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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 . The leftmost cut is selected. Suppose this cut belongs to Alice. So Alice receives the leftmost piece and her value is exactly . The remainder has to be divided between the remaining partners (Bob and Carl). Alice's part is worth ''at most'' and the remainder is worth ''at least'' 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 sure Alice won't envy an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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. Stromquist procedure Simpler version In a simpler version of the problem, a division is regarded as "fair" if all people ("players") are satisfied that each has received at least 1/ n (here n = 3) of the cake. For this version, there is a simple and practical solution, attributed by Steinhaus to Banach and Knaster. Procedure for ther simpler version A referee moves a sword from left to right over the cake, hypothetically dividing it into small left piece and a large right pi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Intermediate Value Theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two important corollaries: # If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). # The image of a continuous function over an interval is itself an interval. Motivation This captures an intuitive property of continuous functions over the real numbers: given ''f'' continuous on ,2/math> with the known values f(1) = 3 and f(2) = 5, then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper. Theorem The intermediate value theorem states the following: Consider the closed interval I = ,b/math> ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
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] [Amazon] |
Pancake Theorem
A pancake, also known as a hotcake, griddlecake, or flapjack, is a flat type of batter bread like cake, often thin and round, prepared from a starch-based batter that may contain eggs, milk, and butter, and then cooked on a hot surface such as a griddle or frying pan. Archaeological evidence suggests that pancakes were probably eaten in prehistoric societies. The pancake's shape and structure varies worldwide. In England, pancakes are often unleavened and resemble a crêpe. In Scotland and North America, a leavening agent is used (typically baking powder) creating a thick fluffy pancake. A ''crêpe'' is a thin pancake of Breton origin cooked on one or both sides in a special pan or crepe maker to achieve a lacelike network of fine bubbles. A well-known variation originating from southeast Europe is palatschinke, a thin moist pancake fried on both sides and filled with jam, cream cheese, chocolate, or ground walnuts, but many other fillings—sweet or savoury—can also be u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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. Fairs showcase a wide range of goods, products, and services, and often include competitions, exhibitions, and educational activities. Fairs can be thematic, focusing on specific industries or interests. Types Variations of fairs include: * Art fairs, including art exhibitions and arts festivals * Book Fairs in communities and schools provide an opportunity for readers, writers, publishers to come together and celebrate literature. * County fair (US) 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 even ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
