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Waste Makes Haste: Bounded Time Protocols for Envy-Free Cake Cutting with Free Disposal
Abstract
We consider the classic problem of envy-free division of a heterogeneous good (aka the cake) among multiple agents. It is well known that if each agent must receive a contiguous piece then there is no finite protocol for the problem, whenever there are 3 or more agents. This impossibility result, however, assumes that the entire cake must be allocated. In this paper we study the problem in a setting where the protocol may leave some of the cake un-allocated, as long as each agent obtains at least some positive value (according to its valuation). We prove that this version of the problem is solvable in a bounded time. For the case of 3 agents we provide a finite and bounded-time protocol that guarantees each agent a share with value at least 1/3, which is the most that can be guaranteed.
Supplementary resource (1)
Data
August 2015
... And what is that number? [11] Erel Segal-Halevi posed the question because he and his coauthors Avinatan Hassidim and Yonatan Aumann used this very procedure as their basic primitive to devise the first discrete and bounded envy-free cake cutting protocol for any number of agents, when it is acceptable to leave some pieces of the cake unassigned (these pieces go to waste) [12,13]. Their work constituted a significant progress on a long-standing open problem; and a key technical lemma in their work uses the algorithms devised in this paper. ...
... Segal-Halevi, Hassidim, and Aumann [12,13] devise a much simpler protocol to find an envy-free division of a cake to n agents when parts of the cake may remain unassigned. Their core method (Algorithm 1 in [13]) is a protocol that allocates contiguous pieces to n agents with the guarantee that each agent values her piece at least 1/2 n−1 of the overall cake. ...
Given a set of n sticks of various (not necessarily different) lengths, what is the largest length so that we can cut k equally long pieces of this length from the given set of sticks? We analyze the structure of this problem and show that it essentially reduces to a single call of a selection algorithm; we thus obtain an optimal linear-time algorithm. This algorithm also solves the related envy-free stick-division problem, which Segal-Halevi et al. (ACM Trans Algorithms 13(1):1–32, 2016. ISSN: 15496325. https://doi.org/10.1145/2988232) recently used as their central primitive operation for the first discrete and bounded envy-free cake cutting protocol with a proportionality guarantee when pieces can be put to waste.
... Brams and Taylor were cognizant of their protocol's drawback and explicitly mentioned the problem of proposing a bounded envy-free protocol even for n = 4. Lindner and Rothe [Lindner and Rothe 2009] write that "the development of finite bounded envy-free cake cutting protocols still appears to be out of reach, and a big challenge for future research." The problem has remained open and has been highlighted in several works [Barbanel and Taylor 1995;Taylor 1995, 1996;Busch et al. 2005;Edmonds and Pruhs 2006;Segal-Halevi et al. 2015;Kurokawa et al. 2013;Procaccia 2013Procaccia , 2016Lindner and Rothe 2015;Robertson and Webb 1998;Saberi and Wang 2009]. Saberi and Wang [Saberi and Wang 2009] term the problem as "one of the most important open problems in the field". ...
... There are positive algorithmic results concerning envy-free cake cutting when agents have restricted valuations functions [Deng et al. 2012;Branzei 2015] or when some part of the cake is left unallocated [Segal-Halevi et al. 2015;Saberi and Wang 2009]. There has also been work on strategyproof cake cutting protocols for restricted valuation functions [Aziz and Ye 2014;Chen et al. 2013;Maya and Nisan 2012] as well as strategic aspects of protocols [Branzei and Miltersen 2015]. ...
We consider the well-studied cake cutting problem in which the goal is to find an envy-free allocation based on queries from n agents. The problem has received attention in computer science, mathematics, and economics. It has been a major open problem whether there exists a discrete and bounded envy-free protocol. We resolve the problem by proposing a discrete and bounded envy-free protocol for any number of agents. The maximum number of queries required by the protocol is . We additionally show that even if we do not run our protocol to completion, it can find in at most queries a partial allocation of the cake that achieves proportionality (each agent gets at least 1/n of the value of the whole cake) and envy-freeness. Finally we show that an envy-free partial allocation can be computed in at most queries such that each agent gets a connected piece that gives the agent at least 1/(3n) of the value of the whole cake.
... The problem of deleting a minimum number of items to obtain envy-freeness was first studied by Brams et al. [4] who gave a polynomial-time algorithm for the case of two agents. 1 In the context of cake cutting, Segal-Halevi et al. [15] proposed the idea of distributing only a portion of the entire cake in order to obtain an envy-free allocation efficiently. For the Hospitals/Residents with Couples problem, Nguyen and Vohra [14] considered another type of control action: they obtained stability by slightly perturbing the capacities of hospitals. ...
We consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preference over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small, since the problem turns out to be W[3]-hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of and k.
... The idea here is to achieve envy-freeness at the cost of discarding some parts of the cake. Aziz and Mackenzie [AM16] (see also [SHHA15]) develop an exponential-time algorithm that finds envy-free divisions, wherein each agent a receives a connected piece of value at least 1 2n times a's value for the entire cake. Here, the proximity to the proportional share is used to quantify the loss incurred due to the disposal. ...
The classic cake-cutting problem provides a model for addressing fair and efficient allocation of a divisible, heterogeneous resource (metaphorically, the cake) among agents with distinct preferences. Focusing on a standard formulation of cake cutting, in which each agent must receive a contiguous piece of the cake, this work establishes algorithmic and hardness results for multiple fairness/efficiency measures. First, we consider the well-studied notion of envy-freeness and develop an efficient algorithm that finds a cake division (with connected pieces) wherein the envy is multiplicatively within a factor of 3. The same algorithm in fact achieves an approximation ratio of 3 for the problem of finding cake divisions with as large a Nash social welfare (NSW) as possible. NSW is another standard measure of fairness and this work also establishes a connection between envy-freeness and NSW: approximately envy-free cake divisions (with connected pieces) always have near-optimal Nash social welfare. Furthermore, we develop an approximation algorithm for maximizing the -mean welfare--this unifying objective, with different values of , interpolates between notions of fairness (NSW) and efficiency (average social welfare). Finally, we complement these algorithmic results by proving that maximizing NSW (and, in general, the -mean welfare) is APX-hard in the cake-division context
... The problem of deleting a minimum number of items to obtain envy-freeness was first studied by Brams et al. [4] who gave a polynomial-time algorithm for the case of two agents. 1 In the context of cake cutting, Segal-Halevi et al. [15] proposed the idea of distributing only a portion of the entire cake in order to obtain an envy-free allocation efficiently. For the Hospitals/Residents with Couples problem, Nguyen and Vohra [14] considered another type of control action: they obtained stability by slightly perturbing the capacities of hospitals. ...
We consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preference over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small, since the problem turns out to be W[3]-hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of and k.
... That is, given a partial envy-free allocation such that each agent dominates n − 2 other agents, we can apply Protocol 1 to extend the allocation to a complete allocation. This provides an alternate proof to a recent paper by Segal-Halevi et al. [25]. We first define a special class of graphs. ...
In the classical cake cutting problem, a resource must be divided among agents with different utilities so that each agent believes they have received a fair share of the resource relative to the other agents. We introduce a variant of the problem in which we model an underlying social network on the agents with a graph, and agents only evaluate their shares relative to their neighbors' in the network. This formulation captures many situations in which it is unrealistic to assume a global view, and also exposes interesting phenomena in the original problem. Specifically, we say an allocation is locally envy-free if no agent envies a neighbor's allocation and locally proportional if each agent values her own allocation as much as the average value of her neighbor's allocations, with the former implying the latter. While global envy-freeness implies local envy-freeness, global proportionality does not imply local proportionality, or vice versa. A general result is that for any two distinct graphs on the same set of nodes and an allocation, there exists a set of valuation functions such that the allocation is locally proportional on one but not the other. We fully characterize the set of graphs for which an oblivious single-cutter protocol-- a protocol that uses a single agent to cut the cake into pieces --admits a bounded protocol with query complexity for locally envy-free allocations in the Robertson-Webb model. We also consider the price of envy-freeness, which compares the total utility of an optimal allocation to the best utility of an allocation that is envy-free. We show that a lower bound of on the price of envy-freeness for global allocations in fact holds for local envy-freeness in any connected undirected graph. Thus, sparse graphs surprisingly do not provide more flexibility with respect to the quality of envy-free allocations.
... All those examples go beyond a pairwise interaction. More specifically, the relation of groups and the possibility of fair allocations is a topic utterly relevant in the context of group buying [19,20], collective bargaining of work contracts or coalition policy making [14] and indeed, there is a growing interest in doing experiments with multiplayer versions of fairness games [3,6,7,9,12,23,39]. A simple extension of UG may turn it adequate to study a wide variety of ubiquitous group encounters. ...
Fairness plays a determinant role in human decisions and definitely shapes social preferences. This is evident when groups of individuals need to divide a given resource. Notwithstanding, computational models seeking to capture the origins and effects of human fairness often assume the simpler case of two person interactions. Here we study a multiplayer extension of the well-known Ultimatum Game. This game allows us to study fair behaviors in a group setting: a proposal is made to a group of Responders and the overall acceptance depends on reaching a minimum number of individual acceptances. In order to capture the effects of different group environments on the human propensity to be fair, we model a population of learning agents interacting through the multiplayer ultimatum game. We show that, contrarily to what would happen with fully rational agents, learning agents coordinate their behavior into different strategies, depending on factors such as the minimum number of accepting Responders (to achieve group acceptance) or the group size. Overall, our simulations show that stringent group criteria leverage fairer proposals. We find these conclusions robust to (i) asynchronous and synchronous strategy updates, (ii) initially biased agents, (iii) different group payoff division paradigms and (iv) a wide range of error and forgetting rates.
... The literature on cake cutting has seen very interesting algorithmic developments in recent years (see, e.g., [26,39,3]), all of which are concerned with additive valuations in the one-dimensional model. Among the few exceptions we mention: [12] studied fair division with additive valuations constrained by a minimum length requirement (PUML), [9] studied externalities in cake cutting, while [37,38], studied cake cutting in two dimensions, where the agents also care about the shape of the pieces that they receive. ...
Single minded agents have strict preferences, in which a bundle is acceptable only if it meets a certain demand. Such preferences arise naturally in scenarios such as allocating computational resources among users, where the goal is to fairly serve as many requests as possible. In this paper we study the fair division problem for such agents, which is harder to handle due to discontinuity and complementarities of the preferences. Our solution concept---the competitive allocation from equal incomes (CAEI)---is inspired from market equilibria and implements fair outcomes through a pricing mechanism. We study the existence and computation of CAEI for multiple divisible goods, cake cutting, and multiple discrete goods. For the first two scenarios we show that existence of CAEI solutions is guaranteed, while for the third we give a succinct characterization of instances that admit this solution; then we give an efficient algorithm to find one in all three cases. Maximizing social welfare turns out to be NP-hard in general, however we obtain efficient algorithms for (i) divisible and discrete goods when the number of different \emph{types} of players is a constant, (ii) cake cutting with contiguous demands, for which we establish an interesting connection with interval scheduling, and (iii) cake cutting with a constant number of players with arbitrary demands. Our solution is useful more generally, when the players have a target set of desired goods, and very small positive values for any bundle not containing their target set.
... Brams and Taylor were cognizant of their protocol's drawback and explicitly mentioned the problem of proposing a bounded envy-free protocol even for n = 4. Lindner and Rothe [2009] write that " even for n = 4, the development of finite bounded envy-free cake-cutting protocols still appears to be out of reach, and a big challenge for future research. " The problem has remained open and has been highlighted in several works [Barbanel and Taylor 1995; Brams and Taylor 1995, 1996; Busch et al. 2005; Edmonds and Pruhs 2006; Segal-Halevi et al. 2015; Kurokawa et al. 2013; Procaccia 2013, 2015; Lindner and Rothe 2015; Robertson and Webb 1998; Saberi and Wang 2009]. Saberi and Wang [2009] term the problem as " one of the most important open problems in the field " and Lindner and Rothe [2015] mentions the case for n = 4 as " the central open problem in the field of cake-cutting " . ...
We consider the well-studied cake cutting problem in which the goal is to
identify a fair allocation based on a minimal number of queries from the
agents. The problem has attracted considerable attention within various
branches of computer science, mathematics, and economics. Although, the elegant
Selfridge-Conway envy-free protocol for three agents has been known since 1960,
it has been a major open problem for the last fifty years to obtain a bounded
envy-free protocol for more than three agents. We propose a discrete and
bounded envy-free protocol for four agents.
... The first and foremost is auctions [4,7,10,17,23,24], but other examples include division of goods in a fair way (e.g. cake cutting [9,39,38,37] studying the effect and utilization of networks [19,26,41], matching markets [5,22], and other topics in social choice [8,2,11,33,43,25,27,18,20]. The assignment problem with couples is a specific case of multi-unit allocation of indivisible objects without transfers (for example, course allocation), and we indeed intend for our treatment to convey a message regarding the more general case. ...
This thesis was submitted as partial fulfillment of the requirements for the
Masters Degree in the Department of Computer Science, Bar-Ilan University.
The final step in getting an Israeli M.D. is performing a year-long
internship in one of the hospitals in Israel. Internships are decided upon by a
lottery, which is known as "The Internship Lottery". In 2014 we redesigned the
lottery, replacing it with a more efficient one. The new method is based on
calculating a tentative lottery, in which each student has some probability of
getting to each hospital. Then a computer program "trades" between the
students, where trade is performed only if it is beneficial to both sides. This
trade creates surplus, which translates to more students getting one of their
top choices. The average student improved his place by 0.91 seats. The new
method can improve the welfare of medical graduates, by giving them more
probability to get to one of their top choices. It can be applied in internship
markets in other countries as well.
This thesis presents the market, the redesign process and the new mechanism
which is now in use. There are two main lessons that we have learned from this
market. The first is the "Do No Harm" principle, which states that (almost) all
participants should prefer the new mechanism to the old one. The second is that
new approaches need to be used when dealing with two-body problems in object
assignment. We focus on the second lesson, and study two-body problems in the
context of the assignment problem. We show that decomposing stochastic
assignment matrices to deterministic allocations is NP-hard in the presence of
couples, and present a polynomial time algorithm with the optimal worst case
guarantee. We also study the performance of our algorithm on real-world and on
simulated data.
... While we do think that Envy-Free Stick Division is worth studying in its own right, it was initially motivated by a recent approach to envy-free cake cutting, i. e., the problem of finding an envy-free assignment of a freely divisible, but inhomogeneous resource (which means that valuations do not only depend on the size of the piece, but also on its position). Segal-Halevi, Hassidim, and Aumann [SHHA15] devise a finite protocol to find an envy-free division of a cake to k agents, where parts of the cake may remain unassigned (waste). ...
Segal-Halevi, Hassidim, and Aumann (AAMAS, 2015) propose the problem of
cutting sticks so that at least k sticks have equal length and no other stick
is longer. This allows for an envy-free allocation of sticks to k players, one
each. The resulting number of sticks should also be minimal.
We analyze the structure of this problem and devise a linearithmic algorithm
for it.
We consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preference over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small, since the problem turns out to be -hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of |I| and k.
Although a finite envy-free cake-cutting protocol has been known for more than twenty years, it had been open whether a protocol exists in which the number of steps taken by the protocol is bounded by a function of the number of agents. In this letter, we report on our recent results on discrete, bounded, and envy-free cake-cutting protocols.
We consider the classic problem of envy-free division of a heterogeneous good ("cake") among several agents. It is known that, when the allotted pieces must be connected, the problem cannot be solved by a finite algorithm for three or more agents. The impossibility result, however, assumes that the entire cake must be allocated. In this article, we replace the entire-allocation requirement with a weaker partial-proportionality requirement: the piece given to each agent must be worth for it at least a certain positive fraction of the entire cake value. We prove that this version of the problem is solvable in bounded time even when the pieces must be connected. We present simple, bounded-time envy-free cake-cutting algorithms for (1) giving each of n agents a connected piece with a positive value; (2) giving each of three agents a connected piece worth at least 1/3; (3) giving each of four agents a connected piece worth at least 1/7; (4) giving each of four agents a disconnected piece worth at least 1/4; and (5) giving each of n agents a disconnected piece worth at least (1 - ε)/n for any positive ε.
We consider the well-studied cake cutting problem in which the goal is to identify an envy-free allocation based on a minimal number of queries from the agents. The problem has attracted considerable attention within various branches of computer science, mathematics, and economics. Although, the elegant Selfridge-Conway envy-free protocol for three agents has been known since 1960, it has been a major open problem to obtain a bounded envy-free protocol for more than three agents. The problem has been termed the central open problem in cake cutting. We solve this problem by proposing a discrete and bounded envy-free protocol for four agents.
Single minded agents have strict preferences, in which a bundle is acceptable only if it meets a certain demand. Such preferences arise naturally in scenarios such as allocating computational resources among users, where the goal is to fairly serve as many requests as possible. In this paper we study the fair division problem for such agents, which is complex due to discontinuity and comple-mentarities of preferences. Our solution concept—the competitive allocation from equal incomes (CAEI)—is inspired from market equilibria and implements fair outcomes through a pricing mechanism. We study existence and computation of CAEI for multiple divisible goods, discrete goods, and cake cutting. Our solution is useful more generally, when the players have a target set of goods, and very small positive values for any bundle not containing their target set.
Segal-Halevi, Hassidim, and Aumann (AAMAS, 2015) propose the problem of
cutting sticks so that at least k sticks have equal length and no other stick
is longer. This allows for an envy-free allocation of sticks to k players, one
each. The resulting number of sticks should also be minimal.
We analyze the structure of this problem and devise a linearithmic algorithm
for it.
We give a randomized algorithm for the well known caking cutting problem that achieves approximate fairness, and has complexity O(n), when all players are honest. The heart of this result involves extending the standard offline multiple-choice balls and bins analysis to the case where the underlying resources/bins/machines have different utilities to different players/balls/jobs
We consider the problem of fairly dividing a heterogeneous cake between a number of players with different tastes. In this setting, it is known that fairness requirements may result in a suboptimal division from the social welfare standpoint. Here, we show that in some cases, discarding some of the cake and fairly dividing only the remainder may be socially preferable to any fair division of the entire cake. We study this phenomenon, providing asymptotically-tight bounds on the social improvement achievable by such discarding.
We give a randomized protocol for the classic cake cutting problem that guarantees approximate proportional fairness, and with high probability uses a linear number of cuts.
We consider the well-known cake cutting problem in which a protocol wants to divide a cake among n ≥ 2 players in such a way that each player believes that they got a fair share. The standard Robertson-Webb model allows the protocol to make two types of queries, Evaluation and Cut, to the players. A deterministic divide-and-conquer protocol with complexity O(n log n) is known. We provide the first an Ω(n log n) lower bound on the complexity of any deterministic protocol in the standard model. This improves previous lower bounds, in that the protocol is allowed to assign to a player a piece that is a union of intervals and only guarantee approximate fairness. We accomplish this by lower bounding the complexity to find, for a single player, a piece of cake that is both rich in value, and thin in width. We then introduce a version of cake cutting in which the players are able to cut with only finite precision. In this case, we can extend the Ω(n log n) lower bound to include randomized protocols.
For decades researchers have struggled with the problem of envy-free cake cutting: how to divide a divisible good between multiple agents so that each agent likes his own allocation best. Although an envy-free cake cutting protocol was ultimately devised, it is unbounded, in the sense that the number of operations can be arbitrarily large, depending on the preferences of the agents. We ask whether bounded protocols exist when the agents' preferences are restricted. Our main result is an envy-free cake cutting protocol for agents with piecewise linear valuations, which requires a number of operations that is polynomial in natural parameters of the given instance. Copyright © 2013, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.
We consider the assignment problem in which agents express ordinal
preferences over m objects and the objects are allocated to the agents based
on the preferences.In a recent paper, Brams, Kilgour, and Klamler (2014)
presented the AL method to compute an envy-free assignment for two agents. The
AL method crucially depends on the assumption that agents have strict
preferences over objects. We generalize the AL method to the case where agents
may express indifferences and prove the axiomatic properties satisfied by the
algorithm. As a result of the generalization, we also get a O(m) speedup on
previous algorithms to check whether a complete envy-free assignment exists or
not.
In the cake cutting problem, n >= 2 players want to cut a cake into n pieces so that every player gets a 'fair' share of the cake by his/her own measure. We prove that in a certain, natural cake cutting model, every fair cake division protocol for n players must use ohm(n log n) cuts and evaluation queries in the worst case. Up to a small constant factor, this lower bound matches a corresponding upper bound in the same model obtained by Even and Paz, from 1984. (c) 2007 Published by Elsevier B.V.
The algorithmic aspects of the following problem are investigated: n (≥2) persons want to cut a cake into n shares so that every person will get at least 1/n of the cake by his own measure and so that the number of cuts made on the cake is minimal. The cutting process is to be governed by a protocol (computer program). It is shown that no deterministic protocol exists which is fair (in a sense defined in the text) and results in at most n − 1 cuts. An O(n log n)-cut deterministic protocol and an O(n)-cut randomized protocol are given explicitly and a deterministic fair protocol with 4 cuts for n=4 is described in the appendix.
The problem of fairly dividing a cake (as a metaphor for a heterogeneous divisible good) has been the subject of much interest since the 1940's, and is of importance in multiagent resource allo- cation. Two fairness criteria are usually consid- ered: proportionality, in the sense that each of the n agents receives at least 1/n of the cake; and the stronger property of envy-freeness, namely that each agent prefers its own piece of cake to the oth- ers' pieces. For proportional division, there are algorithms that require O(n log n) steps, and re- cent lower bounds imply that one cannot do better. In stark contrast, known (discrete) algorithms for envy-free division require an unbounded number of steps, even when there are only four agents. In this paper, we give an Ω(n2) lower bound for the number of steps required by envy-free cake-cutting algorithms. This result provides, for the first time, a true separation between envy-free and proportional division, thus giving a partial explanation for the notorious difficulty of the former problem.
We consider the well-known cake cutting problem in which a protocol wants to divide a cake among n ≥ 2 players in such a way that each player believes that they got a fair share. The standard Robertson-Webb model allows the protocol to make two types of queries, Evaluation and Cut, to the players. A deterministic divide-and-conquer protocol with complexity O(n log n) is known. We provide the first a Ω(n log n) lower bound on the complexity of any deterministic protocol in the standard model. This improves previous lower bounds, in that the protocol is allowed to assign to a player a piece that is a union of intervals and only guarantee approximate fairness. We accomplish this by lower bounding the complexity to find, for a single player, a piece of cake that is both rich in value, and thin in width. We then introduce a version of cake cutting in which the players are able to cut with only finite precision. In this case, we can extend the Ω(n log n) lower bound to include randomized protocols.
Cutting a cake, dividing up the property in an estate, determining the borders in an international dispute - such problems of fair division are ubiquitous. Fair Division treats all these problems and many more through a rigorous analysis of a variety of procedures for allocating goods (or ‘bads’ like chores), or deciding who wins on what issues, when there are disputes. Starting with an analysis of the well-known cake-cutting procedure, ‘I cut, you choose’, the authors show how it has been adapted in a number of fields and then analyze fair-division procedures applicable to situations in which there are more than two parties, or there is more than one good to be divided. In particular they focus on procedures which provide ‘envy-free’ allocations, in which everybody thinks he or she has received the largest portion and hence does not envy anybody else. They also discuss the fairness of different auction and election procedures.
No finite protocol (even if unbounded) can guarantee an envy-free division of a cake among three or more players, if each player is to receive a single connected piece. @InProceedings{stromquist:DSP:2007:1220, author = {Walter Stromquist}, title = {Envy-free cake divisions cannot be found by finite protocols}, booktitle = {Fair Division}, year = {2007}, editor = {Steven Brams and Kirk Pruhs and Gerhard Woeginger}, number = {07261}, series = {Dagstuhl Seminar Proceedings}, ISSN = {1862-4405}, publisher = {Internationales Begegnungs- und Forschungszentrum f{"u}r Informatik (IBFI), Schloss Dagstuhl, Germany}, address = {Dagstuhl, Germany}, URL = {http://drops.dagstuhl.de/opus/volltexte/2007/1220}, annote = {Keywords: Cake cutting, envy free, finite protocol} }
My friend's dilemma was a practical question that mathematics could answer, both elegantly and constructively. He and his housemates were moving to a house with rooms of various sizes and features, and were having trouble deciding who should get which room and for what part of the total rent. He asked, \Do you think there's always a way to partition the rent so that each person will prefer a di erent room?" As we shall see, with mild assumptions, the answer is yes. This rent-partitioning problem is really a kind of fair-division question. It can be viewed as a generaliza-tion of the age-old cake-cutting problem, in which one seeks to divide a cake fairly among several people, and the chore-division problem, posed by Martin Gardner in [6, p. 124], in which one seeks to fairly divide an undesirable entity, such as a list of chores. Lately, there has been much interest in fair division (see, for example, the recent books [3] and [11]), and each of the related problems has been treated before (see [1], [4], [10]). We wish to explain a powerful approach to fair-division questions that uni es these problems and provides new methods for achieving approximate envy-free di-visions, in which each person feels she received the \best" share. This approach was carried out by Forest Simmons [13] for cake-cutting and depends on a sim-ple combinatorial result known as Sperner's lemma. We show that the Sperner's lemma approach can be adapted to treat chore division and rent-partitioning as well, and it generalizes easily to any number of players. From a pedagogical perspective, this approach provides a nice, elementary demon-stration of how ideas from many pure disciplines|combinatorics, topology, and analysis|can combine to address a real-world problem. Better yet, the proofs can be converted into constructive fair-division procedures.