A survey of evolutionary game and resource allocation

Evolutionary game theory involves multiple disciplinary sciences and has enormous scientific value and promising applicability. Collective behavior is an important topic of interdisciplinary study. Ethology has shown the ubiquity of collective behavior and has proven the rationality of evolutionary theory in explaining the emergence of collective behavior. The recent development of complex network theory offers a convenient framework for describing game interactions and competition relationships among individuals. The combination of evolutionary games and complex networks, particularly, evolutionary game theory in a complex network, has been attracting growing interest from different fields. It has undergone substantial development, especially in quantitative analysis of two-strategy competition. Under this framework, the complex network represents the population structure, and the game describes interactions between individuals. On the basis of the methodology from network science, stochastic process, and statistical physics, the framework mainly focuses on how population structures, individual behavior patterns, and interacting environments influence the emergence of collective behavior. In this paper, the mechanisms for the evolution of cooperation were given under the framework of evolutionary game, including kin selection, direct reciprocity, indirect reciprocity, network reciprocity, and group selection. Recently, the effects of individual heterogeneity and environmental feedback on cooperation had attracted growing interest. Next, five main theoretical methods were addressed for analyzing the evolutionary game in complex networks, including the \begin{document}$ \sigma $\end{document}- dominance rule, the coalescing theory, the pairwise approximation, the coalescing random walk theory, and the adaptive dynamics. Particularly, the recently proposed coalescing random walk theory is suitable for analyzing the dynamics of any network structure and any update rule. Then, the studies on the evolution of fairness in ultimatum games were presented, and reasonable resource allocation is the key factor for social stability, economic development, and individual health. Finally, the challenges and further directions of studying ultimatum games in a complex network were summarized.