By specifying behaviour across multiple agents, social norms are a coordination approach to resolving social dilemmas. Decentralized and wide adoption can be achieved by norms whose prescription involves interpreting stochastic signals in the environment. Such signals must have enough correlation to orchestrate mutually beneficial coordination and enough disincentivizing uncertainty about the benefits of exploiting that coordination. Evolutionary game theory of matrix games has been used to describe how, by rational agents comparing and adopting norms, a norm can evolve to become dominant in a population. Morsky & Akçay (2019) classify norms according to a set of rationality criteria. Joint player strategies that adopt norms that are consistent with optimal single-player strategies with respect to expected reward naturally satisfy a correlated, rather than Nash game theoretic equilibrium condition. Here, we present a version of this theory that clarifies the basic ingredients. We formulate it in the more general Markov game setting more commonly used in reinforcement learning theory. We illustrate the theory by mapping norms over the signal and reward space, while also giving a detailed exposition of the underlying mechanics of the approach. Finally, we give a general solution and analysis of replicator dynamics, which Morsky & Akçay (2019) propose as a means by which these norms could emerge.
Social norms cannot be identified from behavior alone: the same cooperative equilibrium may reflect shared expectations, strategic incentives, or simple imitation. Yet in multi-agent large language model systems, prior work largely treats behavioral convergence as evidence of norm emergence. In this work, we introduce an evaluation framework that measures agents' reported empirical and normative expectations in addition to behavioral convergence. Through controlled ablations, we test the effect of expectation elicitation and isolate two collective mechanisms central to theories of norm formation---social learning through interaction and social selection through network-based group formation. We further test the stability of these resulting dynamics under adversarial disruption across four LLM families. We find that eliciting expectations increases cooperative contributions, while social learning stabilizes behavior, and social selection reliably identifies cooperators but provides limited behavioral reinforcement. Following disruption, normative expectations and behavioral coordination recover differently. Together, these results show that similar cooperative outcomes can arise from different underlying social processes. By making expectations observable, our framework allows us to attribute each mechanism's contribution separately, offering designers of multi-agent systems a principled basis for selecting the social processes that sustain cooperation.
In social dilemma games, additional rewards or punishments have been studied as means of promoting cooperation. Therefore, it is important to investigate the ideal situation, in which such an additional payoff would change the game. In this study, we investigated the symmetric solution of the Bellman optimality equation for a repeated harmony game. The calculations showed that three types of symmetric solutions exist. One of them corresponds to the trivial All-C strategy, and another to the Win-stay Lose-shift strategy of the prisoners dilemma game. The nontrivial behavior of the strategy corresponding to the last solution is also discussed in detail. In addition, we numerically investigated which strategy the agents actually learn by the reinforcement learning algorithm.
Standard rational actor models often attribute cooperation failures in social dilemmas to insufficient incentives, overlooking the destabilizing effects of continuous utility maximization. To address this, we propose a framework of will" defined as a mechanism that persistently pursues goals while ignoring local cost-benefit fluctuations. We formalize the Willed Agents as potential minimizers, distinguishing them from cumulative utility maximization. Dynamical analysis of infinite population demonstrates that willed agents shrink the feasible state space, acting as boundary constraints that accelerate convergence in canonical social dilemmas. Through multi-agent simulations in a spatiotemporal Stag Hunt Game, we show that willed agents function as cooperation catalysts", enabling groups to surmount high-risk thresholds where purely utility maximization fails. We find that heterogeneous will strength promotes cooperation, and that agents who autonomously suspend rational re-evaluation can significantly outperform continuous optimizers. These findings suggest that successful cooperation relies on the cognitive capacity to strategically constrain calculation.