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Super Bowl security uses smart Taylor Swift strategy to get giddy pop star from suite to field

Is the Universe a Self-Knowing System? The Case for the Wisdom Field For centuries, we’ve assumed that reality is made of matter and energy—that space-time is the fundamental backdrop of existence. But what if we’ve had it backwards? What if space-time, matter, and even consciousness emerge from something deeper—a fundamental informational field, a Wisdom Field? Quantum mechanics, neuroscience, and complexity science all point in the same direction: information is the foundation of reality. And intelligence, selfhood, and meaning? They’re just progressive refinements of that same informational structure. This article breaks down why this theory makes sense, what evidence supports it, and why it changes everything. The Universe is Information, Not Stuff Physics has long struggled with the weirdness of quantum mechanics. Entangled particles seem to affect each other instantly, violating our sense of space-time. Observing a quantum system collapses its wave function as if reality depends on measurement. But this isn’t strange if we consider a radical idea: Space-time is not fundamental. Information is. Quantum entanglement isn’t “spooky action at a distance” if distance itself is an emergent product of information structure. Selfhood: An Informational Loop Neuroscience is also revealing something profound: consciousness isn’t just a byproduct of neurons firing. It’s an emergent property of information processing. The brain is not just a storage device—it is an active organizer of information, structuring data into meaning. Complexity research shows that consciousness scales with the amount of integrated information. The brain doesn’t generate selfhood—it localizes and refines it. In other words, selfhood is what happens when information loops in on itself in a structured way. Knowledge as Nested Analogies But what about knowledge? How do we “know” anything? It turns out, all knowledge may just be nested analogies. Every idea, every concept, every theory is a structured web of comparisons that grows in complexity over time. Consider how science progresses: Newton’s gravity → Einstein’s relativity → Quantum mechanics → ??? Each step isn’t a rejection of the previous—it’s a deeper refinement of analogy. Over laps on the previous and is understood from its use of historical symbols. Layer upon layer. Knowing emerges as patterns of analogy reinforced and stabilized. The deeper the analogy structure, the greater the knowing. The Wisdom Field: A Self-Knowing Universe If all of this is true, then we’re left with one staggering conclusion: ➡️ The universe isn’t just a collection of things—it is a self-organizing, self-knowing system. ➡️ Wisdom isn’t a property of the universe. Wisdom is the universe. Everything—space-time, matter, intelligence—is just different expressions of a fundamental informational field organizing itself into complexity. And eventually knowledge. All the same stuff but so different. Is the Universe a Self-Knowing System? The Case for the Wisdom Field For centuries, we’ve assumed that reality is made of matter and energy—that space-time is the fundamental backdrop of existence. But what if we’ve had it backwards? What if space-time, matter, and even consciousness emerge from something deeper—a fundamental informational field, a Wisdom Field? Quantum mechanics, neuroscience, and complexity science all point in the same direction: information is the foundation of reality. And intelligence, selfhood, and meaning? They’re just progressive refinements of that same informational structure. This article breaks down why this theory makes sense, what evidence supports it, and why it changes everything. The Universe is Information, Not Stuff The Universe is Information, Not Stuff Physics has long struggled with the weirdness of quantum mechanics. Entangled particles seem to affect each other instantly, violating our sense of space-time. Observing a quantum system collapses its wave function as if reality depends on measurement. But this isn’t strange if we consider a radical idea: Space-time is not fundamental. Information is. Quantum entanglement isn’t “spooky action at a distance” if distance itself is an emergent product of information structure. Selfhood: An Informational Loop Selfhood: An Informational Loop Neuroscience is also revealing something profound: consciousness isn’t just a byproduct of neurons firing. It’s an emergent property of information processing. The brain is not just a storage device—it is an active organizer of information, structuring data into meaning. Complexity research shows that consciousness scales with the amount of integrated information. The brain doesn’t generate selfhood—it localizes and refines it. In other words, selfhood is what happens when information loops in on itself in a structured way. Knowledge as Nested Analogies Knowledge as Nested Analogies But what about knowledge? How do we “know” anything? It turns out, all knowledge may just be nested analogies. Every idea, every concept, every theory is a structured web of comparisons that grows in complexity over time. Consider how science progresses: Newton’s gravity → Einstein’s relativity → Quantum mechanics → ??? Each step isn’t a rejection of the previous—it’s a deeper refinement of analogy. Over laps on the previous and is understood from its use of historical symbols. Layer upon layer. Knowing emerges as patterns of analogy reinforced and stabilized. The deeper the analogy structure, the greater the knowing. The Wisdom Field: A Self-Knowing Universe The Wisdom Field: A Self-Knowing Universe If all of this is true, then we’re left with one staggering conclusion: ➡️ The universe isn’t just a collection of things—it is a self-organizing, self-knowing system. ➡️ Wisdom isn’t a property of the universe. Wisdom is the universe. Everything—space-time, matter, intelligence—is just different expressions of a fundamental informational field organizing itself into complexity. And eventually knowledge. All the same stuff but so different.

Wisdom Field - A Theory of Everything 

Authors:
(1) Nandan Pakhira, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India;
(2) Rajib Mahato, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India and Central Electronics Engineering Research Institute, Pilani, Rajasthan 333031, India. Table of Links Abstract and 1 Introduction II. Mathematical Formalism III. Work Function Distribution IV. Results A. Gaussian work function distribution B. Log-normal work function distribution V. Conclusion, Acknowledgements, and References I. INTRODUCTION Field emission is the process in which electrons from cold surfaces are emitted in the presence of applied strong electric field. This process should be compared agaist the thermo-ionic process in which electrons are emitted from hot metal surface. The field emission forms the back bone of modern semiconductor devices. This effect was first described by Fowler and Nordheim[1]. They considered quantum mechanical tunneling through a triangular potential energy (PE) barrier, created by the application of a constant electric field. Much later Murphy and Good[2] (MG) introduced a more realistic PE barrier by taking into account the induced image charge formed in the presence of emitted electrons. MG calculated the barrier transmission coefficient under semi-classical WKB approximation. More recently, essentially an exact solution of the problem was obtained by Choy et. al.[3]. Also, various cases including finite temperature (thermal emission), tunneling, curvature of th. To the best of our knowledge in all of those studies authors have considered constant local work function. The work function of a material depends on the composition, structure, geometry, local charge distribution etc. of the emitting surface. Assumption of constant work function is only suitable for an atomistic smooth homogeneous surface. For surfaces with inhomogeneities over nano-scale (much smaller than the size of the collectors) assumption of constant work function is no longer valid. Also it has been shown[12] that in a system of nanoparticles there is a distribution of the size of the nanoparticles. Since the work function is more of a property of the surface we naturally can expect that the work function of nano-particles will also have a distribution. The actual microscopic model for work function distribution for a system of nano-particles is beyond the scope of this work. Interestingly, Gamez et. al.[12] using scanning tunneling microscope (STM) have measured the pair distribution function (PDF) for Pt nano-particles and they found that it follows log-normal distribution. In this work, purely as a mathematical model, we choose Gaussian and log-normal distribution for the work function. We then study the field emision current averaged over work function distribution. The organization of the rest of the paper is as follows. In Sec. II we describe the mathematical formalism used to calculate field emission current. In Sec. III we describe the work function distribution used to calculate average current. In Sec. IV we present our results for both the case of Gaussian distribution and log-normal distribution. Finally in Sec. V we conclude. II. MATHEMATICAL FORMALISM We closely follow and summarize the results obtained by Lopes et. al.[13],[14] for field-emission current density, J. In the standard FN-type MG theory the field emission current density is given by the well-known expression[15–18], K(λ) and E(λ) are the complete elliptic integral of the first and second kind, with From the relations above it is quite evident that calculation of transmission current requires evaluation of numerical integrals for complete elliptic integrals. Due to the singularities present in complete elliptic integrals, it is very hard to extract meaningful results purely numerically[19]. Under this circumstances we can consider series expansion for complete elliptic integrals[20] as follows where ls A detailed calculation gives the following form for the field emission current density This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Nandan Pakhira, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India; (2) Rajib Mahato, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India and Central Electronics Engineering Research Institute, Pilani, Rajasthan 333031, India. Authors: Authors: (1) Nandan Pakhira, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India; (2) Rajib Mahato, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India and Central Electronics Engineering Research Institute, Pilani, Rajasthan 333031, India. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction II. Mathematical Formalism II. Mathematical Formalism III. Work Function Distribution III. Work Function Distribution IV. Results A. Gaussian work function distribution A. Gaussian work function distribution B. Log-normal work function distribution B. Log-normal work function distribution V. Conclusion, Acknowledgements, and References V. Conclusion, Acknowledgements, and References I. INTRODUCTION Field emission is the process in which electrons from cold surfaces are emitted in the presence of applied strong electric field. This process should be compared agaist the thermo-ionic process in which electrons are emitted from hot metal surface. The field emission forms the back bone of modern semiconductor devices. This effect was first described by Fowler and Nordheim[1]. They considered quantum mechanical tunneling through a triangular potential energy (PE) barrier, created by the application of a constant electric field. Much later Murphy and Good[2] (MG) introduced a more realistic PE barrier by taking into account the induced image charge formed in the presence of emitted electrons. MG calculated the barrier transmission coefficient under semi-classical WKB approximation. More recently, essentially an exact solution of the problem was obtained by Choy et. al.[3]. Also, various cases including finite temperature (thermal emission), tunneling, curvature of th. To the best of our knowledge in all of those studies authors have considered constant local work function. The work function of a material depends on the composition, structure, geometry, local charge distribution etc. of the emitting surface. Assumption of constant work function is only suitable for an atomistic smooth homogeneous surface. For surfaces with inhomogeneities over nano-scale (much smaller than the size of the collectors) assumption of constant work function is no longer valid. Also it has been shown[12] that in a system of nanoparticles there is a distribution of the size of the nanoparticles. Since the work function is more of a property of the surface we naturally can expect that the work function of nano-particles will also have a distribution. The actual microscopic model for work function distribution for a system of nano-particles is beyond the scope of this work. Interestingly, Gamez et. al.[12] using scanning tunneling microscope (STM) have measured the pair distribution function (PDF) for Pt nano-particles and they found that it follows log-normal distribution. In this work, purely as a mathematical model, we choose Gaussian and log-normal distribution for the work function. We then study the field emision current averaged over work function distribution. The organization of the rest of the paper is as follows. In Sec. II we describe the mathematical formalism used to calculate field emission current. In Sec. III we describe the work function distribution used to calculate average current. In Sec. IV we present our results for both the case of Gaussian distribution and log-normal distribution. Finally in Sec. V we conclude. II. MATHEMATICAL FORMALISM We closely follow and summarize the results obtained by Lopes et. al.[13],[14] for field-emission current density, J. In the standard FN-type MG theory the field emission current density is given by the well-known expression[15–18], K(λ) and E(λ) are the complete elliptic integral of the first and second kind, with From the relations above it is quite evident that calculation of transmission current requires evaluation of numerical integrals for complete elliptic integrals. Due to the singularities present in complete elliptic integrals, it is very hard to extract meaningful results purely numerically[19]. Under this circumstances we can consider series expansion for complete elliptic integrals[20] as follows where ls A detailed calculation gives the following form for the field emission current density 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

Unlocking the Mystery of Nano-Powered Field Emission

How DeepSee Is Changing the Way Scientists Use Data in the Field

Table of Links Abstract and 1 Introduction 2 Related Work 3 Methodology 4 Studying Deep Ocean Ecosystem and 4.1 Deep Ocean Research Goals 4.2 Workflow and Data 4.3 Design Challenges and User Tasks 5 The DeepSea System 5.1 Map View
5.2 Core View 5.3 Interpolation View and 5.4 Implementation 6 Usage Scenarios and 6.1 Scenario: Pre-Cruise Planning 6.2 Scenario: On-the-Fly Decision-Making 7 Evaluation and 7.1 Cruise Deployment 7.2 Expert Interviews 7.3 Limitations 7.4 Lessons Learned 8 Conclusions and Future Work, Acknowledgments, and References 3 METHODOLOGY In this paper, we followed Sedlmair et al.’s methodological framework for design studies [43]. Each phase of the study resulted in a contribution, namely: (1) a characterization of the problem domain via design challenges and user tasks; (2) a novel and validated visualization design; and (3) reflections on the process and lessons learned for improving visualization and interaction design guidelines. We connect these phases throughout the paper by illustrating how our synthesized challenges and tasks informed DeepSee’s design and how DeepSee’s success in addressing the challenges and tasks is reflected in our evaluation. To the best of our knowledge, the problem we describe in Sect. 4 is unique to our collaborators and has not been addressed by other researchers so far. Thus, we posit that a design study based on active participation from collaborators in both the design and evaluation phases is necessary to successfully support domain experts in their highly specialized tasks. We began by establishing a close collaboration between visualization researchers and domain experts, to properly assess the impact of our approach on solving a real-world problem. In line with prior design studies [15, 18, 33], our team comprised all authors and included visualization experts in human-centered computing, interaction design, scientific data visualization and art, as well as scientists and researchers with expertise in environmental microbiology, geochemistry and geology. We developed DeepSee through several user-centered design methodologies including contextual inquiry, mixed-fidelity prototyping, user testing, and design iteration. Over an intensive ten-week period, we worked closely together to develop a shared understanding of the importance and aims of their scientific research (Sect. 4.1), characterize data and workflows (Sect. 4.2), and identify data visualization challenges and user tasks (Sect. 4.3) that would guide our system design. Our process started with contextual inquiry, combining observation and interviews with our domain collaborators to deeply understand their existing tools and workflows, discuss unmet needs and pain points, and identify key technical constraints. After two weeks, we began mixed-fidelity prototyping, starting with paper prototypes, wireframes and small interactive prototypes in code, before moving on to progressively higher-fidelity prototypes. These prototypes were iteratively discussed with the science team over the next seven weeks through semi-structured user testing. Using these prototypes as stimuli, we deepened our understanding of the problem space through on-going contextual inquiry. During this time, systems diagrams and conceptual workflows were also created to help us validate our understanding of the problem space in collaboration with the science team. As a result of this approach, we progressively uncovered insights that guided each subsequent design iteration, leading us to generate a working prototype described in Sect. 5 that addressed our collaborators’ tasks. DeepSee was then deployed in the wild [43] to assess whether our approach solves a real problem – bridging gaps in supporting visual analysis of samples during field research. Deploying DeepSee with the same experts that we designed with allowed us to assess whether domain experts are indeed helped by the new solution, as well as reflect on and refine principles for visualization and interaction design guidelines [43]. Specifically, we investigated the impact of introducing data visualizations into both planning before fieldwork as well as tactical decision making in the field. We learned that visualizing data at multiple scales, integrating 2D and 3D data in the same interface, and modularizing visualizations positively impacted scientists planning workflows the most, and that additional support for integrating data on the fly could better enable in-situ decision making. Methodological details of our evaluation and how we synthesized these lessons learned are provided in Sect. 7. This paper is available on arxiv under CC BY 4.0 DEED license. Authors:
(1) Adam Coscia, Georgia Institute of Technology, Atlanta, Georgia, USA (acoscia6@gatech.edu);
(2) Haley M. Sapers, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, USA (hsapers@caltech.edu);
(3) Noah Deutsch, Harvard University Cambridge, Massachusetts, USA (ndeutsch@mde.harvard.edu);
(4) Malika Khurana, The New York Times Company, New York, New York, USA (malika.khurana@nytimes.com);
(5) John S. Magyar, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (jmagyar@caltech.edu);
(6) Sergio A. Parra, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (sparra@caltech.edu);
(7) Daniel R. Utter, rwipfler@caltech.edu Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (dutter@caltech.edu);
(8) John S. Magyar, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (rwipfler@caltech.edu);
(9) David W. Caress, Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (caress@mbari.org);
(10) Eric J. Martin Jennifer B. Paduan Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (emartin@mbari.org);
(11) Jennifer B. Paduan, Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (paje@mbari.org);
(12) Maggie Hendrie, ArtCenter College of Design, Pasadena, California, USA (maggie.hendrie@artcenter.edu);
(13) Santiago Lombeyda, California Institute of Technology, Pasadena, California, USA (santiago@caltech.edu);
(14) Hillary Mushkin, California Institute of Technology, Pasadena, California, USA (hmushkin@caltech.edu);
(15) Alex Endert, Georgia Institute of Technology, Atlanta, Georgia, USA (endert@gatech.edu);
(16) Scott Davidoff, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA (scott.davidoff@jpl.nasa.gov);
(17) Victoria J. Orphan, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, USA (vorphan@caltech.edu). Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Related Work 2 Related Work 3 Methodology 3 Methodology 4 Studying Deep Ocean Ecosystem and 4.1 Deep Ocean Research Goals 4 Studying Deep Ocean Ecosystem and 4.1 Deep Ocean Research Goals 4.2 Workflow and Data 4.2 Workflow and Data 4.3 Design Challenges and User Tasks 4.3 Design Challenges and User Tasks 5 The DeepSea System 5 The DeepSea System 5.1 Map View 5.2 Core View 5.1 Map View 5.2 Core View 5.3 Interpolation View and 5.4 Implementation 5.3 Interpolation View and 5.4 Implementation 6 Usage Scenarios and 6.1 Scenario: Pre-Cruise Planning 6 Usage Scenarios and 6.1 Scenario: Pre-Cruise Planning 6.2 Scenario: On-the-Fly Decision-Making 6.2 Scenario: On-the-Fly Decision-Making 7 Evaluation and 7.1 Cruise Deployment 7 Evaluation and 7.1 Cruise Deployment 7.2 Expert Interviews 7.2 Expert Interviews 7.3 Limitations 7.3 Limitations 7.4 Lessons Learned 7.4 Lessons Learned 8 Conclusions and Future Work, Acknowledgments, and References 8 Conclusions and Future Work, Acknowledgments, and References 3 METHODOLOGY In this paper, we followed Sedlmair et al.’s methodological framework for design studies [43]. Each phase of the study resulted in a contribution, namely: (1) a characterization of the problem domain via design challenges and user tasks; (2) a novel and validated visualization design; and (3) reflections on the process and lessons learned for improving visualization and interaction design guidelines. We connect these phases throughout the paper by illustrating how our synthesized challenges and tasks informed DeepSee’s design and how DeepSee’s success in addressing the challenges and tasks is reflected in our evaluation. To the best of our knowledge, the problem we describe in Sect. 4 is unique to our collaborators and has not been addressed by other researchers so far. Thus, we posit that a design study based on active participation from collaborators in both the design and evaluation phases is necessary to successfully support domain experts in their highly specialized tasks. We began by establishing a close collaboration between visualization researchers and domain experts, to properly assess the impact of our approach on solving a real-world problem. In line with prior design studies [15, 18, 33], our team comprised all authors and included visualization experts in human-centered computing, interaction design, scientific data visualization and art, as well as scientists and researchers with expertise in environmental microbiology, geochemistry and geology. We developed DeepSee through several user-centered design methodologies including contextual inquiry, mixed-fidelity prototyping, user testing, and design iteration. Over an intensive ten-week period, we worked closely together to develop a shared understanding of the importance and aims of their scientific research (Sect. 4.1), characterize data and workflows (Sect. 4.2), and identify data visualization challenges and user tasks (Sect. 4.3) that would guide our system design. Our process started with contextual inquiry, combining observation and interviews with our domain collaborators to deeply understand their existing tools and workflows, discuss unmet needs and pain points, and identify key technical constraints. After two weeks, we began mixed-fidelity prototyping, starting with paper prototypes, wireframes and small interactive prototypes in code, before moving on to progressively higher-fidelity prototypes. These prototypes were iteratively discussed with the science team over the next seven weeks through semi-structured user testing. Using these prototypes as stimuli, we deepened our understanding of the problem space through on-going contextual inquiry. During this time, systems diagrams and conceptual workflows were also created to help us validate our understanding of the problem space in collaboration with the science team. As a result of this approach, we progressively uncovered insights that guided each subsequent design iteration, leading us to generate a working prototype described in Sect. 5 that addressed our collaborators’ tasks. DeepSee was then deployed in the wild [43] to assess whether our approach solves a real problem – bridging gaps in supporting visual analysis of samples during field research. Deploying DeepSee with the same experts that we designed with allowed us to assess whether domain experts are indeed helped by the new solution, as well as reflect on and refine principles for visualization and interaction design guidelines [43]. Specifically, we investigated the impact of introducing data visualizations into both planning before fieldwork as well as tactical decision making in the field. We learned that visualizing data at multiple scales, integrating 2D and 3D data in the same interface, and modularizing visualizations positively impacted scientists planning workflows the most, and that additional support for integrating data on the fly could better enable in-situ decision making. Methodological details of our evaluation and how we synthesized these lessons learned are provided in Sect. 7. 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 Authors: (1) Adam Coscia, Georgia Institute of Technology, Atlanta, Georgia, USA (acoscia6@gatech.edu); (2) Haley M. Sapers, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, USA (hsapers@caltech.edu); (3) Noah Deutsch, Harvard University Cambridge, Massachusetts, USA (ndeutsch@mde.harvard.edu); (4) Malika Khurana, The New York Times Company, New York, New York, USA (malika.khurana@nytimes.com); (5) John S. Magyar, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (jmagyar@caltech.edu); (6) Sergio A. Parra, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (sparra@caltech.edu); (7) Daniel R. Utter, rwipfler@caltech.edu Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (dutter@caltech.edu); (8) John S. Magyar, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (rwipfler@caltech.edu); (9) David W. Caress, Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (caress@mbari.org); (10) Eric J. Martin Jennifer B. Paduan Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (emartin@mbari.org); (11) Jennifer B. Paduan, Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (paje@mbari.org); (12) Maggie Hendrie, ArtCenter College of Design, Pasadena, California, USA (maggie.hendrie@artcenter.edu); (13) Santiago Lombeyda, California Institute of Technology, Pasadena, California, USA (santiago@caltech.edu); (14) Hillary Mushkin, California Institute of Technology, Pasadena, California, USA (hmushkin@caltech.edu); (15) Alex Endert, Georgia Institute of Technology, Atlanta, Georgia, USA (endert@gatech.edu); (16) Scott Davidoff, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA (scott.davidoff@jpl.nasa.gov); (17) Victoria J. Orphan, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, USA (vorphan@caltech.edu). Authors: Authors: (1) Adam Coscia, Georgia Institute of Technology, Atlanta, Georgia, USA (acoscia6@gatech.edu); (2) Haley M. Sapers, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, USA (hsapers@caltech.edu); (3) Noah Deutsch, Harvard University Cambridge, Massachusetts, USA (ndeutsch@mde.harvard.edu); (4) Malika Khurana, The New York Times Company, New York, New York, USA (malika.khurana@nytimes.com); (5) John S. Magyar, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (jmagyar@caltech.edu); (6) Sergio A. Parra, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (sparra@caltech.edu); (7) Daniel R. Utter, rwipfler@caltech.edu Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (dutter@caltech.edu); (8) John S. Magyar, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (rwipfler@caltech.edu); (9) David W. Caress, Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (caress@mbari.org); (10) Eric J. Martin Jennifer B. Paduan Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (emartin@mbari.org); (11) Jennifer B. Paduan, Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (paje@mbari.org); (12) Maggie Hendrie, ArtCenter College of Design, Pasadena, California, USA (maggie.hendrie@artcenter.edu); (13) Santiago Lombeyda, California Institute of Technology, Pasadena, California, USA (santiago@caltech.edu); (14) Hillary Mushkin, California Institute of Technology, Pasadena, California, USA (hmushkin@caltech.edu); (15) Alex Endert, Georgia Institute of Technology, Atlanta, Georgia, USA (endert@gatech.edu); (16) Scott Davidoff, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA (scott.davidoff@jpl.nasa.gov); (17) Victoria J. Orphan, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, USA (vorphan@caltech.edu).

Making Field Research Easier with DeepSee’s Data Visualization

How DeepSee Bridges Gaps in Ocean Research Visualization for Fieldwork Decisions

The Cyber Resilience Act: A Field Guide for CTOs and CISOs

The Evolution of Apache SeaTunnel’s Technical Architecture and Its Applications in the AI Field 

In modern applications, data security is a top priority, especially when sensitive information may be copied or transferred to external applications. By preventing certain actions like "cut," "copy," and "paste" on text fields within iOS apps, we can safeguard text data, ensuring it remains within a trusted group of applications. This article demonstrates how to achieve this protection using Swift, while maintaining full control over text-related operations. Goals of This Approach Restrict External Text Sharing: By managing text actions, we can prevent unauthorized access to sensitive information.
Safe Text Storage and Management: Creating an internal buffer ensures that sensitive text remains protected within the app.
Controlled Clipboard Access: By limiting access to the system clipboard, we can ensure that only authorized applications within a defined group can access this information. Implementation Overview Our approach centers around the TextActionInterceptor class, which "swizzles" standard text-related methods, replacing them with secure alternatives. This class intercepts and overrides the "copy," "paste," and "cut" methods, with an option to enable or disable swizzling dynamically. Core Components Enum for Text Actions: The TextAction enumeration defines all the actions related to text that we want to control.
Saved Method Structure: The SavedTextAction struct stores original implementations of methods, allowing us to restore them as needed.
Swizzling Methods: The toggleSwizzling method manages the swizzling state, enabling or disabling the replacement of standard implementations. Code Example Let’s go through key code segments that bring this functionality to life. class TextActionInterceptor {

    enum TextAction: String {
        case cut
        case copy
        case paste
        case define
        case translate
        case modify
        case update
        case previewItem
    }

    struct SavedTextAction {
        var methodRef: Method
        var originalImplementation: IMP
        var selector: Selector
    }

    private var isSwizzled = false
    var savedActions: [TextAction: SavedTextAction] = [:]
    
    // Method to intercept the "copy" action
    func interceptCopy(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let text = inputField.text(in: range)
        else { return }

        SecureClipboard.shared.contents = text
    }

    // Similar methods for "cut" and "paste"
    func interceptPaste(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let text = SecureClipboard.shared.contents
        else { return }

        inputField.replace(range, withText: text)
    }

    func interceptCut(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let cutText = inputField.text(in: range)
        else { return }

        SecureClipboard.shared.contents = cutText
        inputField.replace(range, withText: "")
    }
    
    // Method to enable or disable swizzling
    func toggleSwizzling(_ enable: Bool) {
        guard enable != isSwizzled else { return }
        isSwizzled = enable
        savedActions.forEach { (_, action) in
            let swizzledMethod = class_getInstanceMethod(
                object_getClass(self),
                action.selector
            )
            guard enable else {
                method_setImplementation(action.methodRef, action.originalImplementation)
                return
            }
            method_exchangeImplementations(action.methodRef, swizzledMethod!)
        }
    }
} Implementing a Secure Clipboard The SecureClipboard class is responsible for managing text stored in the clipboard. It blocks any attempt to share text data externally, ensuring it stays within the app's defined ecosystem. final class SecureClipboard {

    static let restrictedTextPlaceholder = "Access Restricted"
    static var shared = SecureClipboard()
    private let placeholderItem: [String: String]
    private var clipboard: UIPasteboard
    private(set) var internalBuffer: String?

    init() {
        placeholderItem = ["public.utf8-plain-text": SecureClipboard.restrictedTextPlaceholder]
        clipboard = UIPasteboard.general
    }

    var contents: String? {
        get {
            guard
                let clipboardText = clipboard.string,
                clipboardText != SecureClipboard.restrictedTextPlaceholder
            else { return internalBuffer }
            return clipboardText
        }
        set {
            guard let newText = newValue else {
                internalBuffer = nil
                return
            }
            clipboard.items = [placeholderItem]
            internalBuffer = newText.isEmpty ? nil : newText
        }
    }
} How This Works Swizzling of Methods: We replace the standard text action methods with customized, secure methods.
Secure Copying: The interceptCopy method intercepts "copy" actions and transfers the text to the SecureClipboard instead of the system clipboard.
Flexible Management: Swizzling can be dynamically enabled or disabled, allowing for adaptable data security. Summary and Results Applying this approach achieves multiple levels of data security: Access Control: All text operations are managed within the application, preventing potential data leaks.


Safe Data Storage: Text data is accessible only within a controlled application group.


Flexible configuration: The ability to enable and disable swizzling allows flexible control of data security based on conditions. By using swizzling and an internal clipboard, we ensure a high level of data protection in iOS applications. This approach offers a robust solution for maintaining text security across applications in a defined environment. In modern applications, data security is a top priority, especially when sensitive information may be copied or transferred to external applications. By preventing certain actions like "cut," "copy," and "paste" on text fields within iOS apps, we can safeguard text data, ensuring it remains within a trusted group of applications. This article demonstrates how to achieve this protection using Swift, while maintaining full control over text-related operations. Goals of This Approach Restrict External Text Sharing: By managing text actions, we can prevent unauthorized access to sensitive information. Safe Text Storage and Management: Creating an internal buffer ensures that sensitive text remains protected within the app. Controlled Clipboard Access: By limiting access to the system clipboard, we can ensure that only authorized applications within a defined group can access this information. Restrict External Text Sharing : By managing text actions, we can prevent unauthorized access to sensitive information. Restrict External Text Sharing Safe Text Storage and Management : Creating an internal buffer ensures that sensitive text remains protected within the app. Safe Text Storage and Management Controlled Clipboard Access : By limiting access to the system clipboard, we can ensure that only authorized applications within a defined group can access this information. Controlled Clipboard Access Implementation Overview Our approach centers around the TextActionInterceptor class, which "swizzles" standard text-related methods, replacing them with secure alternatives. This class intercepts and overrides the "copy," "paste," and "cut" methods, with an option to enable or disable swizzling dynamically. Core Components Enum for Text Actions: The TextAction enumeration defines all the actions related to text that we want to control. Saved Method Structure: The SavedTextAction struct stores original implementations of methods, allowing us to restore them as needed. Swizzling Methods: The toggleSwizzling method manages the swizzling state, enabling or disabling the replacement of standard implementations. Enum for Text Actions: The TextAction enumeration defines all the actions related to text that we want to control. Enum for Text Actions: The TextAction enumeration defines all the actions related to text that we want to control. Saved Method Structure: The SavedTextAction struct stores original implementations of methods, allowing us to restore them as needed. Saved Method Structure: The SavedTextAction struct stores original implementations of methods, allowing us to restore them as needed. Swizzling Methods: The toggleSwizzling method manages the swizzling state, enabling or disabling the replacement of standard implementations. Swizzling Methods: The toggleSwizzling method manages the swizzling state, enabling or disabling the replacement of standard implementations. Code Example Let’s go through key code segments that bring this functionality to life. class TextActionInterceptor {

    enum TextAction: String {
        case cut
        case copy
        case paste
        case define
        case translate
        case modify
        case update
        case previewItem
    }

    struct SavedTextAction {
        var methodRef: Method
        var originalImplementation: IMP
        var selector: Selector
    }

    private var isSwizzled = false
    var savedActions: [TextAction: SavedTextAction] = [:]
    
    // Method to intercept the "copy" action
    func interceptCopy(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let text = inputField.text(in: range)
        else { return }

        SecureClipboard.shared.contents = text
    }

    // Similar methods for "cut" and "paste"
    func interceptPaste(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let text = SecureClipboard.shared.contents
        else { return }

        inputField.replace(range, withText: text)
    }

    func interceptCut(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let cutText = inputField.text(in: range)
        else { return }

        SecureClipboard.shared.contents = cutText
        inputField.replace(range, withText: "")
    }
    
    // Method to enable or disable swizzling
    func toggleSwizzling(_ enable: Bool) {
        guard enable != isSwizzled else { return }
        isSwizzled = enable
        savedActions.forEach { (_, action) in
            let swizzledMethod = class_getInstanceMethod(
                object_getClass(self),
                action.selector
            )
            guard enable else {
                method_setImplementation(action.methodRef, action.originalImplementation)
                return
            }
            method_exchangeImplementations(action.methodRef, swizzledMethod!)
        }
    }
} class TextActionInterceptor {

    enum TextAction: String {
        case cut
        case copy
        case paste
        case define
        case translate
        case modify
        case update
        case previewItem
    }

    struct SavedTextAction {
        var methodRef: Method
        var originalImplementation: IMP
        var selector: Selector
    }

    private var isSwizzled = false
    var savedActions: [TextAction: SavedTextAction] = [:]
    
    // Method to intercept the "copy" action
    func interceptCopy(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let text = inputField.text(in: range)
        else { return }

        SecureClipboard.shared.contents = text
    }

    // Similar methods for "cut" and "paste"
    func interceptPaste(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let text = SecureClipboard.shared.contents
        else { return }

        inputField.replace(range, withText: text)
    }

    func interceptCut(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let cutText = inputField.text(in: range)
        else { return }

        SecureClipboard.shared.contents = cutText
        inputField.replace(range, withText: "")
    }
    
    // Method to enable or disable swizzling
    func toggleSwizzling(_ enable: Bool) {
        guard enable != isSwizzled else { return }
        isSwizzled = enable
        savedActions.forEach { (_, action) in
            let swizzledMethod = class_getInstanceMethod(
                object_getClass(self),
                action.selector
            )
            guard enable else {
                method_setImplementation(action.methodRef, action.originalImplementation)
                return
            }
            method_exchangeImplementations(action.methodRef, swizzledMethod!)
        }
    }
} Implementing a Secure Clipboard The SecureClipboard class is responsible for managing text stored in the clipboard. It blocks any attempt to share text data externally, ensuring it stays within the app's defined ecosystem. final class SecureClipboard {

    static let restrictedTextPlaceholder = "Access Restricted"
    static var shared = SecureClipboard()
    private let placeholderItem: [String: String]
    private var clipboard: UIPasteboard
    private(set) var internalBuffer: String?

    init() {
        placeholderItem = ["public.utf8-plain-text": SecureClipboard.restrictedTextPlaceholder]
        clipboard = UIPasteboard.general
    }

    var contents: String? {
        get {
            guard
                let clipboardText = clipboard.string,
                clipboardText != SecureClipboard.restrictedTextPlaceholder
            else { return internalBuffer }
            return clipboardText
        }
        set {
            guard let newText = newValue else {
                internalBuffer = nil
                return
            }
            clipboard.items = [placeholderItem]
            internalBuffer = newText.isEmpty ? nil : newText
        }
    }
} final class SecureClipboard {

    static let restrictedTextPlaceholder = "Access Restricted"
    static var shared = SecureClipboard()
    private let placeholderItem: [String: String]
    private var clipboard: UIPasteboard
    private(set) var internalBuffer: String?

    init() {
        placeholderItem = ["public.utf8-plain-text": SecureClipboard.restrictedTextPlaceholder]
        clipboard = UIPasteboard.general
    }

    var contents: String? {
        get {
            guard
                let clipboardText = clipboard.string,
                clipboardText != SecureClipboard.restrictedTextPlaceholder
            else { return internalBuffer }
            return clipboardText
        }
        set {
            guard let newText = newValue else {
                internalBuffer = nil
                return
            }
            clipboard.items = [placeholderItem]
            internalBuffer = newText.isEmpty ? nil : newText
        }
    }
} How This Works Swizzling of Methods: We replace the standard text action methods with customized, secure methods. Secure Copying: The interceptCopy method intercepts "copy" actions and transfers the text to the SecureClipboard instead of the system clipboard. Flexible Management: Swizzling can be dynamically enabled or disabled, allowing for adaptable data security. Swizzling of Methods : We replace the standard text action methods with customized, secure methods. Swizzling of Methods Secure Copying: The interceptCopy method intercepts "copy" actions and transfers the text to the SecureClipboard instead of the system clipboard. Secure Copying: The interceptCopy method intercepts "copy" actions and transfers the text to the SecureClipboard instead of the system clipboard. Flexible Management : Swizzling can be dynamically enabled or disabled, allowing for adaptable data security. Flexible Management Summary and Results Applying this approach achieves multiple levels of data security: Access Control: All text operations are managed within the application, preventing potential data leaks. Safe Data Storage: Text data is accessible only within a controlled application group. Flexible configuration: The ability to enable and disable swizzling allows flexible control of data security based on conditions. Access Control: All text operations are managed within the application, preventing potential data leaks. Access Control : All text operations are managed within the application, preventing potential data leaks. Access Control Safe Data Storage: Text data is accessible only within a controlled application group. Safe Data Storage : Text data is accessible only within a controlled application group. Safe Data Storage Flexible configuration: The ability to enable and disable swizzling allows flexible control of data security based on conditions. Flexible configuration : The ability to enable and disable swizzling allows flexible control of data security based on conditions. Flexible configuration By using swizzling and an internal clipboard, we ensure a high level of data protection in iOS applications. This approach offers a robust solution for maintaining text security across applications in a defined environment.

Securing Text Fields in iOS Apps: Restricting 'Cut,' 'Copy,' and 'Paste' Operations Using Swift

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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