The fitness landscape of social norms in social dilemmas
Authors: Maximilian Puelma Touzel
Organizations: Mila - Quebec AI Institute Montréal, Canada
Abstract
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.
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.
In social dilemmas self-interested learning agents face the choice between the societal benefit of cooperation and the immediate reward of defection. Significant evidence exists on the benefits of assortment mechanisms such as partner selection for the emergence of cooperation, but this is largely available through agent-based simulations. In this paper, we provide an analytical solution to the problem, studying the policy-gradient dynamics in a multi-agent environment with partner selection. We show how partner selection changes the opponent distribution and hence the reward landscape, and prove this promotes cooperation under simple rules known from the literature. In particular, we find that population variance is a necessary condition for cooperation to emerge. Using a two-dimensional Wiener process, we extend the dynamics to capture the stochastic effects of partner selection and the resulting opponent distribution. We derive a sufficient condition for the population to be cooperation-promoting and prove the existence of a stationary distribution. Simulations confirm that the stochastic model accurately captures the policy-gradient dynamics and clarifies how the learning rate affects the emergence of cooperation.