Competing strategies in an evolutionary game model, or species in a biosystem, can easily form a larger unit which protects them from the invasion of an external actor. Such a defensive alliance may have two, three, four or even more members. But how effective can be such formation against an alternative group composed by other competitors? To address this question we study a minimal model where a two-member and a four-member alliances fight in a symmetric and balanced way. By presenting representative phase diagrams, we systematically explore the whole parameter range which characterizes the inner dynamics of the alliances and the intensity of their interactions. The group formed by a pair, who can exchange their neighboring positions, prevail in the majority of the parameter region. The rival quartet can only win if their inner cyclic invasion rate is significant while the mixing rate of the pair is extremely low. At specific parameter values, when neither of the alliances is strong enough, new four-member solutions emerge where a rock-paper-scissors-like trio is extended by the other member of the pair. These new solutions coexist hence all six competitors can survive. The evolutionary process is accompanied by serious finite-size effects which can be mitigated by appropriately chosen prepared initial states.

翻译：在进化博弈模型中的竞争策略，或生物系统中的物种，很容易形成更大的单元以保护自身免受外部行为者的入侵。这种防御联盟可能由两个、三个、四个甚至更多成员组成。但面对由其他竞争者组成的替代群体时，这种结构的有效性如何？为解答此问题，我们研究了一个最小模型，其中两成员联盟与四成员联盟以对称且均衡的方式对抗。通过呈现代表性相图，我们系统探索了表征联盟内部动态及其相互作用强度的整个参数范围。由可交换相邻位置的一对成员组成的群体在大多数参数区域中占据优势。对手四方联盟仅在内部循环入侵率显著且双方交换率极低时才能获胜。在特定参数值下，当任一联盟均不够强大时，会涌现出新的四成员解，其中类似石头-剪刀-布的三元组被对方联盟的另一成员扩展。这些新解共存，因此所有六名竞争者均能存活。进化过程伴随显著的有限规模效应，但可通过适当选择的初始状态加以缓解。