Egor Ianovski
We examine the history of cake cutting mechanisms and discuss the efficiency of their allocations. In the case of piecewise uniform preferences, we define a game that in the presence of strategic agents has equilibria that are not dominated by the allocations of any mechanism. We identify that the equilibria of this game coincide with the allocations of an existing cake cutting mechanism.
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We characterize methods of dividing a cake between two bidders in a way that is incentive-compatible and Pareto-efficient. In our cake cutting model, each bidder desires a subset of the cake (with a uniform value over this subset), and is allocated some subset. Our characterization proceeds via reducing to a simple one-dimensional version of the problem, and yields, for example, a tight bound on the social welfare achievable.
No, Cake Cutting Really is a Piece of Cake
提出一种确定性蛋糕切割算法，使用线性数量的切割实现比例公平。
We study the envy-free cake-cutting problem for $d+1$ players with $d$ cuts, for both the oracle function model and the polynomial time function model. For the former, we derive a $θ(({1\overε})^{d-1})$ time matching bound for the query complexity of $d+1$ player cake cutting with Lipschitz utilities for any $d> 1$. When the utility functions are given by a polynomial time algorithm, we prove the problem to be PPAD-complete. For measurable utility functions, we find a fully polynomial-time algorithm for finding an approximate envy-free allocation of a cake among three people using two cuts.
Cake-cutting protocols aim at dividing a ``cake'' (i.e., a divisible resource) and assigning the resulting portions to several players in a way that each of the players feels to have received a ``fair'' amount of the cake. An important notion of fairness is envy-freeness: No player wishes to switch the portion of the cake received with another player's portion. Despite intense efforts in the past, it is still an open question whether there is a \emph{finite bounded} envy-free cake-cutting protocol for an arbitrary number of players, and even for four players. We introduce the notion of degree …
The cake-cutting problem involves dividing a heterogeneous, divisible resource fairly between $n$ agents. Brânzei et al. [6] introduced {\em generalised cut and choose} (GCC) protocols, a formal model for representing cake-cutting protocols as trees with "cut" and "choose" nodes corresponding to the agents' actions, and if-else statements. In this paper, we identify an alternative and simpler extensive-form game model for cake-cutting protocols, that we call {\em branch choice} (BC) protocols. We show that the class of protocols we can represent using this model is invariant under certain modi…
We study the query complexity of cake cutting and give lower and upper bounds for computing approximately envy-free, perfect, and equitable allocations with the minimum number of cuts. The lower bounds are tight for computing connected envy-free allocations among n=3 players and for computing perfect and equitable allocations with minimum number of cuts between n=2 players. We also formalize moving knife procedures and show that a large subclass of this family, which captures all the known moving knife procedures, can be simulated efficiently with arbitrarily small error in the Robertson-Webb …
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
本文从测度论视角审视了蛋糕切割（公平分配）问题的模型基础，调研了文献中关于蛋糕片段集合与智能体估值函数的常用定义，剖析其优缺点并提出了最合理的定义建议。
We study the monotonicity properties of solutions in the classic problem of fair cake-cutting --- dividing a heterogeneous resource among agents with different preferences. Resource- and population-monotonicity 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-off (if there is more to share or fewer to share among) or all are worse-off (if there is less to share or more to share among). We formally introduce these concepts to the cake-cut…
This paper extends the classic cake-cutting problem to a situation in which the "cake" is divided among families. Each piece of cake is owned and used simultaneously by all members of the family. A typical example of such a cake is land. We examine three ways to assess the fairness of such a division, based on the classic no-envy criterion: (a) Average envy-freeness means that for each family, the average value of its share (averaged over all family members) is weakly larger than the average value of any other share; (b) Unanimous envy-freeness means that in each family, each member values the…
In the classic cake-cutting problem (Steinhaus, 1948), a heterogeneous resource has to be divided among n agents with different valuations in a proportional way --- giving each agent a piece with a value of at least 1/n of the total. In many applications, such as dividing a land-estate or a time-interval, it is also important that the pieces are connected. We propose two additional requirements: resource-monotonicity (RM) and population-monotonicity (PM). When either the cake or the set of agents changes and the cake is re-divided using the same rule, the utility of all remaining agents must c…
The problem of fair division known as "cake cutting" has been the focus of multiple papers spanning several decades. The most prominent problem in this line of work has been to bound the query complexity of computing an envy-free outcome in the Robertson-Webb query model. However, the root of this problem's complexity is somewhat artificial: the agents' values are assumed to be additive across different pieces of the "cake" but infinitely complicated within each piece. This is unrealistic in most of the motivating examples, where the cake represents a finite collection of homogeneous goods. We…
We initiate the study of multi-layered cake cutting with the goal of fairly allocating multiple divisible resources (layers of a cake) among a set of agents. The key requirement is that each agent can only utilize a single resource at each time interval. Several real-life applications exhibit such restrictions on overlapping pieces; for example, assigning time intervals over multiple facilities and resources or assigning shifts to medical professionals. We investigate the existence and computation of envy-free and proportional allocations. We show that envy-free allocations that are both feasi…
We study the problem of fairly allocating a divisible resource, also known as cake cutting, with an additional requirement that the shares that different agents receive should be sufficiently separated from one another. This captures, for example, constraints arising from social distancing guidelines. While it is sometimes impossible to allocate a proportional share to every agent under the separation requirement, we show that the well-known criterion of maximin share fairness can always be attained. We then provide algorithmic analysis of maximin share fairness in this setting -- for instance…
Cutting Down the Tower: Single-Exponential Envy-Free Cake Cutting
该研究针对无嫉妒蛋糕分割问题，提出查询复杂度为n^{O(1)}2^n的协议，首次实现单指数查询上界，大幅缩小了此前巨大的上下界差距。
We consider the classic cake-cutting problem of producing envy-free allocations, restricted to the case of four agents. The problem asks for a partition of the cake to four agents, so that every agent finds her piece at least as valuable as every other agent's piece. The problem has had an interesting history so far. Although the case of three agents is solvable with less than 15 queries, for four agents no bounded procedure was known until the recent breakthroughs of Aziz and Mackenzie (STOC 2016, FOCS 2016). The main drawback of these new algorithms, however, is that they are quite complicat…
We consider the problem of envy-free cake cutting, which is the distribution of a continuous heterogeneous resource among self interested players such that nobody prefers what somebody else receives to what they get. Existing work has focused on two distinct classes of solutions to this problem - allocations which give each player a continuous piece of cake and allocations which give each player arbitrarily many disjoint pieces of cake. Our aim is to investigate allocations between these two extremes by parameterizing the maximum number of disjoint pieces each player may receive. We characteri…
This work develops algorithmic results for the classic cake-cutting problem in which a divisible, heterogeneous resource (modeled as a cake) needs to be partitioned among agents with distinct preferences. We focus on a standard formulation of cake cutting wherein each agent must receive a contiguous piece of the cake. While multiple hardness results exist in this setup for finding fair/efficient cake divisions, we show that, if the value densities of the agents satisfy the monotone likelihood ratio property (MLRP), then strong algorithmic results hold for various notions of fairness and econom…
We study the cake-cutting problem when agents have single-peaked preferences over the cake. We show that a recently proposed mechanism by Wang-Wu (2019) to obtain envy-free allocations can yield large welfare losses. Using a simplifying assumption, we characterize all Pareto optimal allocations, which have a simple structure: are peak-preserving and non-wasteful. Finally, we provide simple alternative mechanisms that Pareto dominate that of Wang-Wu, and which achieve envy-freeness or Pareto optimality.
We study the classic cake cutting problem from a mechanism design perspective, in particular focusing on deterministic mechanisms that are strategyproof and fair. We begin by looking at mechanisms that are non-wasteful and primarily show that for even the restricted class of piecewise constant valuations there exists no direct-revelation mechanism that is strategyproof and even approximately proportional. Subsequently, we remove the non-wasteful constraint and show another impossibility result stating that there is no strategyproof and approximately proportional direct-revelation mechanism tha…
We study the classic problem of fairly dividing a heterogeneous and divisible resource -- represented by a cake, $[0,1]$ -- among $n$ agents. This work considers an interesting variant of the problem where agents are embedded on a graph. The graphical constraint entails that each agent evaluates her allocated share only against her neighbor's share. Given a graph, the goal is to efficiently find a locally envy-free allocation where every agent values her share to be at least as much as any of her neighbor's share. The best known algorithm (by Aziz and Mackenzie) for finding envy-free cake divi…
Using a lab experiment, we investigate the real-life performance of envy-free and proportional cake-cutting procedures with respect to fairness and preference manipulation. We find that envy-free procedures, in particular Selfridge-Conway, are fairer and also are perceived as fairer than their proportional counterparts, despite the fact that agents very often manipulate them. Our results support the practical use of the celebrated Selfridge-Conway procedure, and more generally, of envy-free cake-cutting mechanisms. We also find that subjects learn their opponents' preferences after repeated in…
We study the recently introduced cake-cutting setting in which the cake is represented by an undirected graph. This generalizes the canonical interval cake and allows for modeling the division of road networks. We show that when the graph is a forest, an allocation satisfying the well-known criterion of maximin share fairness always exists. Our result holds even when separation constraints are imposed, in which case no multiplicative approximation of proportionality can be guaranteed. Furthermore, while maximin share fairness is not always achievable for general graphs, we prove that ordinal r…
A cake has to be divided fairly among $n$ agents. When all agents have equal entitlements, it is known that such a division can be implemented with $n-1$ cuts. When agents may have different entitlements, the paper shows that at least $2 n -2$ cuts may be necessary, and $O(n \log(n))$ cuts are always sufficient.
In cake-cutting, strategy-proofness is a very costly requirement in terms of fairness: for n=2 it implies a dictatorial allocation, whereas for n > 2 it requires that one agent receives no cake. We show that a weaker version of this property recently suggested by Troyan and Morril, called non-obvious manipulability, is compatible with the strong fairness property of proportionality, which guarantees that each agent receives 1/n of the cake. Both properties are satisfied by the leftmost leaves mechanism, an adaptation of the Dubins - Spanier moving knife procedure. Most other classical proporti…
We study the fair division problem on divisible heterogeneous resources (the cake cutting problem) with strategic agents, where each agent can manipulate his/her private valuation in order to receive a better allocation. A (direct-revelation) mechanism takes agents' reported valuations as input and outputs an allocation that satisfies a given fairness requirement. A natural and fundamental open problem, first raised by [Chen et al., 2010] and subsequently raised by [Procaccia, 2013] [Aziz and Ye, 2014] [Branzei and Miltersen, 2015] [Menon and Larson, 2017] [Bei et al., 2017] [Bei et al., 2020]…
We consider the classic problem of fairly dividing a heterogeneous good ("cake") among several agents with different valuations. Classic cake-cutting procedures either allocate each agent a collection of disconnected pieces, or assume that the cake is a one-dimensional interval. In practice, however, the two-dimensional shape of the allotted pieces is important. In particular, when building a house or designing an advertisement in printed or electronic media, squares are more usable than long and narrow rectangles. We thus introduce and study the problem of fair two-dimensional division wherei…
We study the classic problem of \emph{fairly} dividing a heterogeneous and divisible resource -- modeled as a line segment $[0,1]$ and typically called as a \emph{cake} -- among $n$ agents. This work considers an interesting variant of the problem where agents are embedded on a graph. The graphical constraint entails that each agent evaluates her allocated share only against her neighbors' share. Given a graph, the goal is to efficiently find a \emph{locally envy-free} allocation where every agent values her share of the cake to be at least as much as that of any of her neighbors' share. The m…
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.
Cake cutting is a classic fair division problem, with the cake serving as a metaphor for a heterogeneous divisible resource. Recently, it was shown that for any number of players with arbitrary preferences over a cake, it is possible to partition the players into groups of any desired size and divide the cake among the groups so that each group receives a single contiguous piece and every player is envy-free. For two groups, we characterize the group sizes for which such an assignment can be computed by a finite algorithm, showing that the task is possible exactly when one of the groups is a s…
The Incentive Guarantees Behind Nash Welfare in Divisible Resources Allocation
研究可分割资源分配中MNW与PA机制的激励保证，证明MNW在蛋糕切割中激励比率为2，PA介于[e^(1/e),e]间，并给出无嫉妒机制的最优激励比率结果。
We propose an online form of the cake cutting problem. This models situations where agents arrive and depart during the process of dividing a resource. We show that well known fair division procedures like cut-and-choose and the Dubins-Spanier moving knife procedure can be adapted to apply to such online problems. We propose some fairness properties that online cake cutting procedures can possess like online forms of proportionality and envy-freeness. We also consider the impact of collusion between agents. Finally, we study theoretically and empirically the competitive ratio of these online c…
We study classic cake-cutting problems, but in discrete models rather than using infinite-precision real values, specifically, focusing on their communication complexity. Using general discrete simulations of classical infinite-precision protocols (Robertson-Webb and moving-knife), we roughly partition the various fair-allocation problems into 3 classes: "easy" (constant number of rounds of logarithmic many bits), "medium" (poly-logarithmic total communication), and "hard". Our main technical result concerns two of the "medium" problems (perfect allocation for 2 players and equitable allocatio…
In this article we suggest a model of computation for the cake cutting problem. In this model the mediator can ask the same queries as in the Robertson-Webb model but he or she can only perform algebraic operations as in the Blum-Shub-Smale model. All existing algorithms described in the Robertson-Webb model can be described in this new model.We show that in this model there exist explicit couples of measures for which no algorithm outputs an equitable fair division with connected parts.We also show that there exist explicit set of measures for which no algorithm in this model outputs a fair d…
A division of a cake by n people is envy free if everyone thinks they got the biggest pieces. Note that peoples tastes can differ. There is a discrete protocol for envy free division for n=3 which takes at most 5 cuts. For n=4 and beyond there is a protocol but the number of cuts it takes is unbounded. In particular the number of cuts depends on peoples tastes. Given any number N peoples tastes can be set so that the algorithm takes over N cuts. There are three such algorithms known. Which is better? We have devised a way to measure the number of cuts even though it is unbounded. We use ordina…
Fair Division via the Cake-Cutting Share
本文研究将可分物品分配给具有线性估值的智能体的公平分配问题，通过切蛋糕过程定义了考虑嫉妒约束的新公平份额概念，并给出了紧致的近似界限及实证验证。
This paper proposes a cake-cutting protocol using cryptography when the cake is a heterogeneous good that is represented by an interval on a real line. Although the Dubins-Spanier moving-knife protocol with one knife achieves simple fairness, all players must execute the protocol synchronously. Thus, the protocol cannot be executed on asynchronous networks such as the Internet. We show that the moving-knife protocol can be executed asynchronously by a discrete protocol using a secure auction protocol. The number of cuts is n-1 where n is the number of players, which is the minimum.
Cake cutting is one of the most fundamental settings in fair division and mechanism design without money. In this paper, we consider different levels of three fundamental goals in cake cutting: fairness, Pareto optimality, and strategyproofness. In particular, we present robust versions of envy-freeness and proportionality that are not only stronger than their standard counter-parts but also have less information requirements. We then focus on cake cutting with piecewise constant valuations and present three desirable algorithms: CCEA (Controlled Cake Eating Algorithm), MEA (Market Equilibrium…
Verifying Cake-Cutting, Faster
本文改进了蛋糕切割协议验证语言 Slice，通过归约约束公式至线性实数算术理论证明无嫉妒性可高效判定，并设计线性类型系统防止分配重叠，成功验证了首个非平凡的四智能体协议。
We consider the problem of fairly dividing a two dimensional heterogeneous good among multiple players. Applications include division of land as well as ad space in print and electronic media. Classical cake cutting protocols primarily consider a one-dimensional resource, or allocate each player multiple infinitesimally small "pieces". In practice, however, the two dimensional \emph{shape} of the allotted piece is of crucial importance in many applications (e.g. squares or bounded aspect-ratio rectangles are most useful for building houses, as well as advertisements). We thus introduce and stu…
We study the fair division of a continuous resource, such as a land-estate or a time-interval, among pre-specified groups of agents, such as families. Each family is given a piece of the resource and this piece is used simultaneously by all family members, while different members may have different value functions. Three ways to assess the fairness of such a division are examined. (a) Average Fairness means that each family's share is fair according to the "family value function", defined as the arithmetic mean of the value functions of the family members. (b) Unanimous Fairness means that all…
Approximate Envy-Freeness in Graphical Cake Cutting
针对图形蛋糕切割中连通无嫉妒分配未必存在的问题，本文给出一般图和星形图下加性与乘性低嫉妒分配的存在性保证，并提供高效计算算法。
We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where i…
We study the proportional chore division problem where a protocol wants to divide an undesirable object, called chore, among $n$ different players. The goal is to find an allocation such that the cost of the chore assigned to each player be at most $1/n$ of the total cost. This problem is the dual variant of the cake cutting problem in which we want to allocate a desirable object. Edmonds and Pruhs showed that any protocol for the proportional cake cutting must use at least $Ω(n \log n)$ queries in the worst case, however, finding a lower bound for the proportional chore division remained an i…
From Cake-Cutting and Necklace-Splitting to Fair Division of Indivisible Items
本文提出存在性转移框架，将连续公平分配定理转化为不可分物品分配的EFk类保证，结合相关定理得到多种公平分配结果，包括非加性估值下的共识公平分配的EFk类保证等。
Envy-Free Cake-Cutting for Four Agents
本文研究四智能体无嫉妒切蛋糕问题的计算复杂度，首次给出四个单调智能体下寻找连通ε-无嫉妒分配的高效算法，并证明非单调估值下该问题变为PPAD-难且具有高查询与通信复杂度下界。
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 3 or more agents. The impossibility result, however, assumes that the entire cake must be allocated. In this paper we replace the entire-allocation requirement with a weaker \emph{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 …
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 $n^{n^{n^{n^{n^n}}}}$. We additionally show that even if we do not run our protocol to completion, it can find in at mos…
We study the paradigmatic fair division problem of allocating a divisible good among agents with heterogeneous preferences, commonly known as cake cutting. Classical cake cutting protocols are susceptible to manipulation. Do their strategic outcomes still guarantee fairness? To address this question we adopt a novel algorithmic approach, by designing a concrete computational framework for fair division---the class of Generalized Cut and Choose (GCC) protocols}---and reasoning about the game-theoretic properties of algorithms that operate in this model. The class of GCC protocols includes the m…
On Cutting Cakes and Crossing Curves
本文通过揭示蛋糕切割与Jordan曲线问题的新联系，证明了三名代理人情况下无嫉妒蛋糕切割问题的计算复杂性，并首次给出了Jordan曲线问题的查询下界和UEOPL难度。
The classical cake cutting problem studies how to find fair allocations of a heterogeneous and divisible resource among multiple agents. Two of the most commonly studied fairness concepts in cake cutting are proportionality and envy-freeness. It is well known that a proportional allocation among $n$ agents can be found efficiently via simple protocols [16]. For envy-freeness, in a recent breakthrough, Aziz and Mackenzie [5] proposed a discrete and bounded envy-free protocol for any number of players. However, the protocol suffers from high multiple-exponential query complexity and it remains o…
Hardness of Approximate Sperner and Applications to Envy-Free Cake Cutting
本文证明在任意常数维度下寻找仅含三种颜色的单纯形仍是 PPAD-完全的，并将此结论应用于无嫉妒蛋糕切割问题，证明寻找使三个智能体无嫉妒的分配也是 PPAD-完全的。
Cutting a cake is a metaphor for the problem of dividing a resource (cake) among several agents. The problem becomes non-trivial when the agents have different valuations for different parts of the cake (i.e. one agent may like chocolate while the other may like cream). A fair division of the cake is one that takes into account the individual valuations of agents and partitions the cake based on some fairness criterion. Fair division may be accomplished in a distributed or centralized way. Due to its natural and practical appeal, it has been a subject of study in economics. To best of our know…
Truth, Justice, and Secrecy: Cake Cutting Under Privacy Constraints
本文提出首个隐私保护的蛋糕分割协议，在保证无嫉妒和策略证明性的同时，通过密码学技术保护代理的偏好隐私。
The classic cake-cutting problem provides a model for addressing the fair and efficient allocation of a divisible, heterogeneous resource 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 in an offline setting, this work instead focuses on online allocating non-contiguous pieces of cake among agents and establishes algorithmic results for fairness measures. In this regard, we made use of classic adversarial multi-armed bandits to achieve sub-linear Fairness and Revenue Regret at the same t…
We introduce a graphical framework for fair division in cake cutting, where comparisons between agents are limited by an underlying network structure. We generalize the classical fairness notions of envy-freeness and proportionality to this graphical setting. Given a simple undirected graph G, an allocation is envy-free on G if no agent envies any of her neighbor's share, and is proportional on G if every agent values her own share no less than the average among her neighbors, with respect to her own measure. These generalizations open new research directions in developing simple and efficient…
A compositional game to fairly divide homogeneous cake
针对同质蛋糕公平分配问题，提出基于迭代切分组合结构的新博弈与“最大参与者”组合规则，实现纳什均衡下每人1/n连续蛋糕的公平分配，消除简单组合规则的指数级不公与高无政府状态代价，且复杂度为线性。
On Connected Strongly-Proportional Cake-Cutting
本文研究不同权利智能体间连通强比例蛋糕切分的存在性刻画与查询高效算法，并给出匹配下界及饼型资源的扩展判定。
Searching, Sorting, and Cake Cutting in Rounds
本研究将轮次中的比例蛋糕分割归约为带排名查询的排序，并分析无序与有序搜索的查询复杂度，给出随机与确定性算法的精确期望界。
On Hierarchies of Fairness Notions in Cake Cutting: From Proportionality to Super Envy-Freeness
本文研究切蛋糕问题中比例性、无嫉妒性与完美性之间的公平性层级，提出 CHB 和 CLB 两类新概念，并在 Robertson-Webb 模型中给出其查询复杂度的上下界。
We study the existence of fair distributions when we have more guests than pieces to allocate, focusing on envy-free distributions among those who receive a piece. The conditions on the demand from the guests can be weakened from those of classic cake-cutting and rent-splitting results of Stromquist, Woodall, and Su. We extend existing variations of the cake-cutting problem with secretive guests and those that resist the removal of any sufficiently small set of guests.
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