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On the Generic Structure and Stability of
Stackelberg Equilibria
Alberto Bressan and Yilun Jiang
Department of Mathematics, Penn State University.University Park, PA 16802, USA.
e-mails: [email protected], [email protected]
January 19, 2019
Abstract
We consider a non-cooperative Stackelberg game where the two players choose theirstrategies within domains X ⊆ Rm and Y ⊆ Rn. Assuming that the cost functions F,Gfor the two players are sufficiently smooth, we study the structure of the best reply mapfor the follower and the optimal strategy for the leader. Two main cases are considered:either X = Y = [0, 1], or X = R, Y = Rn with n ≥ 1. Using techniques from differentialgeometry, including a multi-jet version of Thom’s transversality theorem, we prove thatfor an open dense set of cost functions F ∈ C2 and G ∈ C3 the Stackelberg equilibrium isunique and is stable w.r.t. small perturbations of the two cost functions.
Key words: Non-cooperative game, Stackelberg equilibrium, stability, generic property.
1 Introduction
In the theory of non-cooperative games, the concept of Stackelberg equilibrium [18] has beenwidely investigated, due to its several applications to economic models [2]. In a basic setting,a game for two players can be formulated as follows.
• Player 1 (the leader) chooses x ∈ X and seeks to minimize his cost F (x, y).
• Player 2 (the follower) chooses y ∈ Y and seeks to minimize his cost G(x, y).
Here X,Y are topological spaces, while F,G : X × Y 7→ R are continuous functions.
For a given x ∈ X, the set of best replies for the follower is defined as
R(x).={y∗ ∈ Y ; G(x, y∗) ≤ G(x, y) for all y ∈ Y
}. (1.1)
1
We say that a couple (x∗, y∗) ∈ X × Y is a Stackelberg equilibrium if y∗ ∈ R(x) and
F (x∗, y∗) ≤ F (x, y) for all x ∈ X and y ∈ R(x). (1.2)
This models a situation where the leading player announces his strategy x ∈ X in advance,and the follower chooses a reply y ∈ Y which minimizes his own cost G(x, y).
In the literature, the existence of a Stackelberg equilibrium is known under fairly generalassumptions [2, 4, 10, 13, 17]. A major related issue is the uniqueness and stability of thisequilibrium. Namely, if the cost functions F,G are slightly perturbed, does the new gamestill have a unique solution, close to the original one? This problem has been investigated in[10, 12, 13], within the general class of continuous cost functions. As pointed out in [10], itis not possible to obtain, under sufficiently general conditions, existence and stability resultsfor the exact Stackelberg solutions. For this reason, in the above papers, a weaker concept ofε-solution was used.
Aim of the present paper is to study stability for the best reply map and for exact Stackelbergsolutions, within a class of smooth functions F,G : Rm × Rn 7→ R. In this setting, examplesof games with multiple equilibria are easy to construct. However, our main results show that,for “most” functions F,G (in a topological sense), the Stackelberg equilibrium is unique andis stable under small perturbations. While the results in [10, 12, 13] are based general topo-logical principles, our stability results rely on completely different techniques, stemming fromdifferential geometry; namely: Sard’s theorem and a multi-jet version of Thom’s transversalitytheorem [3, 7].
The analysis of Stackelberg equilibria can be accomplished in two steps.
(i) Study the graph of the best reply map R(·), namely
Graph(R) = {(x, y) ; y ∈ R(x)} ⊂ Rm × Rn. (1.3)
Show that, for a generic function G ∈ C3(Rm+n), this graph can be expressed in termsof finitely many equalities or inequalities, in generic position.
(ii) Study the constrained minimization problem for the function F , restricted to Graph(R).Show that, for a generic function F ∈ C2(Rm+n), a unique global minimum exists, whichis stable under small perturbations.
We recall that a property is said to be generic if it holds on the intersection of countablymany open dense sets. The main goal of the present paper is to study the stability of theequilibrium (x∗, y∗) under perturbations of the cost functions F,G, in the following sense.
Definition. Given the cost functions F,G, we say that the Stackelberg equilibrium (x∗, y∗) isstrongly stable if, for any ε > 0 there exists δ > 0 such that the condition
‖F − F‖C2 ≤ δ, ‖G−G‖C3 ≤ δ, (1.4)
implies that the perturbed game, with F,G replaced by F , G, has a unique Stackelberg equilib-rium (x, y) ∈ X × Y . Moreover,
|x− x∗| ≤ ε, |y − y∗| ≤ ε. (1.5)
2
We first consider the basic case where X = Y = [0, 1]. Using Thom’s transversality theoremwe prove that the set of couples (F,G) ∈ C2×C3 that yield a unique, strongly stable Stackelbergequilibrium is open and dense. In a later section we prove similar results in the case wherethe strategy of the follower takes values in a multi-dimensional set. Namely: X = R whileY = Rn.
The remainder of the paper is organized as follows. Section 2 collects some basic stabilityresults on the stability of the minimizer of a C2 function F restricted to a set A ⊂ RN which isdefined in terms of finitely many equalities or inequalities. In Section 3 we begin the analysisof the best reply map. When X = Rm, Y = Rn, the necessary conditions for optimality implythat
Graph(R) ⊆ M .= {(x, y) ∈ Rm+n ; ∇yG(x, y) = 0}. (1.6)
As a first step, we observe that, by Sard’s theorem, for a generic function G ∈ C3 the set Min (1.6) is a C2 manifold. Theorem 4.1 in Section 4 describes in detail the generic structureof the best reply map, in the one-dimensional case where X = Y = [0, 1]. In turn, this yieldsthe generic stability of the Stackelberg equilibrium, proved in Section 5.
The analysis in Section 6 shows that similar results on the structure of the best reply mapand on the stability of the Stackelberg equilibrium still hold, in the case where the followerchooses his strategy within a multi-dimensional space. Namely, X = R while Y = Rn.Section 7 contains some concluding remarks. In particular, we discuss the possible extensionof our results to the case where the strategy of the leader also lies in a multi-dimensionalset X ⊆ Rm. Finally, an Appendix collects some basic results from differential geometry. Amulti-jet version of Thom’s transversality theorem is proved, which provides a key ingredientfor our analysis.
In connection with Nash equilibria for special classes of non-cooperative games, generic prop-erties of solutions have been studied in [9, 14].
2 Minima of generic functions on generic sets
In a Stackelberg game, the leader seeks to minimize his own cost F (x, y), restricted to thegraph of the best reply map R(·). As it will be shown in a later section, for a generic functionG in (1.1), this graph can be expressed in terms of a finite set of equalities and inequalities.
We thus consider here a constrained minimization problem of the general form
minx∈A
f(x), (2.1)
under the following assumptions.
(B1) f : RN 7→ R is twice continuously differentiable. Moreover, lim|x|→∞ f(x) = +∞.
(B2) A ⊂ RN is a nonempty closed set, described in terms of finitely many equalities andinequalities:
A ={x ∈ RN ; φi(x) = 0, ψj(x) ≥ 0, i ∈ I, j ∈ J
}. (2.2)
3
(B3) All functions φi, ψj are in C2(RN ). Moreover, for any subset J ′ ⊆ J and any point xsuch that
φi(x) = ψj(x) = 0 for all i ∈ I, j ∈ J ′, (2.3)
the gradients∇φi(x), ∇ψj(x), i ∈ I, j ∈ J ′, (2.4)
are linearly independent.
Roughly speaking, we would like to prove that, for all functions f, φi, ψj in an open denseset of C2, the problem (2.1) admits a unique minimizer, which is stable under perturbations.With this goal in mind, we denote by F∞ the family of all functions f satisfying (B1). SinceF∞ is not a vector space, a suitable topology must first be defined.
We recall that Ck(RN ) is a Banach space with the norm
‖f‖Ck.= sup
|α|≤ksupx∈RN
∣∣∣Dαf(x)∣∣∣ . (2.5)
Using a standard notation (see for example [8]), here
Dαf(x).=
∂|α|
∂α1x1 · · · ∂
αNxN
f(x)
denotes a partial derivative of order |α| = α1 + · · ·+αN . On the set F∞ we now consider thedistance
d2(f, g).= min
{1, ‖f − g‖C2
}. (2.6)
With this distance, (F∞, d2) is a complete metric space. Notice that the convergence fn → fholds iff ‖fn − f‖C2 → 0.
Next, we introduce
Definition 2.1 Under the assumptions (B2)-(B3), we say that the global minimum (2.1) isattained at a point x in generic position if the following conditions hold.
(i) x ∈ A is the unique point where the global minimum is attained.
(ii) Setting J ′ .= {j ∈ J ; ψj(x) = 0}, there exists constants αi, βj, i ∈ I, j ∈ J ′ such that
∇f(x) =∑i∈I
αi∇φi(x) +∑j∈J ′
βj ∇ψj(x), (2.7)
with βj > 0 for all j ∈ J ′.
(iii) There exists ρ, ε > 0 such that
f(x)− f(x) ≥ ε |x− x|2 for all x ∈ A with |x− x| ≤ ρ . (2.8)
4
We remark that, by the assumption (B3), the equations (2.3) define a manifoldMI∪J ′ ⊂ RNof dimension N − |I| − |J ′|. By assumption, the restriction of f to this manifold has alocal minimum at x. According to (2.8), the Hessian matrix D2f(x) has full rank (henceit is strictly positive definite) at x. We remark that this rank condition is invariant undercoordinate changes on the submanifold MI∪J ′ .
As shown by the following theorem, minima attained in generic position (according to Defini-tion 2.1) are stable w.r.t. small C2 perturbations in the cost function f or in the constraintsϕi, ψj .
Theorem 2.1 Let f, φi, ψj ∈ C2(RN ), i ∈ I, j ∈ J , be such that the assumptions (B1)-(B3)hold. Moreover, assume that the global minimum (2.1) is attained at a point x in genericposition. Then there exists δ > 0 such that, if
‖f − f‖C2 ≤ δ, ‖φi − φi‖C2 ≤ δ, ‖ψj − ψj‖C2 ≤ δ (2.9)
for all i ∈ I, j ∈ J , then the corresponding optimization problem
minx∈A
f(x) (2.10)
has a unique minimizer x, also in generic position. Here A is the set defined as in (2.2), withφi, ψj replaced by φi, ψj, respectively. Moreover, for some constant C one has
|x− x| ≤ C ·max{‖f − f‖C2 , ‖φi − φi‖C2 , ‖ψj − ψj‖C2 ; i ∈ I, j ∈ J
}. (2.11)
Proof. 1. As a first step we claim that, for every r > 0, there exists δ > 0 such that theinequalities in (2.9) imply that (2.10) has a minimizer x with
|x− x| ≤ r. (2.12)
Indeed, by (B1)-(B3) and since x is the unique global minimizer of f , by choosing δ1 > 0sufficiently small we have the implication
‖f − f‖C2 ≤ δ1 =⇒ sup|x−x|≤δ1
f(x) < inf|x−x|≥r
f(x). (2.13)
Next, by choosing δ2 > 0 small enough, we have the implication
‖φi − φi‖C2 ≤ δ2, ‖ψj − ψj‖C2 ≤ δ2 for all i ∈ I, j ∈ J
=⇒ A ∩B(x, δ1) 6= ∅ and ψj(x) > 0 for all x ∈ B(x, δ1), j /∈ J ′.(2.14)
Moreover, B(x, δ1) denotes the open ball centered at x with radius δ1.
Now choose any x1 ∈ B(x, δ1) ∩ A. For any x such that |x − x| ≥ r by (2.13) it followsf(x) > f(x1). Hence the global minimum of f cannot be attained outside B(x, r).
5
2. We now prove (2.11). Let x ∈ B(x, δ1) be a global minimizer of f , restricted to A. Thefirst order necessary condition for optimality imply
∇f(x) =∑i∈I
ai∇φi(x) +∑j∈J ]
bj∇ψj(x), (2.15)
for some coefficients ai, bj . Here J ] ⊆ J ′ is the set of indices j ∈ J such that ψj(x) = 0.
By continuity, choosing 0 < δ < min{δ1, δ2} small enough, we obtain
J ] = J ′, bj > 0 for all j ∈ J ′.
3. We now apply the implicit function theorem to the map
Λ : (φ, ψ, x) 7→ (φi(x), ψj(x))i∈I,j∈J ′ , (2.16)
from a space C2 × C2 × RN into R|I|+|J ′|. We choose coordinates
x = (x′, x′′) = (x1, . . . , xν , xν+1, . . . , xN ), (2.17)
with ν = N − |I| − |J ′| such that the equation Λ(φ, ψ, x′, x′′) = 0 defines a C2 function
x′′ = ϕ(x′),
for x′ in a neighborhood of x′.
By the implicit function theorem, one has
Dx′ ϕ = −[∂Λ(φ, ψ)
∂x′′
]−1 ∂Λ(φ, ψ)
∂x′.
By continuity, for all δ > 0 small enough the matrix of partial derivatives ∂Λ(φ,ψ)∂x′′ still has full
rank for all x = (x′, x′′) in a neighborhood of x = (x′, x′′). Hence, the vector equation
Λ(φ, ψ, x′, x′′) = 0,
defined as in (2.16), determines a C2 function
x′′ = ϕ(x′), with Dx′ϕ = −
[∂Λ(φ, ψ)
∂x′′
]−1∂Λ(φ, ψ)
∂x′,
for x′ in a neighborhood of x′.
Computing the second order derivatives of the implicit functions ϕ and ϕ, by (2.9) we obtainan estimate of the form
‖ϕ− ϕ‖C2(B(x′,ρ)) ≤ C0δ, (2.18)
for some constant C0.
4. Now consider the map x′ 7→ F (x′).= f(x′, ϕ(x′)). By the assumption (B1), this has a
strict minimum at x′ = x′. Moreover, by (2.8) the Hessian matrix D2x′F (x′) is strictly positive
definite.
6
By continuity, for δ > 0 small enough, if (2.9) holds then the corresponding function F (x′) =f(x′, ϕ(x′)) has strictly positive definite Hessian matrix D2
x′F (x′) at every point x′ in a neigh-
borhood of x′. By possibly shrinking the value of δ, we conclude that D2x′F (x′) is strictly
positive definite. Observing that (2.15) holds with bj > 0 for all j ∈ J ′, we obtain
f(x)− f(x) ≥ ε |x− x|2 for all x ∈ A with |x− x| ≤ ρ , (2.19)
for some ε, ρ > 0. Hence the global minimum of f on A is attained at a point x in genericposition.
5. It remains to prove the inequality (2.11), showing that the minimizer depends in a Lipschitzcontinuous way on the functions f, φi, ψj , w.r.t. the C2 norm. This follows from the first ordernecessary conditions for optimality, together with the implicit function theorem. Indeed, thepoint x = (x′, x′′) where the constrained minimum is attained is uniquely determined by theN equations
φi(x) = 0, ψj(x) = 0, ∇x′F (x′, ϕ(x′)) = 0. (2.20)
In other words, x can be represented as a zero of the map
Γ : (f, φ, ψ, x) 7→(φi(x), ψj(x), ∂xkF (x′, ϕ(x′))
)∈ RN . (2.21)
Here i ∈ I, j ∈ J ′, k ∈ {1, . . . , ν} with ν = N − |I| − |J ′|, as in (2.17). We regard (2.21) asa map from C2 ×C2 ×C2 ×RN into RN . In order to apply the implicit function theorem on aBanach space [6] and achieve the estimate (2.11), it suffices to check that the N ×N matrix ofpartial derivatives ∂Γ
∂x has full rank in a neighborhood of (f, φ, ψ, x). Indeed, by the condition(2.8) one has
F (x′, ϕ(x′)) ≥ F (x′, ϕ(x′)) + ε|x′ − x′|2
for all x′ in a neighborhood of x′. This implies that the ν × ν Hessian matrix D2x′F (x′, ϕ(x′))
has full rank at x′. Furthermore, since the ν × (N − ν) matrix of partial derivatives(∂[φi, ψj ]
∂x′′
)i∈I,j∈J ′
also has full rank at x′′, the N × N matrix of the partial derivatives ∂Γ∂x has full rank at
(f, φ, ψ, x). By continuity, ∂Γ∂x has full rank in a neighborhood of (f, φ, ψ, x). Observing that
Γ is Lipschitz continuous w.r.t. f, φi, ψj (in the C2 distance), this achieves the proof of (2.11).
We now show that the conditions (i)–(iii) in Definition 2.1 are “generic”. Indeed, they holdfor an open, dense set of C2 functions f, ϕ, ψ.
Theorem 2.2 Let φi, ψj ∈ C2(RN ), i ∈ I, j ∈ J be such that the assumptions (B2)-(B3)hold. Then, for an open dense set of functions f ∈ F∞, the global minimum in (2.1) isattained in generic position.
Proof. 1. Consider any f ∈ F∞. Let x ∈ A be a point where f attains its global minimum.Consider the set of indices
J ′ .= {j ∈ J ; ψj(x) = 0}.
7
Introduce two smooth functions ρ, η : R+ 7→ [0, 1], satisfying
ρ(s) =
{1 if s ∈ [0, r0],
0 if s ≥ 2r0 ,ρ′(s) ≤ 0 for all s > 0, (2.22)
η(s) =
{s2 if s ∈ [0, 1/2],
1 if s ≥ 1 ,η′(s) ≥ 0 for all s > 0, (2.23)
We then define a family of perturbed functions
fε(x) = f(x) + ερ(|x− x|) ·∑j∈J ′
ψj(x) + εη(|x− x|) (2.24)
Choosing r0 > 0 suitably small, it follows that fε(x) > f(x) for all x 6= x and ε > 0.
Thanks to the properties of the cut-off functions ρ, η we have ‖fε − f‖C2 → 0 as ε→ 0.
2. It remains to check that, for every ε > 0, the global minimum of fε, which is attained atthe single point x, is in generic position. If x is a global minimizer of f(·) on A, the first ordernecessary condition (2.7) implies
∇fε(x) = ∇f(x) + ε∑j∈J ′
∇ψj(x) =∑i∈I
αi∇φi(x) +∑j∈J ′
(βj + ε)∇ψj(x).
Therefore, for every ε > 0, the condition (2.7) is satisfied.
On the other hand, the inequality (2.8) follows from
fε(x) ≥ f(x) + ε|x− x|2 ≥ fε(x) + ε|x− x|2 for all x ∈ A with |x− x| ≤ 1/2.
3. The two previous steps show that the set of functions f , for which the minimun is attainedin generic position, is dense on F∞. The fact that it is open follows from Theorem 2.1.
Corollary 2.1 Consider the set F ] of functions (f, φi, ψj)i∈I,j∈J such that either (i) thedomain A in (2.2) is empty, or else (ii) A 6= ∅ and the minimization problem (2.1) has a uniqueminimizer in generic position. Then F ] is open and dense in F∞ × C2(RN )× . . .× C2(RN ).
Proof. 1. The openness of the set of functions (f, φi, ψj) that satisfy the conditions inDefinition 2.1 is again a consequence of Theorem 2.1.
2. To prove that this set is dense in the C2 topology, in view of Theorem 2.2 it suffices to showthat the set of C2 functions (φi, ψj)i∈I,j∈J that satisfy the condition (B3) is dense. Towardthis goal, let functions φi, ψj ∈ C2(RN ) be given. For each subset J ′ ⊂ J consider the map
x 7→ (φi(x), ψj(x))i∈I,j∈J ′
from RN into R|I|+|J ′|. By Sard’s theorem [3, 7], a.e. (y, z) = (yi, zj) ∈ R|I|×R|J ′| is a regularvalue of this map. Hence, taking the perturbations
φi(x) = φi(x) + yi , ψj(x) = ψj(x) + zj ,
one has the alternative:
8
• either (φi(x), ψj(x))i∈I,j∈J ′ 6= (0, . . . , 0) ∈ R|I|+|J ′| ,
• or else (φi(x), ψj(x))i∈I,j∈J ′ = (0, . . . , 0) and the matrix of partial derivatives(∂[φi, ψj ]
∂xk
)i∈I,j∈J ′,1≤k≤N
has rank |I|+ |J ′|.
3. We now observe that the second alternative cannot hold if |I| + |J ′| > N . Since thevector (yi, zj) can be taken arbitrarily small, we conclude that there exists an open dense setof functions (φi, ψj)i∈I,j∈J such that, for any J ′ ⊆ J , the following conditions hold.
• If |I|+ |J ′| > N , then (φi(x), ψj(x))i∈I,j∈J ′ 6= (0, . . . , 0) ∈ R|I|+|J ′| for every x ∈ RN ,
• If (φi(x), ψj(x))i∈I,j∈J ′ = (0, . . . , 0) ∈ R|I|+|J ′|, then the |I| + |J ′| gradients ∇φi(x),∇ψj(x) are linearly independent.
This achieves the proof.
Remark 2.1 Although the family of sets A that can be represented as in (B2)-(B3) is quitegeneral, there are simple examples where the graph of best reply map does not fit within thisframework. For example, let X = Y = [0, 1] and take G(x, y) = y2 − 4xy. Then the graph ofthe best reply map is
graph(R) = {x ∈ [0, 1/2], y = 2x} ∪ {x ∈ [1/2, 1], y = 1}. (2.25)
For this reason, we need to extend the previous results to this slightly more general setting.In this direction, we observe that the set in (2.25) can be equivalently written as
graph(R) = {1− y ≥ 0, 2x− y = 0} ∪ {1− y = 0, 2x− y ≥ 0}. (2.26)
In the following, we shall thus consider more general domains A which admit the followingcharacterization.
(C) There exists a finite open covering RN = V1∪· · ·∪Vm such that, for each k ∈ {1, . . . ,m},the intersection A ∩ Vk admits a representation in terms of finitely many functionsφi ∈ C2(RN ), i ∈ I. Namely, the following properties hold:
(i) For any x ∈ Vk and any subset I ′ ⊆ I, if φi(x) = 0 for all i ∈ I ′ then the gradients∇φi(x), i ∈ I ′, are linearly independent.
(ii) There exists finitely many subsets of indices I1, . . . , Iν ⊆ I, such that
A ∩ Vk.= A1 ∪ · · · ∪ Aν (2.27)
where, for each ` = 1, . . . , ν,
A` = {x ∈ Vk ; φi(x) = 0, φj(x) ≥ 0 for all i ∈ I`, j /∈ I`}. (2.28)
9
Notice that the example (2.26) fits in this framework, with φ1 = 1− y, φ2 = 2x− y.
We now extend Definition 2.1 to this more general setting.
Definition 2.2 Consider a domain A which admits the characterization in (C). We saythat the global minimum (2.1) is attained at a point x in generic position if the followingconditions hold.
(i) x ∈ A is the unique point where the global minimum is attained.
(ii) Let x ∈ Vk, so that (2.27)-(2.28) holds. Then, defining J.= {j ∈ I ; φj(x) = 0}, for
any ` such that x ∈ A`, there exists constants αi, βj, i ∈ I`, j ∈ J \ I` such that
∇f(x) =∑i∈Il
αi∇φi(x) +∑j∈J\Il
βj ∇φj(x), (2.29)
with βj > 0 for all j ∈ J \ I`.
(iii) There exists ρ, ε > 0 such that
f(x)− f(x) ≥ ε |x− x|2 for all x ∈ A with |x− x| ≤ ρ . (2.30)
The results proved in Theorem 2.1 and in Theorem 2.2 can be easily extended to this moregeneral setting.
Corollary 2.2 Let f ∈ C2(RN ) and let A ⊂ RN be a compact domain which admits thecharacterization (C). Moreover, assume that the global minimum (2.1) is attained at a pointx in generic position. Then the conclusions in Theorem 2.1 remain valid.
Proof. Assume that the global minimum is attained at a point x ∈ A ∩ Vk, so that (2.27)-(2.28) holds. If x ∈ A`, then the proof of Theorem 2.1 shows that, for any sufficientlysmall perturbations f , φi, the corresponding minimum is still achieved at a point in A`. Theconclusion is thus obtained by applying Theorem 2.1 to each domain A` which contains x.
Corollary 2.3 Let A ⊂ RN be a compact domain which admits the characterization (C).Then, for an open dense set of functions f ∈ F∞, the global minimum in (2.1) is attained ingeneric position.
Proof. Given the open covering RN = V1 ∪ · · · ∪ Vm, we can find a new covering RN =V ′1 ∪ · · · ∪ V ′m, where each V ′k ⊂ Vk is a closed set. For any given k ∈ {1, . . . ,m}, consider therepresentation (2.27)-(2.28). Using Theorem 2.2 we obtain an open dense set of functions F`,ksuch that, if f ∈ F`,k and the global minimum of f on A is achieved at a point x ∈ A` ∩ V ′k,then it is attained in generic position. Taking the intersection of these finitely many opendense sets of functions, the result is proved.
10
3 Generic properties of the best reply map
Starting with this section, we consider a cost function for the follower G : Rm ×Rn 7→ R, andstudy the structure of the best reply map: x 7→ R(x) ⊂ Rn. We seek a description of thegraph
A .= Graph(R) = {(x, y) ; y ∈ R(x)} (3.1)
for a generic function G ∈ C3. The eventual goal is to show that, for a generic function G,the above graph can be expressed in terms of finitely many equalities or inequalities, as in(2.2)–(2.4). In the following, our basic assumptions will be
(A1) The function G = G(x, y) lies in C3(Rm × Rn).
(A2) There exists ρ > 0 such that
G(x, y) > G(x, 0) for all x ∈ Rm, |y| ≥ ρ, (3.2)
We shall denote by G the family of all functions G = G(x, y) which satisfy (A1)-(A2). Noticethat, if G ∈ G, then by (3.2), for each x ∈ Rm the set of best replies R(x) is a nonemptycompact set contained in the open ball centered at the origin with radius ρ, namely
R(x) ⊆ Bρ ⊂ Rn. (3.3)
As a consequence, the admissible set A in (3.1) is closed. We seek to understand the structureof this set A, for a generic function G ∈ C3(Rm+n). As a preliminary, we observe that thenecessary conditions for optimality imply
A ⊆ M .={
(x, y) ∈ Rm+n ; ∇yG(x, y) = 0}. (3.4)
Here and in the sequel we write ∇yG = (Gy1 , . . . , Gyn). The next lemma establishes thegeneric regularity of M.
Lemma 3.1 Let κ > 0 be given. There exists an open, dense subset G] ⊂ G such that, forevery G ∈ G], the set
Mκ.={
(x, y) ; ∇yG(x, y) = 0, |x| < κ}
(3.5)
is an m-dimensional C2 manifold, embedded in Rm+n.
Proof. 1. Given G ∈ G and ε > 0, by a mollification procedure we can construct g ∈ G ∩ C∞with ‖g −G‖C3 < ε.
2. Consider the map (x, y) 7→ ∇yg(x, y) from Rm+n into Rn. By Sard’s theorem [3, 7], the setof critical values of this map has measure zero. As a consequence, we can find θ = (θ1, . . . , θn)with |θ| < ε such that, at every point where ∇yg(x, y) = θ, the n× (m+n) Jacobian matrixD(∇yg) has full rank, namely
rank
∂x1∂y1g · · · ∂xm∂y1g ∂y1∂y1g . . . ∂yn∂y1g
... · · ·...
∂x1∂yng · · · ∂xm∂yng ∂y1∂yng . . . ∂yn∂yng
= n. (3.6)
11
3. We can now consider a smooth function g : Rm+n 7→ R such that
g(x, y) = g(x, y)−n∑i=1
θiyi |x| ≤ κ, |y| ≤ ρ.
If ε > 0 was chosen sufficiently small, this can be extended to the entire space, still remainingin G.
The above construction yields a function g, arbitrarily close to G in the C3 norm, for whichthe following implication holds:
|x| ≤ κ, |y| ≤ ρ, ∇y g(x, y) = 0 =⇒ rank(D(∇y g)(x, y)
)= n. (3.7)
By the implicit function theorem, this implies that the set
Mκ.={
(x, y) ; ∇gy(x, y) = 0, |x| < κ}
is still a smooth manifold.
4. Let now G] be the set of all functions G ∈ G for which the implication (3.7) holds. By theprevious steps, this set is dense in G. It remains to show that G] is open.
Assume, on the contrary, that there exists a sequence of functions Gk /∈ G], with Gk → G inC3 and G ∈ G]. This implies that, for each k ≥ 1, there exists (xk, yk) with
|xk| ≤ κ, |yk| ≤ ρ, ∇Gk(xk, yk) = 0, rank(D(∇yGk)(xk, yk)
)< n.
Taking a subsequence we can assume the convergence xk → x, yk → y. By continuity, itfollows
|x| ≤ κ, |y| ≤ ρ, ∇G(x, y) = 0, rank(D(∇yG)(x, y)
)< n.
against the assumptions. This contradiction completes the proof.
4 The best reply map with one-dimensional strategies
In this section we study the generic structure of the best reply map, and the stability ofthe Stackelberg equilibrium, starting with a simple one-dimensional framework. Namely, weassume that the strategies x, y for both the leader and the follower range over a closed interval,say
x ∈ X = [0, 1], y ∈ Y = [0, 1]. (4.1)
Given a function G ∈ C3(R2), consider the best reply map
R(x).={y∗ ∈ [0, 1] ; G(x, y∗) ≤ G(x, y) for all y ∈ [0, 1]
}. (4.2)
By Lemma 3.1 there exists an open dense subset G] ⊂ C3(R2) such that, for every G ∈ G], theset
M .={
(x, y) ; Gy(x, y) = 0, |x| < 2, |y| < 2}
(4.3)
12
is a C2 manifold. Indeed, in analogy with (3.7), for a generic function G ∈ C3 we have
|x| ≤ 2, |y| ≤ 2, Gy(x, y) = 0 =⇒ ∇Gy(x, y) 6= 0. (4.4)
We now prove a structure theorem for the best reply map, valid for a generic cost function Gof the follower.
Theorem 4.1 There exists an open dense subset of cost functions G ∈ C3(R2) such that thebest reply map (4.2) has the following structure.
There exists finitely many points 0 = x0 < x1 < · · · < xν = 1, and functions ϕk ∈ C2(R) suchthat {
(x, y) ; y ∈ R(x) , x ∈ [0, 1]}
=ν⋃k=1
{(x, ϕk(x)) ; x ∈ [xk−1, xk]
}. (4.5)
Moreover, either ϕk(xk) 6= ϕk+1(xk), or else
ϕk(xk) = ϕk+1(xk) ∈ {0, 1}, ϕ′k(xk) 6= ϕ′k+1(xk). (4.6)
Proof. 1. We introduce a set of conditions such that, if none of them is satisfied, (for anychoice of x, y, y1, y2, y3 in [0, 1]), then the representation (4.5) holds (Fig. 1). By showing thateach of these conditions is NOT satisfied by all functions G in an open dense subset of C3, thetheorem will be proved.
(i) Gy(x, y) = 0, Gyy(x, y) = 0, and x ∈ {0, 1} or y ∈ {0, 1}.
(ii) Gy(x, y) = 0, Gyx(x, y) = 0, and x ∈ {0, 1} or y ∈ {0, 1}.
(iii) Gy(x, y) = 0, x ∈ {0, 1} and y ∈ {0, 1}.
(iv) Gy(x, y) = Gyy(x, y) = Gyyy(x, y) = 0.
(v) Gy(x, y) = Gyy(x, y) = Gxy(x, y) = 0.
(vi) Gy(x, y1) = Gy(x, y2) = 0, G(x, y1) = G(x, y2), Gyy(x, y1) = 0, for some y1 6= y2.
(vii) Gy(x, y1) = Gy(x, y2) = 0, G(x, y1) = G(x, y2), Gx(x, y1) = Gx(x, y2), for some y1 6= y2.
(viii) Gy(x, y1) = 0, Gyy(x, y1) = 0, G(x, y1) = G(x, y2), y2 ∈ {0, 1}, for some y1 6= y2.
(ix) Gy(x, y1) = 0, G(x, y1) = G(x, y2), Gx(x, y1) = Gx(x, y2), y2 ∈ {0, 1}, for some y1 6= y2.
(x) Gy(x, y1) = Gy(x, y2) = 0, G(x, y1) = G(x, y2), y2 ∈ {0, 1}, for some y1 6= y2.
(xi) Gy(x, y1) = 0, G(x, y1) = G(x, y2), y1, y2 ∈ {0, 1}, for some y1 6= y2.
(xii) G(x, y1) = G(x, y2), Gx(x, y1) = Gx(x, y2), y1, y2 ∈ {0, 1}, for some y1 6= y2.
(xiii) There are three distinct points (x, y1), (x, y2), (x, y3) such that G(x, y1) = G(x, y2) =G(x, y3) and for each i = 1, 2, 3 one has either Gy(x, yi) = 0 or yi ∈ {0, 1}.
13
As the reader will easily check, each of these conditions involves a number of identities whichis strictly larger than the corresponding number of variables. Hence, for “most” functions G,this set of equations will have no solution. As shown in the following steps, a rigorous proofof this fact can be achieved thanks to a multi-jet version of Thom’s transversality theorem.
2. The conditions (i)–(v) are all handled in a similar way. Given a function G ∈ C∞(R2), itsthird order jet prolongation is the vector function whose components are all its derivatives upto order three:
j3G(x, y) = (G, Gx, Gy, Gxx, Gxy, Gyy, Gxxx, Gxxy, Gxyy, Gyyy)(x, y). (4.7)
The map j3G is thus a section of the vector bundle J3(R2,R) of all third order jets of mapsfrom R2 into R.
For each of the conditions in (i)–(v) we shall consider a smooth submanifold W ⊂ J3(R2,R).This will be defined in terms of three independent equalities, hence it will have codimension 3.By Thom’s transversality theorem, there is a dense set of C∞ functions G whose prolongationj3G is transversal to W . Since j3G is a section of J3(R2,R), it is a two-dimensional manifold.In this case, transversality implies that the intersection is empty. In other words, for a denseset of C∞ functions G, the three identities that define W are never simultaneously satisfied.
For example, for condition (i) we consider four distinct sub-manifolds. Each of them is definedby the two identities
Gy = 0, Gyy = 0,
together with one of the four equalities x = 0, x = 1, y = 0, or y = 1. Condition (ii) is entirelysimilar.
For condition (iii) we need again to consider four distinct sub-manifolds. Each of them isdefined by the identity Gy = 0, plus a choice of x ∈ {0, 1} and y ∈ {0, 1}.
To handle condition (iv), it suffices to consider the linear sub-manifold W ⊂ J3(R2;R) con-sisting of all jets such that Gy = 0, Gyy = 0, Gyyy = 0. Finally, condition (v) is handledby defining W ⊂ J2(R2;R) to be the linear manifold of all jets such that Gy = 0, Gyy = 0,Gxy = 0.
3. Conditions (vi)–(xi) refer to the values of G and its first two derivatives at two differ-ent points. For this reason, we shall need a multi-jet transversality theorem, proved in theAppendix. We start by introducing the manifold
Z(2) .={
(x, y1, y2) ; y1 6= y2
}.
On Z(2) we consider the multi-jet bundle J22 (R2,R), consisting of couples of 2-jets of maps
from R2 to R with sources (x, y1), (x, y2), y1 6= y2. Notice that every function G ∈ C∞(R2,R)determines a map
j22G : (x, y1, y2) 7→
(j2G(x, y1) , j2G(x, y2)
). (4.8)
Each of the conditions (vi) and (vii) yields a manifold W ⊂ J22 (R2,R), consisting of multijets
which satisfy the four given identities. By Theorem 8.1, there is a dense set of functionsG ∈ C∞(R2,R) whose second order jet prolongation j2
2G is transversal to W .
14
Since W is defined in terms of four identities, it has codimension 4. On the other hand, thegraph of j2
2G has dimension 3. In this case, transversality implies that the intersection isempty. In other words, for a dense set of functions G, the four conditions in (v) or in (vi) arenot simultaneously satisfied at any couple of distinct points (x, y1), (x, y2).
For each of the conditions (viii), (ix), and (x) we obtain two distinct sub-manifolds W0,W1 ⊂J2
2 (R2,R), imposing the equality y2 = 0 or y2 = 1, respectively. Again, by Theorem 8.1,there is a dense set of functions G ∈ C∞(R2,R) whose second order jet prolongation j2
2G istransversal to W0 or W1, respectively. By dimensionality this implies that, for such functionsG, for every couple of points (x, y1) 6= (x, y2) at least one of the four conditions in (viii) fails.Similarly, at least one of the four conditions in (ix) and at least one in (x) must fail.
Condition (xi) leads to four affine sub-manifolds W , each of codimension 4, depending on thechoices of y1, y2 ∈ {0, 1}. The analysis is entirely similar to the previous cases. Condition (xii)is entirely straightforward.
4. Condition (xiii) refers to the values of G and its first derivatives at three different points.For this reason, we introduce the manifold
Z(3) .= {(x, y1, y2, y3) ; yi 6= yj for i < j}.
On Z(3) we consider the multi-jet bundle J13 (R2,R), consisting of couples of 1-jets of maps
from R2 to R with three distinct sources (x, y1), (x, y2), and (x, y3). Notice that every functionG ∈ C∞(R2,R) determines a map
j13G : (x, y1, y2, y3) 7→
(j1G(x, y1) , j1G(x, y2) , j1G(x, y3)
).
On J13 (R2,R) we consider a finite number of sub-manifolds W , each defined by 5 identities.
The first two identities are
G(x, y1) = G(x, y2), G(x, y1) = G(x, y3).
The remaining three identities are obtained by choosing, for each i = 1, 2, 3, either Gy(x, yi) =0, or yi = 0, or else yi = 1.
By Theorem 8.1, there is a dense set of functions G ∈ C∞(R2,R) whose first order jet prolon-gation j1
3G is transversal to any of the above manifolds W .
We now observe that each W is defined in terms of five identities, and thus has codimension 5.On the other hand, the graph of j1
3G has dimension 4. Once again, transversality implies thatthe intersection is empty. In other words, for a dense set of functions G, the five conditionsin (xiii) are not simultaneously satisfied at any triple of distinct points (x, y1), (x, y2), (x, y3).
5. In this step we prove that the set of functions G ∈ C3(R2,R) for which none of the conditions(i)–(xiii) is satisfied on the domain Q = {(x, y) ∈ [0, 1]× [0, 1]} is open in C3.
Let (G(n))n≥1 be a sequence of functions converging to G in C3. Assume that, for every n, atleast one of the conditions (i)–(xiii) is satisfied, within the domain Q. We need to show thatthe same holds for G.
15
We start with the easy case where, for some sequence (x(n), y(n)) ∈ Q, each G(n) satisfies oneof the conditions (i)–(v). By taking a subsequence, we can assume (x(n), y(n)) → (x, y) ∈ Q.By continuity, the limit function G satisfies the same condition at (x, y), proving our claim.
Concerning the remaining conditions (vi)–(xiii), a more careful analysis is needed, becausethese conditions involve two or three distinct points.
Assume that, for each n ≥ 1, the function G(n) satisfies one of the conditions (vi)–(xiii), for
distinct points (x(n), y(n)i ), i = 1, 2, 3. By taking a subsequence, we can assume the convergence
(x(n), y(n)i ) → (x, yi), i = 1, 2, 3.
If the limit points (x, yi) are distinct, then by continuity G still satisfies the same condition,and we are done. Notice that this is certainly the case for (xi) and (xii), because here we
require y(n)1 , y
(n)2 ∈ {0, 1} with y
(n)1 6= y
(n)2 .
To complete the proof, we need to consider the cases where two of the points (x, yi) coincide,say,
limn→∞
y(n)1 = lim
n→∞y
(n)2 = y ∈ [0, 1]. (4.9)
• If G(n) satisfies all identities in (vi), for every n ≥ 1, then by taking limits we conclude
Gy(x, y) = Gyy(x, y) = Gyyy(x, y) = 0. (4.10)
Hence the limit function G satisfies (iv).
• If G(n) satisfies all identities in (vii), for every n ≥ 1, then
Gy(x, y) = Gyy(x, y) = Gxy(x, y) = 0.
Hence the limit function G satisfies (v).
• Next, assume that G(n) satisfies all identities in(viii), or in (ix), or in (x), for every n ≥ 1.Taking the limit, in all cases we conclude
Gy(x, y) = Gyy(x, y) = 0, y ∈ {0, 1}.
Hence (i) holds.
• Finally, assuming that all functions G(n) satisfy (xiii), different cases need to be consid-ered.
If all three sequences of points y(n)1 , y
(n)2 , y
(n)3 , converge to the same limit y, then the
convergence G(n) → G in C3 implies (4.10). Hence the limit function G satisfies (iv).
The remaining possibility is that
y(n)1 , y
(n)3 → y1 , y
(n)2 → y2 6= y1 . (4.11)
Four sub-cases must be considered.
- If y1 ∈ {0, 1} and y2 ∈ {0, 1}, then
Gy(x, y1) = 0, G(x, y1) = G(x, y2), y1, y2 ∈ {0, 1}.
16
Hence all identities in (xi) hold.
- If y1 /∈ {0, 1} and y2 ∈ {0, 1}, then
Gy(x, y1) = 0, Gyy(x, y1) = 0, G(x, y1) = G(x, y2),
for some y1 6= y2 ∈ {0, 1}. Hence (viii) holds.
- If y1 ∈ {0, 1} and y2 /∈ {0, 1}, then
Gy(x, y1) = Gy(x, y2) = 0, G(x, y1) = G(x, y2),
for some y1 6= y2 ∈ {0, 1}. Hence (x) holds.
- If y1 /∈ {0, 1} and y2 /∈ {0, 1}, then
Gy(x, y1) = Gyy(x, y1) = Gy(x, y2) = 0, G(x, y1) = G(x, y2),
Hence (vi) holds.
0
1
y
1
0 1xx
x
10 4xx
2x
1
ϕ
P
1ϕ
2ϕ
3ϕ
4
3x
31 2x
y
x x
Figure 1: Left: the graph of the “best reply map” (in red), for a generic cost function G. Right: asample of non-generic cases. At x0 the curve where Gy = 0 touches the line y = 1 tangentially, so that(ii) holds. On the whole interval [x1, x2] the function G(x, ·) has two equal minimizers, and (ix) holds.At the point x3 two global minima are attained, where one of these is along a curve where Gy = 0,with vertical tangent. This happens when Gyy = 0, so that (iv) holds. At x4 the function G(x4, ·)achieves the minimum at three distinct points, hence (xiii) holds.
6. To conclude the proof of the theorem, consider a function G ∈ C3(R2,R) in the open, denseset where none of the conditions (i)–(xiii) holds. We claim that, in this case, the best replymap satisfies the conclusions of the theorem.
By the necessary conditions for optimality, the graph of the best reply map is a closed set,contained in the union of the three sets
{(x, y) ; Gy(x, y) = 0, x, y ∈ [0, 1]} ∪ {(x, 0) ; x ∈ [0, 1]} ∪ {(x, 1) ; x ∈ [0, 1]}.
17
Consider any point x ∈ [0, 1]. By (xii), the minimum of the function y 7→ G(x, y) over [0, 1] canbe attained at most at two distinct points. Various cases will be considered in the remainingsteps.
7. We first assume that the global minimum is attained at a single point y. Two main casescan occur.
CASE 1: y ∈ ]0, 1[ . We claim that in this case Gyy(x, y) 6= 0, hence for x in a neighborhoodof x the best reply map is single valued:
R(x) = {φ(x)},
where y = φ(x) is the function implicitly defined by
Gy(x, y) = 0. (4.12)
Indeed, assume on the contrary that Gyy(x, y) = 0. Since we also have Gy(x, y) = 0, by (i) itfollows that x /∈ {0, 1}. Moreover, from (iv) and (v) it follows that
Gxy(x, y) 6= 0, Gyyy(x, y) 6= 0. (4.13)
Hence, by (4.13), the equation (4.12) can be solved in a neighborhood of (x, y) in terms of afunction x = ψ(y), with
ψ(y) = x, ψ′(y) = −Gyy(x, y)
Gxy(x, y)= 0, ψ′′(y) = −Gyyy(x, y)
Gxy(x, y)6= 0.
To fix the ideas, assume ψ′′(y) > 0, the other case being entirely similar. Since x > 0, wecan find a strictly increasing sequence xn ↑ x, with xn > 0 for every n. Call yn ∈ [0, 1]a point where G(xn, ·) attains its global maximum. This implies that either yn ∈ {0, 1} orelse Gy(xn, yn) = 0. Choosing a subsequence, we achieve the convergence (xn, yn) → (x, y∗)for some y∗ ∈ [0, 1]. By continuity, G(x∗, ·) attains its global minimum at y∗. Hence, byuniqueness, y∗ = y. Since y /∈ {0, 1}, we conclude that Gy(xn, yn) = 0 for all sufficiently largen. This yields a contradiction, because the equation (4.12) does not admit any solution withx < x, in a suitably small neighborhood of (x, y).
CASE 2: y = 0. (The case where where y = 1 is entirely similar.)
In this case, a necessary condition is Gy(x, 0) ≥ 0. If Gy(x, 0) > 0, then we immediatelyconclude that R(x) = {0} on an entire neighborhood of x.
The remaining case is where Gy(x, 0) = 0. By (i) and (ii) we then have Gyy(x, 0) 6= 0 andGxy(x, 0) 6= 0. Hence, in a neighborhood of (x, 0), the equation (4.12) uniquely defines afunction y = φ(x), with
φ(x) = 0, φ′(x) = − Gxy(x, 0)
Gyy(x, 0)6= 0.
To fix the ideas, let φ′(x) > 0. Then, in a neighborhood of x, the best reply map is single-valued and has the form
R(x) =
{{0} if x ≤ x,
{φ(x)} if x > x.(4.14)
18
8. Finally, we assume that the global minimum of G(x, ·) is attained at the two distinct pointsy1 6= y2.
CASE 1: y1 = 0, y2 = 1.
Using (xi) and (xii), together with the necessary conditions for optimality, we obtain
Gy(x, 0) > 0, Gy(x, 1) < 0, Gx(x, 0) 6= Gx(x, 1).
To fix the ideas, assume Gx(x, 0) < Gx(x, 1), the other case being similar. Then, for all x ina neighborhood of x, the best reply map has the form
R(x) =
{1} if x < x,
{0, 1} if x = x,
{0} if x > x.
(4.15)
CASE 2: 0 < y1 < y2 = 1. (The case 0 = y1 < y2 < 1 is entirely similar.)
Since Gy(x, y1) = 0 and (viii) fails, we must have Gyy(x, y1) 6= 0. Hence, in a neighborhoodof (x, y1) the equation (4.12) is solved by a function y = φ(x).
At the point x = x we now compute
d
dxG(x, φ(x)) = Gx +Gyφ
′(x) = Gx(x, y) 6= Gx(x, 1)
Note that the last inequality stems from the fact that (ix) fails. To fix the ideas, assumeGx(x, y1) < Gx(x, 1), the other case being similar. Then, for all x in a neighborhood of x, thebest reply map has the form
R(x) =
{1} if x < x,
{y1, 1} if x = x,
{φ(x)} if x > x.
CASE 3: 0 < y1 < y2 < 1.
By the necessary conditions for a minimum, this implies Gy(x, y1) = Gy(x, y2) = 0. Sinceboth (vi) and (vii) fail, this implies
Gyy(x, y1) 6= 0, Gyy(x, y2) 6= 0, (4.16)
Gx(x, y1) 6= Gx(x, y2). (4.17)
By (4.16), near the points (x, y1) and (x, y1) the equation (4.12) implicitly defines two functionsy = φ1(x) and y = φ2(x). By (4.17) at x = x we have
d
dxG(x, φ1(x))
∣∣∣∣x=x
= Gx(x, y1)+Gy(x, y1)φ′1(x) = Gx(x, y1) 6= Gx(x, y2) =d
dxG(x, φ2(x))
∣∣∣∣x=x
.
19
To fix the ideas, assume Gx(x, y1) < Gx(x, y2), the other case being similar. Then, for all xin a neighborhood of x, the best reply map has the form
R(x) =
{φ2(x)} if x < x,
{y1, y2} if x = x,
{φ1(x)} if x > x.
(4.18)
9. By the previous analysis, if G ∈ C3(R2,R) is a function that does not satisfy any of theconditions (i)–(xiii), then for x in a neighborhood of any point x ∈ [0, 1] the best reply maphas the structure described in (4.5)-(4.6). Covering the compact domain [0, 1] with a finitenumber of open interval, the theorem is proved.
5 Generic stability of one-dimensional Stackelberg equilibria
We again consider a noncooperative game, where the players choose strategies in X = Y =[0, 1]. In order to apply the results in Section 2, we shall need
Lemma 5.1 Assume that the graph of the best reply map has the structure described at (4.5)-(4.6) in Theorem 4.1. Then this graph can be also written in the form (2.27)-(2.28), wherethe functions ϕi satisfy property (i) in the characterization (C).
Proof. Let (4.5)-(4.6) hold. By suitably covering the square [0, 1]× [0, 1] with finitely manyopen sets V1, . . . , Vm, for each q ∈ {1, . . . ,m} the intersection Vq ∩ graph(R) takes one of theforms described below.
CASE 1 (see Fig. 2, left). Assume that
A = {y = ϕk(x), a < xk ≤ xk}.
To treat this case, it suffices to consider the function
φ1(x, y).= y − ϕk(x)
and an additional function φ2, with the following properties:{φ2(x, ϕk(x)) > 0 for x ∈ ]a, xk[ ,
φ2(x, ϕk(xk)) = 0 ,(5.1)
d
dxφ2(x, ϕk(x))
∣∣∣∣x=xk
< 0. (5.2)
We then have the representation
A = {(x, y) ; φ1(x, y) = 0, φ2(x, y) ≥ 0}. (5.3)
20
CASE 2 (see Fig. 2, center). Assume that
A = {y = ϕk(x), a < xk ≤ xk} ∪ {y = 1, xk ≤ x < b},
with ϕk(xk) = 1, ϕ′k(xk) > 1. Without loss of generality, we can assume ϕk(x) > 1 for x > xk.
To handle this case we consider the two functions
φ1(x, y).= ϕk(x)− y, φ2(x, y) = 1− y.
We then have the representation
A = A1∪A2 ={
(x, y) ; φ1(x, y) = 0, φ2(x, y) ≥ 0}∪{
(x, y) ; φ1(x, y) ≥ 0, φ2(x, y) = 0}.
All other cases (see for example Fig. 2, right) are entirely similar.
ba
a b
V
qV
q
V
a
q
ϕ
kkx
k−1x x
ϕ
k
kϕ
k
x k
Figure 2: Left and center: the two main cases considered in the proof of Lemma 5.1. Right: anotherconfiguration, entirely similar to the one considered in CASE 2.
As an immediate consequence of Corollaries 2.2 and 2.3 we now obtain the generic stabilityof the Stackelberg equilibrium, in this particular setting (see Fig. 3).
Theorem 5.1 Let G ∈ C3(R2) be a function in the open dense set considered in Theorem 4.1,for which the best reply map R has the structure described at (4.5)-(4.6). Then there existsan open dense set of functions F ⊂ C2(R2) such that, for every F ∈ F , the following holds.
The global minimum of F on A = graph(R) is attained at a point (x, y) in generic position,as defined in (2.2). Moreover there exists constants C, δ > 0 such that, if
‖F − F‖C2 ≤ δ, ‖G−G‖C3 ≤ δ, (5.4)
then the corresponding perturbed optimization problem
min(x,y)∈A
F (x, y) (5.5)
has a unique minimizer (x, y), also in generic position. Here A = graph(R) is the best replymap corresponding to the cost function G. In addition
|x− x|+ |y − y| ≤ C ·(‖F − F‖C2 + ‖G−G‖C3
). (5.6)
21
Proof. Consider a cost function G ∈ C3(R2) in the open dense set where none of the conditions(i)–(xiii) in the proof of Theorem 4.1 are satisfied. Then the corresponding best reply mapR(·) has the generic structure described at in (4.5)-(4.6). In particular (see Fig. 2), there isa finite covering of the square [0, 1] × [0, 1] with open sets Vq, q = 1, . . . , ν, such that therestriction of the graph of R(·) to each set Vq can be described by a finite set of equalities andinequalities involving
• functions of the form φ(x, y) = y − ϕk(x), where ϕk is implicitly defined by the identity∇yG(x, , ϕk(x)) = 0.
• the two functions φ(x, y) = y and φ(x, y) = y − 1,
• functions of the form φ(x, y) = x−xk, where xk is a point where the best reply map hasa jump.
If now G is a another cost function with ‖G − G‖C3 sufficiently small, then by repeatedapplications of the implicit function theorem we check that the best reply map A determinedby G also admits the structure (4.5)-(4.6). Furthermore, all the corresponding functions φi inthe characterization (C) satisfy
‖φi − φi‖C2 ≤ C ‖G− G‖C3 for all i ∈ I,
for some constant C > 0. The estimate (5.6) now follows from Corollaries 2.2 and 2.3.
kϕ
xk−1 k
x
k
k+1
xk−1
xk
ϕ
ϕk
k+1
xk
(x,y)_ _
(x,y)
(x,y)
_ _
_ _
ϕ
ϕ
Figure 3: Examples of a best reply map R(·) with generic structure, according to Theorem 4.1. Threedifferent cases where the function F can attain a global minimum at a point (x, y) on the graph ofR(·), in generic position.
6 Multi-dimensional strategies for the follower
We now extend the analysis to the case where the follower chooses his strategy within a multi-dimensional space. To avoid technicalities associated with the boundary of the sets X,Y , wewill simply assume that x ∈ X = R, y ∈ Y = Rn. The follower will thus solve a minimizationproblem on Rn, depending on a scalar parameter. The generic structure of the best replymap, shown in Fig. 4, can be readily described. As in Section 3, we denote by G the set of allfunctions G : R× Rm 7→ R which satisfy (A1)-(A2).
22
M
1
2 3
φ
1φ
2φ
3
P
P
P
1
2
2 3
3
Q
φ
D
A
x1
x x x
y
C
B
0
A
Figure 4: The generic structure of the best reply map x 7→ R(x) ⊂ Rn, depending on a scalarparameter x ∈ R. As x varies, at the points A,B,C,D, where a new pair of local minima of G(x, ·) iscreated, the value of G must be strictly larger than the global minimum.
Theorem 6.1 Given α > 0, there exists an open dense subset G] ⊂ G of cost functions suchthat, for any G ∈ G], restricted to the interval [−α, α], the best reply map has the followingstructure.
There exists finitely many points −α = x0 < x1 < · · · < xν = α, and functions ϕk ∈ C2(R; Rn)such that
{(x, y) ; y ∈ R(x) , |x| ≤ α} =ν⋃k=1
{(x, ϕk(x)) ; x ∈ [xk, xk+1]
}. (6.1)
Moreover, ϕk(xk) 6= ϕk−1(xk) for all k = 1, . . . , ν − 1.
Proof. 1. Let a function G0 ∈ G be given, together with constants κ, ρ > 0. Using Lemma 3.1,for any ε > 0 we can find a function g ∈ G ∩ C∞ with ‖g −G0‖C3 < ε for which the followingimplication holds:
|x| ≤ κ, |y| ≤ ρ, ∇yg(x, y) = 0, =⇒ rank(D(∇yg)(x, y)
)= n. (6.2)
As a consequence, for every sufficiently small C3 perturbation of g, the implication (6.2) stillholds.
For a given function G ∈ C3, we consider the n× (n+ 1) matrix of partial derivatives
A =(A0
∣∣∣A1
∣∣∣ · · · ∣∣∣An) .=
∂x∂y1G ∂y1∂y1G . . . ∂yn∂y1G
......
...
∂x∂ynG ∂y1∂ynG . . . ∂yn∂ynG
. (6.3)
The condition rank(A) = n at every point in the set
M .={
(x, y) ∈ R1+n ; ∇yG(x, y) = 0}, (6.4)
23
guarantees that M is a 1-dimensional embedded manifold in R1+n. By the implicit functiontheorem, near points where
rank(A1| · · · |An) = n, (6.5)
this manifold M can be represented as the graph of a function x 7→ (y1(x), . . . , yn(x)).
We observe that, by removing any column from the matrix A in (6.3), we obtain an n × nmatrix whose rank can be either n or n− 1.
2. Let g be a smooth function for which (6.1) holds. Given any δ > 0, we claim that thereexists G ∈ C∞ with ‖G−g‖C3 < δ and such that, for |x| ≤ α and |y| ≤ ρ, none of the followingstatements holds true.
(i) There exist three distinct points (x, y1), (x, y2), (x, y3) such that
G(x, y1) = G(x, y2) = G(x, y3), ∇yG(x, y1) = ∇yG(x, y2) = ∇yG(x, y3) = 0.(6.6)
Notice that, by requiring that (i) fails, we rule out the possibility that, for some x1, the globalminimum of G(x1, ·) is attained at three or more distinct points (see Fig. 5, left).
(ii) There exist two distinct points (x, y1), (x, y2) such that
G(x, y1) = G(x, y2), ∇yG(x, y1) = ∇yG(x, y2) = 0, (6.7)
rank(A1| · · · |An
)(x, y1) < n. (6.8)
By requiring that (ii) fails, we preclude the existence of x2 such that the minimum of G(x2, ·)is attained at two distinct points, and at one of these points the tangent vector toM is vertical(see again Fig. 5).
(iii) There exist two distinct points (x, y1), (x, y2) such that
rank(A1| · · · |An
)(x, y1) = rank
(A1| · · · |An
)(x, y2) = n, (6.9)
and moreover
G(x, y1) = G(x, y2), ∇yG(x, y1) = ∇yG(x, y2) = 0,
d
dxG(x, φ1(x)) =
d
dxG(x, φ2(x)).
(6.10)
Here φ1, φ2 are the functions implicitly defined by the system of n equations
∇yG(x, φ(x)) = 0, (6.11)
with φ1(x) = y1 and φ2(x) = y2.
24
Notice that, by (6.9), both functions φ1, φ2 are well defined in a neighborhood of x, and haveC2 regularity. To understand the meaning of (iii), assume that the function G(x3, ·) attainsits global minimum at two distinct points (x3, y
1) and (x3, y2). By (6.9), on a neighborhood
of x3 the equation (6.11) implicitly defines two functions y = φ1(x), y = φ2(x). If (iii) fails,at every point x 6= x3 with |x− x3| small enough, we have
G(x, φ1(x)) 6= G(x, φ2(x)).
In particular, the minimum of G(x, ·) cannot be simultaneously attained at more then onepoint, for all x in a neighborhood of x3 (see Fig. 5).
Our last condition implies that there cannot exist x4 ∈ R such that the 1-dimensional manifoldM has a second order tangency with the vertical hyperplane {(x, y) ; x = x4}, as shown inFig. 5, right.
(iv) There exists a point (x, y) ∈ R1+n such that
∇yG(x, y) = 0, rank(A1|A2| · · · |An)(x, y) = n− 1. (6.12)
Moreover, if s 7→ (x(s), y(s)) is a local arc-length parameterization ofM with (x(0), y(0)) =(x, y), then
d2
ds2x(s)
∣∣∣∣s=0
= 0. (6.13)
x
y
φ
φ2
1
xxx1 2 3
x
y
y y3
y2
y1
1
2y
y
y1
2
M
4
Figure 5: Four non-generic configurations, described at (i)–(iv) of the proof of Theorem 6.1.
Our claims will be proved in the forthcoming steps. Notice that in all cases (i)–(iv), we areconsidering sets of points (x, yi) defined by a number of equations which is strictly larger thanthe dimension of the spaces they live in. Hence, for a generic function G, these sets are empty.
3. In this step we prove that the set of functions G which do not satisfy (i) at any point x ∈ Ris dense. Indeed, consider the open domain
X(3) .={
(x, y1, y2, y3) ∈ R1+3n ; yi 6= yj for 1 ≤ i < j ≤ 3}. (6.14)
25
Call J1(X(3),R) the bundle of all 1-jets of functions f : X(3) 7→ R. Given a smooth mapG : R× Rn 7→ R, this yields a map
j1G(x, y1, y2, y3).=(G(x, y1), G(x, y2), G(x, y3),∇yG(x, y1),∇yG(x, y2),∇yG(x, y3)
).
Consider the set
M1.={
(z1, z2, z3,v1,v2,v3) ∈ R3 × R3n ; z1 − z2 = z2 − z3 = 0, v1 = v2 = v3 = 0}.
(6.15)It is immediate to check that M1 is a smooth submanifold of J1(X(3),R). By the multi-jet version of Thom’s transversality theorem (proved in the Appendix), the set of all G ∈C∞(R1+n ,R) such that j1G is transversal to M1 is residual in the C∞ topology. Observingthat dim(X(3)) = 1+3n while codim(M1) = 2+3n, transversality implies that the intersectionis empty: {
j1G(x, y1, y2, y3) ; (x, y1, y2, y3) ∈ X(3)}∩ M1 = ∅. (6.16)
4. Next, we show that the set of functions G which do not satisfy (ii) at any point (x, y) ∈R× Rn is dense. For this purpose consider the open set
X(2) .={
(x, y1, y2) ∈ R1+2n ; y1 6= y2}. (6.17)
Call J2(X(2),R) the bundle of all 2-jets of functions f : X(2) 7→ R. Given a smooth mapG : R× Rn 7→ R, this yields a map
j2G(x, y1, y2).=(G(x, y1), G(x, y2),∇yG(x, y1),∇yG(x, y2), D2
yG(x, y1), D2yG(x, y2)
).
where D2yG
.= (∂2
yiyjG) denotes the n × n Hessian matrix of second derivatives of G w.r.t.y1, . . . , yn. For each k = 1, 2, . . . , n− 1, consider the set
M2,k.={
(z1, z2,v1,v2, A1, A2) ; z1 − z2 = 0, v1 = v2 = 0, rank(A1) = k}. (6.18)
Using Proposition 3.2.6 in [3], p. 33, we check thatM2 is a smooth submanifold of J2(X(2),R).By a multi-jet version of Thom’s transversality theorem, the set of all G ∈ C∞(R1+n ,R)such that j2G is transversal to M2,k is residual in the C∞ topology. We now observe thatdim(X(2)) = 1+2n while codim(M2,k) = 1+2n+(n−k)2. Hence, for every k = 0, 1, . . . , n−1,transversality implies that the intersection is empty:{
j2G(x, y1, y2) ; (x, y1, y2) ∈ X(2)}∩ M2,k = ∅. (6.19)
5. We now prove that the set of all functions G which do not satisfy (iii) at any point(x, y1, y2) ∈ X(2) is dense.
Call J2(X(2),R) the bundle of all 1-jets of functions f : X(2) 7→ R. Given a smooth mapG : R× Rn 7→ R, consider the map
j1G(x, y1, y2).=(G(x, y1), G(x, y2), Gx(x, y1), Gx(x, y2), ∇yG(x, y1), ∇yG(x, y2)
).
26
Define the set
M3.={
(z1, z2, w1, w2,v1,v2) ∈ R2 × R2 × R2n ; z1 − z2 = 0, w1 − w2 = 0, v1 = v2 = 0}.
It is clear that M3 a smooth submanifold of J1(X(2),R). By a multi-jet version of Thom’stransversality theorem (proved in the Appendix), the set of all G ∈ C∞(R1+n,R) such thatj1G is transversal to M3 is residual in the C∞ topology. Since dim(X(2)) = 1 + 2n whilecodim(M3) = 2 + 2n, transversality implies that the intersection is empty:{
j1G(x, y1, y2); (x, y1, y2) ∈ X(2)}∩M3 = ∅. (6.20)
Notice that the assumption (6.9) guarantees that φ1, φ2 are well defined. Since∇yG(x, φ1(x)) =∇yG(x, φ2(x)) = 0, one has the equivalence
d
dxG(x, φ1(x)) =
d
dxG(x, φ2(x)) ⇐⇒ Gx(x, φ1(x)) = Gx(x, φ2(x)).
Therefore, the set of all functions G ∈ C∞(R1+n,R) which do not satisfy (iii) at any point(x, y1, y2) ∈ X(2) is the intersection of all the set of functions G such that j2G is transversalto M2,k for k = 0, . . . , n− 1, and moreover j1G is transversal to M3. We thus conclude thatthe set of functions G ∈ C∞(R1+n,R) which do not satisfy (iii) at any point (x, y1, y2) ∈ X(2)
is dense in C3(R1+n,R).
6. Finally, we show that the set of functions G which do not satisfy (iv) at any point (x, y) ∈R1+n is dense. For this purpose, we need to show that the family of third order jets of functionsG : R1+n 7→ R, which satisfy all the conditions in (iv), is a smooth manifold with codimensionn− 2.
We recall that the assumption rank(A) = n at every point (x, y) ∈M implies that the setM in(4.3) is a 1-dimensional manifold, embedded in R1+n. The condition rank(A1| · · · |An)(x, y) =n− 1 implies that the tangent vector to M at (x, y) is vertical, i.e.:
d
dsx(s)
∣∣∣∣s=0
= 0. (6.21)
To obtain an expression for the second derivative in (6.13), we first differentiate the identity∇yG(x, y) = 0 and obtain the linear system
(A0
∣∣∣A1
∣∣∣ · · · ∣∣∣An)
dx/dsdy1/ds
...dyn/ds
=
0...0
. (6.22)
Calling A−i the n×n matrix obtained from A = (A0|A1| · · · |An) by deleting the i-th column,the solution to (6.22) can be written as
dx
ds= κ · det(A1|A2| · · · |An),
dyids
= κ · det(A−i), i = 1, . . . , n, (6.23)
where κ = κ(A) is a normalizing factor, chosen in order to achieve(dx
ds
)2
+
n∑i=1
(dyids
)2
= 1.
27
At a point (x, y) where det(A1|A2| · · · |An) = 0, using (6.23) the second derivative (6.13) iscomputed by
d2
ds2x(s) = κ∇y
[det(A1|A2| · · · |An)
]· ddsy(s)
= κn∑i=1
∂yi
[det(A1|A2| · · · |An)
]· dyids
= κ2n∑i=1
n∑j=1
det(Hi,j) det(A−i).
(6.24)
Here
Hi,j.=
∂y1∂y1G · · · ∂yj−1
∂y1G ∂yi∂yj∂y1G ∂yj+1∂y1G . . . ∂yn∂y1G
... · · ·...
∂y1∂ynG · · · ∂yj−1∂ynG ∂yi∂yj∂ynG ∂yj+1
∂ynG . . . ∂yn∂ynG
(6.25)
is the matrix obtained by differentiating the j-th column of (A1| · · · |An) by yi.
Call J3(R1+n;R) the bundle of all 1-jets of functions f : R × Rn 7→ R. Given a smooth mapG : R× Rn 7→ R, its third-order jet at a point (x, y) is
j3G(x, y) =(G(x, y), ∇yG(x, y), D2
yG(x, y), D3yG(x, y)
).
For any (z,v, A, T ) ∈ J3(R1+n;R), denote by Hi,j(A, T ) the n×n matrix obtained from (6.25)by replacing the second and third derivatives of G with the corresponding elements in A andT .
With this notation, the family of third order jets of functions satisfying (iv) can thus beexpressed by
M4.=
{(z,v, A, T ) ∈ J3(R1+n; R) ; v = 0, rank(A1|A2| · · · |An) = n− 1,
n∑i=1
n∑j=1
det(Hi,j(A, T )) det(A−i) = 0
}.
(6.26)
In order to apply Thom’s transversality theorem, we need to show that M4 is a smoothmanifold with codim(M4) = n+2. Indeed, the vector equation v = 0 yields n scalar equations,while the condition
rank(A1|A2| · · · |An) = n− 1, (6.27)
provides one more, independent scalar equation. The last condition in (6.26) can be writtenas
Ψ(A, T ).=
n∑i=1
n∑j=1
det(Hi,j(A, T )
)det(A−i) . (6.28)
We claim that the above equation determines a smooth manifold of codimension 1, transversalto the manifolds determined by the previous equations in (6.26). This will be true if at leastone of the partial derivatives of Ψ does not vanish. Namely
∂
∂Gy1yiyjΨ(A, T ) 6= 0 (6.29)
28
for some indices i, j.
To simplify the computations, we observe that the property (iv) which definesM4 is invariantw.r.t. rotations of the coordinates y = (y1, . . . , yn). Without loss of generality, we can thusassume that the tangent vector to the 1-dimensional manifold M at the point (x, y) is(
dx
ds,dy1
ds,dy2
ds, . . . ,
dynds
)= (0, 1, 0, . . . , 0). (6.30)
In this case, one hasdet(A−2) = · · · = det(A−n) = 0. (6.31)
For notational convenience, we shall here denote by
A = (aij)1≤i,j≤n = (A1| · · · |An)
is the n × n symmetric matrix of second derivatives Gyiyj . Using (6.31), from (6.28) oneobtains
∂
∂Gy1yky`Ψ(A, T ) =
n∑`=1
∂
∂Gy1yky`det(H1,`(A, T )
)=
∂
∂ak`det(A) if i = j,
2 ∂∂ak`
det(A) if i 6= j.
By assumption, A has rank n − 1. Hence there exists a minor Mk` obtained from A byremoving the k-th row and the `-th column, which has rank n− 1.
If k = `, then∂
∂Gy1ykykΨ(A, T ) =
∂
∂akkdet(A) = det(Mkk) 6= 0.
If k 6= `, recalling that A is a symmetric matrix, then by symmetry
∂
∂Gy1yky`Ψ(A, T ) =
(∂
∂ak`+
∂
∂a`k
)det(A) = 2 · (−1)k+` det(Mk`) 6= 0.
In both cases, we find a partial derivative which does not vanish. As a result,M4 is a smoothmanifold with codim(M4) = n+ 2.
We can now use Thom’s transversality theorem and conclude that the set of allG ∈ C∞(R1+n,R)such that j3G is transversal toM4 is residual in the C∞ topology. Since codim(M4) = n+ 2,transversality implies that the intersection is empty:
{j3G(x, y); (x, y) ∈ R1+n} ∩M4 = ∅.
7. In this step we show that the family G] ⊂ C3(R1+n,R) of all cost functions G that donot satisfy any of the conditions (i)–(iv), at any point (x, y) where G(x, ·) attains its globalminimum, is open in the topology induced by C3.
Assume, on the contrary, that there exists a convergent sequence of functions Gk → G in C3,with Gk /∈ G] for every k ≥ 1. We claim that G /∈ G] as well. Since there are four conditionsthat each Gk can satisfy, we consider them separately.
29
CASE 1: Suppose that all functions Gk satisfy condition (i). Hence there exists a sequence ofpoints (xk, y
1k, y
2k, y
3k) satisfying (6.6) for Gk, with
|xk| ≤ κ, |yik| ≤ ρ for i = 1, 2, 3, yik 6= yjk for all 1 ≤ i < j ≤ 3.
By possibly taking a subsequence, one has
xk → x∗, y1k → y1
∗, y2k → y2
∗, y3k → y3
∗ with |x∗| ≤ κ, |y1∗| ≤ ρ, |y2
∗| ≤ ρ, |y3∗| ≤ ρ.
Three sub-cases can arise.
Case 1a: yi∗ 6= yj∗ for all 1 ≤ i < j ≤ 3. In this case, by continuity, G satisfies (i) at(x∗, y
1∗, y
2∗, y
3∗).
Case 1b: Two of the points y1∗, y
2∗, y
3∗ coincide with each other while the third point is differ-
ent. Without loss of generality, suppose y1∗ = y2
∗ 6= y3∗. In this case, since ∇yGk(xk, y1
k) =∇yGk(xk, y2
k) = 0, in a neighborhood of (x∗, y∗) for every k large enough there must be a point
(x]k, y]k) ∈Mk where the tangent vector to Mk is vertical. This means
∇yG(x]k, y]k) = 0, det
(∂yiyjG
)(x]k, y
]k) = 0.
Moreover, (x]k, y]k) → (x∗, y
1∗) as k → ∞. By continuity, we conclude that G satisfies (ii) at
(x∗, y1∗, y
3∗).
Case 1c: y1∗ = y2
∗ = y3∗. As in the previous case, in a neighborhood of (x∗, y∗) for every k
large enough there must be a point (x]k, y]k) ∈Mk where the tangent vector toMk is vertical.
Letting k →∞ we again conclude
det(∂yiyjG
)(x∗, y∗) = 0. (6.32)
By (6.32), the limit manifold M has a vertical tangent at (x∗, y∗). Let s 7→ (x(s), y(s)) be alocal arc-length parameterization of M with (x(0), y(0)) = (x∗, y∗). If (6.13) holds, then thecondition (iv) is verified and we are done (Fig. 6, left). Otherwise, to fix the idea assume
d2
ds2x(s)
∣∣∣∣s=0
> 0.
We claim that (x∗, y∗) cannot be a point of global minimum for G(x∗, ·). Indeed, consider astrictly increasing sequence xν → x∗. Let yν be a value where the function G(xν , ·) attains itsglobal minimum. By taking a subsequence we can assume the convergence (xν , yν) → (x∗, y])for some y] (see Fig. 6, right). Notice that we must have y] 6= y∗, because the manifold Mdoes not contain any point (x, y) with x < x∗, in a neighborhood of (x∗, y∗). By continuity,(x∗, y]) must provide a global minimum to G(x∗, ·). We conclude that all conditions in (ii) arenow satisfied, with
(x, y1) = (x∗, y∗), (x, y2) = (x∗, y]).
CASE 2: Suppose that all Gk satisfy condition (ii). This implies that there exists a sequenceof points (xk, y1,k, y2,k) satisfying (ii) for Gk with
|xk| ≤ κ, |yik| ≤ ρ for i = 1, 2, y1k 6= y2
k .
30
By possibly taking a subsequence, one has
xk → x∗, y1k → y1
∗, y2k → y2
∗, with |x∗| ≤ κ, |y1∗| ≤ ρ, |y2
∗| ≤ ρ.
Two sub-cases must be considered.
Case 2a: y1∗ 6= y2
∗. In this case, by continuity the limit function G satisfies (ii) at (x∗, y1∗, y
2∗).
Case 2b: y1∗ = y2
∗. Since Gk satisfies (ii) at (xk, y1k, y
2k), this case can be handled by the same
arguments as case 1c.
CASE 3: Suppose that all Gk satisfy condition (iii), say at points (xk, y1k, y
2k), with
|xk| ≤ κ, |yik| ≤ ρ for i = 1, 2, y1k 6= y2
k.
By possibly taking a subsequence, one has
xk → x∗, y1k → y1
∗, y2k → y2
∗, with |x∗| ≤ κ, |y1∗| ≤ ρ, |y2
∗| ≤ ρ.
Two sub-cases must be considered.
Case 3a: y1∗ 6= y2
∗. In this case, if both φ1 and φ2 are still well defined at (x∗, y1∗) and (x∗, y
2∗),
then by continuity G satisfies (iii) at (x∗, y1∗, y
2∗). The remaining possibility is that at least
one of the functions φ1 or φ2 that is not well defined. Without loss of generality, assume thatφ1 is not well defined. Namely, det(∂2
yi,yjG(x, y1∗)) = 0 and hence G satisfies (ii). In all cases
we conclude that G /∈ G].
Case 3b: y1∗ = y2
∗ = y∗. In this case, for all k large enough (xk, φ1(xk)) and (xk, φ2(xk)) shouldlie on the same component of a smooth one-dimensional manifoldMk defined by the equation∇yGk(x, y) = 0. In a neighborhood of (x∗, y∗), for every k large enough there must be a point
(x]k, y]k) ∈Mk where the tangent vector to Mk is vertical. Letting k →∞ we again conclude
that (6.32) holds. Assuming that G(x∗, ·) attains its global minimum at (x∗, y∗), the samearguments used in case 1c again imply that either (ii) or (iv) hold.
CASE 4: Suppose that every function Gk satisfies condition (iv) at some point (xk, yk) with
|xk| ≤ κ, |yk| ≤ ρ.
By possibly taking a subsequence, we can assume (xk, yk) → (x∗, y∗). By the convergenceGk → G in C3, we conclude that G satisfies the condition (iv) at the point (x∗, y∗).
8. In this step we claim that, if none of the conditions (i)–(iv) holds true at any point whereG(x, ·) attains its global minimum, then the graph of the best reply map satisfies the conclusionof the theorem.
Indeed, consider any point x. By (i) the minimum of the function y 7→ G(x, y) is attained atmost at two distinct points.
CASE 1: The global minimum is attained at the two points y1 6= y2. Then, since (ii) fails,near (x, y1) and (x, y2) the manifold M coincides with the graph of two functions y = φ1(x)
31
x
y
x xx
yy
x x
y
y
yy
ν
*
Mk
k
*
M
* *
#
M
ν
k
k3
1
k
M
M
k
Figure 6: The two main cases considered in step 7 of the proof of Theorem 6.1.
and y = φ2(x). By the previous steps, we can assume that (iii) fails. To fix the ideas let
d
dxG(x, φ1(x)) >
d
dxG(x, φ2(x)).
This implies that, for some δ > 0 small enough, the global minimum is attained at (x, φ1(x))for x ∈ [x− δ, x] and at (x, φ2(x)) for x ∈ [x, x+ δ].
CASE 2: The global minimum is attained at a single point y. We claim that
det(∂2yiyjG(x, y)
)6= 0, (6.33)
hence in a neighborhood of x the best reply map is single-valued: R(x) = {φ(x)}, for some C2
function φ(·).
Indeed, if (6.33) fails, since G does not satisfy (iv) at (x, y), we can assume that d2xds2|s=0 > 0,
the other case is entirely similar. By the same arguments as case 1c, G(x, ·) achieves the globalminimum at two points y 6= y, reaching a contradiction.
Combining the above two cases, the proof of the theorem is completed.
An application of Theorems 2.1 and 2.2 now yields the generic stability of Stackelberg equi-libria. As in Section 2, we denote by F∞ the family of functions satisfying (B1), with thedistance (2.6).
Theorem 6.2 Consider a generic function G ∈ G] ⊂ C3(R × RN ), satisfying the conclusionof Theorem 6.1. Then there exists an open dense set of functions F ] ⊂ F∞ such that, forevery F ∈ F ], the following holds.
The global minimum of F on A = graph(R) is attained in generic position, as defined in (2.2).Moreover there exists constants C, δ > 0 such that, if
‖F − F‖C2 ≤ δ, ‖G−G‖C3 ≤ δ, (6.34)
32
then the corresponding perturbed optimization problem
min(x,y)∈A
F (x, y) (6.35)
has a unique minimizer (x, y), also in generic position. Here A = graph(R) is the best replymap corresponding to the cost function G. In addition
|x− x|+ |y − y| ≤ C ·(‖F − F‖C2 + ‖G−G‖C3
). (6.36)
Proof. The same arguments used in the proof of Theorem 5.1 apply here as well. We remarkthat, in the present case, the strategy of the follower is not constrained to a closed set, hencethe best reply map has the simpler structure described in Theorem 6.1 (compare Fig. 4 withFig. 2). The conclusion of Theorem 6.2 thus follows already from Theorems 2.1 and 2.2,without using Corollaries 2.2 and 2.3.
7 Concluding remarks
In this paper we used techniques from differential geometry to analyze the generic structureof solutions to noncooperative games. In particular, we described the structure of the bestreply map and proved the uniqueness and stability of the Stackelberg equilibrium, for an opendense set of cost functions to the leader and to the follower.
While our results only cover some specific settings, it is clear that these techniques can havebroader applications. To keep the discussion simple, we considered Stackelberg games wherethe leader’s strategy lies in a one-dimensional space. We expect that a similar analysis canbe performed also in the case where the leader chooses his strategy within a two-dimensionalmanifold, and hence the follower solves a minimization problem depending on two parameters.In this case, the manifoldM defined at (1.6) will then contain not only “fold” but also “cusp”singularities [1, 7]. On the other hand, when the leader’s strategy ranges in a high dimensionalspace, the generic structure of the best reply map will be more difficult to describe.
The main motivation for the present work came from the analysis of infinite-horizon stochasticgames with discrete state space [5]. In that paper, a key role is played by a stability assumptionon the Stackelberg equilibrium solution, when the follower adopts a myopic strategy. Thearguments used in the proof of Theorem 4.1 show that this assumption is not very restrictive.Indeed, it is satisfied by “almost all” cost functionals.
8 Appendix: working tools from differential geometry
Assume f : Rm 7→ Rn
• x ∈ Rm is a critical point of f if the Jacobian matrix Df(x) has rank < n. Equiva-lently: if the differential Df(x) is not surjective.
• y ∈ Rn is a critical value of f if y = f(x) for some critical point x.
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Sard’s Lemma [16].
For any f ∈ C∞(Rm; Rn), the set of critical values has n-dimensional measure zero.
b
x
yf
V
C
Figure 7: An illustration of Sard’s theorem. Here the se of critical points of f is large, but its imagehas measure zero.
8.1 Transversality.
Let f : X 7→ Y be a smooth map of manifolds and let W be a submanifold of Y .
We say that f is transverse to W at a point p, and write f ∩| pW , if
• either f(p) /∈W ,
• or else f(p) ∈W and (df)p(TpX) + Tf(p)W = Tf(p)Y .
We say that f is transverse to W , and write f ∩| W , if f ∩| pW for every p ∈ X.
f(p)
f(p)
Wp
X
Y
f(p)
f
Figure 8: The map shown in the center, whose graph is tangent to W , is not transversal. All otherfunctions are transversal.
Example. Take X = Rm, Y = Rn with m < n. Assume W = {y0} for some y0 ∈ Rn. Inthis case, transversality implies
• either f(x) 6= y0,
• or else f(x) = y0 and rank (Df(x)) = n.
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Since the second alternative is impossible,
f ∩| W ⇐⇒ f(x) 6= y0 for all x ∈ R2
f
X
f(X)
y0
Figure 9: A map f : R2 7→ R3 is transversal to the manifold W = {y0} if and only if its image doesnot contain the point y0.
Transversality lemma. Let X, Θ, and Y be smooth manifolds, W a submanifold of Y . Letθ 7→ φθ be a smooth map which to each θ ∈ Θ associates a function φθ ∈ C∞(X,Y ), and defineΦ : X ×Θ 7→ Y by setting Φ(x, θ) = φθ(x).
If Φ ∩| W , then the set {θ ∈ Θ, ; φθ ∩| W} is dense in Θ.
p
W
Φ
ΘxX
θ
Θ
W
Yφ
0φf =
Y
X0
Figure 10: A family of maps θ 7→ φθ depending on a parameter θ ∈ Θ can also be seen as a single mapΦ from the product space: X ×Θ 7→ Y . If Φ is transversal, then for a.e. θ the map φθ is transversal.
8.2 A multi-jet transversality theorem.
We state here a version of the multi-jet transversality theorem which is used several times inthe paper. In the following, for given integers m,n ≥ 1 we consider maps f : Rm+n 7→ R. LetPk the space of polynomials of degree ≤ k in the m+n variables (x, y) ∈ Rm×Rn. Identifyinga polynomial with its coefficients, it is clear that Pk is a finite dimensional vector space. Theproduct space
Jk(Rm+n; R).= Rm+n × Pk
is a jet bundle over the space Rm+n. Any function f : Rm+n 7→ R determines a section of thisbundle, defined as jkf(x, y) = P (x,y), where the polynomial P (x,y)(·) is the k-th order Taylorapproximation of the function f at the point (x, y).
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Next, for s ≥ 1 we call
Z(s) .= {(x, y1, . . . , ys) ∈ Rm × Rn × · · · × Rn ; yi 6= yj for all 1 ≤ i < j ≤ s}. (8.1)
Notice that Z(s) is an open subset of a vector space of dimension m+sn; hence it is a manifold.The set
Jks (Rm+n; R) ={
(x, y1, . . . , ys, P1, . . . , Ps) ∈ Rm × Rn × · · · × Rn × Pk × · · · × Pk,
yi 6= yj for all 1 ≤ i < j ≤ s}
(8.2)is a k-th order jet bundle over Z(s). Any smooth function f : Rm+n 7→ R determines a sectionof this bundle defined as
jks f(x, y1, . . . , ys) = (Q(x,y1), . . . , Q(x,ys)), (8.3)
where Q(x,yi) is the polynomial of degree ≤ k determined by the k-th order Taylor approxi-mation to f at the point (x, yi). We can now state a version of the multi-jet transversalitytheorem which is used in our paper. The proof is similar to the one on p. 57–59 of [7], withsome simplifications due to the fact that our maps are defined on Euclidean spaces, ratherthan on general manifolds.
Theorem 8.1 Let W be a smooth submanifold of Jks (Rm+n; R). Then the set of functionsf ∈ C∞(Rm+n; R) which are transversal to W is dense, in the C∞ topology.
Proof. 1. Cover the open set Z(s) with countably many open sets Vν , ν ≥ 1, such that
• If (x, y1, . . . , ys) ∈ Vν and (x, y1, . . . , ys) ∈ Vν , then yi 6= yj for all 1 ≤ i < j ≤ s.
Construct C∞ functions φν : Rm+n 7→ [0, 1], ν ≥ 1, with the following properties.
• Supp(φν) ⊂ Vν .
• For each ν ≥ 1, call V ′ν ⊂ Vν the interior of the set where φν = 1. Then⋃ν≥1 V
′ν = Z(s).
2. For each ν ≥ 1, and any polynomials P1, . . . , Ps of degree ≤ k, define the function
f (P )(x, y) = f(x, y) +
s∑`=1
φk(x, y)P`(x, y`). (8.4)
We now consider the map
(x, y1, . . . , ys, P1, . . . , Ps) 7→(jkf (P )(x, y1), . . . jkf (P )(x, ys)
). (8.5)
The right hand sides are the coefficients of the k-th order Taylor approximations to the maps(x, y) 7→ f (P )(x, y) at the points (x, y`). Since φν ≡ 1 on V ′ν , it is clear that the differential ofthe map (8.5) has full rank. Hence this map is transversal to any manifold W , restricted to V ′ν .By the transversality theorem, there is a residual set Sν ⊂ C∞(Rm+n; R) such that, for every
36
f ∈ Sν , the map jks f in (8.3) is transversal to W at every point (x, y1, . . . , ys, P1, . . . , Ps) ∈Wsuch that (x, y1, . . . , ys) ∈ V ′ν .
3. Repeating the same argument for every ν ≥ 1, we obtain a sequence of residual subsetsSν . The intersection S .
=⋂ν≥1 Sν is still residual in C∞(Rm+n; R). By construction, for every
f ∈ S the map jks f is transversal to W .
References
[1] V. Arnold, Catastrophe Theory, Third edition. Springer-Verlag, Berlin-Heidelberg, 1992.
[2] T. Basar and G. J. Olsder, Dynamic Noncooperative Games, Academic Press, NewYork,1982.
[3] J. M. Bloom, The local structure of smooth maps of manifolds, B.A. Thesis, Harvard U.,2004.
[4] A. Bressan, Noncooperative differential games. Milan J. of Mathematics, 79 (2011), 357-427.
[5] A. Bressan and Y. Jiang, Self-consistent feedback Stackelberg solutions for infinite horizonstochastic games, preprint, 2019.
[6] J. Dieudonne, Foundations of Modern Analysis. Academic Press, New York - London,1969.
[7] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. Springer-Verlag, New York, 1973.
[8] L. C. Evans, Partial Differential Equations. Second edition. American Mathematical So-ciety, Providence, RI, 2010.
[9] J. Harsanyi, Oddness of the number of equilibrium points: a new proof. Internat. J. GameTheory 2 (1973), 235–250.
[10] M. B. Lignola and J. Morgan, Topological existence and stability for Stackelberg prob-lems. J. Optim. Theory Appl. 84 (1995), 145–169.
[11] M. B. Lignola and J. Morgan, α-well-posedness for Nash equilibria and for optimizationproblems with Nash equilibrium constraints. J. Global Optim. 36 (2006), 439–459.
[12] L. Mallozzi and J. Morgan, Weak Stackelberg problem and mixed solutions under dataperturbations, Optimization 32 (1995), 269–290.
[13] A. Marhfour, Mixed solutions for weak Stackelberg problems: existence and stabilityresults. J. Optim. Theory Appl. 105 (2000), 417–440.
[14] C. Meroni and C. Pimienta, The structure of Nash equilibria in Poisson games. J. Eco-nomic Theory 169 (2017), 128–144.
[15] P. Olver, Applications of Lie Groups to Differential Equations, Second Edition. Springer-Verlag, New York, 1993.
37
[16] A. Sard, The measure of critical values of differentiable maps. Bull. Amer. Math. Soc. 48(1942), 883–890.
[17] M. Simaan and J. Cruz, On the Stackelberg strategies in nonzero-sum games, J. Optim.Theory Appl. 11 (1973), 533–555.
[18] H. von Stackelberg, The Theory of the Market Economy. Oxford Univ. Press, 1952.
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