Graph continuous function

In game theory, a function is considered graph continuous if it meets specific criteria. This concept was introduced by Partha Dasgupta and Eric Maskin in 1986 as a form of continuity applicable to continuous games. In a game with \( N \) agents, each agent \( i \) has a strategy set \( A_i \subseteq \mathbb{R} \). The payoff function for agent \( i \) is denoted by \( U_i: A_i \rightarrow \mathbb{R} \), where \( A = \prod_{j=1}^N A_j \). A function \( U_i: A \rightarrow \mathbb{R} \) is graph continuous if, for every strategy profile \( \mathbf{a} \in A \), there exists a function \( F_i: A_{-i} \rightarrow A_i \) such that \( U_i(F_i(\mathbf{a}_{-i}), \mathbf{a}_{-i}) \) is continuous at \( \mathbf{a}_{-i} \). This property ensures that the graph of a player's payoff function changes continuously as other players' strategies vary.

A key theorem states that if each agent's strategy set \( A_i \subseteq \mathbb{R}^m \) is non-empty, convex, and compact, and if the payoff function \( U_i \) is quasi-concave in \( a_i \), upper semi-continuous in \( \mathbf{a} \), and graph continuous, then the game possesses a pure strategy Nash equilibrium. This result highlights the importance of graph continuity in ensuring the existence of equilibria in discontinuous economic games.