Time delay effects on evolutionary dynamics of N-player trust game with diverse investment behaviors

Among unrelated individuals, the emergence and maintenance of trust has always been a pressing issue to address, one that has garnered considerable attention through the framework provided by trust games. However, few studies have considered the effects of time delay in trust games. Given that decision-makers in trust games inherently exhibit lagged responses when observing the market, we explore the potential effects of time delay on the evolution of trust and develop an N-player trust game model based on investment behavior with time-delayed expected returns. We note that the system exhibits a stable interior equilibrium point, implying that investors and trustworthy trustees can coexist stably in the population, when the time delay is relatively small. Conversely, when the time delay is moderately large, the system exhibits a stable limit cycle, where the proportion in the population show oscillatory dynamics. Furthermore, we analyze the system's Hopf bifurcation points, identifying all critical thresholds responsible for inducing transitions under various conditions. The above theoretical results have been validated by numerical simulations.