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Selfridge–Conway Procedure
The Selfridge–Conway procedure is a discrete procedure that produces an envy-free cake-cutting for three partners. It is named after John Selfridge and John Horton Conway. Selfridge discovered it in 1960, and told it to Richard Guy, who told many people, but Selfridge did not publish it. John Conway later discovered it independently, and also never published it. This procedure was the first envy-free discrete procedure devised for three partners, and it paved the way for more advanced procedures for ''n'' partners (see envy-free cake-cutting). A procedure is envy-free if each recipient believes that (according to their own measure) no other recipient has received a larger share. The maximal number of cuts in the procedure is five. The pieces are not always contiguous. The Procedure Suppose we have three players P1, P2 and P3. Where the procedure gives a criterion for a decision it means that criterion gives an optimum choice for the player. # P1 divides the cake into three pi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envy-free Cake-cutting
An envy-free cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the envy-free criterion, namely, that every partner feels that their allocated share is at least as good as any other share, according to their own subjective valuation. When there are only two partners, the problem is easy and was solved in antiquity by the divide and choose protocol. When there are three or more partners, the problem becomes much more challenging. Two major variants of the problem have been studied: * Connected pieces, e.g. if the cake is a 1-dimensional interval then each partner must receive a single sub-interval. If there are n partners, only n-1 cuts are needed. * General pieces, e.g. if the cake is a 1-dimensional interval then each partner can receive a union of disjoint sub-intervals. Short history Modern research into the fair cake-cutting problem started in the 1940s. The first fairness criterion studied was proportional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Selfridge
John Lewis Selfridge (February 17, 1927 – October 31, 2010), was an American mathematician who contributed to the fields of analytic number theory, computational number theory, and combinatorics. Education Selfridge received his Ph.D. in 1958 from the University of California, Los Angeles under the supervision of Theodore Motzkin. Career Selfridge served on the faculties of the University of Illinois at Urbana-Champaign and Northern Illinois University from 1971 to 1991 (retirement), chairing the Department of Mathematical Sciences 1972–1976 and 1986–1990. He was executive editor of Mathematical Reviews from 1978 to 1986, overseeing the computerization of its operations. He was a founder of the Number Theory Foundation, which has named its Selfridge prize in his honour. Research In 1962, he proved that 78,557 is a Sierpinski number; he showed that, when ''k'' = 78,557, all numbers of the form ''k''2''n'' + 1 have a factor in the covering set . Fiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius College ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard K
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include " Richie", "Dick", " Dickon", " Dickie", "Rich", " Rick", " Rico", " Ricky", and more. Richard is a common English, German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Catalan "Ricard" and the Italian "Riccardo", among others (see comprehensive variant list below). People named Richard Multiple people with the same name * Richard Andersen (other) * Richard Anderson (other) * Richard Cartwright (disambiguati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Disposal
In various parts of economics, the term free disposal implies that resources can be discarded without any cost. For example, a fair division setting with free disposal is a setting where some resources have to be divided fairly, but some of the resources may be left undivided, discarded or donated. Examples of situations with free disposal are allocation of food, clothes jewels etc. Examples of situations ''without'' free disposal are: * Chore division - since all chores must be done. * Allocation of land with an old structure - since the structure may have to be destructed, and destruction is costly. * Allocation of an old car - since the car may have to be carried away to used cars garage, and moving it may be costly. * Allocation of shares in a firm that may have debts - since the firm cannot be disposed of without paying its debts first. The free disposal assumption may be useful for several reasons: * It enables truthful cake-cutting algorithms: The option to discard some o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brams–Taylor Procedure
The Brams–Taylor procedure (BTP) is a procedure for envy-free cake-cutting. It explicated the first finite procedure to produce an envy-free division of a cake among any positive integer number of players. History In 1988, prior to the discovery of the BTP, Sol Garfunkel contended that the problem solved by the theorem, namely n-person envy-free cake-cutting, was among the most important problems in 20th century mathematics. The BTP was discovered by Steven Brams and Alan D. Taylor. It was first published in the January 1995 issue of American Mathematical Monthly, and later in 1996 in the authors' book. Brams and Taylor hold a joint US patent from 1999 related to the BTP. Description The BTP divides the cake part-by-part. A typical intermediate state of the BTP is as follows: * A part of the cake, say X, is divided in an envy-free way among all partners. * The rest of the cake, say Y, remains undivided, but - * One partner, say Alice, has an Irrevocable Advantage (IA) o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Division Protocols
A fair (archaic: faire or fayre) is a gathering of people for a variety of entertainment or commercial activities. Fairs are typically temporary with scheduled times lasting from an afternoon to several weeks. Types Variations of fairs include: * Art fairs, including art exhibitions and arts festivals * County fair (USA) or county show (UK), a public agricultural show exhibiting the equipment, animals, sports and recreation associated with agriculture and animal husbandry. * Festival, an event ordinarily coordinated with a theme e.g. music, art, season, tradition, history, ethnicity, religion, or a national holiday. * Health fair, an event designed for outreach to provide basic preventive medicine and medical screening * Historical reenactments, including Renaissance fairs and Dickens fairs * Horse fair, an event where people buy and sell horses. * Job fair, event in which employers, recruiters, and schools give information to potential employees. * Regional or state fair, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
