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List Of Unsolved Problems In Fair Division
This page lists notable open problems related to fair division - a field in the intersection of mathematics, computer science, political science and economics. Open problems in fair cake-cutting Query complexity of envy-free cake-cutting In the problem of envy-free cake-cutting, there is a cake modeled as an interval, and ''n'' agents with different value measures over the cake. The value measures are accessible only via queries of the form "evaluate a given piece of cake" or "mark a piece of cake with a given value". With ''n=2'' agents, an envy-free division can be found using two queries, via divide and choose. With ''n>2'' agents, there are several open problems regarding the number of required queries. 1. First, assume that the entire cake must be allocated (i.e., there is ''no disposal''), and pieces may be disconnected. ''How many queries are required?'' * Lower bound: \Omega(n^2); * Upper bound: O\left(n^\right). 2. Next, assume that some cake may be left unal ... [...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] |
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 is 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 first that m identical items have to be al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost All
In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite set, finite, countable set, countable, or null set, null. In contrast, "almost no" means "a negligible quantity"; that is, "almost no elements of X" means "a negligible quantity of elements of X". Meanings in different areas of mathematics Prevalent meaning Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) except for finite set, finitely many". This use occurs in philosophy as well. Similarly, "almost all" can mean "all (elements of an uncountable set) except for countable set, countably many". Examples: * Almost all positive integers are greater than 1012. * Almost all prime numbers are odd (2 is the only ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Competitive Equilibrium
Competitive equilibrium (also called: Walrasian equilibrium) is a concept of economic equilibrium, introduced by Kenneth Arrow and Gérard Debreu in 1951, appropriate for the analysis of commodity markets with flexible prices and many traders, and serving as the benchmark of efficiency in economic analysis. It relies crucially on the assumption of a competitive environment where each trader decides upon a quantity that is so small compared to the total quantity traded in the market that their individual transactions have no influence on the prices. Competitive markets are an ideal standard by which other market structures are evaluated. Definitions A competitive equilibrium (CE) consists of two elements: * A price function P. It takes as argument a vector representing a bundle of commodities, and returns a positive real number that represents its price. Usually the price function is linear - it is represented as a vector of prices, a price for each commodity type. * An allocation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Round-robin Scheduling
Round-robin (RR) is one of the algorithms employed by process and network schedulers in computing. Guowang Miao, Jens Zander, Ki Won Sung, and Ben Slimane, Fundamentals of Mobile Data Networks, Cambridge University Press, , 2016. As the term is generally used, time slices (also known as time quanta) are assigned to each process in equal portions and in circular order, handling all processes without priority (also known as cyclic executive). Round-robin scheduling is simple, easy to implement, and starvation-free. Round-robin scheduling can be applied to other scheduling problems, such as data packet scheduling in computer networks. It is an operating system concept. The name of the algorithm comes from the round-robin principle known from other fields, where each person takes an equal share of something in turn. Process scheduling To schedule processes fairly, a round-robin scheduler generally employs time-sharing, giving each job a time slot or ''quantum'' (its allowa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Price Of Fairness
In the theory of fair division, the price of fairness (POF) is the ratio of the largest economic welfare attainable by a division to the economic welfare attained by a ''fair'' division. The POF is a quantitative measure of the loss of welfare that society has to take in order to guarantee fairness. In general, the POF is defined by the following formula:POF=\frac The exact price varies greatly based on the kind of division, the kind of fairness and the kind of social welfare we are interested in. The most well-studied type of social welfare is '' utilitarian social welfare'', defined as the sum of the (normalized) utilities of all agents. Another type is '' egalitarian social welfare'', defined as the minimum (normalized) utility per agent. Numeric example In this example we focus on the ''utilitarian price of proportionality'', or UPOP. Consider a heterogeneous land-estate that has to be divided among 100 partners, all of whom value it as 100 (or the value is normalized to 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sperner's Lemma
In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an simplex contains a cell whose vertices all have different colors. The initial result of this kind was proved by Emanuel Sperner, in relation with proofs of invariance of domain. Sperner colorings have been used for effective computation of fixed points and in root-finding algorithms, and are applied in fair division (cake cutting) algorithms. According to the Soviet ''Mathematical Encyclopaedia'' (ed. I.M. Vinogradov), a related 1929 theorem (of Knaster, Borsuk and Mazurkiewicz) had also become known as the ''Sperner lemma'' – this point is discussed in the English translation (ed. M. Hazewinkel). It is now commonly known as the Knaster–Kuratowski–Mazurkiewicz lemma. Statement One-dimensional case In one dimension, Spern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envy-cycles Procedure
The envy-graph procedure (also called the envy-cycles procedure) is a procedure for fair item allocation. It can be used by several people who want to divide among them several discrete items, such as heirlooms, sweets, or seats in a class. Ideally, we would like the allocation to be envy-free (EF). i.e., to give each agent a bundle that they prefer over the bundles of all other agents. However, the items are discrete and cannot be cut, so an envy-free assignment might be impossible (for example, consider a single item and two agents). The envy-graph procedure aims to achieve the "next-best" option -- ''envy-freeness up to at most a single good'' (EF1): it finds an allocation in which the envy of every person towards every other person is bounded by the maximum marginal utility it derives from a single item. In other words, for every two people ''i'' and ''j'', there exists an item such that, if that item is removed, ''i'' does not envy ''j''. The procedure was presented by Lipt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximate Competitive Equilibrium From Equal Incomes
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. Unfortunate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envy-free Matching
In economics and social choice theory, an envy-free matching (EFM) is a matching between people to "things", which is envy-free in the sense that no person would like to switch their "thing" with that of another person. This term has been used in several different contexts. In unweighted bipartite graphs In an unweighted bipartite graph G = (''X''+''Y'', ''E''), an envy-free matching is a matching in which no unmatched vertex in ''X'' is adjacent to a matched vertex in ''Y''. Suppose the vertices of ''X'' represent people, the vertices of ''Y'' represent houses, and an edge between a person ''x'' and a house ''y'' represents the fact that ''x'' is willing to live in ''y''. Then, an EFM is a partial allocation of houses to people such that each house-less person does not envy any person with a house, since they do not like any allocated house anyway. Every matching that saturates ''X'' is envy-free, and every empty matching is envy-free. Moreover, if , ''NG''(''X''), ≥ , X, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divide And Choose
Divide and choose (also cut and choose or I cut, you choose) is a procedure for fair division of a continuous resource between two parties. It involves a heterogeneous good or resource and two partners who have different preferences over parts of the cake (both want as much of it as possible). The procedure proceeds as follows: one person divides the resource into two pieces; the other person selects one of the pieces; the cutter receives the remaining piece. Since ancient times some have used the procedure to divide land, food and other resources between two parties. Currently, there is an entire field of research, called fair cake-cutting, devoted to various extensions and generalizations of cut-and-choose. Divide and choose is envy-free in the following sense: each of the two partners can act in a way that guarantees that, according to their own subjective taste, their allocated share is at least as valuable as the other share, regardless of what the other partner does. Desc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Co-NP-complete
In complexity theory, computational problems that are co-NP-complete are those that are the hardest problems in co-NP, in the sense that any problem in co-NP can be reformulated as a special case of any co-NP-complete problem with only polynomial overhead. If P is different from co-NP, then all of the co-NP-complete problems are not solvable in polynomial time. If there exists a way to solve a co-NP-complete problem quickly, then that algorithm can be used to solve all co-NP problems quickly. Each co-NP-complete problem is the complement of an NP-complete problem. There are some problems in both NP and co-NP, for example all problems in P or integer factorization. However, it is not known if the sets are equal, although inequality is thought more likely. See co-NP and NP-complete for more details. Fortune showed in 1979 that if any sparse language is co-NP-complete (or even just co-NP-hard), then , a critical foundation for Mahaney's theorem. Formal definition A decisio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
