Graph continuous function

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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[edit]

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.

The property is interesting in view of the following theorem.

If, for , 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.

References[edit]

  • 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