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Infinite zero-sum two-person games

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Literature cited

  1. R. C. Buck, “Preferred optimal strategies,” Proc. Amer. Math. Soc.,9, 312–314 (1958).

    Google Scholar 

  2. R. Bellman, I. L. Glicksberg, and O. Gross, Some Aspects of the Mathematical Theory of Control Processes, The RAND Corp. Santa Monica, Calif. (1958).

    Google Scholar 

  3. D. H. Blackwell and M. A. Girshick, Theory of Games and Statistical Decisions, Wiley, New York (1954).

    Google Scholar 

  4. F. Bohnenblust, S. Karlin, and L. S. Shapley, “Solutions of discrete two-person games,” in: Contributions to the Theory of Games, Vol. 1, Princeton Univ. Press, Princeton, N. J. (1950), pp. 51–72.

    Google Scholar 

  5. F. Bohnenblust, S. Karlin, and L. S. Shapley, “Games with continuous convex payoff,” in: Contributions to the Theory of Games, Vol. 1, Princeton Univ. Press, Princeton, N. J. (1950), pp. 181–192.

    Google Scholar 

  6. R. H. Brown, “The solution of a certain two-person zero-sum game,” Operations Research,5, 63–67 (1957).

    Google Scholar 

  7. A. Wald, “Statistical decision functions,” Ann. Math. Statist.,20, 165–205 (1949).

    Google Scholar 

  8. N. N. Vorob'ev, “Some methodological problems of game theory,” Vopr. Filosof., No. 1, 93–103 (1966).

    Google Scholar 

  9. N. N. Vorob'ev, “Infinite zero-sum two-person games,” in: Infinite Zero-Sum Two-Person Games [in Russian], Fizmatgiz, Moscow (1963), pp. 7–23.

    Google Scholar 

  10. N. N. Vorob'ev, Mathematical Theory of Games [in Russian], Obshch. po Rasprostr. Polit. i Nauch. Znanii RSFSR, Leningrad. Otdel., Leningrad (1963), 71 pages.

    Google Scholar 

  11. N. N. Vorob'ev, “Application of game theory in the engineering sciences” [in Russian], Fourth Internat. Congr. Applications of Mathematics in the Engineering Sciences, Proc. Vol. 1, Weimar (1967), pp. 411–422.

    Google Scholar 

  12. N. N. Vorob'ev, “State of the art of game theory,” Usp. Matem. Nauk,25, No. 2, 81–140 (1970).

    Google Scholar 

  13. M. A. Woodbury, “Linear-convex games,” J. Soc. Indust. Appl. Math (SIAM),1, 137–142 (1953).

    Google Scholar 

  14. D. Gale and O. Gross, “A note on polynomial separable games,” Pacific J. Math.,8, 735–744 (1958).

    Google Scholar 

  15. M. A. Genin, “Algorithms for the solution of certain convex-concave and linear-convex games,” in: Infinite Zero-Sum Two-Person Games [in Russian], Fizmatgiz, Moscow (1963), pp. 195–204.

    Google Scholar 

  16. D. A. Georgobiani, “Approximation of an infinite game by a matrix game,” Trudy Vychisl. Tsentra Akad. Nauk Gruz. SSR,6, No. 3, 123–131 (1965).

    Google Scholar 

  17. Yu. B. Germeir, Introduction to Operations Research [in Russian], Nauka, Moscow (1971), 383 pages.

    Google Scholar 

  18. I. L. Glicksberg and O. Gross, “Notes on games over the square,” in: Contributions to the Theory of Games, Vol. 2, Princeton Univ. Press, Princeton, N. J. (1953), pp. 173–182.

    Google Scholar 

  19. O. Gross, “A rational game on the square,” in: Contributions to the Theory of Games, Vol. 3, Princeton Univ. Press, Princeton, N. J. (1957), pp. 307–311.

    Google Scholar 

  20. É. G. Davydov, “On games with resources,” in: Cybernetics in the Service of Communism [in Russian], Énergiya, Moscow (1971), pp. 249–263.

    Google Scholar 

  21. J. M. Danskin, “Iterative method for the solution of continuous games,” in: Infinite Zero-Sum Two-Person Games [in Russian], Fizmatgiz, Moscow (1963), pp. 121–132.

    Google Scholar 

  22. V. F. Dem'yanov, “Solution of certain minimax problems (I),” Kibernetika, No. 6, 58–66 (1966).

    Google Scholar 

  23. V. K. Domanskii, “An application of the theory of generalized functions to the theory of games,” Dokl. Akad. Nauk SSSR,199, No. 3, 515–518 (1971).

    Google Scholar 

  24. M. Dresher, “Games of strategy,” Math. Mag.,25, 93–99 (1951).

    Google Scholar 

  25. M. Dresher and S. Karlin, “Solutions of convex games as fixed points,” in: Contributions to the Theory of Games, Vol. 2, Princeton Univ. Press, Princeton, N. J. (1953), pp. 75–86.

    Google Scholar 

  26. M. Dresher, S. Karlin, and L. S. Shapley, “Polynomial games,” in: Contributions to the Theory of Games, Vol. 1, Princeton Univ. Press, Princeton, N. J. (1950).

    Google Scholar 

  27. G. N. Dyubin, “Games on the unit square with a payoff function of the ‘inverted-V’ type,” Teor. Veroyat. i Prim.,13, No. 1, 138–143 (1968).

    Google Scholar 

  28. G. N. Dyubin, “A class of games on the unit square,” Dokl. Akad. Nauk SSSR,168, No. 7, 9–12 (1968).

    Google Scholar 

  29. G. N. Dyubin, “A set of games on the unit square with a unique solution,” Dokl. Akad. Nauk SSSR,184, No. 2, 267–269 (1969).

    Google Scholar 

  30. S. Karlin, “On games described by bell-shaped kernels,” in: Contributions to the Theory of Games, Vol. 3, Princeton Univ. Press, Princeton, N. J. (1957), pp. 365–391.

    Google Scholar 

  31. S. Karlin, “On a class of games,” in: Contributions to the Theory of Games, Vol. 2, Princeton Univ. Press, Princeton, N. J. (1953), pp. 159–171.

    Google Scholar 

  32. S. Karlin, Mathematical Methods and Theory in Games, Programming, and Economics, Addison-Wesley, Reading, Mass. (1959).

    Google Scholar 

  33. S. Karlin, “Operator treatment of minimas principle,” in: Contributions to the Theory of Games, Vol. 1, Princeton Univ. Press, Princeton, N. J. (1950), pp. 133–154.

    Google Scholar 

  34. S. Karlin, “Reduction of certain classes of games to integral equations,” in: Contributions to the Theory of Games, Vol. 2, Princeton Univ. Press, Princeton, N. J. (1953), pp. 125–158.

    Google Scholar 

  35. V. F. Krapivin, “Methods for the solution of continuous zero-sum two-person games,” in: Studies in Cybernetics [in Russian], Sov. Radio, Moscow (1970), pp. 115–129.

    Google Scholar 

  36. V. F. Krapivin and B. S. Fleishman, “Regular method for the solution of games with a piecewise-constant payoff function,” Izv. Akad. Nauk SSSR, Tekh. Kibernet., No. 3, 17–23 (1965).

    Google Scholar 

  37. V. N. Lebedev, “Equivalence of concave-convex zero-sum two-person games to mathematical programming problems,” Izv. Akad. Nauk SSSR, Tekh. Kibernet., No. 2, 32–38 (1966).

    Google Scholar 

  38. V. N. Lebedev and N. T. Tynyanskii, “Theory of the duality of concave-convex games,” Dokl. Akad. Nauk SSSR,174, No. 6, 1264–1267 (1967).

    Google Scholar 

  39. B. A. Mavritskii, “Algorithm for the approximate solution of infinite zero-sum two-person games by random-search methods,” in: Automation and Computer Techniques [in Russian], No. 13, Zinatne, Riga (1966), pp. 63–74.

    Google Scholar 

  40. J. M. McKinsey, Introduction to the Theory of Games, McGraw-Hill, New York (1952).

    Google Scholar 

  41. A. S. Mikhailova, “Certain classes of games on the unit square,” in: Infinite Zero-Sum Two-Person Games [in Russian], Fizmatgiz, Moscow (1963), pp. 426–441.

    Google Scholar 

  42. V. V. Morozov, “A strategy selection principle,” in: Cybernetics in the Service of Communism [in Russian], Vol. 6, Énergiya, Moscow (1971), pp. 185–189.

    Google Scholar 

  43. J. von Neumann and O. Morgenstern, The Theory of Games and Economic Behavior (3rd ed.), Princeton Univ. Press, Princeton, N. J. (1953).

    Google Scholar 

  44. V. F. Ogaryshev, “Generalization of a problem of Gross,” in: Cybernetics in the Service of Communism [in Russian], Vol.6, Énergiya, Moscow (1971), pp. 264–283.

    Google Scholar 

  45. G. Owen, Game Theory, Saunders, Philadelphia, Pa. (1968).

    Google Scholar 

  46. J. E. L. Peck and A. L. Dulmage, “Games on a compact set,” Canad. J. Math.,9, 450–458 (1957).

    Google Scholar 

  47. B. O. Ashkenazi (ed.), Military Applications of Game Theory [in Russian], Sov. Radio, Moscow (1961), 359 pages.

    Google Scholar 

  48. B. N. Pshenichnyi, “The duality method in extremal problems (I, II),” Kibernetika, No. 3, 89–95; No. 4, 64–69 (1965).

  49. I. V. Romanovskii, “Minimax theorems for games with inaccurate information,” Teor. Veroyat. i Prim.,7, No. 1, 89–95 (1962).

    Google Scholar 

  50. M. Sion, “On general minimax theorems,” Pacif. J. Math.,8, 171–176 (1958).

    Google Scholar 

  51. M. Sion and P. Wolfe, “On a game without value,” in: Contributions to the Theory of Games, Vol. 3, Princeton Univ. Press, Princeton, N. J. (1957), pp. 299–306.

    Google Scholar 

  52. A. I. Sobolev, “On the Borel game,” Teor. Veroyat. i Prim.,15, No. 3, 558–561 (1970).

    Google Scholar 

  53. A. G. Sukharev, “On certain games on the unit square,” in: Cybernetics in the Service of Communism [in Russian], Vol. 6, Énergiya, Moscow (1971), pp. 202–206.

    Google Scholar 

  54. N. T. Tynyanskii, Fundamentals of Dual Nonlinear Programming Theory and Differential Games [in Russian], Voennaya Inzh. Akad. im. Dzerzhinskogo, Moscow (1968).

    Google Scholar 

  55. Wen-tsün Wu, “A remark on the fundamental theorem in the theory of games,” Sci. Record (N. S.),3, 229–233 (1959).

    Google Scholar 

  56. Ky Fan, “Minimax theorems,” Proc. Nat. Acad. Sci. U. S.,39, 42–47 (1953).

    Google Scholar 

  57. W. H. Fleming, “On a class of games over function space and related variational problems,” Ann. Math., Ser. 2,60, 578–594 (1954).

    Google Scholar 

  58. W. O. Vogel, “Die Annäherung guter Strategien bei einer gewissen Klasse von Spielen,” Math. Z.,65, 283–308 (1956).

    Google Scholar 

  59. S. Huyberechts, “Sur le problème de l'unicité de la solution des jeux sur le carré-unité,” Acad. Roy. Belg. Bull. Cl. Sci. (5),44, 200–216 (1958).

    Google Scholar 

  60. M. Shiftman, “Games of timing,” in: Contributions to the Theory of Games, Vol. 2, Princeton Univ. Press, Princeton, N. J. (1953), pp. 97–123.

    Google Scholar 

  61. K. J. Arrow, L. Hurwicz, and H. Uzawa, Studies in Linear and Nonlinear Programming (Stanford Mathematical Studies in the Social Sciences, Vol. 2), Stanford Univ. Press, Stanford, Calif. (1958).

    Google Scholar 

  62. E. B. Yanovskaya, “Quasi-invariant kernels in zero-sum two-person games,” Dokl. Akad. Nauk SSSR,151, No. 3 (1963).

  63. E. B. Yanovskaya, “On the uniqueness of optimal strategies in games of timing,” in: Infinite Zero-Sum Two-Person Games [in Russian], Fizmatgiz, Moscow (1963), pp. 295–297.

    Google Scholar 

  64. E. B. Yanovskaya, “Minimax theorems for games on the unit square,” Teor. Veroyat. i Prim.,9, No. 3, 554–555 (1964).

    Google Scholar 

  65. E. B. Yanovskaya, “Zero-sum two-person games played on function spaces,” Liet. Matem. Sborn.,7, No. 3, 547–555 (1964).

    Google Scholar 

  66. E. B. Yanovskaya, “Games of the continuous-fire duel type,” Izv. Akad. Nauk SSSR, Tekh. Kibernet., No. 1 (1969).

  67. E. B. Yanovskaya, “Solution of infinite games in finitely additive strategies,” Teor. Veroyat. i Prim.,15, No. 1 (1970).

  68. A. Auslender, “Recherche des points de selle d'une fonction,” Lect. Notes Math.,132, 37–52 (1970).

    Google Scholar 

  69. R. Bellman, “On ‘Colonel Blotto’ and analogous games,” SIAM Rev.,11, No. 1, 66–68 (1969).

    Google Scholar 

  70. C. Berge, “Sur une convexité reguliére non linéaire et ses applications à la théorie des jeux,” Bull. Soc. Math. France,82, No. 3, 301–319 (1954).

    Google Scholar 

  71. C. Berge, “The most general minimax theorem and conjecture of Sion,” in: Recent Advances in Game Theory, Princeton, N. J. (1962).

  72. D. Bierlein, “Über wesentlich indefinite Spiele,” Lect. Notes Math.,31, 28–35 (1967).

    Google Scholar 

  73. D. Bierlein, “Spiele mit mehr als einen Spielwert,” Arch. Math.,19, No. 3, 330–336 (1968).

    Google Scholar 

  74. D. Bierlein, “Minimal indefinite Randomisierungen von Spielen,” Z. Wahrscheinlichkeitstheor. und verw. Geb.,11, No. 3, 193–207 (1966).

    Google Scholar 

  75. E. Borel, “La théorie du jeu et les équations integrals à noyau symmetrique,” Compt. Rend.,173, 1304–1308 (1921).

    Google Scholar 

  76. E. Burger, “Spieltheoretische Behandlung eines Reklameproblems (Variante eines Spielmodells von Gillman),” Mitteilungsbl. Math. Statist.,6, 39–52 (1954).

    Google Scholar 

  77. E. Cansado, “Dual programming problems as hemi-games,” Management Sci.,15, No. 9, 539–549 (1969).

    Google Scholar 

  78. J. M. Danskin, “A game over spaces of probability distributions,” Naval Res. Logist. Quart.,11, Nos. 2–3, 157–188 (1964).

    Google Scholar 

  79. J. M. Danskin and L. Gillman, “A game over function space,” Riv. Mat. Univ. Parma,4, 83–94 (1953).

    Google Scholar 

  80. M. M. Diaz, “Juegos bipersonales de suma nula con pago convexo,” Trab. Estadist. y Invest. Oper.,20, Nos. 2–3, 3–6 (1969).

    Google Scholar 

  81. M. Dresher, “On an infinite game with a discontinuous game value,” Amer. Math. Soc. Not.,9, No. 2 (1962).

  82. A. L. Dulmage and J. E. L. Peck, “Certain infinite zero-sum two-person games,” Canad. J. Math.,8, No. 3, 412–416 (1956).

    Google Scholar 

  83. Ky Fan, “Fixed-point and minimax theorems in locally convex topological linear spaces,” Proc. Acad. Sci. U. S.,38, 121–126 (1952).

    Google Scholar 

  84. J. E. Fenstad, “Good strategies in general games,” Math. Z.,101, No. 4, 322–330 (1967).

    Google Scholar 

  85. T. S. Ferguson, “On a class of infinite games related to liar dice,” Ann. Math. Statist.,41, No. 2, 353–362 (1970).

    Google Scholar 

  86. F. Forgó, A SUMT Method and Its Application to the Solution of Certain Continuous Games, Publ. Dep. Math. K. Marx Univ. Econ. Budapest, No. 5 (1970), 25 pages.

  87. F. Forgó, Relationship between Continuous Zero-Sum Two-Person Games and Linear Programming, Publ. Dep. Math. K. Marx Univ. Econ. Budapest, No. 1 (1969), 15 pages.

  88. M. Fox and G. S. Kimeldorf, “Noisy duels,” SIAM J. Appl. Math.,17, No. 2, 353–361 (1969).

    Google Scholar 

  89. M. Fox and G. S. Kimeldorf, “Values and shooting times in noisy duels,” J. Amer. Statist. Assoc.,65, No. 329, 422–430 (1970).

    Google Scholar 

  90. M. A. Ghouila-Houri, “Le théorème minimax de Sion,” in: Theory of Games, Engl. Univ. Press, London (1966), pp. 123–129.

    Google Scholar 

  91. P. P. Gillis and S. Huyberechts, “Théorie des jeux le carré-unité,” in: Colloq. Analyse Statist., CBRM, Brussels, 1954 (1955), pp. 159–175.

    Google Scholar 

  92. P. P. Gillis and S. Huyberechts, “Le problème de l'unicité de la solution des jeux infinis de somme nulle à deux jouers,” Theory of Games, Engl. Univ. Press, London (1966), pp. 104–122.

    Google Scholar 

  93. I. Glicksberg, “A derivative test for finite solutions of games,” Proc. Amer. Math. Soc.,4, No. 6, 895–897 (1953).

    Google Scholar 

  94. R. A. Hirschfeld, “On a minimax theorem of K. Fan,” Proc. Koninkl. Nederl. Akad. Wet., Indagationes Math.,A61, No. 4, 470–474 (1958).

    Google Scholar 

  95. S. Huyberechts, “Introduction à l'étude jeux sur des éspaces fonctionneles,” Cahiers Centre Études Rech. Oper.,6, 209–224 (1964).

    Google Scholar 

  96. S. Huyberechts, “Vers une classification des jeux sur le carré-unité,” Cahiers Centre Études Rech. Oper.,4, 172–200 (1962).

    Google Scholar 

  97. S. Karlin, “Continuous games,” Proc. Nat. Acad. Sci. U. S.,37, No. 4, 220–223 (1951).

    Google Scholar 

  98. S. Karlin, “The theory of infinite games,” Ann. Math.,58, No. 2, 371–401 (1953).

    Google Scholar 

  99. H. Kneser, “Sur un théorème fondamental de la théorie des jeux,” Compt. Rend.,234, 2418–2420 (1952).

    Google Scholar 

  100. J. von Neumann, “Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes,” Ergeb. Math. Kolloq.,8, 73–83 (1937).

    Google Scholar 

  101. J. Neuman and E. Pearson, “On the problem of the most efficient test of statistical hypotheses,” Phil. Trans. Roy. Soc.,A231, 289–337 (1933).

    Google Scholar 

  102. H. Nikaido, “On a minimax theorem and its application to functional analysis,” J. Math. Soc. Japan,5, 86–94 (1953).

    Google Scholar 

  103. H. Nikaido, “On von Neumann's minimax theorem,” Pacif J. Math.,4, No. 1, 65–72 (1954).

    Google Scholar 

  104. H. Nikaido, “On a method of proof for the minimax theorem,” Proc. Amer. Math. Soc.,10, No. 2, 205–212 (1959).

    Google Scholar 

  105. T. A. Parthasarathy, “A note on a minimax theorem of T. T. Tie,” Sankhya (Indian J. Statist.),A27, Nos. 2–4, 407–408 (1965).

    Google Scholar 

  106. T. A. Parthasarathy, “On a general minimax theorem,” Math. Student,34, Nos. 3–4, 195–197 (1966).

    Google Scholar 

  107. T. A. Parthasarathy, “On games over the unit square,” SIAM J. Appl. Math.,19, No. 2, 473–476 (1970).

    Google Scholar 

  108. T. A. Parthasarathy and T. E. S. Raghavan, “Some topics in two-person games,” in: Modern Analytic and Computational Methods in Science and Mathematics, No. 22, Amer. Publ. Co., New York (1971).

    Google Scholar 

  109. T. E. S. Raghavan, “Convex cones and finite optimals,” Ann. Math. Statist.,41, No. 2, 702–705 (1970).

    Google Scholar 

  110. R. A. Restrepo, “Tactical problems involving several actions,” Ann. Math. Studies,39, 313–335 (1957).

    Google Scholar 

  111. G. Smith, “A duel with silent-noisy gun versus noisy gun,” Colloq. Math.,17, No. 1, 131–146 (1967).

    Google Scholar 

  112. A. Stefănescu, “Metoda gradientului la jocurile convexe pe patratul unitate,” Ann. Univ. Bucuresti, Mat.-Mec,19, No. 1, 137–143 (1970).

    Google Scholar 

  113. Tjoe-tie Teh, “Minimax theorems on conditionally compact sets,” Ann. Math. Statist.,34, No. 4, 1536–1540 (1963).

    Google Scholar 

  114. J. Ville, “Sur la théorie général de jeux où intervient l'habilité des joueurs,” Traité du Calcul des Probabilités et des ses Applications, par E. Borel et Collaborateurs, Paris,2, No. 5, 105–113 (1938).

    Google Scholar 

  115. A. Wald, “Generalisations of a theorem by von Neumann concerning zero-sum two-person games,” Ann. Math.,46, 281–286 (1945).

    Google Scholar 

  116. A. Wald, “Foundations of a general theory of sequential decision functions,” Econometrica,15, 279–313 (1947).

    Google Scholar 

  117. A. Wald, “An essentially complete class of admissible decision functions,” Econometrica,21 (1947).

  118. A. Wald, “A note on zero-sum two-person games,” Ann. Math.,51, 739–742 (1950).

    Google Scholar 

  119. P. Wolfe, “The strict determinateness of certain infinite games,” Pacific J. Math.,5, No. 1, Suppl. 841–847 (1955).

    Google Scholar 

  120. L. von Wolfersdorf, ‘Zur Berechnung optimaler Strategien für Spiele über dem Einheitsquadrat an der Hauptdiagonalen unstetigen Auszahlungsfunctionen,” Sitzungsber. Sächs. Akad. Wiss. Leipzig, Math.-Naturwiss. Kl.,107, No. 7, 53 pages (1967).

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Translated from Itogi Nauki i Tekhniki (Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika), Vol. 10, pp. 75–106, 1972.

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Yanovskaya, E.B. Infinite zero-sum two-person games. J Math Sci 2, 520–541 (1974). https://doi.org/10.1007/BF01085016

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