Resource-monotonicity and Population-monotonicity in Cake-cutting

Transcription

1 Resource-monotonicity and Population-monotonicity in Cake-cutting Erel Segal-Halevi Bar-Ilan University, Ramat-Gan , Israel. Balázs Sziklai Centre for Economic and Regional Studies, Hungarian Academy of Sciences, December 22, 2015 Abstract We study the monotonicity properties of solutions in the classic problem of fair cake-cutting - dividing a single heterogeneous resource among agents with dierent preferences. Resource- and populationmonotonicity relate to scenarios where the cake, or the number of participants who divide the cake, changes. It is required that the utility of all participants change in the same direction: either all of them are better-o (if there is more to share) or all are worse-o (if there is less to share). We formally introduce these concepts to the cake-cutting setting and present a meticulous axiomatic analysis. We show that classic fair cake-cutting protocols, like the Cut and Choose, Banach-Knaster, Dubins-Spanier and many other fail to be monotonic. We then present monotonic division rules in two utility models. When the utility functions are additive, there are several proportional and Pareto-optimal division rules that satisfy one or both monotonicity axioms. In contrast, when the utility functions have a connectivity constraint such as in land division, no proportional and Pareto-optimal rule can satisfy any monotonicity axiom. Assuming weak-pareto-optimality (no other allocation is better for all agents), we provide a proportional resourcemonotonic protocol for two agents and some population-monotonic rules for n agents. 1

2 1 Introduction Two siblings, Ann and Miriam, inherit a piece of land from their father. Just as soon as they agree on how to share the land, they learn that their father recently bought another strip of land. They re-divide the whole estate. However Miriam ends up with a more valuable piece of land while Ann is worse o. Ann feels that they should try another division protocol. After a little reasoning Ann and Miriam manage to divide the estate. Just as soon as they settle the issue, they discover that they have a little step-brother. They decide to divide the land three-ways by using the same division rule as before. However Ann ends up with a more valuable piece of land while Miriam is worse o. Miriam feels that they should try another division protocol. These two examples motivate the so called resource-monotonicity (RM) and population-monotonicity (PM) axioms. Resource-monotonicity, sometimes known as aggregate monotonicity, requires that when new resources are added, and the same division rule is used consistently, the welfare of the participants should weakly increase. Population-monotonicity is concerned with changes in the number of participants. It requires that when someone leaves the division process and abandons his share, the welfare of the remaining participants should weakly increase. Conversely, when a new agent joins the process, the welfare of the existing participants should weakly decrease. We study the two monotonicity requirements in the framework of the classic fair cake-cutting problem (Steinhaus, 1948), where a single heterogeneous resource - such as land or time - has to be divided fairly. Fair cake-cutting protocols can be applied in inheritance cases and divorce settlements. They can be also used to divide broadcast time of advertisements and priority access time for customers of an Internet service provider (Caragiannis et al., 2011; Procaccia, 2015). The notion of fairness in cake-cutting is commonly restricted to two properties: proportionality means that each of the n agents should receive a value of at least 1/n of the total cake value; envy-freeness means that each agent weakly prefers his share over the share of any other agent. We argue that monotonicity is as important fairness axiom as proportionality and envy-freeness, since the principle stems from solidarity. If there is less to share, some agents will necessarily lose. Hence it is only fair to require that the loss is (weakly) shared by all agents. On the other hand, if there is more to share, it is fair to require that the gain is (weakly) enjoyed by all agents. Additionally, economic growth might be objected if the benets are 2

3 not shared by everyone. 1 Experimental studies show that people value certain fairness criteria more than others. Herreiner and Puppe (2009) demonstrated that people are willing to sacrice Pareto-eciency in order to reach an envy-free allocation. To our knowledge no study was ever conducted to unfold the relationship between monotonicity and eciency or proportionality. However some indirect evidence points toward that monotonicity of the solution is in some cases as important as proportionality. In the apportionment problem there is a parliament with a xed number of seats and administrative regions with dierent number of voters. The question is assuming that electoral districts are of equal size within each of the administrative regions how to distribute the seats among the administrative regions to minimize the dierences between the voters inuences. The problem is analogous to cake-cutting where the cake corresponds to the parliamentary seats which have to be distributed. During the 1880 US census C.W. Seaton, a Chief Clerk of the Census Oce, noted that an enlargement of the House of Representatives from 299 to 300 would result in a loss of seat for State Alabama. This anomaly together with the later discovered population and new state paradoxes pressed the legislators to adopt newer and newer apportionment rules. The currently used seat distribution method is free from such anomalies, however it does not satisfy the so called Hare-quota, a basic guarantee of proportionality (Balinski and Young, 1975). Here we present a systematic analysis of the relationship of resource- and population-monotonicity and classical axioms such as Pareto-eciency and proportionality. To the best of our knowledge, this is the rst paper that studies these properties in a cake-cutting setting Results We survey many traditional cake-cutting protocols and show that they do not satisfy either of the monotonicity axioms. In particular, all methods based on the Cut and Choose scheme violate resource-monotonicity. This motivates a search for division rules that are both fair in the conventional sense and 1 A recurring theme in recent social movements like the "99%" and "Occupy Wall Street" is that although the current economic system indeed brings economic growth, it does only to a small fraction of the population, while the other 99% becomes poorer. 2 We found a single mention of these properties in the concluding comments of Berliant et al. (1992): "...there are a number of important issues that should be tackled next pertaining, in particular, to the existence of selections from the no-envy solution satisfying additional properties, Examples are monotonicity with respect to the amount to be divided (all agents should benet from such an increase), and with respect to changes in the number of claimants (all agents initially present should lose in such circumstances)." 3

4 monotonic. This search is done under two dierent assumptions regarding the agents' utility functions, both of which are common in the cake-cutting literature. In both models, each agent has a value measure dened over the cake. In the additive model, the utility of a piece of cake is just the value measure of that piece; the geometry of the piece has no importance. In the connected model, the utility of a piece of cake is the value of the most valuable connected component of the piece. Connected utility functions make sense in many applications. E.g., when dividing land, a large connected piece can be used for building a house, but a collection of small disconnected pieces is virtually useless. Note that when the agents' utility functions are connected, eciency dictates that each agent should be given a single connected piece, since he will not be able to use the other pieces. In the additive model, our results are positive. There are several Paretooptimal proportional division rules that satisfy one or both monotonicity axioms. The Nash-optimal rule, maximizing the product of values, is envy-free (hence also proportional), resource-monotonic and population-monotonic. The leximin-optimal division rules, which are motivated by the egalitarian justice criterion, are population-monotonic. There are two leximin rules: the rule using relative values is proportional but not resource-monotonic and the rule using absolute values is resource-monotonic but not proportional. In the connected model, the situation is not so positive. Each of the monotonicity properties is incompatible with proportionality and Paretooptimality. That is, no Pareto-optimal proportional division rule can be either resource- or population-monotonic. Thus the fair divider has to choose between Pareto-optimality and monotonicity. While from an economics perspective, Pareto-optimality is crucial, public opinion may not always agree. As explained above, in some cases people are willing to sacrice eciency to get fairness. As a compromise, we suggest several division rules which are proportional and weakly-pareto-optimal (no other allocation is strictly preferred by all agents; see e.g. Varian (1974)) while satisfying one of the monotonicity axioms. The max-equitable-connected rules, which give equally-valuable pieces to each agent while maximizing this value, are both population monotonic. Analogously to the leximin-optimal rules, there are two such rules: the rule equalizing the relative values (normalized such that the entire cake value is 1) is proportional but not resource-monotonic, and the rule equalizing the absolute (not normalized) values is resource-monotonic but not proportional. Additionally, we present a proportional and resource-monotonic division protocol for two agents. It is an open question whether there exists a proportional and resource-monotonic rule for n agents. The equitable and leximin rules belong to the cardinal welfarism frame- 4

5 work (cf. chapter 3 of Moulin (2004)). They rely on inter-agent utility comparison, and make sense if and when such a comparison is feasible. For example, suppose the agents are rms, each of which wants to use the land to a pre-specied purpose (e.g. one rm plans to dig for oil, another rm wants to build housing complexes, etc.). Then, economic models can be used to estimate the monetary utility of each rm for each piece of land and the estimates can be used to calculate leximin/equitable divisions (see the conclusion of Chambers (2005) for further discussion of the additive utility model). Although these two rules have several drawbacks, such as high computational complexity and strategic issues that make them hard to implement in real life, their mere existence shows that it is possible to combine monotonicity with certain fairness axioms. The paper is organized as follows. Section 2 reviews the related literature. Section 3 formally presents the cake-cutting problem and the monotonicity axioms. Section 4 examines classic cake-cutting protocols and shows that they are not monotonic. Section 5 presents our results for agents with additive utilities (and no connectivity constraints). Section 6 presents our results for agents that want connected pieces. Section 7 concludes and presents a table summarizing the various rules' properties. 2 Related Work The cake-cutting problem originates from the work of the polish mathematician Hugo Steinhaus and his students Banach and Knaster (Steinhaus, 1948). Their primary concern was how to divide the cake in a fair way. Since then, game theorists analyzed the strategic issues related to cake-cutting, while computer scientists were focusing mainly on how to implement solutions, i.e. the computational complexity of cake-cutting protocols. Monotonicity issues have been extensively studied with respect to cooperative game theory (Calleja et al., 2012), political representation (Balinski and Young, 1982), computer resource allocation (Ghodsi et al., 2011) and other fair division problems. Extensive reviews of monotonicity axioms can be found in chapters 3, 6 and 7 of (Moulin, 2004) and in chapter 7 of Thomson (2011). Thomson (1997) denes the replacement principle, which requires that, whenever any change happens in the environment, the welfare of all agents not responsible for the change should be aected in the same direction - they should all be made at least as well o as they were initially or they should all be made at most as well o. This is the most general way of expressing the 5

6 idea of solidarity among agents. The PM and RM axioms are special cases of this principle. The consistency axiom (cf. Young (1987) or Thomson (2012)) is related to population-monotonicity, but it is fundamentally dierent as in that case the leaving agents take their fair shares with them. Arzi et al. (2011) study the "dumping paradox" in cake-cutting. They show that, in some cakes, discarding a part of the cake improves the total social welfare of any envy-free division. This implies that enlarging the cake might decrease the total social welfare. This is related to resourcemonotonicity; the dierence is that in our case we are interested in the welfare of the individual agents and not in the total social welfare. Chambers (2005) studies a related cake-cutting axiom called "division independence": if the cake is divided into sub-plots and each sub-plot is divided according to a rule, then the outcome should be identical to dividing the original land using the same rule. He proves that the only rule which satises Pareto-optimality and division independence is the utilitarian-optimal rule - the rule which maximizes the sum of the agents' utilities. The rule is only feasible when the utilities are additive (with no connectivity constraints). Unfortunately, this rule does not satisfy fairness axioms such as proportionality. Walsh (2011) studies the problem of "online cake-cutting", in which agents arrive and depart during the process of dividing the cake. He shows how to adapt classic procedures like cut-and-choose and the Dubins-Spanier in order to satisfy online variants of the fairness axioms. Monotonicity properties are not studied, although the problem is similar in spirit to the concept of population-monotonicity. 3 Model 3.1 Cake-cutting A cake-cutting problem is a triple Γ(N, C, (ˆv C i ) i N ) where: N = {1, 2,..., n} denotes the set of agents who participate in the cakecutting process. In examples with a small number of agents, we often refer to them by names (Alice, Bob, Carl...). C is the cake. For simplicity we assume that C is a interval, C = [0, c] for some real number c. We call a Borel subset of C a slice. 6

7 ˆv i C is the value measure of agent i, which is a real-valued function on the slices of C. When it does not cause confusion we will omit C from the upper index, and simply write ˆv i. A slice with zero Lebesgue-measure is referred to as a zero slice while a slice with positive Lebesgue-measure as a positive slice. A slice that is allotted to an agent is called a piece. As the term "measure" implies, the value measures of all agents are countably additive: the value measures of a union of disjoint slices is the sum of the values of the slices. Moreover, we assume that the value measures are nonnegative and bounded. That is, ˆv i assigns a non-negative, but nite number to each slice of C. We also assume that the value measures are non-atomic: this means that a zero slice has zero value to anyone. In particular any point on the interval is worthless for the agents. Without loss of generality we also assume that every positive slice has positive value to at least one agent. All these assumptions are standard in the cake-cutting literature. The utility of an agent is based on its value measure in one of two ways: In the additive utilities model, the utility is just the value: u i (X) = ˆv i (X) In the connected utilities model, the agent can use only a single connected piece, so the utility of a piece is the value of the most valuable connected interval in that piece: u i (X) = sup ˆv i (I) I X where the supremum is over all connected intervals I that are subsets of X. Both these models are common in the classic cake-cutting literature. In fact, many papers explicitly assume that each agent must be allocated a single connected piece. Our model diverges from the standard cake-cutting setup in that we do not require the value measures to be normalized. That is, the value of the entire cake is not necessarily the same for all agents. This is important because we examine scenarios where the cake changes, so the cake value might become higher or lower. Hence, we dierentiate between absolute and relative value measures: The absolute value measure of the entire cake, ˆv i (C), can be any positive value and it can be dierent for dierent agents. 7

8 The relative value of the entire cake is 1 for all agents. Relative value measures are denoted by vi C and dened by: vi C (S) = ˆv i C (S)/ˆv i C (C). Again, for convenience's sake we will write simply v i (S) when there is only one cake. It is also common to assume that value measures are private information of the agents. This question leads us to whether agents are honest about their preferences. As we noted before cake-cutting problems can be studied from a strategic angle. Here, however, we will not analyze the strategic behavior of the agents. The aim is to divide the cake into n pairwise-disjoint slices. A division rule is a function that takes a cake-cutting problem as input and returns a division X = (X 1,..., X n ), or a set of divisions. Note that a division does not necessarily compose a partition of C (i.e. free disposal is assumed). A division rule R is called essentially single-valued (ESV) if X, Y R(Γ) implies that for all i N, ˆv i (X i ) = ˆv i (Y i ). That is, even if R returns a set of divisions, all agents are indierent between these divisions. The classic requirements of fair cake-cutting are the following. A division X is called: Pareto-optimal (PO) if there is no other division which is weakly better for all agents and strictly better for at least one agent. Weakly-Pareto-optimal (WPO) if there is no other division which is strictly better for all agents. Proportional (PROP) if each agent gets at least 1/n fraction of the cake according to his own evaluation, i.e. for all i N, v i (X i ) 1/n. Note that the denition uses relative values. Envy-free (EF) if each agent gets a piece which is weakly better, for that agent, than all other pieces: for all i, j N, v i (X i ) v i (X j ). Note that here it is irrelevant whether absolute or relative values are used. Note also that PO+EF imply PROP. A division rule is called Pareto-optimal (PO) if it returns only whole divisions. The same applies to weak-pareto-optimality, proportionality and envy-freeness. 3.2 Monotonicity We now dene the two monotonicity properties. In the introduction we dened them informally for the special case in which the division rule returns 8

9 a single division. Our formal denition is more general and applicable to rules that may return a set of divisions. Denition 3.1. Let N be a xed set of agents, C = [0, c], C = [0, c ] two cakes where c < c, and let (ˆv i C ) i N, (ˆv i C ) i N be two value measures such that (ˆv i C ) i N coincides with (ˆv i C ) i N on the slices of C. The cake-cutting problem Γ = (N, C, (ˆv i C ) i N ) is called a cake-enlargement of the problem Γ = (N, C, (ˆv i C ) i N ). By denition the cake is always enlarged on the right hand side. This might be critical for some protocols. For instance in the Dubins-Spanier moving knife protocol the cake is processed from left to right (Dubins and Spanier, 1961). However, most of our results (except that of Subsection 6.4) are valid whenever C C, regardless of whether the cake is enlarged from the left, right or middle. Denition 3.2. (a) A division rule R is called upwards resource-monotonic, if for all pairs (Γ, Γ ), where Γ is a cake-enlargement of Γ, for every division X R(Γ) there exists a division Y R(Γ ) such that ˆv i (Y i ) ˆv i (X i ) for all i N (i.e all agents are weakly better-o in the new division). (b) A division rule R is called downwards resource-monotonic, if for all pairs (Γ, Γ), where Γ is a cake-enlargement of Γ, for every division Y R(Γ ) there exists a division X R(Γ) such that ˆv i (X i ) ˆv i (Y i ) for all i N (i.e all agents are weakly worse-o in the new division). (c) A division rule is resource-monotonic (RM), if it is both upwards and downwards resource-monotonic. Denition 3.3. Let C be a xed cake, N and N two sets of agents such that N N and (ˆv i ) i N their value measures. The cake-cutting problem Γ = (N, C, (ˆv i ) i N ) is called a population-reduction of the problem Γ = (N, C, (ˆv i ) i N ). Denition 3.4. (a) A division rule R is called upwards population-monotonic, if for all pairs (Γ, Γ) such that Γ is a population-reduction of Γ, for every division Y R(Γ ) there exists a division X R(Γ) such that ˆv i (X i ) ˆv i (Y i ) for all i N (all the original agents are weakly worse-o in the new division). (b) A division rule R is called downwards population-monotonic, if for all pairs (Γ, Γ ) such that Γ is a population-reduction of Γ, for every division X R(Γ) there exists a division Y R(Γ ) such that ˆv i (Y i ) ˆv i (X i ) for all i N (all remaining agents are weakly better-o in the new division). (c) A division rule is population-monotonic (PM), if it is both upwards and downwards population-monotonic. 9

10 Remark 3.1. As usual in the literature, the monotonicity axioms care only about absolute values (e.g., if the relative value of an agent decreases when the cake grows, this is not considered a violation of RM). Remark 3.2. For essentially-single-valued solutions, downwards resource (or population) monotonicity implies upwards resource (or population) monotonicity and vice versa. Set valued solutions, however, may satisfy only one direction of these axioms. Remark 3.3. The monotonicity axioms in Thomson (2011) require that all divisions in R(Γ) be weakly better/worse than all divisions in R(Γ ). In contrast, our denition only requires that there exists such a division. This is closer to the denition of aggregate monotonicity, which originates from cooperative game theory Peleg and Sudhölter (2007). The rationale is that even if a set-valued solution is used, only a single allocation will be implemented. Hence, the divider can be faithful to the monotonicity principles even if the rule suggests many non-monotonic allocations as well. Because our monotonicity requirements are weaker, any impossibility result in our model is valid in Thomson's model, too. This is not true in general for positive results; however, the specic positive results in the present paper are all based on essentially-single-valued rules. Hence, by the previous remark, they are valid in Thomson's model, too. 4 Monotonicity and eciency of classical protocols Although resource and population-monotonicity are well established axioms in various elds of fair division, the cake-cutting literature has not adopted these ideas so far. Moreover, classical division methods like the Banach- Knaster (Steinhaus, 1948), Cut and Choose, Dubins-Spanier (Dubins and Spanier, 1961), Even-Paz (Even and Paz, 1984), Fink (Fink, 1964) or the Selfridge-Conway protocol do not satisfy these axioms. A detailed explanation for most of these can be found in (Brams and Taylor, 1996). Admittedly this list is incomplete, still it illustrates our point well. All the counterexamples below feature piecewise homogeneous cakes. These are nite unions of disjoint intervals, such that on each interval the value densities of all agents are constant (although dierent agents may evaluate the same piece dierently). In such cases, the function ˆv i ([0, x]) which displays the value (for Agent i) of the piece which lies left to the point x R is a piecewise-linear function (see Figure 1). We stress that our division rules, 10

11 ˆv ([0, x]) ˆv([0, x]) ˆv A ([0, x]) ˆv B ([0, x]) } {{ } cake x } {{ } cake x Figure 1: Piecewise homogeneous cake with two players. The derivative of ˆv([0, x]) indicates the density of the value. Note that the value measures are not normalized, hence ˆv A ([0, c]) ˆv B ([0, c]). presented in the next section, hold for arbitrary cakes - not only piecewisehomogeneous. Piecewise homogeneous cakes can be represented by a simple table containing the value densities of the agents on the dierent slices. For example the cake in Figure 1 has the following representation. ˆv A ˆv B Resource-monotonicity First let us examine why Cut and Choose is not resource-monotonic. In the next couple of examples the sign over a column indicates the enlargement. ˆv A ˆv B ˆv A ˆv B Table 1: Colored cells indicate the pieces of the agents. Consider the cake described in Table 1. While the extra piece is not present, Agent A (Alice) cuts the cake after the second slice, allowing Agent B (Bob) to choose the piece worth 5 for him. However if we enlarge the cake, Alice cuts after the third slice and no matter which piece Bob chooses, he 11

12 loses utility. This simple counterexample implies that the Banach-Knaster, Dubins-Spanier, Even-Paz and the Fink methods are not resource-monotonic either, as they all produce the same division on the above cake as the Cut and Choose. 3 Note that the fact that we enlarge the cake on the right-hand side is not an issue here. As the next example shows the Dubins-Spanier method violates resource-monotonicity even if we reverse its order, i.e. we move the knives and execute the cuts from right to left. ˆv A ˆv B ˆv C ˆv A ˆv B ˆv C Before the enlargement, Bob got the rightmost slice and Carl got the middle slice, leaving Alice with a piece worth 4 for her. With the extra piece present, Alice picks the rightmost slice 4. Similar reasoning shows that the reverse Even-Paz is not resource-monotonic either. Finally, the Selfridge-Conway protocol - the only envy-free method we survey here - is not resource-monotonic either. Consider the following cake. ˆv A ˆv B ˆv C ˆv A ˆv B ˆv C In the smaller cake, Alice cuts the cake into three parts that worth 6 to her, each made of two adjacent slices. The two most valuable parts for Bob are worth the same (6) so he passes. Then the agents choose in the order Carl, Bob, Alice and each obtain 6 units of utility. In the larger cake, Alice cuts three parts worth 8 to her, made of 3, 2 and 2 slices. The leftmost part is most valuable for Bob, so he trims it to make it equal to the middle part. Carl takes the uncut part, which is worth 4 for him. Now, Carl divides the remainder to 3 equal pieces and the agents choose in order: Bob, Alice, Carl. Regardless of how this is done, Carl ends up with at most A more recent protocol, the Recursive Cut and Choose, proposed by Tasnádi (2003), violates resource-monotonicity for the same reason. 4 Here we rely on the assumption that the value measures are private information, otherwise Alice would be prompted to act strategically. 12

13 4.2 Population-monotonicity Population-monotonicity is not applicable to protocols with xed number of agents, such as Cut and Choose and Selfridge-Conway. For symmetry reasons, reversing the direction of how a protocol processes the cake has no consequence on whether the protocol violates population-monotonicity or not. In the next couple of examples the cells of the leaving player are colored gray. Our rst example shows that the Banach-Knaster, Dubins-Spanier and the Even-Paz protocols are not population monotonic. ˆv A ˆv B ˆv C ˆv A ˆv B ˆv C Suppose we proceed by the Dubins-Spanier method. When all three agents are present Alice is the rst to stop the knife. Then Bob and Carl are the next cutters in that order. Carl manages to obtain a piece that worth 4 to him. However if Bob is not present then Carl will be the rst to cut and he has to content himself with a piece of value 3. Notice that the Even-Paz method and the Banach-Knaster protocol with order Alice-Bob- Carl produces the same allocations. Thus, neither of these three methods are population-monotonic. Consider now the Fink procedure. This procedure was specically designed with upwards-population-monotonicity in mind: when a new agent joins an existing division, he takes a proportional share from each of the existing agents, so all existing agents are weakly worse o (they all participate in supporting the new agent, which is what PM is all about). However, the Fink procedure is not downwards-pm, as the following example shows: ˆv A ˆv B ˆv C ˆv A ˆv B ˆv C Suppose that initially Alice and Bob use Cut and Choose and Bob is the chooser. He is able to salvage the whole cake according to his own evaluation. Now they divide their pieces into three equal parts, and Carl gets to choose one slice from each of them. Hence, Bob ends up with a piece worth at least 8 for him. But if Alice leaves, then Bob and Carl have to redivide the cake using Cut and Choose. Then, no matter who cuts, Bob ends up with only 6. The above example seemingly contradicts our claim that upwards-pm implies downwards-pm and vice versa for single valued solutions. However 13

14 there is a subtle dierence here. The Fink procedure is based on a predened order of the agents, and it is only upwards monotonic if the new agent is the last in the order. An alternative explanation is to treat the Fink rule as a set-valued rule, which returns n! possible allocations, for all n! possible orderings of the agents. Under this denition, the Fink rule is upwards-pm, but not downwards-pm as shown in the example. 4.3 Weak Pareto-optimality It is a well-known fact that the classical protocols that we discuss here are not Pareto-optimal. Now we show that with the exception of Cut and Choose they are not even weakly Pareto-optimal. In the following example the Banach-Knaster, Dubins-Spanier and Even-Paz protocols coincide. ˆv A ˆv B ˆv C ˆv A ˆv B ˆv C Alice gets the rst slice as it composes 1/3 of her cake value, while Bob and Carl perform a Cut and Choose on the rest of the cake. The table on the right presents an alternative allocation (still with connected pieces) which is strictly better for all agents. The Fink method is not contiguous, hence we can obtain an improvement by composing the pieces from more slices. ˆv A ˆv B ɛ ˆv A ˆv B ɛ For two agents the Fink method proceeds as the Cut and Choose: Alice cuts in the middle and Bob chooses the piece on the right. The second cake is a Pareto-improvement with four slices where every agent is strictly better o. The same example shows that the Cut and Choose is not WPO for additive utilities. However it is a contiguous protocol, thus it makes sense to investigate whether we could improve it with connected pieces. Lemma 4.1. The Cut and Choose method is weakly Pareto-optimal whenever the agents have connected utilities. Proof. Suppose Alice cuts at point x [0, c] and Bob chooses. The cake is divided into the left piece (L) and the right piece (R). Without loss of 14

15 generality we may assume that Alice receives the left piece. Alice's utility can be improved in only two ways: (a) Cut at some point x > x and give the left piece to Alice. But then Bob would receive R R, which obviously cannot be strictly better than R. (b) Cut at some point x < x and give Alice the right piece R R. Then Bob receives L L. Since Bob preferred R to L he did not gain utility by this swap. So it is impossible to strictly increase the utility of Alice and Bob simultaneously. Finally we show that the Selfridge-Conway does not satisfy WPO either. Consider the following cake. ˆv A ˆv B ˆv C ˆv A ˆv B ˆv C Alice cuts the cake into three equal parts. Since the two most valuable slices have equal value for Bob he passes. The agents choose in the Carl-Bob- Alice order and obtain the pieces shown by the table on the left. However, as the table on the right shows, there is an allocation where every agent is strictly better o. 5 Additive Utilities In this section we assume that for every agent, the utility of a piece is just that agent's value measure of that piece, regardless of whether the piece is connected or not. With additive utilities, there are several Pareto-optimal fair division rules satisfying monotonicity axioms. 5.1 Leximin-optimal divisions: PO+PM Lexicographic optimization as a solution concept is used in various elds of fair division (Moulin, 2004). Dubins and Spanier (1961) were the rst to employ it in a cake-cutting setting. Later, Dall'Aglio (2001); Dall'Aglio and Hill (2003); Dall'Aglio and Di Luca (2012) presented various properties and approximation algorithms for nding such leximin divisions (they call such divisions Dubins-Spanier-optimal ). The leximin idea is simple: First we narrow down the solution set to divisions that maximize the utility of the poorest agent. Then among those solutions that satisfy this criterion, we select those which maximize the utility of the second poorest agent. We repeat this process with the third, fourth, 15

16 etc. poorest agent. To introduce this concept formally rst we dene lexicographical ordering of real vectors. We say that vector y R n is lexicographically greater than x R n (denoted by x y) if x y and there exists a number 1 j n such that x i = y i if i < j and x j < y j. Denition 5.1. (a) For a cake division X, dene the relative-leximin-welfare vector as a vector of length n which contains the relative values of the agents under division X in a non-decreasing order. (b) A cake division Y is said to be relative-leximin-better than X if the relative-leximin-welfare vector of Y is lexicographically greater than the relative-leximin-welfare vector of X. (c) A cake division X is called relative-leximin-optimal if no other division is relative-leximin-better than X. (d) We dene the relative-leximin division rule as the rule that returns all relative-leximin-optimal divisions of the cake. The terms absolute-leximin-welfare vector, absolute-leximin-better and absoluteleximin-optimal and the absolute-leximin division rule are dened analogously. Example 5.1. In the following cake, the absolute-leximin division rule splits the leftmost slice between Alice and Bob, giving each of them 6. The rightmost slice is given to Carl. Thus, the absolute-leximin-optimal vector is (6,6,9). The relative value vector of the above division is (6/12, 6/12, 9/30). ˆv A 12 0 ˆv B 12 0 ˆv C 21 9 The corresponding relative-leximin welfare vector is (9/30, 6/12, 6/12) = (3/10, 1/2, 1/2), which is not optimal. The relative-leximin rule divides the leftmost slice between all three agents, giving 1/6 to Carl (absolute value 3.5) and 5/12 to Alice and Bob (absolute value 5). The rightmost slice is given to Carl. The relative-leximin-optimal vector is (5/12, 5/12, 5/12). Dubins and Spanier (1961) use the compactness of the space of valuematrices to prove that leximin-optimal cake divisions always exist. The proof is equally valid for absolute and relative leximin-optimal divisions. Hence, both the absolute- and the relative- leximin division rules are well-dened. The leximin rules are closely related to the equitable rules presented in Subsection 6.3. The relations between them are explored by Dall'Aglio 16

17 (2001). In particular, he proves that the smallest value that an agent receives in a leximin-optimal division is equal to the max-equitable value. This means that a leximin-optimal division is a Pareto-improvement of a maxequitable division. In fact, it is a Pareto-optimal improvement. Lemma 5.1. The absolute- and the relative-leximin division rules are both Pareto-optimal. Proof. By denition, if a division Y Pareto-dominates a division X, then both the absolute and the relative leximin-welfare vectors of Y are larger than those of X. Given a division X, we say that agent i is relatively-poorer than agent j if v i (X i ) < v j (X j ) and relatively-richer if v i (X i ) > v j (X j ). We say that i is weakly-relatively-poorer than j if v i (X i ) v j (X j ) and weakly-relativelyricher if v i (X i ) v j (X j ). The terms (weakly-)absolutely-poorer and (weakly- )absolutely-richer are dened analogously based on the absolute values ˆv i, ˆv j. Lemma 5.2. For every absolute/relative-leximin-optimal division X, if agent j is strictly absolutely/relatively poorer than agent i then agent j believes that the piece of agent i is worthless: ˆv j (X i ) = v j (X i ) = 0. Proof. If this were not the case, then we could take a small bit of X i and give it to agent j, thus achieving an absolute/relative-leximin-better division. But this contradicts the leximin-optimality of X. In Example 5.1, in case of the absolute-leximin-optimal division, Carl is absolutely-richer than Alice and Bob, and indeed his share is worthless for both of them. Suppose X is an old division and Y is a new division of the same cake. We say that an agent i conceded a slice to agent j if there is a positive slice that belonged to agent i in X and belongs to agent j in Y (in other words, X i Y j has a positive Lebesgue-measure). If X and Y are both absolute/relativeleximin-optimal, then by Pareto-optimality, X i Y j has positive value to both the agent who concedes the slice (i) and the recipient (j). Hence, we have the following corollary of Lemma 5.2, which is true both for relative- and absolute-values: Corollary 5.1. Let X and Y be two leximin-optimal divisions. If, when switching from X to Y, agent i conceded a slice to agent j, then in division X, agent i is weakly-poorer than j, and in division Y, agent i is weakly-richer than j. 17

18 Lemma 5.3. The absolute and the relative leximin division rules are essentially single-valued. Proof. The proof is the same for the absolute and the relative leximin rules, so the adjectives are omitted. Suppose X and Y are two dierent leximin-optimal divisions. By switching from X to Y, some agents may have lost value; call these agents unlucky. Similarly, call the agents who gained value lucky. We are going to prove that there are neither unlucky nor lucky agents; this implies that all agents have exactly the same value in both divisions. Suppose by contradiction that agent i is unlucky. Then, he must have conceded a slice to at least one other agent, say j. By Corollary 5.1, i is weakly-poorer than j in X and weakly-richer in Y. But this means that j is also unlucky. So, all slices conceded by unlucky agents are held by other unlucky agents. Suppose the unlucky agents take back all the slices that they conceded. This has no eect on the lucky agents, but strictly increases the value of the unlucky agents, since they now have at least the value that they had in X. But this contradicts the optimality of Y. Hence, there are no unlucky agents. If Y had a lucky agent without having any unlucky agent that would contradict the optimality of X. Since X and Y were arbitrary leximin-optimal divisions it follows that all leximin-optimal divisions have the same value vector. Lemma 5.4. The absolute- and relative-leximin division rules are populationmonotonic. Proof. Again the proof is the same for the absolute and the relative leximin rules. Thanks to Lemma 5.3 it is sucient to prove downwards-pm. Let X be a leximin-optimal division of C. Suppose that agent i leaves and abandons his share X i. So X is now a division of C \ X i among the agents N \ {i}. If X i is divided arbitrarily among N \ {i}, the result is X +, a division of C which is weakly leximin-better than X. Let Y be a leximinoptimal division of C among N \ {i}. Y is weakly leximin-better than X + and hence weakly leximin-better than X. The rest of the proof is very similar to the proof of Lemma 5.3. Call the agents that lost value in the move from X to Y, unlucky. Suppose by contradiction that agent i is unlucky. Then he must have conceded a slice to an agent j, who must also be unlucky. All slices conceded by unlucky agents, are held by other unlucky agents. If those took back all the slices that they conceded, then all of them would be strictly better o while the lucky ones would remain unaected. This contradicts the optimality of Y. Hence there are no unlucky agents and PM is proved. 18

19 The following subsections are devoted to uncovering the main dierences between the absolute- and relative- leximin rules Absolute-leximin rule: PO+PM+RM Theorem 5.1. The absolute-leximin division rule is Pareto-optimal, populationmonotonic and resource-monotonic. Proof. PO follows from Lemma 5.1 and PM from Lemma 5.4. The proof of RM is essentially the same as of 5.4. The cake enlargement can be treated as a piece that was acquired from an agent who left the scene. 5 Unfortunately, the absolute-leximin rule is not PROP. For instance, in Example 5.1, the absolute-leximin-optimal division gives Carl a value of 9, which is only 3 of his total cake value. Hence it is also not envy-free Relative-leximin rule: PO+PM+PROP Theorem 5.2. The relative-leximin division rule is proportional, Paretooptimal and population-monotonic. Proof. PO follows from Lemma 5.1 and PM from Lemma 5.4. PROP holds because proportional divisions exist. The relative-leximinwelfare of a proportional division is at least ( 1,..., 1 ), hence the optimal n n relative-leximin-welfare vector must be at least ( 1,..., 1 ). n n Example 5.2. The following cake shows that the relative-leximin rule is not RM. The largest value that can be given to both Alice and Bob is 9. ˆv A ˆv B ˆv C ˆv D ˆv E Hence, in the smaller cake, the optimal relative-leximin-welfare vector is (9/18, 9/18, 10/18, 10/18, 10/18). It is attained by halving the two leftmost 5 Note that this argument does not work for the relative-leximin division rule, from essentially the same reason as in Theorem 6.3. It is possible that the relative-leximinoptimal division of the enlarged cake is lexicographically smaller than the relative-leximinoptimal division of the smaller cake. 19

20 slices between Alice and Bob, and giving the three slices at their right to Carl, David and Eve, in that order. In the larger cake, the largest value that can be given to Carl, David and Eve from the additional slice at the right is 6. So the largest value that can be given to them from the four rightmost slices is 16. However, the total cake value has doubled for them. Hence, if Alice and Bob keep their share of 9, the relative-leximin-welfare vector changes to 16/36, 16/36, 16/36, 9/18, 9/18. Note that Alice and Bob are now relatively-richer than Carl, David and Eve, and their pieces have a positive value for them. Thus, by Lemma 5.2, the division in which Alice and Bob keep their current shares cannot be relative-leximin-optimal. Remark 5.1. The equitable and leximin rules are closely related to the egalitarian-equivalent solution of Kalai and Smorodinsky (1975). It is known that this rule is population-monotonic in other division problems (e.g. Moulin (2004), chapter 7). Above we extended this result to cake-cutting. Remark 5.2. Instead of leximin principle, we could have used another welfare-optimization criterion such as the utilitarian criterion - maximizing the sum of utilities. The absolute- and relative-utilitarian rules are both Pareto-optimal and population-monotonic, and absolute-utilitarian is also resource-monotonic. However, in contrast to our leximin rules, they are not essentially-single-valued and not proportional. Remark 5.3. The leximin rules are not envy-free. Examples can be found in Dall'Aglio and Hill (2003). 5.2 Nash-optimal divisions: EF+PROP+PO+PM+RM The Nash-welfare of a division X is the product of the values of the agents: n v i (X i ) i=1 It is named after the bargaining solution of Nash (1950). The Nash-optimal division rule selects all the divisions in which the Nashwelfare is maximal. Such divisions exist by the Dubins-Spanier compactness theorem (Dubins and Spanier, 1961). 6 6 In the denition of Nash welfare, it is irrelevant whether absolute or relative values are used, since the product of absolute values is maximized if and only if the product of relative values is maximized. 20

21 In general fair division problems, the Nash-optimal rule is considered a compromise between the utilitarian and leximin rules (see e.g. chapter 3 of Moulin (2004)). We found no mentions of this rule in the cake-cutting literature. Surprisingly, this under-represented rule has all the properties that we desire from a division rule except connectivity. Pareto-optimality is obvious. The other properties are proved in a somewhat indirect way: rst, it is proved that a Nash-optimal division is also Competitive Equilibrium with Equal Incomes (CEEI); then, it is proved that any CEEI division is envy-free, population-monotonic and resource-monotonic. Denition 5.2. (Weller, 1985) Let C be a cake and X an allocation of the cake. Let P be a measure on C (called the "price measure") such that P (Z) = 0 when Z is a zero-slice and P (Z) > 0 when Z is a positive-slice. We say that the pair (P, X) is a CEEI if it satises the following conditions: CE: For every agent i, every positive-slice Z i X i and every positiveslice Z C: v i (Z i )/P (Z i ) v i (Z)/P (Z). 7 EI: For every agent i: P (X i ) = 1. CE is Competitive Equilibrium; it says that every agent receives only slices on which his value/price ratio is maximal. EI is Equal Incomes; it says that every agents receives a piece with the same total price. Lemma 5.5. Let X be a Nash-optimal division. Then, there exists a price measure P such that (P, X) is CEEI. Proof. Dene P as follows (Weller, 1985): Z i X i : P (Z i ) = v i(z i ) v i (X i ) Because proportional divisions exist, the optimal Nash-welfare is necessarily positive. Hence, all agents have a strictly positive value in X, so P is well-dened. The value of P on other subsets of C, that are not contained in any X i, can be easily calculated based on the additivity of P. Note that, by the non-atomicity of the v i, the measure P is also nonatomic and the price of zero-slices is 0. Moreover, since X is Nash-optimal, it is Pareto-optimal. Hence, v i (Z i ) is positive when Z i is a positive-slice and the same is true for P (Z i ). The pair (P, X) clearly satises the EI condition, since by denition P (X i ) = v i (X i )/v i (X i ) = 1. 7 Again it does not matter whether absolute or relative values are used, since only values of the same agent are compared. 21

22 It remains to prove that (P, X) satises the CE condition. Suppose by contradiction that CE is not satised. This means that, for some agent i, there is a positive-slice Z i X i and another positive-slice Z C such that: v i (Z i) P (Z i ) < v i(z) P (Z) Note that, by denition of P, the left-hand side of the above inequality is independent of Z i and always equals to v i (X i ). The subset Z can be written as a disjoint union of n subsets: Z = n j=1z j, such that for every j, Z j = Z X j. Hence: v i (X i ) < v i(z) P (Z) = We claim that there exists a k such that n j=1 v i(z j ) n j=1 P (Z j). v i (X i ) < v i(z k ) P (Z k ). (1) By contradiction, suppose that v i (X i ) v i (Z j )/P (Z j ) is true for every j N. Then we get a contradiction with n j=1 v i (X i ) < v i(z j ) n n j=1 P (Z j) j=1 v i(x i )P (Z j ) n j=1 P (Z = v i (X i ). j) So let k be an index for which (1) holds. We now prove that the Nash-welfare (the product of values) can be increased by taking some of Z k from agent k and giving it to agent i; this contradicts the assumption that the division X is Nash-optimal. Because Z k X k, we can apply the denition of P (Z k ) = v k (Z k )/v k (X k ). After some rearrangement, we get: v i (Z k ) v k (X k ) > v i (X i ) v k (Z k ) Hence, there exists a small positive value ɛ 1 such that: 0 < ɛ < v i(z k ) v k (X k ) v i (X i ) v k (Z k ) v i (Z k ) v k (Z k ) By Dubins and Spanier (1961); Stromquist and Woodall (1985), there exists a subset of Z k whose relative value, for both agent i and agent k, is exactly ɛ. Take this subset from agent k and give it to agent i. The new values of these two agents are now: 22

23 The product of the new values is: v k = v k (X k ) ɛ v k (Z k ) v i = v i (X i ) + ɛ v i (Z k ) v k v i = v k (X k ) v i (X i )+ɛ [v i (Z k ) v k (X k ) v i (X i ) v k (Z k ) ɛ (v i (Z k ) v k (Z k ))] By the selection of ɛ, the bracketed term in the right-hand side is positive. Hence: v k v i > v k (X k ) v i (X i ) The values of other agents have not changed. Hence, the Nash-welfare in the division X is not maximal - a contradiction. Example 5.3. In the following cake: ˆv A ˆv B The Nash-optimal solution gives the two central slices to Bob and the four peripheral slices to Alice. The Nash-welfare is 8*8=64. This is also a CEEI allocation, when the price of a central slice is 1/2 and the price of a peripheral slice is 1/4. Lemma 5.6. Every CEEI allocation is envy-free. Proof. Substituting the EI condition into the CE condition yields that for every i, j: v i (X i ) v i (X j ). We are now going to prove that the CEEI rule the rule which returns all CEEI divisions is both essentially-single-valued and monotonic. Let (P, X) be a CEEI on the cake C and (Q, Y ) a CEEI on an enlarged cake C E. By denition of CEEI, the price measures P and Q are absolutely continuous with respect to the Lebesgue measure. Thus, they have Radon-Nikodym derivatives with respect to the Lebesgue measure. Call them p, q, such that for every slice Z C: P (Z) = p(x)dx ; Q(Z) = q(x)dx. x Z x Z 23