Variational inequality formulation of chance-constrained games Joint work with Vikas Singh from IIT Delhi Université Paris Sud XI Computational Management Science Conference Bergamo, Italy May, 2017
Outline of the talk 1 Introduction 2 The model 3 Existence of Nash Equilibrium 4 VI Formulation of NE 5 Generalized Nash Equilibrium 6 VI Formulation of GNE 7 Conclusion
Introduction We consider a chance-constrained game (CCG) where each player has continuous strategy set which does not depend on the strategies of other players. 1 We show that there exists a Nash equilibrium of a CCG. 2 We characterize the set of Nash equilibria of a CCG using the solution set of a variational inequality (VI) problem. We consider the case where the continuous strategy set of each player is defined by a shared constraint set. 1 We show that there exists a generalized Nash equilibrium for a CCG. 2 We characterize the set of a certain types of generalized Nash equilibria of a CCG using the solution set of a VI problem.
Outline of the talk 1 Introduction 2 The model 3 Existence of Nash Equilibrium 4 VI Formulation of NE 5 Generalized Nash Equilibrium 6 VI Formulation of GNE 7 Conclusion
The model We consider an n-player non-cooperative game with random payoffs defined by a tuple pi, px i q ipi, pr i q ipi q. Let I t1, 2,, nu be a set of players. For each i P I, X i Ă R mi denote the set of all strategies of player i, and denote x i its generic element. Let X Ś n i 1 X i Ă R m, m ř n i 1 m i, be the set of all strategy profiles of the game. Denote, X i Ś n j 1;j i X j, and x i P X i be a vector of strategies x j, j i. For each i P I, r i is a random payoff function of player i.
The model For each i P I, let ξ i `ξk li i be a random vector defined by k 1 ξ i : Ω Ñ R li, where pω, F, Pq be a probability space For a given strategy profile x px 1, x 2,, x n q, and for an ω P Ω the realization of random payoff of player i, i P I, is given by r i px, ωq l i ÿ k 1 f i k pxqξ i kpωq where f i k : Rm Ñ R for all k 1, 2,, l i, i P I. Let α i P r0, 1s be the confidence (risk) level of player i, and α pα i q ipi.
The model For a given strategy profile x P X, and a given confidence level vector α the payoff function of player i, i P I, is given by u αi i pxq suptγ P `tω r i px, ωq ě γu ě α i u. We assume that the probability distribution of random vector ξ i, i P I is known to all the players. For a fixed α P r0, 1s n, the payoff function of a player defined above is known to all the players. The above CCG is a non-cooperative game with complete information.
The model A strategy profile x P X is said to be a Nash equilibrium of a CCG at α if for each i P I, the following inequality holds u αi i px i, x i q ě u αi i px i, x i q, @ x i P X i. We can also equivalently write the definition of Nash equilibrium as, x i P argmint u αi i px i, x i q x i P X i u, @ i P I. We have the following general result for the existence of a Nash equilibrium for a CCG.
The model Theorem 1 For each player i P I, and a fixed α P r0, 1s n suppose (i) the strategy set X i is non-empty, convex, and compact, (ii) the payoff function u αi i : R m Ñ R is continuous, (iii) for every x i P X i, u αi i p, x i q is a concave function of x i. Then, there always exists a Nash equilibrium of a CCG at α.
Outline of the talk 1 Introduction 2 The model 3 Existence of Nash Equilibrium 4 VI Formulation of NE 5 Generalized Nash Equilibrium 6 VI Formulation of GNE 7 Conclusion
Existence of Nash Equilibrium We consider the case where ξ i Ellippµ i, Σ i q with Σ i ą 0. For a given x P X, r i pxq pf i pxqq T ξ i follows a univariate elliptically symmetric distribution with parameters µ T i f i pxq and pf i pxqq T Σ i f i pxq. Z S i r i pxq µ T i f i pxq Σ 1{2 i f i pxq, i P I, follows a univariate spherically symmetric distribution with parameters 0 and 1. Let φ 1 Zi S p q be a quantile function of a spherically symmetric distribution. For a given x P X and α, we have,
Existence of Nash Equilibrium u αi i pxq suptγ P `tω r i px, ωq ě γu ě α i u # # + + sup γ P ωˇ r i px, ωq µ T i f i pxq ď γ µt i f i pxq ď 1 α Σ 1{2 i f i pxq Σ 1{2 i i f i pxq! ) sup γ γ ď µ T i f i pxq ` Σ 1{2 i f i pxq φ 1 p1 α i q. Zi S And, u αi i pxq µ T i f i pxq ` Σ 1{2 f i pxq φ 1 p1 α i q, i P I. i Zi S
Existence of Nash Equilibrium Assumption 1 For each player i, i P I, the following conditions hold. 1 X i is a non-empty, convex and compact subset of R mi. 2 fk i : Rm Ñ R is a continuous function of x, for all k 1, 2,, l i. 3 (a) For every x i P X i, fk ip, x i q, k 1, 2,, l i, is an affine function of x i. or (b) For every x i P X i, fk ip, x i q, k 1, 2,, l i, is non-positive and a concave function of x i, and µ i,k ě 0 for all k 1, 2,, l i, and all the elements of Σ i are non-negative. Lemma 2 If Assumption 1 holds, for every x i P X i, u αi i p, x i q, i P I is a concave function of x i for all α i P p0.5, 1s.
Existence of Nash Equilibrium Theorem 3 Consider an n-player non-cooperative game with random payoffs. Let random vector ξ i Ellippµ i, Σ i q, i P I, where Σ i ą 0. If Assumption 1 holds, there always exists a Nash equilibrium of a CCG for all α P p0.5, 1s n.
Outline of the talk 1 Introduction 2 The model 3 Existence of Nash Equilibrium 4 VI Formulation of NE 5 Generalized Nash Equilibrium 6 VI Formulation of GNE 7 Conclusion
Variational Inequality formulation Given a closed, convex set K and a continuous function G, solving the VI(K, G) consists in finding a vector z P K such that py zq T Gpzq ě 0, @ y P K. It is well known that the Nash equilibrium problem of a non-cooperative game can be formulated as a variational inequality problem (See Facchinei 2003) We formulate the Nash equilibrium problem of a CCG as a variational inequality (VI) problem. The continuity and differentiability of the payoff function of each player, in its own strategy, are required in VI formulation. It exists under the following Assumption.
Variational Inequality formulation Assumption 2 For each i P I, the following conditions hold: 1 For every x i P X i, f i k p, x i q is a differentiable function of x i, for all k 1, 2,, l i. 2 The system f i k pxq 0, k 1, 2,, l i, has no solution. If Assumption 2 holds, the gradient of payoff function of player i is given by, x i u α i i px i, x i q `J f qpxq T i p,x i µ i ` `Jf i p,x i qpxq T Σ if i px i, x i qφ 1 Z i S Σ 1{2 i f i px i, x i q where J f i p,x i qpx i q is the Jacobian matrix of f i p, x i q. p1 α iq,
Variational Inequality formulation Define, a function F : R m Ñ R m, F pxq pf 1 pxq, F 2 pxq,, F n pxqq T, where for each i P I, F ipxq `J f i p,x i qpxq T µ i `Jf i p,x i qpxq T Σ if i px i, x i qφ 1 Z i S Σ 1{2 i f i px i, x i q p1 α iq. Theorem 4 Consider an n-player non-cooperative game with random payoffs. Let random vector ξ i Ellippµ i, Σ i q, i P I, where Σ i ą 0. Let Assumptions 1-2 holds. Then, for an α P p0.5, 1s n, x is a Nash equilibrium of a CCG if and only if it is a solution of VI(X, F ).
Outline of the talk 1 Introduction 2 The model 3 Existence of Nash Equilibrium 4 VI Formulation of NE 5 Generalized Nash Equilibrium 6 VI Formulation of GNE 7 Conclusion
Generalized Nash Equilibrium for CCG We consider the case where the strategy set of each player depends on the strategies of other players ùñ Generalized Nash equilibrium. Let X i px i q Ă R mi, i P I, be the strategy set of player i for a given strategy profile x i of other players. A set of all feasible strategy profiles is defined by Xpxq Ś n i 1 X i px i q. A strategy profile x P Xpx q is said to be a generalized Nash equilibrium, for a given α P r0, 1s n, of a CCG, if for each i P I the following inequality holds u αi i px i, x i q ě u αi i px i, x i q, @ x i P X i px i q. We can also equivalently write the definition of generalized Nash equilibrium as x i P argmint u αi i px i, x i q x i P X i px i qu, @ i P I.
Generalized Nash Equilibrium for CCG We assume that a convex and compact set R Ă R m is given, and for each x i the strategy set of player i is defined as, X i px i q tx i P R mi px i, x i q P Ru. This is an important case of generalized Nash equilibrium problem considered in the fundamental paper by Rosen (1965). It appears more often in practice, e.g., when all players share common resources.
Generalized Nash Equilibrium for CCG Theorem 5 Consider an n-player non-cooperative game with random payoffs. Let random vector ξ i Ellippµ i, Σ i q, i P I, where Σ i ą 0. The strategy set of each player is given by X i px i q. If condition 2 and condition 3 of Assumption 1 holds, there always exists a generalized Nash equilibrium for a CCG for all α P p0.5, 1s n. Facchinei (2007) proposed a variational inequality whose solution is a solution of the generalized Nash equilibrium problem considered by Rosen (1965). We show that a solution of VI(R, F ) is a generalized Nash equilibrium of a CCG.
Outline of the talk 1 Introduction 2 The model 3 Existence of Nash Equilibrium 4 VI Formulation of NE 5 Generalized Nash Equilibrium 6 VI Formulation of GNE 7 Conclusion
Variational inequality formulation Theorem 6 Consider an n-player non-cooperative game with random payoffs. Let random vector ξ i Ellippµ i, Σ i q, i P I, where Σ i ą 0. The strategy set of each player is given by X i px i q. Let condition 2 and condition 3 of Assumption 1 and Assumption 2 holds. Then, for an α P p0.5, 1s n, a solution of VI(R, F ) is a generalized Nash equilibrium of a CCG. In general, a generalized Nash equilibrium is not a solution of VI. However, if R is defined by a finite number of convex and continuously differentiable constraints, there exists a set of generalized Nash equilibria which are solutions of a VI. This generalized Nash equilibrium is called a variational equilibrium.
Variational inequality formulation Let R be defined by a finite number of constraints as follows: R tx P R m g k pxq ď 0, @ k 1, 2,, Ku, (1) where all the constraints g i : R m Ñ R, i 1, 2,, K, are convex and continuously differentiable. For an α P p0.5, 1s n, let x be a generalized Nash equilibrium of a CCG corresponding to elliptically symmetric distributed random vector ξ i, i P I, and Assumptions 1-2 hold. Then, for each i P I, x i is a solution of the following convex optimization problem s.t. min x i u αi i px i, x i q g k px i, x i q ď 0, @ k 1, 2,, K. (2)
Variational inequality formulation A point x P R m, where the KKT conditions of (2) corresponding to each player are satisfied simultaneously, is a generalized Nash equilibrium of a CCG if and only if the following conditions hold: x i u αi i px i, x i q ` `J gp,x i qpxq T λ i 0, @ i P I, 0 ď λ i K gpx i, x i q ď 0, @ i P I, where λ i P R K is a vector of Lagrange multipliers corresponding to player i, K implies that at least one side of inequality is equality. If constraint g k is inactive, the corresponding Lagrange multiplier is zero. However, for active constraints the Lagrange multipliers can be different for each player. + (3)
Variational inequality formulation Assume x is a solution of VI(R, F ), where R is defined by (1). Let x satisfies Abadie CQ. Then, the following KKT conditions are necessary and sufficient for the solution of VI(R, F ): F pxq ` `J + gp q pxq T λ 0 0 ď λ K gpxq ď 0. Using the aforementioned definition of F p q, the above KKT conditions can be written as, x 1u α1 1 px, 1, x 1 q `Jgp,x qpxq T x 2u α2 2 px 2, x 2 q 1 `Jgp,x 2 qpxq T. `. λ 0, /. x nun αn px n, x n q `Jgp,x n qpxq T /- 0 ď λ K gpxq ď 0. (4)
Variational inequality formulation Theorem 7 Consider an n-player non-cooperative game with random payoffs. Let random vector ξ i Ellippµ i, Σ i q, i P I, where Σ i ą 0. The strategy set of each player is defined using shared constraints set given by (1). Let condition 2 and condition 3 of Assumption 1, and Assumption 2 holds, and α P p0.5, 1s n. Then, (a) If x is a solution of a VI(R, F ) such that px, λ q satisfy KKT conditions (4), x is a generalized Nash equilibrium of a CCG at which KKT conditions (3) hold with λ 1 λ 2 λ n λ. (b) If x is a generalized Nash equilibrium of a CCG at which KKT conditions (3) hold with λ 1 λ 2 λ n, x is a solution of a VI(R, F ).
Outline of the talk 1 Introduction 2 The model 3 Existence of Nash Equilibrium 4 VI Formulation of NE 5 Generalized Nash Equilibrium 6 VI Formulation of GNE 7 Conclusion
Conclusion We proved the existence of a Nash equilibrium of a CCG for elliptically symmetric distributed random payoffs and continuous strategy set for each player. We characterize the set of Nash equilibria of a CCG using the solution set of a VI problem. For the case of shared constraints, we proved the existence of a generalized Nash equilibrium and give a characterization of the set of a certain types of generalized Nash equilibria using the solution set of a VI problem.