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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 between two parties. It involves a heterogeneous good or resource and two partners who have different preferences over parts of the cake (both want as much of it as possible). The procedure proceeds as follows: one person divides the resource into two pieces; the other person selects one of the pieces; the cutter receives the remaining piece. Since ancient times some have used the procedure to divide land, food 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. Divide and choose is envy-free in the following sense: each of the two partners can act in a way that guarantees that, according to their own subjective taste, their allocated share is at least as valuable as the other share, regardless of what the other partner does. Desc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cake Cutting Division
Cake is a Flour confections, flour confection usually made from flour, sugar, and other ingredients and is usually baking, baked. In their oldest forms, cakes were modifications of bread, but cakes now cover a wide range of preparations that can be simple or elaborate and which share features with Dessert, desserts such as pastry, pastries, meringues, custards, and pies. The most common ingredients include flour, sugar, eggs, fat (such as butter, Vegetable oil, oil, or margarine), a liquid, and a leavening agent, such as baking soda or baking powder. Common additional ingredients include Dried fruit, dried, candied fruit, candied, or fresh fruit, nut (fruit), nuts, cocoa powder, cocoa, and extracts such as vanilla, with numerous substitutions for the primary ingredients. Cakes can also be filled with fruit preserves, nuts, or dessert sauces (like custard, Fruit preserves, jelly, cooked fruit, whipped cream, or syrups), iced with buttercream or other Icing (food), icings, and ... [...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 pointwi ... [...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, until June 1986, Gardner wrote 9 more columns, bringing his total to 297. During this period other authors wrote most of the columns. In 1981, Gardner's column alternated with a new column by Douglas Hofstadter called " Metamagical Themas" (an anagram of "Mathematical Games"). 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 magic, scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis Carroll, L. Frank Baum, and G. K. Chesterton.Martin (2010) He was 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.Martin Gardner obituary (2010) He had a lifelong interest in magic and illusion and in 1999, ''MAGIC'' magazine named him as one of the "10 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even–Paz Protocol
The Even–Paz algorithm is an computationally-efficient algorithm for fair cake-cutting. It involves a certain heterogeneous and divisible resource, such as a birthday cake, and n partners with different preferences over different parts of the cake. It allows the n people to achieve a proportional division. History The first published algorithm for proportional division of a cake was the last diminisher algorithm, published in 1948. Its run-time complexity was O(n^2). In 1984, Shimon Even and Azaria Paz published their improved algorithm, whose run-time complexity is only O(n\log n). Description The algorithm uses a divide-and-conquer strategy, it is possible to achieve a proportional division in time O(n\log n). * Each partner is asked to draw a line dividing the cake into a right and left part such that he believes the ratio is \lfloor n/2\rfloor :\lceil n/2\rceil. The cuts are required to be non-intersecting; a simple way to guarantee this is to allow only horizontal l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Last Diminisher
The last diminisher procedure is a procedure for fair cake-cutting. It involves a certain heterogenous and divisible resource, such as a birthday cake, and ''n'' partners with different preferences over different parts of the cake. It allows the ''n'' people to achieve a proportional division, i.e., divide the cake among them such that each person receives a piece with a value of at least 1/''n'' of the total value according to his own subjective valuation. For example, if Alice values the entire cake as $100 and there are 5 partners then Alice can receive a piece that she values as at least $20, regardless of what the other partners think or do. History During World War II, the Polish-Jewish mathematician Hugo Steinhaus, who was hiding from the Nazis, occupied himself with the question of how to divide resources fairly. Inspired by the divide and choose procedure for dividing a cake between two brothers, he asked his students, Stefan Banach and Bronisław Knaster, to find a proced ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Austin Moving-knife Procedure
The Austin moving-knife procedures are procedures for equitable division of a fair cake-cutting, cake. To each of ''n'' partners, they allocate 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 cake-cutting, 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 pa ... [...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. "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. The central tenet of fair division is that such a division should be performed by the players themselves, without the need for external arbitration, as only the players themselves really know how they value the goods. The name of the procedure comes from the canonical example of the fair division of a cake using a knife. Examples 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Division
Consensus splitting, also called exact 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Veil Of Ignorance
The original position is a hypothetical position from which members of society would consider which principles they would select for the basic structure of their society if they had no knowledge ahead of time regarding the position which they would end up occupying in that society. The idea of having no such knowledge, because everyone is behind a veil of ignorance, represents a thought experiment often associated with the work of the American philosopher John Rawls. In this "original position", their position behind the "veil of ignorance" prevents everyone from knowing their ethnicity, social status, gender, and (crucially in Rawls's formulation) their or anyone else's ideas of how to lead a good life. Ideally, this would force participants acting rationally to adopt an "initial agreement" on the principles impartially. In Rawls's theory the original position plays the same role as the "state of nature" does in the social contract tradition of Thomas Hobbes and John Locke. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arbitration
Arbitration is a formal method of dispute resolution involving a third party neutral who makes a binding decision. The third party neutral (the 'arbitrator', 'arbiter' or 'arbitral tribunal') renders the decision in the form of an 'arbitration award'. An arbitration award is legally binding on both sides and enforceable in local courts, unless all parties stipulate that the arbitration process and decision are non-binding. Arbitration is often used for the resolution of Commercial law, commercial disputes, particularly in the context of International commerce, 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 Class action waiver, a waiver of the right to bring a class action claim. Mandatory consumer and employment arbitration should be distinguished from consensu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mediation
Mediation is a structured, voluntary process for resolving disputes, facilitated by a neutral third party known as the mediator. It is a structured, interactive process where an independent third party, the mediator, assists disputing parties in resolving conflict through the use of specialized communication and negotiation techniques. All participants in mediation are encouraged to participate in the process actively. Mediation is "party-centered," focusing on the needs, interests, and concerns of the individuals involved, rather than imposing a solution from an external authority. 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. Mediation can take different forms, depending on the mediator's approach. In facilitative mediation, the mediator assists parties by fostering communication and helping them understand each other's viewpoints. In evaluative mediation, the mediator may a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
