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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] |
Fair Division
Fair division is the problem in game theory of dividing a set of resources among several people who have an Entitlement (fair division), entitlement to them so that each person receives their due share. The central tenet of fair division is that such a division should be performed by the players themselves, without the need for external arbitration, as only the players themselves really know how they value the goods. There are many different kinds of fair division problems, depending on the nature of goods to divide, the criteria for fairness, the nature of the players and their preferences, and other criteria for evaluating the quality of the division. The archetypal fair division algorithm is divide and choose. The research in fair division can be seen as an extension of this procedure to various more complex settings. Description In game theory, fair division is the problem of dividing a set of resources among several people who have an Entitlement (fair division), entitlem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kneser Graph
Kneser is a surname. Notable people with the surname include: * Adolf Kneser (1862–1930), mathematician * Hellmuth Kneser (1898–1973), mathematician, son of Adolf Kneser * Martin Kneser (1928–2004), mathematician, son of Hellmuth Kneser {{surname ... [...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 (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] |
Pareto-efficient
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 ( non-strictly) prefers A to B, society as a whole also non-strictly prefers A to B. The 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 ''efficiency in production'' vs. ''x-inefficiency'': a set of outputs of goo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Strategyproof
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 col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leximin Order
In mathematics, leximin order is a total preorder on finite-dimensional vectors. A more accurate but less common term is leximin preorder. The leximin order is particularly important in social choice theory and fair division. Definition A vector x = (''x''1, ..., ''x''''n'') is ''leximin-larger'' than a vector y = (''y''1, ..., ''y''''n'') if one of the following holds: * The smallest element of x is larger than the smallest element of y; * The smallest elements of both vectors are equal, and the second-smallest element of x is larger than the second-smallest element of y; * ... * The ''k'' smallest elements of both vectors are equal, and the (''k''+1)-smallest element of x is larger than the (''k''+1)-smallest element of y. Examples The vector (3,5,3) is leximin-larger than (4,2,4), since the smallest element in the former is 3 and in the latter is 2. The vector (4,2,4) is leximin-larger than (5,3,2), since the smallest elements in both are 2, but the second-smallest elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Egalitarian Rule
In social choice and operations research, the egalitarian rule (also called the max-min rule or the Rawlsian rule) is a rule saying that, among all possible alternatives, society should pick the alternative which maximizes the ''minimum utility'' of all individuals in society. It is a formal mathematical representation of the egalitarian philosophy. It also corresponds to John Rawls' principle of maximizing the welfare of the worst-off individual. Definition Let X be a set of possible `states of the world' or `alternatives'. Society wishes to choose a single state from X. For example, in a single-winner election, X may represent the set of candidates; in a resource allocation setting, X may represent all possible allocations. Let I be a finite set, representing a collection of individuals. For each i \in I, let u_i:X\longrightarrow\mathbb be a ''utility function'', describing the amount of happiness an individual ''i'' derives from each possible state. A '' social choice rule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rental Harmony
Rental harmony is a kind of a fair division problem in which indivisible items and a fixed monetary cost have to be divided simultaneously. The housemates problem and room-assignment-rent-division are alternative names to the same problem. In the typical setting, there are n partners who rent together an n-room house for cost fixed by the homeowner. Each housemate may have different preferences — one may prefer a large room, another may prefer a room with a view to the main road, etc. The following two problems should be solved simultaneously: * (a) Assign a room to each partner, * (b) Determine the amount each partner should pay, such that the sum of payments equals the fixed cost. There are several properties that we would like the assignment to satisfy. * Non-negativity (NN): all prices must be 0 or more: no partner should be paid to get a room. * Envy-freeness (EF): Given a pricing scheme (an assignment of rent to rooms), we say that a partner ''prefers'' a given room if he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
Additive Valuation
In economics, additive utility is a cardinal utility function with the sigma additivity property. Additivity (also called ''linearity'' or ''modularity'') means that "the whole is equal to the sum of its parts." That is, the utility of a set of items is the sum of the utilities of each item separately. Let S be a finite set of items. A cardinal utility function u:2^S\to\R, where 2^S is the power set of S, is additive if for any A, B\subseteq S, :u(A)+u(B)=u(A\cup B)+u(A\cap B). It follows that for any A\subseteq S, :u(A)=u(\emptyset)+\sum_\big(u(\)-u(\emptyset)\big). An additive utility function is characteristic of independent goods. For example, an apple and a hat are considered independent: the utility a person receives from having an apple is the same whether or not he has a hat, and vice versa. A typical utility function for this case is given at the right. Notes * As mentioned above, additivity is a property of cardinal utility functions. An analogous property of ordinal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
