1932

Abstract

We review the discontinuous games literature, with a sharp focus on conditions that ensure the existence of pure and mixed strategy Nash equilibria in strategic form games and of Bayes-Nash equilibria in Bayesian games.

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2020-08-02
2024-05-02
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Literature Cited

  1. Allison B, Lepore J. 2014. Verifying payoff security in the mixed extension of discontinuous games. J. Econ. Theory 152:291–303
    [Google Scholar]
  2. Bagh A, Jofre A. 2006. Reciprocal upper semicontinuity and better reply secure games: a comment. Econometrica 74:1715–21
    [Google Scholar]
  3. Balder E. 1994. Lectures on young measures Work. Pap., Univ. Paris-Dauphine Paris: https://basepub.dauphine.fr/handle/123456789/6718
  4. Balder E. 2011. An equilibrium closure result for discontinuous games. Econ. Theory 48:47–65
    [Google Scholar]
  5. Barelli P, Govindan S, Wilson R 2014. Competition for a majority. Econometrica 82:271–314
    [Google Scholar]
  6. Barelli P, Meneghel I. 2013. A note on the equilibrium existence problem in discontinuous games. Econometrica 81:813–24
    [Google Scholar]
  7. Baye MR, Tian G, Zhou J 1993. Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs. Rev. Econ. Stud. 60:935–48
    [Google Scholar]
  8. Berge C. 1954. Sur une convexité régulière non linéaire et ses applications à la théorie des jeux. Bull. Soc. Math. Fr. 82:301–19
    [Google Scholar]
  9. Bertsekas DP, Shreve SE. 1978. Stochastic Optimal Control: The Discrete Time Case London: Academic
  10. Bich P. 2009. Existence of pure Nash equilibria in discontinuous and non quasiconcave games. Int. J. Game Theory 38:395–410
    [Google Scholar]
  11. Bich P. 2019. Strategic uncertainty and equilibrium selection in discontinuous games. J. Econ. Theory 183:786–822
    [Google Scholar]
  12. Bich P, Laraki R. 2012. A unified approach to equilibrium existence in discontinuous strategic games Work. Pap., Univ. Panthéon-Sorbonne Paris:
  13. Bich P, Laraki R. 2017. On the existence of approximate equilibria and sharing rule solutions in discontinuous games. Theor. Econ. 12:79–108
    [Google Scholar]
  14. Carbonell-Nicolau O. 2011. On the existence of pure-strategy perfect equilibrium in discontinuous games. Games Econ. Behav. 71:23–48
    [Google Scholar]
  15. Carbonell-Nicolau O, McLean RP. 2018. On the existence of Nash equilibrium in Bayesian games. Math. Oper. Res. 43:100–29
    [Google Scholar]
  16. Carbonell-Nicolau O, Ok EA. 2007. Voting over income taxation. J. Econ. Theory 134:249–86
    [Google Scholar]
  17. Carmona G. 2005. On the existence of equilibria in discontinuous games: three counterexamples. Int. J. Game Theory 33:181–87
    [Google Scholar]
  18. Carmona G. 2009. An existence result for discontinuous games. J. Econ. Theory 144:1333–40
    [Google Scholar]
  19. Carmona G. 2011. Understanding some recent existence results for discontinuous games. Econ. Theory 48:31–45
    [Google Scholar]
  20. Carmona G. 2013. Existence and Stability of Nash Equilibrium Singapore: World Sci. Publ.
  21. Carmona G, Podczeck K. 2014. Existence of Nash equilibrium in games with a measure space of players and discontinuous payoff functions. J. Econ. Theory 152:130–78
    [Google Scholar]
  22. Carmona G, Podczeck K. 2016. Existence of Nash equilibrium in ordinal games with discontinuous preferences. Econ. Theory 61:457–78
    [Google Scholar]
  23. Carmona G, Podczeck K. 2018a. Invariance of the equilibrium set of games with an endogenous sharing rule. J. Econ. Theory 177:1–33
    [Google Scholar]
  24. Carmona G, Podczeck K. 2018b. The conditions in the existence results for discontinuous games by Reny and by Simon and Zame are incomparable. Games Econ. Behav. 111:16–19
    [Google Scholar]
  25. Cohn DL. 1980. Measure Theory Boston: Birkhauser
  26. Dasgupta P, Maskin E. 1986a. The existence of equilibrium in discontinuous economic games, I: theory. Rev. Econ. Stud. 53:1–26
    [Google Scholar]
  27. Dasgupta P, Maskin E. 1986b. The existence of equilibrium in discontinuous economic games, II: applications. Rev. Econ. Stud. 53:27–41
    [Google Scholar]
  28. de Castro L. 2011. Equilibrium existence and approximation of regular discontinuous games. Econ. Theory 48:67–85
    [Google Scholar]
  29. Duggan J. 2007. Equilibrium existence for zero-sum games and spatial models of elections. Games Econ. Behav. 60:52–74
    [Google Scholar]
  30. Dunford N, Schwartz JT. 1988. Linear Operators Part I: General Theory New York: Wiley
  31. Eilenberg S, Montgomery D. 1946. Fixed point theorems for multi-valued transformations. Am. J. Math. 68:214–22
    [Google Scholar]
  32. Fan K. 1953. Minimax theorems. PNAS 39:42–47
    [Google Scholar]
  33. Glicksberg IL. 1952. A further generalization of the Kakutani fixed point theorem. Proc. Am. Math. Soc. 3:170–74
    [Google Scholar]
  34. He W, Yannelis NC. 2015. Discontinuous games with asymmetric information: an extension of Reny's existence theorem. Games Econ. Behav. 91:26–35
    [Google Scholar]
  35. He W, Yannelis NC. 2016. Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences. Econ. Theory 61:497–513
    [Google Scholar]
  36. Jackson MO. 2009. Non-existence of equilibrium in Vickrey, second-price, and English auctions. Rev. Econ. Des. 13:137–45
    [Google Scholar]
  37. Jackson MO, Simon LK, Swinkels JM, Zame WR 2002. Communication and equilibrium in discontinuous games of incomplete information. Econometrica 70:1711–40
    [Google Scholar]
  38. Jackson MO, Swinkels JM. 2005. Existence of equilibrium in single and double private value auctions. Econometrica 73:93–139
    [Google Scholar]
  39. Kakutani S. 1941. A generalization of Brouwer's fixed point theorem. Duke Math. J. 8:457–59
    [Google Scholar]
  40. Kneser H. 1952. Sur un théorème fundamental de la théorie des jeux. CRAS Paris 234:2418–20
    [Google Scholar]
  41. Kukushkin NS. 2018. Better response dynamics and Nash equilibrium in discontinuous games. J. Math. Econ. 74:68–78
    [Google Scholar]
  42. McLennan A, Monteiro PK, Tourky R 2011. Games with discontinuous payoffs: a strengthening of Reny's existence theorem. Econometrica 79:1643–64
    [Google Scholar]
  43. Milgrom P, Weber R. 1985. Distributional strategies for games with incomplete information. Math. Oper. Res. 10:619–32
    [Google Scholar]
  44. Monteiro P, Page F. 2007. Uniform payoff security and Nash equilibrium in compact games. J. Econ. Theory 134:566–75
    [Google Scholar]
  45. Monteiro P, Page F. 2008. Catalog competition and Nash equilibrium in nonlinear pricing games. Econ. Theory 34:503–24
    [Google Scholar]
  46. Munkres J. 1975. Topology: A First Course Englewood Cliffs, NJ: Prentice-Hall
  47. Nash J. 1950. Equilibrium points in n-person games. PNAS 36:48–49
    [Google Scholar]
  48. Nash J. 1951. Non-cooperative games. Ann. Math. 54:286–95
    [Google Scholar]
  49. Nessah R. 2011. Generalized weak transfer continuity and Nash equilibrium. J. Math. Econ. 47:659–62
    [Google Scholar]
  50. Nessah R. 2013. Weakly continuous security in discontinuous and nonquasiconcave games: existence and characterization Work. Pap., IESEG Sch. Manag. Lille, Fr.:
  51. Nessah R, Tian G. 2008. Existence of equilibria in discontinuous and nonconvex games Work. Pap., IESEG Sch. Manag Lille, Fr.:
  52. Nessah R, Tian G. 2016. On the existence of Nash equilibrium in discontinuous games. Econ. Theory 61:515–40
    [Google Scholar]
  53. Nishimura K, Friedman J. 1981. Existence of Nash equilibrium in n-person games without quasiconcavity. Int. Econ. Rev. 22:637–48
    [Google Scholar]
  54. Olszewski W, Siegel R. 2016. Large contests. Econometrica 84:835–54
    [Google Scholar]
  55. Olszewski W, Siegel R. 2019. Bid caps in large contests. Games Econ. Behav. 115:101–12
    [Google Scholar]
  56. Olszewski W, Siegel R. 2020. Performance-maximizing large contests. Theor. Econ. 15:57–88
    [Google Scholar]
  57. Prokopovych P. 2011. On equilibrium existence in payoff secure games. Econ. Theory 48:5–16
    [Google Scholar]
  58. Prokopovych P. 2013. The single deviation property in games with discontinuous payoffs. Econ. Theory 53:383–402
    [Google Scholar]
  59. Prokopovych P. 2016. Majorized correspondences and equilibrium existence in discontinuous games. Econ. Theory 61:541–52
    [Google Scholar]
  60. Prokopovych P, Yannelis NC. 2014. On the existence of mixed strategy Nash equilibria. J. Math. Econ. 52:87–97
    [Google Scholar]
  61. Reny PJ. 1996. Local payoff security and the existence of pure and mixed strategy Nash equilibria in discontinuous games Work. Pap., Univ. Pittsburgh Pittsburgh, PA:
  62. Reny PJ. 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67:1029–56
    [Google Scholar]
  63. Reny PJ. 2009. Further results on the existence of Nash equilibria in discontinuous games Work. Pap., Univ. Chicago Chicago:
  64. Reny PJ. 2011. Strategic approximations of discontinuous games. Econ. Theory 48:17–29
    [Google Scholar]
  65. Reny PJ. 2016a. Equilibrium in discontinuous games without complete or transitive preferences. Econ. Theory Bull. 4:1–4
    [Google Scholar]
  66. Reny PJ. 2016b. Nash equilibrium in discontinuous games. Econ. Theory 61:553–69
    [Google Scholar]
  67. Rothstein P. 2007. Discontinuous payoffs, shared resources and games of fiscal competition: existence of pure strategy Nash equilibrium. J. Public Econ. Theory 9:335–68
    [Google Scholar]
  68. Scalzo V. 2013. Essential equilibria of discontinuous games. Econ. Theory 54:27–44
    [Google Scholar]
  69. Scalzo V. 2019a. Equilibrium existence in games: slight single deviation property and Ky Fan minimax inequality. J. Math. Econ. 82:197–201
    [Google Scholar]
  70. Scalzo V. 2019b. Continuity properties of the Nash equilibrium correspondence in a discontinuous setting. J. Math. Anal. Appl. 473:1270–79
    [Google Scholar]
  71. Scalzo V. 2020. On the uniqueness of Nash equilibrium in discontinuous ordinal and normal form games. Econ. Theory Bull. 8:16368
    [Google Scholar]
  72. Shafer W, Sonnenschein H. 1975. Equilibrium in abstract economies without ordered preferences. J. Math. Econ. 2:345–48
    [Google Scholar]
  73. Sonnenschein H. 1971. Demand theory without transitive preferences, with applications to the theory of competitive equilibrium. Preferences, Utility and Demand Theory J Chipman, L Hurwicz, MK Richter, HF Sonnenschein 215–23 New York: Harcourt Brace Jovanovich
    [Google Scholar]
  74. Simon L. 1987. Games with discontinuous payoffs. Rev. Econ. Stud. 54:569–97
    [Google Scholar]
  75. Simon L, Zame W. 1990. Discontinuous games and endogenous sharing rules. Econometrica 58:861–72
    [Google Scholar]
  76. Simon RS. 2003. Games of incomplete information, ergodic theory, and the measurability of equilibria. Isr. J. Math. 138:73–92
    [Google Scholar]
  77. Sion M. 1958. On general minimax theorems. Pac. J. Math. 8:171–76
    [Google Scholar]
  78. Tian G. 1992a. Generalizations of the FKKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarity. J. Math. Anal. Appl. 170:457–71
    [Google Scholar]
  79. Tian G. 1992b. Existence of equilibrium in abstract economies with discontinuous payoffs and non-compact choice spaces. J. Math. Econ. 21:379–88
    [Google Scholar]
  80. Tian G. 1992c. On the existence of equilibria in generalized games. Int. J. Game Theory 20:247–54
    [Google Scholar]
  81. Tian G, Zhou J. 1992. The maximum theorem and the existence of Nash equilibrium of (generalized) games without lower semicontinuities. J. Math. Anal. Appl. 166:351–64
    [Google Scholar]
  82. Tian G, Zhou J. 1995. Transfer continuities, generalizations of the Weierstrass and maximum theorems: a full characterization. J. Math. Econ. 24:281–303
    [Google Scholar]
  83. Zhou J, Chen G. 1988. Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities. J. Math. Anal. Appl. 132:213–25
    [Google Scholar]
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