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Group Envy-freeness
Group envy-freeness (also called: coalition fairness) is a criterion for fair division. A group-envy-free division is a division of a resource among several partners such that every group of partners feel that their allocated share is at least as good as the share of any other group with the same size. The term is used particularly in problems such as fair resource allocation, fair cake-cutting and fair item allocation. Group-envy-freeness is a very strong fairness requirement: a group-envy-free allocation is both envy-free and Pareto efficient, but the opposite is not true. Definitions Consider a set of ''n'' agents. Each agent ''i'' receives a certain allocation ''Xi'' (e.g. a piece of cake or a bundle of resources). Each agent ''i'' has a certain subjective preference relation <''i'' over pieces/bundles (i.e. means that agent ''i'' prefers piece ''X'' to piece ''Y''). Consider a group ''G'' of the agents, with its current allocation |
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] |
Resource Allocation
In economics, resource allocation is the assignment of available resources to various uses. In the context of an entire economy, resources can be allocated by various means, such as markets, or planning. In project management, resource allocation or resource management is the scheduling of activities and the resources required by those activities while taking into consideration both the resource availability and the project time. Economics In economics, the field of public finance deals with three broad areas: macroeconomic stabilization, the distribution of income and wealth, and the allocation of resources. Much of the study of the allocation of resources is devoted to finding the conditions under which particular mechanisms of resource allocation lead to Pareto efficient outcomes, in which no party's situation can be improved without hurting that of another party. Strategic planning In strategic planning, resource allocation is a plan for using available resources, for exampl ... [...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 that he or she believes 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 already in the book of Genesis. It solves the fair division problem for two people. The modern ... [...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] |
Pareto Efficient
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Competitive Equilibrium
Competitive equilibrium (also called: Walrasian equilibrium) is a concept of economic equilibrium introduced by Kenneth Arrow and Gérard Debreu in 1951 appropriate for the analysis of commodity markets with flexible prices and many traders, and serving as the benchmark of efficiency in economic analysis. It relies crucially on the assumption of a competitive environment where each trader decides upon a quantity that is so small compared to the total quantity traded in the market that their individual transactions have no influence on the prices. Competitive markets are an ideal standard by which other market structures are evaluated. Definitions A competitive equilibrium (CE) consists of two elements: * A price function P. It takes as argument a vector representing a bundle of commodities, and returns a positive real number that represents its price. Usually the price function is linear - it is represented as a vector of prices, a price for each commodity type. * An allocation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Measure
In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only. Definitions and first consequences Given a field of sets (\Omega, \mathcal F) and a Banach space X, a finitely additive vector measure (or measure, for short) is a function \mu:\mathcal \to X such that for any two disjoint sets A and B in \mathcal one has \mu(A\cup B) =\mu(A) + \mu (B). A vector measure \mu is called countably additive if for any sequence (A_i)_^ of disjoint sets in \mathcal F such that their union is in \mathcal F it holds that \mu = \sum_^\mu(A_i) with the series on the right-hand side convergent in the norm of the Banach space X. It can be proved that an additive vector measure \mu is countably additive if and only if for any sequence (A_i)_^ as above one has where \, \cdot\, is the norm on X. Countably additive vector measur ... [...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] |
Proportional Division
A proportional division is a kind of fair division in which a resource is divided among ''n'' partners with subjective valuations, giving each partner at least 1/''n'' of the resource by his/her own subjective valuation. Proportionality was the first fairness criterion studied in the literature; hence it is sometimes called "simple fair division". It was first conceived by Steinhaus. Example Consider a land asset that has to be divided among 3 heirs: Alice and Bob who think that it's worth 3 million dollars, and George who thinks that it's worth $4.5M. In a proportional division, Alice receives a land-plot that she believes to be worth at least $1M, Bob receives a land-plot that ''he'' believes to be worth at least $1M (even though Alice may think it is worth less), and George receives a land-plot that he believes to be worth at least $1.5M. Existence A proportional division does not always exist. For example, if the resource contains several indivisible items and the number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utilitarian Cake-cutting
Utilitarian cake-cutting (also called maxsum cake-cutting) is a rule for dividing a heterogeneous resource, such as a cake or a land-estate, among several partners with different cardinal utility functions, such that the ''sum'' of the utilities of the partners is as large as possible. It is a special case of the utilitarian social choice rule. Utilitarian cake-cutting is often not "fair"; hence, utilitarianism is often in conflict with fair cake-cutting. Example Consider a cake with two parts: chocolate and vanilla, and two partners: Alice and George, with the following valuations: The utilitarian rule gives each part to the partner with the highest utility. In this case, the utilitarian rule gives the entire chocolate to Alice and the entire Vanilla to George. The maxsum is 13. The utilitarian division is not fair: it is not proportional since George receives less than half the total cake value, and it is not envy-free since George envies Alice. Notation The cake is calle ... [...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] |
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