1 Pedestrian macroscopic models: game-theoretic vs mechanistic viewpoints Pierre Degond Department of Mathematics, Imperial College London pdegond@imperial.ac.uk (see http://sites.google.com/site/degond/) ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
’PEDIGREE’ ANR Collaboration 2 Toulouse Math. Institute (IMT) P. Degond, J. Fehrenbach, J. Hua S. Motsch (ASU) Animal Cognition Lab, Toulouse (CRCA) M. Moussaid (Berlin), G. Theraulaz, M. Moreau Theoretical Physics Lab, Orsay (LPT) C. Appert-Rolland, A. Jelic (ICTP) INRIA project Lagadic, Rennes S. Donikian, S. Lemercier, J. Pettr´ e ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Summary 3 1. Issues & context 2. The Heuristic-Based Model (HBM) 3. Mean-field models 4. Macroscopic model 5. Relation to game theory 6. Conclusion ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
4 1. Issues & context ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Issues 5 Safety Avoid crowd disasters e.g. Duisburg love parade Cambodia water festival Demonstration control Design, comfort, efficiency Terminals, shopping malls, etc. ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Pedestrian models 6 Individual-Based Models (IBM) Each individual followed in time Social force model [Helbing & Molnar, Phys. Rev. E51, 1995] Analogy with physics: Attractive/repulsive forces others ... Cellular automata [Burstedde et al, Physica A 295, 2001] ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Pedestrian models 7 Macroscopic models Inspired by gas kinetics [Henderson, Transp. Res. 8, 1974] Static/dynamic field ( ∼ chemotaxis) [Hughes et al, Transp. Res. B36, 2002] Inspired from road traffic [Colombo et al, MMAS 28, 2005] ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
8 2. The Heuristic-Based Model (HBM) ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Heuristic-Based Model (HBM) 9 [Moussa¨ ıd, Helbing, Theraulaz, PNAS 2011] ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Experiments 10 Motion capture system Sensors reflect infra-red light Reflection point camera recorded Triangulation → coordinates Circular arena Avoids boundary effects ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Experiments vs model 11 Lane formation Lane definition by clustering method Cluster lifetime statistics p ( t ) dt = probability that lifetime ∈ [ t, t + dt ] p ( t ) = p 0 e at k , Streched exponential k = 0 . 4 In insert: results of model (See [Moussaid et al, PlosCB 2012]) ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Perception phase 12 Pedestrians have constant speed Evaluation assumes pedestrians move on straight lines x j Distance to Interaction (DTI) v j Minimal Distance (MD) x i int DTI x i v i MD In case of multiple encounters x j int x j Take the minimal DTI v j DTI ( k ) DTI ( j ) x i v i v k x k ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Decision phase 13 Optimisation: Discrete time step New cruising direction u ′ chosen such that Estimation X E ( u ′ ) minimizes distance to target X T � X E ( w ) − X T � 2 among test directions w Target direction a New cruising direction u ′ X T X E ( u ′ ) Closest distance between X T and the curve w → X E ( w ) Direction w X E ( w ) u The vision cone C u ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Time continuous model 14 N Particles (pedestrians) i = 1 , . . . , N Position x i ( t ) , velocity u i ( t ) , Target direction a i ( t ) i.e. u i , a i ∈ S 1 with | u i ( t ) | = 1 , | a i ( t ) | = 1 , x i = cu i , ˙ √ du i = F i dt + P u ⊥ i ( 2 d ◦ dB i ( t )) Speed c , noise intensity d , Stratonowich sense ◦ Force F i ⊥ u i , i maintains | u i | = 1 P u ⊥ ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Force 15 Test velocity directions w ∈ S 1 → Potential Φ i ( w, t ) Φ i ( w, t ) = k 2 | D i ( w ) w − La i | 2 Reaction rate k , horizon L D i ( w ) maximal walkable distance in direction w Force F i ( t ) defined by steepest descent of Φ i F i ( t ) = −∇ w Φ i ( u i ( t ) , t ) ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Maximal walkable distance D i ( w ) 16 DTI of ’ i ’ against ’ j ’ when ’ i ’ walks in direction w : D ij ( w ) D i ( w ) = ” min ” D ij ( w ) j For continuum model, replace ’min’ by average e.g. harmonic average in some interaction region xj vj vj − vi xi vi MD = DTI = this distance this distance ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
17 3. The Mean-Field Model ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Mean-Field Model 18 x ∈ R 2 , u , a ∈ S 1 Distribution function f ( x, u, a, t ) Probability to find pedestrians at x with velocity u and target velocity a at time t ∂ t f + ∇ x · ( cuf ) + ∇ u · ( Ff ) = d ∆ u f F = −∇ w Φ ( x,a,t ) ( u ) Φ ( x,a,t ) ( w ) = k 2 | D ( x,t ) ( w ) w − La | 2 D ( x,t ) ( w ) walkable distance of subject at x in direction w : functional of f ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Case: local interactions / no blind zone 19 Supposes interaction region ”very small” � ( v,b ) ∈ T 2 K ( | v − w | ) f ( x, v, b, t ) dv db D − 1 ( x,t ) ( w ) = � ( v,b ) ∈ T 2 f ( x, v, b, t ) dv db where K is analytically known (related to the DTI) If blind zone, K = K ( u, | v − w | ) Then D = D ( x,u,t ) ( x ) and Φ = Φ ( x,u,a,t ) ( w ) Dependence of Φ on u problematic Subsequent macroscopic theory cannot be developed Other closures can be done ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
20 4. Macroscopic model ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Hydrodynamic scaling 21 Let D ( u ) be arbitrary and define Q D ( f ) = −∇ u · ( F D f ) + d ∆ u f Φ D ( u, a ) = k 2 | D ( u ) u − La | 2 F D ( u, a ) = −∇ u Φ D ( u, a ) , For f ( u, a ) arbitrary, define � ( v,b ) ∈ T 2 K ( | v − u | ) f ( v, b ) dv db D − 1 f ( u ) = � ( v,b ) ∈ T 2 f ( x, v, b, t ) dv db Then mean-field model can be written ∂ t f + ∇ x · ( cuf ) = 1 εQ D f ( f ) ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
’Generalized’ Von-Mises (GVM) distributions 22 For given D ( u ) , solutions f of Q D ( f ) = 0 are of the form f ( u, a ) = ρ ( a ) M D ( u, a ) with ρ ( a ) arbitrary and 1 − Φ D ( u, a ) � � M D ( u, a ) = Z D ( a ) exp d Direction of U Direction of the global minimum of Φ D where Z D ( a ) is s.t. uy � M D ( u, a ) du = 1 LTE distribution MD ( u ) ux Direction of a local minimum of Φ D Potential Φ D ( u ) ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Equilibria 23 Solutions f of Q D f ( f ) = 0 : are GVM f = ρ ( a ) M D ( u, a ) such that D = D ρM D Leads to a fixed point equation � ( v,b ) ∈ T 2 K ( | v − u | ) ρ ( b ) M D ( v, b ) dv db D − 1 ( u ) = � ( v,b ) ∈ S 1 ρ ( b ) db Mathematical theory open Here we assume that for any function ρ ( a ) : there exists a ’distinguished’ solution D ρ ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Hydrodynamic limit 24 When ε → 0 , formally we have f ε → ρ ( x,t ) ( a ) M D ρ ( x,t ) ( u, a ) where ρ ( x,t ) ( a ) satisfies the continuity eq. ∂ t ρ ( x,t ) ( a ) + ∇ x · ( cρ ( x,t ) ( a ) U ρ ( x,t ) ( a )) = 0 and U ρ ( x,t ) ( a ) is the mean equilibrium velocity � U ρ ( a ) = u ∈ S 1 M D ρ ( u, a ) u du ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
25 5. Relation to game theory ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Game definition 26 Spatially homogeneous case: For probability f ( u, a ) , introduce the ’cost function’ µ f ( u, a ) = Φ D f ( u, a ) + d ln f ( u, a ) Non-cooperative anonymous game with a continuum of players (aka ’Mean-Field Game [Lasry & Lions]) each pedestrian (player) tries to minimize its cost by acting on its own decision variable u ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
Nash equilibrium 27 f NE is a Nash Equilibrium if No player can reduce its cost by acting on its control variable u f NE is a Nash Equilibrium iff ∃ K s.t. µ f NE ( u, a ) = K , ∀ ( u, a ) ∈ Supp ( f NE ) ∀ ( u, a ) ∈ T 2 µ f NE ( u, a ) ≥ K , The following statements are equivalent: f is an equilibrium of the kinetic model and is therefore a GVM distribution f is a Nash Equilibrium for the Mean-Field Game defined by cost function µ f ↑ Pierre Degond - Pedestrians macroscopic models - ICERM, Brown Univ., 21/08/2017 ↓
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