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| {{Orphan|date=February 2009}}
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| ''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 <math>a^j_k</math> amount of the ''kth'' resource. The products can be sold at a given market price <math>\vec{c}</math> while the resources themselves can not. Each of the ''N'' players is given a vector <math>\vec{b^i}=(b^i_1,...,b^i_m)</math> 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 <math>P(S)</math> as follows.
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| {| border="1" cellspacing="0" cellpadding="5" align="center"
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| |<math> v(S)=\max_{x\geq 0} (c_1x_1+...+c_nx_n) </math>
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| <math>s.t. \quad a^1_jx_1+a^2_jx_2+...+a^n_jx_n\leq \sum_{i\in S}b^i_j \quad \forall j=1,2,...,m</math>
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| == The core of the LP game ==
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| Every LP game ''v'' is a [[totally balanced game]]. So every subgame of ''v'' has a non-empty [[Core (game theory)|core]]. One [[imputation]] can be computed by solving the [[dual problem]] of <math>P(N)</math>. Let <math>\alpha</math> be the optimal dual solution of <math>P(N)</math>. The payoff to player'' i'' is <math>x^i=\sum_{k=1}^m\alpha_k b^i_k</math>. It can be proved by the [[Duality (mathematics)|duality]] theorems that <math>\vec{x}</math> is in the core of ''v''.
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| An important interpretation of the imputation <math>\vec{x}</math> is that under the current market, the value of each resource ''j'' is exactly <math>\alpha_j</math>, although it is not valued in themselves. So the payoff one player i should receive is the total value of the resources he possesses.
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| However, not all the [[imputation]]s in the core can be obtained from the optimal dual solutions. There are a lot of discussions on this problem. One of the mostly widely used method is to consider the [[r-fold replication]] of the original problem. It can be shown that if an imputation ''u'' is in the core of the r-fold replicated game for all r, then ''u'' can be obtained from the optimal dual solution.
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| == References ==
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| * {{Citation | last1=OWEN | first1=Guillermo | title=[[On the Core of Linear Production Games]] | publisher=[[Mathematical Programming ]] | year=1975 | journal=Mathematical Programming | volume=9 | pages=358–370 | doi=10.1007/BF01681356}}
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| [[Category:Game theory]]
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Latest revision as of 09:38, 23 February 2014
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