# Graph continuous function

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{{Short description|Concept in game theory}}
In [mathematics](/source/mathematics), particularly in [game theory](/source/game_theory) and [mathematical economics](/source/mathematical_economics), a function is '''graph continuous''' if its [graph](/source/Graph_(function))—the set of all input-output pairs—is a closed set in the [product topology](/source/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 property](/source/Closed_graph_theorem) in [functional analysis](/source/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](/source/Partha_Dasgupta) and [Eric Maskin](/source/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 [standard continuity](/source/Continuous_function), 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](/source/auction_theory), [oligopoly](/source/oligopoly) models, and [location competition](/source/Location_theory), where payoff discontinuities naturally arise.

==Notation and preliminaries==
Consider a [game](/source/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](/source/Convex_function), and [compact](/source/compact_set); and if <math>U_i:A\longrightarrow\mathbb{R}</math> is [quasi-concave](/source/quasi-concave_function) in <math>a_i</math>, [upper semi-continuous](/source/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](/source/pure_strategy) [Nash equilibrium](/source/Nash_equilibrium).

==References==
{{Reflist}}
* [Partha Dasgupta](/source/Partha_Dasgupta) and [Eric Maskin](/source/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

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Adapted from the Wikipedia article [Graph continuous function](https://en.wikipedia.org/wiki/Graph_continuous_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Graph_continuous_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.