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 agents with agent having strategy ; write for an N-tuple of actions (i.e. ) and as the vector of all agents' actions apart from agent .

Let be the payoff function for agent .

A game is defined as . If a graph is continuous you should connect it if it's not then don't connect it.

Definition

Function is graph continuous if for all there exists a function such that is continuous at .

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.