Existence and Uniqueness Theorem of Continuous and Monotone Bayesian Nash Equilibrium and Stability Analysis

References

• [1] Aliprantis CD, Border KC (2006) Infinite Dimensional Analysis: A Hitchhiker’s Guide (Springer, Berlin).Google Scholar
• [2] Anderson SP, De Palma A, Thisse JF (1992) Discrete Choice Theory of Product Differentiation (MIT Press, Cambridge, MA).Crossref, Google Scholar
• [3] Athey S (2001) Single crossing properties and the existence of pure strategy equilibria in games of incomplete information. Econometrica 69(4):861–889.Crossref, Google Scholar
• [4] Athey S (2002) Monotone comparative statics under uncertainty. Quart. J. Econom. 117(1):187–223.Crossref, Google Scholar
• [5] Berge C (1963) Topological Spaces: Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity (Oliver & Boyd, Edinburgh, UK).Google Scholar
• [6] Bertsekas D (2012) Dynamic Programming and Optimal Control, vol. II (Athena Scientific, Nashua, NH).Google Scholar
• [7] Chemseddine J, Hagemann P, Steidl G, Wald C (2025) Conditional Wasserstein distances with applications in Bayesian OT flow matching. J. Machine Learn. Res. 26(141):1–47.Google Scholar
• [8] Dontchev AL, Rockafellar RT (2014) Implicit Functions and Solution Mappings (Springer, Berlin).Crossref, Google Scholar
• [9] Gao J, Iyer K, Topaloglu H (2017) Price competition under linear demand and finite inventories: Contraction and approximate equilibria. Oper. Res. Lett. 45(4):382–387.Crossref, Google Scholar
• [10] Gibbs AL, Su FE (2002) On choosing and bounding probability metrics. Internat. Statist. Rev. 70(3):419–435.Crossref, Google Scholar
• [11] Graf L, Harks T, Markl M (2024) Stochastic prediction equilibrium for dynamic traffic assignment. Preprint, submitted September 19, https://arxiv.org/abs/2409.12650v1.Google Scholar
• [12] Guo S, Xu H, Zhang L (2021) Existence and approximation of continuous Bayesian Nash equilibria in games with continuous type and action spaces. SIAM J. Optim. 31(4):2481–2507.Crossref, Google Scholar
• [13] Harsanyi JC (1967) Games with incomplete information played by “Bayesian” players, I–III part I. The basic model. Management Sci. 14(3):159–182. [Link, Google Scholar
• [14] Harsanyi JC (1968) Games with incomplete information played by “Bayesian” players part II. Bayesian equilibrium points. Management Sci. 14(5):320–334.Link, Google Scholar
• [15] Harsanyi JC (1968) Games with incomplete information played by “Bayesian” players, part III. The basic probability distribution of the game. Management Sci. 14(7):486–502.Link, Google Scholar
• [16] Liu S (2026) Games with incomplete information played by risk-revising players. Preprint, submitted March 20, https://arxiv.org/abs/2603.19738.Google Scholar
• [17] Liu J, Su Z, Xu H (2025) Bayesian distributionally robust Nash equilibrium and its application. Pacific J. Optim. 21(4):571–599.Crossref, Google Scholar
• [18] Liu Y, Xu H, Yang SJS, Zhang J (2018) Distributionally robust equilibrium for continuous games: Nash and Stackelberg models. Eur. J. Oper. Res. 265(2):631–643.Crossref, Google Scholar
• [19] Mason R, Valentinyi A (2007) The existence and uniqueness of monotone pure strategy equilibrium in Bayesian games. Discussion Papers in Economics and Econometrics No. 710, University of Southampton, Southampton, UK.Google Scholar
• [20] Mathevet L (2010) A contraction principle for finite global games. Econom. Theory 42(3):539–563.Crossref, Google Scholar
• [21] McAdams D (2003) Isotone equilibrium in games of incomplete information. Econometrica 71(4):1191–1214.Crossref, Google Scholar
• [22] Meirowitz A (2003) On the existence of equilibria to Bayesian games with non-finite type and action spaces. Econom. Lett. 78(2):213–218.Crossref, Google Scholar
• [23] Milgrom P, Shannon C (1994) Monotone comparative statics. Econometrica 62(1):157–180.Crossref, Google Scholar
• [24] Quah JKH, Strulovici B (2012) Aggregating the single crossing property. Econometrica 80(5):2333–2348.Crossref, Google Scholar
• [25] Reny PJ (2011) On the existence of monotone pure-strategy equilibria in Bayesian games. Econometrica 79(2):499–553.Crossref, Google Scholar
• [26] Rudin W (1991) Functional Analysis, International Series in Pure and Applied Mathematics (McGraw-Hill, New York).Google Scholar
• [27] Su Z, Xu H (2025) Continuous and monotone Bayesian Nash equilibrium with incomplete information about player’s risk preferences. Preprint, submitted January 31, http://dx.doi.org/10.2139/ssrn.5118754.Google Scholar
• [28] Su Z, Xu H (2025) Existence and uniqueness theorem of continuous and monotone Bayesian Nash equilibrium and stability analysis. Preprint, submitted November 19, https://arxiv.org/abs/2511.15457v1.Google Scholar
• [29] Tao Y, Xu H (2025) Generalized Bayesian Nash equilibrium with continuous type and action spaces. SIAM J. Optim. 35(2):789–817.Crossref, Google Scholar
• [30] Topkis DM (1978) Minimizing a submodular function on a lattice. Oper. Res. 26(2):305–321.Link, Google Scholar
• [31] Topkis DM (1998) Supermodularity and Complementarity (Princeton University Press, Princeton, NJ).Crossref, Google Scholar
• [32] Ui T (2016) Bayesian Nash equilibrium and variational inequalities. J. Math. Econom. 63:139–146.Crossref, Google Scholar
• [33] Van Long N, Soubeyran A (2000) Existence and uniqueness of Cournot equilibrium: A contraction mapping approach. Econom. Lett. 67(3):345–348.Crossref, Google Scholar
• [34] Van Zandt T, Vives X (2007) Monotone equilibria in Bayesian games of strategic complementarities. J. Econom. Theory 134(1):339–360.Crossref, Google Scholar
• [35] Vives X (1990) Nash equilibrium with strategic complementarities. J. Math. Econom. 19(3):305–321.Crossref, Google Scholar
• [36] Zeidler E (1985) Nonlinear Functional Analysis and Its Applications: Fixed-Point Theorems (Springer, Berlin).Crossref, Google Scholar