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Online Fair Division
Online fair division is a class of fair division problems in which the resources, or the people to whom they should be allocated, or both, are not all available when the allocation decision is made. Some situations in which not all resources are available include: * Allocating food donations to charities (the "food bank" problem). Each donation must be allocated immediately when it arrives, before future donations arrive. * Allocating donated blood or organs to patients. Again, each donation must be allocated immediately, and it is not known when and what future donations will be. Some situations in which not all participants are available include: * Dividing a cake among people in a party. Some people come early and want to get a cake when they arrive, but other people may come later. * Dividing the rent and rooms among tenants in a rented apartment, when one or more of them are not available during the allocation. The online nature of the problem requires different techniques ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Division
Fair division is the problem in game theory of dividing a set of resources among several people who have an entitlement to them so that each person receives their due share. That problem arises in various real-world settings such as division of inheritance, partnership dissolutions, divorce settlements, electronic frequency allocation, airport traffic management, and exploitation of Earth observation satellites. It is an active research area in mathematics, economics (especially social choice theory), dispute resolution, etc. The central tenet of fair division is that such a division should be performed by the players themselves, maybe using a mediator but certainly not an arbiter as only the players really know how they value the goods. The archetypal fair division algorithm is divide and choose. It demonstrates that two agents with different tastes can divide a cake such that each of them believes that he got the best piece. The research in fair division can be seen as an exten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envy-freeness
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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distributions
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Price Of Fairness
In the theory of fair division, the price of fairness (POF) is the ratio of the largest economic welfare attainable by a division to the economic welfare attained by a ''fair'' division. The POF is a quantitative measure of the loss of welfare that society has to take in order to guarantee fairness. In general, the POF is defined by the following formula: :POF=\frac The exact price varies greatly based on the kind of division, the kind of fairness and the kind of social welfare we are interested in. The most well-studied type of social welfare is '' utilitarian social welfare'', defined as the sum of the (normalized) utilities of all agents. Another type is '' egalitarian social welfare'', defined as the minimum (normalized) utility per agent. Numeric example In this example we focus on the ''utilitarian price of proportionality'', or UPOP. Consider a heterogeneous land-estate that has to be divided among 100 partners, all of whom value it as 100 (or the value is normalized t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong NP-completeness
In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. A general computational problem may have numerical parameters. For example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters. A problem is said to be strongly NP-complete (NP-complete in the strong sense), if it remains NP-complete even when all of its numerical parameters are bounded by a polynomial in the length of the input. A problem is said to be strongly NP-hard if a strongly NP-complete problem has a polynomial reduction to it; in combinatorial optimization, particularly, the phrase "strongly NP-hard" is reserved for problems that are not known to have a polynomial reduction to another strongly NP-complete problem. Normally numerical parameters to a problem are given in positional notation, so a problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Price Of Fairness
In the theory of fair division, the price of fairness (POF) is the ratio of the largest economic welfare attainable by a division to the economic welfare attained by a ''fair'' division. The POF is a quantitative measure of the loss of welfare that society has to take in order to guarantee fairness. In general, the POF is defined by the following formula: :POF=\frac The exact price varies greatly based on the kind of division, the kind of fairness and the kind of social welfare we are interested in. The most well-studied type of social welfare is '' utilitarian social welfare'', defined as the sum of the (normalized) utilities of all agents. Another type is '' egalitarian social welfare'', defined as the minimum (normalized) utility per agent. Numeric example In this example we focus on the ''utilitarian price of proportionality'', or UPOP. Consider a heterogeneous land-estate that has to be divided among 100 partners, all of whom value it as 100 (or the value is normalized t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ex-ante
The term ''ex-ante'' (sometimes written ''ex ante'' or ''exante'') is a phrase meaning "before the event". Ex-ante or notional demand refers to the desire for goods and services that is not backed by the ability to pay for those goods and services. This is also termed as 'wants of people'. ''Ex-ante'' is used most commonly in the commercial world, where results of a particular action, or series of actions, are forecast (or intended). The opposite of ''ex-ante'' is ''ex-post'' (actual) (or ''ex post''). Buying a lottery ticket loses you money ex ante (in expectation), but if you win, it was the right decision ex post. Examples: * In the financial world, the ''ex-ante return'' is the expected return of an investment portfolio. * In the recruitment industry, ''ex-ante'' is often used when forecasting resource requirements on large future projects. The ''ex-ante'' (and ''ex-post'') reasoning in economic topics was introduced mainly by Swedish economist Gunnar Myrdal in his 1927– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pareto Efficiency
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related: * Given an initial situation, a Pareto improvement is a new situation where some agents will gain, and no agents will lose. * A situation is called Pareto-dominated if there exists a possible Pareto improvement. * A situation is called Pareto-optimal or Pareto-efficient if no change could lead to improved satisfaction for some agent without some other agent losing or, equivalently, if there is no scope for further Pareto improvement. The Pareto front (also called Pareto frontier or Pareto set) is the set of all Pareto-efficient situations. Pareto originally used the word "optimal" for t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
With High Probability
In mathematics, an event that occurs with high probability (often shortened to w.h.p. or WHP) is one whose probability depends on a certain number ''n'' and goes to 1 as ''n'' goes to infinity, i.e. the probability of the event occurring can be made as close to 1 as desired by making ''n'' big enough. Applications The term WHP is especially used in computer science, in the analysis of probabilistic algorithms. For example, consider a certain probabilistic algorithm on a graph with ''n'' nodes. If the probability that the algorithm returns the correct answer is 1-1/n, then when the number of nodes is very large, the algorithm is correct with a probability that is very near 1. This fact is expressed shortly by saying that the algorithm is correct WHP. Some examples where this term is used are: * Miller–Rabin primality test: a probabilistic algorithm for testing whether a given number ''n'' is prime or composite. If ''n'' is composite, the test will detect ''n'' as composite WHP. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joint Probability Distribution
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distributions, i.e. the distributions of each of the individual random variables. It also encodes the conditional probability distributions, which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable(s). In the formal mathematical setup of measure theory, the joint distribution is given by the pushforward measure, by the map obtained by pairing together the given random variables, of the sample space's probability measure. In the case of real-valued random variables, the joint distribution, as a particular multivariate distribution, may be expressed by a multivariate cumulativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrepancy Theory
In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of ''classical'' discrepancy theory, namely distributing points in some space such that they are evenly distributed with respect to some (mostly geometrically defined) subsets. The discrepancy (irregularity) measures how far a given distribution deviates from an ideal one. Discrepancy theory can be described as the study of inevitable irregularities of distributions, in measure-theoretic and combinatorial settings. Just as Ramsey theory elucidates the impossibility of total disorder, discrepancy theory studies the deviations from total uniformity. A significant event in the history of discrepancy theory was the 1916 paper of Weyl on the uniform distribution of sequences in the unit interval. __NOTOC__ Theorems Discrepancy theory is based on the following classic theorems: * T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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