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| | Neurologist Benton Fritter from Shippagan, loves to spend some time snooker, riot points and warhammer. Last month just made a journey to Garajonay National Park.<br><br>My homepage: [http://wyevalleyfreeschool.co.uk/?attachment_id=1205 rp gratuits league of legends] |
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| In [[mathematics]], and in particular the study of [[game theory]], a [[function (mathematics)|function]] is '''graph continuous''' if it exhibits the following properties. The concept was originally defined by [[Partha Dasgupta]] and [[Eric Maskin]] in 1986 and is a version of [[continuous function|continuity]] that finds application in the study of [[continuous game]]s.
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| ==Notation and preliminaries==
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| Consider a [[game]] with <math>N</math> agents with agent <math>i</math> having strategy <math>A_i\subseteq\Bbb{R}</math>; write <math>\mathbf{a}</math> for an N-tuple of actions (i.e. <math>\mathbf{a}\in\prod_{j=1}^NA_j</math>) and <math>\mathbf{a}_{-i}=(a_1,a_2,\ldots,a_{i-1},a_{i+1},\ldots,a_N)</math> as the vector of all agents' actions apart from agent <math>i</math>.
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| Let <math>U_i:A_i\longrightarrow\Bbb{R}</math> be the payoff function for agent <math>i</math>.
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| A '''game''' is defined as <math>[(A_i,U_i); i=1,\ldots,N]</math>.
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| If a graph is continuous you should connect it if it's not then don't connect it.
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| ==Definition==
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| Function <math>U_i:A\longrightarrow\Bbb{R}</math> is '''graph continuous''' if for all <math>\mathbf{a}\in A</math> there exists a function <math>F_i:A_{-i}\longrightarrow A_i</math> such that <math>U_i(F_i(\mathbf{a}_{-i}),\mathbf{a}_{-i})</math> is continuous at <math>\mathbf{a}_{-i}</math>.
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| Dasgupta and Maskin named this property "graph continuity" because, if one plots a graph of a player's payoff as a function of his own strategy (keeping the other players' strategies fixed), then a graph-continuous payoff function will result in this graph changing continuously as one varies the strategies of the other players.
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| The property is interesting in view of the following theorem.
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| If, for <math>1\leq i\leq N</math>, <math>A_i\subseteq\Bbb{R}^m</math> is non-empty, [[Convex function|convex]], and [[compact set|compact]]; and if <math>U_i:A\longrightarrow\Bbb{R}</math> is [[quasi-concave function|quasi-concave]] in <math>a_i</math>, [[upper semi-continuous]] in <math>\mathbf{a}</math>, and graph continuous, then the game <math>[(A_i,U_i); i=1,\ldots,N]</math> possesses a [[pure strategy]] [[Nash equilibrium]].
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| ==References==
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| * [[Partha Dasgupta]] and [[Eric Maskin]] 1986. ''The existence of equilibrium in discontinuous economic games, I: theory''. The Review of Economic Studies, 53(1):1-26
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| {{DEFAULTSORT:Graph Continuous Function}}
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| [[Category:Game theory]]
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Neurologist Benton Fritter from Shippagan, loves to spend some time snooker, riot points and warhammer. Last month just made a journey to Garajonay National Park.
My homepage: rp gratuits league of legends