Approximate consensus of networked evolutionary matrix game: A finite-cost event-triggered mechanism

In scenarios such as voting and elections, the endeavor to achieve a certain degree of consensus is widespread. Consequently, in this paper, the approximate consensus problem of networked evolutionary matrix games is proposed to better describe such situations. First, the verification of (locally) approximate consensus is given from both algebraic and topological perspectives. Secondly, taking the consumption for operators and the compensation for puppet players into account, a more reasonable and cost-effective finite-cost event-triggered mechanism, which is composed of non-functional control and asynchronous control by different operators, is proposed to facilitate approximate consensus. Under this module, a necessary and sufficient condition is derived to verify the existence of this mechanism. Then, the shortest time and minimum cost for achieving approximate consensus, as well as the corresponding event-triggered mechanisms, are obtained by breadth-first search algorithms. Finally, the conclusions are extended to the scenarios including groups and unfixed approximate consensus and the theoretical result is demonstrated effectively by an illustrative example.