Games without winners: Catching-up with asymmetric spillovers

Dynamic game with changing leader is studied on the example of R&D co-opetition structure. The leader benefits from higher followers' innovations rate and followers are enjoying a spillover from the leader. Leadership changes because of asymmetric efficiency of investments of players. It is demonstrated that under sufficiently asymmetric players there is no long-run leader in this game and all players act as followers. Moreover this outcome may be the socially optimal one. In decentralised setting additional complex types of dynamics are observed: permanent uctuations around symmetric (pseudo)equilibrium and chaotic dynamics. This last is possible only once strategies of players are interdependent. Cooperative solution is qualitatively similar for any number of players while market solution is progressively complex given all players are asymmetric. Results are extended to an arbitrary linear-quadratic multi-modal differential game with spillovers and the structure necessary for the onset of non-deterministic chaos is discussed.