Cooperation is a defining attribute of life as we know it, from the delicate interactions of intracellular components to social behavior in groups. However, defection and exploitation are at least as ubiquitous. Evolutionary game theory is a successful tool for investigating how cooperation may be maintained despite large advantages for defection. The Prisoners Dilemma is one such game where spatial structure can maintain cooperation, but only if the benefit-to-cost ratio (b/c) is greater than some threshold, which appears to be the average number of neighbors (k). However, this inequality was tested only for regular spatial and irregular non-spatial networks. In this paper, we use networks in Cartesian space that are based on radii of interactions. We investigate whether the b/c > k threshold holds for these irregular spatial networks, and we use a much broader range of k than previously studied. We find that this rule, and other related inequalities, hold well for the larger radii even when there is noise in the expected neighborhood size. As the expected neighborhood size increases, so does the variation in the empirical edge distribution. However, the variation in the threshold for cooperation decreases. This paper is a first step in a broad investigation of how uncertainty affects the outcome of game theoretic simulations.

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