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Envy-free Item Assignment
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] |
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] |
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] |
Exact Division
Exact division, also called consensus division, is a partition of a continuous resource (" cake") into some ''k'' pieces, such that each of ''n'' people with different tastes agree on the value of each of the pieces. For example, consider a cake which is half chocolate and half vanilla. Alice values only the chocolate and George values only the vanilla. The cake is divided into three pieces: one piece contains 20% of the chocolate and 20% of the vanilla, the second contains 50% of the chocolate and 50% of the vanilla, and the third contains the rest of the cake. This is an exact division (with ''k''=3 and ''n''=2), as both Alice and George value the three pieces as 20%, 50% and 30% respectively. Several common variants and special cases are known by different terms: * Consensus halving – the cake should be partitioned into two pieces (''k''=2), and all agents agree that the pieces have equal values. *Consensus 1/''k''-division, for any constant ''k''>1 - the cake should be partition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pairwise Maximin Share
Maximin share (MMS) is a criterion of fair item allocation. Given a set of items with different values, the ''1-out-of-n maximin-share'' is the maximum value that can be gained by partitioning the items into ''n'' parts and taking the part with the minimum value. An allocation of items among ''n'' agents with different valuations is called MMS-fair if each agent gets a bundle that is at least as good as his/her 1-out-of-''n'' maximin-share. MMS fairness was invented by Eric Budish as a relaxation of the criterion of proportionality - each agent gets a bundle that is at least as good as the equal split (1/''n'' of every resource). Proportionality can be guaranteed when the items are divisible, but not when they are indivisible, even if all agents have identical valuations. In contrast, MMS fairness can always be guaranteed to identical agents, so it is a natural alternative to proportionality even when the agents are different. Motivation and examples Identical items. Suppose fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Groupwise Maximin Share
Maximin share (MMS) is a criterion of fair item allocation. Given a set of items with different values, the ''1-out-of-n maximin-share'' is the maximum value that can be gained by partitioning the items into ''n'' parts and taking the part with the minimum value. An allocation of items among ''n'' agents with different valuations is called MMS-fair if each agent gets a bundle that is at least as good as his/her 1-out-of-''n'' maximin-share. MMS fairness was invented by Eric Budish as a relaxation of the criterion of proportionality - each agent gets a bundle that is at least as good as the equal split (1/''n'' of every resource). Proportionality can be guaranteed when the items are divisible, but not when they are indivisible, even if all agents have identical valuations. In contrast, MMS fairness can always be guaranteed to identical agents, so it is a natural alternative to proportionality even when the agents are different. Motivation and examples Identical items. Suppose fi ... [...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] |
Leximin
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] |
Simmons–Su Protocols
The Simmons–Su protocols are several protocols for envy-free division. They are based on Sperner's lemma. The merits of these protocols is that they put few restrictions on the preferences of the partners, and ask the partners only simple queries such as "which piece do you prefer?". Protocols were developed for solving several related problems: Cake cutting In the envy-free cake-cutting problem, a "cake" (a heterogeneous divisible resource) has to be divided among ''n'' partners with different preferences over parts of the cake. The cake has to be divided to ''n'' pieces such that: (a) each partner receives a single connected piece, and (b) each partner believes that his piece is (weakly) better than all other pieces. A protocol for solving this problem was developed by Forest Simmons in 1980, in a correspondence with Michael Starbird. It was first publicized by Francis Su in 1999. Given a cut-set (i.e. a certain partition of the cake to ''n'' pieces), we say that a partne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Efficient Approximately-fair Item Allocation
When allocating objects among people with different preferences, two major goals are Pareto efficiency and fairness. Since the objects are indivisible, there may not exist any fair allocation. For example, when there is a single house and two people, every allocation of the house will be unfair to one person. Therefore, several common approximations have been studied, such as ''maximin-share fairness'' (MMS)'', envy-freeness up to one item'' (EF1), '' proportionality up to one item'' (PROP1), and equitability up to one item (EQ1). The problem of efficient approximately-fair item allocation is to find an allocation that is both Pareto-efficient (PE) and satisfies one of these fairness notions. The problem was first presented at 2016 and has attracted considerable attention since then. Setting There is a finite set of objects, denoted by ''M''. There are ''n'' agents. Each agent ''i'' has a value-function ''Vi'', that assigns a value to each subset of objects. The goal is to partitio ... [...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] |
A-CEEI Mechanism
Approximate Competitive Equilibrium from Equal Incomes (A-CEEI) is a procedure for fair item assignment. It was developed by Eric Budish. Background CEEI (Competitive Equilibrium from Equal Incomes) is a fundamental rule for fair division of divisible resources. It divides the resources according to the outcome of the following hypothetical process: * Each agent receives a single unit of fiat money. This is the Equal Incomes part of CEEI. * The agents trade freely until the market attains a Competitive Equilibrium. This is a price-vector and an allocation, such that (a) each allocated bundle is optimal to its agent given his/her income - the agent cannot purchase a better bundle with the same income, and (b) the market clears - the sum of all allocations exactly equals the initial endowment. The equilibrium allocation is provably envy free and Pareto efficient. Moreover, when the agents have linear utility functions, the CEEI allocation can be computed efficiently. Unfortunately ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marginal Utility
In economics, utility is the satisfaction or benefit derived by consuming a product. The marginal utility of a Goods (economics), good or Service (economics), service describes how much pleasure or satisfaction is gained by consumers as a result of the increase or decrease in Consumption (economics), consumption by one unit. There are three types of marginal utility. They are positive, negative, or zero marginal utility. For instance, you like eating pizza, the second piece of pizza brings you more satisfaction than only eating one piece of pizza. It means your marginal utility from purchasing pizza is positive. However, after eating the second piece you feel full, and you would not feel any better from eating the third piece. This means your marginal utility from eating pizza is zero. Moreover, you might feel sick if you eat more than three pieces of pizza. At this time, your marginal utility is negative. In other words, a negative marginal utility indicates that every unit of good ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |