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Leximin Cake-cutting
Egalitarian cake-cutting is a kind of fair cake-cutting in which the fairness criterion is the egalitarian rule. The ''cake'' represents a continuous resource (such as land or time), that has to be allocated among people with different valuations over parts of the resource. The goal in egalitarian cake-cutting is to maximize the smallest value of an agent; subject to this, maximize the next-smallest value; and so on. It is also called leximin cake-cutting, since the optimization is done using the leximin order on the vectors of utilities. The concept of egalitarian cake-cutting was first described by Dubins and Spanier, who called it "optimal partition". Existence Leximin-optimal allocations exist whenever the set of allocations is a compact space. This is always the case when allocating discrete objects, and easy to prove when allocating a finite number of continuous homogeneous resources. Dubins and Spanier proved that, with a continuous ''heterogeneous'' resource (" cake") ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Cake-cutting
Fair cake-cutting is a kind of fair division problem. The problem involves a ''heterogeneous'' resource, such as a cake with different toppings, that is assumed to be ''divisible'' – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible. The division should be ''unanimously'' fair - each person should receive a piece that he or she believes to be a fair share. The "cake" is only a metaphor; procedures for fair cake-cutting can be used to divide various kinds of resources, such as land estates, advertisement space or broadcast time. The prototypical procedure for fair cake-cutting is divide and choose, which is mentioned already in the book of Genesis. It solves the fair division problem for two people. The modern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proportional Cake-cutting
A proportional cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the proportionality criterion, namely, that every partner feels that his allocated share is worth at least 1/''n'' of the total. Two assumptions are usually made when proportionality is discussed: * The valuations of the partners are ''non-atomic'', i.e., there are no indivisible elements with positive value. * The valuations of the partners are ''additive'', i.e., when a piece is divided, the value of the piece is equal to the sum of its parts. Formal definitions The cake is denoted by C. There are n people. Each person i has a value function V_i. A partition of the cake, X_1\sqcup \cdots \sqcup X_n = C, is called ''proportional'' if:V_i(X_i) \ge V_i(C)/n for every person i \in \. Procedures For two people, divide and choose is the classic solution. One person divides the resource into what they believe are equal halves, and the other person ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Egalitarian Equivalence
Egalitarian equivalence (EE) is a criterion of fair division. In an egalitarian-equivalent division, there exists a certain "reference bundle" Z such that each agent feels that his/her share is equivalent to Z. The EE fairness principle is usually combined with Pareto efficiency. A PEEEA is an allocation that is both Pareto efficient and egalitarian-equivalent. Definition A set of resources are divided among several agents such that every agent i receives a bundle X_i. Every agent i has a subjective preference relation \succeq_i which is a total order over bundle. These preference relations induce an equivalence relation in the usual way: X \sim_i Y iff X \succeq_i Y \succeq_i X. An allocation is called ''egalitarian-equivalent'' if there exists a bundle Z such that, for all i: :::X_i \sim_i Z An allocation is called ''PEEEA'' if it is both Pareto-efficient and egalitarian-equivalent. Motivation The EE criterion was introduced by Elisha Pazner and David Schmeidler in 1978. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equitable Cake-cutting
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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utilitarian Cake-cutting
Utilitarian cake-cutting (also called maxsum cake-cutting) is a rule for dividing a heterogeneous resource, such as a cake or a land-estate, among several partners with different cardinal utility functions, such that the ''sum'' of the utilities of the partners is as large as possible. It is a special case of the utilitarian social choice rule. Utilitarian cake-cutting is often not "fair"; hence, utilitarianism is often in conflict with fair cake-cutting. Example Consider a cake with two parts: chocolate and vanilla, and two partners: Alice and George, with the following valuations: The utilitarian rule gives each part to the partner with the highest utility. In this case, the utilitarian rule gives the entire chocolate to Alice and the entire Vanilla to George. The maxsum is 13. The utilitarian division is not fair: it is not proportional since George receives less than half the total cake value, and it is not envy-free since George envies Alice. Notation The cake is calle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Distribution
In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit ''a'', upper limit ''b'' and mode ''c'', where ''a'' < ''b'' and ''a'' ≤ ''c'' ≤ ''b''. Special cases Mode at a bound The distribution simplifies when ''c'' = ''a'' or ''c'' = ''b''. For example, if ''a'' = 0, ''b'' = 1 and ''c'' = 1, then the and CDF become: : : |
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] |
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] |
Resource Monotonic
Resource monotonicity (RM; aka aggregate monotonicity) is a principle of fair division. It says that, if there are more resources to share, then all agents should be weakly better off; no agent should lose from the increase in resources. The RM principle has been studied in various division problems. Allocating divisible resources Single homogeneous resource, general utilities Suppose society has m units of some homogeneous divisible resource, such as water or flour. The resource should be divided among n agents with different utilities. The utility of agent i is represented by a function u_i; when agent i receives y_i units of resource, he derives from it a utility of u_i(y_i). Society has to decide how to divide the resource among the agents, i.e, to find a vector y_1,\dots,y_n such that: y_1+\cdots+y_n = m. Two classic allocation rules are the egalitarian rule - aiming to equalize the utilities of all agents (equivalently: maximize the minimum utility), and the utilitari ... [...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] |
Population Monotonic
Population monotonicity (PM) is a principle of consistency in allocation problems. It says that, when the set of agents participating in the allocation changes, the utility of all agents should change in the same direction. For example, if the resource is good, and an agent leaves, then all remaining agents should receive at least as much utility as in the original allocation. The term "population monotonicity" is used in an unrelated meaning in the context of apportionment of seats in the congress among states. There, the property relates to the population of an individual state, which determines the state's ''entitlement.'' A population-increase means that a state is entitled to more seats. This different property is described in the page ''state-population monotonicity''. In fair cake cutting In the fair cake-cutting problem, classic allocation rules such as divide and choose are not PM. Several rules are known to be PM: * When the pieces may be ''disconnected'', any function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Cake-cutting
Fair cake-cutting is a kind of fair division problem. The problem involves a ''heterogeneous'' resource, such as a cake with different toppings, that is assumed to be ''divisible'' – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible. The division should be ''unanimously'' fair - each person should receive a piece that he or she believes to be a fair share. The "cake" is only a metaphor; procedures for fair cake-cutting can be used to divide various kinds of resources, such as land estates, advertisement space or broadcast time. The prototypical procedure for fair cake-cutting is divide and choose, which is mentioned already in the book of Genesis. It solves the fair division problem for two people. The modern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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