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Mean-Field Type Games between Two Players Driven by Backward Stochastic Differential Equations
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
2018 (English)In: Games, ISSN 2073-4336, E-ISSN 2073-4336, Vol. 9, no 5Article in journal (Refereed) Published
Abstract [en]

In this paper, mean-field type games between two players with backward stochastic dynamics are defined and studied. They make up a class of non-zero-sum, non-cooperating, differential games where the players’ state dynamics solve backward stochastic differential equations (BSDE) that depend on the marginal distributions of player states. Players try to minimize their individual cost functionals, also depending on the marginal state distributions. Under some regularity conditions, we derive necessary and sufficient conditions for existence of Nash equilibria. Player behavior is illustrated by numerical examples, and is compared to a centrally planned solution where the social cost, the sum of playercosts, is minimized. The inefficiency of a Nash equilibrium, compared to socially optimal behavior, is quantified by the so-called price of anarchy. Numerical simulations of the price of anarchy indicate how the improvement in social cost achievable by a central planner depends on problem parameters.

Place, publisher, year, edition, pages
2018. Vol. 9, no 5
Keywords [en]
Backward stochastic differential equations; Cooperative game; Linear-quadratic stochastic control; Mean-field type game; Non-zero-sum differential game; Price of anarchy; Social cost
National Category
Probability Theory and Statistics Other Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-248531DOI: 10.3390/g9040088Scopus ID: 2-s2.0-85056271574OAI: oai:DiVA.org:kth-248531DiVA, id: diva2:1303364
Note

QC 20190514

Available from: 2019-04-09 Created: 2019-04-09 Last updated: 2019-11-12Bibliographically approved
In thesis
1. Topics in the mean-field type approach to pedestrian crowd modeling and conventions
Open this publication in new window or tab >>Topics in the mean-field type approach to pedestrian crowd modeling and conventions
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of five appended papers, primarily addressingtopics in pedestrian crowd modeling and the formation of conventions.The first paper generalizes a pedestrian crowd model for competingsubcrowds to include nonlocal interactions and an arbitrary (butfinite) number of subcrowds. Each pedestrian is granted a ’personalspace’ and is effected by the presence of other pedestrians within it.The interaction strength may depend on subcrowd affinity. The paperinvestigates the mean-field type game between subcrowds and derivesconditions for the reduction of the game to an optimization problem.The second paper suggest a model for pedestrians with a predeterminedtarget they have to reach. The fixed and non-negotiablefinal target leads us to formulate a model with backward stochasticdifferential equations of mean-field type. Equilibrium in the game betweenthe tagged pedestrians and a surrounding crowd is characterizedwith the stochastic maximum principle. The model is illustrated by anumber of numerical examples.The third paper introduces sticky reflected stochastic differentialequations with boundary diffusion as a means to include walls andobstacles in the mean-field approach to pedestrian crowd modeling.The proposed dynamics allow the pedestrians to move and interactwhile spending time on the boundary. The model only admits a weaksolution, leading to the formulation of a weak optimal control problem.The fourth paper treats two-player finite-horizon mean-field typegames between players whose state trajectories are given by backwardstochastic differential equations of mean-field type. The paper validatesthe stochastic maximum principle for such games. Numericalexperiments illustrate equilibrium behavior and the price of anarchy.The fifth paper treats the formation of conventions in a large populationof agents that repeatedly play a finite two-player game. Theplayers access a history of previously used action profiles and form beliefson how the opposing player will act. A dynamical model wheremore recent interactions are considered to be more important in thebelief-forming process is proposed. Convergence of the history to acollection of minimal CURB blocks and, for a certain class of games,to Nash equilibria is proven.

Abstract [sv]

Den här avhandlingen består av fem artiklar som behandlar några utvalda problem inom matematisk modellering av folkmassors rörelse och uppkomsten av konventioner. Den första artikeln generaliserar en modell för växelverkan mellan grupper av fotgängare. Varje fotgängare (agent) ges ett ’personligtutrymme’ och påverkas av andra agenter som befinner sig i dess utrymme. I artikeln analyseras situationen som ett matematiskt spel av medelfältstyp och villkor för när spelet kan reduceras till ett optimeringsproblem härleds.I den andra artikeln modelleras fotgängare med ett mål som de är tvungna att nå efter en bestämd (ändlig) tid. Detta ej förhandlingsbara mål leder oss till stokastiska differentialekvationer med ändvillkor. Med den stokastiska maximumprincipen härleds nödvändiga villkor för jämvikt i ett matematisk spel där fotgängarna och en omgivande folkmassa växelverkar i tävlan om den bästa färdvägen. Modellen illustreras med flera numeriska exempel. I den tredje artikeln introducerar vi reflekterande stokastiska differentialekvationer med limaktiga randvillkor och randdiffusion som ett verktyg för att modellera hur fotgängaren påverkas av väggar och andra fasta hinder. Den föreslagna dynamiska modellen tillåter fotgängarna att spendera tid vid väggar och då också växelverka med omgivningen. Ekvationerna kan endast lösas i en svag mening och därför formuleras modellen som ett styrproblem för fotgängarnas statistiska fördelning. Artikel fyra behandlar ett spel av medelfältstyp med två spelare vars tillstånd beskrivs av ett system av stokastiska differentialekvationer med ändvillkor. Med den stokastiska maximumprincipen härleds nödvändiga villkor för spelets jämvikt och en numerisk simulering visar på skillnaden i utfall mellan konkurrens och samarbete, alltså mellanspelet och en relaterad styrmodell. Den femte artikeln handlar om uppkomsten av konventioner i en stor population av agenter som upprepade gånger spelar ett ändligt spel med två roller. När agenterna ska välja strategi har de en historik av tidigare spelade strategier till hjälp. Artikeln introducerar en speldynamik där den senare historiken antas vara viktigare än den tidigare. Vi bevisar konvergens av historiken till strategier i minimala CURBblock och, för en specifik klass av spel, till Nashjämvikter.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2019. p. 245
Series
TRITA-SCI-FOU ; 2019;47
Keywords
pedestrian crowds, stochastic differential equations, mean field, stochastic control, games, backward dynamics, sticky boundary, stochastic maximum principle, social conventions, folkmassor, stokastiska differentialekvationer, medelfält, stokastisk styrning, limaktiga randvillkor, stokastiska maximumprincipen, dynamik med ändvillkor, spel, konventioner
National Category
Probability Theory and Statistics Other Mathematics
Identifiers
urn:nbn:se:kth:diva-263759 (URN)978-91-7873-350-7 (ISBN)
Public defence
2019-12-16, Kollegiesalen, Brinellvägen 8, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20191112

Available from: 2019-11-12 Created: 2019-11-12 Last updated: 2019-12-04Bibliographically approved

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