Skip to main content
SpringerLink
Log in

Complementarity and Diagonal Dominance in Discounted Stochastic Games

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

We consider discounted stochastic games characterized by monotonicity, supermodularity and diagonal dominance assumptions on the reward functions and the transition law. A thorough novel discussion of the scope and limitations of this class of games is provided. Existence of a Markov-stationary equilibrium for the infinite-horizon game, proved by Curtat (1996), is summarized. Uniqueness of Markov equilibrium and dominance solvability of the finite-horizon game are established. In both cases, the equilibrium strategies and the corresponding value functions are nondecreasing Liptschitz-continuous functions of the state vector. Some specific economic applications are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Amir, On stochastic games with uncountable state and action spaces, in: Stochastic Games and Related Topics, eds. T. Raghavan, T. Ferguson, T. Parthasarathy and O. Vrieze (Kluwer, Dordrecht, 1991) pp. 149-159.

    Google Scholar 

  2. R. Amir, Continuous stochastic games of capital accumulation with convex transitions, Games and Economic Behavior 16 (1996) 111-131.

    Google Scholar 

  3. R. Amir, Sensitivity analysis of multi-sector optimal economic dynamics, Journal of Mathematical Economics 25 (1996) 123-141.

    Google Scholar 

  4. R. Amir, Strategic intergenerational bequests with stochastic convex technology, Economic Theory 8 (1996) 367-376.

    Google Scholar 

  5. R. Amir and H. Hadim, On some structured deterministic dynamic programs arising in economics, Computers and Mathematics, with Applications 24 (1992) 209-218.

    Google Scholar 

  6. R. Amir, L. Mirman and W. Perkins, One-sector nonclassical optimal growth: Optimality conditions and comparative dynamics, International Economic Review 32 (1991) 625-644.

    Google Scholar 

  7. S. Athey, Characterizing properties of stochastic objective functions, mimeograph, MIT (1998).

  8. S. Athey, Monotone comparative statics under uncertainty, Quarterly Journal of Economics (2001), to appear.

  9. T. Basar and G. Olsder, Dynamic Noncooperative Game Theory (SIAM, 1999).

  10. D. Bernheim and D. Ray, Economic growth with intergenerational altruism, Review of Economic Studies 54 (1987) 227-242.

    Google Scholar 

  11. D. Bertsekas and S. Shreve, Stochastic Optimal Control: The Discrete-Time Case (Academic Press, New York, 1978).

    Google Scholar 

  12. W. Brock and L. Mirman, Optimal growth under uncertainty: The discounted case, Journal of Economic Theory 4 (1972) 479-513.

    Google Scholar 

  13. L. Curtat, Markov equilibria of stochastic games with complementarities, Games and Economic Behavior 17 (1996) 177-199.

    Google Scholar 

  14. M. Datta and L. Mirman, Externalities, market power, and resource extraction, Journal of Environmental Economics and Management 37 (1999) 233-255.

    Google Scholar 

  15. P. Diamond, Aggregate demand management in search equilibrium, Journal of Political Economy 90 (1982) 881-894.

    Google Scholar 

  16. P. Dutta and R. Sundaram, Markovian equilibrium in a class of stochastic games: Existence theorems for discounted and undiscounted models, Economic Theory 2 (1992) 197-214. 56 AMIR

    Google Scholar 

  17. A. Fedegruen, On N-person stochastic game with denumerable state space, Advances in Applied Probability 10 (1978) 452-471.

    Google Scholar 

  18. R. Fischer and L. Mirman, The compleat fish wars: Biological and dynamic interactions, Journal of Environmental Economics and Management 30 (1996) 34-42.

    Google Scholar 

  19. F. Granot and A. Veinott, Substitutes, complements, and ripples in network flows, Mathematics of Operations Research 10 (1985) 471-497.

    Google Scholar 

  20. E. Hewitt and K. Stromberg, Real and Abstract Analysis (Springer, New York, 1965).

    Google Scholar 

  21. D. Heyman and M. Sobel, Stochastic Models in Operations Research, Vol. 2 (McGraw-Hill, New York, 1984).

    Google Scholar 

  22. H. Hopenhayn and E. Prescott, Stochastic monotonicity and stationary distributions for dynamic economies, Econometrica 60 (1992) 1387-1406.

    Google Scholar 

  23. W. Leininger, Existence of perfect equilibrium in a model of growth with altruism between generations, Review of Economic Studies 53 (1986) 349-67.

    Google Scholar 

  24. D. Levhari and L. Mirman, The great fish war: An example using a dynamic Cournot-Nash solution, Bell Journal of Economics (1980) 322-344.

  25. R. Mendelsohn and M. Sobel, Capital accumulation and the optimization of renewable resources, Journal of Economic Theory 23 (1981) 243-260.

    Google Scholar 

  26. J.-F. Mertens and T. Parthasarathy, Equilibria for discounted stochastic games, in: Stochastic Games, eds. A. Neyman and S. Sorin, NATO Advanced Science Institutes Series D: Behavioural and Social Sciences (Kluwer in cooperation with NATO Scientific Affairs, Dordrecht and Boston, 2002), to appear (also, CORE DP # 87-50, Belgium).

    Google Scholar 

  27. P. Milgrom and J. Roberts, Rationalizability, learning, and equilibrium in games with strategic complementarities, Econometrica 58 (1990) 1255-1278.

    Google Scholar 

  28. P. Milgrom and J. Roberts, Comparing equilibria, American Economic Review 84 (1994) 441-459.

    Google Scholar 

  29. P. Milgrom and C. Shannon, Monotone comparative statics, Econometrica 62 (1994) 157-180.

    Google Scholar 

  30. A. Neyman and S. Sorin, eds., Stochastic Games, NATO Advanced Science Institutes Series D: Behavioural and Social Sciences (Kluwer in cooperation with NATO Scientific Affairs, Dordrecht and Boston, 2002), to appear.

    Google Scholar 

  31. A. Nowak, Existence of equilibrium stationary strategies in discounted noncooperative stochastic games with uncountable state space, Journal of Optimization Theory and Applications 45 (1985) 591-602.

    Google Scholar 

  32. U. Rieder, Equilibrium plans for nonzero-sum Markov games, in: Game Theory and Related Topics, eds. H. Moeschlin and D. Pallaschke (North-Holland, Amsterdam, 1978) pp. 91-101.

    Google Scholar 

  33. J. Rosen, Existence and uniqueness of equilibrium points for concave n-person games, Econometrica 33 (1965) 520-53

    Google Scholar 

  34. R. Serfozo, Monotone optimal policies for Markov decision processes, Mathematical Programming Study 6 (1976) 202-215.

    Google Scholar 

  35. L. Shapley, Stochastic games, Proceedings of the National Academy of Sciences of the USA 39 (1953) 1095-1100.

    Google Scholar 

  36. A. Tarski, A latice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics 5 (1955) 285-309.

    Google Scholar 

  37. D. Topkis, Ordered optimal solutions, Ph.D. thesis, Stanford University (1968).

  38. D. Topkis, Minimizing a submodular function on a lattice, Operations Research 26 (1978) 305-321.

    Google Scholar 

  39. D. Topkis, Equilibrium points in nonzero-sum submodular games, SIAM Journal of Control and Optimization 17 (1979) 773-787.

    Google Scholar 

  40. D. Topkis, Supermodularity and Complementarity (Princeton University Press, Princeton, NJ, 1998).

    Google Scholar 

  41. X. Vives, Nash equilibrium with strategic complementarities, Journal of Mathematical Economics 19 (1990) 305-321.

    Google Scholar 

  42. W. Whitt, Represention and approximation of noncooperative sequential games, SIAM Journal of Control and Optimization 18 (1980) 33-48.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Economics, U.C.L., CORE and, 1348, Louvain-la-Neuve, Belgium

    Rabah Amir

Authors
  1. Rabah Amir
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Amir, R. Complementarity and Diagonal Dominance in Discounted Stochastic Games. Annals of Operations Research 114, 39–56 (2002). https://doi.org/10.1023/A:1021097716583

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021097716583

Keywords

Search

Navigation