Mean Field Games and Stochastic Growth Modeling Minyi Huang School - PowerPoint PPT Presentation

Brief overview Stochastic Growth Capital Accumulation Game Continuous Time Mean Field Games and Stochastic Growth Modeling Minyi Huang School of Mathematics and Statistics Carleton University, Ottawa Conference on Optimization, Transportation and Equilibrium in Economics Fields Institute, September 2014 Minyi Huang Mean Field Games and Stochastic Growth Modeling

Brief overview Stochastic Growth Mean field games and stochastic growth Capital Accumulation Game Continuous Time Background ◮ Mean field games: Competitive decision with a large no. of agents ◮ “An interacting N -particle system”. Then let N → ∞ ◮ Caines, Huang, and Malham´ e (03, 06, ...); Lasry and Lions (06, 07, ...); an overview by Bensoussan et. al. (2012); Buckdahn et. al. (2011); a survey by Gomes and Sa´ ude (2013) ◮ Early ideas in economic literature: Jovanovic and Rosenthal (Anonymous sequential games, 1988); continuum population modeling, finite MDP ◮ Stochastic growth theory ◮ Optimal control of a whole sector of an economy ◮ The pioneering work (Brock and Mirman, J. Econ. Theory, 1972); a nice survey (Olson and Roy, 2006) ◮ Continuous time (Merton, 1975) ◮ More generally: Nash games of N producers (e.g., Amir, Games Econ. Behav., 1996). Example: several firms in the fishery industry Minyi Huang Mean Field Games and Stochastic Growth Modeling

Brief overview Stochastic Growth Mean field games and stochastic growth Capital Accumulation Game Continuous Time The start of growth theory: deterministic root Frank Ramsey (1903-1930) ◮ F. P. Ramsey. A mathematical theory of saving. The Economic Journal , vol. 38, no. 152, pp. 543-559, 1928. Minyi Huang Mean Field Games and Stochastic Growth Modeling

Brief overview Stochastic Growth Mean field games and stochastic growth Capital Accumulation Game Continuous Time Early motivation in engineering ◮ N wireless users; x i : channel gain (in dB); p i : power. Continuous time channel modeling: Charalambous et al (1999) ◮ objective for SIR (signal-to-interference ratio): e x i p i j ̸ = i e x j p j + σ 2 ≈ γ target ∑ α N σ 2 : thermal noise; 1 N is due to using a spreading gain whose length is proportional to the user number ◮ Dynamic game dx i = a ( µ − x i ) dt + CdW i dp i = u i dt ∫ T {[ ] 2 } + ru 2 e x i p i − γ target ( α j ̸ = i e x j p j + σ 2 ) J i = E ∑ dt i N 0 Minyi Huang Mean Field Games and Stochastic Growth Modeling

Brief overview Stochastic Growth Mean field games and stochastic growth Capital Accumulation Game Continuous Time Early motivation from engineering Nonlinear dynamic game dx i = a ( µ − x i ) dt + CdW i dp i = u i dt ∫ T {[ ] 2 } + ru 2 e x i p i − γ target ( α k ̸ = i e x k p k + σ 2 ) J i = E ∑ dt i N 0 = ⇒ Linear-Quadratic-Gaussian mean field game theory dx i = ( a i x i + bu i ) dt + CdW i  2    ∫ T   x i − γ ( 1    ∑ + ru 2 J i = E x j + η ) dt  i N 0  j ̸ = i    Even such a simple model is interesting enough! (HCM’03, 04, 07) Minyi Huang Mean Field Games and Stochastic Growth Modeling

Brief overview Stochastic Growth Mean field games and stochastic growth Capital Accumulation Game Continuous Time Early motivation from engineering Linear-Quadratic-Gaussian mean field game theory dx i = ( a i x i + bu i ) dt + CdW i ∫ T { [ x i − γ ( 1 x j + η )] 2 + ru 2 ∑ J i = E i } dt N 0 j ̸ = i Fundamental issues: ◮ Existing theory yields Nash strategies of the form u i ( t , x 1 , . . . , x N ) ◮ Informational requirement is too high! ◮ Hope to design strategies of the form u i ( t , “local state” x i , “macoroscopic effect”) ◮ How well such decentralized strategies perform in the original N player game? Minyi Huang Mean Field Games and Stochastic Growth Modeling

Brief overview Stochastic Growth Mean field games and stochastic growth Capital Accumulation Game Continuous Time Mean field game: one against the MASS Mass influence u z i i Play against mass i m(t) ◮ Everyone plays against m t (freeze it!), giving optimal responses ◮ m t can appear as a measure, first order statistic (mean), etc. ◮ The optimal responses regenerate m t when no. of players N → ∞ Minyi Huang Mean Field Games and Stochastic Growth Modeling

Brief overview Stochastic Growth Mean field games and stochastic growth Capital Accumulation Game Continuous Time The basic framework of MFGs P 0 — Game with N players ; Example   dx i = f ( x i , u i , δ ( N ) Coupled Hamilton - Jacobi - Bellman system  ) dt + σ ( · · · ) dw i  solution  x u i = u i ( t , x 1 , . . . , x N ) , 1 ≤ i ≤ N ∫ T −− → 0 l ( x i , u i , δ ( N ) J i ( u i , u − i ) = E ) dt x Centralized strategy !   δ ( N )  : empirical distribution of ( x j ) N  x j =1 ↓ construct ↖ performance ? ↓ N → ∞  u i ( t , x i ) : optimal response ˆ   HJB ( v ( T , · ) given ) : P ∞ — Limiting problem , 1 player     − v t = inf u i ( f T v x i + l + 1 2 Tr [ σσ T v x i x i ])   dx i = f ( x i , u i , µ t ) dt + σ ( · · · ) dw i   solution   ∫ T −− → Fokker - Planck - Kolmogorov : ¯ J i ( u i ) = E 0 l ( x i , u i , µ t ) dt   p t = − div ( fp ) + ∑ (( σσ T Freeze µ t , as approx . of δ ( N )   ) jk p ) x j   x  2 i x k   i  Coupled via µ t ( w . density p t , p 0 given )  ◮ The consistency based approach (red) is more popular; related to ideas in statistical physics (McKean-Vlasov equation), FPK may appear as MV-SDE ◮ When a major player or common noise appears, new tools (stochastic mean field dynamics, master equation, etc) are needed Minyi Huang Mean Field Games and Stochastic Growth Modeling

Brief overview Stochastic Growth Mean field games and stochastic growth Capital Accumulation Game Continuous Time Further major issues ◮ Major-minor players instead of peers in the mean field game ◮ Motivation: institutional traders, large corporations, power generators (with respect to residential consumers), etc ◮ Mean field teams (cooperative social optimization) instead of games ◮ Robustness with model uncertainty ◮ . . . Minyi Huang Mean Field Games and Stochastic Growth Modeling

Brief overview Stochastic Growth Mean field games and stochastic growth Capital Accumulation Game Continuous Time Application of MFGs to economic growth, finance, ... ◮ Gu´ eant, Lasry and Lions (2011): human capital optimization ◮ Lucas and Moll (2011): Knowledge growth and allocation of time (JPE in press) ◮ Carmona and Lacker (2013): Investment of n brokers ◮ Espinosa and Touzi (2013): Optimal investment with relative 1 performance concern (depending on ∑ j ̸ = X j ) N − 1 ◮ Chan and Sircar (2014): Bertrand and Cournot MFGs (coupling via average prices or quantities) ◮ Jaimungal (2014): Optimal execution with major-minor agents in trading (liquidation). ◮ ...... Minyi Huang Mean Field Games and Stochastic Growth Modeling

Brief overview Stochastic Growth Mean field games and stochastic growth Capital Accumulation Game Continuous Time Organization of the talk ◮ Discrete time ◮ We extend the neo-classical growth model (pioneered by Brock and Mirman 1972; see a comprehensive survey by Olson and Roy, 2006) to the mean field setting ◮ Continuous time ◮ The classical SDE modeling by Merton (1975) ◮ Stochastic depreciation: Walde (J. Econ. Dyn. Control, 2011); Feicht and Stummer (2010) ◮ Our mean field modeling is based on the above works (Huang and Nguyen, to be presented at IEEE CDC’14) Minyi Huang Mean Field Games and Stochastic Growth Modeling

Brief overview Stochastic Growth Capital Accumulation Game Continuous Time Classical stochastic growth model: Review The one-sector economy at stage t involves two basic quantities: ◮ κ t : the capital stock (used for production); c t : consumption The next stage output y t +1 : y t +1 = f ( κ t , r t ) , t = 0 , 1 , . . . , ◮ f ( · , · ): production function; r t : random disturbance; y 0 : given ◮ κ t + c t = y t Objective: maximize the utility functional E ∑ ∞ t =0 ρ t ν ( c t ); ν ( c t ): utility from consumption, usually concave on [0 , ∞ ) Brock and Mirman (J. Econ. Theory, 1972) pioneered stochastic growth theory. Minyi Huang Mean Field Games and Stochastic Growth Modeling

Brief overview Stochastic Growth Capital Accumulation Game Continuous Time Notation in the mean field model Keep track of the notation (for the main part): u i t : control (allocation for capital stock) X i t : state (production output) N : number of players in the game c i t : consumption V i ( x , t ): value function G ( p , W ) , g : growth coefficient in production W : white noise p : aggregate capital stock γ : HARA utility exponent Minyi Huang Mean Field Games and Stochastic Growth Modeling