COMPUTATION OF GENERALIZED NASH EQUILIBRIA WITH

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1 COMPUTATION OF GENERALIZED NASH EQUILIBRIA WITH THE GNE PACKAGE C. Dutang 1,2 1 ISFA - Lyon, 2 AXA GRM - Paris, 1/19 12/08/2011 user! 2011

2 OUTLINE 1 GAP FUNCTION MINIZATION APPROACH 2 FIXED-POINT APPROACH 3 KKT REFORMULATION 2/19 12/08/2011 user! 2011

3 WHAT IS A NASH EQUILIBRIUM (NE)? Some history, An equilibrium concept was first introduced by Cournot in But mathematical concepts had widely spread with The theory of games and economic behavior of von Neumann & Morgenstern book in John F. Nash introduces the NE equilibrium concept in a 1949 seminar. 3/19 12/08/2011 user! 2011

4 WHAT IS A NASH EQUILIBRIUM (NE)? Some history, An equilibrium concept was first introduced by Cournot in But mathematical concepts had widely spread with The theory of games and economic behavior of von Neumann & Morgenstern book in John F. Nash introduces the NE equilibrium concept in a 1949 seminar. Various names for various games Which type of players? Cooperative vs. non-cooperative Which type of for the strategy set? Finite vs. infinite games e.g. {squeal, silent} for Prisoner s dilemna vs. [0, 1] cake cutting game Is there any time involved? Static vs. dynamic games e.g. battle of the sexes vs. pursuit-evasion game Is it stochastic? Deterministic payoff vs. random payoff 3/19 12/08/2011 user! 2011

5 NON-COOPERATIVE GAMES Idea : to model competition among players Characterization by a pair of a set of stategies S for the n players a payoff function f such that f i (s) is the payoff for player i given the overall strategy s 4/19 12/08/2011 user! 2011

6 NON-COOPERATIVE GAMES Idea : to model competition among players Characterization by a pair of a set of stategies S for the n players a payoff function f such that f i (s) is the payoff for player i given the overall strategy s Nash equilibrium is defined as a strategy in which no player can improve unilateraly his payoff. 4/19 12/08/2011 user! 2011

7 NON-COOPERATIVE GAMES Idea : to model competition among players Characterization by a pair of a set of stategies S for the n players a payoff function f such that f i (s) is the payoff for player i given the overall strategy s Nash equilibrium is defined as a strategy in which no player can improve unilateraly his payoff. Duopoly example: Consider two profit-seeking firms sell an identical product on the same market. Each firm will choose its production rates. Let x i R + be the production of firm i and d, λ and ρ be constants. The market price is p(x) = d ρ(x 1 + x 2), from which we deduce the ith firm profit p(x)x i λx i. For d = 20, λ = 4, ρ = 1, we get x = (16/3, 16/3). 4/19 12/08/2011 user! 2011

8 EXTENSION TO GENERALIZED NE (GNE) DEFINITION (NEP) The Nash equilibrium problem NEP(N, θ ν, X) for N players consists in finding x solving N sub problems x ν solves min y ν X ν θ ν(y ν, x ν), ν = 1,..., N, where X ν is the action space of player ν. 5/19 12/08/2011 user! 2011

9 EXTENSION TO GENERALIZED NE (GNE) DEFINITION (NEP) The Nash equilibrium problem NEP(N, θ ν, X) for N players consists in finding x solving N sub problems x ν solves min y ν X ν θ ν(y ν, x ν), ν = 1,..., N, where X ν is the action space of player ν. GNE was first introduced by Debreu, [Deb52]. DEFINITION (GNEP) The generalized NE problem GNEP(N, θ ν, X ν) consists in finding x solving N sub problems x ν solves min y ν θ ν(y ν, x ν) such that x ν X ν(x ν), ν = 1,..., N, where X ν(x ν) is the action space of player ν given others player actions x ν. Remark: we study a jointly convex case X = {x, g(x) 0, ν, h ν(x ν) 0}. 5/19 12/08/2011 user! 2011

10 OUTLINE 1 GAP FUNCTION MINIZATION APPROACH 2 FIXED-POINT APPROACH 3 KKT REFORMULATION 6/19 12/08/2011 user! 2011

11 REFORMULATION AS A GAP DEFINITION (GP) Let ψ be a gap function. We define a merit function as ˆV (x) = sup ψ(x, y). y X(x) From [vhk09], the gap problem GP(N, X, ψ) is x X(x ) and ˆV (x ) = 0. 7/19 12/08/2011 user! 2011

12 REFORMULATION AS A GAP DEFINITION (GP) Let ψ be a gap function. We define a merit function as ˆV (x) = sup ψ(x, y). y X(x) From [vhk09], the gap problem GP(N, X, ψ) is x X(x ) and ˆV (x ) = 0. PROPOSITION Assuming continuity, ˆV (x) 0 and so x solves the GP(N, X, ψ) is equivalent to min ˆV (x) and ˆV (x ) = 0. x X(x) 7/19 12/08/2011 user! 2011

13 REFORMULATION AS A GAP DEFINITION (GP) Let ψ be a gap function. We define a merit function as ˆV (x) = sup ψ(x, y). y X(x) From [vhk09], the gap problem GP(N, X, ψ) is x X(x ) and ˆV (x ) = 0. PROPOSITION Assuming continuity, ˆV (x) 0 and so x solves the GP(N, X, ψ) is equivalent to min ˆV (x) and ˆV (x ) = 0. x X(x) Example: the Nikaido-Isoda function NX ψ(x, y) = [θ(x ν, x ν) θ(y ν, x ν)] α 2 x y 2. ν=1 7/19 12/08/2011 user! 2011

14 THE M I NGA P FUNCTION The gap function minimization consists in minimizing a gap function min V (x). The function mingap provides two optimization methods. Barzilai-Borwein method x n+1 = x n + t nd n, where direction is d n = V (x n) and stepsize BFGS method t n = d T n 1 d T n 1d n 1 ( V (xn) V (xn 1)). x n+1 = x n + t nh 1 n d n, where H n approximates the Hessian by a symmetric rank two update of the type H n+1 = H n + auu T + bvv T and t n is line searched. 8/19 12/08/2011 user! 2011

15 DUOPOLY EXAMPLE Let x i R + be the production of firm i and d = 20, λ = 4 and ρ = 1 be constants. The ith firm profit θ i(x) = (d ρ(x 1 + x 2))x i λx i. > mingap(c(10,20), V, GV, method="bb",...) **** k 0 x_k **** k 1 x_k **** k 2 x_k **** k 3 x_k **** k 4 x_k $par [1] $outer.counts Vhat gradvhat 5 9 $outer.iter [1] 3 $code [1] 0 > 9/19 12/08/2011 user! 2011

16 DUOPOLY EXAMPLE Let x i R + be the production of firm i and d = 20, λ = 4 and ρ = 1 be constants. The ith firm profit θ i(x) = (d ρ(x 1 + x 2))x i λx i. > mingap(c(10,20), V, GV, method="bb",...) **** k 0 x_k **** k 1 x_k **** k 2 x_k **** k 3 x_k **** k 4 x_k $par [1] Outer iterations 3 Inner iterations - V 141 Inner iterations - V 269 $outer.counts Vhat gradvhat 5 9 $outer.iter [1] 3 $code [1] 0 > 9/19 12/08/2011 user! 2011

17 DUOPOLY EXAMPLE Let x i R + be the production of firm i and d = 20, λ = 4 and ρ = 1 be constants. The ith firm profit θ i(x) = (d ρ(x 1 + x 2))x i λx i. > mingap(c(10,20), V, GV, method="bb",...) **** k 0 x_k **** k 1 x_k **** k 2 x_k **** k 3 x_k **** k 4 x_k $par [1] $outer.counts Vhat gradvhat 5 9 $outer.iter [1] 3 Outer iterations 3 Inner iterations - V 141 Inner iterations - V 269 Function calls - V 5 leading to calls - ψ 978 leading to calls - ψ 276 Function calls - V 9 leading to calls - ψ 2760 leading to calls - ψ 652 $code [1] 0 > 9/19 12/08/2011 user! 2011

18 OUTLINE 1 GAP FUNCTION MINIZATION APPROACH 2 FIXED-POINT APPROACH 3 KKT REFORMULATION 10/19 12/08/2011 user! 2011

19 REFORMULATION OF THE GNE PROBLEM AS A FIXED-POINT DEFINITION (REGULARIZED NIF) The regularized Nikaido-Isoda function is NX NIF α(x, y) = [θ(x ν, x ν) θ(y ν, x ν)] α 2 x y 2, ν=1 with a regularization parameter α > 0. Let y α : R n R n be the equilibrium response defined as y α(x) = (yα(x), 1..., yα N (x)), where yα(x) i = arg max NIF α(x, y). y i X i (x i ) 11/19 12/08/2011 user! 2011

20 THE F I X E D P O I N T FUNCTION Fixed-point methods for f(x) = 0 pure fixed point method Polynomial methods Relaxation algorithm x n+1 = f(x n). x n+1 = λ nf(x n) + (1 λ n)x n, where (λ n) n is either constant, decreasing or line-searched. RRE and MPE method x n+1 = x n + t n(f(x n) x n) where r n = f(x n) x n and v n = f(f(x n)) 2f(x n) + x n, with t n equals to < v n, r n > / < v n, v n > for RRE1, < r n, r n > / < v n, r n > for MPE1. Squaring method It consists in applying twice a cycle step to get the next iterate given t n such as RRE and MPE. Epsilon algorithms: not implemented x n+1 = x n 2t nr n + t 2 n vn, 12/19 12/08/2011 user! 2011

21 DUOPOLY EXAMPLE Let x i R + be the production of firm i and d = 20, λ = 4 and ρ = 1 be constants. The ith firm profit θ i(x) = (d ρ(x 1 + x 2))x i λx i. > fixedpoint(c(0,0), method="pure",...) $par [1] $outer.counts yfunc Vfunc $outer.iter [1] 24 $code [1] 0 > 13/19 12/08/2011 user! 2011

22 DUOPOLY EXAMPLE Let x i R + be the production of firm i and d = 20, λ = 4 and ρ = 1 be constants. The ith firm profit θ i(x) = (d ρ(x 1 + x 2))x i λx i. > fixedpoint(c(0,0), method="pure",...) $par [1] Total calls ψ ψ Outer iter Gap - BB $outer.counts yfunc Vfunc $outer.iter [1] 24 $code [1] 0 > 13/19 12/08/2011 user! 2011

23 DUOPOLY EXAMPLE Let x i R + be the production of firm i and d = 20, λ = 4 and ρ = 1 be constants. The ith firm profit θ i(x) = (d ρ(x 1 + x 2))x i λx i. > fixedpoint(c(0,0), method="pure",...) $par [1] $outer.counts yfunc Vfunc $outer.iter [1] 24 Total calls ψ ψ Outer iter Gap - BB Pure FP Decr. FP Line search FP $code [1] 0 > 13/19 12/08/2011 user! 2011

24 DUOPOLY EXAMPLE Let x i R + be the production of firm i and d = 20, λ = 4 and ρ = 1 be constants. The ith firm profit θ i(x) = (d ρ(x 1 + x 2))x i λx i. > fixedpoint(c(0,0), method="pure",...) $par [1] $outer.counts yfunc Vfunc $outer.iter [1] 24 $code [1] 0 > Total calls ψ ψ Outer iter Gap - BB Pure FP Decr. FP Line search FP RRE FP MPE FP /19 12/08/2011 user! 2011

25 DUOPOLY EXAMPLE Let x i R + be the production of firm i and d = 20, λ = 4 and ρ = 1 be constants. The ith firm profit θ i(x) = (d ρ(x 1 + x 2))x i λx i. > fixedpoint(c(0,0), method="pure",...) $par [1] $outer.counts yfunc Vfunc $outer.iter [1] 24 $code [1] 0 > Total calls ψ ψ Outer iter Gap - BB Pure FP Decr. FP Line search FP RRE FP MPE FP SqRRE FP SqMPE FP /19 12/08/2011 user! 2011

26 OUTLINE 1 GAP FUNCTION MINIZATION APPROACH 2 FIXED-POINT APPROACH 3 KKT REFORMULATION 14/19 12/08/2011 user! 2011

27 KKT SYSTEM REFORMULATION DEFINITION (KKT) From [FFP09], the first-order necessary conditions for ν subproblem states, there exists a Lagrangian multiplier λ ν R m such that L(x, λ) = xν θ xν (x) + X 1 j m λ νj xν g j(x) = 0 ( R nν ). 0 λ ν g(x) 0 ( R m ) 15/19 12/08/2011 user! 2011

28 KKT SYSTEM REFORMULATION DEFINITION (KKT) From [FFP09], the first-order necessary conditions for ν subproblem states, there exists a Lagrangian multiplier λ ν R m such that L(x, λ) = xν θ xν (x) + X 1 j m λ νj xν g j(x) = 0 ( R nν ). 0 λ ν g(x) 0 ( R m ) PROPOSITION The extended KKT system can be reformulated as a system of equations using complementarity functions. Let φ be a complementarity function. «L(x, λ) F φ (x, λ) = 0 where F φ (x, λ) =, φ.( g(x), λ) where φ. is the component-wise version of φ. Example: φ(a, b) = min(a, b). 15/19 12/08/2011 user! 2011

29 THE NE W T O NKKT FUNCTION The problem is Φ(z) = 0, with z = (x T λ T ) T. It is solved by an iterative scheme z n+1 = z n + d n, where the direction d n is computed in two different ways: Newton method: The direction solves the system with V n Φ(x n). V nd = Φ(x n), The Levenberg-Marquardt method: The direction solves the system (V T n V n + λ k I)d = V T n Φ(x n), where I denotes the identity matrix, λ k is the LM parameter, e.g. λ n = Φ(z n) 2. 16/19 12/08/2011 user! 2011

30 DUOPOLY EXAMPLE Let x i R + be the production of firm i and d = 20, λ = 4 and ρ = 1 be constants. The ith firm profit θ i(x) = (d ρ(x 1 + x 2))x i λx i. > NewtonKKT(rep(0, 4), "Leven",...) $par [1] e-08-3e-08 $value [1] e-08 $counts phi jacphi $iter [1] 11 > 17/19 12/08/2011 user! 2011

31 DUOPOLY EXAMPLE Let x i R + be the production of firm i and d = 20, λ = 4 and ρ = 1 be constants. The ith firm profit θ i(x) = (d ρ(x 1 + x 2))x i λx i. > NewtonKKT(rep(0, 4), "Leven",...) $par [1] e-08-3e-08 $value [1] e-08 $counts phi jacphi Total calls ψ ψ Outer iter Gap - BB Decr. FP MPE FP $iter [1] 11 > 17/19 12/08/2011 user! 2011

32 DUOPOLY EXAMPLE Let x i R + be the production of firm i and d = 20, λ = 4 and ρ = 1 be constants. The ith firm profit θ i(x) = (d ρ(x 1 + x 2))x i λx i. > NewtonKKT(rep(0, 4), "Leven",...) $par [1] e-08-3e-08 $value [1] e-08 $counts phi jacphi $iter [1] 11 > Total calls ψ ψ Outer iter Gap - BB Decr. FP MPE FP Newton LM /19 12/08/2011 user! 2011

33 CONCLUSION The GNE package 1 provides base tools to compute generalized Nash equilibria for static infinite noncooperative games. Three methods : Gap function minimization: mingap, Fixed-point methods: fixedpoint, KKT reformulation: NewtonKKT. 1 hosted on R-forge. 18/19 12/08/2011 user! 2011

34 CONCLUSION The GNE package 1 provides base tools to compute generalized Nash equilibria for static infinite noncooperative games. Three methods : Gap function minimization: mingap, Fixed-point methods: fixedpoint, KKT reformulation: NewtonKKT. Future developments: extends to non jointly convex case for FP and GP methods, develop further the KKT reformulation, later, extends to finite noncooperative games, e.g. Lemke-Howson algorithm, far later, extends to cooperative games! 1 hosted on R-forge. 18/19 12/08/2011 user! 2011

35 References REFERENCES Gerard Debreu, A social equilibrium existence theorem, Proc. Nat. Acad. Sci. U.S.A. (1952). Francisco Facchinei, Andreas Fischer, and Veronica Piccialli, Generalized Nash equilibrium problems and Newton methods, Math. Program., Ser. B 117 (2009), Anna von Heusinger and Christian Kanzow, Optimization reformulations of the generalized Nash equilibrium problem using the Nikaido-Isoda type functions, Computational Optimization and Applications 43 (2009), no. 3. Thank you for your attention! 19/19 12/08/2011 user! 2011

Solving a Class of Generalized Nash Equilibrium Problems

Journal of Mathematical Research with Applications May, 2013, Vol. 33, No. 3, pp. 372 378 DOI:10.3770/j.issn:2095-2651.2013.03.013 Http://jmre.dlut.edu.cn Solving a Class of Generalized Nash Equilibrium