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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] |
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 divi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walter Stromquist
Walter may refer to: People * Walter (name), both a surname and a given name * Little Walter, American blues harmonica player Marion Walter Jacobs (1930–1968) * Gunther (wrestler), Austrian professional wrestler and trainer Walter Hahn (born 1987), who previously wrestled as "Walter" * Walter, standard author abbreviation for Thomas Walter (botanist) ( – 1789) Companies * American Chocolate, later called Walter, an American automobile manufactured from 1902 to 1906 * Walter Energy, a metallurgical coal producer for the global steel industry * Walter Aircraft Engines, Czech manufacturer of aero-engines Films and television * ''Walter'' (1982 film), a British television drama film * Walter Vetrivel, a 1993 Tamil crime drama film * ''Walter'' (2014 film), a British television crime drama * ''Walter'' (2015 film), an American comedy-drama film * ''Walter'' (2020 film), an Indian crime drama film * ''W*A*L*T*E*R'', a 1984 pilot for a spin-off of the TV series ''M*A*S*H'' * ''W ... [...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] |
Referee
A referee is an official, in a variety of sports and competition, responsible for enforcing the rules of the sport, including sportsmanship decisions such as ejection. The official tasked with this job may be known by a variety of other titles depending on the sport, including umpire, judge, arbiter (chess), commissaire, or technical official (by the International Olympic Committee). Referees may be assisted by umpires, linesmen, timekeepers, touch judges, or video review officials. Football (association) Originally team captains would consult each other in order to resolve any dispute on the pitch. Eventually this role was delegated to an ''umpire''. Each team would bring their own partisan umpire allowing the team captains to concentrate on the game. Later, the referee, a third "neutral" official was added; this ''referee'' would be "referred to" if the umpires could not resolve a dispute. The referee did not take his place on the pitch until 1891, when the umpires ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chore Division
Chore division is a fair division problem in which the divided resource is undesirable, so that each participant wants to get as little as possible. It is the mirror-image of the fair cake-cutting problem, in which the divided resource is desirable so that each participant wants to get as much as possible. Both problems have heterogeneous resources, meaning that the resources are nonuniform. In cake division, cakes can have edge, corner, and middle pieces along with different amounts of frosting. Whereas in chore division, there are different chore types and different amounts of time needed to finish each chore. Similarly, both problems assume that the resources are divisible. Chores can be infinitely divisible, because the finite set of chores can be partitioned by chore or by time. For example, a load of laundry could be partitioned by the number of articles of clothing and/or by the amount of time spent loading the machine. The problems differ, however, in the desirability of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Pie-cutting
The fair pie-cutting problem is a variation of the fair cake-cutting problem, in which the resource to be divided is circular. As an example, consider a birthday cake shaped as a disk. The cake should be divided among several children such that no child envies another child (as in a standard cake-cutting problem), with the additional constraint that the cuts must be radial, so that each child receives a circular sector. A possible application of the pie model might be for dividing an island’s shoreline into connected lots. Another possible application is in division of periodic time, such as dividing a daily cycle into "on-call" periods. Model A pie is usually modeled as the 1-dimensional interval ,2π(or ,1, in which the two endpoints are identified. This model was introduced in 1985 and later in 1993. Every procedure for fair cake-cutting can also be applied to cutting a pie by just ignoring the fact that the two endpoints are identified. For example, if the cake-cuttin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robertson–Webb Rotating-knife Procedure
The Robertson–Webb rotating-knife procedure is a procedure for envy-free cake-cutting of a two-dimensional cake among three partners. It makes only two cuts, so each partner receives a single connected piece. Its main advantage over the earlier Stromquist moving-knives procedure and the later Barbanel–Brams moving-knives procedure is that it requires only a single moving-knife. This advantage uses the two-dimensional nature of the cake. Procedure Initially, each partner makes a vertical cut such that the cake to its left is worth for him exactly 1/3. The leftmost cut is selected. Suppose this cut belongs to Alice. So Alice receives the leftmost piece and her value is exactly 1/3. The remainder has to be divided between the remaining partners (Bob and Carl). Note that Alice's part is worth ''at most'' 1/3 and the remainder is worth ''at least'' 2/3 for Bob and Carl. So, if Bob and Carl each receive at least half of the remainder, they do not envy. The challenge is to make sur ... [...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] |
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, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
