A novel rule for cooperation in the spatial public goods game: Second-order social learning

In real-world contexts, sustaining cooperation is often challenging; however, the learning rule by which individuals learn from their peers may dramatically reshape evolutionary dynamics. In reality, imitation-based traditional social learning (TSL) is inherently constrained by its reliance on copying only immediate neighbors, which limits the depth of information sampling. To overcome this, we introduce a novel second-order social learning (SOSL) rule that broadens the horizon of social information sampling. Specifically, SOSL employs a two-step information sampling process: instead of directly copying the strategy of the successful neighbor, individual adopts the strategy that their successful neighbor has identified as optimal through TSL rule. Using Monte Carlo simulations of the public goods game on a lattice-structured population, compared with the TSL rule, we find that the SOSL rule markedly lowers the conditions of the critical synergy factor for the emergence of cooperation and full cooperation. Even when the synergy factor is too low for cooperation to emerge under the TSL rule, the SOSL rule sustains cooperative clusters and accelerates the transition from defection to widespread cooperation. Furthermore, by introducing a strategy-preference coevolutionary framework that allows individuals to adapt their learning mode, we show that the population converges overwhelmingly to the SOSL rule under moderate synergy factors. Finally, the cooperative advantage of the SOSL rule holds across diverse network structures and persists even under noisy parameter conditions.