An optimal stopping mean-field game of resource sharing Geraldine Bouveret 1 Roxana Dumitrescu 2 Peter Tankov 3 1 Oxford University 2 King’s College London 3 CREST–ENSAE IMS–FIPS Workshop London, September 10, 2018 Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 1 / 31
Introduction Outline 1 Introduction 2 Mean-field games 3 MFG of optimal stopping 4 MFG of optimal stopping: the relaxed control approach 5 Back to the game of resource sharing Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 2 / 31
Introduction How do economic agents adapt to climate change? • Water security is one of the most tangible and fastest-growing social, political and economic challenges faced today • The coal industry is an important consumer of freshwater resources and is responsible for 7% of all water withdrawal globally • Cooling power plants are responsible for the greatest demand in fresh water Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 3 / 31
Introduction A model for producers competing for a scarce resource • Consider N producers sharing a resource whose supply per unit of time is limited (e.g., fresh water) and denoted by � Z t ; Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 4 / 31
Introduction A model for producers competing for a scarce resource • Consider N producers sharing a resource whose supply per unit of time is limited (e.g., fresh water) and denoted by � Z t ; • Each producer initially uses technology 1 requiring fresh water, and can switch to technology 2 (not using water) at some future date τ i ; Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 4 / 31
Introduction A model for producers competing for a scarce resource • Consider N producers sharing a resource whose supply per unit of time is limited (e.g., fresh water) and denoted by � Z t ; • Each producer initially uses technology 1 requiring fresh water, and can switch to technology 2 (not using water) at some future date τ i ; • Each producer faces demand level M i t and can produce up to M i t if the water supply allows: • With technology 1, one unit of water is required to produce one unit of good; • With technology 2, no water is required. Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 4 / 31
Introduction A model for producers competing for a scarce resource • Consider N producers sharing a resource whose supply per unit of time is limited (e.g., fresh water) and denoted by � Z t ; • Each producer initially uses technology 1 requiring fresh water, and can switch to technology 2 (not using water) at some future date τ i ; • Each producer faces demand level M i t and can produce up to M i t if the water supply allows: • With technology 1, one unit of water is required to produce one unit of good; • With technology 2, no water is required. • In case of shortage of water, the available supply is shared among producers according to their demand levels. Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 4 / 31
Introduction A model for producers competing for a scarce resource • The demand of i -th producer follows the dynamics dM i t = µ dt + σ dW i M i 0 = m i . t , M i t where W 1 , . . . , W N are independent Brownian motions. Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 5 / 31
Introduction A model for producers competing for a scarce resource • The demand of i -th producer follows the dynamics dM i t = µ dt + σ dW i M i 0 = m i . t , M i t where W 1 , . . . , W N are independent Brownian motions. • With technology 2, the output is M i t and with technology 1 the output is N � M i if � M j t , Z t ≥ t 1 τ j > t i =1 Q i t = � Z t M i otherwise . � N t j =1 M j t 1 τ j > t ⇒ Q i t = ω N t M i t , where ω N t is the proportion of demand which may be satisfied � Z t ω N t = ∧ 1 . � N j =1 M j t 1 τ j > t Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 5 / 31
Introduction Cost function of producers The cost function of the producer is given by � τ i � τ i � ∞ e − ρ t ˆ e − ρ t ˜ e − ρ t pQ i t ) dt − e − ρτ i K + p ( M i t − Q i pM i t dt − t dt 0 0 τ i � τ i � τ i � ∞ e − ρ t ˆ e − ρ t ˜ e − ρ t p ω N t M i p (1 − ω N t ) M i t dt − e − ρτ i K + pM i = t dt − t dt 0 0 τ i where we assume that ρ > µ . p is the gain from producing with technology 1; p is the penalty paid for not meeting the demand; ˆ K is the cost of switching the technology; p is the gain from producing with technology 2. ˜ Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 6 / 31
Mean-field games Outline 1 Introduction 2 Mean-field games 3 MFG of optimal stopping 4 MFG of optimal stopping: the relaxed control approach 5 Back to the game of resource sharing Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 7 / 31
Mean-field games Mean-field games Introduced by Lasry and Lions (2006,2007) and Huang, Caines and Malham´ e (2006) to describe large-population games with symmetric interactions. t ∈ R d by taking an action α i • Each player controls its state X i t ∈ A ⊂ R k : µ N − 1 µ N − 1 dX i t = b ( t , X i , α i t ) dt + σ ( t , X i , α i t ) dW i t , ¯ t , ¯ t , X − i X − i t t W i are independent and ¯ µ N − 1 is the empirical distribution of other players. X − i t Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 8 / 31
Mean-field games Mean-field games Introduced by Lasry and Lions (2006,2007) and Huang, Caines and Malham´ e (2006) to describe large-population games with symmetric interactions. t ∈ R d by taking an action α i • Each player controls its state X i t ∈ A ⊂ R k : µ N − 1 µ N − 1 dX i t = b ( t , X i , α i t ) dt + σ ( t , X i , α i t ) dW i t , ¯ t , ¯ t , X − i X − i t t W i are independent and ¯ µ N − 1 is the empirical distribution of other players. X − i t • Each player minimises the cost �� T � J i ( α f ( t , X i µ N − 1 , α i t ) dt + g ( X i µ N − 1 α α ) = E t , ¯ T , ¯ T ) , X − i X − i t 0 α ∈ A N : ∀ i , ∀ α i ∈ A , J i (ˆ α − i ). α ) ≤ J i ( α i , ˆ • We look for a Nash equilibrium: ˆ α α α α α α Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 8 / 31
Mean-field games Mean-field games µ N − 1 As N → ∞ , it is natural to assume that ¯ converges to a deterministic X − i t distribution; Nash equilibrium is described as follows (Carmona and Delarue ’17): • The representative player controls its state X α depending on the deterministic flow ( µ t ) 0 ≤ t ≤ T : dX α t = b ( t , X α t , µ t , α t ) dt + σ ( t , X α t , µ t , α t ) dW t . • It minimises the cost �� T � α ∈ A J µ ( α ) , J µ ( α ) = E f ( t , X α t , µ t , α t ) dt + g ( X α inf T , µ T ) ( ∗ ) 0 Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 9 / 31
Mean-field games Mean-field games µ N − 1 As N → ∞ , it is natural to assume that ¯ converges to a deterministic X − i t distribution; Nash equilibrium is described as follows (Carmona and Delarue ’17): • The representative player controls its state X α depending on the deterministic flow ( µ t ) 0 ≤ t ≤ T : dX α t = b ( t , X α t , µ t , α t ) dt + σ ( t , X α t , µ t , α t ) dW t . • It minimises the cost �� T � α ∈ A J µ ( α ) , J µ ( α ) = E f ( t , X α t , µ t , α t ) dt + g ( X α inf T , µ T ) ( ∗ ) 0 X µ is • We look for a flow ( µ t ) 0 ≤ t ≤ T such that L ( ˆ t ) = µ t , t ∈ [0 , T ], where ˆ X µ the solution to ( ∗ ). Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 9 / 31
Mean-field games The analytic approach The stochastic control problem is characterized as the solution to a HJB equation � � f ( t , x , µ t , α ) + b ( t , x , µ t , α ) ∂ x V + 1 2 σ 2 ( t , x , µ t , α ) ∂ 2 ∂ t V + max xx V = 0 α with the terminal condition V ( T , x ) = g ( x , µ T ). The flow of densities solves the Fokker-Planck equation ∂ t µ t − 1 2 ∂ 2 xx ( σ 2 ( t , x , µ t , ˆ α t ) µ t ) + ∂ x ( b ( t , x , µ t , ˆ α t ) µ t ) = 0 , with the initial condition µ 0 = δ X 0 , where ˆ α is the optimal feedback control. ⇒ A coupled system of a Hamilton-Jacobi-Bellman PDE (backward) and a Fokker-Planck PDE (forward) Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 10 / 31
MFG of optimal stopping Outline 1 Introduction 2 Mean-field games 3 MFG of optimal stopping 4 MFG of optimal stopping: the relaxed control approach 5 Back to the game of resource sharing Peter Tankov (ENSAE) A mean-field game of resource sharing June 25–29, 2018 11 / 31
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