Competing or collaborating, with no symmetrical behaviour: Leadership opportunities and winning strategies under stability

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.