In this paper, a new dynamic mathematical model describing leadership emergence or disappearance in agent based networks is proposed. Through a generalised Verhulst–Lotka–Volterra model, a triad of agents operates in a market where each agent detains a quota. The triad is composed of a leader, who leads communication, and two followers. Communications flows both ways from leader to followers and vice versa. Competition, collaboration and cheating are allowed. Stability solutions are investigated analytically through a fixed point analysis. Various solutions exist depending on the type of behavioural interactions. Results show that communication counts: survival of the leader is a condition for stability. All configurations with the leader out of the market are unstable. Conversely, the two followers position is highly difficult. The three agents cannot all survive unless they behave under mutual collaboration and in very special conditions. For followers, cheating the leader, especially if the leader is collaborating, can be a disaster. By the way, collaboration with the leader may not always ensure market survival. However, this can be a strategy to survive and even share the leadership, in particular when the other agent cheats (or is cheated by) the leader. Cheating is a cause of instability. In fact, only a few cases reach stability: this occurs when cheating comes from the leader and the leader always wins. The leader may be interested in cheating if she does not want to share the leadership with a follower, that is to get the monopoly of the market.

本文提出了一种新的动态数学模型，该模型描述了基于代理的网络中领导者的出现或消失。通过广义的Verhulst–Lotka–Volterra模型，三人代理在一个每个代理都有配额的市场中运作。这个三合会由一个领导沟通的领导者和两个追随者组成。沟通从领导者流向跟随者，反之亦然。允许比赛，合作和作弊。通过定点分析对稳定性解决方案进行了分析研究。根据行为交互的类型，存在各种解决方案。结果表明，沟通很重要：领导者的生存是稳定的条件。领先者退出市场的所有配置都是不稳定的。相反，两个追随者的职位非常困难。除非他们在相互合作和非常特殊的条件下行事，否则这三个特工都无法生存。对于追随者而言，欺骗领导者（尤其是领导者正在协作）可能是一场灾难。顺便说一句，与领导者的合作可能并不总是确保市场生存。但是，这可以是生存甚至分享领导者的策略，尤其是当其他特工欺骗领导者（或被其欺骗）时。作弊是不稳定的原因。实际上，只有少数情况可以达到稳定：这种情况发生在领导者作弊，而领导者总是获胜时。如果领导者不想与追随者分享领导权，那就可能有兴趣作弊，那就是垄断市场。欺骗领导者，特别是在领导者进行协作的情况下，可能是一场灾难。顺便说一句，与领导者的合作可能并不总是确保市场生存。但是，这可以是生存甚至分享领导者的策略，尤其是当其他特工欺骗领导者（或被其欺骗）时。作弊是不稳定的原因。实际上，只有少数情况可以达到稳定：这种情况发生在领导者作弊，而领导者总是获胜时。如果领导者不想与追随者分享领导权，那就可能有兴趣作弊，那就是垄断市场。欺骗领导者，尤其是在领导者进行协作的情况下，可能是一场灾难。顺便说一句，与领导者的合作可能并不总是确保市场生存。但是，这可以是生存甚至分享领导者的策略，尤其是当其他特工欺骗领导者（或被其欺骗）时。作弊是不稳定的原因。实际上，只有少数情况可以达到稳定：这种情况发生在领导者作弊，而领导者总是获胜时。如果领导者不想与追随者分享领导权，那就可能有兴趣作弊，那就是垄断市场。特别是当其他特工欺骗领导者（或被其欺骗）时。作弊是不稳定的原因。实际上，只有少数情况可以达到稳定：这种情况发生在领导者作弊，而领导者总是获胜时。如果领导者不想与追随者分享领导权，那就可能有兴趣作弊，那就是垄断市场。特别是当其他特工欺骗领导者（或被其欺骗）时。作弊是不稳定的原因。实际上，只有少数情况可以达到稳定：这种情况发生在领导者作弊，而领导者总是获胜时。如果领导者不想与追随者分享领导权，那就可能有兴趣作弊，那就是垄断市场。