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We introduce a game-theoretic model for network formation inspired by earlier stochastic models that mix localized and long-distance connectivity. In this model, players may purchase edges at distance d at a cost of dα, and wish to minimize the sum of their edge purchases and their average distance to other players. In this model, we show there is a striking “small world” threshold phenomenon: in two dimensions, if α < 2 then every Nash equilibrium results in a network of constant diameter (independent of network size), and if α > 2 then every Nash equilibrium results in a network whose diameter grows as a root of the network size, and thus is unbounded. We contrast our results with those of Kleinberg  in a stochastic model, and empirically investigate the “navigability” of equilibrium networks. Our theoretical results all generalize to higher dimensions.