Nudge-driven cooperation emerges through adaptive learning in evolutionary games

Nudge-based interventions are widely regarded as effective tools for promoting cooperative behavior, yet their impact is often studied under static assumptions that neglect adaptive decision-making and spatial interactions. This work shows that the effectiveness of nudges depends critically on both their structural form and the underlying adaptation mechanism. A generalized evolutionary game is considered in which cooperation is influenced through modifications of perceived payoffs, implemented via constant, state-dependent, and time-decaying nudges. These interventions reshape the effective incentive landscape without altering the baseline payoff structure, enabling a systematic investigation of behavioral influence in structured populations. The results demonstrate that different nudge formulations generate qualitatively distinct evolutionary outcomes. Constant nudges produce a uniform shift of phase boundaries, predictably expanding cooperative regions. In contrast, state-dependent nudges introduce nonlinear feedback, leading to threshold effects and rapid amplification of cooperation once a critical mass is reached. Time-decaying nudges further enrich the dynamics by generating path dependence and delayed transitions through competition between early reinforcement and long-term selection. It is further revealed that evolutionary outcomes depend strongly on the mode of adaptation. Under imitation dynamics, the system exhibits rapid transitions and clear phase separation. By contrast, reinforcement learning introduces memory effects that smooth temporal evolution, prolong transient coexistence, and stabilize mixed or metastable states. Overall, cooperation emerges from the interplay among incentive design, learning, spatial structure, and temporal adaptation.