The Size of the Stag Determines the Level of Cooperation

Human societies have expanded from small bands to large nation-states over the past 12,000 years. Yet, how groups scale up cooperation and why cooperation varies widely between societies, remains a central puzzle. We present a theoretical model that addresses these puzzles by extending the classic Stag Hunt game to (a) multiple players, (b) multiple rewards ("stags") of different sizes, and (c) endogenous population growth. This framework reveals a "cooperation ladder" where each rung corresponds to a reward that requires a threshold number of cooperators. As cooperation increases, larger rewards become attainable. Securing a larger reward raises carrying capacity (e.g. by providing more food or energy), enabling subsequent population growth and unlocking the possibility of further cooperative gains. However, between these thresholds, cooperation can stagnate or reverse, effectively incentivizing free-riders at intermediate levels. We show that history matters. Early cooperation and population growth can set groups on divergent, path-dependent trajectories. Our model predicts multiple stable equilibria, with surges in cooperation when a new threshold is within reach, and stalls when higher rewards seem unattainable. This framework helps explain key patterns including why cooperation can sometimes accelerate rapidly, why some societies get stuck at smaller scales, and how seemingly selfish behavior can persist in cooperative groups. Expanding the scale of cooperation may depend on how incentives correspond to both environmental conditions, existing levels of cooperation, and population dynamics, offering a new lens on historical transitions in social complexity and insights for modern coordination challenges such as climate change. Significance StatementSmall groups can sometimes grow into large-scale cooperative societies, while other societies stagnate at smaller scales. Our theoretical model reveals that population growth and cooperative thresholds for accessing resources or energy create a "cooperation ladder." As a groups population increases, new larger-scale payoffs become attainable, which in turn fuels further growth. However, between these critical thresholds, cooperation tends to stall and some free-riding is tolerated. This mechanism offers a new explanation for historical bursts in population and cooperation (such as the agricultural and Industrial revolutions), why some societies remain small, and how selfish behavior can persist in cooperative groups. It also provides insights into global coordination challenges like climate change by highlighting the material conditions needed to sustain and expand large-scale cooperation.