From the master equation to mean field game asymptotics Daniel - PowerPoint PPT Presentation

From the master equation to mean field game asymptotics From the master equation to mean field game asymptotics Daniel Lacker Division of Applied Mathematics, Brown University June 16, 2017 Joint work with Francois Delarue and Kavita Ramanan

From the master equation to mean field game asymptotics Overview Overview A mean field game (MFG) will refer to a game with a continuum of players. In various contexts, we know rigorously that the MFG arises as the limit of n -player games as n → ∞ .

From the master equation to mean field game asymptotics Overview Overview A mean field game (MFG) will refer to a game with a continuum of players. In various contexts, we know rigorously that the MFG arises as the limit of n -player games as n → ∞ . This talk: Refined MFG asymptotics in the form of a central limit theorem and large deviation principle, as well as non-asymptotic concentration bounds. Key idea: Use the master equation to quantitatively relate n -player equilibrium to n -particle system of McKean-Vlasov type, building on idea of Cardaliaguet-Delarue-Lasry-Lions ’15.

From the master equation to mean field game asymptotics Interacting diffusion models Interacting diffusions Suppose particles i = 1 , . . . , n interact through their empirical measure according to n � t = 1 dX i t = b ( X i ν n t ) dt + dW i ν n t , ¯ t , ¯ δ X k t , n k =1 where W 1 , . . . , W n are independent Brownian motions.

From the master equation to mean field game asymptotics Interacting diffusion models Interacting diffusions Suppose particles i = 1 , . . . , n interact through their empirical measure according to n � t = 1 dX i t = b ( X i ν n t ) dt + dW i ν n t , ¯ t , ¯ δ X k t , n k =1 where W 1 , . . . , W n are independent Brownian motions. ν n Under “nice” assumptions on b , we have ¯ t → ν t , where ν t solves the McKean-Vlasov equation, dX t = b ( X t , ν t ) dt + dW t , ν t = Law ( X t ) ,

From the master equation to mean field game asymptotics Interacting diffusion models Interacting diffusions Suppose particles i = 1 , . . . , n interact through their empirical measure according to n � t = 1 dX i t = b ( X i ν n t ) dt + dW i ν n t , ¯ t , ¯ δ X k t , n k =1 where W 1 , . . . , W n are independent Brownian motions. ν n Under “nice” assumptions on b , we have ¯ t → ν t , where ν t solves the McKean-Vlasov equation, dX t = b ( X t , ν t ) dt + dW t , ν t = Law ( X t ) , or dt � ν t , ϕ � = � ν t , b ( · , ν t ) ∇ ϕ ( · ) + 1 d 2∆ ϕ ( · ) � .

From the master equation to mean field game asymptotics Interacting diffusion models Empirical measure limit theory ν n There is a rich literature on asymptotics of ¯ t : ν n → ν , where ν solves a McKean-Vlasov equation. 1. LLN: ¯ (Oelschl¨ ager ’84, G¨ artner ’88, Sznitman ’91, etc.)

From the master equation to mean field game asymptotics Interacting diffusion models Empirical measure limit theory ν n There is a rich literature on asymptotics of ¯ t : ν n → ν , where ν solves a McKean-Vlasov equation. 1. LLN: ¯ (Oelschl¨ ager ’84, G¨ artner ’88, Sznitman ’91, etc.) 2. Fluctuations: √ n (¯ ν n t − ν t ) converges to a distribution-valued process driven by space-time Brownian motion. (Tanaka ’84, Sznitman ’85, Kurtz-Xiong ’04, etc.)

From the master equation to mean field game asymptotics Interacting diffusion models Empirical measure limit theory ν n There is a rich literature on asymptotics of ¯ t : ν n → ν , where ν solves a McKean-Vlasov equation. 1. LLN: ¯ (Oelschl¨ ager ’84, G¨ artner ’88, Sznitman ’91, etc.) 2. Fluctuations: √ n (¯ ν n t − ν t ) converges to a distribution-valued process driven by space-time Brownian motion. (Tanaka ’84, Sznitman ’85, Kurtz-Xiong ’04, etc.) ν n has an explicit LDP. 3. Large deviations: ¯ (Dawson-G¨ artner ’87, Budhiraja-Dupius-Fischer ’12)

From the master equation to mean field game asymptotics Interacting diffusion models Empirical measure limit theory ν n There is a rich literature on asymptotics of ¯ t : ν n → ν , where ν solves a McKean-Vlasov equation. 1. LLN: ¯ (Oelschl¨ ager ’84, G¨ artner ’88, Sznitman ’91, etc.) 2. Fluctuations: √ n (¯ ν n t − ν t ) converges to a distribution-valued process driven by space-time Brownian motion. (Tanaka ’84, Sznitman ’85, Kurtz-Xiong ’04, etc.) ν n has an explicit LDP. 3. Large deviations: ¯ (Dawson-G¨ artner ’87, Budhiraja-Dupius-Fischer ’12) 4. Concentration: Finite- n bounds are available for ν n , ν ) > ǫ ), for various metrics d . P ( d (¯ (Bolley-Guillin-Villani ’07, etc.)

From the master equation to mean field game asymptotics Interacting diffusion models Empirical measure limit theory ν n There is a rich literature on asymptotics of ¯ t : ν n → ν , where ν solves a McKean-Vlasov equation. 1. LLN: ¯ (Oelschl¨ ager ’84, G¨ artner ’88, Sznitman ’91, etc.) 2. Fluctuations: √ n (¯ ν n t − ν t ) converges to a distribution-valued process driven by space-time Brownian motion. (Tanaka ’84, Sznitman ’85, Kurtz-Xiong ’04, etc.) ν n has an explicit LDP. 3. Large deviations: ¯ (Dawson-G¨ artner ’87, Budhiraja-Dupius-Fischer ’12) 4. Concentration: Finite- n bounds are available for ν n , ν ) > ǫ ), for various metrics d . P ( d (¯ (Bolley-Guillin-Villani ’07, etc.) The idea: Use the more tractable McKean-Vlasov system to analyze the large- n -particle system.

From the master equation to mean field game asymptotics Mean field games A class of mean field games Agents i = 1 , . . . , n have state process dynamics dX i t = α i t dt + dW i t , with W 1 , . . . , W n independent Brownian, ( X 1 0 , . . . , X n 0 ) i.i.d.

From the master equation to mean field game asymptotics Mean field games A class of mean field games Agents i = 1 , . . . , n have state process dynamics dX i t = α i t dt + dW i t , with W 1 , . . . , W n independent Brownian, ( X 1 0 , . . . , X n 0 ) i.i.d. Agent i chooses α i to minimize �� T � � � t ) + 1 J n i ( α 1 , . . . , α n ) = E f ( X i µ n 2 | α i t | 2 dt + g ( X i µ n t , ¯ T , ¯ T ) , 0 n � t = 1 µ n ¯ δ X k t . n k =1

From the master equation to mean field game asymptotics Mean field games A class of mean field games Agents i = 1 , . . . , n have state process dynamics dX i t = α i t dt + dW i t , with W 1 , . . . , W n independent Brownian, ( X 1 0 , . . . , X n 0 ) i.i.d. Agent i chooses α i to minimize �� T � � � t ) + 1 J n i ( α 1 , . . . , α n ) = E f ( X i µ n 2 | α i t | 2 dt + g ( X i µ n t , ¯ T , ¯ T ) , 0 n � t = 1 µ n ¯ δ X k t . n k =1 Say ( α 1 , . . . , α n ) form an ǫ -Nash equilibrium if J n i ( α 1 , . . . , α n ) ≤ ǫ + inf β J n i ( . . . , α i − 1 , β, α i +1 , . . . ) , ∀ i = 1 , . . . , n

From the master equation to mean field game asymptotics Mean field games The n -player HJB system The value function v n i ( t , ① ), for ① = ( x 1 , . . . , x n ), for agent i in the n -player game solves n � i ( t , ① ) + 1 i ( t , ① ) + 1 i ( t , ① ) | 2 ∂ t v n ∆ x k v n 2 | D x i v n 2 k =1 � � n � � x i , 1 D x k v n k ( t , ① ) · D x k v n + i ( t , ① ) = f δ x k . n k � = i k =1

From the master equation to mean field game asymptotics Mean field games The n -player HJB system The value function v n i ( t , ① ), for ① = ( x 1 , . . . , x n ), for agent i in the n -player game solves n � i ( t , ① ) + 1 i ( t , ① ) + 1 i ( t , ① ) | 2 ∂ t v n ∆ x k v n 2 | D x i v n 2 k =1 � � n � � x i , 1 D x k v n k ( t , ① ) · D x k v n + i ( t , ① ) = f δ x k . n k � = i k =1 A Nash equilibrium is given by α i t = − D x i v n i ( t , X 1 t , . . . , X n t ) .

From the master equation to mean field game asymptotics Mean field games The n -player HJB system The value function v n i ( t , ① ), for ① = ( x 1 , . . . , x n ), for agent i in the n -player game solves n � i ( t , ① ) + 1 i ( t , ① ) + 1 i ( t , ① ) | 2 ∂ t v n ∆ x k v n 2 | D x i v n 2 k =1 � � n � � x i , 1 D x k v n k ( t , ① ) · D x k v n + i ( t , ① ) = f δ x k . n k � = i k =1 A Nash equilibrium is given by α i t = − D x i v n i ( t , X 1 t , . . . , X n t ) . But v n i is generally hard to find, especially for large n .

From the master equation to mean field game asymptotics Mean field games Mean field limit n → ∞ ? The problem Given a Nash equilibrium ( α n , 1 , . . . , α n , n ) for each n , can we µ n describe the asymptotics of (¯ t ) t ∈ [0 , T ] ?

From the master equation to mean field game asymptotics Mean field games Mean field limit n → ∞ ? The problem Given a Nash equilibrium ( α n , 1 , . . . , α n , n ) for each n , can we µ n describe the asymptotics of (¯ t ) t ∈ [0 , T ] ? Previous results, limited to LLN Lasry/ Lions ’06, Feleqi ’13, Fischer ’14, L. ’15, Cardaliaguet-Delarue-Lasry-Lions ’15, Cardaliaguet ’16...