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Envy-graph Procedure
The envy-graph procedure (also called the envy-cycles procedure) is a procedure for fair item allocation. It can be used by several people who want to divide among them several discrete items, such as heirlooms, sweets, or seats in a class. Ideally, we would like the allocation to be envy-free (EF). i.e., to give each agent a bundle that he/she prefers over the bundles of all other agents. However, the items are discrete and cannot be cut, so an envy-free assignment might be impossible (for example, consider a single item and two agents). The envy-graph procedure aims to achieve the "next-best" option -- ''envy-freeness up to at most a single good'' (EF1): it finds an allocation in which the envy of every person towards every other person is bounded by the maximum marginal utility it derives from a single item. In other words, for every two people ''i'' and ''j'', there exists an item such that, if that item is removed, ''i'' does not envy ''j''. The procedure was presented by Lip ... [...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] |
Envy-free
Envy-freeness, also known as no-envy, is a criterion for fair division. It says that, when resources are allocated among people with equal rights, each person should receive a share that is, in their eyes, at least as good as the share received by any other agent. In other words, no person should feel envy. General definitions Suppose a certain resource is divided among several agents, such that every agent i receives a share X_i. Every agent i has a personal preference relation \succeq_i over different possible shares. The division is called envy-free (EF) if for all i and j: :::X_i \succeq_i X_j Another term for envy-freeness is no-envy (NE). If the preference of the agents are represented by a value functions V_i, then this definition is equivalent to: :::V_i(X_i) \geq V_i(X_j) Put another way: we say that agent i ''envies'' agent j if i prefers the piece of j over his own piece, i.e.: :::X_i \prec_i X_j :::V_i(X_i) 2 the problem is much harder. See envy-free cake-cutting. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Utility
In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value u(x_i) of one index ''u'', occurring at any quantity x_i of the goods bundle being evaluated, the corresponding value v(x_i) of the other index ''v'' satisfies a relationship of the form :v(x_i) = au(x_i) + b\!, for fixed constants ''a'' and ''b''. Thus the utility functions themselves are related by :v(x) = au(x) + b. The two indices differ only with respect to scale and origin. Thus if one is concave, so is the other, in which case there is often said to be diminishing marginal utility. Thus the use of cardinal utility imposes the assumption that levels of absolute satisfaction exist, so that the magnitudes of increments to satisfaction can be compared across different situations. In consumer choice theory, ordinal utility with its weaker assumption ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonic Function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is called ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent Goods
Independent goods are goods that have a zero cross elasticity of demand. Changes in the price of one good will have no effect on the demand for an independent good. Thus independent goods are neither complements nor substitutes. For example, a person's demand for nails is usually independent of his or her demand for bread, since they are two unrelated types of goods. Note that this concept is subjective and depends on the consumer's personal utility function. A Cobb-Douglas utility function implies that goods are independent. For goods in quantities ''X''1 and ''X''2, prices ''p''1 and ''p''2, income ''m'', and utility function parameter ''a'', the utility function : u(X_1, X_2) = X_1^a X_2^, when optimized subject to the budget constraint that expenditure on the two goods cannot exceed income, gives rise to this demand function for good 1: X_1= am/p_1, which does not depend on ''p''2. See also * Consumer theory * Good (economics and accounting) In economics, goods are i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Cycle
Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''Director'' (Avant album) (2006) * ''Director'' (Yonatan Gat album) Occupations and positions Arts and design * Animation director * Artistic director * Creative director * Design director * Film director * Music director * Music video director * Sports director * Television director * Theatre director Positions in other fields * Director (business), a senior level management position * Director (colonial), head of chartered company's colonial administration in a territory * Director (education), head of a university or other educational body * Company director * Cruise director * Executive director * Finance director or chief financial officer * Funeral director * Managing director * Non-executive director * Technical director * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pair where * ''V'' is a set whose elements are called '' vertices'', ''nodes'', or ''points''; * ''A'' is a set of ordered pairs of vertices, called ''arcs'', ''directed edges'' (sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A''), ''arrows'', or ''directed lines''. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called ''edges'', ''links'' or ''lines''. The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Depth-first Search
Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures. The algorithm starts at the root node (selecting some arbitrary node as the root node in the case of a graph) and explores as far as possible along each branch before backtracking. Extra memory, usually a stack, is needed to keep track of the nodes discovered so far along a specified branch which helps in backtracking of the graph. A version of depth-first search was investigated in the 19th century by French mathematician Charles Pierre Trémaux as a strategy for solving mazes. Properties The time and space analysis of DFS differs according to its application area. In theoretical computer science, DFS is typically used to traverse an entire graph, and takes time where , V, is the number of vertices and , E, the number of edges. This is linear in the size of the graph. In these applications it also uses space O(, V, ) in the worst case to store the stack of vertices on th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Round-robin Item Allocation
Round robin is a procedure for fair item allocation. It can be used to allocate several indivisible items among several people, such that the allocation is "almost" envy-free: each agent believes that the bundle he received is at least as good as the bundle of any other agent, when at most one item is removed from the other bundle. In sports, the round-robin procedure is called a draft. Setting There are ''m'' objects to allocate, and ''n'' people ("agents") with equal rights to these objects. Each person has different preferences over the objects. The preferences of an agent are given by a vector of values - a value for each object. It is assumed that the value of a bundle for an agent is the sum of the values of the objects in the bundle (in other words, the agents' valuations are an additive set function on the set of objects). Description The protocol proceeds as follows: # Number the people arbitrarily from 1 to n; # While there are unassigned objects: #* Let each per ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Assignment Valuation
In economics, assignment valuation is a kind of a utility function on sets of items. It was introduced by Shapley and further studied by Lehmann, Lehmann and Nisan, who use the term OXS valuation (not to be confused with XOS valuation). 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 whol ... in this setting was studied by Benabbou, Chakraborty, Elkind, Zick and Igarashi. Assignment valuations correspond to preferences of groups. In each group, there are several individuals; each individual attributes a certain numeric value to each item. The assignment-valuation of the group to a set of items ''S'' is the value of the maximum weight matching of the items in ''S'' to the individuals in the group. The assignment valuations are a subset of the submodular valuations. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envy-free Item Allocation
Envy-free (EF) item allocation is a fair item allocation problem, in which the fairness criterion is envy-freeness - each agent should receive a bundle that they believe to be at least as good as the bundle of any other agent. Since the items are indivisible, an EF assignment may not exist. The simplest case is when there is a single item and at least two agents: if the item is assigned to one agent, the other will envy. One way to attain fairness is to use monetary transfers; see Fair allocation of items and money. When monetary transfers are not allowed or not desired, there are allocation algorithms providing various kinds of relaxations. Finding an envy-free allocation whenever it exists Preference-orderings on bundles: envy-freeness The undercut procedure finds a complete EF allocation for two agents, if-and-only-if such allocation exists. It requires the agents to rank bundles of items, but it does not require cardinal utility information. It works whenever the agents' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
