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Max-min Item Allocation
Egalitarian item allocation, also called max-min item allocation is a fair item allocation problem, in which the fairness criterion follows the egalitarian rule. The goal is to maximize the minimum value of an agent. That is, among all possible allocations, the goal is to find an allocation in which the smallest value of an agent is as large as possible. In case there are two or more allocations with the same smallest value, then the goal is to select, from among these allocations, the one in which the second-smallest value is as large as possible, and so on (by the leximin order). Therefore, an egalitarian item allocation is sometimes called a leximin item allocation. The special case in which the value of each item ''j'' to each agent is either 0 or some constant ''vj'' is called the santa claus problem: santa claus has a fixed set of gifts, and wants to allocate them among children such that the least-happy child is as happy as possible. Some related problems are: * ''Multiwa ... [...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] |
Fractional Matching
In graph theory, a fractional matching is a generalization of a matching in which, intuitively, each vertex may be broken into fractions that are matched to different neighbor vertices. Definition Given a graph ''G'' = (''V'', ''E''), a fractional matching in ''G'' is a function that assigns, to each edge ''e'' in ''E'', a fraction ''f''(''e'') in , 1 such that for every vertex ''v'' in ''V'', the sum of fractions of edges adjacent to ''v'' is at most 1: \forall v\in V: \sum_f(e)\leq 1 A matching in the traditional sense is a special case of a fractional matching, in which the fraction of every edge is either 0 or 1: ''f''(''e'') = 1 if ''e'' is in the matching, and ''f''(''e'') = 0 if it is not. For this reason, in the context of fractional matchings, usual matchings are sometimes called ''integral matchings''. The size of an integral matching is the number of edges in the matching, and the matching number \nu(G) of a graph ''G'' is the largest size of a matching in ''G''. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead. Combinatorial optimization is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, VLSI, applied mathematics and theoretical computer science. Some research literature considers discrete o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid Rank Function
In the mathematical theory of matroids, the rank of a matroid is the maximum size of an independent set in the matroid. The rank of a subset ''S'' of elements of the matroid is, similarly, the maximum size of an independent subset of ''S'', and the rank function of the matroid maps sets of elements to their ranks. The rank function is one of the fundamental concepts of matroid theory via which matroids may be axiomatized. Matroid rank functions form an important subclass of the submodular set functions. The rank functions of matroids defined from certain other types of mathematical object such as undirected graphs, matrices, and field extensions are important within the study of those objects. Examples In all examples, ''E'' is the base set of the matroid, and ''B'' is some subset of ''E''. * Let ''M'' be the free matroid, where the independent sets are all subsets of ''E''. Then the rank function of ''M'' is simply: ''r''(''B'') = , ''B'', . * Let ''M'' be a uniform matroid, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
EFX 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] |
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] |
Proportional Item Allocation
Proportional item allocation is a fair item allocation problem, in which the fairness criterion is proportionality - each agent should receive a bundle that they value at least as much as 1/''n'' of the entire allocation, where ''n'' is the number of agents. Since the items are indivisible, a proportional 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 have a value of 0, which is less than 1/2. Therefore, the literature considers various relaxations of the proportionality requirement. Proportional allocation An allocation of objects is called proportional (PROP) if every agent ''i'' values his bundle at least 1/''n'' of the total. Formally, for all ''i'' (where ''M'' is the set of all goods): * V_i(X_i) \geq V_i(M)/n. A proportional division may not exist. For example, if the number of people is larger than the number of items, then some people will get no item at all and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simultaneous Eating Algorithm
A simultaneous eating algorithm (SE) is an algorithm for allocating divisible objects among agents with ordinal preferences. "Ordinal preferences" means that each agent can rank the items from best to worst, but cannot (or does not want to) specify a numeric value for each item. The SE allocation satisfies SD-efficiency - a weak ordinal variant of Pareto-efficiency (it means that the allocation is Pareto-efficient for ''at least one'' vector of additive utility functions consistent with the agents' item rankings). SE is parametrized by the "eating speed" of each agent. If all agents are given the same eating speed, then the SE allocation satisfies SD-envy-freeness - a strong ordinal variant of envy-freeness (it means that the allocation is envy-free for ''all'' vectors of additive utility functions consistent with the agents' item rankings). This particular variant of SE is called the Probabilistic Serial rule (PS). SE was developed by Hervé Moulin and Anna Bogomolnaia as a sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decreasing Demand Procedure
The Decreasing Demand procedure is a procedure for fair item allocation. It yields a Pareto-efficient division that maximizes the rank of the agent with the lowest rank. This corresponds to the Rawlsian justice criterion of taking care of the worst-off agent. The procedure was developed by Dorothea Herreiner and Clemens Puppe. Description Each agent is supposed to have a linear ranking on all bundles of items. The agents are queried in a round-robin fashion: each agent, in turn, reports his next bundle in the ranking, going from the best to the worst. After each report, the procedure checks whether it is possible to construct a complete partition of the items based on the reports made so far. If it is possible, then the procedure stops and returns one such partition. If there is more than one partition, then a Pareto-efficient one is returned. The procedure produces "balanced" allocations, that is, allocations which maximize the rank in the preference ordering of the bund ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Utilities
In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to ask ''how much'' better it is or how good it is. All of the theory of consumer decision-making under conditions of certainty can be, and typically is, expressed in terms of ordinal utility. For example, suppose George tells us that "I prefer A to B and B to C". George's preferences can be represented by a function ''u'' such that: :u(A)=9, u(B)=8, u(C)=1 But critics of cardinal utility claim the only meaningful message of this function is the order u(A)>u(B)>u(C); the actual numbers are meaningless. Hence, George's preferences can also be represented by the following function ''v'': :v(A)=9, v(B)=2, v(C)=1 The functions ''u'' and ''v'' are ordinally equivalent – they represent George's preferences equally well. Ordinal utility contrasts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submodular Set Function
In mathematics, a submodular set function (also known as a submodular function) is a set function whose value, informally, has the property that the difference in the incremental value of the function that a single element makes when added to an input set decreases as the size of the input set increases. Submodular functions have a natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory (as functions modeling user preferences) and electrical networks. Recently, submodular functions have also found immense utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, active learning, sensor placement, image collection summarization and many other domains. Definition If \Omega is a finite set, a submodular function is a set function f:2^\rightarrow \mathbb, where 2^\Omega denotes the power set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjusted Winner Procedure
Adjusted Winner (AW) is a procedure for envy-free item allocation. Given two agents and some goods, it returns a partition of the goods between the two agents with the following properties: # Envy-freeness: Each agent believes that his share of the goods is at least as good as the other share; # Equitable division, Equitability: The "relative happiness levels" of both agents from their shares are equal; # Pareto-optimality: no other allocation is better for one agent and at least as good for the other agent; # At most one good has to be shared between the agents. For two agents, Adjusted Winner is the only Pareto optimal and equitable procedure that divides at most a single good. The procedure can be used in divorce settlements and partnership dissolutions, as well as international conflicts. The procedure was designed by Steven Brams and Alan D. Taylor. It was first published in their book on fair division and later in a stand-alone book. The algorithm has been commercialized t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
