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

Two-person pie-cutting: The fairest cuts

Barbanel, Brams, and Stromquist (2009) asked whether there exists a two-person moving-knife procedure that yields an envy-free, undominated, and equitable allocation of a pie. We present two procedures: One yields an envy-free, almost undominated, and almost equitable allocation, whereas the second yields an allocation with the two “almosts” removed. The latter, however, requires broadening the definition of a “procedure," which raises philosophical, as opposed to mathematical, issues. An analogous approach for cakes fails because of problems in eliciting truthful preferences.mechanism design; fair division; divisible good; cake-cutting; pie-cutting

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

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

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.Fair division; cake-cutting; pie-cutting; divisible good; envy-freeness; allocative efficiency

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

N-Person cake-cutting: there may be no perfect division

A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envy-free, and equitable. We give an example of a cake in which it is impossible to divide it among three players such that these three properties are satisfied, however many cuts are made. 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 is not envy-free, is more desirable (a 2-cut division can at best satisfy either envy-freeness or equitability but not both). We prove that no perfect division exists for an extension of the example for three or more players

A Discrete and Bounded Envy-free Cake Cutting Protocol for Four Agents

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

A cluster randomised control trial to evaluate the effectiveness and cost-effectiveness of the Italian medicines use review (I-MUR) for asthma patients

Background The economic burden of asthma, which relates to the degree of control, is €5 billion annually in Italy. Pharmacists could help improve asthma control, reducing this burden. This study aimed to evaluate the effectiveness and cost-effectiveness of Medicines Use Reviews provided by community pharmacists in asthma. Methods This cluster randomised, multi-centre, controlled trial in adult patients with asthma was conducted in 15 of the 20 regions of Italy between September 2014 and July 2015. After stratification by region, community pharmacists were randomly allocated to group A (trained in and delivered the intervention at baseline) or B (training and delivery 3 months later), using computerised random number generation in blocks of 10. Each recruited up to five patients, with both groups followed for 9 months. The intervention consisted of a systematic, structured face-to-face consultation with a pharmacist, covering asthma symptoms, medicines used, attitude towards medicines and adherence, recording pharmacist-identified pharmaceutical care issues (PCIs). The primary outcome was asthma control, assessed using the Asthma-Control-Test (ACT) score (ACT ≥ 20 represents good control). Secondary outcomes were: number of active ingredients, adherence, cost-effectiveness compared with usual care. Although blinding was not possible for either pharmacists or patients, assessment of outcomes was conducted by researchers blind to group allocation. Results Numbers of pharmacists and patients enrolled were 283 (A = 136; B = 147) and 1263 (A = 600; B = 663), numbers completing were 201 (A = 97; B = 104) and 816 (A = 400; B = 416), respectively. Patients were similar in age and gender and 56.13% (458/816) had poor/partial asthma control. Pharmacists identified 1256 PCIs (mean 1.54/patient), mostly need for education, monitoring and potentially ineffective therapy. Median ACT score at baseline differed between groups (A = 19, B = 18; p < 0.01). Odds ratio for improved asthma control was 1.76 (95% CI 1.33–2.33) and number needed to treat 10 (95% CI 6–28). Number of active ingredients reduced by 7.9% post-intervention (p < 0.01). Adherence improved by 35.4% 3 months post-intervention and 40.0% at 6 months (p < 0.01). The probability of the intervention being more cost-effective than usual care was 100% at 9 months. Conclusions This community pharmacist-based intervention demonstrated both effectiveness and cost-effectiveness. It has since been implemented as the first community pharmacy cognitive service in Italy