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Spatial structure often inhibits the evolution of cooperation in the snowdrift game

Abstract

Understanding the emergence of cooperation is a fundamental problem in evolutionary biology1. Evolutionary game theory2,3 has become a powerful framework with which to investigate this problem. Two simple games have attracted most attention in theoretical and experimental studies: the Prisoner's Dilemma4 and the snowdrift game (also known as the hawk–dove or chicken game)5. In the Prisoner's Dilemma, the non-cooperative state is evolutionarily stable, which has inspired numerous investigations of suitable extensions that enable cooperative behaviour to persist. In particular, on the basis of spatial extensions of the Prisoner's Dilemma, it is widely accepted that spatial structure promotes the evolution of cooperation6,7,8. Here we show that no such general predictions can be made for the effects of spatial structure in the snowdrift game. In unstructured snowdrift games, intermediate levels of cooperation persist. Unexpectedly, spatial structure reduces the proportion of cooperators for a wide range of parameters. In particular, spatial structure eliminates cooperation if the cost-to-benefit ratio of cooperation is high. Our results caution against the common belief that spatial structure is necessarily beneficial for cooperative behaviour.

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Figure 1: Frequency of cooperators as a function of the cost-to-benefit ratio r = c/(2b - c) in the snowdrift game for different lattice geometries.
Figure 2: Snapshots of equilibrium configurations of cooperators (black) and defectors (white) in the spatial Prisoner's Dilemma and spatial snowdrift game on a square lattice with N = 4 neighbours near the extinction threshold of cooperators.
Figure 3: Average mixed strategy at stochastic equilibrium in the spatial hawk–dove game as a function of the parameter r = β/γ for different lattice geometries.

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Acknowledgements

We thank M. Ackermann and G. Szabó for comments. C.H. acknowledges support from the Swiss National Science Foundation. M.D. was supported by the National Science and Engineering Research Council (NSERC), Canada, and by the James S. McDonnell Foundation, USA.

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Correspondence to Christoph Hauert.

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Supplementary information

Supplementary Information

This provides a detailed account of the method of pair approximation applied to game theory in spatially structured populations together with results for alternative update rules. Variations in the updating reflect different approaches to modelling the odds of successful reproduction of competing individuals. (PDF 376 kb)

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Hauert, C., Doebeli, M. Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428, 643–646 (2004). https://doi.org/10.1038/nature02360

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