Followers’ Strategy in Stackelberg Equilibrium Problems on Curved Strategy Sets

Followers’ Strategy in Stackelberg Equilibrium Problems on Curved Strategy Sets

Alexandru Krist´aly

Obuda University, Institute of Applied Mathematics, Budapest, Hungary´ Babes¸-Bolyai University, 400591 Cluj-Napoca, Romania

Email address:alexandru.kristaly@econ.ubbcluj.ro

Szil´ard Nagy

Babes¸-Bolyai University, 400591 Cluj-Napoca, Romania Email address:szilard.nagy@econ.ubbcluj.ro

Abstract: We study the existence of Stackelberg equilibrium points on strategy sets which are geodesic convex in certain Riemannian manifolds by using metric projection arguments.

The present results extend those obtained in Nagy [J. Global Optimization (2013)] in the Euclidean context.

Keywords: Stackelberg model; curved spaces; variational analysis

1 Introduction

Recently, the second author obtained certain existence and location results for the Stackelberg equilibria in the Euclidean framework, see [9]. More precisely, the existence of solutions for the leader-follower game has been obtained via the study of certain variational inequalities defined on the strategy sets by using the variational backward induction method.

The purpose of the present study is to extend the analytical results from [9] to games defined on strategy sets which are embedded in a geodesic convex manner into cer- tain Riemannian manifolds. Similar studies can be found in the literature, where certain variational arguments are applied to study equilibrium problems on Rieman- nian manifolds, see [4], [7], [11], [10] and references therein.

For simplicity, in the present paper we shall consider only two players although our arguments can be extended to several players as well. Let K₁⊂M₁andK₂⊂M₂ be two sets in the Riemannian manifolds(M₁,g1)and(M₂,g2), respectively, and let h1,h2:M1×M2→Rbe the payoff functions for the two players. As we already

know from the backward induction method, the first step (for the follower) is to find the response set

RSE(x₁) ={x₂∈K2:h2(x₁,y)−h₂(x₁,x2)≥0, ∀y∈K2}

for every fixedx₁∈K₁. If RSE(x₁)6=/0 for everyx₁∈K₁, the next step (for the leader) is to minimize the map x7→h₁(x,r(x))onK₁whereris a fixed selection function of the set-valued mapx7→RSE(x); more precisely, the objective of the first player is to determine the set

SSE ={x₁∈K₁:h₁(x,r(x))−h₁(x₁,r(x₁))≥0,∀x∈K₁}.

Since the location of the setsRSE(x₁)andSSE is not an easy task, we shall intro- duce further sets related to them by variational inequalities defined on the Rieman- nian manifolds. Let us assume thath₂:M₁×M₂→Ris a function of classC¹; for everyx₁∈K₁, we introduce the set

RSV(x₁) =

x2∈K2:g2

∂h₂

∂x₂(x₁,x2),exp⁻¹_x

2 (y)

≥0, ∀y∈K2

. Here and in the sequel, exp denotes the usual exponential function in Riemannian geometry. According to [4] and [5], it is more easier to determine the setRSV(x₁) thanRSE(x₁).Moreover, usually we have thatRSE(x₁)⊂RSV(x₁),thus we shall choose the appropriate Stackelberg equilibrium candidates from the elements of the latter set. Finally, by imposing further curvature assumptions on the Riemannian manifolds we are working on, we are able to characterize the elements of the set RSV(x₁)by the fixed points of a suitable set-valued map which involves the metric projection map into the setK2. In fact, we shall assume that the strategy sets are em- bedded into non-positively curved Riemannian manifolds where two basic proper- ties of the metric projection will be deeply exploited; namely, the non-expansiveness and the so-called Moskovitz-Dines property (see [8]); for further details, see Sec- tion 2. Having this fixed-point characterization, we will be able to apply various fixed point theorems on (acyclic) metric spaces in order to find elements of the set RSV(x₁). We emphasize that projection-like methods for Nash equilibria have been developed in the Euclidean context in [1], [15], [16].

We assume finally that h₁:M₁×M₂→Ris a function of classC¹and for every x₁∈K₁we have that RSV(x₁)6=/0. If we are able to choose aC¹-class selection r:K₁→K₁of the set-valued mapRSV, we also introduce the set

SSV =

x₁∈K₁:g₁ ∂h₁

∂x₁(x₁,r(x₁)),exp⁻¹_x

1 (y)

≥0,∀y∈K₁

. In particular,SSV contains the optimal strategies of the leader, i.e., the minimizers for the mapx7→h₁(x,r(x))onK₁.

Section 2 contains some basic notions and results from Riemannian geometry which are needed for our investigations: geodesics, curvature, metric projections, Moskovitz- Dines property, etc. Finally, in Section 3 we present the main results of the paper concerning the strategy of the follower.

2 Preliminaries

2.1 Elements from Riemannian manifolds

Let (M,g) be a connected m-dimensional Riemannian manifold, m≥2, and let T M=∪_p∈M(p,TpM)andT^∗M=∪_p∈M(p,T_p^∗M)be the tangent and cotangent bun- dles toM.Ifξ ∈T_p^∗Mthen there exists a uniqueW_ξ ∈TpMsuch that

hξ,Vi_g,p=g_p(W_ξ,V)for allV∈T_pM. (1)

Due to (1), the elementsξ andW_ξ are identified. The norms onTpMandT_p^∗M are defined by

kξk_g=kW_ξk_g=q

g(W_ξ,W_ξ).

It is clear that for everyV∈TpMandξ∈T_p^∗M,

|hξ,Vi_g| ≤ kξk_gkVk_g. (2) Let h:M→Rbe aC¹function at p∈M; the differential of hat p, denoted by dh(p), belongs toT_p^∗Mand is defined by

hdh(p),Vi_g=g(gradh(p),V)for allV∈T_pM.

Let γ :[0,r]→M be aC¹ path, r>0. The length of γ is defined by L_g(γ) = Rr

0kγ(t)k˙ _gdt.For any two pointsp,q∈M, let

d_g(p,q) =inf{L_g(γ):γis aC¹path joiningpandqinM}.

The functiond_g:M×M→Rclearly verifies the properties of the metric function.

For everyp∈Mandr>0, the open ball of centerp∈Mand radiusr>0 is defined by

B_g(p,r) ={q∈M:d_g(p,q)<r}.

AC^∞parameterized pathγ is a geodesic in(M,g)if its tangent ˙γ is parallel along itself, i.e.,∇γ˙γ˙=0. Here,∇is the Levi-Civita connection. The geodesic segment γ :[a,b]→M is called minimizing if L_g(γ) =d_g(γ(a),γ(b)).From the theory of ODE we have that for everyV ∈T_pM, p∈M, there exists an open intervalI_V 30 and a unique geodesicγ_V :I_V →M withγ_V(0) =pand ˙γ_V(0) =V.On account of [2, p. 64], we introduce the exponential map exp_p:T_pM→Mas exp_p(V) =γ_V(1).

Moreover,

dexp_p(0) =id_TpM.

In particular, for every two pointsq1,q2∈Mwhich are close enough to each other, we have

kexp⁻¹_q

1 (q₂)k_g=d_g(q₁,q₂). (3)

LetK⊂Mbe a non-empty set. Let

P_K(q) ={p∈K:d_g(q,p) =inf

z∈Kd_g(q,z)}

be the set of metric projectionsof the pointq∈M to the setK. According to the theorem of Hopf-Rinow, if(M,g)is complete, then for any closed setK⊂Mwe have that card(P_K(q))≥1 for everyq∈M. The mapP_Kisnon-expansiveif

d_g(p₁,p₂)≤d_g(q₁,q₂) for allq₁,q₂∈Mandp₁∈P_K(q₁),p₂∈P_K(q₂).

In particular, whenP_Kis non-expansive, thenKis a Chebishev set, i.e., card(P_K(q)) = 1 for everyq∈M.

The setK⊂Misgeodesic convexif every two pointsq1,q2∈Kcan be joined by a unique minimizing geodesic whose image belongs toK.Clearly, relation (3) holds for everyq₁,q₂∈Kin a geodesic convex setK since exp⁻¹_q

i is well-defined onK, i∈ {1,2}. The function f :K→Risconvex, if f◦γ:[0,1]→Ris convex in the usual sense for every geodesicγ:[0,1]→KonceK⊂Mis a geodesic convex set.

A non-empty closed set K⊂M verifies theMoskovitz-Dines propertyif for fixed q∈Mandp∈Kthe following two statements are equivalent:

(MD₁) p∈P_K(q);

(MD₂) Ifγ:[0,1]→Mis the unique minimal geodesic fromγ(0) =p∈Ktoγ(1) = q, then for every geodesicσ:[0,δ]→K(δ≥0)emanating from the pointp, we haveg(γ(0),˙ σ˙(0))≤0.

A Riemannian manifold (M,g)is aHadamard manifoldif it is complete, simply connected and its sectional curvature is non-positive. We recall that on a Hadamard manifold(M,g), ifh(p) =d_g²(p,p₀),p₀∈Mis fixed, then

gradh(p) =−2 exp⁻¹_p (p₀). (4)

It is well-known that on a Hadamard manifold(M,g)every geodesic convex set is a Chebyshev set. Moreover, we have

Proposition 1. ([3], [13])Let(M,g)be a finite-dimensional Hadamard manifold, K⊂M be a closed set. The following statements hold true:

(i) If K⊂M is geodesic convex, it verifies the Moskovitz-Dines property;

(ii) P_K is non-expansive if and only if K⊂M is geodesic convex.

2.2 Basic properties of the response sets

In the sequel we shall establish some basic properties of the response sets by using some elements from the theory of variational inequalities on Riemannian manifolds.

Lemma 1. Let(M_i,g_i)be Riemannian manifolds, h_i:M₁×M₂→Rbe functions of class C¹, and K_i⊂M_iclosed, geodesic convex sets, i=1,2.Then the following assertions hold:

(i) RSE(x₁)⊆RSV(x₁)for every x₁∈K₁;

(ii) RSE(x₁) =RSV(x₁)when h2(x₁,·)is convex on K2for some x1∈K1;

(iii) SSE⊆SSVwhen x7→RSV(x)is a single-valued function which has a C¹−extension to an arbitrary open neighborhood D₁⊂M₁of K₁.

Proof.(i) Letx₂∈RSE(x₁)be an arbitrarily fixed element, i.e.,h₂(x₁,y)≥h₂(x₁,x₂) for ally∈K₂. By definition, we have that

g₂ ∂h₂

∂x₂(x₁,x₂),exp⁻¹_x

2 (y)

= lim

t→0⁺

h₂(x₁,exp_x

2(texp⁻¹_x

2 (y))−h₂(x₁,x₂)

t , ∀y∈K₂.

SinceK₂is geodesic convex, the element exp_x

2(texp⁻¹_x

2 (y)∈K₂for everyt∈[0,1]

whenevery∈K₂.By the above expression one has that for everyy∈K, g2

∂h2

∂x₂(x₁,x2),exp⁻¹_x

2 (y)

≥0, which implies thatRSE(x₁)⊆RSV(x₁)for allx1∈K1.

(ii) Since the functionh₂(x₁, .)is convex and of classC¹, one has h₂(x₁,y)−h₂(x₁,x₂)≥g₂

∂h₂

∂x₂(x₁,x₂),exp⁻¹_x

2 (y)

for ally∈K₂,see [14]. Taking into account thatx₂∈RSV(x₁), one has that g₂

∂h2

∂x₂(x₁,x₂),exp⁻¹_x

2 (y)

≥0

for all y∈K2. Thus, one has h2(x₁,y)−h2(x₁,x2)≥0 for ally∈K2, i.e., x2∈ RSE(x₁).

(iii) The proof is similar to (i). 4

In the sequel, we shall prove that the elements of the setRSV(x₁)can be obtained as the fixed points of a carefully choosen map. More precisely, for a fixedx₁∈K₁ andα>0, letFα^x1:K₂→K₂be defined by

Fα^x1(x) =P_K2

exp_x

−α∂h₂

∂x2

(x₁,x)

. (5)

Theorem1. Let(M₁,g₁)be a Riemannian manifold, and(M₂,g₂)be a Hadamard manifold. Let h₂:M₁×M₂→Rbe a function of classC¹ andK_i⊂M_i closed, geodesic convex sets,i=1,2.Letx₁∈K₁. The following statements are equivalent:

(i) x₂∈RSV(x₁);

(ii) Fα^x1(x₂) =x₂for allα>0;

(iii) Fα^x1(x₂) =x₂for someα>0.

Proof. Let us fixx2∈RSV(x₁)arbitrarily, wherex1∈K1. By definition, we have that

g₂

−α∂h₂

∂x₂(x₁,x₂),exp⁻¹_x

2 (y)

≤0, ∀y∈K₂,

for all/someα >0.Letγ,σ:[0,1]→M₂be the unique minimal geodesics defined by

γ(t) =exp_x

2(−tα∂h₂

∂x₂(x₁,x₂)) and

σ(t) =exp_x

2(texp⁻¹_x

2 (y))

for any fixed α >0 and y∈K2. SinceK2 is geodesic convex in (M₂,g2), then Imσ⊂K2and

g₂(γ˙(0),σ˙(0)) =g₂

−α∂h₂

∂x2

(x₁,x₂),exp⁻¹_x

2 (y)

,

i.e.,(MD₂)holds. By the Moskovitz-Dines property, see Proposition 1, one has that x₂=γ(0)∈P_K2(γ(1)) =P_K2

exp_x

2

−α∂h2

∂x₂(x₁,x₂)

=Fα^x1(x₂).

Since card(Fα^x1(x₂)) =1, the proof is complete. 4 Remark. Note that for allα>0,

RSV(x₁) =

x₂∈K₂:P_K2

exp_x

2

−α∂h₂

∂x2

(x₁,x₂)

=x₂

.

3 Follower strategy: existence of equilibria

3.1 Compact case

Theorem2. (Compact case)Let(M_i,g_i)be Hadamard manifolds,h_i:M₁×M₂→R be functions of classC¹andK_i⊂M_icompact, geodesic convex sets,i=1,2.Then the following statements hold:

(i) RSV(x₁)6=/0 for everyx1∈K1;

(ii) SSV 6=/0, whenever RSV(x₁)is a singleton for everyx1∈K1 and the map x7→RSV(x)has aC¹−extension to an arbitrary open neighborhoodD₁⊂M₁ ofK₁.

Proof. (i) Fixx₁∈K₁andα>0.SinceK₂is a Chebishev set andP_K2 is globally Lipschitz, we see thatFα^x1:K₂→K₂is a single-valued continuous function; in par- ticular,Fα^x1 :K₂→K₂has a closed graph. Moreover, sinceK₂is geodesic convex, it is contractible, thus an acyclic set. Now, we may apply the fixed point theorem of Begle on the compact setK₂, obtaining thatFα^x1 has at least a fixed pointx₂∈K₂. Due to Theorem 1,x₂∈RSV(x₁), which concludes the proof of (i).

(ii) For someβ>0, we introduce the mapGβ :K₁→K₁defined by Gβ(x) =P_K1

exp_x

−β∂h₁

∂x(x,RSV(x))

.

Since card(RSV(x)) =1 for every x∈K₁ and the map x7→RSV(x) has aC¹- extension to an arbitraryD₁⊂M₁ofK₁, the functionGβ is well-defined for every β >0. By the hypotheses, the functionGβ is also continuous, thus on account of the Belge fixed point theorem, there exits at least x₁∈K₁ such thatGβ(x₁) =x₁. Since(M₁,g1)is a Hadamard manifold where the Moskovitz-Dines property holds, an analogous argument as in Theorem 1 shows that Gβ(x₁) =x1is equivalent to

x1∈SSV. The proof is complete. 4

3.2 Non-compact case

When the strategy sets are non-compact, certain growth assumptions are needed on the payoff functions in order to guarantee the existence of Stackelberg equilibria.

We first assume that for somex₁∈K₁one has (H_x^h₁²)There existsx₂∈K₂such that

Lx₁,x₂= lim sup

dg2(x,x₂)→∞,x∈K₂

g2

∂h₂

∂x₂(x1,x),exp⁻¹_x (x2) +g2

_∂h

2

∂x₂(x1,x2),exp⁻¹_x2 (x)

dg₂(x,x₂) <

<−

∂h2

∂x₂(x₁,x₂) g2

.

Theorem3. Let(M₁,g₁)be a Riemannian manifold, and(M₂,g₂)be a Hadamard manifold. Let h₂:M₁×M₂→Rbe a function of classC¹ andK_i⊂M_i closed, geodesic convex sets,i=1,2.Letx₁∈K₁and assume that hypothesis(H_x^h₁²)holds true. ThenRSV(x₁)6=/0.

Proof.LetE₀∈Rsuch that

L_x1,x2<−E₀<−

∂h₂

∂x₂(x₁,x₂) _g

2

.

On account of hypothesis(H_x^h₁²)there existsR>0 large enough such that for every x∈K₂withd_g2(x,x₂)≥R, we have

g2

∂h2

∂x₂(x₁,x),exp⁻¹_x (x₂)

+g₂ ∂h2

∂x₂(x₁,x2),exp⁻¹_x

2 (x)

≤ −E₀d_g2(x,x2).

Clearly, one may assume thatK₂∩B_g2(x₂,R)6=/0.In particular, from (3) and (2), for everyx∈K₂withd_g2(x,x₂)≥R, the above relation yields

g2

∂h2

∂x2

(x₁,x),exp⁻¹_x (x₂)

≤ −E0dg2(x,x2) +

∂h2

∂x2

(x1,x2) _g

2

kexp⁻¹_x2 (x)kg2 (6)

= −E0+

∂h2

∂x2

(x1,x2) _g

2

!

dg2(x,x2)

< 0.

LetK_R=K₂∩B_g2(x₂,R).It is clear thatK_Ris a geodesic convex, compact subset of(M₂,g₂). Due to Theorem 2, we immediately have that there exists ˜x₂∈K_Rsuch that

g₂ ∂h₂

∂x2

(x₁,x˜₂),exp⁻¹_x˜

2 (y)

≥0 for ally∈K_R. (7)

Note thatd_g2(x˜₂,x₂)<R.By assuming the contrary, from (6) withx=x˜₂we have that

g₂ ∂h₂

∂x2

(x₁,x˜₂),exp⁻¹_x˜

2 (x₂)

<0, by contradicting relation (7).

Let us choosez∈K₂arbitrarily. From the fact thatd_g2(x˜₂,x₂)<R,forε>0 small enough, the element y=exp_x˜

2(εexp⁻¹_x˜

2 (z))belongs both toK2∩Bg₂(x₂,R) =KR. By replacingyinto (7), we obtain that

g₂ ∂h₂

∂x₂(x₁,x˜₂),exp⁻¹_x˜

2 (z)

≥0.

Sincez∈K₂is arbitrarily fixed, one has that ˜x₂∈RSV(x₁), which ends the proof.

4

In the sequel, we are dealing with another class of functions. For a fixedx1∈K1, α>0 and 0<ρ<1 we introduce the hypothesis:

(H_x^α,ρ

1 ): dg₂

exp_x

−α∂h₂

∂x₂(x₁,x)

,exp_y

−α∂h₂

∂x₂(x₁,y)

≤

≤(1−ρ)d_g2(x,y)for allx,y∈K₂.

For fixedx₁∈K₁andα>0, we consider the following two dynamical systems:

(a) let(DDS)_x1 be the discrete differential system in the form yn+1=Fα^x1(P_K2(y_n)), n≥0,

y₀∈M₂;

(b) Let(CDS)_x1be the continuous differential system in the form ( _dy

dt =exp⁻¹_y(t)(Fα^x1(P_K2(y(t)))), y(0) =x₂∈M₂.

The main result of the present section is the following theorem.

Theorem4 (Non-compact case). Let(M₁,g₁)be a Riemannian manifold, and(M₂,g₂) be a Hadamard manifold. Let h₂:M₁×M₂→R be a function of classC¹ and K_i⊂M_iclosed, geodesic convex sets,i=1,2.Letx₁∈K₁and assume that hypoth- esis (H_x^α,ρ₁ )holds true. ThenRSV(x₁)is a singleton and both dynamical systems, (DDS)_x1 and(CDS)_x1, exponentially converge to the unique element ofRSV(x₁).

Proof.Since(M₂,g₂)is a Hadamard manifold, for the geodesic convex setK₂⊂M₂ we have thatP_K2is non-expansive. Therefore, by(H_x^α,ρ₁ ), one has for everyx,y∈K₂ that

d_g2(Fα^x1(x),Fα^x1(y))

= d_g2

P_K2

exp_x

−α∂h2

∂x₂(x₁,x)

,P_K2

exp_y

−α∂h2

∂x₂(x₁,y)

≤ dg₂

exp_x

−α∂h₂

∂x₂(x₁,x)

,exp_y

−α∂h₂

∂x₂(x₁,y)

≤ (1−ρ)d_g2(x,y).

Consequently, the functionFα^x1 is a(1−ρ)−contraction onK₂.

(a) The system (DDS)_x1. We shall apply the Banach fixed point theorem to the functionFα^x1 :K₂→K₂, by guaranteeing the existence of the unique fixed point of Fα^x1 for everyx₁∈K₁. Moreover, every iterated sequence in the dynamical system (DDS)_x1 converges exponentially to the unique fixed point x₂∈K₂ ofFα^x1. Due to Theorem 1 the setRSV(x₁)is a singleton with the elementx₂. Moreover, for all k∈Nwe have that

d_g2(y_k,x₂)≤(1−ρ)^k

ρ d_g2(y₁,y₀).

(b)The system(CDS)_x1. First of all, standard ODE theory shows that(CDS)_x1 has a (local) solution in[0,T).We actually prove that T = +∞. To see this fact, we assume thatT <+∞,and we introduce the Lyapunov function which has the form

h_x1(t) =1

2d_g2(y(t),x₂)². Note that for a.e.t∈[0,T), we have

d

dthx₁(t) = −g₂

exp⁻¹_y(t)(x₂),dy dt

= −g₂

exp⁻¹_y(t)(x₂),exp⁻¹_y(t)(Fα^x1(P_K2(y(t))))

= −g2

exp⁻¹_y(t)(x₂),exp⁻¹_y(t)(Fα^x1(P_K2(y(t))))−exp⁻¹_y(t)(x₂)

−g₂

exp⁻¹_y(t)(x₂),exp⁻¹_y(t)(x₂))

≤ kexp⁻¹_y(t)(Fα^x1(P_K2(y(t))))−exp⁻¹_y(t)(x₂)k_g2kexp⁻¹_y(t)(x₂)k_g2

−kexp⁻¹_y(t)(x₂)k²_g

2.

By using the fact that(M₂,g₂)is a Hadamard manifold, a Rauch comparison theo- rem and further straightforward estimates show that

kexp⁻¹_y(t)(Fα^x1(P_K2(y(t))))−exp⁻¹_y(t)(x₂)k_g2 ≤d_g2(Fα^x1(P_K2(y(t))),x₂).

Therefore, by (3) and the non-expansiveness ofP_K2, we have d

dthx₁(t) ≤ dg₂(Fα^x1(P_K2(y(t))),x2)d_g2(y(t),x2)−dg₂(y(t),x2)²

= dg₂(Fα^x1(P_K2(y(t))),Fα^x1(x₂))d_g2(y(t),x2)−d_g2(y(t),x2)²

≤ (1−ρ)d_g2(P_K2(y(t)),x2)d_g2(y(t),x₂)−dg₂(y(t),x2)²

≤ (1−ρ)d_g2(y(t),x2)²−dg₂(y(t),x2)²

= −ρd_g2(y(t),x2)²

= −2ρh_x1(t),a.e.t∈[0,T).

Therefore, one has d

dt[h_x1(t)e^2ρt] = d

dth_x1(t) +2ρh_x1(t)

e^2ρt≤0.

In particular, the function t7→h_x1(t)e^2ρt is non-increasing; therefore, for allt ∈ [0,T)one has that h_x1(t)e^2ρt ≤h_x1(0). Consequently, t7→y(t)can be extended beyondT, contradicting our assumption. Therefore,T= +∞.

The above estimate gives that for everyt≥0,h_x1(t)≤h_x1(0)e^−2ρt.In particular, it yields that

d_g2(y(t),x₂)≤d_g2(y₀,x₂)e^−ρt.

The proof is concluded. 4

Remark. Assume that M_i =R^mi, i=1,2 and ^∂f2

∂x₂(x₁,·) is an λ−Lipschitz and σ−strictly monotone function for somex₁∈K₁, i.e.,

• k^∂f2

∂x2(x₁,x)−^∂f2

∂x2(x₁,y)k ≤λkx−yk,

• h^∂f2

∂x₂(x₁,x)−^∂f2

∂x₂(x₁,y),x−yi ≥σkx−yk², ∀x,y∈R^m2. In this case,(H_x^ε,ρ₁ )holds true with

0<ε<σ−p

(σ²−λ²)₊ λ² and

ρ=1−p

1−2ε σ+ε²λ²∈(0,1).

Remark. Very recently, Krist´aly and Repovs [6] proved that the Moskovitz-Dines property on a generic Riemannian manifold implies the non-positiveness of the sec- tional curvature. Consequently, in order to develop the aforementioned results on

’curved’ spaces, the non-positiveness of the sectional curvature seems to be a natural requirement.

Remark. By following the non-smooth critical point theory of Szulkin [12], it would be interesting to guarantee not only the existence of Stackelberg equilibrium points but also some multiplicity results. Here, the indicator function of geodesic convex

sets as well as the Fr´echet subdifferential of the indicator function (as the normal cone to the geodesic convex set) seem to play crucial roles which will be investigated in a forthcoming paper.

Acknowledgement

The work of Szil´ard Nagy is supported by the grant PCCE-55/2008.

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