IDEAS home Printed from
   My bibliography  Save this paper

Equilibrium and Guaranteeing Solutions in Evolutionary Nonzero Sum Games


  • A.V. Kryazhimskii
  • A.M. Tarasyev


Advanced methods of theory of optimal guaranteeing control and techniques of generalized (viscosity, minimax) solutions of Hamilton-Jacobi equations are applied to nonzero game interaction between two large groups (coalitions) of agents (participants) arising in economic and biological evolutionary models. Random contacts of agents from different groups happen according to a control dynamical process which can be interpreted as Kolmogorov's differential equations in which coefficients describing flows are not fixed a priori and can be chosen on the feedback principle. Payoffs of coalitions are determined by the functionals of different types on infinite horizon. The notion of a dynamical Nash equilibrium is introduced in the class of control feedbacks. A solution feedbacks maximizing with the guarantee the own payoffs (guaranteeing feedback) is proposed. Guaranteeing feedbacks are constructed in the framework of the the theory of generalized solutions of Hamilton-Jacobi equations. The analytical formulas are obtained for corresponding value functions. The equilibrium trajectory is generated by guaranteeing feedbacks and its properties are investigated. The considered approach provides new qualitative results for the equilibrium trajectory in evolutionary models. The first striking result consists in the fact that the designed equilibrium trajectory provides better (in some bimatrix games strictly better) index values for both coalitions than trajectories which converge to static Nash equilibria (as, for example, trajectories of classical models with the replicator dynamics). The second principle result implies both evolutionary properties of the equilibrium trajectory: evolution takes place in the characteristic domains of Hamilton-Jacobi equations and revolution at switching curves of guaranteeing feedbacks. The third specific feature of the proposed solution is "positive" nature of guaranteeing feedbacks which maximize the own payoff unlike the "negative" nature of punishing feedbacks which minimize the opponent payoff and lead to static Nash equilibrium. The fourth concept takes into account the foreseeing principle in constructing feedbacks due to the multiterminal character of payoffs in which future |states are also evaluated. The fifth idea deals with the venturous factor of the equilibrium trajectory and prescribes the risk barrier surrounding it. These results indicate promising applications of theory of guaranteeing control for constructing solutions in evolutionary models.

Suggested Citation

  • A.V. Kryazhimskii & A.M. Tarasyev, 1998. "Equilibrium and Guaranteeing Solutions in Evolutionary Nonzero Sum Games," Working Papers ir98003, International Institute for Applied Systems Analysis.
  • Handle: RePEc:wop:iasawp:ir98003

    Download full text from publisher

    File URL:
    Download Restriction: no

    File URL:
    Download Restriction: no


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. F. de Vries, 1999. "The Behavioral Firm and Its Internal Game: Evolutionary Dynamics of Decision Making," Working Papers ir99036, International Institute for Applied Systems Analysis.
    2. A.V. Kryazhimskii & A. Nentjes & S. Shibayev & A.M. Tarasyev, 1998. "Searching Market Equilibria under Uncertain Utilities," Working Papers ir98007, International Institute for Applied Systems Analysis.

    More about this item


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wop:iasawp:ir98003. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Thomas Krichel). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.