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Linear Production Game
Linear production game (LP Game) is a N-person game in which the value of a coalition can be obtained by solving a linear programming problem. It is widely used in the context of resource allocation and payoff distribution. Mathematically, there are ''m'' types of resources and ''n'' products can be produced out of them. Product ''j'' requires a^j_k amount of the ''kth'' resource. The products can be sold at a given market price \vec while the resources themselves can not. Each of the ''N'' players is given a vector \vec=(b^i_1,...,b^i_m) of resources. The value of a coalition ''S'' is the maximum profit it can achieve with all the resources possessed by its members. It can be obtained by solving a corresponding linear programming problem P(S) as follows. Core Every LP game ''v'' is a totally balanced game. So every subgame of ''v'' has a non-empty core. One imputation can be computed by solving the dual problem In mathematical optimization theory, duality or the duality pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coalition
A coalition is a group formed when two or more people or groups temporarily work together to achieve a common goal. The term is most frequently used to denote a formation of power in political or economical spaces. Formation According to ''A Guide for Political Parties'' published by National Democratic Institute and The Oslo Center for Peace and Human Rights, there are five steps of coalition-building: # Developing a party strategy: The first step in coalition-building involves developing a party strategy that will prepare for successful negotiation. The more effort parties place on this step, the more likely they are to identify strategic partners, negotiate a good deal and avoid some of the common mistakes associated with coalition-building. # Negotiating a coalition: Based on the strategy that each party has prepared, in step 2 the parties come together to negotiate and hopefully reach agreement on the terms for the coalition. Depending on the context and objectives of the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Balanced Game
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Core (game Theory)
In cooperative game theory, the core is the set of feasible allocations that cannot be improved upon by a subset (a ''coalition'') of the economy's agents. A coalition is said to ''improve upon'' or ''block'' a feasible allocation if the members of that coalition are better off under another feasible allocation that is identical to the first except that every member of the coalition has a different consumption bundle that is part of an aggregate consumption bundle that can be constructed from publicly available technology and the initial endowments of each consumer in the coalition. An allocation is said to have the ''core property'' if there is no coalition that can improve upon it. The core is the set of all feasible allocations with the core property. Origin The idea of the core already appeared in the writings of , at the time referred to as the ''contract curve''. Even though von Neumann and Morgenstern considered it an interesting concept, they only worked with zero-sum ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imputation (game Theory)
In fully cooperative games players will opt to form coalitions when the value of the payoff is equal to or greater than if they were to work alone. The focus of the game is to find acceptable distributions of the payoff of the grand coalition. Distributions where a player receives less than it could obtain on its own, without cooperating with anyone else, are unacceptable - a condition known as ''individual rationality''. Imputations are distributions that are efficient and are individually rational. Theory For 2-player games the set of imputations coincides with the core, a popularly studied concept due to its stability against group deviations. The core is a solution concept of cooperative games and consists of multiple imputations, a set of distributions as a result of a game. The core cannot be improved upon by any coalition. However, problems will arise when it comes to selecting a set of imputations, it will require bargaining. Solutions Nash bargaining theory, a type of c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Problem
In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem. Therefore, the solution to the primal is an upper bound to the solution of the dual, and the solution of the dual is a lower bound to the solution of the primal. This fact is called weak duality. In general, the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition. This fact is called strong duality. Dual problem Usually the term "dual problem" refers to the ''Lagrangian dual problem'' but other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (mathematics)
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the dual of is . Such involutions sometimes have fixed points, so that the dual of is itself. For example, Desargues' theorem is self-dual in this sense under the ''standard duality in projective geometry''. In mathematical contexts, ''duality'' has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, ''linear algebra duality'' corresponds in this way to bilinear maps from pairs of vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On The Core Of Linear Production Games
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Mathematical Programming
''Mathematical Programming'' is a peer-reviewed scientific journal that was established in 1971 and is published by Springer Science+Business Media. It is the official journal of the Mathematical Optimization Society and consists of two series: ''A'' and ''B''. The "A" series contains general publications, the "B" series focuses on topical mathematical programming areas. The editor-in-chief of Series A is Jon Lee ( U Michigan); for Series B this is Sven Leyffer ( Argonne). History The journal has been published by Springer since January 1999. ''Mathematical Programming Studies'' is the predecessor of the Series B part of this journal. Abstracting and indexing ''Mathematical Programming'' is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2010 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
