We analyze a graph process (or network creation game) where the vertices as players can establish mutual relations between each other at a fixed price. Each vertex receives income from every other vertex, exponentially decreasing with their distance. To establish an edge, both players have to make a consent acting selfishly. This process has originially been proposed in economics to analyse social networks of cooperation. Though the exponential payoff is a desirable principle to model the benefit of distributed systems, it has so far been an obstacle for analysis. We show that the process has a positive probability to cycle. We reduce the creation rule with payoff functions to graph theoretic criteria. Moreover, these criteria can be evaluated locally. This allows us to thoroughly reveal the structure of all stable states. In addition, the question for the price of anarchy can be reduced to counting the maximum number of edges of a stable graph. This together with a probabilistic argument allows to determine the price of anarchy exactly.
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