ON MEAN FIELD GAMES. Pierre-Louis LIONS. Collège de France, Paris. (joint project with Jean-Michel LASRY)

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1 Collège de France, Paris (joint project with Jean-Michel LASRY)

2 I INTRODUCTION II A REALLY SIMPLE EXAMPLE III GENERAL STRUCTURE IV TWO PARTICULAR CASES V OVERVIEW AND PERSPECTIVES

3 I. INTRODUCTION New class of models for the average (Mean Field) behavior of small agents (Games) started in the early 2000 s by J-M. Lasry and P-L. Lions. Requires new mathematical theories. Numerous applications : economics, finance, social networks, crowd motions... Independent introduction of a particular class of MFG models by M. Huang, P.E. Caines and R.P. Malhamé in A research community in expansion : mathematics, economics, finance. Some written references but most of the existing mathematical material to be found in the Collège de France videotapes (4 18h) that can be downloaded...!

4 Combination of Mean Field theories (classical in Physics and Mechanics) and the notion of Nash equilibria in Games theory. Nash equilibria for continua of small players : a single heterogeneous group of players (adaptations to several groups... ). Interpretation in particular cases (but already general enough!) like process control of McKean-Vlasov... Each generic player is rational i.e. tries to optimize (control) a criterion that depends on the others (the whole group) and the optimal decision affects the behavior of the group (however, this interpretation is limited to some particular situations... ). Huge class of models : agents particles, no dep. on the group are two extreme particular cases.

5 II. A REALLY SIMPLE EXAMPLE Simple example, not new but gives an idea of the general class of models (other simple exs later on). E metric space, N players (1 i N) choose a position x i E according to a criterion F i (X ) where X = (x 1,..., x N ) E N. Nash equilibrium : X = ( x 1,..., x N ) if for all 1 i N x i min over E of F i ( x 1,..., x i 1, x i, x i+1,... x N ). Usual difficulties with the notion N? simpler? Indistinguishable players : F i (X ) = F (x i, {x j } j i ), F sym. in (x j ) j i

6 Part of the mathematical theories is about N : F i = F (x, m) x E, m P(E) where x = x i, m = 1 N 1 Thm : Nash equilibria converge, as N, to solutions of (MFG) x Supp m, F (x, m) = inf y E j i Facts : i) general existence and stability results ii) uniqueness if (m F (, m)) monotone δ xj F (y, m) iii) If F = Φ (m), then (min Φ) yields one solution of MFG. P(E)

7 ( ) Example : E = R d #{j/ xi x j < ε}, F i (X ) = f (x i ) + g (N 1) B ε g aversion crowds, g like crowds F (x, m) = f (x) + g(m 1 Bε (x)( B ε 1 ) (MFG) ε 0 supp m Arg min F (x, m) = f (x) + g(m(x)) ( ) f (x) + g(m(x)) g uniqueness, g non uniqueness { min fm + } Z G(m)/m P(E), G = f (s)ds 0 m = 1 explicit solution if g : m = g 1 (λ f ), λ R s.t.

8 III. GENERAL STRUCTURE Particular case : dynamical problem, horizon T, continuous time and space, Brownian noises (both indep. and common), no intertemporal preference rate, control on drifts (Hamiltonian H), criterion dep. only on m U(x, m, t) (x R d, m P(R d ) or M + (R d ), t [0, T ] and H(x, p, m) (convex in p R d ) MFG master equation U (ν + α) xu + H(x, x U, m)+ + U H m, (ν + α) m + div ( p m) + α U ( m, m) + 2α m 2 m xu, m = 0 and U t=0 = U 0 (x, m) (final cost) ν amount of ind. rand., α amount of common rand.

9 d problem! If ν = 0 (ind) : Nash N special case using x = x i, m = 1 N 1 δ x Ñj j i Aggregation/decentralization : IF H(x, p, m) = H(x, p) + F (m) and U 0 = Φ Φ 0 (m), then U = m solves MFG if Φ solves HJB on P(E) for the optimal control of a SPDE Particular case : many extensions and variants...

10 IV. TWO PARTICULAR CASES d problem in general but reductions to finite d in two cases 1. Indep. noises (α = 0) int. along caract. in m yields where m is given u ν u + H(x, u, m) = 0 (MFGi) u t=0 = U 0 (x, m(0)), m t=t = m m H + ν m + div ( p m) = 0 FORWARD BACKWARD system! contains as particular cases : HJB, heat, porous media, FP, V, B, Hartree, semilinear elliptic, barotropic Euler...

11 (MFGf) 2. Finite state space (i i k) U + (F (x, U). ) U = G(x, U), U t=0 = U 0 (no common noise here to simplify... ) x R k, U R k, F and G : R 2k R k non-conservative hyperbolic system Example : If F = F (U) = H (U), G 0 and if U 0 = ϕ 0 (ϕ 0 R) then solve HJ ϕ + H( ϕ) = 0, ϕ t=0 = ϕ 0 take U = ϕ, U solves (MFGf) in this case

12 V. OVERVIEW AND PERSPECTIVES Lots of questions, partial results exist, many open problems Existence/regularity : (MFGi) simple if H smooth in m (or if H almost linear... ), OK if monotone (Zoom 1) (MFGf) OK if (G, F ) mon. on R 2k or small time (Zoom 2) Uniqueness : OK if monotone or T small... Non existence, non uniqueness, non regularity (!) Qualitative properties, stationary states and stability, comparison, cycles... N (see above) Numerical methods (currently, 3 general methods and some particular cases) Variants : other noises, several populations... random heterogeneity, partial info... applications (MFG Labs... )

13 intertemporal preference rates (+λ effective models macroscopic limits? Beyond MFG? (fluctuations, LD, transitions) Two more S. examples : at which time will the meeting start? the (mexican) wave

14 ZOOM 1 u ν u + H(x, u) = f (x, m) (MFGi) u t=0 = U 0 (x), m t=t = m m H + ν m + div ( p m) = 0 m f (, m) smoothing operator regular solution uniqueness if operator monotone or if T small f (m(x)) :! regular solution ν > 0 f (m(x)) : if ν = 0 m = f 1 ( u + H(x, u)) equation in m becomes quasilinear elliptic equation of second order (x Q, t [0, T ]) with elliptic boundary conditions u t=0 = U 0 (x), u + H( u) = f ( m) if t = T

15 ZOOM 2 (MFGf) { u + (F (x, U). ) U = G(x, U) x Rd U R d, U t=0 = U 0 (x) shocks (discontinuities of U) in finite time in general well-posed problem on [0, T max ) (T max + )!regular solution monotone in x if U 0 monotone and (G, F ) monotone of R 2,k in R 2k (+...) + change of unknown functions : ex. : U + (F (U). )U = 0

16 then V = F (U) solves max class of regularity V + (V. )V = 0 δ > 0, inf dist(sp(dv 0(x)), (, δ]) > 0 x R d (V 0 = F (U 0 ) gives the maximum class of regularity composed of 2 monotone applications) Rem. : gives new results of regularity for Hamilton-Jacobi equations of the first order.