[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-123111-en":3,"doc-seo-123111-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},123111,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","A Machine Learning Method for Stackelberg Mean Field Games - Paper Summary","A Machine Learning Method for Stackelberg Mean Field Games develops a single-level numerical framework for Stackelberg mean field game problems. It reformulates the intrinsically bi-level interaction between a principal and an agent mean field into a mean field optimal control problem using penalization, then proves convergence to the original formulation. The work further introduces a machine learning solution based on feed-forward and recurrent neural networks, demonstrating performance on multiple literature examples. ","arXiv :2302 . 10440v2 [math .OC] 23 Apr 2024  \nA Machine Learning Method for Stackelberg Mean Field Games  \nG¨ok¸ce Dayanıklı∗ Mathieu Lauri`ere †  \nAbstract  \nWe propose a single-level numerical approach to solve Stackelberg mean field game (MFG) problems. In Stackelberg MFG, an infinite population of agents play a non-cooperative game and choose their controls to optimize their individual objectives while interacting with the principal and other agents through the population distribution. The principal can influence the mean field Nash equilibrium at the population level through policies, and she optimizesher own objective, which depends on the population distribution. This leads to a bi-level problem between the principal and mean field of agents that cannot be solved using traditional methods for MFGs. We propose a reformulation of this problem as a single-level mean field optimal control problem through a penalization approach. We prove convergence of thereformulated problem to the original problem. We propose a machine learning method based on (feed-forward and recurrent) neural networks and illustrate it on several examples from the literature.  \nKeywords. mean field games, Stackelberg equilibrium, Nash equilibrium, contract theory, deep learning  \nAMS subject classifications. 49N90, 91A13, 91A15, 62M45 .  \n1 Introduction.  \nIn policy making, finding optimal policies to solve socioeconomic problems is the ultimate goal. However, this problem has an underlying complexity: Individualistic nature of humankind prevents policymakers to directly control the behavior of people. Instead, the policymakers should take into account the reaction of the society – consisting of non-cooperative agents – to a policy while deciding on the best one. One approach to understand the emergent reactions to any given policy can be to use simulation techniques such as agent-based simulation, which is commonly used for modeling complex interactions among agents. However, it lacks the tractability of the solutions. This lack of tractability issue prevents us from solving for optimal policies. Instead, with this approach only the outcomes of different policies can be compared through simulations. Therefore, in order to attain tractability of the solutions when there are many agents interacting with each other, we can utilize game theoretical tools such as mean field games (MFGs) . Intuitively, in the MFG setup, we focus on a game among a large number of indistinguishable agents (i.e., players) that are interacting symmetrically. Then, we study the equilibrium  \n∗ Department of Statistics, University of Illinois at Urbana-Champaign, Champaign, IL 61820, USA ([gokced@illinois.edu](gokced@illinois.edu)).  \n†Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning; NYU-ECNU Institute of Mathematical Sciences, NYU Shanghai, Shanghai, 200126, People’s Republic of China ([mathieu.lauriere@nyu.edu](mathieu.lauriere@nyu.edu)).  \nML for Stackelberg MFG  \nbetween a representative player and the distribution of the other players’ states (and possibly actions) instead of focusing on the interactions of every player with each other. With this simplification, we can characterize an approximate Nash equilibrium in the society given an policy (i.e., incentive or contract) by using forward-backward differential equations. Even if this forward-backward system could be hard to analyze, mean field equilibria are simpler to identify and compute than equilibria of populations with finite but large number of agents because of the curse of dimensionality that results from the exponentially increasing number of interactions between the agents when the number of agents increases. Therefore, the mean field equilibria provide approximate Nash equilibria for games with a large but finite number of players.  \nAfter finding a tractable solution for the non-cooperative game (i.e., Nash equilibrium) of large number of agents given any policy by using MFG appr","cbCaitp5pt7hWAVe","https://ap.wps.com/l/cbCaitp5pt7hWAVe","pdf",1070869,1,47,"English","en",105,"# Introduction\n## Mean field games and tractability challenges\n## Nash equilibrium computation via forward-backward equations\n## From standard MFG to Stackelberg MFG (principal + agents)\n## Single-level reformulation and learning-based solution","[{\"question\":\"What problem does the paper address in Stackelberg mean field games?\",\"answer\":\"It addresses the difficulty of solving a bi-level Stackelberg MFG, where the principal influences the mean field Nash equilibrium and optimizes an objective depending on the population distribution.\"},{\"question\":\"How does the proposed method convert the bi-level problem into a single-level one?\",\"answer\":\"It reformulates the Stackelberg MFG as a single-level mean field optimal control problem using a penalization approach.\"},{\"question\":\"What machine learning model is used to solve the reformulated problem?\",\"answer\":\"The method uses feed-forward and recurrent neural networks and illustrates results on several examples from the literature.\"}]","A Machine Learning Method for Stackelberg Mean Field Games - 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