|
Austin Moving-knife Procedure
The Austin moving-knife procedures are procedures for equitable division of a fair cake-cutting, cake. To each of ''n'' partners, they allocate a piece of the cake which this partner values as ''exactly'' 1/n of the cake. This is in contrast to proportional division procedures, which give each partner ''at least'' 1/n of the cake, but may give more to some of the partners. When n=2, the division generated by Austin's procedure is an exact division and it is also envy-free cake-cutting, envy-free. Moreover, it is possible to divide the cake to any number ''k'' of pieces which both partners value as exactly 1/''k''. Hence, it is possible to divide the cake between the partners in any fraction (e.g. give 1/3 to Alice and 2/3 to George). When n>2, the division is neither exact nor envy-free, since each partner only values his own piece as 1/n, but may value other pieces differently. The main mathematical tool used by Austin's procedure is the intermediate value theorem (IVT). Two pa ... [...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 i ... [...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 believed 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 in the book of Book of Genesis, Genesis to resolve Abraham and Lot's conflict. This procedure solves the fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proportional Division
A proportional division is a kind of fair division in which a resource is divided among ''n'' partners with subjective valuations, giving each partner at least 1/''n'' of the resource by his/her own subjective valuation. Proportionality was the first fairness criterion studied in the literature; hence it is sometimes called "simple fair division". It was first conceived by Steinhaus. Example Consider a land asset that has to be divided among 3 heirs: Alice and Bob who think that it's worth 3 million dollars, and George who thinks that it's worth $4.5M. In a proportional division, Alice receives a land-plot that she believes to be worth at least $1M, Bob receives a land-plot that ''he'' believes to be worth at least $1M (even though Alice may think it is worth less), and George receives a land-plot that he believes to be worth at least $1.5M. Existence A proportional division does not always exist. For example, if the resource contains several indivisible items and the number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Division
Consensus splitting, also called exact 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 ... [...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 d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intermediate Value Theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two important corollaries: # If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). # The image of a continuous function over an interval is itself an interval. Motivation This captures an intuitive property of continuous functions over the real numbers: given ''f'' continuous on ,2/math> with the known values f(1) = 3 and f(2) = 5, then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper. Theorem The intermediate value theorem states the following: Consider the closed interval I = ,b/math> ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entitlement (fair Division)
In fair division, a person's entitlement is the value of the goods they are owed or deserve, i.e. the total value of the goods or resources that a player would ideally receive. For example, in party-list proportional representation, a party's seat entitlement (sometimes called its seat quota) is equal to its share of the vote, times the number of seats in the legislature. Dividing money Even when only money is to be divided and some fixed amount has been specified for each recipient, the problem can be complex. The amounts specified may be more or less than the amount of money, and the profit or loss will then need to be shared out. The proportional rule is normally used in law nowadays, and is the default assumption in the theory of bankruptcy. However, other rules can also be used. For example: * The Shapley value is one common method of deciding bargaining power, as can be seen in the airport problem. * Welfare economics on the other hand tries to determine allocations de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fink Protocol
The Fink protocol (also known as Successive Pairs or Lone Chooser) is a protocol for proportional division of a cake Cake is a flour confection usually made from flour, sugar, and other ingredients and is usually baked. In their oldest forms, cakes were modifications of bread, but cakes now cover a wide range of preparations that can be simple or elabor .... Its main advantage is that it can work in an online fashion, without knowing the number of partners in advance. When a new partner joins the party, the existing division is adjusted to give a fair share to the newcomer, with minimal effect on existing partners. Its main disadvantage is that, instead of giving each partner a single connected piece, it gives each partner a large number of "crumbs". Protocol We describe the protocol inductively for an increasing number of partners. When partner #1 enters the party, he just takes the entire cake. His value is thus 1. When partner #2 comes, partner #1 cuts the cake t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Near-exact Division
Consensus splitting, also called exact 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brams–Taylor–Zwicker Procedure
The Brams–Taylor–Zwicker procedure is a protocol for 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 sh ... among 4 partners. The procedure uses a variation of Austin's procedure for two partners and general fractions. That procedure allows two partners to divide an entire cake to k pieces, each of which is worth exactly 1/k for both of them. The main procedure works as follows. A. Use Austin's procedure with k=4 and partners #1 and #2. So we have 4 pieces that the first two partners believe to be of exactly identical value of 1/4. B. Partner #3 trims one piece to create a two-way tie for the largest; the partners now choose pieces in reverse order (#4, #3, #2, #1). Either #4 or #3 must take the trimmed piece. This creates an envy-free division for the entir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Division Protocols
A fair (archaic: faire or fayre) is a gathering of people for a variety of entertainment or commercial activities. Fairs are typically temporary with scheduled times lasting from an afternoon to several weeks. Fairs showcase a wide range of goods, products, and services, and often include competitions, exhibitions, and educational activities. Fairs can be thematic, focusing on specific industries or interests. Types Variations of fairs include: * Art fairs, including art exhibitions and arts festivals * Book Fairs in communities and schools provide an opportunity for readers, writers, publishers to come together and celebrate literature. * County fair (US) or county show (UK), a public agricultural show exhibiting the equipment, animals, sports and recreation associated with agriculture and animal husbandry. * Festival, an event ordinarily coordinated with a theme e.g. music, art, season, tradition, history, ethnicity, religion, or a national holiday. * Health fair, an even ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
