Examine individual changes

'{{Short description|Concept in game theory}} 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. ==Notation and preliminaries== Consider a [[game]] with <math>N</math> agents with agent <math>i</math> having strategy <math>A_i\subseteq\mathbb{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>. Let <math>U_i:A_i\longrightarrow\mathbb{R}</math> be the payoff function for agent <math>i</math>. A '''game''' is defined as <math>[(A_i,U_i); i=1,\ldots,N]</math>. ==Definition== Function <math>U_i:A\longrightarrow\mathbb{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>. 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. The property is interesting in view of the following theorem. If, for <math>1\leq i\leq N</math>, <math>A_i\subseteq\mathbb{R}^m</math> is non-empty, [[Convex function|convex]], and [[compact set|compact]]; and if <math>U_i:A\longrightarrow\mathbb{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]]. ==References== * [[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 {{DEFAULTSORT:Graph Continuous Function}} [[Category:Game theory]] [[Category:Theory of continuous functions]] EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE<ref><ref></ref></ref>'