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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] |
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-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] |
Draft (sports)
A draft is a process used in some countries (especially in North America) and sports (especially in closed leagues) to allocate certain players to teams. In a draft, teams take turns selecting from a pool of eligible players. When a team selects a player, the team receives exclusive rights to sign that player to a contract, and no other team in the league may sign the player. The process is similar to round-robin item allocation. The best-known type of draft is the entry draft, which is used to allocate players who have recently become eligible to play in a league. Depending on the sport, the players may come from college, high school or junior teams, or teams in other countries. An entry draft is intended to prevent expensive bidding wars for young talent and to ensure that no team can sign contracts with all of the best young players and make the league uncompetitive. To encourage parity, teams that do poorly in the previous season usually get to choose first in the postseas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Set Function
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of ''k'' disjoint sets (where ''k'' is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely-additive set function (the terms are equivalent). However, a finitely-additive set function might not have the additivity property for a union of an ''infinite'' number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, \mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n). Additivity and sigma-additivity are particularly important properties of measures. They are abstrac ... [...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] |
Weakly Additive
In fair division, a topic in economics, a preference relation is weakly additive if the following condition is met: : If A is preferred to B, and C is preferred to D (and the contents of A and C do not overlap) then A together with C is preferable to B together with D. Every additive utility function is weakly-additive. However, additivity is applicable only to cardinal utility functions, while weak additivity is applicable to ordinal utility functions. Weak additivity is often a realistic assumption when dividing up goods between claimants, and simplifies the mathematics of certain fair division problems considerably. Some procedures in fair division do not need the value of goods to be additive and only require weak additivity. In particular the adjusted winner procedure only requires weak additivity. Cases where weak additivity fails Case where the assumptions might fail would be either *The value of A and C together is the less than the sum of their values. For instance ... [...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] |
Assignment Problem
The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: :The problem instance has a number of ''agents'' and a number of ''tasks''. Any agent can be assigned to perform any task, incurring some ''cost'' that may vary depending on the agent-task assignment. It is required to perform as many tasks as possible by assigning at most one agent to each task and at most one task to each agent, in such a way that the ''total cost'' of the assignment is minimized. Alternatively, describing the problem using graph theory: :The assignment problem consists of finding, in a weighted bipartite graph, a matching of a given size, in which the sum of weights of the edges is minimum. If the numbers of agents and tasks are equal, then the problem is called ''balanced assignment''. Otherwise, it is called ''unbalanced assignment''. If the total cost of the assignment for all tasks is equal to the sum of the costs for each agent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denoting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Division Among Groups
Fair division among groups (or families) is a class of fair division problems, in which the resources are allocated among ''groups'' of agents, rather than among individual agents. After the division, all members in each group consume the same share, but they may have different preferences; therefore, different members in the same group might disagree on whether the allocation is fair or not. Some examples of group fair division settings are: * Several siblings inherited some houses from their parents and have to divide them. Each sibling has a family, whose members may have different opinions regarding which house is better. * A partnership is dissolved, and its assets should be divided among the partners. The partners are firms; each firm has several stockholders, who might disagree regarding which asset is more important. *The university management wants to allocate some meeting-rooms among its departments. In each department there are several faculty members, with differing opi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approval Voting
Approval voting is an electoral system in which voters can select many candidates instead of selecting only one candidate. Description Approval voting ballots show a list of the options of candidates running. Approval voting lets each voter indicate support for one or more candidates. Final tallies show how many votes each candidate received, and the winner is the candidate with the most support. Effect on elections Approval voting advocates Steven Brams and Dudley R. Herschbach predict that approval voting should increase voter participation, prevent minor-party candidates from being spoilers, and reduce negative campaigning. FairVote published a position paper arguing that approval voting has three flaws that undercut it as a method of voting and political vehicle (the group instead advocates for Instant-runoff voting). They argue that it can result in the defeat of a candidate who would win an absolute majority in a plurality election, can allow a candidate to win who ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
