Cooperation, Correlation, and Competition in Ergodic N-Player Games and Mean-Field Games of Singular Controls: A Case Study

References

• [1] Achdou Y, Buera FJ, Lasry J-M, Lions P-L, Moll B (2014) Partial differential equation models in macroeconomics. Philos. Trans. A Math. Phys. Engrg. Sci. 372(2028):20130397.Google Scholar
• [2] Adlakha S, Johari R (2013) Mean field equilibrium in dynamic games with strategic complementarities. Oper. Res. 61(4):971–989.Link, Google Scholar
• [3] Albeverio S, De Vecchi FC, Romano A, Ugolini S (2022) Mean-field limit for a class of stochastic ergodic control problems. SIAM J. Control Optim. 60(1):479–504.Crossref, Google Scholar
• [4] Alvarez LHR (2001) Reward functionals, salvage values, and optimal stopping. Math. Methods Oper. Res. 54(2):315–337.Crossref, Google Scholar
• [5] Alvarez LHR (2018) A class of solvable stationary singular stochastic control problems. Preprint, submitted March 9, https://arxiv.org/abs/1803.03464.Google Scholar
• [6] Alvarez F, Lippi F, Souganidis P (2023) Price setting with strategic complementarities as a mean field game. Econometrica 91(6):2005–2039.Crossref, Google Scholar
• [7] Alvarez FE, Argente D, Lippi F, Méndez E, Van Patten D (2023) Strategic complementarities in a dynamic model of technology adoption: P2P digital payments. NBER Working Paper No. 31280, National Bureau of Economic Research, Cambridge, MA.Google Scholar
• [8] Arapostathis A, Biswas A, Carroll J (2017) On solutions of mean field games with ergodic cost. J. Mathématiques Pures Appliquées 107(2):205–251.Crossref, Google Scholar
• [9] Aumann RJ (1974) Subjectivity and correlation in randomized strategies. J. Math. Econom. 1(1):67–96.Crossref, Google Scholar
• [10] Aumann RJ (1987) Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55(1):1–18.Crossref, Google Scholar
• [11] Bank P (2005) Optimal control under a dynamic fuel constraint. SIAM J. Control Optim. 44(4):1529–1541.Crossref, Google Scholar
• [12] Bank P, Riedel F (2001) Optimal consumption choice with intertemporal substitution. Ann. Appl. Probab. 11(3):750–788.Crossref, Google Scholar
• [13] Bao X, Tang S (2023) Ergodic control of McKean–Vlasov SDEs and associated Bellman equation. J. Math. Anal. Appl. 527(1):127404.Crossref, Google Scholar
• [14] Bardi M, Kouhkouh H (2024) Long-time behavior of deterministic mean field games with nonmonotone interactions. SIAM J. Math. Anal. 56(4):5079–5098.Crossref, Google Scholar
• [15] Bardi M, Priuli FS (2014) Linear-quadratic N-person and mean-field games with ergodic cost. SIAM J. Control Optim. 52(5):3022–3052.Crossref, Google Scholar
• [16] Basei M, Cao H, Guo X (2022) Nonzero-sum stochastic games and mean-field games with impulse controls. Math. Oper. Res. 47(1):341–366.Link, Google Scholar
• [17] Bayraktar E, Kara AD (2024) Infinite horizon average cost optimality criteria for mean-field control. SIAM J. Control Optim. 62(5):2776–2806.Crossref, Google Scholar
• [18] Beneš VE, Shepp LA, Witsenhausen HS (1980) Some solvable stochastic control problems. Anal. Optim. Stochastic Systems Proc. Internat. Conf. Univ. Oxford (Academic Press, London), 3–10.Google Scholar
• [19] Bonesini O (2023) Four essays in between probability theory and financial mathematics. PhD thesis, Università degli Studi di Padova, Padua, Italy.Google Scholar
• [20] Bonesini O, Campi L, Fischer M (2025) Correlated equilibria for mean field games with progressive strategies. Math. Oper. Res. 50(2):1072–1111.Link, Google Scholar
• [21] Borodin AN, Salminen P (2002) Handbook of Brownian Motion—Facts and Formulae, Probability and Its Applications, 2nd ed. (Birkhäuser Verlag, Basel, Switzerland).Crossref, Google Scholar
• [22] Calvia A, Federico S, Ferrari G, Gozzi F (2024) Existence and uniqueness results for a mean-field game of optimal investment. Preprint, submitted April 3, https://arxiv.org/abs/2404.02871.Google Scholar
• [23] Campi L, Fischer M (2022) Correlated equilibria and mean field games: A simple model. Math. Oper. Res. 47(3):2240–2259.Link, Google Scholar
• [24] Campi L, Cannerozzi F, Cartellier F (2025) Coarse correlated equilibria in linear quadratic mean field games and application to an emission abatement game. Appl. Math. Optim. 91(1):8.Crossref, Google Scholar
• [25] Campi L, Cannerozzi F, Fischer M (2024) Coarse correlated equilibria for continuous time mean field games in open loop strategies. Electronic J. Probab. 29(196):1–56.Google Scholar
• [26] Campi L, De Angelis T, Ghio M, Livieri G (2022) Mean-field games of finite-fuel capacity expansion with singular controls. Ann. Appl. Probab. 32(5):3674–3717.Crossref, Google Scholar
• [27] Cannerozzi F (2025) Coarse correlated equilibria in continuous-time mean field games. PhD thesis, Università degli Studi di Milano, Milan, Italy.Google Scholar
• [28] Cao H, Guo X (2022) MFGs for partially reversible investment. Stochastic Processes Appl. 150:995–1014.Crossref, Google Scholar
• [29] Cao H, Dianetti J, Ferrari G (2023) Stationary discounted and ergodic mean field games with singular controls. Math. Oper. Res. 48(4):1871–1898.Abstract, Google Scholar
• [30] Cardaliaguet P, Lasry J-M, Lions P-L, Porretta A (2012) Long time average of mean field games. Networks Heterogeneous Media 7(2):279–301.Crossref, Google Scholar
• [31] Carmona R (2016) Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games with Financial Applications, Financial Mathematics, vol. 1 (Society for Industrial and Applied Mathematics, Philadelphia).Crossref, Google Scholar
• [32] Carmona R, Delarue F (2015) Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics. Ann. Probab. 43(5):2647–2700.Crossref, Google Scholar
• [33] Carmona R, Delarue F, Lacker D (2016) Mean field games with common noise. Ann. Probab. 44(6):3740–3803.Crossref, Google Scholar
• [34] Carmona R, Delarue F, Lacker D (2017) Mean field games of timing and models for bank runs. Appl. Math. Optim. 76(1):217–260.Crossref, Google Scholar
• [35] Cecchin A, Conforti G, Durmus A, Eichinger K (2024) The exponential turnpike phenomenon for mean field game systems: Weakly monotone drifts and small interactions. Preprint, submitted September 13, https://arxiv.org/abs/2409.09193.Google Scholar
• [36] Chiarolla MB, Haussmann UG (2005) Explicit solution of a stochastic, irreversible investment problem and its moving threshold. Math. Oper. Res. 30(1):91–108.Link, Google Scholar
• [37] Christensen S, Neumann BA, Sohr T (2021) Competition versus cooperation: A class of solvable mean field impulse control problems. SIAM J. Control Optim. 59(5):3946–3972.Crossref, Google Scholar
• [38] Cirant M (2016) Stationary focusing mean-field games. Comm. Partial Differential Equations 41(8):1324–1346.Crossref, Google Scholar
• [39] Cohen A, Sun C (2025) Existence of optimal stationary singular controls and mean field game equilibria. Math. Oper. Res., ePub ahead of print July 7, https://doi.org/10.1287/moor.2024.0549.Link, Google Scholar
• [40] Cohen A, Hening A, Sun C (2022) Optimal ergodic harvesting under ambiguity. SIAM J. Control Optim. 60(2):1039–1063.Crossref, Google Scholar
• [41] Dellacherie C, Meyer P-A (1982) Probabilities and Potential B: Theory of Martingales, North-Holland Mathematics Studies, vol. 72 (North-Holland Publishing Co., Amsterdam).Google Scholar
• [42] Denkert R, Horst U (2025) Extended mean-field games with multidimensional singular controls and nonlinear jump impact. SIAM J. Control Optim. 63(2):1374–1406.Crossref, Google Scholar
• [43] Dianetti J (2025) Strong solutions to submodular mean field games with common noise and related McKean–Vlasov FBSDEs. Ann. Appl. Probab. 35(3):1622–1667.Crossref, Google Scholar
• [44] Dianetti J, Ferrari G, Tzouanas I (2023) Ergodic mean-field games of singular control with regime-switching (extended version). Preprint, submitted July 22, https://arxiv.org/abs/2307.12012.Google Scholar
• [45] Dianetti J, Federico S, Ferrari G, Floccari G (2025) Multiple equilibria in mean-field game models of firm competition with strategic complementarities. Quant. Finance 25(3):343–357.Crossref, Google Scholar
• [46] Dianetti J, Ferrari G, Fischer M, Nendel M (2021) Submodular mean field games: Existence and approximation of solutions. Ann. Appl. Probab. 31(6):2538–2566.Crossref, Google Scholar
• [47] Dianetti J, Ferrari G, Fischer M, Nendel M (2023) A unifying framework for submodular mean field games. Math. Oper. Res. 48(3):1679–1710.Link, Google Scholar
• [48] Dokka T, Moulin H, Ray I, SenGupta S (2023) Equilibrium design in an n-player quadratic game. Rev. Econom. Design 27(2):419–438.Crossref, Google Scholar
• [49] Dragoni F, Feleqi E (2018) Ergodic mean field games with Hörmander diffusions. Calculus Variations Partial Differential Equations 57(5):116.Crossref, Google Scholar
• [50] Federico S, Ferrari G, Riedel F, Röckner M (2021) On a class of infinite-dimensional singular stochastic control problems. SIAM J. Control Optim. 59(2):1680–1704.Crossref, Google Scholar
• [51] Feleqi E (2013) The derivation of ergodic mean field game equations for several populations of players. Dynam. Games Appl. 3(4):523–536.Crossref, Google Scholar
• [52] Ferrari G, Salminen P (2016) Irreversible investment under Lévy uncertainty: An equation for the optimal boundary. Adv. Appl. Probab. 48(1):298–314.Crossref, Google Scholar
• [53] Fu G (2023) Extended mean field games with singular controls. SIAM J. Control Optim. 61(1):283–312.Crossref, Google Scholar
• [54] Fu G, Horst U (2017) Mean field games with singular controls. SIAM J. Control Optim. 55(6):3833–3868.Crossref, Google Scholar
• [55] Fuhrman M, Rudà S (2025) Ergodic control of McKean–Vlasov systems on the Wasserstein space. SIAM J. Control Optim. 63(6):4018–4043.Google Scholar
• [56] Gilboa I, Zemel E (1989) Nash and correlated equilibria: Some complexity considerations. Games Econom. Behav. 1(1):80–93.Crossref, Google Scholar
• [57] Guo X, Pham H (2005) Optimal partially reversible investment with entry decision and general production function. Stochastic Processes Appl. 115(5):705–736.Crossref, Google Scholar
• [58] Guo X, Xu R (2019) Stochastic games for fuel follower problem: N versus mean field game. SIAM J. Control Optim. 57(1):659–692.Crossref, Google Scholar
• [59] Hannan J (1957) Approximation to Bayes risk in repeated play. Dresher M, Tucker AW, Wolfe P, eds. Contributions to the Theory of Games, vol. III (Princeton University Press, Princeton, NJ), 97–139.Google Scholar
• [60] Hart S, Mas-Colell A (2003) Regret-based continuous-time dynamics. Games Econom. Behav. 45(2):375–394.Crossref, Google Scholar
• [61] Huang M, Malhamé RP, Caines PE (2006) Large population stochastic dynamic games: Closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Comm. Inform. Systems 6(3):221–251.Crossref, Google Scholar
• [62] Jack A, Zervos M (2006) A singular control problem with an expected and a pathwise ergodic performance criterion. Internat. J. Stochastic Anal. 2006:082538.Crossref, Google Scholar
• [63] Kallenberg O (2002) Foundations of Modern Probability, Probability and Its Applications, 2nd ed. (Springer-Verlag, New York).Crossref, Google Scholar
• [64] Karatzas I (1983) A class of singular stochastic control problems. Adv. Appl. Probab. 15(2):225–254.Crossref, Google Scholar
• [65] Karatzas I, Shreve SE (1984) Connections between optimal stopping and singular stochastic control. I. Monotone follower problems. SIAM J. Control Optim. 22(6):856–877.Crossref, Google Scholar
• [66] Karatzas I, Shreve SE (1991) Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, vol. 113, 2nd ed. (Springer-Verlag, New York).Google Scholar
• [67] Lasry J-M, Lions P-L (2007) Mean field games. Japanese J. Math. 2(1):229–260.Crossref, Google Scholar
• [68] Laurière M, Perrin S, Pérolat J, Girgin S, Muller P, Élie R, Geist M, Pietquin O (2024) Learning in mean field games: A survey. Preprint, submitted July 26, https://arxiv.org/abs/2205.12944.Google Scholar
• [69] Maschler M, Solan E, Zamir S (2013) Game Theory (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• [70] Moulin H, Vial J-P (1978) Strategically zero-sum games: The class of games whose completely mixed equilibria cannot be improved upon. Internat. J. Game Theory 7(3–4):201–221.Crossref, Google Scholar
• [71] Moulin H, Ray I, Gupta SS (2014) Coarse correlated equilibria in an abatement game. Cardiff Economics Working Paper No. E2014/24, Cardiff University, Cardiff, UK.Google Scholar
• [72] Moulin H, Ray I, Sen Gupta S (2014) Improving Nash by coarse correlation. J. Econom. Theory 150:852–865.Crossref, Google Scholar
• [73] Muller P, Rowland M, Elie R, Piliouras G, Perolat J, Lauriere M, Marinier R, Pietquin O, Tuyls K (2021) Learning equilibria in mean-field games: Introducing mean-field PSRO. Preprint, submitted November 16, https://arxiv.org/abs/2111.08350.Google Scholar
• [74] Muller P, Elie R, Rowland M, Lauriere M, Perolat J, Perrin S, Geist M, Piliouras G, Pietquin O, Tuyls K (2022) Learning correlated equilibria in mean-field games. Preprint, submitted August 22, https://arxiv.org/abs/2208.10138.Google Scholar
• [75] Neyman A (1997) Correlated equilibrium and potential games. Internat. J. Game Theory 26(2):223–227.Crossref, Google Scholar
• [76] Pham H (2006) Explicit solution to an irreversible investment model with a stochastic production capacity. Kabanov Y, Liptser R, Stoyanov J, eds. From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift (Springer, Berlin), 547–565.Crossref, Google Scholar
• [77] Pilipenko A (2014) An Introduction to Stochastic Differential Equations with Reflection, Lectures in Pure and Applied Mathematics (Universitätsverlag Potsdam, Potsdam, Germany).Google Scholar
• [78] Protter PE (2005) Stochastic Integration and Differential Equations, Stochastic Modelling and Applied Probability, vol. 21, 2nd ed. (Springer-Verlag, Berlin).Crossref, Google Scholar
• [79] Riedel F, Su X (2011) On irreversible investment. Finance Stochastics 15(4):607–633.Crossref, Google Scholar
• [80] Roughgarden T (2016) Twenty Lectures on Algorithmic Game Theory (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• [81] Rudà S (2025) Infinite time horizon optimal control of McKean–Vlasov SDEs. Preprint, submitted March 26, https://arxiv.org/abs/2503.20572.Google Scholar
• [82] Steg J-H (2012) Irreversible investment in oligopoly. Finance Stochastic 16(2):207–224.Crossref, Google Scholar
• [83] Vives X (1990) Nash equilibrium with strategic complementarities. J. Math. Econom. 19(3):305–321.Crossref, Google Scholar
• [84] Vives X (2005) Complementarities and games: New developments. J. Econom. Literature 43(2):437–479.Crossref, Google Scholar
• [85] Vives X (2005) Games with strategic complementarities: New applications to industrial organization. Internat. J. Indust. Organ. 23(7–8):625–637.Crossref, Google Scholar
• [86] Vives X (2018) Strategic complementarities in oligopoly. Corchon LC, Marini MA, eds. Handbook of Game Theory and Industrial Organization, vol. I (Edward Elgar Publishing, Cheltenham, UK), 9–39.Crossref, Google Scholar
• [87] Weerasinghe APN (2002) Stationary stochastic control for Itô processes. Adv. Appl. Probab. 34(1):128–140.Crossref, Google Scholar