A switched model for mixed cooperative-competitive social dynamics

A multi-agent continuous-time nonlinear model of social behaviour allowing for both competition and cooperation is presented and analysed. The state of each agent is represented by its payoff, which the agent aims at maximising. The role of control variables is played by the model parameters, which account for the agents' decisions to either cooperate with or boycott the other agents and can vary in time within assigned intervals. Alliances and enmities can be established at any time, according to either a greedy or a longsighted criterion. The general nonlinear case is first considered. It is proved that, under realistic assumptions, the system evolution is bounded positive (no extinction) and there is a unique globally-stable equilibrium point. As is somehow expected, the optimal decision for all agents corresponds to full cooperation (decision parameters kept at their positive maximum value) in the case of both shortsighted and farsighted criteria. This is not true if some parameters have negative upper bounds (meaning that some agents systematically boycott some others). Then, in the linear case, it is shown that the system is stable for arbitrarily-varying decision parameters, provided that a Metzler matrix associated with full cooperation is Hurwitz. A characterisation of the long-term behaviour of the linear system is also provided. In particular, it is proved that, under stability conditions, a Nash equilibrium exists if a steady strategy is adopted.