Cake Division with Minimal Cuts: Envy-Free Procedures for 3 Person, 4 Persons, and Beyond

The minimal number of parallel cuts required to divide a cake into n pieces is n-1. A new 3-person procedure, requiring 2 parallel cuts, is given that produces an envy- free division, whereby each person thinks he or she receives at least a tied- for- largest piece. An extension of this procedure leads to a 4-person division, us ing 3 parallel cuts, that makes at most one player envious. Finally, a 4-person envy-free procedure is given, but it requires up to 5 parallel cuts, and some pieces may be disconnected. All these procedures improve on extant procedures by using fewer moving knives, making fewer people envious, or using fewer cuts. While the 4-person, 5-cut procedure is complex, endowing people with more information about others' preferences, or allowing them to do things beyond stopping moving knives, may yield simpler procedures for making envy- free divisions with minimal cuts, which are known always to existFAIR DIVISION; CAKE CUTTING; ENVY-FREENESS; MAXIMIN

Cutting A Pie Is Not A Piece Of Cake

Is there a division among n players of a cake using n-1 parallel vertical cuts, or of a pie using n radial cuts, that is envy-free (each player thinks he or she receives a largest piece and so does not envy another player) and undominated (there is no other allocation as good for all players and better for at least one)? David Gale first asked this question for pies. We provide complete answers for both cakes and pies. The answers depend on the number of players (two versus three or more players) and whether the players' preferences satisfy certain continuity assumptions. We also give some simple algorithms for cutting a pie when there are two or more players, but these algorithms do not guarantee all the properties one might desire in a division, which makes pie-cutting harder than cake-cutting. We suggest possible applications and conclude with two open questions

Games That End in a Bang or a Whimper

Using truels, or three-person duels, as an example, we show that how players perceive a multiple-round game will end can make a big difference in whether it ends non-cooperatively (producing a "bang") or just peters out (producing a "whimper"): 1. If the players view the number of rounds as bounded-reasonable, because the game must end in a finite number of rounds-they will shoot from the start. 2. If the players view the number of rounds as unbounded-reasonable, because the horizon of the game is infinite-then a cooperative equilibrium, involving no shooting, can also occur. Real- life examples are given of players with bounded and unbounded outlooks in truel- like situations. Unbounded outlooks encourage cooperative play, foster hope, and lead to more auspicious outcomes. These outcomes are facilitated by institutions that put no bounds on play-including reprisals-thereby allowing for a day of reckoning for those who violate established norms. Eschatological implications of the analysis, especially for thinking about the future and how it might end, are also discussed.TRUELS; BACKWARD INDUCTION; INFINITE-HORIZON GAMES; ESCHATOLOGY

Cooperative vs. Non-Cooperative Truels: Little Agreement, But Does That Matter?

It is well-known that non-cooperative and cooperative game theory may yield different solutions to games. These differences are particularly dramatic in the case of truels, or three-person duels, in which the players may fire sequentially or simultaneously, and the games may be one-round or n-round. Thus, it is never a Nash equilibrium for all players to hold their fire in any of these games, whereas in simultaneous one-round and n-round truels such cooperation, wherein everybody survives, is in both the alpha-core and beta-core. On the other hand, both cores may be empty, indicating a lack of stability, when the unique Nash equilibrium is one survivor. Conditions under which each approach seems most applicable are discussed. Although it might be desirable to subsume the two approaches within a unified framework, such unification seems unlikely since the two approaches are grounded in fundamentally different notions of stability.COOPERATIVE GAME; NON-COOPERATIVE GAME; TRUEL; NASH EQUILIBRIUM; CORE

Single-Peakedness and Disconnected Coalitions

Ordinally single-peaked preferences are distinguished from cardinally singlepeaked preferences, in which all players have a similar perception of distances in some one-dimensional ordering. While ordinal single-peakedness can lead to disconnected coalitions that have a "hole" in the ordering, cardinal single-peakedness precludes this possibility, based on two models of coalition formation: ¥ Fallback (FB): Players seek coalition partners by descending lower and lower in their preference rankings until a majority coalition forms. ¥ Build-Up (BU): Similar to FB, except that when nonmajority subcoalitions form, they fuse into composite players, whose positions are defined cardinally and who are treated as single players in the convergence process. FB better reflects the unconstrained, or nonmyopic, possibilities of coalition formation, whereas BU-because all subcoalition members must be included in any majority coalition that forms-restricts combinatorial possibilities and tends to produce less compact majority coalitions. The "strange bedfellows" frequently observed in legislative coalitions and military alliances suggest that even when players agree on, say, a left-right ordering, their perceptions of exactly where players stand in this ordering may differ substantially. If so, a player may be acceptable to a coalition but may not find every member in it acceptable, causing that player not to join and possibly creating a disconnected coalition.COALITION FORMATION; SINGLE-PEAKEDNESS; LEGISLATURES; ALLIANCES

Paradoxes of Fair Division

Two or more players are required to divide up a set of indivisible items that they can rank from best to worst. They may, as well, be able to indicate preferences over subsets, or packages, of items. The main criteria used to assess the fairness of a division are efficiency (Pareto-optimality) and envy-freeness. Other criteria are also suggested, including a Rawlsian criterion that the worst-off player be made as well off as possible and a scoring procedure, based on the Borda count, that helps to render allocations as equal as possible. Eight paradoxes, all of which involve unexpected conflicts among the criteria, are described and classified into three categories, reflecting (1) incompatibilities between efficiency and envy-freeness, (2) the failure of a unique efficient and envy-free division to satisfy other criteria, and (3) the desirability, on occasion, of dividing up items unequally. While troublesome, the paradoxes also indicate opportunities for achieving fair division, which will depend on the fairness criteria one deems important and the trade-offs one considers acceptable.FAIR DIVISION; ALLOCATION OF INDIVISIBLE ITEMS; ENVY-FREENESS; PARETO- OPTIMALITY; RAWLSIAN JUSTICE; BORDA COUNT.

THE WAIT-AND-SEE OPTION IN ASCENDING PRICE AUCTIONS

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 of guaranteed envy-freeness (DGEF) as a measure of how good a cake-cutting protocol can approximate the ideal of envy-freeness while keeping the protocol finite bounded (trading being disregarded). We propose a new finite bounded proportional protocol for any number n \geq 3 of players, and show that this protocol has a DGEF of 1 + \lceil (n^2)/2 \rceil. This is the currently best DGEF among known finite bounded cake-cutting protocols for an arbitrary number of players. We will make the case that improving the DGEF even further is a tough challenge, and determine, for comparison, the DGEF of selected known finite bounded cake-cutting protocols.Comment: 37 pages, 4 figure

Fair Division of Indivisible Items

This paper analyzes criteria of fair division of a set of indivisible items among people whose revealed preferences are limited to rankings of the items and for whom no side payments are allowed. The criteria include refinements of Pareto optimality and envy-freeness as well as dominance-freeness, evenness of shares, and two criteria based on equally-spaced surrogate utilities, referred to as maxsum and equimax. Maxsum maximizes a measure of aggregate utility or welfare, whereas equimax lexicographically maximizes persons' utilities from smallest to largest. The paper analyzes conflicts among the criteria along possibilities and pitfalls of achieving fair division in a variety of circumstances.FAIR DIVISION; ALLOCATION OF INDIVISIBLE ITEMS; PARETO OPTIMALITY; ENVY-FREENESS; LEXICOGRAPHIC MAXIMUM

Swap Bribery

In voting theory, bribery is a form of manipulative behavior in which an external actor (the briber) offers to pay the voters to change their votes in order to get her preferred candidate elected. We investigate a model of bribery where the price of each vote depends on the amount of change that the voter is asked to implement. Specifically, in our model the briber can change a voter's preference list by paying for a sequence of swaps of consecutive candidates. Each swap may have a different price; the price of a bribery is the sum of the prices of all swaps that it involves. We prove complexity results for this model, which we call swap bribery, for a broad class of election systems, including variants of approval and k-approval, Borda, Copeland, and maximin.Comment: 17 page