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On the Variational Equilibrium as a Refinement of the Generalized Nash Equilibrium Ankur A. Kulkarni

Uday V. Shanbhag

Department of Industrial and Enterprise Systems Engineering University of Illinois, Urbana-Champaign Urbana IL, U.S.A.

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Introduction Set of players N = {1, 2, . . . , N }; strategy “xi ”; objective “ϕi ”. Let x = (x1 , x2 , . . . , xN ) and x−i = (x1 , . . . , xi−1 , xi+1 , . . . , xN ) and (yi , x−i ) = (x1 , . . . , xi−1 , yi , xi+1 , . . . , xN ). Further requirement, “shared constraint”: x ∈ C ⊆ Ki (x−i ) := {yi ∈

Rm

i

| (yi , x−i ) ∈ C},

Rm .

K(x) :=

Y

Ki (x−i )

i∈N

Generalized Nash game with shared constraints “G” Ai (x−i )

minimize xi

subject to

ϕi (xi ; x−i ) xi ∈ Ki (x−i ).

Definition (Generalized Nash equilibrium (GNE)) A strategy tuple x ≡ (x1 , x2 , . . . , xN ) is a generalized Nash equilibrium of G if xi ∈ SOL(Ai (x−i )) for all i ∈ N .

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Variational Equilibrium

` Let F (x) = ∇1 ϕ1 (x)T

...

Find x ∈ K(x) s.t.

∇N ϕN (x)T

´T

. x is a GNE iff x solves

F (x)T (y − x) ≥ 0 ∀y ∈ K(x).

(QVI(K, F ))

A related VI is the following

Find x ∈ C s.t.

F (x)T (y − x) ≥ 0 ∀y ∈ C.

(VI(C, F ))

Definition (Variational equilibrium (VE)) If x is a solution of VI(C, F ) then x is said to be a variational equilibrium of G. Our goal: Study the VE as a refinement of the GNE

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Refinement of an Equilibrium

Definition (Refinement) A refinement of the set of equilibria of a game is a subset satisfying a certain rule where this rule has the property that any game with a nonempty set of equilibria also possesses an equilibrium satisfying this rule. Refined equilibria have some additional properties that make them more attractive Provide a way of selecting one or few of the equilibria Refinements of Nash equilibria: trembling hand perfect equilibria [Selten], proper equilibria [Myerson] etc

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What do we know about the VE?

Every VE is a GNE. i.e. SOL(V I(C, F )) ⊆ SOL(QV I(K, F ))

Suppose C = {x | c(x) ≥ 0, x ≥ 0}, c :

[Facchinei et al. ]

Rm → Rn , .then GNE x solves

0 ≤ xi ⊥ ∇i ϕi (x) − λi ∇i c(x) ≥ 0 0 ≤ λi ⊥ c(x) ≥ 0,

(KKTi )

GNE = equilibrium with non-shared multipliers VE = equilibrium with shared multipliers,

cf. Facchinei et al.

To show that the VE is a refinement of the GNE we need to show: SOL(QV I(K, F )) 6= ∅ =⇒ SOL(V I(C, F )) 6= ∅.

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Motivation – theory of games Such games often admit large number of GNEs For a modeller, the GNE by itself provides no clarity about the outcome of the game Example Ai (x−i )

minimize xi

subject to where X = are

P

i∈N

xi `(X) X = α

: λi ,

xi . The KKT conditions characterizing the GNE, x∗ , of this game ∗

∗

0

(xi `(X )) = λi ,

∗

X =α

∀i ∈ N .

Clearly, every point in the set C = {x | X = α} is a GNE of this game. Is there an equilibrium which is better than the rest?

If λi ’s are prices, it is tempting to think that x s.t. λi := λ for all i is a “better” equilibrium Does there always exist such a “better” equilibrium?

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Motivation – prices in a network game

Example Consider a bandwith sharing game. The Lagrange multipliers can be interpreted as the “price” charged on a player by an administrator. The equilibrium with non-shared multipliers is an equilibrium resulting from “discriminatory prices” But often the situation modeled makes it unrealistic for the administrator to be able to distinguish between various users The VE is really the “right” equilibrium. But . . . Does a VE always exist? Can an administrator charge a uniform price across all users to enforce equilibrium? If the VE is a refinement of the GNE, whenever an equilibrium with discriminatory prices exists, if and only if one with uniform prices also exists.

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Motivation – computation of equilibria

Example Imagine a game for which we know a GNE exists and our goal is to compute any GNE. A GNE is hard to compute since it involves the solution of a QVI A VE is comparatively easier to compute But do we know if a VE exists? When is the existence of a GNE sufficient for the existence of a VE? Computation of GNEs has often been restricted to those classes of games, S, for which a one can claim the existence of a VE, independently of the existence of a GNE. Eg games with compact C or coercive F . Is there a class of games, S 0 , S 0 ⊃ S, with properties weaker than these, that are not by themselves sufficient for the existence of a VE, but given that a GNE exists, imply the existence of the VE ?

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Structure of K

Lemma Let C be a closed set in m . Q mi 1 If C = not necessarily convex sets, then K(x) = C for i∈N Ci , where Ci ⊆ every x in C and is empty otherwise.

R

R

2

For any C, not necessarily convex, x ∈ K(x) if and only if x ∈ C.

3

x ∈ int(C) if and only if x ∈ int(K(x)).

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A counter-example for the refinement Example (Game with unique GNE and no VE) Let C = {(x1 , x2 ) | x2 ≥ e−x1 , x1 ≥ 0}, and K(x) = {(y1 , y2 ) | y2 ≥ e Let F (x) = (1 + x1 −

1 x2

−x1 ,

x1 ≥ 0, y1 ≥ 0, x2 ≥ e

−y1

}.

∗

, 1). x = (0, 1) is the only GNE.

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Contributions

Our goal: find conditions on F, C such that SOL(QV I(K, F )) 6= ∅ =⇒ SOL(V I(C, F )) 6= ∅.

1

Are there games other than those in S satisfying the above?

2

Is there a neat unifying criterion that can identify these games?

1

Yes – pseudo monotone F and some nice properties of the recession cone of C

2

Brouwer degree is seen to play an important role in relating the VE and the GNE and in our analysis.

And... Extend the above to primal-dual setting Prove that VE is a refinement for many games where we expect it to be Comparison with Harker and Facchinei et al.

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Brouwer degree theory

R

Let Ω ⊂ m be an open bounded set, f : Ω → p ∈ m \f (∂Ω).

R

Rm be continuous and

deg(f, Ω, p), is well defined if and only if p ∈ / f (∂Ω) Properties: 1

(Normalization) deg(1, Ω, p) = 1 if and only if p ∈ Ω.

2

(Solvability) deg(f, Ω, p) 6= 0 then f (x) = p for some x ∈ Ω.

3

(Homotopy invariance) deg(H(·, t), Ω, p) is independent of t ∈ [0, 1] for any continuous function H : Ω × [0, 1] → m and p ∈ m such that p∈ / ∪t∈[0,1] H(∂Ω, t).

R

R

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Primal results Theorem Let Ω be an open bounded set such that Ω ⊆ dom(K) and suppose K is continuous on e nat (∂Ω), then Ω. If 0 ∈ /F K

nat

nat

e , Ω, 0) = deg(F , Ω, 0). deg(F K C Theorem Consider game G and suppose K is continuous. Consider the following statements: (C0) G admits a GNE. (C1) There exists an open bounded set, Ω, with Ω ⊆ dom(K) such that Ω contains a GNE of G, and has no GNE of G on its boundary. e nat , Ω, 0) is well defined and (C2) For an open bounded set, Ω with Ω ⊆ dom(K), deg(F K nonzero. (C3) G admits a VE. Then, (a) we have (C2) =⇒ (C3). Consequently if (C2) holds for G, then G admits a GNE and a VE. (b) If G has the property that (C0) =⇒ (C1) =⇒ (C2), then G admits VE and GNE. 13 / 23

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Primal results – special cases

Games that admit VE independently of existence of GNE (i.e. S) are special cases Lemma (Brouwer’s fixed point theorem) Let C be compact and suppose Ω is an open bounded set large enough to strictly contain C. Then deg(Fnat C , Ω, 0) = 1 and the game admits a VE and a GNE. Lemma Suppose there exists an xref ∈ C such that lim

F (x)T (x − xref ) > 0.

(1)

x∈C,kxk→∞

There exists an open bounded set Ω such that deg(Fnat C , Ω, 0) = 1 and this game has a VE and GNE.

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Primal results – special cases Theorem (Facchinei and Pang)

Let K : m → 2R be a closed-valued and convex-valued point-to-set map. Let F : m → m be a continuous function. Suppose there exist a bounded open set e nat (∂Ω) and Ω ⊂ dom(K) and a vector xref ∈ Ω such that 0 ∈ /F K

R

R

m

R

1

K is continuous on Ω

2

xref belongs to K(x) for every x ∈ Ω

3

the following holds {x ∈ K(x) : (x − xref )T F (x) < 0} ∩ ∂Ω = ∅. e nat , Ω, 0) 6= 0. Then deg(F K Notice that {x ∈ K(x) : (x−xref )T F (x) < 0}∩∂Ω = {x ∈ C : (x − xref )T F (x) < 0} ∩ ∂Ω 6= ∅ | {z } =⇒ deg(Fnat ,Ω,0)6=0 C

Unifies existence results for QVI(K, F ) and VI(C, F ). 15 / 23

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Primal results – other classes of games Theorem Suppose F is pseudo-monotone and xref is a GNE of G. Consider the following conditions: 1

Either F (xref ) = 0 or −F (xref ) ∈ int(N (xref ; K(xref ))),

1’. T (xref ; K(xref ))∗ ⊆ int(C∗∞ ), 2

C∞ ⊆ T (xref ; K(xref )).

If G has the property that condition (2) holds and either of conditions (1) or (1’) holds then G admits a VE. Essentially says that for a certain class of games (C0) implies that either (C3) holds or (C2) holds, and hence (C0) =⇒ (C3). Pseudo-monotonicity of F and the properties of C in (1’),(2) are by themselves insufficient for the existence of a VE of G. But given that a GNE xref exists, the above theorem provides sufficient conditions for G to have a VE. There do exist games in S 0 other than those in S

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Primal results – other classes of games Example 1

Consider G where F is psuedo-monotone and C = {x | Ax ≥ b, x ≥ 0} for some nonnegative b ∈ n and n × m matrix A with nonnegative elements. Let xref be a GNE such that Axref = b. If (1) holds, this game also admits a VE.

R

2

Hard to get expressions for C∞ for non-affine C

3

Non-affine example in

R2

Figure: Example where (1’) holds 17 / 23

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Primal-dual results 0

x1 − Π+ (x1 − ∇1 ϕ1 (x) + λ1 ∇1 c(x)) . . . ` ´ − Π+ xN − ∇N ϕN (x) + λN ∇N c(x) λ1 − Π+ (λ1 − c(x))

B B Bx nat N G (x, Λ) := B B B @ 0

. . . λN − Π+ (λN − c(x))

x1 − Π+ (x1 − ∇1 ϕ1 (x) + λ1 ∇1 c(x))

B B BxN B Jnat (x, Λ) := B B B @

Theorem Let Ω be an open bounded set in

. . . ` ´ − Π+ xN − ∇N ϕN (x) + λN ∇N c(x) λ1 − Π+ (λ1 − c(x)) λ2 − λ1 . . . λN − λ1

1 C C C C, C C A

1 C C C C C, C C A

Rm+N such that 0 ∈/ Gnat (∂Ω). Then

deg(Gnat , Ω, 0) = deg(Jnat , Ω, 0).

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Primal-dual results ∇1 ϕ1 (x) − λ1 ∇1 c(x) B . B . B . B B∇N ϕN (x) − λN ∇N c(x) ψ(x, Λ) := B c(x) B B B . . @ . c(x) 0

1

(N times)

C C C C C C C C C A

Theorem Let ψ be a P0 function and let z ∗ = (x∗ , Λ∗ ) ∈ m+N be a zero of Gnat . If there exists an open bounded set Ω ⊂ m+N containing z ∗ such that 0 ∈ / Gnat (∂Ω), then deg(Gnat , Ω, 0) = 1.

R

R

Ψ := ∇ψ =

„ H C

« B , 0

Lemma Let c : m → be a concave function in C 2 . Assuume that for all x, Λ ≥ 0 H(x, Λ) is a block diagonal positive definite matrix with blocks H1,1 , . . . , HN,N where for each i ∈ N , the submatrix Hi,i is a positive definite matrix in mi ×mi . Then Ψ(x, Λ) is a P0 matrix.

R

R

R

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Example

Ai (x−i )

minimize xi

subject to where a ∈

R

m

,b ∈

R and U

i

ϕi (xi ; x−i ) = Ui (xi ) aT x ≥ b x ≥ 0,

: λi ,

∈ C 1 , ∇2 Ui  0.

Shared constraint game with C = {x |aT x ≥ b} F is strictly monotone, H  0. Ordinarily, this is not sufficient to claim that VI(C, F ) has a solution and that this game has a VE. But if given that this game has a bounded and nonempty set of GNEs (in the primal-dual space), one may use above result to conclude that a VE exists for this game. Conforms to the economic intuition and provides a formal justification for it.

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Comparison with Harker and Facchinei et al.

Theorem (Harker 1991) Let F and L be respectively point-to-point and point-to-set mappings from itself. Suppose that there exists a nonempty closed, convex set A such that

R

m

to

(i) L(x) ⊆ A for all x ∈ A and (ii) x ∈ L(x) for all x ∈ A. Then any solution to the variational inequality VI(A, F ) is a solution of QVI(L, F ) Theorem (Facchinei et al. 2007) If x is a solution of V I(C, F ) then x is a solution of QVI(K, F ). e nat , Ω, 0) = deg(Fnat , Ω, 0) does not imply either of these results. deg(F K C None of these results help say any thing about SOL(QV I(K, F )) 6= ∅ =⇒ SOL(V I(C, F )) 6= ∅. Harker’s result is hard to use [Facchinei et al. ] Harker’s result does not cover generalized Nash games with shared constraints [Facchinei et al. ]

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Comparison with Harker and Facchinei et al. Theorem Let C ⊆ m be closed and convex with nonempty interior. Let A ⊆ consider the following three conditions:

R

R

m

and

(i) A is closed, convex, and has nonempty interior (ii) K(x) ⊆ A for all x ∈ A (iii) x ∈ K(x) for all x ∈ A. If A satisfies (i), (ii), and (iii), then A = C. Therefore, (a) if C satisfies the above three conditions, it is the only set satisfying these conditions. (b) If C does not satisfy these conditions, there does not exist any set that satisfies them. Furthermore, Theorem SOL(QV I(K, F )) = SOL(QV I(K ∩ C, F )). Applying Harker’s theorem to L(x) = K(x) ∩ C, we see A = C satisfies (i), (ii). Thus SOL(V I(C, F )) ⊆ SOL(QV I(K ∩ C, F )) = SOL(QV I(K, F )), i.e. SOL(V I(C, F )) ⊆ SOL(QV I(K, F )), i.e. the result of Facchinei et al..

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Thank you!

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