1 On the interplay between kinetic theory and game theory Pierre Degond Department of mathematics, Imperial College London pdegond@imperial.ac.uk http://sites.google.com/site/degond/ Joint work with J. G. Liu (Duke) and C. Ringhofer (ASU) ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Summary 2 1. Motivation 2. Nash equilibria vs kinetic equilibria 3. Hydrodynamics driven by local Nash equilibria 4. Wealth distribution 5. Conclusion ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
3 1. Motivation ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Particles vs rational agents 4 Social or biological agents can be mechanical particles subject to forces: kinetic theory rational agents trying to optimize a goal: game theory Our goal: try to reconcile these viewpoints show that kinetic theory can deal with rational agents incorporate time-dynamics in game theory Applications: Pedestrians with C. Appert-Rolland . . . & G. Theraulaz, JSP 2013 & KRM 2013, based on D. Helbing, . . . & G. Theraulaz, PNAS 2011 Social herding behavior with J-G. Liu & C. Ringhofer, JNLS 2014 Economics with J-G. Liu & C. Ringhofer, JSP 2014 and PTRS A 2014 ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
5 2. Nash equilibria vs kinetic equilibria P. D., J-G. Liu, C. Ringhofer, J. Nonlinear Sci. 24 (2014), pp. 93-115 ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Game with a finite number of players 6 N players j = 1 , . . . , N Each player can play a strategy Y j in strategy space Y The cost function of player j playing strategy Y j in the presence of the other players playing strategy ˆ φ j ( Y j , ˆ Y j = ( Y 1 , . . . , Y j − 1 , Y j +1 , . . . , Y N ) is Y j ) Players try to minimize their cost function by acting on their strategy Y j , not touching the others’ strategies ˆ Y j Nash equilibrium Strategy Y = ( Y 1 , . . . , Y N ) such that no player can improve on its cost function by acting on his own strategy variable Y Nash equilibrium ⇐ ⇒ Z j φ j ( Z j , ˆ φ j ( Y ) = min Y j ) , ∀ j = 1 , . . . , N ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Best reply strategy 7 Describe behavior of the agents in time Agents march towards the local optimum by acting on their own strategy variable assuming the other agents will not change theirs Y j ( t ) = −∇ Y j φ j ( Y j , ˆ ˙ Y j ) , ∀ j = 1 , . . . , N Add noise to account for uncertainties √ dY j ( t ) = −∇ Y j φ j ( Y j , ˆ 2 d dB j Y j ) dt + t , ∀ j = 1 , . . . , N ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Continuum of players 8 Anonymous game with a continuum of player Players with the same strategy cannot be distinguished Agents described by strategy probability distribution dF ( y ) Non-atomic: dF ( y ) = f ( y ) dy is absolutely continuous Cost function is φ ( y ; f ) General framework of Non-Cooperative, Non-Atomic, Anonymous game with a Continuum of Players (NCNAACP) Aumann, Mas Colell, Schmeidler, Shapiro & Shapley Mean-field games Lasry & Lions, Cardaliaguet ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Nash Equilibrium for a continuum of players 9 The probability distribution f NE is a Nash Equilibrium (NE) iff φ ( y ; f NE ) = K, ∀ y ∈ Supp f NE , ∃ K s. t. φ ( y ; f NE ) ≥ K, ∀ y This is equivalent to the following “mean-field equation” � � φ ( y ; f NE ) f NE dy = inf φ ( y ; f NE ) f dy f ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Best reply strategy for continuum of players 10 Distribution of players f ( y, t ) satisfies kinetic eq. � � ∂ t f − ∇ y · ∇ y φ f f − d ∆ y f = 0 , φ f = φ ( · ; f ) Define: “collision operator” Q : � � Q ( f ) = ∇ y · ∇ y φ f f + d ∆ y f Kinetic Equilibria (KE) are solutions of Q ( f ) = 0 For a given potential φ ( y ) , define Gibbs measure M φ M φ ( y ) = 1 − φ ( y ) � � � exp , M φ ( y ) dy = 1 Z φ d ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Kinetic Equilibria 11 Write Q as � f � �� Q ( f ) = d ∇ y · M φ f ∇ y M φ f Implies: � � f f � � 2 �� Q ( f ) dy = − d M φ f dy � ∇ y � � M φ f M φ f � Theorem: f KE Kinetic Equilibrium (and normalized, i.e. � f KE = 1 ) iff f KE is a solution of the fixed point eq. f = M φ f or equivalently f KE = M φ K E with φ KE a solution of the fixed point eq. φ = φ M φ ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Kinetic Equilibria vs Nash Equilibria (I) 12 Let a NCNAACP - game be defined by the cost function µ f = φ f + d log f Theorem: Suppose φ f continuous; ∀ f . Then, f KE Kinetic Equilibrium associated to Q ( f ) iff it is Nash Equilibrium of this game Proof: “ ⇒ ”: φ f is locally finite ∀ f . So, M φ f ( y ) = Z − 1 φ f exp( − φ f ( y ) /d ) > 0 , ∀ y, and, µ M φf = − d log Z φ f = Constant , ∀ y. So, if f = M φ f , i.e. if f = f KE Kinetic Equilibrium then, it is a Nash Equilibrium for the game with cost function µ f ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Kinetic Equilibria vs Nash Equilibria (II) 13 Proof (cont): “ ⇐ ”: Suppose f = f NE Nash Equilibrium. Then f > 0 , ∀ y . Otherwise ∃ y s.t. f ( y ) = 0 and µ f ( y ) = −∞ ≥ K . Then K = −∞ and f ≡ 0 : contradiction � with f = 1 . Therefore, µ f = K , ∀ y , which implies f = M φ f , implying that f is a Kinetic Equilibrium. Special case: potential games (Monderer & Shapley) Suppose ∃ a functional U ( f ) s.t. φ f = δ U δf Define free energy: � F ( f ) = U ( f ) + d f log f dy. Then, Cost function µ f is a “Chemical potential”: µ f = δ F δf ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Potential games 14 In general: � � Q ( f ) = ∇ y · f ∇ y µ f If potential game, leads to gradient flow: � δ F � � � ∂ t f = ∇ y · ∇ y f δf Free-energy dissipation: d � δ F � 2 � �� dt F ( f ) = −D ( f ) < 0 , D ( f ) = f dy � ∇ y � � δf � We have the equivalence (i) ⇔ (ii): � (i) f critical point of F subject to the constraint f dy = 1 (ii) f Nash equilibrium Ground state, metastable equilibria, phase transition, hysteresis ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
15 3.Hydrodynamics driven by local Nash equilibria P. D., J-G. Liu, C. Ringhofer, J. Nonlinear Sci. 24 (2014), pp. 93-115 ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Games with configuration variable 16 Add configuration (aka “type”) variable X j (e.g. space) Motion depends on both type X j and strategy Y j ˙ X j = V ( X j , Y j ) , ∀ j = 1 , . . . , N Cost function depends also on types X = ( X j ) j =1 ,...,N √ dY j ( t ) = −∇ Y j φ j ( Y j , ˆ 2 d dB j Y j , X ) dt + t , ∀ j = 1 , . . . , N ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Continuum of players 17 Probability distribution depends on type x and strategy y : f = f ( x, y, t ) Satisfies space-dependent Kinetic Eq.: � � ∂ t f + ∇ x · ( V ( x, y ) f ) − ∇ y · ∇ y φ f f − d ∆ y f = 0 with φ f = φ f ( t ) ( x, y ) Goal of this work: Provide continuum model for moments of f wrt strategy y such as agent density ρ f ( x, t ) or mean strategy ¯ Υ f ( x, t ) � � ρ ¯ ρ f ( x, t ) = f ( x, y, t ) dy, Υ f ( x, t ) = f ( x, y, t ) y dy ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Mean-field game approach (Lasry & Lions) 18 Mean-field game approach directly provides continuum eq. Without Kinetic Eq. step Relies on an optimal control approach within a finite horizon time [0 , T ] and terminal data R d × (0 , T ) , − ∂ t Υ − ν ∆Υ + H ( x, ρ, D Υ) = 0 , in R d × (0 , T ) , ∂ t ρ − ν ∆ ρ − div ( D p H ( x, ρ, D Υ) ρ ) = 0 , in ρ ( x, 0) = ρ 0 ( x ) , Υ( x, T ) = G ( x, ρ ( T )) In this model H ∼ cost function G = cost function for reaching target at terminal time T ρ satisfies convection-diffusion in field determined by H Υ acts as a control variable and satisfies backwards eq. ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Link with our approach 19 Best reply strategy can be recovered from MGF Through receding horizon (aka model predictive control) Chop [0 , T ] into small intervals of size ∆ t Control defined by one step Euler discretization of HJB [PD., M. Herty, J. G. Liu, Comm. Math. Sci. 15 (2017) 1403-1411] ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
Main hypothesis of our work 20 Scale separation hypothesis Variation of strategy y much faster than that of type x Fast equilibration of strategy leads to slow evolution of type ↑ Pierre Degond - Interplay between kinetic and game theories - Cemracs, 18/07/2017 ↓
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