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.
Consider a game with
{\displaystyle N}
agents with agent
i
{\displaystyle i}
having strategy
⊆
; write
for an N-tuple of actions (i.e.
as the vector of all agents' actions apart from agent
be the payoff function for agent
A game is defined as
Function
is graph continuous if for all
there exists a function
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.
The property is interesting in view of the following theorem.
is non-empty, convex, and compact; and if
is quasi-concave in
, upper semi-continuous in
, and graph continuous, then the game
possesses a pure strategy Nash equilibrium.