Bias and Overtaking Equilibria for Zero-Sum Stochastic Differential Games


This paper deals with zero-sum stochastic differential games with long-run average payoffs. Our main objective is to give conditions for existence and characterization of bias and overtaking optimal equilibria. To this end, first we characterize the family of optimal average payoff strategies. Then, within this family, we impose suitable conditions to determine the subfamilies of bias and overtaking equilibria. A key step to obtain these facts is to show the existence of solutions to the average payoff optimality equations. This is done by the usual “vanishing discount” approach. Finally, a zero-sum game associated to a certain manufacturing process illustrates our results.

This is a preview of subscription content, access via your institution.


  1. 1.

    Borkar, V.S., Ghosh, M.K.: Stochastic differential games: occupation measure based approach. J. Optim. Theory Appl. 73, 359–385 (1992). Correction: 88, 251–252 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Kushner, H.J.: Numerical approximations for stochastic differential games: the ergodic case. SIAM J. Control Optim. 42, 1911–1933 (2003)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Ramsey, F.P.: A mathematical theory of savings. Econ. J. 38, 543–559 (1928)

    Article  Google Scholar 

  4. 4.

    Atsumi, H.: Neoclassical growth and the efficient program of capital accumulation. Rev. Econ. Stud. 32, 127–136 (1965)

    Article  Google Scholar 

  5. 5.

    von Weizsäcker, C.C.: Existence of optimal programs of accumulation for an infinite horizon. Rev. Econ. Stud. 32, 85–104 (1965)

    Article  Google Scholar 

  6. 6.

    Brock, W.A.: Differential games with active and passive variables. In: Henn, R., Moeschlin, O. (eds.) Mathematical Economics and Game Theory: Essays in Honor of Oscar Morgenstern, pp. 34–52. Springer, Berlin (1977)

    Google Scholar 

  7. 7.

    Rubinstein, A.: Equilibrium in supergames with the overtaking criterion. J. Econ. Theory 21, 1–9 (1979)

    MATH  Article  Google Scholar 

  8. 8.

    Carlson, D.: Normalized overtaking Nash equilibrium for a class of distributed parameter dynamic games. In: Nowak, A.S., Swajowski, K. (eds.) Advances in Dynamic Games. Birkhauser, Boston (2005)

    Google Scholar 

  9. 9.

    Carlson, D., Haurie, A.: A turnpike theory for infinite horizon open-loop differential games with decoupled controls. In: Olsder, G.J. (ed.) New Trends in Dynamic Games and Applications. Annals of the ISDG, vol. 3, pp. 353–376. Birkhäuser, Boston (1995)

    Google Scholar 

  10. 10.

    Nowak, A.S.: Equilibrium in a dynamic game of capital accumulation with the overtaking criterion. Econ. Lett. 99, 233–237 (2008)

    Article  Google Scholar 

  11. 11.

    Nowak, A.S., Vega-Amaya, O.: A counterexample on overtaking optimality. Math. Methods Oper. Res. 49, 435–439 (1998)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Nowak, A.S.: Sensitive equilibria for ergodic stochastic games with countable state spaces. Math. Methods Oper. Res. 50, 65–76 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Nowak, A.S.: Optimal strategies in a class of zero-sum ergodic stochastic games. Math. Methods Oper. Res. 50, 399–419 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Jasso-Fuentes, H., Hernández-Lerma, O.: Characterizations of overtaking optimality for controlled diffusion processes. Appl. Math. Optim. 57, 349–369 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Prieto-Rumeau, T., Hernández-Lerma, O.: Bias and overtaking equilibria for zero-sum continuous-time Markov games. Math. Methods Oper. Res. 61, 437–454 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Arapostathis, A., Borkar, V.S.: Uniform recurrence properties of controlled diffusions and applications to optimal control. SIAM J. Control Optim. 48, 4181–4223 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Oksendal, B.: Stochatic Differential Equations: An Introduction with Applications. Springer, New York (1995)

    Google Scholar 

  18. 18.

    Ghosh, M.K., Arapostathis, A., Marcus, S.I.: Optimal control of switching diffusions with applications to flexible manufacturing systems. SIAM J. Control Optim. 31, 1183–1204 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Ghosh, M.K., Arapostathis, A., Marcus, S.I.: Ergodic control of switching diffusions. SIAM J. Control Optim. 35, 1962–1988 (1997)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Bhatt, A.G., Karandikar, R.L.: Invariant measures and evolution equations for Markov processes. Ann. Probab. 21, 2246–2268 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Klebaner, F.C.: Introduction to Stochastic Calculus with Applications, 2nd edn. Imperial College Press, London (2005)

    Google Scholar 

  22. 22.

    Fan, K.: Minimax theorems. Proc. Natl. Acad. Sci. USA 39, 32–47 (1953)

    Article  Google Scholar 

  23. 23.

    Sion, M.: On general minimax theorems. Pac. J. Math. 8, 171–176 (1958)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Bensoussan, A.: Perturbation Methods in Optimal Control. Wiley, New York (1998)

    Google Scholar 

  25. 25.

    Morimoto, H., Okada, M.: Some results on the Bellman equation of ergodic control. SIAM J. Control Optim. 38, 159–174 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    López-Barrientos, J.D., Escobedo-Trujillo, B.A., Hernández-Lerma, O.: Zero-sum stochastic differential games with discounted payoffs. Submitted (2011)

  27. 27.

    Schäl, M.: Conditions for optimality and for the limit of n-stage optimal policies to be optimal. Z. Wahrs. Verw. Gerb. 32, 179–196 (1975)

    MATH  Article  Google Scholar 

  28. 28.

    Borkar, V.S., Ghosh, M.K.: Ergodic control of multidimensional diffusions II: adaptive control. Appl. Math. Optim. 21, 191–220 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Hashemi, S.N., Heunis, A.J.: On the Poisson equation for singular diffusions. Stochastics 77, 155–189 (2005)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Jasso-Fuentes, H., Hernández-Lerma, O.: Blackwell optimality for controlled diffusion processes. J. Appl. Probab. 46, 372–391 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Akella, R., Kumar, P.R.: Optimal control of production rate in a failure prone manufacturing system. IEEE Trans. Autom. Control 31, 116–126 (1985)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Heidelberg (1998). Reprinted version

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Beatris Escobedo-Trujillo.

Additional information

Communicated by Negash G. Medhin.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Escobedo-Trujillo, B., López-Barrientos, D. & Hernández-Lerma, O. Bias and Overtaking Equilibria for Zero-Sum Stochastic Differential Games. J Optim Theory Appl 153, 662–687 (2012).

Download citation


  • Average (or ergodic) payoff criteria
  • Bias optimality
  • Zero-sum stochastic differential games
  • Overtaking optimality