Graph continuous function: Difference between revisions

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{{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.

In [[mathematics]], particularly in [[game theory]] and [[mathematical economics]], a function is '''graph continuous''' if its [[Graph (function)|graph]]—the set of all input-output pairs—is a closed set in the [[product topology]] of the ___domain and codomain. In simpler terms, if a sequence of points on the graph converges, its limit point must also belong to the graph. This concept, related to the [[Closed graph theorem|closed graph property]] in [[functional analysis]], allows for a broader class of discontinuous payoff functions while enabling equilibrium analysis in economic models.

Graph continuity gained prominence through the work of [[Partha Dasgupta]] and [[Eric Maskin]] in their 1986 paper on the existence of equilibria in discontinuous economic games.<ref>{{cite journal |last1=Dasgupta |first1=Partha |last2=Maskin |first2=Eric |year=1986 |title=The Existence of Equilibrium in Discontinuous Economic Games, I: Theory |journal=The Review of Economic Studies |volume=53 |issue=1 |pages=1–26 |doi=10.2307/2297588}}</ref> Unlike [[Continuous function|standard continuity]], which requires small changes in inputs to produce small changes in outputs, graph continuity permits certain well-behaved discontinuities. This property is crucial for establishing equilibria in settings such as [[auction theory]], [[oligopoly]] models, and [[Location theory|___location competition]], where payoff discontinuities naturally arise.

===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 (iei.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:AA_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===

{{Reflist}}

* [[Partha Dasgupta]] and [[Eric Maskin]] 1986. ''"The existence of equilibrium in discontinuous economic games, I: theory''". ''The Review of Economic Studies'', 53(1):1-261–26

{{DEFAULTSORT:Graph Continuous Function}}

[[Category:Game theory]]

[[Category:Theory of continuous functions]]