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Function <math>U_i:A\longrightarrow\Bbb{R}</math> is '''graph continuous''' if for all <math>\mathbf{a}\in A</math> there exists a function <math>F_i:A_{-i}\longrightarrow A_i</math> such that <math>U_i(F_i(\mathbf{a}_{-i}),\mathbf{a}_{-i})</math> is continuous at <math>\mathbf{a}
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.
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