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Individual Pieces Set
In the theory of fair cake-cutting, the individual-pieces set (IPS) is a geometric object that represents all possible utility vectors in cake partitions. Example Suppose we have a cake made of four parts. There are two people, Alice and George, with different tastes: each person values the different parts of the cake differently. The table below describes the parts and their values. The cake can be divided in various ways. Each division (Alice's-piece, George's-piece) yields a different utility vector (Alice's utility, George's utility). The IPS is the set of utility vectors of all possible partitions. The IPS for the example cake is shown on the right. Properties The IPS is a convex set and a compact set. This follows from the Dubins–Spanier theorems. With two agents, the IPS is symmetric across the middle point (in this case it is the point (15,15)). Take some int (x,y) on the IPS. This point comes from some partition. Swap the pieces between Alice and George. Then, ... [...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 believed 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 in the book of Book of Genesis, Genesis to resolve Abraham and Lot's conflict. This procedure solves the fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Individual Pieces Set
In the theory of fair cake-cutting, the individual-pieces set (IPS) is a geometric object that represents all possible utility vectors in cake partitions. Example Suppose we have a cake made of four parts. There are two people, Alice and George, with different tastes: each person values the different parts of the cake differently. The table below describes the parts and their values. The cake can be divided in various ways. Each division (Alice's-piece, George's-piece) yields a different utility vector (Alice's utility, George's utility). The IPS is the set of utility vectors of all possible partitions. The IPS for the example cake is shown on the right. Properties The IPS is a convex set and a compact set. This follows from the Dubins–Spanier theorems. With two agents, the IPS is symmetric across the middle point (in this case it is the point (15,15)). Take some int (x,y) on the IPS. This point comes from some partition. Swap the pieces between Alice and George. Then, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary (topology), boundary of a convex set in the plane is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval (mathematics), interval with the property that its epigraph (mathematics), epigraph (the set of points on or above the graph of a function, graph of the function) is a convex set. Convex minimization is a subfield of mathematical optimization, optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. One su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dubins–Spanier Theorems
The Dubins–Spanier theorems are several theorems in the theory of fair cake-cutting. They were published by Lester Dubins and Edwin Spanier in 1961. Although the original motivation for these theorems is fair division, they are in fact general theorems in measure theory. Setting There is a set U, and a set \mathbb which is a sigma-algebra of subsets of U. There are n partners. Every partner i has a personal value measure V_i: \mathbb \to \mathbb. This function determines how much each subset of U is worth to that partner. Let X a partition of U to k measurable sets: U = X_1 \sqcup \cdots \sqcup X_k. Define the matrix M_X as the following n\times k matrix: :M_X ,j= V_i(X_j) This matrix contains the valuations of all players to all pieces of the partition. Let \mathbb be the collection of all such matrices (for the same value measures, the same k, and different partitions): :\mathbb = \ The Dubins–Spanier theorems deal with the topological properties of \mathbb. State ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pareto Frontier
In multi-objective optimization, the Pareto front (also called Pareto frontier or Pareto curve) is the set of all Pareto efficient solutions. The concept is widely used in engineering. It allows the designer to restrict attention to the set of efficient choices, and to make tradeoffs within this set, rather than considering the full range of every parameter. Definition The Pareto frontier, ''P''(''Y''), may be more formally described as follows. Consider a system with function f: X \rightarrow \mathbb^m, where ''X'' is a compact set of feasible decisions in the metric space \mathbb^n, and ''Y'' is the feasible set of criterion vectors in \mathbb^m, such that Y = \. We assume that the preferred directions of criteria values are known. A point y^ \in \mathbb^m is preferred to (strictly dominates) another point y^ \in \mathbb^m, written as y^ \succ y^. The Pareto frontier is thus written as: : P(Y) = \. Marginal rate of substitution A significant aspect of the Pareto fronti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pareto Efficient
In welfare economics, a Pareto improvement formalizes the idea of an outcome being "better in every possible way". A change is called a Pareto improvement if it leaves at least one person in society better off without leaving anyone else worse off than they were before. A situation is called Pareto efficient or Pareto optimal if all possible Pareto improvements have already been made; in other words, there are no longer any ways left to make one person better off without making some other person worse-off. In social choice theory, the same concept is sometimes called the unanimity principle, which says that if ''everyone'' in a society ( non-strictly) prefers A to B, society as a whole also non-strictly prefers A to B. The Pareto front consists of all Pareto-efficient situations. In addition to the context of efficiency in ''allocation'', the concept of Pareto efficiency also arises in the context of ''efficiency in production'' vs. '' x-inefficiency'': a set of outputs of go ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weller's Theorem
Weller's theorem is a theorem in economics. It says that a heterogeneous resource ("cake") can be divided among ''n'' partners with different valuations in a way that is both Pareto-efficient (PE) and envy-free (EF). Thus, it is possible to divide a cake fairly without compromising on economic efficiency. Moreover, Weller's theorem says that there exists a price such that the allocation and the price are a competitive equilibrium (CE) with equal incomes (EI). Thus, it connects two research fields which were previously unrelated: fair cake-cutting and general equilibrium. Background Fair cake-cutting has been studied since the 1940s. There is a heterogeneous divisible resource, such as a cake or a land-estate. There are ''n'' partners, each of whom has a personal value-density function over the cake. The value of a piece to a partner is the integral of his value-density over that piece (this means that the value is a nonatomic measure over the cake). The envy-free cake-cutting pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julius Barbanel
Julius may refer to: People * Julius (name), a masculine given name and surname (includes a list of people with the name) * Julius (nomen), the name of a Roman family (includes a list of Ancient Romans with the name) ** Julius Caesar (100–44 BC), Roman military and political leader and one of the most influential men of classical antiquity * Julius (judge royal) (fl. before 1135), noble in the Kingdom of Hungary * Julius, Count of Lippe-Biesterfeld (1812–1884), German noble * Julius, Duke of Brunswick-Lüneburg (1528–1589), German noble Arts and entertainment * Julius (''Everybody Hates Chris''), a character from the American sitcom * "Julius" (song), by Phish, 1994 Other uses * Julius (chimpanzee), a chimpanzee at Kristiansand Zoo and Amusement Park in Norway * Julius (month), the month of the ancient Roman calendar originally called ''Quintilis'' and renamed for Julius Caesar * Julius (restaurant), a tavern in Greenwich Village, New York City * Julius (software), a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radon–Nikodym Set
In the theory of fair cake-cutting, the Radon–Nikodym set (RNS) is a geometric object that represents a cake, based on how different people evaluate the different parts of the cake. Example Suppose we have a cake made of four parts. There are two people, Alice and George, with different tastes: each person values the different parts of the cake differently. The table below describes the parts and their values; the last row, "RNS Point", is explained afterwards. The "RNS point" of a piece of cake describes the relative values of the partners to that piece. It has two coordinates – one for Alice and one for George. For example: * The partners agree on the values for the chocolate part, so the coordinates of its RNS point are also equal (they are normalized such that their sum is 1). * The lemon part is only valuable for Alice, so in its RNS point, only Alice's coordinate is 1 while George's coordinate is 0. * In both the vanilla and the cherries part, the ratio between Alice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cake-cutting
Cake-cutting may refer to: * 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 ..., a mathematical problem of fairly dividing a heterogenous resource among people with different preferences ** Efficient cake-cutting, a similar division problem in economics and computer science * Wedding-cake cutting, the habit of cutting the wedding cake and distributing it to the guests, as a symbol of fertility {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
