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### N-Person Cake-Cutting: There May Be No Perfect Division

Steven J. Brams, Michael A. Jones, Christian Klamler
2011 Social Science Research Network
We prove that no perfect division exists for an extension of the example for three or more players.  ...  It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which  ...  (We show later, however, that our results for a cake also apply to a pie. ) We focus in this paper on n-person cake-cutting.  ...

### N-Person Cake-Cutting: There May Be No Perfect Division

Steven J. Brams, Michael A. Jones, Christian Klamler
2013 The American mathematical monthly
We prove that no perfect division exists for an extension of the example for three or more players.  ...  It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which  ...  (We show later, however, that our results for a cake also apply to a pie. ) We focus in this paper on n-person cake-cutting.  ...

### Online Cake Cutting [chapter]

Toby Walsh
2011 Lecture Notes in Computer Science
. . " We will formulate cake cutting as dividing the unit interval, [0, 1] between n agents.  ...  A division of the cake amongst n agents is a partition of some cutting of {[0, 1]} into n subsets.  ...  ., 2010] , we say that a cake cutting procedure is truthful iff there are no valuations where an agent will do better by lying.  ...

### Fair Multi-Cake Cutting [article]

Erel Segal-Halevi
2020 arXiv   pre-print
In the classic problem of fair cake-cutting, a single interval ("cake") has to be divided among n agents with different value measures, giving each agent a single sub-interval with a value of at least  ...  The paper presents a polynomial-time algorithm that guarantees to each agent at least min(1/n, k/(m+n-1)) of the total value, and shows that this is the largest fraction that can be guaranteed.  ...  Some relatively up-to-date surveys can be found in the Wikipedia pages "Fair division", "Fair cakecutting", "Proportional cake-cutting", "Envy-free cake-cutting" and "Fair item allocation".  ...

### Cake-Cutting Algorithms: Be Fair if You Can

Francis Edward Su, Jack Robertson, William Webb
2000 The American mathematical monthly
each person feels that no one else received a larger piece.  ...  . , X n to be assigned to players P 1 , P 2 , . . . , P n , we may construct a bipartite graph G with the players and objects as vertices; there is an edge between P i and X j if P i thinks X j is at least  ...

### Two-Person Cake Cutting: The Optimal Number of Cuts

Julius B. Barbanel, Steven J. Brams
2014 The Mathematical intelligencer
When two players divide such a good, there is always a perfect division-one that is efficient (Pareto-optimal), envyfree, and equitable-which can be effected with a finite number of cuts under certain  ...  ), that yields a perfect division of multiple homogenous goods.  ...  If there are n > 2 players, there may be no perfect division of a cake (Brams, Jones, and Klamler, 2011b) .  ...

### Divide-and-Conquer: A Proportional, Minimal-Envy Cake-Cutting Algorithm

Steven J. Brams, Michael A. Jones, Christian Klamler
2011 SIAM Review
We analyze a class of proportional cake-cutting algorithms that use a minimal number of cuts (n − 1 if there are n players) to divide a cake that the players value along one dimension.  ...  However, D&C may not allow players to obtain proportional, connected pieces if they have unequal entitlements. Possible applications of D&C to land division are briefly discussed.  ...  An "ideal" division might be an envy-free division that is as close as possible to being equitable, but there is no general n-person algorithm that yields even an envyfree division, much less one that  ...

### Fair Division (Dagstuhl Seminar 16232)

Yonatan Aumann, Jérôme Lang, Ariel D. Procaccia, Marc Herbstritt
2016 Dagstuhl Reports
This report documents the program and the outcomes of Dagstuhl Seminar 16232 "Fair Division".  ...  We deal with a simple problem: There are n units of two types that need to be allocated among n people, one per person. Preferences are stochastic.  ...  Land-value data: For land, in contrast to cake, there exist detailed value maps, which can be used to test the performance of cake-cutting algorithms.  ...

### Lecture Notes on Fair Division [article]

Ulle Endriss
2018 arXiv   pre-print
Second, we give an introduction to cake-cutting procedures as an example of methods for fairly dividing a single divisible resource amongst a group of individuals.  ...  While the classical literature on fair division has largely developed within Economics, these Notes are specifically written for readers with a background in Computer Science or similar, and who may be  ...  For the domain of cake-cutting discussed in Section 4 there can be no general (finitary) language for describing all preferences that agents may have.  ...

### Don't cry to be the first!Symmetric fair division exist [article]

Guillaume Chèze
2019 arXiv   pre-print
In this article we study a cake cutting problem.  ...  In the second part, we give a proportional and symmetric fair division algorithm with a complexity in O(n 3) in the Robertson-Webb model of complexity. This algorithm is based on Kuhn's algorithm.  ...  When we design a cake cutting algorithm, we have to precise what is the meaning of a fair division. Indeed, there exists different notions of fair division.  ...

### Geometric division with a fixed point: Not half the cake, but at least 4/9

Andreas Wagener
2006 Group Decision and Negotiation
We study a two-person problem of cutting a homogeneous cake where one player is disadvantaged from the outset: Unlike under the divide-and-choose rule he may only choose a point on the cake through which  ...  Problem, Player 1 is disadvantaged, both relative to Player 2 and to the standard cake-cutting problem.  ...  The cake-cutting problem is probably the most familiar example, but problems of allocating burdens (chore division) and also mixed, rent partitioning problems are widely discussed. 1 In two-player settings  ...

### Envy-free Matchings in Bipartite Graphs and their Applications to Fair Division [article]

2021 arXiv   pre-print
We show how envy-free matchings can be used in various fair division problems with either continuous resources ("cakes") or discrete ones.  ...  In particular, we propose a symmetric algorithm for proportional cake-cutting, an algorithm for 1-out-of-(2n-2) maximin-share allocation of discrete goods, and an algorithm for 1-out-of-floor(2n/3) maximin-share  ...  In the general case, there may be several different minimum-cost envy-free matchings, Algorithm 5 A symmetric algorithm for proportional cake-cutting. Based on Chèze [12] .  ...

### What Is Social Software? [chapter]

Rohit Parikh, Marc Pauly
2012 Lecture Notes in Computer Science
• A division algorithm is envy-free if each player feels she has done at least as well as any of the others. • Does the Parikh analysis of Knaster Tarski cake cutting establish that the algorithm is envy-free  ...  The key ingredient of the procedure is a loop operation: continue to trim the piece until there are no further objections about the size.  ...  Example: Inheritance Division • An inheritance has to be divided among N heirs.  ...

### FIXP-membership via Convex Optimization: Games, Cakes, and Markets [article]

Aris Filos-Ratsikas, Kristoffer Arnsfelt Hansen, Kasper Høgh, Alexandros Hollender
2021 arXiv   pre-print
Our technique constructs a "pseudogate" which can be used as a black box when building FIXP circuits. This pseudogate, which we term the "OPT-gate", can solve most convex optimization problems.  ...  Using the OPT-gate, we prove new FIXP-membership results, and we generalize and simplify several known results from the literature on fair division, game theory and competitive markets.  ...  We argue by contradiction, so suppose that x is not an envy-free division. This means that there is no perfect matching in the bipartite graph described at the beginning of this subsection.  ...

### Slices and Lumps

Lee Anne Fennell
2008 Social Science Research Network
TAYLOR, FAIR DIVISION: FROM CAKE-CUTTING TO DISPUTE RESOLUTION 6-7 & n.2 (discussing the judgment and citing additional work providing formal analyses); Richard R.W.  ...  cake-cutting examples).  ...
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