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| {{Orphan|date=May 2012}}
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| A '''bandwidth-sharing game''' is a type of [[resource allocation]] game designed to model the real-world allocation of [[Bandwidth_(computing)|bandwidth]] to many users in a network. The game is popular in [[game theory]] because the conclusions can be applied to real-life networks. The game is described as follows:
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| == The game ==
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| * <math>n</math> players
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| * each player <math>i</math> has utility <math>U_i(x)</math> for amount <math>x</math> of bandwidth
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| * user <math>i</math> pays <math>w_i</math> for amount <math>x</math> of bandwidth and receives net utility of <math>U_i(x)-w_i</math>
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| * the total amount of bandwidth available is <math>B</math>
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| We also use assumptions regarding <math>U_i(x)</math>
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| * <math>U_i(x)\ge0</math>
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| * <math>U_i(x)</math> is increasing and concave
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| * <math>U(x)</math> is continuous
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| The game arises from trying to find a price <math>p</math> so that every player individually optimizes their own welfare. This implies every player must individually find <math>argmax_xU_i(x)-px</math>. Solving for the [[maximum yields]] <math>U_i^'(x)=p</math>.
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| == The problem ==
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| With this maximum condition, the game then becomes a matter of finding a price that satisfies an equilibrium. Such a price is called a [[market clearing price]].
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| == A possible solution ==
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| A popular idea to find the price is a method called fair sharing.<ref>{{cite web|last=Tsitsiklis|first=Johari|title=Qualitative Properties of a-Fair Policies in Bandwidth-Sharing Networks|url=http://www.mit.edu/~jnt/Papers/P-11-alpha-fair.pdf|publisher=Massachusetts Institute of Technology|accessdate=15 May 2012}}</ref> In this game, every player <math>i</math> is asked for amount they are willing to pay for the given resource denoted by <math>w_i</math>. The resource is then distributed in <math>x_i</math> amounts by the formula <math>x_i=(\frac{w_i}{\sum_jw_j})*(B)</math>. This method yields an effective price <math>p=\frac{\sum_jw_j}{B}</math>.
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| This price can proven to be market clearing thus the distribution <math>x_1,...,x_n</math> is optimal. The proof is as so:
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| == Proof ==
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| <math>argmax_{x_i}U_i(x_i)-w_i</math>
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| <math>\implies argmax_{w_i}U_i(\frac{w_i}{\sum_jw_j}*B)-w_i</math><br />
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| <math>\implies U^'_i(\frac{w_i}{\sum_jw_j}*B)(\frac{1}{\sum_jw_j}*B-\frac{w_i}{(\sum_jw_j)^2}*B)-1=0</math><br />
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| <math>\implies U^'_i(x_i)(\frac{1}{p}-\frac{1}{p}*\frac{x_i}{B})-1=0</math><br />
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| <math>\implies U^'_i(x_i)(1-\frac{x_i}{B})=p </math>
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| Comparing this result to the equilibrium condition above, we see that when <math>\frac{x_i}{B}</math> is very small, the two conditions equal each other and thus, the fair sharing game is almost optimal.
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| ==References==
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| <references />
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| [[Category:Game theory]]
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