Christian Hilbe

People often cooperate out of concern for their reputation. The corresponding theory of indirect reciprocity predicts that such cooperation can only evolve if people’s opinions of each other are sufficiently correlated. This correlation, however, can be difficult to achieve when individuals form their opinions independently from one another, as in private assessment models. In that case, errors can generate disagreements that propagate throughout a population. Here, we identify a simple mechanism that mitigates this problem. Most prior work assumes that observers revise only the reputations of donors—the individuals who choose whether to cooperate or defect. We instead allow observers to also revise the reputations of recipients—the individuals affected by the donors’ choices. Using analytical calculations and simulations, we show that such dual reputation updates help synchronize opinions and facilitate cooperation. To demonstrate this approach, we focus on a particularly simple rule, which we call Recipient Image Scoring (RIS). Under RIS, recipients are assessed as good whenever donors choose to cooperate with them. We show that this rule for judging recipients effectively complements the classical leading-eight rules for judging donors. Our results establish dual assessment as a simple mechanism for cooperation that relies only on directly observable information.

People do not just cooperate—they invest in becoming better cooperators. Partners learn to manage conflict, professionals train for specialized roles, and newcomers acquire the practices of their communities. Unlike cooperative acts, which directly benefit others, these efforts shape the individual, lowering their future cooperation costs. Why do people make such investments? Using a mathematical model and evolutionary simulations, we show that cooperation investments serve a dual function. First, they build cooperative capacity, enabling cooperation where it would otherwise be too costly. Second, when investments are observable and sufficiently demanding, they can honestly signal cooperative intent. When someone takes the time to work through personal conflict or learn a community's language, they do more than build capacity—they reveal an intention to cooperate.

People make strategic decisions many times a day—during negotiations, when coordinating actions with others, or when choosing partners for cooperation. The resulting dynamics can be studied with learning theory and evolutionary game theory. These frameworks explore how people adapt their decisions over time, in light of how effective their strategies have been. The outcomes of such learning processes depend on how sensitive individuals are to the performance of their strategies. When they are more sensitive, they systematically favor strategies they deem more successful. When they are less sensitive, their learning process is noisier and more erratic. Traditionally, most models treat this sensitivity as a fixed parameter—like the “selection strength” parameter in evolutionary models. Instead, we study how strategies and sensitivities coevolve. We find that the coevolutionary endpoints depend on both the type of strategic interaction and the learning rule employed. In prisoner’s dilemmas, we often observe sensitivities to increase indefinitely. But in snowdrift and stag-hunt games, sensitivities often converge to a finite value, or we observe evolutionary branching altogether. These results shed light on how evolution might shape learning mechanisms for social behavior. They suggest that noisy learning does not need to be a by-product of cognitive constraints. Instead, it can serve as a means to gain strategic advantages.

Reply to Sacco: Complete spaces as outcomes of evolutionary optimization

Repeated games and stochastic games are important frameworks to study direct reciprocity. Individuals react strategically to their coplayers’ previous behavior. While strategies in such games can be arbitrarily complex, explorations of evolutionary dynamics are often done in specific strategy spaces. Individuals may consider a fixed number of past rounds, or only some of the partner’s previous actions. Such restrictions can make the interpretation of the results difficult. Strategies found to be superior within a restricted set may lose stability when more complex strategies are permitted. We discuss two notions of completeness that rule out this possibility. If a strategy space, S, is best-reply-complete, then any strategy in S is guaranteed to have a best reply in S. If a space, S, is payoff-complete, then any strategy playing against an opponent in S can be replaced by an equivalent strategy within S without affecting either player’s payoff. Sufficient conditions for best-reply-completeness have been given in a seminal paper by Levínský et al. Here, we show that for strategies of bounded memory, the same conditions are sufficient for payoff-completeness. Furthermore, using those conditions, we illustrate how to construct many complete spaces for simple games. Taken together, our findings highlight the importance of complete strategy spaces, which are particularly useful when interpreting evolutionary simulations and determining best responses.

When people collaborate in groups, they routinely face collective action problems: for the group effort to succeed, individuals need to cooperate despite any incentives to defect. These problems can be modeled with public goods games. To facilitate such an analysis, many studies assume the game is symmetric. Group members have the same means to cooperate (equal endowments), and contributions of different group members are equally effective (equal productivities). Studies that allow for some inequality tend to focus on one kind of inequality only. In practice, however, people can be unequal in many ways. The effect of these inequalities may in turn depend on the specific public goods game considered. To explore these issues, we combine a large-sample experiment with extensive theoretical work. We systematically vary four aspects: the group members’ endowments, their productivities, group size, and whether the public goods game exhibits linear returns or returns given by a threshold function. By exploring all four aspects, we obtain a unique dataset to explore the effect of asymmetry on cooperation. Based on this dataset, we study whether there is an advantage of “aligned inequality”: whether groups achieve a better surplus if more productive individuals have a larger endowment. For public goods games with linear returns, we find such an advantage, thereby corroborating previous research. If returns follow a threshold function, however, aligned inequality results in inferior payoffs. These results show that the effect of inequality on cooperation depends on the kind of public goods game considered.

Indirect reciprocity is a key mechanism for large-scale cooperation. This mechanism captures the insight that in part, people help others to build and maintain a good reputation. To enable such cooperation, appropriate social norms are essential. They specify how individuals should act based on each others’ reputations, and how reputations are updated in response to individual actions. Although previous work has identified several norms that sustain cooperation, a complete analytical characterization of all evolutionarily stable norms remains lacking, especially when assessments or actions are noisy. In this study, we provide such a characterization for the public assessment regime. This characterization reproduces known results, such as the leading eight norms, but it extends to more general cases, allowing for various types of errors and additional actions including costly punishment. We also identify norms that impose a fixed payoff on any mutant strategy, analogous to the zero-determinant strategies in direct reciprocity. These results offer a rigorous foundation for understanding the evolution of cooperation through indirect reciprocity and the critical role of social norms.

Cooperation at scale is critical for achieving a sustainable future for humanity. However, achieving collective, cooperative behavior—in which intelligent actors in complex environments jointly improve their well-being—remains poorly understood. Complex systems science (CSS) provides a rich understanding of collective phenomena, the evolution of cooperation, and the institutions that can sustain both. Yet, much of the theory in this area fails to fully consider individual-level complexity and environmental context—largely for the sake of tractability and because it has not been clear how to do so rigorously. These elements are well captured in multiagent reinforcement learning (MARL), which has recently put focus on cooperative (artificial) intelligence. However, typical MARL simulations can be computationally expensive and challenging to interpret. In this perspective, we propose that bridging CSS and MARL affords new directions forward. Both fields can complement each other in their goals, methods, and scope. MARL offers CSS concrete ways to formalize cognitive processes in dynamic environments. CSS offers MARL improved qualitative insight into emergent collective phenomena. We see this approach as providing the necessary foundations for a proper science of collective, cooperative intelligence. We highlight work that is already heading in this direction and discuss concrete steps for future research.

Multiagent learning is challenging when agents face mixed-motivation interactions, where conflicts of interest arise as agents independently try to optimize their respective outcomes. Recent advancements in evolutionary game theory have identified a class of “zero-determinant” strategies, which confer an agent with significant unilateral control over outcomes in repeated games. Building on these insights, we present a comprehensive generalization of zero-determinant strategies to stochastic games, encompassing dynamic environments. We propose an algorithm that allows an agent to discover strategies enforcing predetermined linear (or approximately linear) payoff relationships. Of particular interest is the relationship in which both payoffs are equal, which serves as a proxy for fairness in symmetric games. We demonstrate that an agent can discover strategies enforcing such relationships through experience alone, without coordinating with an opponent. In finding and using such a strategy, an agent (“enforcer”) can incentivize optimal and equitable outcomes, circumventing potential exploitation. In particular, from the opponent’s viewpoint, the enforcer transforms a mixed-motivation problem into a cooperative problem, paving the way for more collaboration and fairness in multiagent systems.

Social dilemmas are collective-action problems where individual interests are at odds with group interests. Such dilemmas occur frequently at all scales of human interactions. When dealing with collective-action problems, people often act reciprocally. They adjust their behavior to match the previous behavior of the recipient. The literature distinguishes two kinds of reciprocity. According to direct reciprocity, individuals react to their immediate experiences with the recipient. They are more likely to cooperate if the recipient previously cooperated with them. According to indirect reciprocity, individuals react to the recipient’s general behavior, irrespectively of whether or not they benefited directly. In practice, the two kinds of reciprocity are often intertwined; people typically base their decisions on both direct experiences and indirect observations. Yet only recently have researchers begun to explore how the two kinds of reciprocity interact. So far, this research only addresses a single type of social dilemma, the donation game, where the effects of individual behaviors are independent. Instead, here we allow for all pairwise social dilemmas. By applying novel techniques to generalize the theory of zero-determinant strategies, we establish an important proof of principle: In all social dilemmas, socially optimal outcomes can be sustained as an equilibrium, using either direct or indirect reciprocity, or arbitrary mixtures thereof. These results neither require games to be repeated infinitely often, nor that individual opinions are synchronized. In this way, we considerably generalize the scope of models of reciprocity, and we build further bridges between the literatures on direct and indirect reciprocity.