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On the Computation of Equilibria in Discounted Stochastic Dynamic Games

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Dynamic Games and Applications in Economics

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 265))

Abstract

Various algorithms for numerical solutions of discounted stochastic games are presented. For zero-sum two-person games, the existing algorithms are compared on randomly generated games and a hybrid algorithm is proposed. For general-sum N-person games, a new mathematical programming formulation which permits the numerical solution of a game by using a non-linear programming code Is presented.

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References

  • Basar, T. and G.J. Olsder (1982), Dynamic NoncooperatIve Game Theory, Academic Press.

    Google Scholar 

  • Bertsekas, D.P. and S.E. Shreve (1978), Stochastic Optimal Control: The Discrete Time Case, Academic Press.

    Google Scholar 

  • Blackwell, D. and T.S. Ferguson (1968), «The Big Match», Ann, of Math. Stat., vol. 39, pp. 159–163.

    Article  Google Scholar 

  • Boel, R. and P. Varayia (1977), «Optimal Control of Jump Process», SIAM Journal on Control and Opt imlzat Ion, vol. 15, pp. 385–394.

    Google Scholar 

  • Brown, G.W. (1951), «Iterative Solution of Games by Fictitious Play» in: Activity Analysis of Production and Allocation, Koopmans, T.C. (ed.) John Wiley, N.Y., pp. 374–376.

    Google Scholar 

  • Case, J.H. (1969), «Toward a Theory of Many Player Differential Games», SIAM Journal on Control, vol. 7, no. 2, pp. 179–197

    Article  Google Scholar 

  • Clemhout, S. and H.Y. Wang (1974), «A Class of Trilinear Differential Games», Journ. of Opt. Theory and Applications, vol. 14, pp. 419–424.

    Article  Google Scholar 

  • Denardo, E.V. (1967), «Contraction Mappings in the Theory Underlying Dynamic Programming», Slam Rev, vol. 9, pp. 165–177.

    Article  Google Scholar 

  • Filar, J.A. (1985), «Quadratic Programming and the Single-Controller Stochastic Game», to appear in Journal of Math. Analysis and Appl.

    Google Scholar 

  • Filar, J.A. and T. Schultz (1985), «Bilinear Programming and Structured Stochastic Games», to appear In Journ. of Opt. Theory and Applications.

    Google Scholar 

  • Fink, A.M. (1964), «Equilibrium in a Stochastic n-Person Game», J. Sci. Hiroshima Univ. Ser., vol. Al;28, pp. 89–93.

    Google Scholar 

  • Friedman, A. (1971), Differential Games, Wi1ey-Interscience, N.Y.

    Google Scholar 

  • Gill, P.E., W. Murray, M.A. Saunders and M.H. Wright (1983), «Users’ Guide for SOL/NPSOL: A Fortran Package for Non Linear Programming», Technical Report SOL 83-12, Stanford University, Stanford, California.

    Google Scholar 

  • Haurle, A. and P. L’Écuyer (1982), «A Stochastic Control Approach to Group Preventive Replacement In a Multi-Component System», IEEE Trans, on Automatic Control, vol. AC-27, pp. 387–393.

    Article  Google Scholar 

  • Haurle, A. and P. L’Écuyer (1985), «Approximation and bounds In Discrete Event Dynamic Programming», to appear In IEEE Transactions.

    Google Scholar 

  • Hoffman, A. and R. Karp (1966), «On Non-TermlnatIng Stochastic Games», Man. Science vol. 12, pp. 359–370.

    Google Scholar 

  • Hordljk, L. and Kallenberg, L.C.M. (1981), «Linear Programming and Markov Games I», In: Game Theory and Mathematical Economics, O. Moeschlin, D. Pallaschke (eds.), North-Holland, pp. 291–305.

    Google Scholar 

  • Hordljk, L. and Kallenberg, L.C.M. (1981), «Linear Programming and Markov Games II», In: Game Theory and Mathematical Economics, O. Moeschlin, D. Pallaschke (eds.), North-Holland, pp. 307–319.

    Google Scholar 

  • Hordljk, L., O.J. Vrleze and G.L. Wanrooij (1982), «Semi Markov Strategies in Stochastic Games», Intl. Journ. of Game Theory, vol. 12, pp. 81–89.

    Article  Google Scholar 

  • Isaacs, R. (1965), Differential Games, Wiley, New-York.

    Google Scholar 

  • Karmadlan, S. (1976), Fixed points: Algorithms and ApplIcations, Academic Press, N.Y.

    Google Scholar 

  • Lemke, C.E. and J.T. Howson (1964), «Equilibrium Points of Blmatrlx Games», SIAM Journal on Applied Math., vol. 12, pp. 413–423.

    Article  Google Scholar 

  • Mohan, S.R. and Raghavan, T.E.S. (1983), An Algorithm for Switching Control Discounted Stochastic Games, Tech. Report,, University of I11inols at Chicago.

    Google Scholar 

  • Parthasarathy, T., S.H. Tijs and O.J. Vrieze (1984), «Stochastic Games with State Indépendant Transitions and Separable Rewards», In: Selected Topics In OR and Math. Economics, G. Hammer and D. Pollaschke (eds) Springer-Verlag Notes Series #226.

    Google Scholar 

  • Parthasarathy, T. and T.E.S. Raghavan (1981), «An Ordered Field Property for Stochastic Games when One Player Controls Transition Probabilities», JOTA, vol. 33, pp. 375–392.

    Article  Google Scholar 

  • Pollatschek, M. and B. Avi-Itzhak (1969), «Algorithms for Stochastic Games with Geometrical Interpretation», Man. Science vol. 15, pp. 399–415.

    Google Scholar 

  • Raghavan, T.E.S., (1984), Algorithms for Stochastic Games, A Survey, Technical Report, University of Illinois at Chicago.

    Google Scholar 

  • Rao, S., R. Chandrasekaran and K. Nair (1973), «Algorithms for discounted Stochastic Games», JOTA vol. 11, pp. 627–637.

    Article  Google Scholar 

  • Rishel, R. (1977), «Optimality for Completely Observed Controlled Jump Processes», IEEE Trans. Autom. Cont., vol. AC-22, pp. 906–908.

    Article  Google Scholar 

  • Robinson, J. (1950), «An Iterative Method of Solving a Game», Ann, of Math., vol. 54, pp. 296–301.

    Article  Google Scholar 

  • Rogers, P.D. (1969), Nonzero-sum Stochastic Games, Ph. D. Thesis, University of California at Berkeley.

    Google Scholar 

  • Shapley, L.S. (1953), «Stochastic Games», Proc. Nat. Acad. Sci. USA, vol. 39, pp. 1095–1100.

    Article  Google Scholar 

  • Sobel, M.J. (1971), «Noncooperative Stochastic Games», Annals of Math. Stat. vol. 42, pp. 1930–1935.

    Article  Google Scholar 

  • Van der Wal, J. (1978), «Discounted Markov Games: Generalized Policy Iteration Method», JOTA vol. 25, pp. 125–138.

    Article  Google Scholar 

  • Vrieze, O.J. (1981), «Linear Programming and Undlscounted Stochastic Games in Which One Player Controls Transitions», OR Spektrum, vol. 3, pp. 29–35.

    Article  Google Scholar 

  • Vrieze, O.J. (1983), Stochastic Games with Finite State and Action Spaces, Ph. D. Thesis, Free University of Amsterdam, The Netherlands.

    Google Scholar 

  • Vrieze, O.J. and S.H. Tijs (1980), «Fictitious Play Applied to Sequences of Games and Discounted Stochastic Games», Int1. Journ. of Game Theory vol. 11, pp. 71–85.

    Article  Google Scholar 

  • Vrieze, O.J., S.H. Tijs, T.E.S. Raghavan and J.A. Filar (1983), «A Finite Algorithm for the Switching Control Stochastic Game», OR Spektrum, vol. 5, pp. 15–24.

    Article  Google Scholar 

  • Whitt, W. (1978), «Approximations of Dynamic Programs I», Math, of Oper. Res, vol. 2, pp. 297–302.

    Google Scholar 

  • Whitt, W. (1979), «Approximations of Dynamic Programs II», Math, of Oper. Res, vol. 3, pp. 231–243.

    Article  Google Scholar 

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© 1986 Springer-Verlag Berlin Heidelberg

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Breton, M., Filar, J.A., Haurle, A., Schultz, T.A. (1986). On the Computation of Equilibria in Discounted Stochastic Dynamic Games. In: Başar, T. (eds) Dynamic Games and Applications in Economics. Lecture Notes in Economics and Mathematical Systems, vol 265. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61636-5_4

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  • DOI: https://doi.org/10.1007/978-3-642-61636-5_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16435-7

  • Online ISBN: 978-3-642-61636-5

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