature are finite. Thus we are led to consider unions of countably many pieces and the valuation of such countable unions, see also property in Definition 2.2. To deal with such situations, measure theory provides the right tools.

We will now introduce some basics from measure theory, which we need in the subsequent discussion of the cake-cutting literature. Our standard references for measure theory are the monographs by schi:b:measures and schi:b:counterexamples, where also further background information can be found.

Definition 2.9.

Let be an arbitrary abstract cake. A subset is called a -algebra over if is an algebra over and, for all sequences with , the countable union is in , too.

Let us return to the standard cake . Every algebra in containing finitely many sets is automatically a -algebra. On the other hand, is both an algebra and a -algebra, whereas the family is an algebra, but not a -algebra: For instance, the Cantor dust (cf. Example 1.1) is not in . Recall that we defined to be the smallest algebra containing all (finite unions of) intervals in ; thus it is natural to consider the smallest -algebra containing all (finite unions of) intervals in .

To see that this is well-defined, we need a bit more notation. Recall that stands for any interval of . We denote by

the family of all intervals within .

Moreover, if is any family, then denotes the smallest -algebra containing . This can be a fairly complicated object and its existence is not really obvious. To get an idea as to why makes sense, we note that , that is a -algebra, and that the intersection of any number of -algebras is still a -algebra.

The next lemma is a standard result from measure theory.

Lemma 2.10.

Let denote any of the four families of open intervals, closed intervals, left-open intervals, or right-open intervals within . It holds that

The fact that coincides with the -algebra generated by all closed intervals in is often used in connection with abstract cakes, which carry a topology, hence a family of open and of closed sets. The thus generated “topological” -algebra plays a special role and has a special name.

Definition 2.11.

We denote by the smallest -algebra on containing all closed intervals from and call it the Borel or topological -algebra over .

The following definition is well-known as well:

Definition 2.12.

Let be a cake and a -algebra on . A (positive) measure on is a map satisfying that and is -additive.

It is useful to see a measure as a function defined on the sets. If the set-function is additive, then -additivity is, in fact, a continuity requirement on , as it allows to interchange the limiting process in the infinite union of pairwise disjoint sets with a limiting process in the sum. To wit:

since all terms are positive, there is no convergence issue. Equivalently, we can state -additivity as , then (for any measure ) or as , then (for finite measures ).

Sometimes (and a bit provocatively) it is claimed that there are essentially only two measures on the line (or on ): Lebesgue measure and Dirac measure , where is a fixed point. Let us briefly discuss these two extremes and explain as to why the claim is incorrect but still sensible.

Dirac Measure.

Let be a fixed point and set or according to or , respectively. This definition works for any , and it is easy to see that this set-function is indeed a measure (in the sense of Definition 2.12 on the -algebra  – or any smaller -algebra over .

Dirac’s measure is the derivative of the physicists’ “Delta function”: Indeed, the integral can be shown to yield , which makes (the derivative is understood in a distributional sense) a “function” such that if , , and  – the trouble being that cannot be defined pointwise for each . This is best understood if we use and consider the distribution function: This is the Heaviside function on , which is zero on , one on with a jump of size , and exploding differential quotient at .

We call the support of since, by definition, charges only sets such that . If we compare Dirac measure with Lebesgue’s measure, the problem is that the support of is a degenerate interval of length zero, see below.

Lebesgue Measure.

The idea behind Lebesgue measure is to have a set-function in (or in or ) with all properties of the familiar volume from geometry; in particular, we want a volume that is additive and invariant under shifts and rotations. Thus it is natural to define for a simple set like an interval (or an -dimensional “cube” )

Invariance under shifts together with the -additivity allow us to exhaust (“triangulate”) more complicated shapes like a circle with countably many disjoint sets such that with , we have . The restriction to countable unions is natural, as we exhaust a given shape by nontrivial sets , having nonempty interior: Each of them contains a rational point ; hence, there are at most countably many nonoverlapping .

There are immediate questions with this approach: Which types of sets can be “measured”? Is the procedure unique? Is the process of measuring more complicated sets constructive? At this point we encounter a problem: General sets are way too complicated to get a well-defined and unique extension of from the rectangles to . In dimension and for the standard cake , the Cantor sets from Example 1.1 were already challenging, but the Vitali set from Example 1.2 shows that the cocktail of shift invariance and -additivity becomes toxic.

The way out is the notion of measurable sets and Carathéodory’s extension theorem (stated as Theorem 2.13 further down). This works as follows: In view of the -additivity property of , it makes sense to consider the -algebra which contains the intervals (respectively, cubes). Thus we naturally arrive at the notion of the Borel -algebra as the canonical domain of Lebesgue measure. Unfortunately, there are so many Borel sets that we cannot build them constructively from rectangles – we would need transfinite induction for this – and this is one of the reasons why cutting a cake is not always a piece of cake.

The question of whether every set has a unique geometric volume (in the above sense) is dimension-dependent. If or , we can extend the notion of length and area to all sets, but not in a unique way. In dimension and higher, we’ll end up with contradictory statements (such as the Banach–Tarski paradox; <see, e.g.,>wag:b:banach-tarski-paradox) if we try to have a finitely additive geometric volume for all sets. This conundrum can be resolved by looking at the Borel sets or the Lebesgue sets – these are the Borel sets enriched by all subsets of Borel sets with Lebesgue measure zero.

General Measures.

Let us return to the assertion that and are “essentially the only measures” on . To keep things simple, we discuss here only the standard cake .

Lebesgue’s decomposition theorem shows that all -additive measures on with the Borel -algebra are of the form where “ac,” “sc,” and “d” stand for absolutely continuous, singular continuous, and discontinuous. This is best explained by looking at the distribution function . Since is increasing, it is either continuous or discontinuous (with at most countably many discontinuities), accounting for the parts () and , respectively. At the points where is continuous, we have again two possibilities: is either differentiable () or it isn’t, yielding “ac” vs. “sc.” From Lebesgue’s differentiation theorem it is known that the points with “sc” or “d” must have Lebesgue measure zero. Thus, we finally arrive at the decomposition

where are the at most countably many discontinuities (jump points) of and . Intuitively, this decomposition refers to a cake which, given a valuation is homogeneous (this is the part with ), contains “raisins” of size , and some valuable dust which is fractal-like and divisible but not homogeneous.

Let us close this section with the central result on the extension of valuations defined on an algebra to measures on the -algebra generated by . We state it only for the standard cake; the formulation for more abstract cakes is obvious.

Theorem 2.13 (Carathéodory’s extension theorem).

Let be a valuation on and denote by the algebra of admissible pieces of cake. If is additive and -additive relative to , i.e., satisfies , then there is a unique extension of , defined on , which is a -additive measure on .

2.3 Abstract Cakes

Let us briefly discuss more general cakes than . In this section, will be a general set, an algebra of admissible pieces, and the -algebra generated by . The definition and the properties of a valuation (cf. Definition 2.2) still work in this general setting, but since is abstract, there may not be (an equivalent of) a distribution function; this means that the connection between divisibility and -additivity, cf. Lemma 2.7 and Corollary 2.8, might fail in an abstract setting.

We begin with a new definition of for finitely additive valuations on abstract cakes.

Definition 2.14.

A finitely additive valuation on an abstract cake and an algebra of admissible pieces has the property if for every and , there is an increasing sequence of sets , , such that and .

If is a -additive valuation and a -algebra, then is again in , and, because of -additivity, we see that . Thus the properties and are indeed equivalent for -additive valuations (or, in view of Corollary 2.8, for finitely additive valuations on the standard cake and ).

We will also need the opposite of the property ; to this end, recall Definition 2.4 of an atom. If and are atoms, then we have either or ; in the latter case, if , we call the atoms equivalent. If and are nonequivalent, then and are still nonequivalent and disjoint. Iterating this procedure, we can always assume that countably many nonequivalent atoms are disjoint: Just replace the atoms by .

Since , a finitely additive valuation can have at most nonequivalent atoms such that , and so there are at most countably many atoms. Comparing Definition 2.14 which defines property  with Definition 2.4 of an atom, it is clear that implies that has no atoms. We will see in Theorem 2.16 that the converse implication holds as well.

Definition 2.15.

Let be a finitely additive valuation on the algebra over . The valuation is sliceable if for any , there are finitely many disjoint sets , , , such that and .

A set is -sliceable if the set-function is sliceable.

We will now see that a sliceable finitely additive valuation enjoys property , and vice versa, i.e., sliceability, atom-freeness, and property  are pairwise equivalent for finitely additive valuations.

Theorem 2.16.

Let be a finitely additive valuation on an algebra over an abstract cake . The conditions , is sliceable,” and has no atoms” are pairwise equivalent.

Proof.

We start by showing that atom-freeness implies sliceability. Fix .

Step 1: Let be any subset, and assume that there is some , , such that . Define

We claim that for the special choice the family is not empty.

Since is not an atom, there is some , , with .

If , then , and we are done.

If , we assume, to the contrary that there is no subset , , with . Since cannot be an atom, there is a subset with and . Iterating this with furnishes a sequence of disjoint sets with for all . This is impossible since . So we can find some with , i.e., is not empty.

Step 2: Define a(n obviously monotone) set-function for any ; as usual, . Since is not empty, we can pick some such that .

If , we set ; otherwise, we can pick some such that .

In general, if , we set ; otherwise, we pick

(1)

We are done if this procedure stops after finitely many steps; otherwise, we get a sequence of disjoint sets satisfying (1). Define . This set need not be in , but we still have, because of (1),

since the series

converges. In particular, .

Using again the convergence of the series , we find some such that , hence and are the desired small pieces of . This completes the proof that is sliceable.

We now show that sliceability implies condition . Let with . Since the “relative” finitely additive valuation inherits the nonatomic property from , it is clearly enough to show that for every , there is an increasing sequence

which is the property relative to the full cake only.

Since is sliceable, there are mutually disjoint sets , where , , and .

Let . Set , where is the unique number such that

By construction, . Thus, we can iterate this procedure, considering and constructing a set that satisfies

For , we get .

The sequence , , satisfies , i.e., is the sequence of sets we need to have property .

As mentioned earlier, implies atom-freeness, which completes this proof. ∎

Since for a -additive valuation on a -algebra , properties and are equivalent, we immediately get:

Corollary 2.17.

Let be a -additive valuation on a -algebra over an abstract cake . The conditions , , is sliceable,” and has no atoms” are pairwise equivalent.

If is an enumeration of the nonequivalent atoms of the -additive valuation , then , and we can restate Corollary 2.17 in the form of a decomposition theorem.

Corollary 2.18.

Let be a -additive valuation on a -algebra over an abstract cake . Then can be written as a disjoint union of a -sliceable set and at most countably many atoms .

3 Five Possible Definitions from the Literature

In the cake-cutting literature, a great variety of different definitions have been used for the set of admissible pieces of cake. We first collect the most commonly used definitions for , along with the corresponding references and discuss them in detail. Then we show several relations among these definitions and discuss what this implies for a most reasonable choice of .

Typical choices for the set containing all admissible pieces of a standard cake are

  1. all finite unions of intervals from , i.e., the family defined earlier on page 2.1;

  2. all countable unions of intervals from , i.e., ;

  3. the Borel -algebra over , i.e., ;

  4. the set of all Lebesgue-measurable sets over , i.e., ;222Recall that a set is Lebesgue-measurable if, and only if, there is a Borel-measurable set such that the symmetric difference is contained in a Borel-measurable set with Lebesgue measure . We will see in Theorem 3.1 that there are indeed Lebesgue-measurable sets that are not Borel-measurable. or

  5. the power set of .

Assuming is common among papers that consider only finite cake-cutting protocols. Such protocols can make only a finite number of cuts, thus producing a finite set of contiguous pieces, i.e., intervals, to be evaluated by the players. Authors that make this assumption and use include woe-sga:j:complexity-cake-cutting, str:j:finite-protocols-cannot-ef, lin-rot:c:dgef, pro:c:though-shalt-covet, wal:c:online-cake-cutting, coh-lai-par-pro:c:optimal-envy-free-cake-cutting, bei-che-hua-tao-yan:c:optimal-connected-cake-cutting, cec-pil:j:computability-equitable-divisions, bra-fel-lai-mor-pro:c:maxsum-fair-cake-divisions, cec-dob-pil:j:existence-equitable-divisions, che-lai-par-pro:j:truth-justice-cake, bra-mil:c:equilibrium-analysis-cake-cutting, azi-mac:c:discrete-bounded-ef-protocol-four-players,azi-mac:c:discrete-bounded-envy-free-cake-cutting-protocol,azi-mac:j:bounded-envy-free-cake-cutting-algorithm, edm-pru:c:no-piece-of-cake, and azi-mac:c:discrete-bounded-envy-free-cake-cutting-protocol.

As a special case, valuation functions may even be restricted to single intervals, which is done by cec-pil:j:near-equitable-2-person-cake-cutting-algorithm and aum-dom:c:efficiency-connected-pieces. Even though the restriction to finite unions of intervals is sensible from a practical perspective, it may artificially constrain results that could hold also in a more general setting.

bra-pro-zha:c:externalities-cake-cutting extend to contain countably infinite unions of intervals, i.e., .

Authors assuming include str-woo:j:measures-agree, den-qi-sab:t:complexity-envy-free, and seg-nit-has-aum:j:two-dimensional-cake-cutting.

Works using include those by rei-pot:j:finding-ef-pareto-optimal-division, arz-aum-dom:c:throw-cake, and rob-web:j:near-exact-and-envy-free-cake-division. Additionally, several authors do not explicitly make the assumption

, but they define valuation functions based on (Lebesgue-)measurable sets only, most prominently, a valuation function is often defined as the integral of a given probability density function on 

. This or a similar assumption is made by bra-jon-kla:perfect-cake-cutting-with-money,bra-jon-kla:j:better-cut-cake,bra-jon-kla:j:pie-cutting,bra-jon-kla:j:there-may-be-no-perfect-division, rob-web:b:cake-cutting-algorithms-be-fair-if-you-can, web:j:minimal-cuts, aum-dom-has:c:socially-efficient-cake-divisions, bra-car-kur-pro:t:strategic-fair-division, and car-lai-pro:c:more-expressive-cake-cutting.

Papers that assume include those by mac-mar:j:cut-pizza-fairly, sga-woe:j:linear-approximation-cake-cutting, sab-wan:c:envy-free-cake-cutting-for-five, man-oka:c:meta-envy-free-cake-cutting-protocols, and aum-dom-has:c:truthful-cake-auctions.

Finally, several works, including those by dub-spa:j:cut-cake-fairly, bar:j:game-theoretic-algorithms-for-cake-divisions,bar:j:super-envy-free-divisions, zen:c:approximate-envy-free-procedures, and bra-tay:j:envy-free-protocol, define the set of admissible pieces of cake to be some (-)algebra (not necessarily Borel) over .

Note that each of the sets , , , and is an algebra over , and all, except , are also -algebras over . That is an algebra is shown in Lemma 4.4 below. However, is not an algebra, as the proof of the following theorem shows.

Having introduced all the different approaches currently used in the literature, we will now prove the strict inclusions among these sets stated in the following theorem.

Theorem 3.1.

.

Proof.

We start with proving (a): . Obviously, is true, as every finite union of intervals is a countable union of intervals. To see that the two sets are not equal, look at . It is clear that is true, as is a countable union of intervals. However, it holds that , i.e., cannot be written as a finite union of intervals, as all these subintervals are pairwise disjoint. Hence, , so , and we have shown (a).

In Lemma 2.10 and Definition 2.11, we have seen that where is the family of all intervals within . Since a -algebra is stable under (finite and countable) unions, we get . Using again the stability of a -algebra under countable unions, we arrive at .

Since, however, is true, as can be written as a countable union of intervals that each contain one element, it must hold that by the definition of a -algebra. However, the irrational numbers in cannot be written as a countable union of intervals, since every interval containing more than one element immediately contains a rational number. Therefore, is not an algebra and holds, proving (b).

The inclusion holds by definition, as all Borel sets are Lebesgue-measurable. However, there are Lebesgue-measurable sets that are not Borel-measurable: Observe that the cardinality of is the cardinality of (which is ), whereas there are only continuum-many (i.e., , the cardinality of