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Unchecked strategy diversification and collapse in continuous voluntary public good games

Cooperation or defection and participation or withdrawal are well-known options of behavior in game-like activities in free societies, yet the co-evolutionary dynamics of these behavioral traits in the individual level are not well understood. Here we investigate the continuous voluntary public good game, in which individuals have two types of continuous-valued options: a probability of joining the public good game and a level of cooperative investment in the game. Our numerical results reveal hitherto unreported phenomena: (i) The evolutionary dynamics are initially characterized by oscillations in individual cooperation and participation levels, in contrast to the population-level oscillations that have previously been reported. (ii) Eventually, the population's average cooperation and participation levels converge to and stabilize at a center. (iii) Then, a most peculiar phenomenon unfolds: The strategies present in the population diversify and give rise to a "cloud" of tinkering individuals who each tries out a different strategy, and this process continues unchecked as long as the population's cooperation and participation levels remain balanced. Over time, however, imbalances build up as a consequence of random drift and there is a sudden and abrupt collapse of the strategy-diversity cloud. The process then repeats again in a cyclic manner. To understand the three aforementioned phenomena, we investigate the system analytically using adaptive-dynamics techniques. Our analysis casts light on the mechanisms which underpin the unexpected and surprising evolutionary dynamics.