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Envy-free Pricing
Envy-free pricing is a kind of fair item allocation. There is a single seller that owns some items, and a set of buyers who are interested in these items. The buyers have different valuations to the items, and they have a quasilinear utility function; this means that the utility an agent gains from a bundle of items equals the agent's value for the bundle minus the total price of items in the bundle. The seller should determine a price for each item, and sell the items to some of the buyers, such that there is Envy-freeness, ''no envy''. Two kinds of envy are considered: * ''Agent envy'' means that some agent assigns a higher utility (a higher difference value-price) to a bundle allocated to another agent. * ''Market envy'' means that some agent assigns a higher utility (a higher difference value-price) to any bundle. The no-envy conditions guarantee that the market is stable and that the buyers do not resent the seller. By definition, every market envy-free allocation is also agent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Item Allocation
Fair item allocation is a kind of a fair division problem in which the items to divide are ''discrete'' rather than continuous. The items have to be divided among several partners who value them differently, and each item has to be given as a whole to a single person. This situation arises in various real-life scenarios: * Several heirs want to divide the inherited property, which contains e.g. a house, a car, a piano and several paintings. * Several lecturers want to divide the courses given in their faculty. Each lecturer can teach one or more whole courses. *White elephant gift exchange parties The indivisibility of the items implies that a fair division may not be possible. As an extreme example, if there is only a single item (e.g. a house), it must be given to a single partner, but this is not fair to the other partners. This is in contrast to the fair cake-cutting problem, where the dividend is divisible and a fair division always exists. In some cases, the indivisibility pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractionally Subadditive
A set function is called fractionally subadditive (or XOS) if it is the maximum of several additive set functions. This valuation class was defined, and termed XOS, by Noam Nisan, in the context of combinatorial auctions. The term fractionally-subadditive was given by Uriel Feige. Definition There is a finite base set of items, M := \. There is a function v which assigns a number to each subset of M. The function v is called ''fractionally-subadditive'' (or XOS) if there exists a collection of set functions, \, such that: * Each a_j is additive, ''i.e.'', it assigns to each subset X\subseteq M, the sum of the values of the items in X. * The function v is the pointwise maximum of the functions a_j. I.e, for every subset X\subseteq M: :v(X) = \max_^l a_j(X) Equivalent Definition The name fractionally subadditive comes from the following equivalent definition: a set function v is ''fractionally subadditive'' if, for any S\subseteq M and any collection \_^k with \alpha_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demand Oracle
In algorithmic game theory, a branch of both computer science and economics, a demand oracle is a function that, given a price-vector, returns the demand of an agent. It is used by many algorithms related to pricing and optimization in online market. It is usually contrasted with a value oracle, which is a function that, given a set of items, returns the value assigned to them by an agent. Demand The demand of an agent is the bundle of items that the agent most prefers, given some fixed prices of the items. As an example, consider a market with three objects and one agent, with the following values and prices. Suppose the agent's utility function is additive (= the value of a bundle is the sum of values of the items in the bundle), and quasilinear (= the utility of a bundle is the value of the bundle minus its price). Then, the demand of the agent, given the prices, is the set , which gives a utility of (4+6)-(3+1) = 6. Every other set gives the agent a smaller utility. For exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unique Coverage
Unique primarily refers to: * Uniqueness, a state or condition wherein something is unlike anything else *In mathematics and logic, a unique object is the only object with a certain property, see Uniqueness quantification Unique may also refer to: Companies *Unique Art, an American toy company *Unique Broadcasting Company, a former name of UBC Media Group, based in London *Unique Business News a television news channel in Taiwan *Unique Mobility, a former name of UQM Technologies, a manufacturing company based in the United States *Unique Pub Company, a pub company based in the United Kingdom, acquired by Enterprise Inns * Unique Theater, a theater in Minneapolis, Minnesota, United States *Unique Group, a conglomerate in Bangladesh Music * ''Unique'' (DJ Encore album) * ''Unique'' (Juliette Schoppmann album) * Unique (band), a musical group from New York City *Unique (also known as Darren Styles), British musician *Unique Records, a former name of RKO/Unique Records * Unique ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truthful Mechanism
In game theory, an asymmetric game where players have private information is said to be strategy-proof or strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information, i.e. given no information about what the others do, you fare best or at least not worse by being truthful. SP is also called truthful or dominant-strategy-incentive-compatible (DSIC), to distinguish it from other kinds of incentive compatibility. An SP game is not always immune to collusion, but its robust variants are; with group strategyproofness no group of people can collude to misreport their preferences in a way that makes every member better off, and with strong group strategyproofness no group of people can collude to misreport their preferences in a way that makes at least one member of the group better off without making any of the remaining members worse off. Examples Typical examples of SP mechanisms are majority voting between two alternatives, second- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prior-free Mechanism
A prior-free mechanism (PFM) is a mechanism in which the designer does not have any information on the agents' valuations, not even that they are random variables from some unknown probability distribution. A typical application is a seller who wants to sell some items to potential buyers. The seller wants to price the items in a way that will maximize his profit. The optimal prices depend on the amount that each buyer is willing to pay for each item. The seller does not know these amounts, and cannot even assume that the amounts are drawn from a probability distribution. The seller's goal is to design an auction that will produce a reasonable profit even in worst-case scenarios. PFMs should be contrasted with two other mechanism types: * Bayesian-optimal mechanisms (BOM) assume that the agents' valuations are drawn from a known probability distribution. The mechanism is tailored to the parameters of this distribution (e.g, its median or mean value). * Prior-independent mechanis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terminology of both linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory. Definition There are many equivalent ( cryptomorphic) ways to define a (finite) matroid.A standard source for basic definitions and results about matroids is Oxley (1992). An older standard source is Welsh (1976). See Brylawsk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independence System
In combinatorics, combinatorial mathematics, an independence system is a pair (V, \mathcal), where is a finite Set (mathematics), set and is a collection of subsets of (called the independent sets or feasible sets) with the following properties: # The empty set is independent, i.e., \emptyset\in\mathcal. (Alternatively, at least one subset of is independent, i.e., \mathcal\neq\emptyset.) # Every subset of an independent set is independent, i.e., for each Y\subseteq X, we have X\in\mathcal\Rightarrow Y\in\mathcal. This is sometimes called the hereditary property, or downward-closedness. Another term for an independence system is an abstract simplicial complex. Relation to other concepts * A pair (V, \mathcal), where is a finite Set (mathematics), set and is a collection of subsets of is also called a hypergraph. When using this terminology, the elements in the set are called vertices and elements in the family are called hyperedges. So an independence system can be defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle Inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If , , and are the lengths of the sides of the triangle, with no side being greater than , then the triangle inequality states that :z \leq x + y , with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths ( norms): :\, \mathbf x + \mathbf y\, \leq \, \mathbf x\, + \, \mathbf y\, , where the length of the third side has been replaced by the vector sum . When and are real numbers, they can be viewed as vectors in , and the trian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric (mathematics)
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Demand
In economics, a unit demand agent is an agent who wants to buy a single item, which may be of one of different types. A typical example is a buyer who needs a new car. There are many different types of cars, but usually a buyer will choose only one of them, based on the quality and the price. If there are ''m'' different item-types, then a unit-demand valuation function is typically represented by ''m'' values v_1,\dots,v_m, with v_j representing the subjective value that the agent derives from item j. If the agent receives a set A of items, then his total utility is given by: :u(A)=\max_v_j since he enjoys the most valuable item from A and ignores the rest. Therefore, if the price of item j is p_j, then a unit-demand buyer will typically want to buy a single item – the item j for which the net utility v_j - p_j is maximized. Ordinal and cardinal definitions A unit-demand valuation is formally defined by: * For a preference relation: for every set B there is a subset A\subsete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
