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The DeGroot model is one of the most classical models in the field of opinion dynamics. The standard DeGroot model assumes that agents are homogeneous and update their opinions in the direction of a weighted average of their neighbors’ opinions. One natural question is whether a second type of agents could significantly change the main properties of the model. The authors address this question by introducing rebels, who update their opinions toward the opposite of their neighbors’ weighted average. The authors find that the existence of rebels remarkably affects the opinion dynamics. Under certain mild conditions, the existence of a few rebels will lead the group opinion to the golden mean, regardless of the initial opinions of the agents and the structure of the learning network. This result is completely different from that of the standard DeGroot model, where the final consensus opinion is determined by both the initial opinions and the learning topology. The study then provides new insights into understanding how heterogeneous individuals in a group reach consensus and why the golden mean is so common in human society.

Recent studies show that conditional dissociation, a.k.a. post-interaction partner-refusal, can promote the emergence and stability of cooperation. However, in most of these studies, players’ strategies are restricted to pure ones, which is obviously inconsistent with many biological and economic situations. Another concern with line of these studies is that conditional dissociation is often combined with other mechanisms. These mechanisms may favor cooperation per se, leaving it unclear whether conditional dissociation is indeed a key factor. In this paper, we study a clean model, pruning all the factors other than conditional dissociation that may favor cooperation. We find that conditional dissociation, which could be viewed as a variant of peer punishment, does promote cooperation, no matter whether mixed strategies are allowed or not. This confirms the previous findings in the literature. In addition, compared with the pure strategy scenario, cooperators are less competitive when mixed strategies are allowed. Our main finding is supported by both the numerical simulations and the theoretical analysis of Neutrally Stable Strategy. We also find that cooperative behavior is favored when waiting time and/or the population’s lifespan are longer.

We study the Stackleberg variant of the repeated Sequential Prisoner's Dilemma (SPD). The game goes in two stages, and the two players, the leader and the follower, are asymmetric in both stages. In the first stage of the game, the leader chooses a strategy (for the repeated SPD of the second stage), which is immediately known to the follower. In the second stage, they play repeated SPD: In each round the follower moves after observing the leader's action. Assuming complete rationality, we find some extraordinary properties of this model. (i) The (subgame perfect) equilibrium payoff profile is unique, which lies on the corner of the region predicted by classical folk theorems: It is best for the leader and at the same time worst for the follower, (ii) the leader has simple optimal strategies that are one-step memory and stationary. These features are in great contrast with classical results, where either uniqueness cannot be guaranteed and equilibrium strategies are often quite complicated, or bounded rationality is required. Although full cooperation, i.e., the outcome is always (cooperate, cooperate), is not attainable in our model, at least a half of the optimal social welfare can be guaranteed. We also do a non-equilibrium analysis which makes the usual equilibrium analysis more convincing.

Leaving is usually an option for individuals if they cannot tolerate their defective partners. In a two-player game, when a player chooses to leave, both she and her opponent become single players. However, in a multi-player game, the same decision may have different consequences depending on whether group cohesion exists. Players who choose not to leave would still be united together rather than be separated into singletons if there is cohesion among them. Considering this difference, we study two leaving mechanisms in public goods games. In the first mechanism, every player would be single once any of the group members leaves. In the second, we assume group cohesion exists that members who don't leave form a union. In our model, each player adopts a trigger strategy characterized by a threshold: she leaves if the number of defectors in her group exceeds the threshold. We find that under both mechanisms, when the expected lifespan of individuals is long enough, cooperators with zero tolerance toward defection succeed in the evolution. Moreover, when cohesion exists in groups, cooperation is better promoted because the cooperators have a higher chance to play together. That is, group cohesion facilitates positive assortment and therefore promotes cooperation.