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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] |
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] |
Envy-free Cake-cutting
An envy-free cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the envy-free criterion, namely, that every partner feels that their allocated share is at least as good as any other share, according to their own subjective valuation. When there are only two partners, the problem is easy and was solved in antiquity by the divide and choose protocol. When there are three or more partners, the problem becomes much more challenging. Two major variants of the problem have been studied: * Connected pieces, e.g. if the cake is a 1-dimensional interval then each partner must receive a single sub-interval. If there are n partners, only n-1 cuts are needed. * General pieces, e.g. if the cake is a 1-dimensional interval then each partner can receive a union of disjoint sub-intervals. Short history Modern research into the fair cake-cutting problem started in the 1940s. The first fairness criterion studied was proportional divi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Fair Division Experiments
Various experiments have been made to evaluate various procedures for fair division, the problem of dividing resources among several people. These include case studies, computerized simulations, and lab experiments. Case studies Allocating indivisible heirlooms 1. Flood describes a division of a gift containing 5 parcels: whiskey, prunes, eggs, suitcase, etc. The division was done using the Knaster auction. The resulting division was fair, but in retrospect it was found that coalitions could gain from manipulation. 2. When Mary Anna Lee Paine Winsor died at the age of 93, her estate included two trunks of silver, that had to be divided among her 8 grandchildren. It was divided using a decentralized, fair and efficient allocation procedure, which combined market equilibrium and a Vickrey auction. Although most participants did not fully understand the algorithm or the preference information desired, it handled the major considerations well and was regarded as equitable. Allo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 |
Envy-free Item Allocation
Envy-free (EF) item allocation is a fair item allocation problem, in which the fairness criterion is envy-freeness - each agent should receive a bundle that they believe to be at least as good as the bundle of any other agent. Since the items are indivisible, an EF assignment may not exist. The simplest case is when there is a single item and at least two agents: if the item is assigned to one agent, the other will envy. One way to attain fairness is to use monetary transfers; see Fair allocation of items and money. When monetary transfers are not allowed or not desired, there are allocation algorithms providing various kinds of relaxations. Finding an envy-free allocation whenever it exists Preference-orderings on bundles: envy-freeness The undercut procedure finds a complete EF allocation for two agents, if-and-only-if such allocation exists. It requires the agents to rank bundles of items, but it does not require cardinal utility information. It works whenever the agents' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Efficient Envy-free Division
Efficiency and fairness are two major goals of welfare economics. Given a set of resources and a set of agents, the goal is to divide the resources among the agents in a way that is both Pareto efficient (PE) and envy-free (EF). The goal was first defined by David Schmeidler and Menahem Yaari. Later, the existence of such allocations has been proved under various conditions. Existence of PEEF allocations We assume that each agent has a preference-relation on the set of all bundles of commodities. The preferences are complete, transitive, and closed. Equivalently, each preference relation can be represented by a continuous utility function. Weakly-convex preferences ''Theorem 1 (Varian):'' ''If the preferences of all agents are convex and strongly monotone, then PEEF allocations exist.'' ''Proof'': The proof relies on the existence of a competitive equilibrium with equal incomes. Assume that all resources in an economy are divided equally between the agents. I.e, if the total e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Fair Cake-cutting
Symmetric fair cake-cutting is a variant of the fair cake-cutting problem, in which fairness is applied not only to the final outcome, but also to the assignment of roles in the division procedure. As an example, consider a birthday cake that has to be divided between two children with different tastes, such that each child feels that his/her share is "fair", i.e., worth at least 1/2 of the entire cake. They can use the classic divide and choose procedure: Alice cuts the cake into two pieces worth exactly 1/2 in her eyes, and George chooses the piece that he considers more valuable. The outcome is always fair. However, the procedure is not symmetric: while Alice always gets a value of exactly 1/2 of her value, George may get much more than 1/2 of his value. Thus, while Alice does not envy George's share, she does envy George's role in the procedure. In contrast, consider the alternative procedure in which Alice and George both make half-marks on the cake, i.e., each of them marks t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Super Envy-freeness
A super-envy-free division is a kind of a fair division. It is a division of resources among ''n'' partners, in which each partner values his/her share at strictly ''more'' than his/her due share of 1/''n'' of the total value, and simultaneously, values the share of every other partner at strictly less than 1/''n''. Formally, in a super-envy-free division of a resource ''C'' among ''n'' partners, each partner ''i'', with value measure ''Vi'', receives a share ''Xi'' such that:V_i(X_i) > V_i(C)/n ~~ \text ~~ \forall j\neq i:V_i(X_j) < V_i(C)/n .This is a strong fairness requirement: it is stronger than both and super-proportionality. Existence Super envy-freeness was introduced by Julius Barbanel in 199 ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Random Assignment
Fair random assignment (also called probabilistic one-sided matching) is a kind of a fair division problem. In an ''assignment problem'' (also called '' house-allocation problem'' or '' one-sided matching''), there ''m'' objects and they have to be allocated among ''n'' agents, such that each agent receives at most one object. Examples include the assignment of jobs to workers, rooms to housemates, dormitories to students, time-slots to users of a common machine, and so on. In general, a fair assignment may be impossible to attain. For example, if Alice and Batya both prefer the eastern room to the western room, only one of them will get it and the other will be envious. In the random assignment setting, fairness is attained using a lottery. So in the simple example above, Alice and Batya will toss a fair coin and the winner will get the eastern room. History Random assignment is mentioned already in the Bible: a lottery was used to allocate the lands of Canaan among the Tribes o ... [...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 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] |
No Justified Envy
In economics and social choice theory, a no-justified-envy matching is a matching in a two-sided market, in which no agent prefers the assignment of another agent and is simultaneously preferred by that assignment. Consider, for example, the task of matching doctors for residency in hospitals. Each doctor has a preference relation on hospitals, ranking the hospitals from best to worst. Each hospital has a preference relation on doctors, ranking the doctors from best to worst. Each doctor can work in at most one hospital, and each hospital can employ at most a fixed number of doctors (called the ''capacity'' of the hospital). The goal is to match doctors to hospitals, without monetary transfers. ''Envy'' is a situation in which some doctor ''d''1, employed in some hospital ''h''1, prefers some other hospital ''h''2, which employs some other doctor ''d''2 (we say that ''d1 envies d2''). The envy is ''justified'' if, at the same time, ''h''2 prefers ''d''1 over ''d''2. Note that, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
