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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 (ie <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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If, for <math>1\leq i\leq N</math>, <math>A_i\subseteq\Bbb{R}^m</math> is non-empty, [[convex]], and [[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]].
* [[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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