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

Kingmakers and Leaders in Coalition Formation

Assume that players strictly rank each other as coalition partners. We propose a procedure whereby they “fall back” on their preferences, yielding internally compatible, or coherent, majority coalition(s), which we call fallback coalitions. If there is more than one fallback coalition, the players common to them, or kingmakers, determine which fallback coalition will form. The players(s) who are the first to be acceptable to all other members of a fallback coalition are the leader(s) of that coalition. The effects of different preference assumptions—particularly, different kinds of single-peakedness—and of player weights on the number of coherent coalitions, their connectedness, and which players become kingmakers and leaders are investigated. The fallback procedure may be used (i) empirically to identify kingmakers and leaders or (ii) normatively to select them.Coalition, Fallback Process, Kingmaker Leader, Cardinally Single-peaked, Ordinally Single-peaked

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

Two-person cake-cutting: the optimal number of cuts

A cake is a metaphor for a heterogeneous, divisible good. When two players divide such a good, there is always a perfect division—one that is efficient (Pareto-optimal), envy-free, and equitable—which can be effected with a finite number of cuts under certain mild conditions; this is not always the case when there are more than two players (Brams, Jones, and Klamler, 2011b). We not only establish the existence of such a division but also provide an algorithm for determining where and how many cuts must be made, relating it to an algorithm, “Adjusted Winner” (Brams and Taylor, 1996, 1999), that yields a perfect division of multiple homogenous goods.Cake-cutting; fair division; envy-freeness; adjusted winner; heterogeneous good

Stabilizing Power Sharing

Power sharing is modeled as a duel over some prize. Each of two players may either share the prize in some ratio or fire at the other player—either in sequence or simultaneously—and eliminate it with a specified probability. If one player eliminates the other without being eliminated itself, it captures the entire prize, but the prize is damaged over time when there is shooting. Simultaneous shooting, which is more damaging than sequential shooting, tends to induce the players to share the prize and expand their opportunities for sharing it. It was effectively implemented by the superpowers with the doctrine of “launch on warning” during the Cold War, and it was strengthened by the development of second-strike capability. Deterring terrorism has proved a different matter, because terrorists are difficult to detect and present few targets that can be damaged.power sharing, game, duel, deterrence, terrorism

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

Dividing the Indivisible: Procedures for Allocating Cabinet Ministries to Political Parties in a Parliamentary System

Political parties in Northern Ireland recently used a divisor method of apportionment to choose, in sequence, ten cabinet ministries. If the parties have complete information about each others' preferences, we show that it may not be rational for them to act sincerely by choosing their most-preferred ministry that is available. One consequence of acting sophisticatedly is that the resulting allocation may not be Pareto-optimal, making all the parties worse off. Another is nonmonotonicty—choosing earlier may hurt rather than help a party. We introduce a mechanism that combines sequential choices with a structured form of trading that results in sincere choices for two parties. Although there are difficulties in extending this mechanism to more than two parties, other approaches are explored, such as permitting parties to making consecutive choices not prescribed by an apportionment method. But certain problems, such as eliminating envy, remain.Proportional Representation, apportionment, divisor methods, Sincere and Sophisticated Choices, Envy Free Allocations, Sports Drafts

The Instability of Power Sharing

Three models are presented in which two players agree to share power in a particular ratio, but either player may subsequently “fire” at the other, as in a duel, to try to eliminate it. The players have positive probabilities of eliminating each other by firing. If neither is successful, the agreement stays in place; if one is successful, that player obtains all the power; if each eliminates the other, both players get nothing. In Model I, the game is played once, and in Model II it is repeated, with discounting of future payoffs. Although there are conditions under which each player would prefer not to shoot, satisfying these conditions for one player precludes satisfying them for the other, so at least one player will always have an incentive to shoot. In anticipation, its rival would prefer to shoot, too, so there will be a race to preempt. In Model III, a damage factor caused by shooting, whether successful or not, is introduced into Model II. This mitigates the incentive to shoot but does not eliminate it entirely. The application of the models to conflicts, especially civil wars, is discussed.power sharing, repeated game, duel, civil wars

Narrowing the field in elections: the next-two rule

We suggest a new approach to narrowing the field in elections, based on the deservingness of candidates to be contenders in a runoff, or to be declared one of several winners. Instead of specifying some minimum percentage (e.g., 50) that the leading candidate must surpass to avoid a runoff (usually between the top two candidates), we propose that the number of contenders depend on the distribution of votes among candidates. Divisor methods of apportionment proposed by Jefferson and Webster, among others, provide measures of deservingness, but they can prescribe a runoff even when one candidate receives more than 50 percent of the vote. We propose a new measure of deservingness, called the Next-Two rule, which compares the performance of candidates to the two that immediately follow them. It never prescribes a runoff when one candidate receives more than 50 percent of the vote. More generally, it identifies as contenders candidates who are bunched together near the top and, unlike the Jefferson and Webster methods, never declares that all candidates are contenders. We apply the Next-Two rule to several empirical examples, including one (elections to major league baseball’s Hall of Fame) in which more than one candidate can be elected.voting; contenders in elections; runoffs; apportionment; fairness