Consensus towards Partially Cooperative Strategies in Self-Regulated Evolutionary Games on Networks

Cooperation is widely recognized to be fundamental for the well-balanced development of human societies. Several different approaches have been proposed to explain the emergence of cooperation in populations of individuals playing the Prisoner’s Dilemma game, characterized by two concurrent natural mechanisms: the temptation to defect and the fear to be betrayed by others. Few results are available for analyzing situations where only the temptation to defect (Chicken game) or the fear to be betrayed (Stag-Hunt game) is present. In this paper, we analyze the emergence of full and partial cooperation for these classes of games. We find the conditions for which these Nash equilibria are asymptotically stable, and we show that the partial one is also globally stable. Furthermore, in the Chicken and Stag-Hunt games, partial cooperation has been found to be more rewarding than the full one of the Prisoner’s Dilemma game. This result highlights the importance of such games for understanding and sustaining different levels of cooperation in social networks.

Consensus towards Partially Cooperative Strategies in Self-regulated Evolutionary Games on Networks

Cooperation is widely recognized to be challenging for the well-balanced development of human societies. The emergence of cooperation in populations has been largely studied in the context of the Prisoner's Dilemma game, where temptation to defect and fear to be betrayed by others often activate defective strategies. In this paper we analyze the decision making mechanisms fostering cooperation in the two-strategy Stag-Hunt and Chicken games, which include the mixed strategy Nash equilibrium, describing partially cooperative behavior. We find the conditions for which cooperation is asymptotically stable in both full and partial cases, and we show that the partially cooperative steady state is also globally stable in the simplex. Furthermore, we show that the last can be more rewarding than the first, thus making the mixed strategy effective, although people cooperate at a lower level with respect to the maximum allowed, as it is reasonably expected in real situations. Our findings highlight the importance of Stag-Hunt and Chicken games in understanding the emergence of cooperation in social networks.

Collaboration and Gender Focality in Stag Hunt Bargaining

Knowing the gender of a counterpart can be focal in the willingness to collaborate in team settings that resemble the classic coordination problem. This paper explores whether knowing a co-worker’s gender affects coordination on the mutually beneficial outcome in a socially risky environment. In an experimental setting, subjects play a one-shot stag hunt game framed as a collaborative task in which they can “work together” or “work alone.” We exogenously vary whether workers know the gender of their counterparts pre-play. When gender is revealed, female players tend to gravitate to collaboration and efficient coordination regardless of the knowledge. Males, when knowingly paired with another male, tend to collaborate less, and thus, are less likely to coordinate on the Pareto optimal outcome. These results demonstrate one way that gender focality can lead to inefficient outcomes and provide insight for organizations looking to induce collaboration among workers.

Free neighborhood choice boosts socially optimal outcomes in stag-hunt coordination problem

Situations where independent agents need to align their activities to achieve individually and socially beneficial outcomes are abundant, reaching from everyday situations like fixing a time for a meeting to global problems like climate change agreements. Often such situations can be described as stag-hunt games, where coordinating on the socially efficient outcome is individually optimal but also entails a risk of losing out. Previous work has shown that in fixed interaction neighborhoods agents’ behavior mostly converges to the collectively inefficient outcome. However, in the field, interaction neighborhoods often can be self-determined. Theoretical work investigating such circumstances is ambiguous in whether the efficient or inefficient outcome will prevail. We performed an experiment with human subjects exploring how free neighborhood choice affects coordination. In a fixed interaction treatment, a vast majority of subjects quickly coordinates on the inefficient outcome. In a treatment with neighborhood choice, the outcome is dramatically different: behavior quickly converges to the socially desirable outcome leading to welfare gains 2.5 times higher than in the environment without neighborhood choice. Participants playing efficiently exclude those playing inefficiently who in response change their behavior and are subsequently included again. Importantly, this mechanism is effective despite that only few exclusions actually occur.

The Size of the Stag Determines the Level of Cooperation

ABSTRACTIn the last 12,000 years, human societies have scaled up from small bands to large states of millions and even billions. Many modern societies and even groups of societies cooperate on large-scale projects with relatively low levels of conflict, but the scale and intensity of cooperation varies dramatically between societies. Here we attempt to formalize dynamics that may be driving this rapid increase in cooperation and the differences we see between societies. Our model extends an N-person stag hunt to include population growth dynamics, “stags” with different sized payoffs, and competition for these stags. An increasing number of cooperators is required to access larger stags. The payoff from these stags in turn increases carrying capacity, which increases competition for the stag. As population size increases, new cooperative thresholds are attainable, and as population size shrinks, previously attainable thresholds fall out of reach. Among other predictions, we show that when a new threshold is accessible to a population, the level of cooperation will increase to reach this threshold. However, when the next threshold is out of reach, cooperation decreases as individuals refrain from costly cooperation, preferring a smaller stag. This model offers a framework for understanding the rapid increase in the scale of human cooperation and decline of violence, differences between societies, and challenges to future cooperation.

Pairwise imitation and evolution of the social contract

This paper studies the evolution of conventions in Stag Hunt games when agents’ behaviour depends on pairwise payoff comparisons. The results of two imitative decision rules are compared with each other and with those obtained when agents myopically best respond to the distribution of play. These rules differ in terms of their rationale, their requirements, and the extent to which they make individuals learn from others. Depending on payoffs and the interaction process being considered, best response learning can cause either the rewarding All Stag equilibrium or the inefficient All Hare equilibrium to emerge as the long-run convention. In contrast, pairwise imitation favours the emergence of the Pareto-inferior equilibrium. This result is robust to assuming assortative matching and some heterogeneity in decision rules.

The differential impact of friendship on cooperative and competitive coordination

Friendship is commonly assumed to reduce strategic uncertainty and enhance tacit coordination. However, this assumption has never been tested across two opposite poles of coordination involving either strategic complementarity or substitutability. We had participants interact with friends or strangers in two classic coordination games: the stag-hunt game, which exhibits strategic complementarity and may foster “cooperation”, and the entry game, which exhibits strategic substitutability and may foster “competition”. Both games capture a frequent trade-off between a potentially high paying but uncertain option and a low paying but safe alternative. We find that, relative to strangers, friends are more likely to choose options involving uncertainty in stag-hunt games, but the opposite is true in entry games. Furthermore, in stag-hunt games, friends “tremble” less between options, coordinate better and earn more, but these advantages are largely decreased or lost in entry games. We further investigate how these effects are modulated by risk attitudes, friendship qualities, and interpersonal similarities.