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Divide And Choose
Divide and choose (also Cut and choose or I cut, you choose) is a procedure for fair division of a continuous resource, such as a cake, between two parties. It involves a heterogeneous good or resource ("the cake") and two partners who have different preferences over parts of the cake. The protocol proceeds as follows: one person ("the cutter") cuts the cake into two pieces; the other person ("the chooser") selects one of the pieces; the cutter receives the remaining piece. The procedure has been used since ancient times to divide land, cake and other resources between two parties. Currently, there is an entire field of research, called fair cake-cutting, devoted to various extensions and generalizations of cut-and-choose. History Divide and choose is mentioned in the Bible, in the Book of Genesis (chapter 13). When Abraham and Lot come to the land of Canaan, Abraham suggests that they divide it among them. Then Abraham, coming from the south, divides the land to a "left" (western) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Division
Fair division is the problem in game theory of dividing a set of resources among several people who have an entitlement to them so that each person receives their due share. That problem arises in various real-world settings such as division of inheritance, partnership dissolutions, divorce settlements, electronic frequency allocation, airport traffic management, and exploitation of Earth observation satellites. It is an active research area in mathematics, economics (especially social choice theory), dispute resolution, etc. The central tenet of fair division is that such a division should be performed by the players themselves, maybe using a mediator but certainly not an arbiter as only the players really know how they value the goods. The archetypal fair division algorithm is divide and choose. It demonstrates that two agents with different tastes can divide a cake such that each of them believes that he got the best piece. The research in fair division can be seen as an exten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Map
In algebra, an additive map, Z-linear map or additive function is a function f that preserves the addition operation: f(x + y) = f(x) + f(y) for every pair of elements x and y in the domain of f. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. For a specific case of this definition, see additive polynomial. More formally, an additive map is a \Z-module homomorphism. Since an abelian group is a \Z-module, it may be defined as a group homomorphism between abelian groups. A map V \times W \to X that is additive in each of two arguments separately is called a bi-additive map or a \Z-bilinear map. Examples Typical examples include maps between rings, vector spaces, or modules that preserve the additive group. An additive map does not necessarily preserve any other structure of the object; for example, the product operation of a ring. If f and g are additive maps, then the map f + g (defined pointwise) is additiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Games Column
Over a period of 24 years (January 1957 – December 1980), Martin Gardner wrote 288 consecutive monthly "Mathematical Games" columns for ''Scientific American'' magazine. During the next years, through June 1986, Gardner wrote 9 more columns, bringing his total to 297, as other authors wrote most of the "Mathematical Games" columns. The table below lists Gardner's columns. Twelve of Gardner's columns provided the cover art for that month's magazine, indicated by "over in the table with a hyperlink to the cover. Other articles by Gardner Gardner wrote 5 other articles for ''Scientific American''. His flexagon article in December 1956 was in all but name the first article in the series of ''Mathematical Games'' columns and led directly to the series which began the following month. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis Carroll, L. Frank Baum, and G. K. Chesterton.Martin (2010) He was also a leading authority on Lewis Carroll. ''The Annotated Alice'', which incorporated the text of Carroll's two Alice books, was his most successful work and sold over a million copies. He had a lifelong interest in magic and illusion and in 1999, MAGIC magazine named him as one of the "100 Most Influential Magicians of the Twentieth Century". He was considered the doyen of American puzzlers. He was a prolific and versatile author, publishing more than 100 books. Gardner was best known for creating and sustaining interest in recreational mathematicsand by extension, mathematics in generalthroughout the latter half of the 20th century, principally through his "Mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Last Diminisher
A last is a mechanical form shaped like a human foot. It is used by shoemakers and cordwainers in the manufacture and repair of shoes. Lasts typically come in pairs and have been made from various materials, including hardwoods, cast iron, and high-density plastics. The term is derived from the Proto-Germanic *''laistaz'' ("track, trace, footprint"); cognates include Swedish ''läst'', Danish ''læste'', German ''Leisten''. Production Lasts come in many styles and sizes, depending on the exact job they are designed for. Common variations include simple one-size lasts used for repairing soles and heels, durable lasts used in modern mass production, and custom-made lasts used in the making of bespoke footwear. Though a last is made approximately in the shape of a human foot, the precise shape is tailored to the kind of footwear being made. For example, a boot last would be designed to hug the instep for a close fit. Modern last shapes are typically designed using dedicated c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Austin Moving-knife Procedure
The Austin moving-knife procedures are procedures for equitable division of a cake. They allocate each of ''n'' partners, a piece of the cake which this partner values as ''exactly'' 1/n of the cake. This is in contrast to proportional division procedures, which give each partner ''at least'' 1/n of the cake, but may give more to some of the partners. When n=2, the division generated by Austin's procedure is an exact division and it is also envy-free. Moreover, it is possible to divide the cake to any number ''k'' of pieces which both partners value as exactly 1/''k''. Hence, it is possible to divide the cake between the partners in any fraction (e.g. give 1/3 to Alice and 2/3 to George). When n>2, the division is neither exact nor envy-free, since each partner only values his own piece as 1/n, but may value other pieces differently. The main mathematical tool used by Austin's procedure is the intermediate value theorem (IVT). Two partners and half-cakes The basic procedures ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moving-knife Procedure
In the mathematics of social science, and especially game theory, a moving-knife procedure is a type of solution to the fair division problem. The canonical example is the division of a cake using a knife. The simplest example is a moving-knife equivalent of the I cut, you choose scheme, first described by A.K.Austin as a prelude to his own procedure: * One player moves the knife across the cake, conventionally from left to right. * The cake is cut when ''either'' player calls "stop". * If each player calls stop when he or she perceives the knife to be at the 50-50 point, then the first player to call stop will produce an envy-free division if the caller gets the left piece and the other player gets the right piece. (This procedure is not necessarily efficient.) Generalizing this scheme to more than two players cannot be done by a discrete procedure without sacrificing envy-freeness. Examples of moving-knife procedures include * The Stromquist moving-knives procedur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Division
Exact division, also called consensus division, is a partition of a continuous resource (" cake") into some ''k'' pieces, such that each of ''n'' people with different tastes agree on the value of each of the pieces. For example, consider a cake which is half chocolate and half vanilla. Alice values only the chocolate and George values only the vanilla. The cake is divided into three pieces: one piece contains 20% of the chocolate and 20% of the vanilla, the second contains 50% of the chocolate and 50% of the vanilla, and the third contains the rest of the cake. This is an exact division (with ''k''=3 and ''n''=2), as both Alice and George value the three pieces as 20%, 50% and 30% respectively. Several common variants and special cases are known by different terms: * Consensus halving – the cake should be partitioned into two pieces (''k''=2), and all agents agree that the pieces have equal values. *Consensus 1/''k''-division, for any constant ''k''>1 - the cake should be partition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Veil Of Ignorance
The original position (OP), often referred to as the veil of ignorance, is a thought experiment used for reasoning about the principles that should structure a society based on mutual dependence. The phrases ''original position'' and ''veil of ignorance'' were coined by the American philosopher John Rawls, but the thought experiment itself was developed by William Vickrey and John Harsanyi in earlier writings. In the original position, you are asked to consider which principles you would select for the basic structure of society, but you must select as if you had no knowledge ahead of time what position you would end up having in that society. This choice is made from behind a "veil of ignorance", which prevents you from knowing your ethnicity, social status, gender and, crucially in Rawls' formulation, your or anyone else's idea of how to lead a good life. Ideally, this would force participants to select principles impartially and rationally. In Rawls's theory the original posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arbitration
Arbitration is a form of alternative dispute resolution (ADR) that resolves disputes outside the judiciary courts. The dispute will be decided by one or more persons (the 'arbitrators', 'arbiters' or 'arbitral tribunal'), which renders the 'arbitration award'. An arbitration decision or award is legally binding on both sides and enforceable in the courts, unless all parties stipulate that the arbitration process and decision are non-binding. Arbitration is often used for the resolution of commercial disputes, particularly in the context of international commercial transactions. In certain countries such as the United States, arbitration is also frequently employed in consumer and employment matters, where arbitration may be mandated by the terms of employment or commercial contracts and may include a waiver of the right to bring a class action claim. Mandatory consumer and employment arbitration should be distinguished from consensual arbitration, particularly commercial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mediation
Mediation is a structured, interactive process where an impartial third party neutral assists disputing parties in resolving conflict through the use of specialized communication and negotiation techniques. All participants in mediation are encouraged to actively participate in the process. Mediation is a "party-centered" process in that it is focused primarily upon the needs, rights, and interests of the parties. The mediator uses a wide variety of techniques to guide the process in a constructive direction and to help the parties find their optimal solution. A mediator is facilitative in that she/he manages the interaction between parties and facilitates open communication. Mediation is also evaluative in that the mediator analyzes issues and relevant norms ("reality-testing"), while refraining from providing prescriptive advice to the parties (e.g., "You should do..."). Mediation, as used in law, is a form of alternative dispute resolution resolving disputes between two o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
