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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 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] |
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] |
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] |
Barbanel–Brams Moving-knives Procedure
The Barbanel–Brams rotating-knife procedure is a procedure for envy-free cake-cutting of a cake among three partners.Section 2 in It makes only two cuts, so each partner receives a single connected piece. Its main advantage over the earlier Stromquist moving-knives procedure is that it requires only two moving knives, instead of four. The earlier Robertson–Webb rotating-knife procedure requires only one moving knife, but it works only for a two-dimensional cake, while the Barbanel–Brams procedure works also for a one-dimensional cake. Procedure Initially, each partner marks a point such that the cake to its left is worth for them exactly 1/3. The leftmost mark is selected. Suppose this mark belongs to Alice. Alice is then asked to mark another point such that the cake to its left is worth for her exactly 2/3. So now the cake is divided to three pieces that are equal for Alice. Bob and Carl are asked to evaluate the two rightmost pieces. There are several cases: # Each o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 an interval I = ,b/math> of real n ... [...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] |
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] |
Pancake Theorem
In mathematical measure theory, for every positive integer the ham sandwich theorem states that given measurable "objects" in -dimensional Euclidean space, it is possible to divide each one of them in half (with respect to their measure, e.g. volume) with a single -dimensional hyperplane. This is even possible if the objects overlap. It was proposed by Hugo Steinhaus and proved by Stefan Banach (explicitly in dimension 3, without taking the trouble to state the theorem in the -dimensional case), and also years later called the Stone–Tukey theorem after Arthur H. Stone and John Tukey. Naming The ham sandwich theorem takes its name from the case when and the three objects to be bisected are the ingredients of a ham sandwich. Sources differ on whether these three ingredients are two slices of bread and a piece of ham , bread and cheese and ham , or bread and butter and ham . In two dimensions, the theorem is known as the pancake theorem to refer to the flat nature of the two ... [...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] |
