Games Without Winners: Catching-up with Asymmetric Spillovers

Multi-modal differential R&D game with asymmetric players is studied. It is demonstrated that under sufficiently asymmetric players there is no long-run ‘winner’ in this game in terms of developed technologies and all players try to imitate each other. Moreover, this outcome may be the only equilibrium in the cooperative game. In decentralized setting, additional complex types of dynamics are observed: permanent fluctuations around symmetric (pseudo)equilibrium and chaotic dynamics. This last is possible only once strategies of players are interdependent. These new emergent dynamics types call for additional regulation tools which are shortly discussed. It is shown that cooperative solution is qualitatively similar for any number of players, while non-cooperative solution is progressively complex given 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.