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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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Social Science
Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of society", established in the 19th century. In addition to sociology, it now encompasses a wide array of academic disciplines, including anthropology, archaeology, economics, human geography, linguistics, management science, communication science and political science. Positivist social scientists use methods resembling those of the natural sciences as tools for understanding society, and so define science in its stricter modern sense. Interpretivist social scientists, by contrast, may use social critique or symbolic interpretation rather than constructing empirically falsifiable theories, and thus treat science in its broader sense. In modern academic practice, researchers are often eclectic, using multiple methodologies (for instance, by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathema ... [...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] |
Cake
Cake is a flour confection made from flour, sugar, and other ingredients, and is usually 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 desserts such as pastries, meringues, custards, and pies. The most common ingredients include flour, sugar, eggs, fat (such as butter, oil or margarine), a liquid, and a leavening agent, such as baking soda or baking powder. Common additional ingredients include dried, candied, or fresh fruit, nuts, 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, jelly, cooked fruit, whipped cream or syrups), iced with buttercream or other icings, and decorated with marzipan, piped borders, or candied fruit. Cake is often served as a celebratory dish on ceremonial occasions, such as wedd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knife
A knife ( : knives; from Old Norse 'knife, dirk') is a tool or weapon with a cutting edge or blade, usually attached to a handle or hilt. One of the earliest tools used by humanity, knives appeared at least 2.5 million years ago, as evidenced by the Oldowan tools. Originally made of wood, bone, and stone (such as flint and obsidian), over the centuries, in step with improvements in both metallurgy and manufacturing, knife blades have been made from copper, bronze, iron, steel, ceramic, and titanium. Most modern knives have either fixed or folding blades; blade patterns and styles vary by maker and country of origin. Knives can serve various purposes. Hunters use a hunting knife, soldiers use the combat knife, scouts, campers, and hikers carry a pocket knife; there are kitchen knives for preparing foods (the chef's knife, the paring knife, bread knife, cleaver), table knives (butter knives and steak knives), weapons (daggers or switchblades), knives for throwing or juggling, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
I Cut, You 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] |
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] |
Envy-free
Envy-freeness, also known as no-envy, is a criterion for fair division. It says that, when resources are allocated among people with equal rights, each person should receive a share that is, in their eyes, at least as good as the share received by any other agent. In other words, no person should feel envy. General definitions Suppose a certain resource is divided among several agents, such that every agent i receives a share X_i. Every agent i has a personal preference relation \succeq_i over different possible shares. The division is called envy-free (EF) if for all i and j: :::X_i \succeq_i X_j Another term for envy-freeness is no-envy (NE). If the preference of the agents are represented by a value functions V_i, then this definition is equivalent to: :::V_i(X_i) \geq V_i(X_j) Put another way: we say that agent i ''envies'' agent j if i prefers the piece of j over his own piece, i.e.: :::X_i \prec_i X_j :::V_i(X_i) 2 the problem is much harder. See envy-free cake-cutting. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Efficiency (economics)
In microeconomics, economic efficiency, depending on the context, is usually one of the following two related concepts: * Allocative or Pareto efficiency: any changes made to assist one person would harm another. * Productive efficiency: no additional output of one good can be obtained without decreasing the output of another good, and production proceeds at the lowest possible average total cost. These definitions are not equivalent: a market or other economic system may be allocatively but not productively efficient, or productively but not allocatively efficient. There are also other definitions and measures. All characterizations of economic efficiency are encompassed by the more general engineering concept that a system is efficient or optimal when it maximizes desired outputs (such as utility) given available inputs. Standards of thought There are two main standards of thought on economic efficiency, which respectively emphasize the distortions created by ''governments'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stromquist Moving-knives Procedure
The Stromquist moving-knives procedure is a procedure for envy-free cake-cutting among three players. It is named after Walter Stromquist who presented it in 1980. This procedure was the first envy-free moving knife procedure devised for three players. It requires four knives but only two cuts, so each player receives a single connected piece. There is no natural generalization to more than three players which divides the cake without extra cuts. The resulting partition is not necessarily efficient. Procedure A referee moves a sword from left to right over the cake, hypothetically dividing it into small left piece and a large right piece. Each player moves a knife over the right piece, always keeping it parallel to the sword. The players must move their knives in a continuous manner, without making any "jumps".The importance of this continuity is explained here: When any player shouts "cut", the cake is cut by the sword and by whichever of the players' knives happens to be the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Austin Moving-knife Procedures
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] |
