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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Irrigation game ： ウィキペディア英語版 Irrigation game Irrigation games are cooperative games which model cost sharing problems on networks (more precisely, on rooted trees). The irrigation game is a transferable utility game assigned to a cost-tree problem. A common example of this cost-tree problems are the irrigation networks. The irrigation ditch is represented by a graph, its nodes are water users, the edges are sections of the ditch. There is a cost of maintaining the ditch (each section has an own maintenance cost), and we are looking for the fair division of the costs among the users. The irrigation games are mentioned first by Aadland and Kolpin 1998, but the formal concept and the characterization of the game class is introduced by Márkus et al. 2011. == Mathematical definition == The definition of Márkus et al. 2011 is the following: A ''graph'' is a pair $G\; =(V;A\; )$, where the elements of $V$ are called vertices or nodes, and $A$ stands for the ordered pairs of vertices, called arcs or edges. A rooted tree is a graph in which any two vertices are connected by exactly one simple path, and one vertex has been designated the root, in which case the edges have a natural orientation, away from the root. The tree-order is the partial ordering on the vertices of a rooted tree with $i\; \backslash leq\; j$, if the unique path from the root to $j$ passes through $i$. For any $e\; \backslash in\; A,$ $e\; =\; \backslash overline$ means $e$ is an edge between vertices $i;\; j\; \backslash in\; V$ such that $i\; \backslash leq\; j$. Let $c\; :\; A\; \backslash rightarrow\; \backslash mathbb\_+$. Then $c$ and $(G;\; c$) are called ''cost function'' and ''cost-tree'' respectively: for any $e\; \backslash in\; A,$ $e\; =\; \backslash overline,$ $c\_e$ is the cost of joining player $j$ to player $i$. Assume that the cost-tree problems have fixed, at least two, number of players ($\backslash \#\; V\; \backslash geq\; 3,\; \backslash \#\; N\backslash geq\; 2$). Let $(G;\; c)$ be a cost-tree, and $N$ be the set of the players (the vertices but the root). Consider an $S\backslash subseteq\; N$ non-empty coalition, then the cost of connecting the players of $S$ to the root is given by the cost of the minimal rooted tree which covers coalition $S$. By this method for each cost-tree we can define a game, called ''irrigation game''. Formally: Definition (Irrigation game): For any cost-tree $(G;\; c)$, let $N=V\backslash setminus\backslash$ be the player set, and for any coalition $S$ (the empty sum is 0) let $v\_(S)=\backslash sum\; \backslash limits\_\; c(e).$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Irrigation game」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.