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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] |
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] |
Additive Map
In algebra, an additive map, Z-linear map or additive function is a function f that preserves the addition operation: f(x + y) = f(x) + f(y) for every pair of elements x and y in the domain of f. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. For a specific case of this definition, see additive polynomial. More formally, an additive map is a \Z-module homomorphism. Since an abelian group is a \Z-module, it may be defined as a group homomorphism between abelian groups. A map V \times W \to X that is additive in each of two arguments separately is called a bi-additive map or a \Z-bilinear map. Examples Typical examples include maps between rings, vector spaces, or modules that preserve the additive group. An additive map does not necessarily preserve any other structure of the object; for example, the product operation of a ring. If f and g are additive maps, then the map f + g (defined pointwise) is additiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Austin Moving-knife Procedure
The Austin moving-knife procedures are procedures for equitable division of a cake. They allocate each of ''n'' partners, 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. 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 partners and half-cakes The basic procedures ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Pie-cutting
The fair pie-cutting problem is a variation of the fair cake-cutting problem, in which the resource to be divided is circular. As an example, consider a birthday cake shaped as a disk. The cake should be divided among several children such that no child envies another child (as in a standard cake-cutting problem), with the additional constraint that the cuts must be radial, so that each child receives a circular sector. A possible application of the pie model might be for dividing an island’s shoreline into connected lots. Another possible application is in division of periodic time, such as dividing a daily cycle into "on-call" periods. Model A pie is usually modeled as the 1-dimensional interval ,2π(or ,1, in which the two endpoints are identified. This model was introduced in 1985 and later in 1993. Every procedure for fair cake-cutting can also be applied to cutting a pie by just ignoring the fact that the two endpoints are identified. For example, if the cake-cuttin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levmore–Cook Moving-knives Procedure
The Levmore–Cook moving-knives procedure is a procedure for envy-free cake-cutting among three partners. It is named after Saul X. Levmore and Elizabeth Early Cook who presented it in 1981. It assumes that the cake is two-dimensional. It requires a referee, two knives and four cuts, so some partners may receive disconnected pieces. Procedure We name the partners Alice, Bob and Carl. Initially, Alice cuts the cake to three pieces equal in her eyes. Bob and Carl each point to their favorite piece. Easy case: Bob and Carl point to different pieces. Each receives his favorite piece and Alice the remaining piece. Hard case: Bob and Carl point to the same piece. Say this is piece X and the other pieces are Y and Z. Now the referee (ideally it could be Alice) takes one knife and moves it horizontally over piece X, and Alice takes the other knife and moves it vertically over piece X: * Knife #1 is moved ''horizontally'' and ''continuously'' by the referee from the left of piece X to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Disposal
In various parts of economics, the term free disposal implies that resources can be discarded without any cost. For example, a fair division setting with free disposal is a setting where some resources have to be divided fairly, but some of the resources may be left undivided, discarded or donated. Examples of situations with free disposal are allocation of food, clothes jewels etc. Examples of situations ''without'' free disposal are: * Chore division - since all chores must be done. * Allocation of land with an old structure - since the structure may have to be destructed, and destruction is costly. * Allocation of an old car - since the car may have to be carried away to used cars garage, and moving it may be costly. * Allocation of shares in a firm that may have debts - since the firm cannot be disposed of without paying its debts first. The free disposal assumption may be useful for several reasons: * It enables truthful cake-cutting algorithms: The option to discard some of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipschitz-continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the ''Lipschitz constant'' of the function (or '' modulus of uniform continuity''). For instance, every function that has bounded first derivatives is Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. We have the following chain of strict inclus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PPAD
In computer science, PPAD ("Polynomial Parity Arguments on Directed graphs") is a complexity class introduced by Christos Papadimitriou in 1994. PPAD is a subclass of TFNP based on functions that can be shown to be total by a parity argument. The class attracted significant attention in the field of algorithmic game theory because it contains the problem of computing a Nash equilibrium: this problem was shown to be complete for PPAD by Daskalakis, Goldberg and Papadimitriou with at least 3 players and later extended by Chen and Deng to 2 players.*. Definition PPAD is a subset of the class TFNP, the class of function problems in FNP that are guaranteed to be total. The TFNP formal definition is given as follows: :A binary relation P(''x'',''y'') is in TFNP if and only if there is a deterministic polynomial time algorithm that can determine whether P(''x'',''y'') holds given both ''x'' and ''y'', and for every ''x'', there exists a ''y'' such that P(''x'',''y'') holds. Subclas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robertson–Webb Query Model
In computer science, the Robertson–Webb (RW) query model is a model of computation used by algorithms for the problem of fair cake-cutting. In this problem, there is a resource called a "cake", and several agents with different value measures on the cake. The goal is to divide the cake among the agents such that each agent will consider his/her piece as "fair" by his/her personal value measure. Since the agents' valuations can be very complex, they cannot - in general - be given as inputs to a fair division algorithm. The RW model specifies two kinds of ''queries'' that a fair division algorithm may ask the agents: Eval and Cut. Informally, an Eval query asks an agent to specify his/her value to a given piece of the cake, and a Cut query (also called a Mark query) asks an agent to specify a piece of cake with a given value. Despite the simplicity of the model, many classic cake-cutting algorithms can be described only by these two queries. On the other hand, there are fair cake-cut ... [...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] |
Robertson–Webb Rotating-knife Procedure
The Robertson–Webb rotating-knife procedure is a procedure for envy-free cake-cutting of a two-dimensional cake among three partners. It makes only two cuts, so each partner receives a single connected piece. Its main advantage over the earlier Stromquist moving-knives procedure and the later Barbanel–Brams moving-knives procedure is that it requires only a single moving-knife. This advantage uses the two-dimensional nature of the cake. Procedure Initially, each partner makes a vertical cut such that the cake to its left is worth for him exactly 1/3. The leftmost cut is selected. Suppose this cut belongs to Alice. So Alice receives the leftmost piece and her value is exactly 1/3. The remainder has to be divided between the remaining partners (Bob and Carl). Note that Alice's part is worth ''at most'' 1/3 and the remainder is worth ''at least'' 2/3 for Bob and Carl. So, if Bob and Carl each receive at least half of the remainder, they do not envy. The challenge is to make sur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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