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Proportional Cake-cutting
A proportional cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the proportionality criterion, namely, that every partner feels that his allocated share is worth at least 1/''n'' of the total. Two assumptions are usually made when proportionality is discussed: * The valuations of the partners are ''non-atomic'', i.e., there are no indivisible elements with positive value. * The valuations of the partners are ''additive'', i.e., when a piece is divided, the value of the piece is equal to the sum of its parts. Formal definitions The cake is denoted by C. There are n people. Each person i has a value function V_i. A partition of the cake, X_1\sqcup \cdots \sqcup X_n = C, is called ''proportional'' if:V_i(X_i) \ge V_i(C)/n for every person i \in \. Procedures For two people, divide and choose is the classic solution. One person divides the resource into what they believe are equal halves, and the other person ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Cake-cutting
Fair cake-cutting is a kind of fair division problem. The problem involves a ''heterogeneous'' resource, such as a cake with different toppings, that is assumed to be ''divisible'' – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible. The division should be ''unanimously'' fair - each person should receive a piece that he or she believes to be a fair share. The "cake" is only a metaphor; procedures for fair cake-cutting can be used to divide various kinds of resources, such as land estates, advertisement space or broadcast time. The prototypical procedure for fair cake-cutting is divide and choose, which is mentioned already in the book of Genesis. It solves the fair division problem for two people. The modern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proportional Cake-cutting With Different Entitlements
In the fair cake-cutting problem, the partners often have different entitlements. For example, the resource may belong to two shareholders such that Alice holds 8/13 and George holds 5/13. This leads to the criterion of ''weighted proportionality'' (WPR): there are several weights w_i that sum up to 1, and every partner i should receive at least a fraction w_i of the resource by their own valuation. In contrast, in the simpler proportional cake-cutting setting, the weights are equal: w_i=1/n for all i Several algorithms can be used to find a WPR division. Cloning Suppose all the weights are rational numbers, with common denominator D. So the weights are p_1/D,\dots,p_n/D, with p_1+\cdots+p_n=D. For each player i, create p_i clones with the same value-measure. The total number of clones is D. Find a proportional cake allocation among them. Finally, give each partner i the pieces of his p_i clones. Robertson and Webb show a simpler procedure for two partners: Alice cuts the cake ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Division
Exact division, also called consensus 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 ''k''>1 - the cake should be partition ... [...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] |
Super-proportional Division
A strongly-proportional division (sometimes called super-proportional division) is a kind of a fair division. It is a division of resources among ''n'' partners, in which the value received by each partner is strictly more than his/her due share of 1/''n'' of the total value. Formally, in a strongly-proportional division of a resource ''C'' among ''n'' partners, each partner ''i'', with value measure ''Vi'', receives a share ''Xi'' such thatV_i(X_i) > V_i(C)/n.Obviously, a strongly-proportional division does not exist when all partners have the same value measure. The best condition that can ''always'' be guaranteed is V_i(X_i) \geq V_i(C)/n, which is the condition for a plain proportional division. However, one may hope that, when different agents have different valuations, it may be possible to use this fact for the benefit of all players, and give each of them strictly more than their due share. Existence In 1948, Hugo Steinhaus conjectured the existence of a super-proportiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Division
Exact division, also called consensus 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 ''k''>1 - the cake should be partition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Economically Efficient
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] |
Fat Object (geometry)
In geometry, a fat object is an object in two or more dimensions, whose lengths in the different dimensions are similar. For example, a square is fat because its length and width are identical. A 2-by-1 rectangle is thinner than a square, but it is fat relative to a 10-by-1 rectangle. Similarly, a circle is fatter than a 1-by-10 ellipse and an equilateral triangle is fatter than a very obtuse triangle. Fat objects are especially important in computational geometry. Many algorithms in computational geometry can perform much better if their input consists of only fat objects; see the applications section below. Global fatness Given a constant ''R''≥1, an object ''o'' is called ''R''-fat if its "slimness factor" is at most ''R''. The "slimness factor" has different definitions in different papers. A common definition is: :\frac where ''o'' and the cubes are ''d''-dimensional. A 2-dimensional cube is a square, so the slimness factor of a square is 1 (since its smallest enclosing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fat Rectangle
In geometry, a fat object is an object in two or more dimensions, whose lengths in the different dimensions are similar. For example, a square is fat because its length and width are identical. A 2-by-1 rectangle is thinner than a square, but it is fat relative to a 10-by-1 rectangle. Similarly, a circle is fatter than a 1-by-10 ellipse and an equilateral triangle is fatter than a very obtuse triangle. Fat objects are especially important in computational geometry. Many algorithms in computational geometry can perform much better if their input consists of only fat objects; see the applications section below. Global fatness Given a constant ''R''≥1, an object ''o'' is called ''R''-fat if its "slimness factor" is at most ''R''. The "slimness factor" has different definitions in different papers. A common definition is: :\frac where ''o'' and the cubes are ''d''-dimensional. A 2-dimensional cube is a square, so the slimness factor of a square is 1 (since its smallest enclos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hill–Beck Land Division Problem
The following variant of the fair cake-cutting problem was introduced by Ted Hill in 1983. There is a territory ''D'' adjacent to ''n'' countries. Each country values the different subsets of ''D'' differently. The countries would like to divide ''D'' fairly among them, where "fair" means a proportional division. Additionally, ''the share allocated to each country must be connected and adjacent to that country''. This geographic constraint distinguishes this problem from classic fair cake-cutting. Formally, every country ''Ci'' must receive a disjoint piece of ''D'', marked ''Di'', such that a portion of the border between ''Ci'' and ''D'' is contained in the interior of ''Ci ∪ Di''. Impossibility and possibility There are cases in which the problem cannot be solved: # If there is a single point to which two countries assign all their value (e.g. a holy place), then obviously the territory cannot be divided proportionally. To prevent such situations, we assume that all count ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Mapping
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in U if it preserves angles between directed curves through u_0, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix ( orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix. For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Super-proportional Division
A strongly-proportional division (sometimes called super-proportional division) is a kind of a fair division. It is a division of resources among ''n'' partners, in which the value received by each partner is strictly more than his/her due share of 1/''n'' of the total value. Formally, in a strongly-proportional division of a resource ''C'' among ''n'' partners, each partner ''i'', with value measure ''Vi'', receives a share ''Xi'' such thatV_i(X_i) > V_i(C)/n.Obviously, a strongly-proportional division does not exist when all partners have the same value measure. The best condition that can ''always'' be guaranteed is V_i(X_i) \geq V_i(C)/n, which is the condition for a plain proportional division. However, one may hope that, when different agents have different valuations, it may be possible to use this fact for the benefit of all players, and give each of them strictly more than their due share. Existence In 1948, Hugo Steinhaus conjectured the existence of a super-proportiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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