Effect of memory, intolerance, and second-order reputation on cooperation

STP-based control of networked evolutionary games with multi-channel structure

The channel delay in the game process has an important influence on its evolutionary dynamics. This paper aims to optimize the strategy game with general information delays, including the state delay in the previous work, and the control delay that is introduced for the first time to depict the time consumed by strategy propagation in reality. Specifically, the dynamics of networked evolutionary games is transformed into an algebraic form by use of the newly proposed semi-tensor product of matrices, which extends the ordinary matrix multiplication. Subsequently, according to the values of control and state delays, the strategy optimization problem can be divided into six different cases, and then via the constructed algebraic equation, we can obtain the sufficient and necessary conditions for the existence of the strategy optimization. Meanwhile, based on a reachable set method, the corresponding feedback controllers are further designed. Last, one illustrative example is taken to demonstrate the feasibility of our model. The results of this paper will be helpful to investigate the game-based control issues in the complex networked environment.

Cooperation and social organization depend on weighing private and public reputations

To avoid exploitation by defectors, people can use past experiences with others when deciding to cooperate or not ('private information'). Alternatively, people can derive others' reputation from 'public' information provided by individuals within the social network. However, public information may be aligned or misaligned with one's own private experiences and different individuals, such as 'friends' and 'enemies', may have different opinions about the reputation of others. Using evolutionary agent-based simulations, we examine how cooperation and social organization is shaped when agents (1) prioritize private or public information about others' reputation, and (2) integrate others' opinions using a friend-focused or a friend-and-enemy focused heuristic (relying on reputation information from only friends or also enemies, respectively). When agents prioritize public information and rely on friend-and-enemy heuristics, we observe polarization cycles marked by high cooperation, invasion by defectors, and subsequent population fragmentation. Prioritizing private information diminishes polarization and defector invasions, but also results in limited cooperation. Only when using friend-focused heuristics and following past experiences or the recommendation of friends create prosperous and stable populations based on cooperation. These results show how combining one's own experiences and the opinions of friends can lead to stable and large-scale cooperation and highlight the important role of following the advice of friends in the evolution of group cooperation.

Heterogeneous decision-making dynamics of threshold-switching agents on complex networks

In the classical two-player decision-making scenario, individuals may have different tendencies to take a certain action, given that there exists a sufficient number of neighbors adopting a particular option. This is ubiquitous in many real-life contexts including traffic congestion, crowd evacuation, and minimal vertex cover problem. Under best-response dynamics, we investigate the decision-making behaviors of heterogeneous agents on complex networks. Results of the networked games are twofold: for networks of uniform degree distribution (e.g., the lattice) and fraction of the strategy is of a linear function of the threshold setting. Moreover, the equilibrium analysis is provided and the relationship between the equilibrium dynamics and the change of the threshold value is given quantitatively. Next, if the games are played on networks with non-uniform degree distribution (e.g., random regular and scale-free networks), influence of the threshold-switching will be weakened. Robust experiments indicate that it is not the value of the average degree, but the degree distribution that influences how the strategy evolves affected by the threshold settings. Our result shows that the decision-making behaviors can be effectively manipulated by tuning the parameters in the utility function (i.e., thresholds) of some agents for more regular network structures.

Reputation and reciprocity

Reputation and reciprocity are key mechanisms for cooperation in human societies, often going hand in hand to favor prosocial behavior over selfish actions. Here we review recent researches at the interface of physics and evolutionary game theory that explored these two mechanisms. We focus on image scoring as the bearer of reputation, as well as on various types of reciprocity, including direct, indirect, and network reciprocity. We review different definitions of reputation and reciprocity dynamics, and we show how these affect the evolution of cooperation in social dilemmas. We consider first-order, second-order, as well as higher-order models in well-mixed and structured populations, and we review experimental works that support and inform the results of mathematical modeling and simulations. We also provide a synthesis of the reviewed researches along with an outlook in terms of six directions that seem particularly promising to explore in the future.

Evolutionary stability of cooperation in indirect reciprocity under noisy and private assessment

Indirect reciprocity is a mechanism that explains large-scale cooperation in humans. In indirect reciprocity, individuals use reputations to choose whether or not to cooperate with a partner and update others' reputations. A major question is how the rules to choose their actions and the rules to update reputations evolve. In the public reputation case where all individuals share the evaluation of others, social norms called Simple Standing (SS) and Stern Judging (SJ) have been known to maintain cooperation. However, in the case of private assessment where individuals independently evaluate others, the mechanism of maintenance of cooperation is still largely unknown. This study theoretically shows for the first time that cooperation by indirect reciprocity can be evolutionarily stable under private assessment. Specifically, we find that SS can be stable, but SJ can never be. This is intuitive because SS can correct interpersonal discrepancies in reputations through its simplicity. On the other hand, SJ is too complicated to avoid an accumulation of errors, which leads to the collapse of cooperation. We conclude that moderate simplicity is a key to stable cooperation under the private assessment. Our result provides a theoretical basis for the evolution of human cooperation.

Towards preferential selection in the prisoner's dilemma game

In previous works, the choice of learning neighbor for an individual has generally obeyed pure random selection or preferential selection rules. In this paper, we introduce a tunable parameter ε to characterize the strength of preferential selection and focus on the transition towards preferential selection in the spatial evolutionary game by controlling ε to guide the system from pure random selection to preferential selection. Our simulation results reveal that the introduction of preferential selection can hugely alleviate social dilemmas and enhance network reciprocity. A larger ε leads to a higher critical threshold of the temptation b for the extinction of cooperators. Moreover, we provide some intuitive explanations for the above results from the perspective of strategy transition and cooperative clusters. Finally, we examine the robustness of the results for noise K and different topologies, find that qualitative features of the results are unchanged.

Incorporating social payoff into reinforcement learning promotes cooperation

Reinforcement learning has been demonstrated to be an effective approach to investigate the dynamic of strategy updating and the learning process of agents in game theory. Most studies have shown that Q-learning failed to resolve the problem of cooperation in well-mixed populations or homogeneous networks. To this aim, we investigate the self-regarding Q-learning's effect on cooperation in spatial prisoner's dilemma games by incorporating the social payoff. Here, we redefine the reward term of self-regarding Q-learning by involving the social payoff; that is, the reward is defined as a monotonic function of the individual payoff and the social payoff represented by its neighbors' payoff. Numerical simulations reveal that such a framework can facilitate cooperation remarkably because the social payoff ensures agents learn to cooperate toward socially optimal outcomes. Moreover, we find that self-regarding Q-learning is an innovative rule that ensures cooperators coexist with defectors even at high temptations to defection. The investigation of the emergence and stability of the sublattice-ordered structure shows that such a mechanism tends to generate a checkerboard pattern to increase agents' payoff. Finally, the effects of Q-learning parameters are also analyzed, and the robustness of this mechanism is verified on different networks.

Evolution of cooperation through cumulative reciprocity

Reciprocity is a simple principle for cooperation that explains many of the patterns of how humans seek and receive help from each other. To capture reciprocity, traditional models often assume that individuals use simple strategies with restricted memory. These memory-1 strategies are mathematically convenient, but they miss important aspects of human reciprocity, where defections can have lasting effects. Here we instead propose a strategy of cumulative reciprocity. Cumulative reciprocators count the imbalance of cooperation across their previous interactions with their opponent. They cooperate as long as this imbalance is sufficiently small. Using analytical and computational methods, we show that this strategy can sustain cooperation in the presence of errors, that it enforces fair outcomes and that it evolves in hostile environments. Using an economic experiment, we confirm that cumulative reciprocity is more predictive of human behaviour than several classical strategies. The basic principle of cumulative reciprocity is versatile and can be extended to a range of social dilemmas.

Defectors in bad circumstances possessing higher reputation can promote cooperation

In nature and human society, social relationships and behavior patterns are usually unpredictable. In any interaction, individuals will constantly have to deal with prior uncertainty. The concept of "reputation" can provide some information to mitigate such uncertainty. In previous studies, researchers have considered that only cooperators are able to maintain a high reputation; no matter the circumstances of a defector, they are classified as a faithless individual. In reality, however, some individuals will be forced to defect to protect themselves against exploitation. Therefore, it makes sense that defectors in bad circumstances could also obtain higher reputations, and cooperators can maintain higher reputations in comfortable circumstances. In this work, the reputations of individuals are calculated using the fraction of their neighbors who have the same strategy. In this way, some defectors in a population may obtain higher reputations than some cooperators. We introduce this reputation rule using heterogeneous investments in public goods games. Our numerical simulation results indicate that this reputation rule and heterogeneous investments can better stimulate cooperation. Additionally, stronger investment heterogeneity can further increase the level of cooperation. To explain this phenomenon, dynamical evolution is observed in Monte Carlo simulations. We also investigated the effects of the noise intensity of the irrational population and the original proportion of cooperation in the population. The robustness of this cooperation model was also considered with respect to the network structure and total investment, and we found that the conclusions remained the same.

Two-dimensional adaptive dynamics of evolutionary public goods games: finite-size effects on fixation probability and branching time

Public goods games (PGGs) describe situations in which individuals contribute to a good at a private cost, but others can free-ride by receiving a share of the public benefit at no cost. The game occurs within local neighbourhoods, which are subsets of the whole population. Free-riding and maximal production are two extremes of a continuous spectrum of traits. We study the adaptive dynamics of production and neighbourhood size. We allow the public good production and the neighbourhood size to coevolve and observe evolutionary branching. We explain how an initially monomorphic population undergoes evolutionary branching in two dimensions to become a dimorphic population characterized by extremes of the spectrum of trait values. We find that population size plays a crucial role in determining the final state of the population. Small populations may not branch or may be subject to extinction of a subpopulation after branching. In small populations, stochastic effects become important and we calculate the probability of subpopulation extinction. Our work elucidates the evolutionary origins of heterogeneity in local PGGs among individuals of two traits (production and neighbourhood size), and the effects of stochasticity in two-dimensional trait space, where novel effects emerge.

Strategically positioning cooperators can facilitate the contagion of cooperation

The spreading of cooperation in structured population is a challenging problem which can be observed at different scales of social and biological organization. Generally, the problem is studied by evaluating the chances that few initial invading cooperators, randomly appearing in a network, can lead to the spreading of cooperation. In this paper we demonstrate that in many scenarios some cooperators are more influential than others and their initial positions can facilitate the spreading of cooperation. We investigate six different ways to add initial cooperators in a network of cheaters, based on different network-based measurements. Our research reveals that strategically positioning the initial cooperators in a population of cheaters allows to decrease the number of initial cooperators necessary to successfully seed cooperation. The strategic positioning of initial cooperators can also help to shorten the time necessary for the restoration of cooperation. The optimal ways in which the initial cooperators should be placed is, however, non-trivial in that it depends on the degree of competition, the underlying game, and the network structure. Overall, our results show that, in structured populations, few cooperators, well positioned in strategically chosen places, can spread cooperation faster and easier than a large number of cooperators that are placed badly.

Complex networks identification using Bayesian model with independent Laplace prior

Identification of complex networks from limited and noise contaminated data is an important yet challenging task, which has attracted researchers from different disciplines recently. In this paper, the underlying feature of a complex network identification problem was analyzed and translated into a sparse linear programming problem. Then, a general framework based on the Bayesian model with independent Laplace prior was proposed to guarantee the sparseness and accuracy of identification results after analyzing influences of different prior distributions. At the same time, a three-stage hierarchical method was designed to resolve the puzzle that the Laplace distribution is not conjugated to the normal distribution. Last, the variational Bayesian was introduced to improve the efficiency of the network reconstruction task. The high accuracy and robust properties of the proposed method were verified by conducting both general synthetic network and real network identification tasks based on the evolutionary game dynamic. Compared with other five classical algorithms, the numerical experiments indicate that the proposed model can outperform these methods in both accuracy and robustness.