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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 they 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 p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fair Item Allocation
Fair item allocation is a kind of the fair division problem in which the items to divide are ''discrete'' rather than continuous. The items have to be divided among several partners who potentially 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 ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 (economics), 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 e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 sports, 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 (sports), parity, teams that do poorly in the previous season usually get ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Additive Set Function
In mathematics, an additive set function is a function \mu 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 ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Pareto Efficiency
In welfare economics, a Pareto improvement formalizes the idea of an outcome being "better in every possible way". A change is called a Pareto improvement if it leaves at least one person in society better off without leaving anyone else worse off than they were before. A situation is called Pareto efficient or Pareto optimal if all possible Pareto improvements have already been made; in other words, there are no longer any ways left to make one person better off without making some other person worse-off. In social choice theory, the same concept is sometimes called the unanimity principle, which says that if ''everyone'' in a society (strict inequality, non-strictly) prefers A to B, society as a whole also non-strictly prefers A to B. The Pareto frontier, Pareto front consists of all Pareto-efficient situations. In addition to the context of efficiency in ''allocation'', the concept of Pareto efficiency also arises in the context of productive efficiency, ''efficiency in prod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 graph, weighted bipartite graph, a Matching (graph theory), matching of maximum 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, and the graph-theoretic version is called minimum-cost perfect matching. Otherwise, it is called unbalanced assig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), 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 cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, 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 Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Truthful Mechanism
In mechanism design, a strategyproof (SP) mechanism is a game form in which each player has a weakly- dominant strategy, so that no player can gain by "spying" over the other players to know what they are going to play. When the players have private information (e.g. their type or their value to some item), and the strategy space of each player consists of the possible information values (e.g. possible types or values), a truthful mechanism is a game in which revealing the true information is a weakly-dominant strategy for each player. An SP mechanism is also called dominant-strategy-incentive-compatible (DSIC), to distinguish it from other kinds of incentive compatibility. A SP mechanism is immune to manipulations by individual players (but not by coalitions). In contrast, in a group strategyproof mechanism, no group of people can collude to misreport their preferences in a way that makes every member better off. In a strong group strategyproof mechanism, no group of people can c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 opini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Approval Voting
Approval voting is a single-winner rated voting system where voters can approve of all the candidates as they like instead of Plurality voting, choosing one. The method is designed to eliminate vote-splitting while keeping election administration simple and Summability criterion, easy-to-count (requiring only a single score for each candidate). Approval voting has been used in both organizational and political elections to improve representativeness and voter satisfaction. Critics of approval voting have argued the simple ballot format is a disadvantage, as it forces a Dichotomous preferences, binary choice for each candidate (instead of the expressive grades of other rated voting rules). Effect on elections Research by Social choice theory, social choice theorists Steven Brams and Dudley R. Herschbach found that approval voting would increase voter participation, prevent minor-party candidates from being spoiler effect, spoilers, and reduce negative campaigning. Brams' researc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
