[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81501-en":3,"doc-seo-81501-105":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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81501,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","A Variational Formulation of a Multi-Population Mean Field Games with Non-local Interactions","A quadratic multi-population Mean Field Games framework is developed with non-local interaction terms defined through a given potential. The formulation yields coupled Hamilton–Jacobi–Bellman and Fokker–Planck PDE systems featuring N interacting populations. An Eulerian variational principle is introduced to handle non-convex interaction effects while still producing weak solutions. A corresponding Lagrangian entropy minimization problem is derived, and numerical solutions are obtained via a Sinkhorn-like scheme, with simulations distinguishing repulsive versus attractive non-local interactions.","arXiv :2408 .03118v4 [math .AP] 10 Jul 2026  \nA variational formulation of a Multi-Population Mean Field Games with non-local interactions  \nLuigi De Pascale∗, Luca Nenna†  \nJuly 13, 2026  \nWe propose a MFG model with quadratic Hamiltonian involving N populations. This results in a system of N Hamilton-Jacobi-Bellman and N Fokker-Planck equations with non-local interactions. As in the classical case we introduce an Eulerian variational formulation which, despite the non convexity of the interaction, still gives a weak solution to the MFG model. The problem can be reformulated in Lagrangian terms and solved numerically by a Sinkhorn-like scheme. We present numerical results based on this approach, these simulations exhibit different behaviors depending on the nature (repulsive or attractive) of the non-local interaction.  \nKeywords. Mean field games, optimal transport, multi-marginal optimal transport, entropic regularization, convex analysis.  \n2020 Mathematics Subject Classification. Primary: 49L25 ; Secondary: 49K40, 49N15, 65K10, 91A14 .  \n1. Introduction  \nThe aim of this paper is to establish a variational formulation (both Lagrangian and Eulerian), as well as a suitable numerical methods, for quadratic second-order Mean Field Games which involve N different populations interacting through a given nonlocal functional. Let d be the dimension of the space; then for i = 1 , ... , N we consider  \n∗ Dipartimento di Matematica ed Informatica, Universit´a di Firenze, Viale Morgagni, 67/a - 50134 Firenze, Italy.  \n†Universit´e Paris-Saclay, CNRS, Laboratoire de math´ematiques d’Orsay, ParMA, Inria Saclay, 91405, Orsay, [France. email:](France. email:) [luca.nenna@universite-paris-saclay.fr](luca.nenna@universite-paris-saclay.fr)  \nthe following coupled PDE system.  \n􀀼􀀸􀀾􀀾􀀾􀀾􀀾􀀾 ttiu−i −1i +di12v(|uuiiρ| 2i)==,i ZRd 􀀒 Vi,j (x − y) + Vj,i(x − y)􀀓 ρjdy,  \n􀀾 2  \n􀀺 ρi(0, x) = ρi0(x), ui(T, x) = gi(x);  \n(1.1)  \nwhere ρi0 ∈ Pac(Rd ) and gi ∈ C0 (Rd ) and Vi,j is a given positive and lower semicontinuous potential, the full set of assumptions is given at the end of this introduction. In the paper, we will mostly use the more compact notation ρ t for ρ (t, x) . Since t →7 ρtwill be an absolutely continuous curve with values in the space of probability measure equipped with a Wasserstein distance (see section 2) the initial datum ρ0 will be assumed in a straightforward sense.  \nMean-field games involving several populations have been studied recently by [20, 21, 16, 3, 9, 23, 32, 27]; for some more details about the theory of MFG we refer the reader to the seminal work by Lasry and Lions [24], the book [18] or the lecture notes by Cardaliaguet [14] . Notice that the interaction of the populations can be expressed via amore general functional, but we will not discuss here the extra difficulties, or via some Optimal Transport coupling as done in [3] . We will show that the above system can be seen as optimality conditions for the following Eulerian variational problem .  \ninf 􀀚 J (ρ1 , v 1 ; ... ;ρN , v N ) | ∂tρit − 12∆ρit + div(ρitvit) = 0, ρi(0, x) = ρi0 , ∀i􀀛  \nwhere  \nJ (ρ1 , v 1 ; ... ;ρN , v N ) :=X  \ni  \nZ[0,T]×Rd |~~ ~~vi2|~~ ~~2 dρitdt+  \n1j≤N Z0 T ZRd ×Rd 12 􀀒 Vi,j (x − y) + Vj,i(x − y)􀀓 dρit ⊗ dρjtdt + ZRd gi(x)dρi(T, x) . (1.2) j i  \nMoreover, we will also relate the above minimization problem with a Lagrangian relative entropy minimization problem, that is  \nmin{J(Q1 ,..., QN ) : eQi = ρi0}  \n(1.3)  \nwhere  \nJ (Q1 ,..., QN ) := XH (Qi|Ri) + X  \ni 1≤i≤N  \nj i  \nZ0 T ZRd ×Rd 12 􀀒 Vi,j (x − y) + Vj,i(x − y)􀀓 deQi ⊗ eQjdt  \n+ Xi  \nZΩ gi(ω(T))dQi ,  \nwhere Qi ∈ P(Ω) with Ω = C([0, T];Rd ), Ri is a well-chosen reference measure which will be defined in Section 3 and H (P |R) is the Boltzmann-Shannon entropy, that is  \nH (γ |π) = Z ρlog ρdπ, if γ = ρπ .  \nWhen the reference measure π in the entropy is non indicated, is intended the Lebesgue measure.  \nWe denote the N-uple by ρ := (ρ1 ,...,ρN ) .  \nThe existence and uniqueness of weak so","cbCaihQUcfZTkX1Z","https://ap.wps.com/l/cbCaihQUcfZTkX1Z","pdf",701585,2,1,33,"English","en",105,"# Introduction\n## Variational (Eulerian and Lagrangian) formulation\n## Coupled PDE system for multi-population games\n## Assumptions and well-posedness context\n## Numerical approach and observed behaviors","[{\"question\":\"What coupled PDE system does the proposed multi-population Mean Field Games model lead to?\",\"answer\":\"The model produces N Hamilton–Jacobi–Bellman equations and N Fokker–Planck equations, coupled through a non-local interaction functional between populations.\"},{\"question\":\"How is a variational formulation used in this work?\",\"answer\":\"An Eulerian variational formulation is introduced, and despite the interaction’s non-convexity it still yields a weak solution. The problem is also connected to a Lagrangian relative-entropy minimization formulation.\"},{\"question\":\"How are the problems solved numerically and what behaviors are reported?\",\"answer\":\"The Lagrangian problem is reformulated to enable numerical solution using a Sinkhorn-like scheme. Simulations show qualitatively different behaviors depending on whether the non-local interaction is repulsive or attractive.\"}]",1784173835,83,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-variational-formulation-of-a-multi-population-mean-field-games-with-non-local-interactions","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-variational-formulation-of-a-multi-population-mean-field-games-with-non-local-interactions/81501/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What coupled PDE system does the proposed multi-population Mean Field Games model lead to?","Question",{"text":75,"@type":76},"The model produces N Hamilton–Jacobi–Bellman equations and N Fokker–Planck equations, coupled through a non-local interaction functional between populations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is a variational formulation used in this work?",{"text":80,"@type":76},"An Eulerian variational formulation is introduced, and despite the interaction’s non-convexity it still yields a weak solution. The problem is also connected to a Lagrangian relative-entropy minimization formulation.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the problems solved numerically and what behaviors are reported?",{"text":84,"@type":76},"The Lagrangian problem is reformulated to enable numerical solution using a Sinkhorn-like scheme. Simulations show qualitatively different behaviors depending on whether the non-local interaction is repulsive or attractive.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]