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Field

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Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar B Derivation of Marginal Likelihood with Stochastic Field Amplitude The full signal in time space is given by Using the series representation of the Bessel function, together with Gamma function identities, the 5 random variables can be integrated out analytically. We arrive at the following marginalized (and normalized) likelihood: where This likelihood can be split into three individual likelihoods for the sum/difference peaks and the Compton peak, as given in Eq. (3.2). The form of the likelihoods for the sum and difference peaks is equivalent. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar B Derivation of Marginal Likelihood with Stochastic Field Amplitude The full signal in time space is given by Using the series representation of the Bessel function, together with Gamma function identities, the 5 random variables can be integrated out analytically. We arrive at the following marginalized (and normalized) likelihood: where This likelihood can be split into three individual likelihoods for the sum/difference peaks and the Compton peak, as given in Eq. (3.2). The form of the likelihoods for the sum and difference peaks is equivalent. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Derivation of Marginal Likelihood with Stochastic Field Amplitude

Authors:
(1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing;
(2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing;
(3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 5 Conclusion In conclusion, this paper proposes a novel generalized fictitious play algorithm for computing mixed strategy equilibria in OSMFGs. The key innovations include leveraging an iterative process of solving pure strategy systems to approximate mixed equilibria, as well as expanding the design flexibility for the learning rate parameter. Rigorous convergence results are provided, and finite difference schemes are constructed to efficiently solve the obstacle and Fokker-Planck equations during each iteration. Future work includes extensions to problems with common noise, where the equilibria consist of randomized stopping times that depend on the realized common noise path. The generalized fictitious play framework could also be applied to other competitive games involving optimal stopping decisions. Additionally, further analysis on quantifying the convergence rate and computational complexity could provide deeper theoretical insights. Overall, this paper introduces a novel algorithm and analysis to overcome the limitations of current OSMFG methods, opening the door for handling broader classes of large-scale dynamic games with optimal stopping. Acknowledgement ZZ is supported by the National Key R&D Program of China, Project Number 2021YFA1001200, and the NSFC, grant Number 12031013, 12171013. YL is supported by the NSFC, grant Number 12090022. We thank Xu’an Dou, Jian-Guo Liu and Jiajun Tong for helpful discussions. References [1] Y. Achdou, F. Camilli, and I. C. Dolcetta, Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2012), pp. 2585–2612. [2] ________, Mean field games: Numerical methods for the planning problem, SIAM J. Control. Optim., 50 (2012), pp. 77–109. [3] Y. Achdou and I. C. Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), pp. 1136–1162. [4] R. Aid, R. Dumitrescu, and P. Tankov, The entry and exit game in the electricity markets: A mean-field game approach, Journal of Dynamics & Games, (2020). [5] C. Bertucci, Optimal stopping in mean field games, an obstacle problem approach, Journal de Math´ematiques Pures et Appliqu´ees, (2017). [6] ________, A remark on uzawa’s algorithm and an application to mean field games systems, ESAIM: Mathematical Modelling and Numerical Analysis, (2018). [7] G. Bouveret, R. Dumitrescu, and P. Tankov, Mean-field games of optimal stopping: A relaxed solution approach, SIAM J. Control. Optim., 58 (2018), pp. 1795–1821. [8] L. Brugnano and A. Sestini, Iterative solution of piecewise linear systems for the numerical solution of obstacle problems 12, arXiv: Numerical Analysis, (2009). [9] P. Cardaliaguet and S. Hadikhanloo, Learning in mean field games: The fictitious play, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2015), pp. 569–591. [10] R. Dumitrescu, M. Leutscher, and P. Tankov, Linear programming fictitious play algorithm for mean field games with optimal stopping and absorption, ESAIM: Mathematical Modelling and Numerical Analysis, (2022). [11] X. Fernandez-Real and X. Ros-Oton ´ , Regularity theory for elliptic pde, 2022. [12] R. H. W. Hoppe, Multigrid algorithms for variational inequalities, SIAM Journal on Numerical Analysis, 24 (1987), pp. 1046–1065. [13] J. Huang and T. Xie, A class of mean-field games with optimal stopping and its applications, Asian Journal of Control, (2023). [14] J. M. Lasry and P. L. Lions, Jeux `a champ moyen. i – le cas stationnaire, Comptes Rendus Mathematique, 343 (2006), pp. 619–625. [15] ________, Jeux `a champ moyen. ii – horizon fini et contrˆole optimal, Comptes Rendus Mathematique, 343 (2006), pp. 679–684. [16] ________, Mean field games, Japanese Journal of Mathematics, 2 (2007), pp. 229–260. [17] M. Lauri`ere, Numerical methods for mean field games and mean field type control, ArXiv, abs/2106.06231 (2021). [18] P. Lee, T. W. Kim, and S. Kim, Accurate and efficient numerical solutions for elliptic obstacle problems, Journal of Inequalities and Applications, 2017 (2017). [19] S. Perrin, J. P´erolat, M. Lauri`ere, M. Geist, R. Elie, and O. Pietquin ´ , Fictitious play for mean field games: Continuous time analysis and applications, ArXiv, abs/2007.03458 (2020). [20] G. Tran, H. Schaeffer, W. M. Feldman, and S. Osher, An l1 penalty method for general obstacle problems, SIAM J. Appl. Math., 75 (2014), pp. 1424–1444. [21] X. Yang, G. Wang, and X. Gu, Numerical solution for a parabolic obstacle problem with nonsmooth initial data, Numerical Methods for Partial Differential Equations, 30 (2014). This paper is available on arxiv under CC 4.0 license. Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Authors: Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 5 Conclusion, Acknowledgement, and References 5 Conclusion In conclusion, this paper proposes a novel generalized fictitious play algorithm for computing mixed strategy equilibria in OSMFGs. The key innovations include leveraging an iterative process of solving pure strategy systems to approximate mixed equilibria, as well as expanding the design flexibility for the learning rate parameter. Rigorous convergence results are provided, and finite difference schemes are constructed to efficiently solve the obstacle and Fokker-Planck equations during each iteration. Future work includes extensions to problems with common noise, where the equilibria consist of randomized stopping times that depend on the realized common noise path. The generalized fictitious play framework could also be applied to other competitive games involving optimal stopping decisions. Additionally, further analysis on quantifying the convergence rate and computational complexity could provide deeper theoretical insights. Overall, this paper introduces a novel algorithm and analysis to overcome the limitations of current OSMFG methods, opening the door for handling broader classes of large-scale dynamic games with optimal stopping. Acknowledgement ZZ is supported by the National Key R&D Program of China, Project Number 2021YFA1001200, and the NSFC, grant Number 12031013, 12171013. YL is supported by the NSFC, grant Number 12090022. We thank Xu’an Dou, Jian-Guo Liu and Jiajun Tong for helpful discussions. References [1] Y. Achdou, F. Camilli, and I. C. Dolcetta, Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2012), pp. 2585–2612. [2] ________, Mean field games: Numerical methods for the planning problem, SIAM J. Control. Optim., 50 (2012), pp. 77–109. [3] Y. Achdou and I. C. Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), pp. 1136–1162. [4] R. Aid, R. Dumitrescu, and P. Tankov, The entry and exit game in the electricity markets: A mean-field game approach, Journal of Dynamics & Games, (2020). [5] C. Bertucci, Optimal stopping in mean field games, an obstacle problem approach, Journal de Math´ematiques Pures et Appliqu´ees, (2017). [6] ________, A remark on uzawa’s algorithm and an application to mean field games systems, ESAIM: Mathematical Modelling and Numerical Analysis, (2018). [7] G. Bouveret, R. Dumitrescu, and P. Tankov, Mean-field games of optimal stopping: A relaxed solution approach, SIAM J. Control. Optim., 58 (2018), pp. 1795–1821. [8] L. Brugnano and A. Sestini, Iterative solution of piecewise linear systems for the numerical solution of obstacle problems 12, arXiv: Numerical Analysis, (2009). [9] P. Cardaliaguet and S. Hadikhanloo, Learning in mean field games: The fictitious play, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2015), pp. 569–591. [10] R. Dumitrescu, M. Leutscher, and P. Tankov, Linear programming fictitious play algorithm for mean field games with optimal stopping and absorption, ESAIM: Mathematical Modelling and Numerical Analysis, (2022). [11] X. Fernandez-Real and X. Ros-Oton ´ , Regularity theory for elliptic pde, 2022. [12] R. H. W. Hoppe, Multigrid algorithms for variational inequalities, SIAM Journal on Numerical Analysis, 24 (1987), pp. 1046–1065. [13] J. Huang and T. Xie, A class of mean-field games with optimal stopping and its applications, Asian Journal of Control, (2023). [14] J. M. Lasry and P. L. Lions, Jeux `a champ moyen. i – le cas stationnaire, Comptes Rendus Mathematique, 343 (2006), pp. 619–625. [15] ________, Jeux `a champ moyen. ii – horizon fini et contrˆole optimal, Comptes Rendus Mathematique, 343 (2006), pp. 679–684. [16] ________, Mean field games, Japanese Journal of Mathematics, 2 (2007), pp. 229–260. [17] M. Lauri`ere, Numerical methods for mean field games and mean field type control, ArXiv, abs/2106.06231 (2021). [18] P. Lee, T. W. Kim, and S. Kim, Accurate and efficient numerical solutions for elliptic obstacle problems, Journal of Inequalities and Applications, 2017 (2017). [19] S. Perrin, J. P´erolat, M. Lauri`ere, M. Geist, R. Elie, and O. Pietquin ´ , Fictitious play for mean field games: Continuous time analysis and applications, ArXiv, abs/2007.03458 (2020). [20] G. Tran, H. Schaeffer, W. M. Feldman, and S. Osher, An l1 penalty method for general obstacle problems, SIAM J. Appl. Math., 75 (2014), pp. 1424–1444. [21] X. Yang, G. Wang, and X. Gu, Numerical solution for a parabolic obstacle problem with nonsmooth initial data, Numerical Methods for Partial Differential Equations, 30 (2014). This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Conclusion and References

Authors:
(1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing;
(2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing;
(3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 4.2 A Local OSMFG Example Setup. In this example, the state of the representative agent belongs to the same domain as the previous one. The state’s law of motion follows the following SDE: The initial distribution is a sum of two Gaussian functions: The running cost f(x, t, m) is defined as and the stopping cost is defined as We have used the first-order upwind scheme to discretize the drift term. Other settings are identical to the previous example. In this example, the dynamics drive the agent to concentrate near the origin. However, the agent dislikes crowded states, and an excessively crowded state would cause an agent to quit. On the other hand, if too many agents quit at a certain state, the running cost becomes attractive again at that location. Therefore, in this example, there doesn’t seem to exist a pure strategy equilibrium. It can be observed that no agents quit initially, and as more agents gather around x = 0, some begin to quit at around t = 0.25. It is worth pointing out that, unlike the previous example, agents do not quit with 100% certainty upon reaching the exiting boundary in this case. This is evidenced by the fact that the density remains positive for the exiting region. This clearly demonstrates that our solution is a mixed strategy equilibrium. This paper is available on arxiv under CC 4.0 license. Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Authors: Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 5 Conclusion, Acknowledgement, and References 4.2 A Local OSMFG Example Setup . In this example, the state of the representative agent belongs to the same domain as the previous one. The state’s law of motion follows the following SDE: Setup The initial distribution is a sum of two Gaussian functions: The running cost f(x, t, m) is defined as f(x, t, m) and the stopping cost is defined as We have used the first-order upwind scheme to discretize the drift term. Other settings are identical to the previous example. In this example, the dynamics drive the agent to concentrate near the origin. However, the agent dislikes crowded states, and an excessively crowded state would cause an agent to quit. On the other hand, if too many agents quit at a certain state, the running cost becomes attractive again at that location. Therefore, in this example, there doesn’t seem to exist a pure strategy equilibrium. It can be observed that no agents quit initially, and as more agents gather around x = 0, some begin to quit at around t = 0.25. It is worth pointing out that, unlike the previous example, agents do not quit with 100% certainty upon reaching the exiting boundary in this case. This is evidenced by the fact that the density remains positive for the exiting region. This clearly demonstrates that our solution is a mixed strategy equilibrium. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: A Local OSMFG Example

Authors:
(1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing;
(2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing;
(3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 4 Numerical Experiments In this section, we conduct several numerical experiments to demonstrate the effectiveness of the proposed semi-implicit finite difference algorithm (Algorithm 2). Through these experiments, we examine the convergence properties of our algorithm highlighting the implementation of the fictitious play. We demonstrate that our requirement for δn in (3.3) is a sufficient yet unnecessary condition. However, in certain cases where the pure strategy equilibrium may not exist, the iteration method may not converge if the condition (3.3) is violated. 4.1 A Non-local OSMFG Example Setup. In this example, the state of the representative agent belongs to the domain [0, 1]×R. It dynamic is a Brownian motion, i.e. Intuitively, as the agent aims to minimize expected cost, the running cost encourages the agent to remain in the game when its state is below the average state of other remaining agents, while the stopping cost encourages the agent to continue playing for a longer duration. It is easy to check that the cost functions above are equivalent with In Figure 2, log-log plots of error versus iteration numbers are shown for different learning rates δn. For this example, taking δn = 1 leads to the fastest convergence, although with δn = 1 no fictitious play is implemented. The reason for this is that the convergence requirement for the learning rate δn in Definition 3.1 is only a sufficient condition, rather than a necessary one. For problems with a pure strategy equilibrium, the condition in definition 3.1 can be relaxed and a more aggressive learning rate can be taken to obtain faster convergence. However, we would like to stress that we could not know a priori whether a pure strategy equilibrium exists, but the proposed algorithm, which is based on fictitious play, always produces convergence results. This paper is available on arxiv under CC 4.0 license. Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Authors: Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 5 Conclusion, Acknowledgement, and References 4 Numerical Experiments In this section, we conduct several numerical experiments to demonstrate the effectiveness of the proposed semi-implicit finite difference algorithm (Algorithm 2). Through these experiments, we examine the convergence properties of our algorithm highlighting the implementation of the fictitious play. We demonstrate that our requirement for δ n in (3.3) is a sufficient yet unnecessary condition. However, in certain cases where the pure strategy equilibrium may not exist, the iteration method may not converge if the condition (3.3) is violated. n 4.1 A Non-local OSMFG Example Setup . In this example, the state of the representative agent belongs to the domain [0, 1]×R. It dynamic is a Brownian motion, i.e. Setup Intuitively, as the agent aims to minimize expected cost, the running cost encourages the agent to remain in the game when its state is below the average state of other remaining agents, while the stopping cost encourages the agent to continue playing for a longer duration. It is easy to check that the cost functions above are equivalent with In Figure 2, log-log plots of error versus iteration numbers are shown for different learning rates δn. For this example, taking δn = 1 leads to the fastest convergence, although with δn = 1 no fictitious play is implemented. The reason for this is that the convergence requirement for the learning rate δn in Definition 3.1 is only a sufficient condition, rather than a necessary one. For problems with a pure strategy equilibrium, the condition in definition 3.1 can be relaxed and a more aggressive learning rate can be taken to obtain faster convergence. However, we would like to stress that we could not know a priori whether a pure strategy equilibrium exists, but the proposed algorithm, which is based on fictitious play, always produces convergence results. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Numerical Experiments

Authors:
(1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing;
(2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing;
(3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 3.4 Numerical Analysis In this part, our goal is to prove the convergence of algorithm 2 when implicit scheme (3.25) and (3.26) are applied. The convergence analysis mirrors the proof for Theorem 3.1, requiring only adapting the arguments to a discretized version. Definition 3.4 (implicit discretized system for mixed strategy equilibrium) We define the complementary condition will be weaker than the following one Before stating the main result, we present a property of the implicit scheme for obstacle equations: the discretized solution u continuously depends on the discretized source term f. Lemma 3.1 Consider the following discrete obstacle problem: Now we can state the main convergence result in this section. Proof The spirit of the proof is analog to the one in theorem 3.1. We divide the proof into 3 steps just parallel to the proof of theorem 3.1. with equality if and only if 3. We conclude that any cluster point (u∗, m∗) is a solution to (3.29). We first verify that u∗ will satisfy the discretized obstacle problem as follows: This paper is available on arxiv under CC 4.0 license. Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Authors: Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 5 Conclusion, Acknowledgement, and References 3.4 Numerical Analysis In this part, our goal is to prove the convergence of algorithm 2 when implicit scheme (3.25) and (3.26) are applied. The convergence analysis mirrors the proof for Theorem 3.1, requiring only adapting the arguments to a discretized version. Definition 3.4 (implicit discretized system for mixed strategy equilibrium) We define Definition 3.4 (implicit discretized system for mixed strategy equilibrium) We define the complementary condition the complementary condition will be weaker than the following one will be weaker than the following one Before stating the main result, we present a property of the implicit scheme for obstacle equations: the discretized solution u continuously depends on the discretized source term f . f Lemma 3.1 Consider the following discrete obstacle problem: Lemma 3.1 Consider the following discrete obstacle problem: Now we can state the main convergence result in this section. Proof The spirit of the proof is analog to the one in theorem 3.1. We divide the proof into 3 steps just parallel to the proof of theorem 3.1. Proof with equality if and only if 3. We conclude that any cluster point (u∗, m∗) is a solution to (3.29). We first verify that u∗ will satisfy the discretized obstacle problem as follows: This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Numerical Analysis

Authors:
(1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing;
(2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing;
(3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 3.3 Algorithm Based on Fictitious Play Theorem 3.1 provides the convergence result of the fictitious play as in Definition 3.1. We can turn the fictitious play into the following algorithm for finding the mixed strategy equilibrium. In the remainder of this section, we will introduce the discretization method for the obstacle equation (2.2) and the Fokker-Planck equation (2.3) in algorithm 1. Assuming the spatial discretization grid size is h However, iteration are unavoidable when numerically solving equations (3.25) and (3.26), regardless of the method used. The application of nonlinear solvers renders the implicit scheme computationally inefficient. Therefore, in practice, semi-implicit schemes are preferred to reduce the computational cost. The semi-implicit scheme for u can be written as follows: And the semi-implicit scheme for m can be written as: Semi-implicit schemes for u and m can be viewed as a two-step method that decouples the linear and nonlinear parts of each equation. In the first step, u and m are evolved using a standard implicit scheme on the whole domain. In the second step, a cutoff is applied to u and m respectively to account for the free boundary effects. We only need to solve a sparse system of linear equations for each time step in the semi-implicit schemes (3.27) and (3.28. Now we can summarize the finite difference algorithm into the following algorithm. This paper is available on arxiv under CC 4.0 license. Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Authors: Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 5 Conclusion, Acknowledgement, and References 3.3 Algorithm Based on Fictitious Play Theorem 3.1 provides the convergence result of the fictitious play as in Definition 3.1. We can turn the fictitious play into the following algorithm for finding the mixed strategy equilibrium. In the remainder of this section, we will introduce the discretization method for the obstacle equation (2.2) and the Fokker-Planck equation (2.3) in algorithm 1. Assuming the spatial discretization grid size is h h However, iteration are unavoidable when numerically solving equations (3.25) and (3.26), regardless of the method used. The application of nonlinear solvers renders the implicit scheme computationally inefficient. Therefore, in practice, semi-implicit schemes are preferred to reduce the computational cost. The semi-implicit scheme for u can be written as follows: And the semi-implicit scheme for m can be written as: m Semi-implicit schemes for u and m can be viewed as a two-step method that decouples the linear and nonlinear parts of each equation. In the first step, u and m are evolved using a standard implicit scheme on the whole domain. In the second step, a cutoff is applied to u and m respectively to account for the free boundary effects. We only need to solve a sparse system of linear equations for each time step in the semi-implicit schemes (3.27) and (3.28. Now we can summarize the finite difference algorithm into the following algorithm. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Fictitious Play Algorithm

Authors:
(1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing;
(2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing;
(3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium To prove the convergence of the generalized fictitious play (3.1), we first need to ensure that the iteration scheme is well-defined. We introduce the following definition: Additionally, we require some additional technical assumptions on the functions f and ψ. A common assumption is that f and ψ are variations of potential functions. Games that satisfy this property are known as potential games, of which we present the formal definition as follows. Definition 3.3 (potential games) We call an optimal stopping mean field game a potential game if there exist potential functions F, Ψ : C → R such that Now we present the main convergence result. Without loss of generality, we assume the generator of the diffusion process is simply the Laplacian operator ∆ in the subsequent analysis. From step 1 we know that Therefore, we have We define that By the learning rate condition (3.3), which we recall here for convenience we can choose N such that Hence, for all n ≥ M, we have the following inequality: This is a contradiction with the continuity of Φ and the assumption that m∗ is a cluster point. Thus the cluster point m∗ should satisfy for all m ∈ T. 3. We conclude that any cluster point (u ∗, m∗) is a mixed strategy equilibrium in this step. We first verify that u ∗ will satisfy the obstacle problem as following: It remains to verify and Define m∗∗ as in (3.13). By (3.21) and the relation we know that From (3.11) and (3.12), this equality implies (3.23) and (3.24). Hence we have finished the proof. This paper is available on arxiv under CC 4.0 license. Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Authors: Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 5 Conclusion, Acknowledgement, and References 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium To prove the convergence of the generalized fictitious play (3.1), we first need to ensure that the iteration scheme is well-defined. We introduce the following definition: Additionally, we require some additional technical assumptions on the functions f and ψ . A common assumption is that f and ψ are variations of potential functions. Games that satisfy this property are known as potential games, of which we present the formal definition as follows. f ψ f ψ Definition 3.3 (potential games) We call an optimal stopping mean field game a potential game if there exist potential functions F, Ψ : C → R such that Definition 3.3 (potential games) We call an optimal stopping mean field game a potential game if there exist potential functions F, Ψ : C → R such that Now we present the main convergence result. Without loss of generality, we assume the generator of the diffusion process is simply the Laplacian operator ∆ in the subsequent analysis. From step 1 we know that Therefore, we have We define that By the learning rate condition (3.3), which we recall here for convenience we can choose N such that N Hence, for all n ≥ M, we have the following inequality: This is a contradiction with the continuity of Φ and the assumption that m∗ is a cluster point. Thus the cluster point m∗ should satisfy for all m ∈ T. 3. We conclude that any cluster point (u ∗, m∗) is a mixed strategy equilibrium in this step. We first verify that u ∗ will satisfy the obstacle problem as following: It remains to verify and Define m∗∗ as in (3.13). By (3.21) and the relation we know that From (3.11) and (3.12), this equality implies (3.23) and (3.24). Hence we have finished the proof. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Convergence of Fictitious Play

Authors:
(1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing;
(2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing;
(3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 3 Algorithm Construction In this section, we introduce an iterative algorithm for solving the mixed strategy equilibrium system 2.5). As mentioned previously, directly solving system (2.5) is intractable. Instead, we obtain the mixed strategy equilibrium by solving the more tractable pure strategy system (2.4) using fictitious play as follows. First, we introduce a generalized fictitious play and build the convergence result from the pure strategy system to the mixed strategy system through fictitious play for potential games. Next, we present the finite difference schemes used to numerically solve the obstacle and Fokker-Planck equations. Finally, we show the implicit scheme preserves the convergence property from pure to mixed strategies. 3.1 Fictitious Play Fictitious play is a learning procedure in which, at each step, every agent chooses the best response strategy based on the average of the strategies previously employed by other agents. It would seem natural, therefore, to expect that the best response strategy would converge to the strategy corresponding to the Nash equilibrium of the game. While fictitious play provides a simple approach to approximate the Nash equilibrium, it does not always work. In classical MFG models, it has been proven in [9] that the sequence obtained via fictitious play converges to the MFG solution. When applied to OSMFG models, it has been proven that the linear programming method utilizing fictitious play converges to the relaxed Nash equilibrium under a measure flow framework, as demonstrated in [10]. However, fictitious play has not yet been directly applied to the PDE systems representing the Nash equilibrium for OSMFG models. Our goal is to adapt the fictitious play to the PDE systems in order to find the mixed strategy equilibrium solutions for OSMFG models. The main challenge is to investigate the convergence of the fictitious play of the pure strategy system (2.4) to the solution of the mixed strategy system (2.5). This requires novel theoretical advances since the current results on fictitious play for classic MFG systems analyzed in [9] cannot be applied to OSMFG models, as classic MFG systems do not need the notion of mixed strategy equilibrium. Recall that fictitious play approximates a mixed strategy equilibrium through iterative pure strategies. An agent decides whether to quit the game when using a pure strategy, whereas he specifies the likelihood of quitting the game when using a mixed strategy. This is the main difference between the two notions of equilibrium. In a fictitious play for OSMFG, agents repeatedly play the game, applying a pure strategy that is the best response to the average historic distribution of agents in each instance. Over repeated plays, agents may apply different pure strategies at a given state (x, t). Therefore, at some state (x, t), an agent may quit during some rounds of plays, but for other rounds he may not. Overall, he quits at (x, t) with a probability no less than 0 but no greater than 1. This illustrates the idea behind a mixed strategy. Each iteration of a fictitious play for OSMFG can be divided into the following three steps: Finding the Best Response. Agents calculate an optimal pure strategy response based on the so-called updated distribution of agents, which encodes the historical information from previous iterations.


Propagating the Proposed Distribution. The proposed distribution of agents is determined based on the strategies obtained in the previous step.


Calculating the Updated Distribution. The agents calculate the updated distribution by incorporating the proposed distribution with a certain updating rule. The updated distribution will be used to calculate controls in the next iteration. The procedure can be formulated in the definition below. Definition 3.1 (generalized fictitious play) Given the initial distribution m0. The following iteration is called a generalized fictitious play. where n = 0, 1, 2, ... is the iteration round, and This paper is available on arxiv under CC 4.0 license. Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Authors: Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 5 Conclusion, Acknowledgement, and References 3 Algorithm Construction In this section, we introduce an iterative algorithm for solving the mixed strategy equilibrium system 2.5). As mentioned previously, directly solving system (2.5) is intractable. Instead, we obtain the mixed strategy equilibrium by solving the more tractable pure strategy system (2.4) using fictitious play as follows. First, we introduce a generalized fictitious play and build the convergence result from the pure strategy system to the mixed strategy system through fictitious play for potential games. Next, we present the finite difference schemes used to numerically solve the obstacle and Fokker-Planck equations. Finally, we show the implicit scheme preserves the convergence property from pure to mixed strategies. 3.1 Fictitious Play Fictitious play is a learning procedure in which, at each step, every agent chooses the best response strategy based on the average of the strategies previously employed by other agents. It would seem natural, therefore, to expect that the best response strategy would converge to the strategy corresponding to the Nash equilibrium of the game. While fictitious play provides a simple approach to approximate the Nash equilibrium, it does not always work. In classical MFG models, it has been proven in [9] that the sequence obtained via fictitious play converges to the MFG solution. When applied to OSMFG models, it has been proven that the linear programming method utilizing fictitious play converges to the relaxed Nash equilibrium under a measure flow framework, as demonstrated in [10]. However, fictitious play has not yet been directly applied to the PDE systems representing the Nash equilibrium for OSMFG models. Our goal is to adapt the fictitious play to the PDE systems in order to find the mixed strategy equilibrium solutions for OSMFG models. The main challenge is to investigate the convergence of the fictitious play of the pure strategy system (2.4) to the solution of the mixed strategy system (2.5). This requires novel theoretical advances since the current results on fictitious play for classic MFG systems analyzed in [9] cannot be applied to OSMFG models, as classic MFG systems do not need the notion of mixed strategy equilibrium. Recall that fictitious play approximates a mixed strategy equilibrium through iterative pure strategies. An agent decides whether to quit the game when using a pure strategy, whereas he specifies the likelihood of quitting the game when using a mixed strategy. This is the main difference between the two notions of equilibrium. In a fictitious play for OSMFG, agents repeatedly play the game, applying a pure strategy that is the best response to the average historic distribution of agents in each instance. Over repeated plays, agents may apply different pure strategies at a given state (x, t). Therefore, at some state (x, t), an agent may quit during some rounds of plays, but for other rounds he may not. Overall, he quits at (x, t) with a probability no less than 0 but no greater than 1. This illustrates the idea behind a mixed strategy. Each iteration of a fictitious play for OSMFG can be divided into the following three steps: Finding the Best Response. Agents calculate an optimal pure strategy response based on the so-called updated distribution of agents, which encodes the historical information from previous iterations. Propagating the Proposed Distribution. The proposed distribution of agents is determined based on the strategies obtained in the previous step. Calculating the Updated Distribution. The agents calculate the updated distribution by incorporating the proposed distribution with a certain updating rule. The updated distribution will be used to calculate controls in the next iteration. Finding the Best Response. Agents calculate an optimal pure strategy response based on the so-called updated distribution of agents, which encodes the historical information from previous iterations. Finding the Best Response. Agents calculate an optimal pure strategy response based on the so-called updated distribution of agents, which encodes the historical information from previous iterations. Finding the Best Response. Propagating the Proposed Distribution. The proposed distribution of agents is determined based on the strategies obtained in the previous step. Propagating the Proposed Distribution. The proposed distribution of agents is determined based on the strategies obtained in the previous step. Propagating the Proposed Distribution. Calculating the Updated Distribution. The agents calculate the updated distribution by incorporating the proposed distribution with a certain updating rule. The updated distribution will be used to calculate controls in the next iteration. Calculating the Updated Distribution . The agents calculate the updated distribution by incorporating the proposed distribution with a certain updating rule. The updated distribution will be used to calculate controls in the next iteration. Calculating the Updated Distribution The procedure can be formulated in the definition below. Definition 3.1 (generalized fictitious play) Given the initial distribution m0. The following iteration is called a generalized fictitious play. Definition 3.1 (generalized fictitious play) Given the initial distribution m0. The following iteration is called a generalized fictitious play. where n = 0, 1, 2, ... is the iteration round, and where n = 0, 1, 2, ... is the iteration round, and This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Algorithm Construction

Authors:
(1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing;
(2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing;
(3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 2.4 Mixed Strategy Equilibrium for OSMFG If we relax the constraint forcing agents to uniformly exit at {u = ψ}, mixed strategy equilibrium can be defined. With multiple optimal choices available, mixed equilibria allow agents to randomize over strategies freely, rather than requiring identical decisions. Assume that the cost functions f and ψ depend only on the crowd distribution m, the PDE system governing the mixed strategy equilibrium for OSMFG is as follows: We refer to [5] for more discussions of the mixed strategy equilibrium. The last equation of (2.5) shows that is the region where agents can choose to exit or to stay. It can be derived by standard stochastic calculus that agents would have the same cost whenever exiting or staying at (x, t) when (x, t) ∈ {f(m)+ (∂t+L)ψ(m) = 0}. This is the main relaxation compared to pure strategy equilibrium. The following existence result shows that such relaxation is sufficient, and hence mixed strategy equilibrium is a better notion for mean field games with optimal stopping time. In order to state the theorem rigorously, we first introduce two spaces and some technical assumptions for f and ψ. Definition 2.1 We define function spaces A, B and C as: Assumption 2.1 The running cost f(·, ·, m) and the stopping cost ψ(·, ·, m) satisfy: The map m → f(·, ·, m) is continuous from C to itself;


The map m → ψ(·, ·, m) is continuous from C to A A. Then the existence result can be formulated in the following theorem. Theorem 2.2 Suppose that Assumption 2.1 holds. Then there exists at least one solution (u, m) ∈ A × B for (2.5) We refer the readers to Theorem 2.1 of [5] for the proof of the theorem above. Although the existence of the mixed strategy equilibrium can be guaranteed, it is yet difficult to design a simple and efficient numerical PDE algorithm for (2.5) directly due to its complicated form. The coupling between u and m is more intricate due to the complementary condition In the next section, the primary goal is to construct an iterative algorithm that provides an accurate approximation of the mixed strategy equilibria for OSMFGs via a generalized fictitious play. This paper is available on arxiv under CC 4.0 license. Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Authors: Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 5 Conclusion, Acknowledgement, and References 2.4 Mixed Strategy Equilibrium for OSMFG If we relax the constraint forcing agents to uniformly exit at {u = ψ}, mixed strategy equilibrium can be defined. With multiple optimal choices available, mixed equilibria allow agents to randomize over strategies freely, rather than requiring identical decisions. Assume that the cost functions f and ψ depend only on the crowd distribution m , the PDE system governing the mixed strategy equilibrium for OSMFG is as follows: f ψ m We refer to [5] for more discussions of the mixed strategy equilibrium. The last equation of (2.5) shows that is the region where agents can choose to exit or to stay. It can be derived by standard stochastic calculus that agents would have the same cost whenever exiting or staying at ( x, t ) when ( x, t ) ∈ { f ( m )+ ( ∂t+L ) ψ ( m ) = 0}. This is the main relaxation compared to pure strategy equilibrium. The following existence result shows that such relaxation is sufficient, and hence mixed strategy equilibrium is a better notion for mean field games with optimal stopping time. In order to state the theorem rigorously, we first introduce two spaces and some technical assumptions for f and ψ. x, t x, t f m ∂t+L ψ m Definition 2.1 We define function spaces A, B and C as: Definition 2.1 We define function spaces A, B and C as: Assumption 2.1 The running cost f(·, ·, m) and the stopping cost ψ(·, ·, m) satisfy: Assumption 2.1 The running cost f(·, ·, m) and the stopping cost ψ(·, ·, m) satisfy: The map m → f(·, ·, m) is continuous from C to itself; The map m → ψ(·, ·, m) is continuous from C to A The map m → f(·, ·, m) is continuous from C to itself; The map m → f(·, ·, m) is continuous from C to itself; The map m → f(·, ·, m) is continuous from C to itself; The map m → ψ(·, ·, m) is continuous from C to A The map m → ψ(·, ·, m) is continuous from C to A The map m → ψ(·, ·, m) is continuous from C to A A. Then the existence result can be formulated in the following theorem. Theorem 2.2 Suppose that Assumption 2.1 holds. Then there exists at least one solution (u, m) ∈ A × B for (2.5) Theorem 2.2 Suppose that Assumption 2.1 holds. Then there exists at least one solution (u, m) ∈ A × B for (2.5) We refer the readers to Theorem 2.1 of [5] for the proof of the theorem above. Although the existence of the mixed strategy equilibrium can be guaranteed, it is yet difficult to design a simple and efficient numerical PDE algorithm for (2.5) directly due to its complicated form. The coupling between u and m is more intricate due to the complementary condition In the next section, the primary goal is to construct an iterative algorithm that provides an accurate approximation of the mixed strategy equilibria for OSMFGs via a generalized fictitious play. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Fictitious Play for Mixed Strategy Equilibria in Mean Field Games: Mixed Strategy Equilibrium

Authors:
(1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing;
(2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing;
(3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 2.3 Pure Strategy Equilibrium for OSMFG As for pure strategy equilibrium, we require in addition that all agents choose the same control when they have multiple optimal controls. This is the simplest notion of equilibrium. Now we construct the pure strategy equilibrium in detail. Suppose that all agents apply the same optimal control, i.e. exit when (x, t) ∈ {u = ψ} and stay when (x, t) ∈ {u < ψ}, the actual crowd propagation will satisfy the following Fokker-Planck equation: When the actual crowd propagation matches the guessed one (i.e. m = µ), we can combine (2.2) and (2.3) to obtain the PDE system of the Nash equilibrium with pure strategy: We have mentioned that in some regions of the set {u = ψ}, agents are indifferent between exiting the game and staying in the game. And we have required all agents to leave in such cases. In other words, when an agent faces choices that bring the same cost, they should all choose a specific control rather than individually make a random choice. Therefore, we refer to (2.4) as the pure strategy equilibrium of mean field games with optimal stopping. However, the pure strategy equilibrium may not exist for mean field game models. In fact, we have the following result: Theorem 2.1 There exists f and ψ such that there is no solution for (2.4). We refer to [5] for the proof of such a non-existence result for the stationary case. The construction of f and ψ for the time-dependent case (2.4) is similar and we omit the details here. Instead of elaborating on the proof, we provide an economic intuition for such non-existence. Consider a game with a population of agents facing an optimal stopping problem, where agents experience crowd aversion. As we will show, there exists no pure strategy Nash equilibrium in this setting. Suppose all other agents follow the pure strategy of exiting at time t. For any individual agent, the cost of staying past t will be low due to crowd aversion. Thus, the agent’s best response is to deviate from the crowd and continue playing. On the other hand, if all agents use the pure strategy of staying past t, an individual agent will prefer exiting at t to avoid the crowd. In either case, no single pure strategy can be sustained in equilibrium, as agents have an incentive to deviate. The underlying issue is that the definition of pure strategy equilibrium requires all agents to make exactly the same decision, even when indifferent. If we relax this and allow some agents to exit at t while others stay, or have agents randomize between the two actions, an equilibrium may exist. This leads to the concept of mixed strategy equilibrium, where agents follow randomized policies that make opponents indifferent across actions. By permitting randomization, mixed equilibria can capture stable outcomes in games without pure strategy solutions. This provides an avenue to model strategic interactions in mean field games with crowd aversion and optimal stopping. This paper is available on arxiv under CC 4.0 license. Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Authors: Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 5 Conclusion, Acknowledgement, and References 2.3 Pure Strategy Equilibrium for OSMFG As for pure strategy equilibrium, we require in addition that all agents choose the same control when they have multiple optimal controls. This is the simplest notion of equilibrium. Now we construct the pure strategy equilibrium in detail. Suppose that all agents apply the same optimal control, i.e. exit when (x, t) ∈ {u = ψ} and stay when (x, t) ∈ {u < ψ}, the actual crowd propagation will satisfy the following Fokker-Planck equation: When the actual crowd propagation matches the guessed one (i.e. m = µ), we can combine (2.2) and (2.3) to obtain the PDE system of the Nash equilibrium with pure strategy: We have mentioned that in some regions of the set {u = ψ}, agents are indifferent between exiting the game and staying in the game. And we have required all agents to leave in such cases. In other words, when an agent faces choices that bring the same cost, they should all choose a specific control rather than individually make a random choice. Therefore, we refer to (2.4) as the pure strategy equilibrium of mean field games with optimal stopping. However, the pure strategy equilibrium may not exist for mean field game models. In fact, we have the following result: Theorem 2.1 There exists f and ψ such that there is no solution for (2.4). Theorem 2.1 There exists f and ψ such that there is no solution for (2.4). We refer to [5] for the proof of such a non-existence result for the stationary case. The construction of f and ψ for the time-dependent case (2.4) is similar and we omit the details here. Instead of elaborating on the proof, we provide an economic intuition for such non-existence. f Consider a game with a population of agents facing an optimal stopping problem, where agents experience crowd aversion. As we will show, there exists no pure strategy Nash equilibrium in this setting. Suppose all other agents follow the pure strategy of exiting at time t . For any individual agent, the cost of staying past t will be low due to crowd aversion. Thus, the agent’s best response is to deviate from the crowd and continue playing. On the other hand, if all agents use the pure strategy of staying past t , an individual agent will prefer exiting at t to avoid the crowd. In either case, no single pure strategy can be sustained in equilibrium, as agents have an incentive to deviate. t t t t The underlying issue is that the definition of pure strategy equilibrium requires all agents to make exactly the same decision, even when indifferent. If we relax this and allow some agents to exit at t while others stay, or have agents randomize between the two actions, an equilibrium may exist. This leads to the concept of mixed strategy equilibrium, where agents follow randomized policies that make opponents indifferent across actions. By permitting randomization, mixed equilibria can capture stable outcomes in games without pure strategy solutions. This provides an avenue to model strategic interactions in mean field games with crowd aversion and optimal stopping. t This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

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