Graph continuous function

In mathematics, and in particular the study of game theory, a 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 continuity that finds application in the study of continuous games.

Notation and preliminaries

Consider a game with ${\displaystyle N}$ agents with agent ${\displaystyle i}$ having strategy ${\displaystyle A_i\subseteq \mathbb{R}}$; write ${\displaystyle \mathbf{a}}$ for an N-tuple of actions (i.e. ${\displaystyle \mathbf{a}\in \prod _{j=1}^N{A}_j}$) and ${\displaystyle {\mathbf{a}}_{-i}=(a_1,a_2,\dots ,a_{i-1},a_{i+1},\dots ,a_N)}$ as the vector of all agents' actions apart from agent ${\displaystyle i}$.

Let ${\displaystyle U_i:A_i⟶\mathbb{R}}$ be the payoff function for agent ${\displaystyle i}$.

A game is defined as ${\displaystyle [(A_i,U_i);i=1,\dots ,N]}$.

Definition

Function ${\displaystyle U_i:A⟶\mathbb{R}}$ is graph continuous if for all ${\displaystyle \mathbf{a}\in A}$ there exists a function ${\displaystyle F_i:A_{-i}⟶A_i}$ such that ${\displaystyle U_i(F_i({\mathbf{a}}_{-i}),{\mathbf{a}}_{-i})}$ is continuous at ${\displaystyle {\mathbf{a}}_{-i}}$.

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 ${\displaystyle 1\le i\le N}$, ${\displaystyle A_i\subseteq {\mathbb{R}}^m}$ is non-empty, convex, and compact; and if ${\displaystyle U_i:A⟶\mathbb{R}}$ is quasi-concave in ${\displaystyle a_i}$, upper semi-continuous in ${\displaystyle \mathbf{a}}$, and graph continuous, then the game ${\displaystyle [(A_i,U_i);i=1,\dots ,N]}$ 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

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