Cyclical oscillations and absorbing-state probabilities in optional public goods games: Interplay of reward and group size

The optional public goods game (OPGG) is a three-strategy model in which individuals can cooperate, defect, or not participate. Despite its simplicity, this model effectively captures various social dilemmas, including those involving public services, environmental sustainability, and broader societal issues. In this study, we investigate how the reward (r) and group size of potential players (S) of public goods games influence the steady-state coexistence of these strategies or the alternation of their dominance in a rock-paper-scissors dynamic. The OPGG is simulated using Monte Carlo in a nonspatial scenario, meaning there is no topology connecting the agents, allowing any player to interact with any other player. We show that under sufficiently noisy conditions, the system consistently evolves to an absorbing state, with the prevailing strategy determined by the values of r and S. In the range 2≤r≤S, the system shows multiple stable absorbing states, with groups of size S=4 exhibiting more pronounced and transient rock-paper-scissors dynamics with longer average absorbing times. We present a thorough analysis of our results in terms of the fraction of time the system spends in rock-paper-scissor cycles, the number of cycles, and the average probability that the system relaxes to each possible absorbing state, including scenarios where the system does not reach an absorbing state at all.