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Topics in the mean-field type approach to pedestrian crowd modeling and conventions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0003-4051-5546
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 [en]
pedestrian crowds, stochastic differential equations, mean field, stochastic control, games, backward dynamics, sticky boundary, stochastic maximum principle, social conventions
Keywords [sv]
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: urn:nbn:se:kth:diva-263759ISBN: 978-91-7873-350-7 (print)OAI: oai:DiVA.org:kth-263759DiVA, id: diva2:1369577
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
List of papers
1. Mean-field type modeling of nonlocal crowd aversion in pedestrian crowd dynamics
Open this publication in new window or tab >>Mean-field type modeling of nonlocal crowd aversion in pedestrian crowd dynamics
2018 (English)In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 56, no 1, p. 434-455Article in journal (Refereed) Published
Abstract [en]

We extend the class of pedestrian crowd models introduced by Lachapelle and Wolfram [Transp. Res. B: Methodol., 45 (2011), pp. 1572–1589] to allow for nonlocal crowd aversion and arbitrarily but finitely many interacting crowds. The new crowd aversion feature grants pedestrians a “personal space” where crowding is undesirable. We derive the model from a particle picture and treat it as a mean-field type game. Solutions to the mean-field type game are characterized via a Pontryagin-type maximum principle. The behavior of pedestrians acting under nonlocal crowd aversion is illustrated by a numerical simulation.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics Publications, 2018
Keywords
Crowd aversion, Crowd dynamics, Interacting populations, Mean-field approximation, Mean-field type game, Optimal control
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-224585 (URN)10.1137/17M1119196 (DOI)000426744900020 ()2-s2.0-85043490666 (Scopus ID)
Funder
Swedish Research Council, 2016-04086
Note

QC 20180320

Available from: 2018-03-20 Created: 2018-03-20 Last updated: 2019-11-12Bibliographically approved
2. Modeling tagged pedestrian motion: A mean-field type game approach
Open this publication in new window or tab >>Modeling tagged pedestrian motion: A mean-field type game approach
2019 (English)In: Transportation Research Part B: Methodological, ISSN 0191-2615, E-ISSN 1879-2367, Vol. 121, p. 168-183Article in journal (Refereed) Published
Abstract [en]

This paper suggests a model for the motion of tagged pedestrians: Pedestrians moving towards a specified targeted destination, which they are forced to reach. It aims to be a decision-making tool for the positioning of fire fighters, security personnel and other services in a pedestrian environment. Taking interaction with the surrounding crowd into account leads to a differential nonzero-sum game model where the tagged pedestrians compete with the surrounding crowd of ordinary pedestrians. When deciding how to act, pedestrians consider crowd distribution-dependent effects, like congestion and crowd aversion. Including such effects in the parameters of the game, makes it a mean-field type game. The equilibrium control is characterized, and special cases are discussed. Behavior in the model is studied by numerical simulations.

Place, publisher, year, edition, pages
Elsevier, 2019
Keywords
Backward-forward stochastic differential equations, Congestion, Crowd aversion, Evacuation planning, Mean-field type games, Pedestrian dynamics
National Category
Transport Systems and Logistics
Identifiers
urn:nbn:se:kth:diva-246491 (URN)10.1016/j.trb.2019.01.011 (DOI)2-s2.0-85060867885 (Scopus ID)
Note

QC 20190319

Available from: 2019-03-19 Created: 2019-03-19 Last updated: 2019-11-12Bibliographically approved
3. Behavior near walls in the mean-field approach to crowd dynamics
Open this publication in new window or tab >>Behavior near walls in the mean-field approach to crowd dynamics
(English)Manuscript (preprint) (Other academic)
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-263687 (URN)
Note

QC 20191205

Available from: 2019-11-08 Created: 2019-11-08 Last updated: 2019-12-05Bibliographically approved
4. Mean-Field Type Games between Two Players Driven by Backward Stochastic Differential Equations
Open this publication in new window or tab >>Mean-Field Type Games between Two Players Driven by Backward Stochastic Differential Equations
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.

Keywords
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:nbn:se:kth:diva-248531 (URN)10.3390/g9040088 (DOI)2-s2.0-85056271574 (Scopus ID)
Note

QC 20190514

Available from: 2019-04-09 Created: 2019-04-09 Last updated: 2019-11-12Bibliographically approved
5. Stochastic stability of mixed equilibria
Open this publication in new window or tab >>Stochastic stability of mixed equilibria
(English)Manuscript (preprint) (Other academic)
National Category
Other Mathematics
Identifiers
urn:nbn:se:kth:diva-263688 (URN)
Note

QC 20191122

Available from: 2019-11-08 Created: 2019-11-08 Last updated: 2019-11-22Bibliographically approved

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23456785 of 18
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