Graph continuous function: Difference between revisions

Revision as of 15:43, 29 March 2014

Revision as of 13:34, 11 May 2019

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==Notation and preliminaries==

Consider a [[game]] with <math>N</math> agents with agent <math>i</math> having strategy <math>A_i\subseteq\Bbbmathbb{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\Bbbmathbb{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>.

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==Definition==

Function <math>U_i:A\longrightarrow\Bbbmathbb{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.

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The property is interesting in view of the following theorem.

If, for <math>1\leq i\leq N</math>, <math>A_i\subseteq\Bbbmathbb{R}^m</math> is non-empty, [[Convex function|convex]], and [[compact set|compact]]; and if <math>U_i:A\longrightarrow\Bbbmathbb{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==