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Fair Allocation Of Items And Money
Fair allocation of items and money is a class of fair item allocation problems in which, during the allocation process, it is possible to give or take money from some of the participants. Without money, it may be impossible to allocate indivisible items fairly. For example, if there is one item and two people, and the item must be given entirely to one of them, the allocation will be unfair towards the other one. Monetary payments make it possible to attain fairness, as explained below. Two agents and one item With two agents and one item, it is possible to attain fairness using the following simple algorithm (which is a variant of cut and choose): * Alice says a price ''p'' that she is willing to pay for the item. * George chooses whether to take the item and pay ''p'', or leave the item to Alice so that Alice pays ''p''. This algorithm assumes that the agents have quasilinear utilities, that is, their utility is the value of items plus the amount of money that they have. If ... [...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] |
Utilitarian Rule
In social choice and operations research, the utilitarian rule (also called the max-sum rule) is a rule saying that, among all possible alternatives, society should pick the alternative which maximizes the ''sum of the utilities'' of all individuals in society. It is a formal mathematical representation of the utilitarian philosophy. 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 of the resource. 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'' is a mechanism which uses the data (u_i)_ to select some element(s) from X which are `best' for s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superadditive Utility
In mathematics, a function f is superadditive if f(x+y) \geq f(x) + f(y) for all x and y in the domain of f. Similarly, a sequence \left\, n \geq 1, is called superadditive if it satisfies the inequality a_ \geq a_n + a_m for all m and n. The term "superadditive" is also applied to functions from a boolean algebra to the real numbers where P(X \lor Y) \geq P(X) + P(Y), such as lower probabilities. Properties If f is a superadditive function, and if 0 is in its domain, then f(0) \leq 0. To see this, take the inequality at the top: f(x) \leq f(x+y) - f(y). Hence f(0) \leq f(0+y) - f(y) = 0. The negative of a superadditive function is subadditive. Fekete's lemma The major reason for the use of superadditive sequences is the following lemma due to Michael Fekete. :Lemma: (Fekete) For every superadditive sequence \left\, n \geq 1, the limit \lim a_n/n is equal to \sup a_n/n. (The limit may be positive infinity, for instance, for the sequence a_n = \log n!.) For example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equitable Division
Equitable (EQ) cake-cutting is a kind of a fair cake-cutting problem, in which the fairness criterion is equitability. It is a cake-allocation in which the subjective value of all partners is the same, i.e., each partner is equally happy with his/her share. Mathematically, that means that for all partners and : :V_i(X_i) = V_j(X_j) Where: *X_i is the piece of cake allocated to partner ; *V_i is the value measure of partner . It is a real-valued function that, for every piece of cake, returns a number that represents the utility of partner from that piece. Usually these functions are normalized such that V_i(\emptyset)=0 and V_i(EntireCake)=1 for every . See the page on equitability for examples and comparison to other fairness criteria. Finding an equitable cake-cutting for two partners One cut - full revelation When there are 2 partners, it is possible to get an EQ division with a single cut, but it requires full knowledge of the partners' valuations. Assume that the cake ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Makespan
In operations research, the makespan of a project is the length of time that elapses from the start of work to the end. This type of multi-mode resource constrained project scheduling problem (MRCPSP) seeks to create the shortest logical project schedule, by efficiently using project resources, adding the lowest number of additional resources as possible to achieve the minimum makespan. ', Afshar-Nadjafi, B, ''in Applied Computing and Informatics (2014) The term commonly appears in the context of |
Envy-free Pricing
Envy-free pricing is a kind of fair item allocation. There is a single seller that owns some items, and a set of buyers who are interested in these items. The buyers have different valuations to the items, and they have a quasilinear utility function; this means that the utility an agent gains from a bundle of items equals the agent's value for the bundle minus the total price of items in the bundle. The seller should determine a price for each item, and sell the items to some of the buyers, such that there is Envy-freeness, ''no envy''. Two kinds of envy are considered: * ''Agent envy'' means that some agent assigns a higher utility (a higher difference value-price) to a bundle allocated to another agent. * ''Market envy'' means that some agent assigns a higher utility (a higher difference value-price) to any bundle. The no-envy conditions guarantee that the market is stable and that the buyers do not resent the seller. By definition, every market envy-free allocation is also agent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Budget-balanced Mechanism
In mechanism design, a branch of economics, a weakly-budget-balanced (WBB) mechanism is a mechanism in which the total payment made by the participants is 0. This means that the mechanism operator does not incur a deficit, i.e., does not have to subsidize the market. Weak budget balance is considered a necessary requirement for the economic feasibility of a mechanism. A strongly-budget-balanced (SBB) mechanism is a mechanism in which the total payment made by the participants is ''exactly'' 0. This means that all payments are made among the participants - the mechanism has neither a deficit nor a surplus. The term budget-balanced mechanism is sometimes used as a shorthand for WBB, and sometimes as a shorthand for SBB. Weak budget balance A simple example of a WBB mechanism is the Vickrey auction, in which the operator wants to sell an object to one of ''n'' potential buyers. Each potential buyer bids a value, the highest bidder wins an object and pays the second-highest bid. As ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-dimensional Matching
In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs, which consist of hyperedges each of which contains 3 vertices (instead of edges containing 2 vertices in a usual graph). 3-dimensional matching, often abbreviated as 3DM, is also the name of a well-known computational problem: finding a largest 3-dimensional matching in a given hypergraph. 3DM is one of the first problems that were proved to be NP-hard. Definition Let ''X'', ''Y'', and ''Z'' be finite sets, and let ''T'' be a subset of ''X'' × ''Y'' × ''Z''. That is, ''T'' consists of triples (''x'', ''y'', ''z'') such that ''x'' ∈ ''X'', ''y'' ∈ ''Y'', and ''z'' ∈ ''Z''. Now ''M'' ⊆ ''T'' is a 3-dimensional matching if the following holds: for any two distinct triples (''x''1, ''y''1, ''z''1) ∈ ''M'' a ... [...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] |
Maximum Weight Matching
In computer science and graph theory, the maximum weight matching problem is the problem of finding, in a weighted graph, a matching in which the sum of weights is maximized. A special case of it is the assignment problem, in which the input is restricted to be a bipartite graph, and the matching constrained to be have cardinality that of the smaller of the two partitions. Another special case is the problem of finding a maximum cardinality matching on an unweighted graph: this corresponds to the case where all edge weights are the same. Algorithms There is a O(V^E) time algorithm to find a maximum matching or a maximum weight matching in a graph that is not bipartite; it is due to Jack Edmonds, is called the ''paths, trees, and flowers'' method or simply Edmonds' algorithm, and uses bidirected edges. A generalization of the same technique can also be used to find maximum independent set In graph theory, an independent set, stable set, coclique or anticlique is a set ... [...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 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] |
