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	Irrigation games are [[cooperative game]]s which model cost sharing problems on networks (more precisely, on [[rooted tree]]s). 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\|Aadland and Kolpin 1998]], but the formal concept and the characterization of the game class is introduced by [[#MJPMPRAR\|Márkus et al. 2011]].

	~~== Mathematical definition ==~~
	~~The definition of [[#MJPMPRAR\|Márkus et al. 2011]] is the following:~~
	A ''graph'' is a pair <math>G =(V;A )</math>, where the elements of <math>V</math> are called vertices or nodes, and <math>A</math> 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 <math>i \leq j</math>, if the unique path from the root to <math>j</math> passes through <math>i</math>. For any <math>e \in A,</math> <math>e = \overline{ij}</math> means <math>e</math> is an edge between vertices <math>i; j \in V</math> such that <math>i \leq j</math>.

	~~Let~~ <~~math>c : A \rightarrow \mathbb{R}_+</math~~>. ~~Then~~ <~~math>c</math~~> and ~~<math>(G; c</math>) are called ''cost function'' and ''cost-tree'' respectively: for any <math>e \~~in A,</~~math> <math>e = \overline{ij},<~~/~~math> <math>c_e<~~/~~math>~~ is ~~the cost~~ of ~~joining player~~ <~~math~~>j<~~/math~~> to ~~player <math>i</math>~~.		Out of China<br>Cruising from Chongqing to Shanghai, you will not only recognize the most elite sections of Yangtze scenery, but also knowledge the visits to the wonder Mt.<br>Huangshan and cultural Nanjing metropolis.It plays an important role in Chinese historical past.Among the so numerous towns in China, China could be the most feathered one particular for its [http://tinyurl.com/pch83be ugg boots sale] exceptional customs and society.Arriving and departing, no matter whether by air, prepare or bus, is a sleek course of action.<br><br>The immense valley is home to more than a dozen ethnic groups. Some, like the minority individuals, are identified only in this article. The marketplaces deliver the mountain tribes with each other. To proceed his expeditions, Rock experienced to get his complete entourage across the large rivers in China.The city highlight the actual daily life of the folks and the ancient society further than your anticipations.One issue I want to remark on about all of Best China tours was the wonderful function of our tutorial, Marco.<br><br>You could come across that there are considerably too quite a few issues to see in this article in a single short trip.<br>In superb colour he recorded the plant life he observed on special photographic glass plates. Sending 1000's of specimens back to the West, the Plant Hunters changed the gardens of the earth eternally. The sky of is blue, the clouds float across the planet can only be explained as a clean lofty.<br><br>The trees and the bamboos seemed as if they were being all gilded. The floor was protected with tender grasses and the beaded dewdrops stood on their suggestions and reflected the sunshine. Everything was incredibly nicely organised from start to end. <br>Its central part is a windswept and freezing wilderness the measurement of China. But [http://tinyurl.com/pch83be ugg boots] this demanding spot is property to unbelievable wildlife.Outside the airport, there are mini buses to get travelers to the scenic places [http://tinyurl.com/pch83be ugg boots usa] of China (about two yuan).Touring to China is like traveling to one more world.<br><br>Its several well known scenic sights will captivate you with their exclusive charms. However, it is generally not sufficient for a traveler [http://tinyurl.com/pch83be ugg boots usa] only to go to these scenic spots.I like this city incredibly much for China [http://tinyurl.com/pch83be http://tinyurl.com/pch83be] has a lot of exceptional points I by no means discovered in other locations.<br><br>For anyone intrigued in Chinese historic record and wishing to know a lot more on it, China is one city that need to not be missed.All sorts of [http://Www.Bbc.Co.uk/search/?q=displays displays] are held through the four seasons. A golden route that illustrates the elegance of [http://tinyurl.com/pch83be cheap ugg boots] the eastern China incorporates the visits to the contemporary society of Shanghai, classical yard city of Suzhou and well known water town Zhouzhuang.What a fantastic excursion we just had our expectations had been exceeded, the foods the cleanness of all metropolitan areas and rural parts of China and we could not have envision a superior time if we tried out.

	~~Assume that~~ the ~~cost-tree problems have fixed~~, ~~at least two~~, ~~number~~ of ~~players (<math>\# V \geq 3~~, ~~\# N\geq 2</math>)~~. ~~Let~~ <~~math~~>~~(G; c)~~<~~/math~~> be a ~~cost-tree, and~~ <~~math~~>~~N</math> be~~ the ~~set~~ of the ~~players (~~the ~~vertices but~~ the ~~root)~~. ~~Consider an <math>S\subseteq N</math> non-empty coalition~~, ~~then~~ the ~~cost of connecting~~ the ~~players of~~ <~~math~~>S<~~/math~~> to the ~~root is given by~~ the ~~cost of~~ the ~~minimal rooted tree which covers coalition~~ <~~math~~>~~S</math>. By this method for each cost-tree we can define~~ a ~~game, called ''irrigation game''~~. ~~Formally~~:

	~~'''Definition''' (Irrigation game): For any cost-tree <math>(G; c)<~~/~~math>, let <math>N=V\setminus\{root\}<~~/~~math> be~~ the ~~player set~~, ~~and for any coalition <math>S</math> (~~the ~~empty sum is 0) let~~

	~~<center><math>v_{(G; c)}(S)=\sum \limits_{e\in A_s} c(e).<~~/~~math><~~/~~center>~~

	~~== Example ==~~
	~~[[File:Example of irrigation game~~.~~png~~]]

	~~The above graph represents a simple irrigation game played by three players. <math>G=~~(~~V,A~~)~~</math>~~ is ~~as follows:~~ <~~math~~>~~V=\{root,1,2,3\}, A=\{\overline{root1},\overline{root2}, \overline{23}\}~~<~~/math~~>, ~~and the cost function <math>c<~~/~~math> is defined as <math>c(\overline{root1})=11,\ \ c(\overline{root2})=5<~~/~~math> and <math>c(\overline{23})=7<~~/~~math>~~.

	~~Then the irrigation game is the following~~: ~~<math>v_{(G; c)}(\emptyset)=0<~~/~~math>, <math>v_{(G; c)}(\{1\})=11<~~/~~math>, <math>v_{(G; c)}(\{2\})=5<~~/~~math>, <math>v_{(G; c)}(\{3\})=7<~~/~~math>,~~ <~~math~~>~~v_{(G; c)}(\{1,2\})=16~~<~~/math~~>, ~~<math>v_{(G; c)}(\{1,3\})=23<~~/~~math>, <math>v_{(G; c)}(\{2,3\})=12<~~/~~math>, <math>v_{(G; c)}(\{1,2,3\})=23~~.</~~math>~~

	~~== Properties and Results~~ ==
	~~[[#MJPMPRAR\|Márkus et al. 2011~~]~~] reported several important properties of~~ the ~~class of the irrigation games~~. ~~First,~~ the ~~well-known class~~ of ~~the airport games is a proper subset of the class of irrigation games, since the airport games are defined on special rooted trees, on chains (see~~: ~~[[Airport problem~~]~~]). Second,~~ the ~~class of irrigation games is a non-convex cone which is a proper subset of~~ the ~~finite convex cone spanned by~~ the ~~duals~~ of ~~the unanimity games~~, ~~therefore every irrigation game is concave~~.

	~~[[#MJPMPRAR\|Márkus et al. 2011]] also extended the [[Lloyd Shapley\|Shapley]] and Young axiomatizations of~~ the ~~[[Shapley value]] to~~ the ~~class~~ of ~~irrigation games~~ and ~~showed that the Shapley value~~ of ~~an irrigation game is always in the [[Core (economics)\|core]] of the game.~~

	~~== References ==~~
	~~{{wikicite \|id=Aadland \|reference=Aadland D, Kolpin V (1998). Shared irrigation costs: An empirical~~ and ~~axiomatic analysis. ''Mathematica Social Sciences'' 35:203-218}}~~

	~~{{wikicite \|id=MJPMPRAR \|reference=Márkus J, Pintér M, Radványi A (2011)~~. ~~The Shapley value for airport and~~
	~~irrigation games. ''MPRA Paper No. 30031'' http://mpra.ub.uni-muenchen.de/30031/}}~~



	~~[[Category:Cooperative games]]~~

en>Dthomsen8: clean up, typos fixed: et al → et al. (2) using AWB

2012-06-22T13:31:42Z

clean up, typos fixed: et al → et al. (2) using AWB

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Irrigation games are [[cooperative game]]s which model cost sharing problems on networks (more precisely, on [[rooted tree]]s). 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|Aadland and Kolpin 1998]], but the formal concept and the characterization of the game class is introduced by [[#MJPMPRAR|Márkus et al. 2011]].

== Mathematical definition ==
The definition of [[#MJPMPRAR|Márkus et al. 2011]] is the following:
A ''graph'' is a pair <math>G =(V;A )</math>, where the elements of <math>V</math> are called vertices or nodes, and <math>A</math> 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 <math>i \leq j</math>, if the unique path from the root to <math>j</math> passes through <math>i</math>. For any <math>e \in A,</math> <math>e = \overline{ij}</math> means <math>e</math> is an edge between vertices <math>i; j \in V</math> such that <math>i \leq j</math>.

Let <math>c : A \rightarrow \mathbb{R}_+</math>. Then <math>c</math> and <math>(G; c</math>) are called ''cost function'' and ''cost-tree'' respectively: for any <math>e \in A,</math> <math>e = \overline{ij},</math> <math>c_e</math> is the cost of joining player <math>j</math> to player <math>i</math>.

Assume that the cost-tree problems have fixed, at least two, number of players (<math>\# V \geq 3, \# N\geq 2</math>). Let <math>(G; c)</math> be a cost-tree, and <math>N</math> be the set of the players (the vertices but the root). Consider an <math>S\subseteq N</math> non-empty coalition, then the cost of connecting the players of <math>S</math> to the root is given by the cost of the minimal rooted tree which covers coalition <math>S</math>. By this method for each cost-tree we can define a game, called ''irrigation game''. Formally:

'''Definition''' (Irrigation game): For any cost-tree <math>(G; c)</math>, let <math>N=V\setminus\{root\}</math> be the player set, and for any coalition <math>S</math> (the empty sum is 0) let

<center><math>v_{(G; c)}(S)=\sum \limits_{e\in A_s} c(e).</math></center>

== Example ==
[[File:Example of irrigation game.png]]

The above graph represents a simple irrigation game played by three players. <math>G=(V,A)</math> is as follows: <math>V=\{root,1,2,3\}, A=\{\overline{root1},\overline{root2}, \overline{23}\}</math>, and the cost function <math>c</math> is defined as <math>c(\overline{root1})=11,\ \ c(\overline{root2})=5</math> and <math>c(\overline{23})=7</math>.

Then the irrigation game is the following: <math>v_{(G; c)}(\emptyset)=0</math>, <math>v_{(G; c)}(\{1\})=11</math>, <math>v_{(G; c)}(\{2\})=5</math>, <math>v_{(G; c)}(\{3\})=7</math>, <math>v_{(G; c)}(\{1,2\})=16</math>, <math>v_{(G; c)}(\{1,3\})=23</math>, <math>v_{(G; c)}(\{2,3\})=12</math>, <math>v_{(G; c)}(\{1,2,3\})=23.</math>

== Properties and Results ==
[[#MJPMPRAR|Márkus et al. 2011]] reported several important properties of the class of the irrigation games. First, the well-known class of the airport games is a proper subset of the class of irrigation games, since the airport games are defined on special rooted trees, on chains (see: [[Airport problem]]). Second, the class of irrigation games is a non-convex cone which is a proper subset of the finite convex cone spanned by the duals of the unanimity games, therefore every irrigation game is concave.

[[#MJPMPRAR|Márkus et al. 2011]] also extended the [[Lloyd Shapley|Shapley]] and Young axiomatizations of the [[Shapley value]] to the class of irrigation games and showed that the Shapley value of an irrigation game is always in the [[Core (economics)|core]] of the game.

== References ==
{{wikicite |id=Aadland |reference=Aadland D, Kolpin V (1998). Shared irrigation costs: An empirical and axiomatic analysis. ''Mathematica Social Sciences'' 35:203-218}}

{{wikicite |id=MJPMPRAR |reference=Márkus J, Pintér M, Radványi A (2011). The Shapley value for airport and
irrigation games. ''MPRA Paper No. 30031'' http://mpra.ub.uni-muenchen.de/30031/}}

[[Category:Cooperative games]]