Graph continuous function

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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

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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  .

Definition

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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

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  • 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