1 Introduction

We consider a Stackelberg game with multiple leaders \(N=\{1,\ldots ,n\}\), \(n\ge 2\), and a continuum of followers represented by the interval [0, 1]Footnote 1 The followers’ problem constitutes a (nonatomic) singleton congestion game, that is, each follower chooses one resource and the cost of the resource depends on the congestion of this resource. The leaders of the game each control one resource, and decide about a price which is charged to the followers for the usage of the resource, as well as a capacity which influences the slope of the congestion cost of her resource. Concretely, each leader \(i\in N\) chooses a price \(p_i\in [0,C_i]\), where \(C_i>0\) is a given price cap, and a capacity \(z_i\ge 0\). The resulting effective cost function of leader i’s resource - from the perspective of a follower - is the sum of the congestion cost function \(\ell _i(x_i,z_i)\) and the price \(p_i\) charged, where \(x_i\) denotes the volume of followers who have chosen leader i. We assume that the congestion cost \(\ell _i(x_i,z_i)\) is infinite if no capacity is installed (i.e., \(z_i=0\)), and else it depends linearly on the volume of followers and inverse-linearly on the installed capacity, that is,

$$\begin{aligned} \ell _i(x_i,z_i):={\left\{ \begin{array}{ll} \frac{a_i x_i}{z_i}+b_i, &{} \text { for }\quad z_i>0, \\ \infty , &{} \text { for }\quad z_i=0, \end{array}\right. } \end{aligned}$$

where \(a_i>0\) and \(b_i\ge 0\) are given parameters for \(i\in N\). As is common in the transportation science literature (see, e.g., [18, 28] and the references mentioned therein), the capacity \(z_i\) is not a strict bound on the admissible flow, but instead influences the congestion dependent cost. The case \(z_i=0\) can be interpreted as if the resource controlled by leader i is not present in the followers’ congestion game. Given a capacity vector \(z=(z_1,\ldots ,z_n)\) with \(\sum _{i\in N}{z_i}>0\), i.e., there is at least one resource in the followers’ congestion game, and a price vector \(p=(p_1,\ldots ,p_n)\), the followers choose rationally the most attractive resource in terms of the effective cost. That is, the outcome \(x\in P:=\{x\in {\mathbb {R}}^n_{\ge 0}|\sum _{i\in N}x_i=1\}\) of the followers’ congestion game is a Wardrop equilibrium, described by the following Wardrop equilibrium conditions:

$$\begin{aligned} c_i(x,z,p):=\ell _i(x_i,z_i)+p_i\le \ell _j(x_j,z_j)+p_j=:c_j(x,z,p) \end{aligned}$$

holds for all \(i,j\in N\) with \(x_i>0\). Note that for given capacities \(z\ne 0\) and prices p, there is exactly one \(x\in P\) satisfying the Wardrop equilibrium conditions (see, e.g., [10]). Call this flow \(x=x(z,p)\) the Wardrop flow induced by (zp).Footnote 2 In particular, there is a constant \(K\ge 0\) such that \(c_i(x,z,p)=K\) holds for each \(i\in N\) with \(x_i>0\), and \(c_i(x,z,p) \ge K\) holds for each \(i\in N\) with \(x_i=0\). For a Wardrop flow x, call the corresponding constant K the (routing) cost of x. We assume that each leader \(i\in N\) seeks to maximize her own profit function, which is defined as

$$\begin{aligned} \Pi _i(z,p):={\left\{ \begin{array}{ll} p_ix_i(z,p)-\gamma _iz_i, &{} \text { for }\quad \sum _{i\in N}{z_i}>0, \\ 0, &{} \text { else}, \end{array}\right. } \end{aligned}$$

where \(\gamma _i>0\) is a given installation cost parameter for leader i. This completes the description of our game, which we denote a Stackelberg pricing game.

Note that the introduced Stackelberg pricing game captures many aspects of realistic oligopolistic markets where firms offer a service to customers, like build-operate schemes in traffic networks or competition among WIFI-providers (see, e.g., [1, 9, 20, 21, 26, 34]).

In this paper, we analyze the stable states of Stackelberg pricing games, that is, pure Nash(–Stackelberg) equilibria as defined in the following. For each leader \(i\in N\), let \(S_i:=\{s_i=(z_i,p_i): 0\le z_i, 0\le p_i \le C_i\}\) be her strategy set. A vector s consisting of strategies \(s_i=(z_i,p_i) \in S_i\) for all \(i\in N\) is called a strategy profile, and \(S:=\times _{i \in N}S_i\) denotes the set of strategy profiles. Usually, we will write a strategy profile \(s \in S\) in the form \(s=(z,p)\), where z denotes the vector consisting of all capacities \(z_i\) for \(i\in N\), and p is the vector of prices \(p_i\) for \(i\in N\). The profit of leader i for a strategy profile \(s=(z,p)\) is then defined as \(\Pi _i(s):=\Pi _i(z,p)\). Furthermore, we write \(x(s):=x(z,p)\) for the Wardrop-flow induced by \(s=(z,p)\) and \(K(s):=K(z,p)\) for the routing cost of x(s). For leader i, denote by \(s_{-i}=(z_{-i},p_{-i})\in S_{-i}:=\times _{j\in N \setminus \{i\}}{S_j}\) the vector consisting of strategies \(s_j=(z_j,p_j)\in S_j\) for all \(j\in N\setminus \{i\}\). We then write \((s_i,s_{-i})=((z_i,p_i),(z_{-i},p_{-i}))\) for the strategy profile where leader i chooses \(s_i=(z_i,p_i)\in S_i\), and the other leaders choose \(s_{-i}=(z_{-i},p_{-i})\in S_{-i}\). Moreover, we use the simplified notation \(\Pi _i((s_i,s_{-i}))=\Pi _i(s_i,s_{-i})\) and \(x((s_i,s_{-i}))=x(s_i,s_{-i})\). A strategy profile \(s=(s_i,s_{-i})\) is a pure Nash(–Stackelberg) equilibrium (PNE), if for each leader \(i\in N\):

$$\begin{aligned} \Pi _i( s_i,s_{-i} )\ge \Pi _i( s_i',s_{-i} ) \text { for all } s_i'\in S_i. \end{aligned}$$

For given strategies \(s_{-i}\in S_{-i}\) of the other leaders, the best response correspondence of leader i is defined by

$$\begin{aligned} \text {BR}_i(s_{-i}):=\arg \max \{\Pi _i( s_i,s_{-i} ) \vert s_i\in S_i\}. \end{aligned}$$

If \(s_{-i}\) is clear from the context, we just write \(\text {BR}_i\) instead of \(\text {BR}_i(s_{-i})\). Clearly, the strategy profile \(s=(s_i,s_{-i})\) is a PNE if and only if \(s_i\in \text {BR}_i(s_{-i})\) is fulfilled for each \(i\in N\).

The next subsection summarizes our results in terms of existence, uniqueness and quality of PNE for Stackelberg pricing games.

1.1 Our results and proof techniques

As our main result, we show existence of PNE for the introduced Stackelberg pricing game. This result requires a completely new proof approach compared to previous approaches, since the leaders’ profit functions are not continuous. Therefore, best responses do not always exist and standard fixed-point arguments à la Kakutani are not directly applicable. For the existence proof, we first completely characterize in Theorem 1 the continuity of the joint profit function (consisting of all leaders’ profits). Using this, we completely characterize the structure of best response correspondences of leaders; including the possibility of non-existence of a best response (Theorem 2). We then establish the existence of equilibria (Theorem 4) using the concept of C-security introduced by [29], which in turn resembles ideas of [31]. A game is C-secure at a given strategy profile, if each player has a pure strategy guaranteeing a certain utility value, even if the other players play some perturbed strategy within a (small enough) neighborhood, and furthermore, for each slightly perturbed strategy profile, there is a player whose perturbed strategy can in some sense be strictly separated from her securing strategies. Intuitively, the concept of securing strategies means that those strategies are robust to other players’ small deviations. The result of [29] states that a game with compact, convex strategy sets and bounded profit functions admits an equilibrium, if every non-equilibrium profile is C-secure. It is important to note that the concept of C-security does not rely on quasi-concavity or continuity of profit functions. With our characterization of best response correspondences at hand, we show that the considered Stackelberg pricing game fulfills the conditions of [29] and thus admits PNE.

As our second main result, we show that the equilibrium is essentially unique (Theorem 5). While the general proof approach is related to that of Johari et al. [21] (see also the related work in the following subsection), our model allows for price caps thus requiring additional ideas. In particular, the set of leaders having positive capacity needs to be decomposed, where the decomposition is related to the property whether the price of a leader is equal to its cap, or strictly smaller.

We finally study the efficiency of the unique equilibrium compared to a natural benchmark, in which we relax the equilibrium conditions of the leaders, but not the equilibrium conditions of the followers. We show that the unique equilibrium might be arbitrarily inefficient (Theorem 6), by presenting a family of instances such that the quality of the equilibrium gets arbitrarily bad. Furthermore, for instances with identical leaders, we derive a closed-form expression for the equilibrium quality (Theorem 7).

1.2 Related work

Johari et al. [21] study existence, uniqueness and worst-case quality of PNE assuming that the demand of the followers is elastic: The volume of followers participating in the congestion game decreases with increasing combined cost of congestion and price. As a consequence of the elastic demand assumption, best responses of leaders do always exist in their model, and Kakutani’s fixed point theorem can be applied to show existence of PNE. As already noted, this is not possible for the model with inelastic demand that we consider in this paper. We discuss in detail the motivation for assuming inelastic demand in Sect. 1.3. Johari et al. also consider the case of inelastic follower’s demand, assuming homogeneous leaders (that is, all leaders have the same parameters). As shown in their paper, homogeneity (together with some assumptions on the congestion costs) implies that there is only one symmetric equilibrium candidate profile. For this specific symmetric strategy profile, they directly prove stability using concavity arguments. This proof technique is clearly not applicable in the general non-homogeneous case (with fixed demand). Further differences between our model and Johari et al. are that they do not consider price caps, but on the other hand, allow more general congestion cost functions.

Acemoglu et al. [1] study a model in which the capacities represent “hard” capacities bounding the admissible customer volume for a leader. They observe that equilibria do not exist in this model. Subsequently they study a model in which capacities and prices are not chosen simultaneously anymore, but the leaders first determine capacities, and only after the chosen capacities became apparent, set prices. For this model, they investigate existence and worst-case quality of equilibria (see also [23] for earlier work on the two-stage model).

Schmand et al. [33] study a network investment game in which the leaders invest in edges of a series-parallel graph (but do not directly set prices).

Further related models are used in the papers of Harks et al. [20] and Correa et al. [9]. There, leaders do not choose capacities, but only prices, and the prices are upper-bounded by caps (equal for all leaders in [20], leader-specific in [9]). The two papers consider the problem of a system designer who chooses the cap(s) in order to minimize total congestion.

There are also numerous works analyzing Stackelberg games with a single leader. For example, Labbé et al. [24] study a model where a single leader sets prices in order to maximize her profit in a subsequent network routing game without congestion effects. Situations where the leader determines capacities or prices in order to reduce the total congestion (plus investments for the case of capacities) of the resulting Wardrop equilibria are for example studied in [18, 28] for setting capacities, and [4, 35] for setting prices. Castiglioni et al. [7] and Marchesi et al. [27] consider Stackelberg games with an underlying (atomic) congestion game, where the single leader participates in the same congestion game than the followers.

Finally, network design problems with congestion dependent costs have also been studied from a purely optimization point of view, for example in the context of energy-efficient networks [3, 14]. There one wants to minimize the congestion-dependent cost under certain network connectivity requirements.

1.3 Motivation for inelastic demand

We focus on the case of inelastic demand, that is, there is a fixed volume of followers in the congestion game. This assumption is made by many works in the transportation science and algorithmic game theory literature (see, e.g., [6, 11,12,13, 17, 35] and [2, 8, 16, 32], respectively) and usually considered as a fundamental base case. As already noted, in terms of equilibrium existence, the case of inelastic demand is much more complicated for our model compared to the seemingly more general case of elastic demand: best responses do not always exist in the inelastic case, putting standard fixed-point approaches out of reach.

Besides this theoretical aspect, we think that a thorough analysis of the inelastic demand case will be helpful in understanding realistic demand scenarios which are not captured by the literature for the elastic demand. To make this point clear it is worth recalling the underlying assumptions made in the literature for the elastic demand case. In Johari et al. [21], the demand function can be described by a differentiable, strictly decreasing and concave function. Liu et al. [26] assume instead of concavity that \(d\cdot B(d)\) is concave, where B(d) denotes the inverse demand function.Footnote 3 The strict monotonicity of the demand function implies that every follower particle has its own unique valuation w.r.t. participating in the game. These assumptions do not cover the realistic case that commuters stick to a travel mode (private car) for some cost range, and only if the travel cost exceeds that of an alternative travel mode (public transport), they switch mode. A demand function of this type would be piecewise constant, or piecewise linear, but in either case not strictly decreasing, and perhaps not even continuous.

Further examples of a rather inelastic demand appear for higher value travel (as business or commute travel, in particular “urban peak-period trips”), in case the public transport alternatives are sparse (rural areas) or if commuters with higher income are considered, see also Litman [25], where various factors influencing travel demand are discussed.

1.4 Outline of the paper

The following sections contain the technical presentation of our results. Concretely, we characterize in Sects. 2 and 3 the continuity of the leaders’ profit functions, as well as the best response correspondences. These results are then used to show existence (Sect. 4) and uniqueness (Sect. 5) of PNE. Finally, we analyze the quality of PNE in Sect. 6.

2 Continuity of the profits

In this section, we prove a fundamental result about the continuity of the profits. We will use Theorem 1, which completely characterizes the strategy profiles s having the property that all profit functions \(\Pi _i\), \(i\in N\), are continuous at s, several times during the rest of the paper.

Theorem 1

Let \(s=(z,p)\in S\). Then: The profit function \(\Pi _i\) is continuous at \(s=(z,p)\) for all \(i\in N\) if and only if \(z\ne 0\) or \((z,p)=(0,0)\).

Proof

We start with the strategy profile \(s=(z,p)=(0,0)\) and show that for each leader i, her profit function \(\Pi _i\) is continuous at s. Let \(i\in N\). For \(\varepsilon >0\), define \(\delta :=\min \{\varepsilon /(2\gamma _i), \varepsilon /2\}>0\), and let \(s'\in S\) with \(||s'||=||s'-s||<\delta \) (where \(||\cdot ||\) denotes the Euclidean norm). If \(z_i'=0\), then \(\Pi _i(s')=\Pi _i(s)=0\). Otherwise, the following holds, showing that \(\Pi _i\) is continuous at s:

$$\begin{aligned} |\Pi _i(s')-\Pi _i(s)|&=|x_i(s')p_i'-\gamma _iz_i'|\le p_i'+\gamma _iz_i'\le ||s'||+\gamma _i||s'|| <\delta (1+\gamma _i)\le \varepsilon \end{aligned}$$

Now consider \(s=(z,p)\in S\) with \(z\ne 0\). We again need to show that all profit functions \(\Pi _i\), \(i\in N\), are continuous at s. Since \(z\ne 0\), we get that \(N^+:=\{j\in N: z_j>0\}\ne \emptyset \). Furthermore, for \(\delta _1>0\) sufficiently small, \(N^+\subseteq \{j\in N: z_j'>0\}=:N^+(s')\) holds for all \(s'=(z',p')\in S\) with \(||s-s'||<\delta _1\). Write \(s'=(s'_1,s'_2)\in S\), where \(s_1'\) denotes the strategies of the leaders in \(N^+\), and \(s_2'\) denotes the strategies of the leaders in \(N\setminus N^+\). Now let \(i\in N\). We need to show that \(\Pi _i\) is continuous at \(s=(z,p)\). For all \(s'\in S\) with \(||s-s'||<\delta _1\), leader i’s profit is \(\Pi _i(s')=x_i(s')p_i'-\gamma _iz_i'\). Thus it is sufficient to show that \(x_i\) is continuous at \(s=(s_1,s_2)\), where \(s_1\) and \(s_2\) denote the strategies of the leaders in \(N^+\) and \(N\setminus N^+\), respectively. The idea of the proof is to show that, for a slightly perturbed strategy profile \((s_1',s_2')\), the difference between \(x_i(s)\) and \(x_i( s_1',s_2 )\), as well as the difference between \(x_i( s_1',s_2 )\) and \(x_i( s_1',s_2' )\), is small. In the following, let \(s'\in S\) with \(||s-s'||<\delta _1\). It is well known ([4], compare also [10]) that \(x(s')\) is the unique optimal solution of the following optimization problem \(\text {Q}=\text {Q}(s')\):

$$\begin{aligned}&(\text {Q}) \quad \min \ \ \sum _{j\in N}{\int _{0}^{x_j}{\left( \ell _j(t,z_j')+p_j'\right) \text {d}t}} \ \\&\qquad \text { s.t.}\sum _{j\in N}{x_j}=1, \ x_j\ge 0 \ \forall j\in N. \end{aligned}$$

Furthermore, \(x_j(s')=0\) for \(j\notin N^+(s')\). Therefore, the values \((x_j(s'))_{j\in N^+(s')}\) are the unique optimal solution of

$$\begin{aligned}&\max -\sum _{j\in N^+(s')}{(a_j/(2z_j') \cdot x_j^2+(b_j+p_j')x_j)} \\&\qquad \text { s.t.} \sum _{j\in N^+(s')}{x_j}=1, \ x_j\ge 0 \ \forall j\in N^+(s'). \end{aligned}$$

By Berge’s theorem of the maximum [5], for all \(\varepsilon >0\) there is \(0<\delta _2=\delta _2(\varepsilon )<\delta _1\) such that \(||x(s)-x( s_1',s_2 )||<\varepsilon \) for all \((s_1',s_2)\in S\) with \(||s-(s_1',s_2)||<\delta _2\). That is, x is continuous at s if we only allow changes in \(s_1\), but not in \(s_2\). Furthermore, if \(q(s')\) denotes the optimal objective function value of \(\text {Q}(s')\), and if only changes in \(s_1\) are allowed, q is also continuous in s, i.e., for all \(\varepsilon >0\) there is \(0<\delta _3=\delta _3(\varepsilon )<\delta _1\) such that \(|q(s)-q( s_1',s_2 )|<\varepsilon \) for all \((s_1',s_2)\in S\) with \(||s-(s_1',s_2)||<\delta _3\). We now distinguish between \(z_i>0\) and \(z_i=0\).

First consider \(z_i=0\), i.e., \(i\notin N^+\), and let \(\varepsilon >0\). Note that \(x_i(s)=0\), thus we need to find \(\delta >0\) such that \(|x_i(s)-x_i(s')|=x_i(s')<\varepsilon \) for all \(s'\in S\) with \(||s-s'||<\delta \). To this end, define

$$\begin{aligned} \delta =\delta (i,\varepsilon ):={\left\{ \begin{array}{ll} \delta _3(1), &{} \text { if } q(s)+1\le b_i\varepsilon ,\\ \min \{\delta _3(1), \frac{a_i \varepsilon ^2}{2(q(s)+1-b_i\varepsilon )}\}, &{} \text { else,} \end{array}\right. } \end{aligned}$$

and let \(s'\in S\) with \(||s-s'||<\delta \). In particular, \(|z_i-z_i'|=z_i'<\delta \). Furthermore, \(q(s')\le q( s_1',s_2 )\le q(s)+1\) holds since \(||s-(s_1',s_2)||\le ||s'-s||<\delta \le \delta _3(1)\). If \(z_i'=0\), we immediately get \(x_i(s')=0<\varepsilon \). Thus assume \(z_i'>0\) and assume, by contradiction, that \(x_i(s')\ge \varepsilon \). Then, by definition of \(\delta \), we get the following contradiction:

$$\begin{aligned} q(s)+1\ge & {} q(s')\ge \frac{a_i}{2z_i'}x_i(s')^2+(b_i+p_i')x_i(s')\ge \frac{a_i}{2z_i'}\varepsilon ^2+b_i\varepsilon \\> & {} \frac{a_i}{2\delta }\varepsilon ^2+b_i\varepsilon \ge q(s)+1. \end{aligned}$$

Therefore, \(x_i(s')< \varepsilon \) holds, showing that \(x_i\) is continuous at s if \(i\notin N^+\).

Now consider the case \(i\in N^+\), i.e. \(z_i>0\). For \(\varepsilon >0\), we need to find \(\delta >0\) such that \(|x_i(s)-x_i(s')|<\varepsilon \) for all \(s'\in S\) with \(||s-s'||<\delta \). To this end, define

$$\begin{aligned} \delta :=\min \left\{ \min \left\{ \delta \left( j,\frac{\varepsilon }{2n}\right) : j\notin N^+\right\} , \delta _2\left( \frac{\varepsilon }{2}\right) \right\} \end{aligned}$$

and let \(s'\in S\) with \(||s-s'||<\delta \). In particular, \(||s-(s_1',s_2)||<\delta \le \delta _2(\frac{\varepsilon }{2})\), thus \(|x_i(s)-x_i( s_1',s_2 )|<\varepsilon /2\). Furthermore, since \(\delta \le \delta (j,\frac{\varepsilon }{2n})\), we get \(x_j(s')\le \frac{\varepsilon }{2n}\) for all \(j\notin N^+\). If \(x_j(s')=0\) for all \(j\notin N^+\), we get \(x_i(s')=x_i( s_1',s_2 )\) and thus \(|x_i(s)-x_i(s')|=|x_i(s)-x_i( s_1',s_2 )|<\varepsilon /2<\varepsilon \), as desired. Otherwise, there is \(j\notin N^+\) with \(0<x_j(s')\le \frac{\varepsilon }{2n}\). In particular, \(z_j'>0\). We now use a result about the sensitivity of Wardrop flows [15, Theorem 2]. They show that if the followers are not able to choose leader j’s resource anymore (we say that leader j is deleted from the followers’ game), the resulting change in the Wardrop flow can be bounded by the flow that j received. More formally, if \(x\in [0,1]^n\) is the Wardrop flow for the game with leaders N, and \(x'\in [0,1]^{n-1}\) is the Wardrop flow if leader j is deleted from the followers’ game, then \(|x_k-x_k'|\le x_j\) for all \(k\in N\setminus \{j\}\). Obviously, changing leader j’s capacity from \(z_j'>0\) to \(z_j=0\) has the same effect on the Wardrop flow as deleting leader j. Therefore, if we change, one after another, the capacities of all leaders \(j\notin N^+\) having \(z_j'>0\) to \(z_j=0\), we get \(|x_i(s')-x_i( s_1',s_2 )|\le (n-1)\varepsilon /(2n)<\varepsilon /2\) (note that the flow values for \(j\notin N^+\) are always upper-bounded by \(\varepsilon /(2n)\) due to our choice of \(\delta \) and the analysis of the case \(z_i=0\)). Using this, we now get the desired inequality:

$$\begin{aligned} |x_i(s)-x_i(s')|\le |x_i(s')-x_i( s_1',s_2 )|+|x_i(s)-x_i( s_1',s_2 )|<\varepsilon /2+\varepsilon /2=\varepsilon . \end{aligned}$$

Altogether we showed that all profit functions \(\Pi _i\) are continuous at \(s=(z,p)\) if \(z\ne 0\).

To complete the proof, it remains to show that if all profit functions \(\Pi _i\), \(i \in N\), are continuous at \(s=(z,p)\), then \(z\ne 0\) or \((z,p)=(0,0)\) holds. We show this by contraposititon, thus assume that \(s=(z,p)\) fulfills \(z=0\) and \(p\ne 0\). We need to show that there is a leader i such that \(\Pi _i\) is not continuous at s. To this end, let \(i\in N\) with \(p_i>0\). Define the sequence of strategy profiles \(s^n\) by \((z_j^n,p_j^n):=(z_j,p_j)\) for all \(j\ne i\), and \((z_i^n,p_i^n):=(1/n,p_i)\). Obviously, \(s^n \rightarrow s\) for \(n \rightarrow \infty \). But for the profits, we get

$$\begin{aligned} \Pi _i(s^n)=x_i(s^n)p_i^n-\gamma _iz_i^n=p_i-\gamma _i/n \rightarrow _{n\rightarrow \infty }p_i>0. \end{aligned}$$

Since \(\Pi _i(s)=0\), this shows that \(\Pi _i\) is not continuous at s. \(\square \)

3 Characterization of best responses

The aim of this section is to derive a complete characterization of the best-response correspondences of the leaders. We will make use of this characterization in all our main results, i.e., existence, uniqueness and quality of PNE. Given a leader \(i\in N\) and fixed strategies \(s_{-i}=(z_{-i},p_{-i})\in S_{-i}\) for the other leaders, we characterize the set \(\text {BR}_i=\text {BR}_i(s_{-i})\) of best responses of leader i to \(s_{-i}\). To this end, we distinguish between the two cases that \(z_{-i}=0\) (Sect. 3.1) and \(z_{-i}\ne 0\) (Sect. 3.2). Section 3.3 then contains the derived complete characterization. In Sect. 3.4, we discuss how our results about the best responses influence the applicability of Kakutani’s fixed point theorem.

3.1 The case \(z_{-i}=0\)

In this subsection, assume that the strategies \(s_{-i}=(z_{-i},p_{-i})\) of the other leaders fulfill \(z_{-i}=0\). Under this assumption, leader i does not have a best response to \(s_{-i}\):

Lemma 1

If \(z_{-i}=0\), then \(\text {BR}_i(z_{-i},p_{-i})=\emptyset \).

Proof

Whenever leader i chooses a strategy \((z_i,p_i)\) with \(z_i>0\), then \(x_i=1\) holds for the induced Wardrop-flow x, thus leader i’s profit is \(p_i-\gamma _iz_i\). On the other hand, any strategy \((z_i,p_i)\) with \(z_i=0\) yields a profit of 0. Thus, leader i’s profit depends solely on her own strategy \((z_i,p_i)\), and can be stated as follows:

$$\begin{aligned} \Pi _i(z_i,p_i):= {\left\{ \begin{array}{ll} p_i-\gamma _iz_i, &{} \text { for }\quad z_i>0, \\ 0, &{} \text { for }\quad z_i=0. \end{array}\right. } \end{aligned}$$

Obviously, \(\Pi _i(z_i,p_i)<C_i\) holds for each \((z_i,p_i)\in S_i\), i.e. for \(z_i\ge 0\) and \(0\le p_i\le C_i\). On the other hand, by \((z_i,p_i)=(\varepsilon ,C_i)\) for \(\varepsilon >0\), leader i gets a profit of \(C_i-\gamma _i\cdot \varepsilon \) arbitrarily near to \(C_i\), that is, \(\sup \{\Pi _i(z_i,p_i): (z_i,p_i)\in S_i\}=C_i\). This shows \(\text {BR}_i(z_{-i},p_{-i})=\emptyset \). \(\square \)

3.2 The case \(z_{-i}\ne 0\)

In this subsection, assume that the strategies \(s_{-i}=(z_{-i},p_{-i})\) of the other leaders fulfill \(z_{-i}\ne 0\). For a strategy \(s_i=(z_i,p_i)\) of leader i, write \(\Pi _i(z_i,p_i):=\Pi _i( s_i,s_{-i} )\) for leader i’s profit function, \(x(z_i,p_i):=x( s_i,s_{-i} )\) for the Wardrop-flow induced by \((s_i,s_{-i})\) and \(K(z_i,p_i):=K( s_i,s_{-i} )\) for the corresponding routing cost.

For \((z_i,p_i)\in S_i\), leader i’s profit is \(\Pi _i(z_i,p_i)=x_i(z_i,p_i)p_i-\gamma _iz_i\). It is clear that each strategy \((z_i,p_i)\) with \(z_i=0\) yields \(x_i(z_i,p_i)=0\), and thus \(\Pi _i(z_i,p_i)=0\). On the other hand, each strategy \((z_i,p_i)\) with \(z_i>C_i/\gamma _i\) yields negative profit since \(\Pi _i(z_i,p_i)=x_i(z_i,p_i)p_i -\gamma _iz_i < C_i -\gamma _i \cdot C_i/\gamma _i=0\). Therefore, each best response \((z_i,p_i)\) fulfills \(z_i\le C_i/\gamma _i\) since it yields nonnegative profit. Thus, \(\text {BR}_i\) can be described as the set of optimal solutions of the problem

Due to the theorem of Weierstrass, \((\text {P}_i)\) has an optimal solution: The feasible set of \((\text {P}_i)\) is compact and nonempty, and \(\Pi _i\) is continuous at \((z_i,p_i)\) for all feasible \((z_i,p_i)\) (Theorem 1). Since \(\text {BR}_i\) can be described as the set of optimal solutions of \((\text {P}_i)\), we get \(\text {BR}_i\ne \emptyset \).

Note that \((\text {P}_i)\) is a bilevel optimization problem (since \(x(z_i,p_i)\) can be described as the optimal solution of a minimization problem [4]), and these problems are known to be notoriously hard to solve. The characterization of \(\text {BR}_i\) that we derive here has the advantage that it only uses ordinary optimization problems, namely the following two (1-dimensional) optimization problems in the variable \(K\in {\mathbb {R}}\),

where

$$\begin{aligned} {\overline{x}}_i(K):=1-\sum _{j \in N(K)}{\frac{(K-b_j-p_j)z_j}{a_j}} \end{aligned}$$

with \(N^+:=\{j\in N\setminus \{i\}:z_j>0\}\) and \(N(K):=\{j\in N^+: b_j+p_j<K\}\).

Note that \({\overline{x}}_i: {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function which is equal to 1 for \(K\le \min \{b_j+p_j: j\in N^+\}\), and strictly decreasing for \(K\ge \min \{b_j+p_j: j\in N^+\}\ge 0\) with \(\lim _{K\rightarrow \infty }{\overline{x}}_i(K)=-\infty \). Therefore, there is a unique constant \(K_i^{\max }>0\) with the property \({\overline{x}}_i(K_i^{\max })=0\). Obviously, \(\overline{x}_i(K)>0\) if and only if \(K<K_i^{\max }\). Furthermore, the function \({\overline{x}}_i\) is closely related to Wardrop-flows, as described in the following lemma.

Lemma 2

  1. 1.

    If \((z_i,p_i)\in S_i\) with \(x_i(z_i,p_i)>0\), then \({\overline{x}}_i(K)=x_i(z_i,p_i)\) for \(K:=K(z_i,p_i)\).

  2. 2.

    If \(K\ge 0\) with \({\overline{x}}_i(K)>0\), and \((z_i,p_i)\in S_i\) fulfills \(z_i>0\) and \(a_i{\overline{x}}_i(K)/z_i+b_i+p_i=K\), then \(x_i(z_i,p_i)={\overline{x}}_i(K)\) and \(K(z_i,p_i)=K\).

Proof

We start with statement 1. of the lemma, so let \((z_i,p_i)\in S_i\) with \(x_i(z_i,p_i)>0\). By definition of \(K(z_i,p_i)=:K\), we get \(\ell _j(x_j(z_i,p_i), z_j)+p_j=K\) for all \(j\in N\) with \(x_j(z_i,p_i)>0\), and \(\ell _j(x_j(z_i,p_i), z_j)+p_j\ge K\) for all \(j\in N\) with \(x_j(z_i,p_i)=0\). Since

$$\begin{aligned} \ell _j(x_j(z_i,p_i), z_j)+p_j={\left\{ \begin{array}{ll} a_jx_j(z_i,p_i)/z_j+b_j+p_j, &{} \text { for }\quad j \in N \text { with } z_j>0, \\ \infty , &{} \text { for }\quad j\in N \text { with } z_j=0, \end{array}\right. } \end{aligned}$$

we get that \(\{j \in N: x_j(z_i,p_i)>0\}=\{j \in N^+: b_j+p_j<K\}\cup \{i\}=N(K)\cup \{i\}\). Therefore, for each \(j\in N(K)\), we get \(a_jx_j(z_i,p_i)/z_j+b_j+p_j=K\), which is equivalent to \(x_j(z_i,p_i)=(K-b_j-p_j)z_j/a_j\). Using \(\sum _{j\in N}{x_j(z_i,p_i)}=1\) yields

$$\begin{aligned} x_i(z_i,p_i)=1-\sum _{j\in N \setminus \{i\}}{x_j(z_i,p_i)}=1-\sum _{j \in N(K)}{(K-b_j-p_j)z_j/a_j}={\overline{x}}_i(K). \end{aligned}$$

Now we turn to statement 2. of the lemma, so let \(K\ge 0\) with \({\overline{x}}_i(K)>0\) and let \((z_i,p_i)\in S_i\) be a strategy with \(z_i>0\) and \(a_i{\overline{x}}_i(K)/z_i+b_i+p_i=K\). Consider \(x\in [0,1]^n\) defined by

$$\begin{aligned} x_j:={\left\{ \begin{array}{ll}{\overline{x}}_i(K), &{} j=i, \\ (K-b_j-p_j)z_j/a_j, &{} j\in N^+ \text { with } b_j+p_j <K, \\ 0, &{} j\in N^+ \text { with } b_j+p_j \ge K \text { or } j\in N\setminus (N^+\cup \{i\}). \end{array}\right. } \end{aligned}$$

It is clear that \(x_j>0\) holds for all \(j\in N^+\) with \(b_j+p_j<K\), and \(x_i={\overline{x}}_i(K)>0\). Furthermore, the definition of \({\overline{x}}_i(K)\) yields \(\sum _{j\in N}{x_j}=1\). Finally, x fulfills the Wardrop conditions:

$$\begin{aligned} c_j(x,z,p) ={\left\{ \begin{array}{ll} a_i{\overline{x}}_i(K)/z_i +b_i+p_i=K, &{} j=i, \\ a_j(K-b_j-p_j)z_j/(a_jz_j)+b_j+p_j=K, &{} j\in N^+ \text { with } b_j+p_j<K, \\ b_j+p_j \ge K, &{} j\in N^+ \text { with } b_j+p_j\ge K, \\ \infty >K, &{} j \in N \setminus (N^+\cup \{i\}). \end{array}\right. } \end{aligned}$$

The uniqueness of the Wardrop-flow now implies \(x=x(z_i,p_i)\), and \(K(z_i,p_i)=K\) follows from \(x_i(z_i,p_i)={\overline{x}}_i(K)>0\) and \(K(z_i,p_i)=c_i(x,z,p)=K\), completing the proof.\(\square \)

In the following lemmata, we analyze the connection between \((\text {P}_i^1)\) and \((\text {P}_i^2)\) and the optimal solutions of \((\text {P}_i)\).

Lemma 3

  1. 1.

    If K is feasible for problem \((\text {P}_i^1)\), the tuple \((z_i,p_i):=(\sqrt{a_i/\gamma _i}\cdot {\overline{x}}_i(K),K-\sqrt{a_i\gamma _i}-b_i)\) is feasible for \((\text {P}_i)\), and fulfills \(z_i>0\) and \(\Pi _i(z_i,p_i)=f_i^1(K)\).

  2. 2.

    If K is feasible for problem \((\text {P}_i^2)\), the tuple \((z_i,p_i):=(a_i {\overline{x}}_i(K)/(K-b_i-C_i),C_i)\) is feasible for \((\text {P}_i)\), and fulfills \(z_i>0\) and \(\Pi _i(z_i,p_i)=f_i^2(K)\).

Proof

We start with statement 1. of the lemma, thus assume that K is feasible for problem \((\text {P}_i^1)\). Let \(z_i:=\sqrt{a_i/\gamma _i}\cdot {\overline{x}}_i(K)\) and \(p_i:=K-\sqrt{a_i\gamma _i}-b_i\) as stated in 1. The feasibility of K for \((\text {P}_i^1)\) yields \({\overline{x}}_i(K)>0\) and \(2\sqrt{a_i\gamma _i}+b_i\le K\le \sqrt{a_i\gamma _i}+b_i+C_i\). From this we conclude \(z_i>0\) and \(0<p_i=K-\sqrt{a_i\gamma _i}-b_i\le C_i\), thus \((z_i,p_i)\in S_i\). Furthermore, \(a_i{\overline{x}}_i(K)/z_i +b_i+p_i= a_i\overline{x}_i(K)/(\sqrt{a_i/\gamma _i}{\overline{x}}_i(K)) +b_i+K-\sqrt{a_i\gamma _i}-b_i =K\) holds, thus we get \(x_i(z_i,p_i)={\overline{x}}_i(K)\) from statement 2. of Lemma 2. Using this, we can now show that leader i’s profit for \((z_i,p_i)\) equals the objective function value of K for \((\text {P}_i^1)\):

$$\begin{aligned} \Pi _i(z_i,p_i)&=p_i x_i(z_i,p_i)-\gamma _i z_i =(K-\sqrt{a_i\gamma _i}-b_i)\cdot {\overline{x}}_i(K)-\gamma _i\cdot \sqrt{a_i/\gamma _i}\cdot {\overline{x}}_i(K) \\&= {\overline{x}}_i(K)\cdot (K-2\sqrt{a_i\gamma _i}-b_i)=f_i^1(K) \end{aligned}$$

Note that the feasibility of K for \((\text {P}_i^1)\) yields \(f_i^1(K)\ge 0\). It remains to show that \((z_i,p_i)\) is feasible for \((\text {P}_i)\). We already know that \(z_i>0\) and \(0<p_i\le C_i\) holds. The remaining inequality \(z_i\le C_i/\gamma _i\) follows from the nonnegativity of \(\Pi _i(z_i,p_i)=f_i^1(K)\ge 0\) and the fact that any strategy with \(z_i>C_i /\gamma _i\) yields negative profit for leader i.

Now turn to statement 2. Assume that K is feasible for \((\text {P}_i^2)\), and let \(z_i:=a_i \overline{x}_i(K)/(K-b_i-C_i)\) and \(p_i:=C_i\). The feasibility of K for \((\text {P}_i^2)\) implies \(K>\sqrt{a_i\gamma _i}+b_i+C_i>b_i+C_i\) and \({\overline{x}}_i(K)>0\), thus \(z_i>0\) holds and this yields \((z_i,p_i)\in S_i\). Furthermore, \(a_i{\overline{x}}_i(K)/z_i+b_i+p_i=K\) holds, thus we get \(x_i(z_i,p_i)={\overline{x}}_i(K)\) from 2. of Lemma 2. The profit of leader i thus is

$$\begin{aligned} \Pi _i(z_i,p_i)&=p_ix_i(z_i,p_i)-\gamma _iz_i =C_i {\overline{x}}_i(K)- \frac{\gamma _ia_i {\overline{x}}_i(K)}{K-b_i-C_i} \\&= {\overline{x}}_i(K) (C_i-\frac{a_i\gamma _i}{K-b_i-C_i})=f_i^2(K). \end{aligned}$$

Note that \(f_i^2(K)\ge 0\) holds due to the feasibility of K for \((\text {P}_i^2)\). As in the proof of statement 1. of the lemma, this implies \(z_i\le C_i/\gamma _i\). Thus \((z_i,p_i)\) is feasible for \((\text {P}_i)\). \(\square \)

In particular, Lemma 3 shows that any optimal solution of \((\text {P}_i^1)\) or \((\text {P}_i^2)\) yields a feasible strategy for \((\text {P}_i)\) with the same objective fuction value. The next lemma shows that for certain optimal solutions of \((\text {P}_i)\), the converse is also true.

Lemma 4

Let \((z_i^*,p_i^*)\) be an optimal solution of \((\text {P}_i)\) and \(K^*:=K(z_i^*,p_i^*)\). If \(z_i^*>0\), then exactly one of the following two cases holds:

  1. 1.

    \((z_i^*,p_i^*)=(\sqrt{a_i/\gamma _i} {\overline{x}}_i(K^*), K^*-\sqrt{a_i\gamma _i}-b_i)\); \(K^*\) optimal for \((\text {P}_i^1)\) with \(f_i^1(K^*)=\Pi _i(z_i^*,p_i^*)\).

  2. 2.

    \((z_i^*,p_i^*)=(a_i {\overline{x}}_i(K^*)/(K^*-b_i-C_i),C_i)\); \(K^*\) optimal for \((\text {P}_i^2)\) with \(f_i^2(K^*)=\Pi _i(z_i^*,p_i^*)\).

Proof

Let \((z_i^*,p_i^*)\) with \(z_i^*>0\) and \(K^*\) as in the lemma statement, and define \(x^*:=x(z_i^*,p_i^*)\). Since \(\Pi _i(z_i^*,p_i^*)\ge 0\) and \(z_i^*>0\) holds, \(0\le \Pi _i(z_i^*,p_i^*)=p_i^*x_i^*-\gamma _iz_i^*<p_i^*x_i^*\) follows, which implies \(x_i^*>0\) and \(p_i^*>0\). Therefore \(K^*=a_ix_i^*/z_i^* +b_i+p_i^*\) holds and \((z_i^*,p_i^*)\) is an optimal solution for the following problem (with variables \(z_i\) and \(p_i\)):

$$\begin{aligned} \text { (P) } \qquad \max&\quad p_i x_i^*-\gamma _iz_i \quad \text {s.t.:}&a_ix_i^*/z_i +b_i+p_i=K^*, \quad 0< z_i, \quad 0< p_i\le C_i. \end{aligned}$$

Note that the optimal solutions of (P) correspond to all best responses for leader i such that \(x^*\) remains the Wardrop flow. Reformulating the equality constraint in (P) yields \(p_i=K^*-a_ix_i^*/z_i-b_i\). The constraints \(0< p_i\le C_i\) then become (note that \(K^*>b_i+p_i^*\ge b_i\) holds) \(0<K^*-a_ix_i^*/z_i -b_i \Leftrightarrow z_i>a_ix_i^*/(K^*-b_i)\) and \(K^*-a_i x_i^*/z_i-b_i\le C_i \Leftrightarrow 1/z_i\ge (K^*-b_i-C_i)/(a_ix_i^*)\). Thus (P) is equivalent to the following problem (with variable \(z_i\)):

Let f be the objective function of (P\('\)) and consider the derivative \(f'(z_i)=a_i{x_i^*}^2/(z_i^2)-\gamma _i\). We get that f is strictly increasing for \(0<z_i<\sqrt{a_i/\gamma _i} \cdot x_i^*\) and strictly decreasing for \(z_i>\sqrt{a_i/ \gamma _i} \cdot x_i^*\). We now distinguish between the cases that \(z_i=\sqrt{a_i/\gamma _i}\cdot x_i^*\) is feasible for (P\('\)), or not. As we will see, the former case leads to statement 1. of the lemma, and the latter case to statement 2. Note that in either case, \({\overline{x}}_i(K^*)=x_i^*\) holds (by statement 1. of Lemma 2).

If \(z_i=\sqrt{a_i/\gamma _i}\cdot x_i^*\) is feasible for (P\('\)), it is the unique optimal solution of (P\('\)). But since \(z_i^*\) is also optimal for (P\('\)), we get

$$\begin{aligned} z_i^*=\sqrt{a_i/\gamma _i}\cdot {\overline{x}}_i(K^*) \text { and } p_i^*=K^*-a_i{\overline{x}}_i(K^*)/z_i^* -b_i=K^*-\sqrt{a_i\gamma _i}-b_i. \end{aligned}$$

For the profit of leader i, we get

$$\begin{aligned} \Pi _i(z_i^*,p_i^*)= p_i^*x_i^*-\gamma _iz_i^*=(K^*-\sqrt{a_i\gamma _i}-b_i) \overline{x}_i(K^*)-\gamma _i\sqrt{a_i/\gamma _i} {\overline{x}}_i(K^*) =f_i^1(K^*). \end{aligned}$$

It remains to show that \(K^*\) is optimal for \((\text {P}_i^1)\). For feasibility, we need \(2\sqrt{a_i\gamma _i}+b_i\le K^*\le \sqrt{a_i\gamma _i}+b_i+C_i\) and \({\overline{x}}_i(K^*)>0\). The last inequality follows directly from \({\overline{x}}_i(K^*)=x_i^*>0\). Using this and \(\overline{x}_i(K^*)\cdot (K^*-2\sqrt{a_i\gamma _i}-b_i)=\Pi _i(z_i^*,p_i^*)\ge 0\) yields \(K^*\ge 2\sqrt{a_i\gamma _i}+b_i\). Finally, the feasibility of \(z_i^*=\sqrt{a_i/\gamma _i}\cdot x_i^*\) for (P\('\)) implies \((K^*-b_i-C_i)/(a_ix_i^*) \le \sqrt{\gamma _i}/(\sqrt{a_i}x_i^*)\), and thus \(K^*\le \sqrt{a_i\gamma _i}+b_i+C_i\) is fulfilled. Therefore, \(K^*\) is feasible for \((\text {P}_i^1)\). The optimality follows from Lemma 3 and the optimality of \((z_i^*,p_i^*)\) for \((\text {P}_i)\).

Now turn to the case that \(z_i=\sqrt{a_i/\gamma _i}\cdot x_i^*\) is not feasible for (P\('\)). We show that statement 2. of the lemma holds. Since (P\('\)) has an optimal solution (namely \(z_i^*\)), we get that \(0<a_ix_i^*/(K^*-b_i-C_i)<\sqrt{a_i/\gamma _i}\cdot x_i^*\) holds, and therefore \(z_i=a_ix_i^*/(K^*-b_i-C_i)\) is the unique optimal solution for (P\('\)). This shows

$$\begin{aligned} z_i^*=a_i{\overline{x}}_i(K^*)/(K^*-b_i-C_i) \ \text { and } \ p_i^*=K^*-a_i{\overline{x}}_i(K^*)/z_i^* -b_i=C_i. \end{aligned}$$

The profit of leader i becomes

$$\begin{aligned} \Pi _i(z_i^*,p_i^*)= & {} C_i\cdot {\overline{x}}_i(K^*)-\gamma _i\cdot \frac{a_i{\overline{x}}_i(K^*)}{K^*-b_i-C_i}=\overline{x}_i(K^*)\cdot \left( C_i-\frac{a_i\gamma _i}{K^*-b_i-C_i}\right) \\= & {} f_i^2(K^*). \end{aligned}$$

Since \({\overline{x}}_i(K^*)=x_i^*>0\) and the profit is nonnegative, \(C_i-a_i\gamma _i/(K^*-b_i-C_i) \ge 0\), thus \(K^* \ge a_i\gamma _i/C_i+b_i+C_i \) holds. Finally, \(z_i^*=a_ix_i^*/(K^*-b_i-C_i)<\sqrt{a_i/\gamma _i}x_i^* \Leftrightarrow K^*>\sqrt{a_i\gamma _i}+b_i+C_i\), which completes the proof since we showed that \(K^*\) is a feasible solution of problem \((\text {P}_i^2)\) (optimality follows from Lemma 3 and the optimality of \((z_i^*,p_i^*)\) for \((\text {P}_i)\)). \(\square \)

In the next lemma, we analyze the existence of optimal solutions for the problems \((\text {P}_i^1)\) and \((\text {P}_i^2)\), as well as properties of such solutions.

Lemma 5

  1. 1.

    If \((\text {P}_i^1)\) is feasible, it has a unique optimal solution.

  2. 2.

    Assume that \((\text {P}_i^2)\) is feasible.

  • If \(C_i\le \sqrt{a_i\gamma _i}\), then \((\text {P}_i^2)\) has a unique optimal solution.

  • If \(C_i> \sqrt{a_i\gamma _i}\), then \((\text {P}_i^2)\) has at most one optimal solution.

  • If \(K^*_2\) is optimal for \((\text {P}_i^2)\), then \(f_i^2(K^*_2)>0\).

  1. 3.

    If \(K^*_1\) is optimal for \((\text {P}_i^1)\) and \(K^*_2\) is optimal for \((\text {P}_i^2)\), then \(f_i^1(K^*_1)< f_i^2(K^*_2)\).

Proof

We start with statement 1. of the lemma, so assume that \((\text {P}_i^1)\) is feasible. Note that the feasible set \(I_1\) of \((\text {P}_i^1)\) either is of the form \(I_1=[2\sqrt{a_i\gamma _i}+b_i,\sqrt{a_i\gamma _i}+b_i+C_i]\), or \(I_1=[2\sqrt{a_i\gamma _i}+b_i, K_i^{\max })\), depending on whether \(\sqrt{a_i\gamma _i}+b_i+C_i< K_i^{\max }\) holds or not. Furthermore note that the objective function

$$\begin{aligned} f_i^1(K) = \left( 1-\sum _{j \in N^+: b_j+p_j<K}{\frac{(K-b_j-p_j)z_j}{a_j}}\right) \cdot \left( K-b_i-2\sqrt{a_i\gamma _i}\right) \end{aligned}$$

of \((\text {P}_i^1)\) is continuous (over \({\mathbb {R}}\)). From this, we can conclude that \((\text {P}_i^1)\) has at least one optimal solution: For \(I_1=[2\sqrt{a_i\gamma _i}+b_i,\sqrt{a_i\gamma _i}+b_i+C_i]\), this follows directly from the theorem of Weierstrass (\(f_i^1\) is continuous and \(I_1\) is nonempty and compact). For \(I_1=[2\sqrt{a_i\gamma _i}+b_i, K_i^{\max })\), the theorem of Weierstrass yields that \(f_i^1\) attains its maximum over the closure of \(I_1\), that is, over \([2\sqrt{a_i\gamma _i}+b_i, K_i^{\max }]\). But since \(f_i^1(K_i^{\max })=0\) \((=f_i^1(2\sqrt{a_i\gamma _i}+b_i))\), and any \(K\in (2\sqrt{a_i\gamma _i}+b_i, K_i^{\max })\) fulfills \(f_i^1(K)>0\), the maximum is not attained for \(K=K_i^{\max }\), and we conclude that \(f_i^1\) also attains its maximum over \(I_1\). Thus \((\text {P}_i^1)\) has an optimal solution for both cases. To complete the proof of statement 1. of the lemma, it remains to show that there is also at most one optimal solution. We prove this by showing the following monotonicity behaviour of \(f_i^1\) over \(I_1\): Either \(f_i^1\) is strictly increasing over \(I_1\), or strictly decreasing over \(I_1\), or strictly increasing up to a unique point, and strictly decreasing afterwards. In all three cases, we obviously get the desired statement, namely that \((\text {P}_i^1)\) has at most one optimal solution. To prove the described monotonicity behaviour, we distinguish between three cases according to the value of \(\min \{b_j+p_j: j \in N^+\}\). The first case is \(\sqrt{a_i\gamma _i}+b_i+C_i\le \min \{b_j+p_j: j \in N^+\}\), which implies \(N(K)=\{j \in N^+: b_j+p_j<K\}=\emptyset \) and \(\overline{x}_i(K)=1\) for all \(K\in I_1\). We conclude that \(f_i^1(K)=K-b_i-2\sqrt{a_i\gamma _i}\) is strictly increasing over \(I_1\) (in particular, \(f_i^1\) reaches its unique maximum over \(I_1\) at \(K=\sqrt{a_i\gamma _i}+b_i+C_i\)). Next, consider the case that \(\min \{b_j+p_j: j \in N^+\}<2\sqrt{a_i\gamma _i}+b_i\). This implies that \(N(K)\ne \emptyset \) for all \(K\in I_1\). Note that \(f_i^1\) is twice differentiable on any open interval where N(K) is constant, and the first and second derivative of \(f_i^1\) then are

$$\begin{aligned} (f_i^1)'(K)&= 1-\sum _{j \in N(K)}{\frac{(2K-b_j-p_j-b_i-2\sqrt{a_i\gamma _i})z_j}{a_j}} \quad \text { and} \\ (f_i^1)''(K)&=-2\sum _{j \in N(K)}{\frac{z_j}{a_j}}. \end{aligned}$$

Since \(N(K)\ne \emptyset \) for all \(K\in I_1\), we conclude that for all \(K\in I_1\) where \((f_i^1)''(K)\) exists, \((f_i^1)''(K)<0\) holds. If N(K) is constant on the complete interior of \(I_1\) (that is, on \((2\sqrt{a_i\gamma _i}+b_i,\sqrt{a_i\gamma _i}+b_i+C_i)\) or \((2\sqrt{a_i\gamma _i}+b_i, K_i^{\max })\), depending on the two possible cases for \(I_1\)), the desired monotonicity behaviour of \(f_i^1\) over \(I_1\) follows. Otherwise, let \(\alpha _1<\alpha _2<\cdots < \alpha _k\) denote the different values of \(b_j+p_j, j \in N^+\) which lie in the interior of \(I_1\). Define \(\alpha _0:=2\sqrt{a_i\gamma _i}+b_i\) and \(\alpha _{k+1}:=\sup {I_1}\) (that is, \(\alpha _{k+1}=\sqrt{a_i\gamma _i}+b_i+C_i\) or \(\alpha _{k+1}=K_i^{\max }\)). Then, N(K) is constant on the intervals \((\alpha _{\ell -1},\alpha _{\ell }]\) for \(\ell \in \{1,\ldots ,k+1\}\). For each \(\ell \in \{1,\ldots ,k\}\), the set N(K) increases immediately after \(\alpha _{\ell }\), that is, \(N(\alpha _{\ell })\subsetneq N(\alpha _{\ell }+\varepsilon )\) holds for any \(\varepsilon >0\). In particular, \(N(\alpha _{\ell }+\varepsilon )=N(\alpha _{\ell })\cup \{j \in N^+: b_j+p_j=\alpha _{\ell }\}\) holds for all \(0<\varepsilon \le \alpha _{\ell +1}-\alpha _{\ell }\). We now show that for any \(\ell \in \{1,\ldots ,k\}\), the slope of \(f_i^1\) decreases at \(\alpha _{\ell }\), whereby we mean that \((f_i^1)_{+}'(\alpha _{\ell })<(f_i^1)_{-}'(\alpha _{\ell })\) holds, with \((f_i^1)_{+}'(\alpha _{\ell })\) and \((f_i^1)_{-}'(\alpha _{\ell })\) denoting the right and left derivative of \(f_i^1\) at \(\alpha _{\ell }\), respectively. This implies the desired monotonicity behaviour of \(f_i^1\) over \(I_1\). Analyzing the left and right derivative of \(f_i^1\) at \(\alpha _{\ell }\) yields

$$\begin{aligned} (f_i^1)_{-}'(\alpha _{\ell })&=1-\sum _{j \in N(\alpha _{\ell })}{\frac{(2\alpha _{\ell }-b_j-p_j-b_i-2\sqrt{a_i\gamma _i})z_j}{a_j}} \quad \text { and } \\ (f_i^1)_{+}'(\alpha _{\ell })&=1-\sum _{j \in N(\alpha _{\ell })\cup \{j\in N^+: b_j+p_j=\alpha _{\ell }\}}{\frac{(2\alpha _{\ell }-b_j-p_j-b_i-2\sqrt{a_i\gamma _i})z_j}{a_j}}\\&=(f_i^1)_{-}'(\alpha _{\ell }) -\sum _{j\in N^+: b_j+p_j=\alpha _{\ell }}{\frac{(2\alpha _{\ell }-\alpha _{\ell }-b_i-2\sqrt{a_i\gamma _i})z_j}{a_j}} \\&=(f_i^1)_{-}'(\alpha _{\ell })-(\alpha _{\ell }-b_i-2\sqrt{a_i\gamma _i})\cdot \sum _{j \in N^+: b_j+p_j=\alpha _{\ell }}{\frac{z_j}{a_j}}. \end{aligned}$$

Since \(\alpha _{\ell }\) lies in the interior of \(I_1\), we get \(\alpha _{\ell }> 2\sqrt{a_i\gamma _i}+b_i\) and therefore the desired inequality \((f_i^1)_{+}'(\alpha _{\ell })<(f_i^1)_{-}'(\alpha _{\ell })\), completing the proof for the case \(\min \{b_j+p_j: j \in N^+\}<2\sqrt{a_i\gamma _i}+b_i\). The remaining case is \(2\sqrt{a_i\gamma _i}+b_i \le \min \{b_j+p_j: j \in N^+\} <\sqrt{a_i\gamma _i}+b_i+C_i\), which implies that \(N(K)=\emptyset \) holds in \(I_1\) for \(K \le \min \{b_j+p_j: j \in N^+\}\), and \(N(K)\ne \emptyset \) holds in \(I_1\) for \(K >\min \{b_j+p_j: j \in N^+\}\). We can now obviously combine the arguments of the other two cases to obtain the desired monotonicity behaviour of \(f_i^1\) also for this case. This completes the proof of statement  1. of the lemma.

Now we turn to statement 2., thus we assume that \((\text {P}_i^2)\) is feasible. Let \(I_2\) denote the feasible set of \((\text {P}_i^2)\). Then, either \(I_2=[a_i\gamma _i/C_i+b_i+C_i, K_i^{\max })\) or \(I_2=(\sqrt{a_i\gamma _i}+b_i+C_i,K_i^{\max })\) holds, depending on whether \(C_i< \sqrt{a_i\gamma _i}\) holds or not. But in both cases, \(I_2\) is an interval with positive length, so that there exists \(K\in I_2\) with \(K>a_i\gamma _i/C_i+b_i+C_i\), which implies \(f_i^2(K)>0\). Therefore, if \((\text {P}_i^2)\) has an optimal solution \(K^*_2\), we get \(f_i^2(K^*_2)>0\). Furthermore note that the objective function

$$\begin{aligned} f_i^2(K) =\left( 1-\sum _{j \in N(K)}{\frac{(K-b_j-p_j)z_j}{a_j}}\right) \cdot \left( C_i-\frac{a_i\gamma _i}{K-b_i-C_i}\right) \end{aligned}$$

of \((\text {P}_i^2)\) is continuous over \((b_i+C_i,{\mathbb {R}})\). Using this, we can show that \((\text {P}_i^2)\) has at least one optimal solution, if \(C_i\le \sqrt{a_i\gamma _i}\) holds: Due to the theorem of Weierstrass, \(f_i^2\) attains its maximum over \([a_i\gamma _i/C_i+b_i+C_i,K_i^{\max }]\), the closure of \(I_2\). But since \(f_i^2(a_i\gamma _i/C_i+b_i+C_i)=0=f_i^2(K_i^{\max })\), and \(f_i^2(K)>0\) for any \(K \in (a_i\gamma _i/C_i+b_i+C_i, K_i^{\max })\), the maximum is not attained at \(K=a_i\gamma _i/C_i+b_i+C_i\) or \(K=K_i^{\max }\), which shows that \(f_i^2\) also attains its maximum over \(I_2\). Thus, if \(C_i\le \sqrt{a_i\gamma _i}\) holds, \((\text {P}_i^2)\) has at least one optimal solution. To complete the proof of statement 2, it remains to show that \((\text {P}_i^2)\) has at most one optimal solution (in the general case). As in the proof of statement 1 of the lemma, we prove this by showing that \(f_i^2\) exhibits a certain monotonicity behaviour over \(I_2\), namely: Either \(f_i^2\) is strictly decreasing over \(I_2\), or strictly increasing up to a unique point, and strictly decreasing afterwards. Note that \(f_i^2\) cannot be strictly increasing over \(I_2\), due to the continuity of \(f_i^2\) and the fact that \(f_i^2(K_i^{\max })=0<f_i^2(K)\) holds for any K in the interior of \(I_2\). The remaining proof is very similar to the proof of statement 1. First, \(f_i^2\) is twice differentiable on any open interval where N(K) is constant. First and second derivative of \(f_i^2\) then are

$$\begin{aligned} (f_i^2)'(K)&=-\sum _{j \in N(K)}{\frac{z_j}{a_j}}\cdot \left( C_i-\frac{a_i\gamma _i}{K-b_i-C_i}\right) + {\overline{x}}_i(K)\cdot \frac{a_i\gamma _i}{(K-b_i-C_i)^2} \quad \text {and}\\ (f_i^2)''(K)&= -\frac{2a_i\gamma _i}{(K-b_i-C_i)^2}\cdot \left( \sum _{j \in N(K)}{\frac{z_j}{a_j}}+\frac{{\overline{x}}_i(K)}{K-b_i-C_i}\right) . \end{aligned}$$

Since \({\overline{x}}_i(K)>0\) for all \(K\in I_2\), we conclude that for all \(K\in I_2\) where \((f_i^2)''(K)\) exists, \((f_i^2)''(K)<0\) holds. If N(K) is constant on the complete interior of \(I_2\), the desired monotonicity behaviour of \(f_i^2\) over \(I_2\) follows, otherwise let \(\beta _1<\beta _2<\cdots < \beta _k\) denote the different values of \(b_j+p_j, j \in N^+\) which lie in the interior of \(I_2\). We show that the slope of \(f_i^2\) decreases at \(\beta _{\ell }\), i.e. \( (f_i^2)_+'(\beta _{\ell })< (f_i^2)_-'(\beta _{\ell }) \) holds, which implies the desired monotonicity behaviour of \(f_i^2\) over \(I_2\). Analyzing the left and right derivative yields

$$\begin{aligned} (f_i^2)_-'(\beta _{\ell })&=-\sum _{j \in N(\beta _{\ell })}{\frac{z_j}{a_j}}\cdot \left( C_i-\frac{a_i\gamma _i}{\beta _{\ell }-b_i-C_i}\right) + {\overline{x}}_i(\beta _{\ell }) \cdot \frac{a_i\gamma _i}{(\beta _{\ell }-b_i-C_i)^2} \\&\quad \text { and} \\ (f_i^2)_+'(\beta _{\ell })&=-\sum _{j \in N(\beta _{\ell })\cup \{j\in N^+: b_j+p_j=\beta _{\ell }\}}{\frac{z_j}{a_j}}\cdot \left( C_i-\frac{a_i\gamma _i}{\beta _{\ell }-b_i-C_i}\right) \\&\quad + {\overline{x}}_i(\beta _{\ell }) \cdot \frac{a_i\gamma _i}{(\beta _{\ell }-b_i-C_i)^2} \\&=(f_i^2)_-'(\beta _{\ell }) -\sum _{j\in N^+: b_j+p_j=\beta _{\ell }}{\frac{z_j}{a_j}}\cdot \left( C_i-\frac{a_i\gamma _i}{\beta _{\ell }-b_i-C_i}\right) \end{aligned}$$

Since \(\beta _{\ell }\) lies in the interior of \(I_2\), we get \(C_i-a_i\gamma _i/(\beta _{\ell }-b_i-C_i) > 0\), and thus the desired inequality \((f_i^2)_+'(\beta _{\ell })< (f_i^2)_-'(\beta _{\ell })\), completing the proof of statement 2. of the lemma.

Finally we show statement 3. Assume that \(K^*_1\) and \(K^*_2\) are the optimal solutions of \((\text {P}_i^1)\) and \((\text {P}_i^2)\). Since \((\text {P}_i^1)\) and \((\text {P}_i^2)\) have to be feasible, \(\sqrt{a_i\gamma _i}\le C_i\) and \(\sqrt{a_i\gamma _i}+b_i+C_i<K_i^{\max }\) holds. This implies that the feasible set of \((\text {P}_i^1)\) is \(I_1=[2\sqrt{a_i\gamma _i}+b_i,\sqrt{a_i\gamma _i}+b_i+C_i]\) and the feasible set of \((\text {P}_i^2)\) is \(I_2=(\sqrt{a_i\gamma _i}+b_i+C_i,K_i^{\max })\). Let \(\bar{K}:=\sqrt{a_i\gamma _i}+b_i+C_i\). Then, \(f_i^1({\bar{K}})=f_i^2({\bar{K}})\) holds. If additionally the slope of \(f_i^1\) in \({\bar{K}}\) is greater than or equal to the slope of \(f_i^2\) in \({\bar{K}}\), whereby we mean that \( (f_i^1)_-'({\bar{K}})\ge (f_i^2)_+'({\bar{K}}) \) holds, we get \(f_i^1(K^*_1)=f_i^1({\bar{K}})<f_i^2(K^*_2)\) from our analysis of \(f_i^1\) and \(f_i^2\) in the proofs of the statements 1. and 2. (note that \(f_i^2\) is strictly increasing on \((\sqrt{a_i\gamma _i}+b_i+C_i,K_2^*]\)). The remaining inequality for the slopes follows from

$$\begin{aligned} (f_i^1)_-'({\bar{K}})&= 1-\sum _{j \in N({\bar{K}})}{\frac{(2\bar{K}-b_j-p_j-b_i-2\sqrt{a_i\gamma _i})z_j}{a_j}} \\&= 1-\sum _{j \in N(\bar{K})}{\frac{(2C_i+b_i-b_j-p_j)z_j}{a_j}} \qquad \text {and} \end{aligned}$$
$$\begin{aligned} (f_i^2)_+'({\bar{K}})&= -\sum _{j \in N({\bar{K}})\cup \{j\in N^+: b_j+p_j={\bar{K}}\}}{\frac{z_j}{a_j}}\left( C_i-\frac{a_i\gamma _i}{{\bar{K}}-b_i-C_i}\right) \\&\quad + {\overline{x}}_i({\bar{K}}) \frac{a_i\gamma _i}{({\bar{K}}-b_i-C_i)^2} \\&= -\sum _{j \in N({\bar{K}})\cup \{j\in N^+: b_j+p_j={\bar{K}}\}}{\frac{z_j}{a_j}}\cdot (C_i-\sqrt{a_i\gamma _i})\\&\quad + 1-\sum _{j \in N({\bar{K}})}{\frac{({\bar{K}}-b_j-p_j)z_j}{a_j}} \\&=1-\sum _{j \in N({\bar{K}})}{\frac{({\bar{K}}-b_j-p_j+C_i-\sqrt{a_i\gamma _i})z_j}{a_j}}\\&\quad -\sum _{j\in N^+: b_j+p_j={\bar{K}}}{\frac{z_j(C_i-\sqrt{a_i\gamma _i})}{a_j}} \\&=1-\sum _{j \in N({\bar{K}})}{\frac{(2C_i+b_i-b_j-p_j)z_j}{a_j}}-\sum _{j\in N^+: b_j+p_j={\bar{K}}}{\frac{z_j(C_i-\sqrt{a_i\gamma _i}) }{a_j}}\\&= (f_i^1)_-'({\bar{K}}) -\sum _{j\in N^+: b_j+p_j=\bar{K}}{\frac{z_j(C_i-\sqrt{a_i\gamma _i})}{a_j}} \le (f_i^1)_-'({\bar{K}}), \end{aligned}$$

where the inequality follows from \(\sqrt{a_i\gamma _i}\le C_i\). \(\square \)

3.3 The characterization

The following theorem provides a complete characterization of the best response correspondence. We will make use of this characterization several times during the rest of the paper.

Theorem 2

For a leader \(i\in N\) and fixed strategies \(s_{-i}=(z_{-i},p_{-i}) \in S_{-i}\) of the other leaders, the set \(\text {BR}_i=\text {BR}_i(s_{-i})\) of best responses of leader i to \(s_{-i}\) is given as indicated in Table 1, where the first column contains \(\text {BR}_i\) and the second column contains the conditions on \(s_{-i}\) under which \(\text {BR}_i\) has the stated form. For \(j=1,2\), \(K^*_j\) denotes the unique optimal solution of problem \((\text {P}_i^j)\), if this problem has an optimal solution.

Furthermore, if \(\text {BR}_i(s_{-i})\) consists of a unique best response \(s_i=(z_i,p_i)\) of leader i to \(s_{-i}\), we get \(z_i>0\) and \(\Pi _i( s_i,s_{-i} )>0\).

Table 1 Characterization of \(\text {BR}_i\)

Proof

Note that if \(\text {BR}_i(s_{-i})\) consists of a unique best response \(s_i=(z_i,p_i)\) of leader i to \(s_{-i}\), then \(z_i>0\) and \(\Pi _i( s_i,s_{-i} )>0\) hold: Otherwise, any strategy \((z_i',p_i') \in \{0\}\times [0,C_i]\) is a best response, too, contradicting the uniqueness assumption.

Now turn to the proof of the characterization. We show that the case distinction covers all possible cases, and that the given representation for \(\text {BR}_i\) is correct for each case. If \(z_{-i}=0\), Lemma 1 shows \(\text {BR}_i=\emptyset \). For the rest of the proof, assume \(z_{-i}\ne 0\). Then, leader i has at least one best response to \(s_{-i}\), since \(\text {BR}_i\) can be described as the set of optimal solution of the problem \((\text {P}_i)\), and this problem has an optimal solution (as shown in the beginning of Sect. 3.2). If \((\text {P}_i^1)\) and \((\text {P}_i^2)\) are both infeasible, Lemma 4 implies that each best response \((z_i,p_i)\) fulfills \(z_i=0\). Therefore, \(\text {BR}_i=\{0\}\times [0,C_i]\). For the remaining proof, assume that at least one of \((\text {P}_i^1)\) and \((\text {P}_i^2)\) is feasible.

First consider the case that \((\text {P}_i^2)\) has an optimal solution. It follows from 2. of Lemma 5 that the solution is unique, and, if \(K^*_2\) denotes this solution, that \(f_i^2(K^*_2)>0\). Let \((z_i,p_i)\) be an arbitrary best response of player i to \(s_{-i}\). We need to show that \((z_i,p_i)=(a_i{\overline{x}}_i(K_2^*)/(K_2^*-b_i-C_i), C_i)\) holds. First, statement 2. of Lemma 3 shows that \(\Pi _i(z_i,p_i)\ge f_i^2(K^*_2)>0\). Thus \(z_i>0\) holds, since \(z_i=0\) yields a profit of 0. Note that either (\((\text {P}_i^1)\)) is infeasible, or it has a unique optimal solution \(K^*_1\) with \(f_i^1(K^*_1)<f_i^2(K^*_2)\le \Pi _i(z_i,p_i)\) (see 1. and 3. of Lemma 5). In both cases, Lemma 4 yields \((z_i,p_i)=(a_i{\overline{x}}_i(K_2^*)/(K_2^*-b_i-C_i), C_i)\).

Now assume that \((\text {P}_i^2)\) does not have an optimal solution (either \((\text {P}_i^2)\) is infeasible, or it is feasible, but the maximum is not attained). We first show that \((\text {P}_i^1)\) is feasible. If \((\text {P}_i^2)\) is infeasible, \((\text {P}_i^1)\) is feasible since we assumed that at least one of the two problems is feasible. Otherwise \((\text {P}_i^2)\) is feasible, but does not have an optimal solution. Then, \(C_i>\sqrt{a_i\gamma _i}\) follows from 2. of Lemma 5, and \(\sqrt{a_i\gamma _i}+b_i+C_i<K_i^{\max }\) follows since \((\text {P}_i^2)\) is feasible. Together, \(2\sqrt{a_i\gamma _i}+b_i<\sqrt{a_i\gamma _i}+b_i+C_i<K_i^{\max }\) holds, showing that \((\text {P}_i^1)\) is feasible. By 1. of Lemma 5 we then get that \((\text {P}_i^1)\) has a unique optimal solution \(K^*_1\). Furthermore, each best response \((z_i,p_i)\) with \(z_i>0\) fulfills \((z_i,p_i)=(\sqrt{a_i/\gamma _i}\cdot {\overline{x}}_i(K_1^*), K_1^*-\sqrt{a_i\gamma _i}-b_i)\) (see Lemma 4). To complete the proof, we need to show that there is no best response \((z_i,p_i)\) with \(z_i=0\). This follows from Lemma 3 if \(f_i^1(K^*_1)>0\). Thus it remains to show that \(f_i^1(K^*_1)>0\) holds. Assume, by contradiction, that \(f_i^1(K^*_1)=0\). This implies that \(K^*_1=2\sqrt{a_i\gamma _i}+b_i\) is the only feasible solution for \((\text {P}_i^1)\), which in turn yields \(\sqrt{a_i\gamma _i}=C_i\) and \(\sqrt{a_i\gamma _i}+b_i+C_i<K_i^{\max }\). But this implies that \((\text {P}_i^2)\) is feasible and has an optimal solution (by 2. of Lemma 5), contradicting our assumption that \((\text {P}_i^2)\) does not have an optimal solution. \(\square \)

3.4 Discussion

We now briefly discuss consequences of the characterization of the best reponses with respect to applying Kakutani’s fixed point theorem (see [22]). Kakutani’s theorem in particular requires that for each leader i and each vector \(s_{-i}=(z_{-i},p_{-i})\) of strategies of the other leaders, the set \(\text {BR}_i(s_{-i})\) of best responses is nonempty and convex. But as we have seen in Lemma 1, the set \(\text {BR}_i(s_{-i})\) can be empty, namely if \(z_{-i}=0\). On the other hand, a profile with \(z_{-i}=0\) for some leader i will of course never be a PNE.

A first natural approach to overcome the problem of empty best responses is the following. Given a strategy profile \(s=(z,p)\) such that \(z_{-i}=0\) for some leader i, redefine, for each such leader i, the set \(\text {BR}_i(s_{-i})\) by some suitable nonempty convex set. “Suitable” here means that the correspondence \(\text {BR}_i\) has a closed graph, and at the same time, s must not be a fixed point of the global best response correspondence \(\text {BR}\) (where \(\text {BR}(s):=\{s'\in S: s_i'\in \text {BR}_i(s_{-i}) \text { for each } i\in N\}\)). But unfortunately, these two goals are not compatible: For the strategy profile \(s=(z,p)\) with \((z_i,p_i)=(0,C_i)\) for all leaders i, the closed graph property requires \((0,C_i)\in \text {BR}_i(s_{-i})\) for all i, which implies that s is a fixed point of \(\text {BR}\).

Another intuitive idea is to consider a game in which each leader has an initial capacity of some \(\varepsilon >0\). If this game has a PNE for each \(\varepsilon \), the limit for \(\varepsilon \) going to zero should be a PNE for our original Stackelberg pricing game. For the game with at least \(\varepsilon \) capacity, one can also characterize the best response correspondences (now by optimal solutions of three optimization problems), but the main problem is that an analogue of Lemma 5 may not hold anymore. As a consequence, it is not clear if the best responses are always convex, and we again do not know how to apply Kakutani’s theorem. Instead, we show existence of PNE by using a result of McLennan et al. [29], see Sect. 4.Footnote 4

4 Existence of equilibria

In this section, we show that each Stackelberg pricing game has a PNE. A frequently used tool to show existence of PNE is Kakutani’s fixed point theorem. But, as discussed in Sect. 3.4, we cannot directly apply this result to show existence of PNE. Furthermore, the existence theorem of [31] can also not be used, since a Stackelberg pricing game is not quasiconcave in general (it is not difficult to construct instances which are not quasiconcave). Instead, we turn to another existence result due to McLennan et al. [29]. They introduced a concept called C-security and they showed that if the game is C-secure at each strategy profile which is not a PNE, then a PNE exists. Informally, the game is C-secure at a strategy profile s if there is a vector \(\alpha \in {\mathbb {R}}^n\) satisfying the following two properties: First, each leader i has some securing strategy for \(\alpha _i\) which is robust to small deviations of the other leaders, i.e. leader i always achieves a profit of at least \(\alpha _i\) by playing this strategy even if the other leaders slightly deviate from their strategies in \(s_{-i}\). The second property requires for each slightly perturbed strategy profile \(s'\) resulting from s, that there is at least one leader i such that her perturbed strategy \(s_i'\) can (in some sense) be strictly separated from all strategies achieving a profit of \(\alpha _i\), so in particular from all her securing strategies. One can think of leader i being “not happy” with her perturbed strategy \(s_i'\) since she could achieve a higher profit. This already indicates the connection between a strategy profile which is not a PNE, and C-security. We will see that for certain strategy profiles, \(\alpha _i\) can be chosen as the profit that leader i gets by playing a best response to \(s_{-i}\).Footnote 5 Then, leader i’s securing strategies for \(\alpha _i\) are related to her set of best responses, and we need to “strictly separate” these best responses from \(s_i\). At this point, our characterization of best responses in Theorem 2 becomes useful.

We now formally describe McLennan et al.’s result in our context. First of all, note that they consider games with compact convex strategy sets and bounded profit functions. In a Stackelberg pricing game, the strategy set \(S_i=\{(z_i,p_i): 0\le z_i, 0\le p_i\le C_i\}\) of leader i is not compact a priori. But since \(z_i\) will never be larger than \(C_i/\gamma _i\) in any best response, and thus in any PNE (see the discussion at the beginning of Sect. 3.2), we can redefine \(S_i:=\{(z_i,p_i): 0 \le z_i \le C_i/\gamma _i, 0 \le p_i \le C_i\}\) without changing the set of PNE of the game, for any leader i. Furthermore, this also does not change the best responses, so Theorem 2 continues to hold. Using the redefined strategies, for any leader i and any strategy profile s, the profit of leader i is bounded by \(-C_i\le \Pi _i(s)\le C_i\). For a strategy profile \(s\in S\), leader \(i \in N\) and \(\alpha _i \in {\mathbb {R}}\) let

$$\begin{aligned} B_i(s,\alpha _i):=\{s_i'\in S_i: \Pi _i( s_i',s_{-i} )\ge \alpha _i\} \ \text { and } \ C_i(s,\alpha _i):=\text {conv}B_i(s,\alpha _i), \end{aligned}$$

where \(\text {conv}B_i(s,\alpha _i)\) denotes the convex hull of \(B_i(s,\alpha _i)\).

Definition 1

A leader i can secure a profit \(\alpha _i \in {\mathbb {R}}\) on \(S' \subseteq S\), if there is some \(s_i \in S_i\) such that \(s_i \in B_i(s',\alpha _i)\) for all \(s' \in S'\). We say that leader i can secure \(\alpha _i\) at \(s \in S\), if she can secure \(\alpha _i\) on \(U\cap S\) for some open set U with \(s \in U\).

Definition 2

The game is C-secure on \(S' \subseteq S\), if there is an \(\alpha \in {\mathbb {R}}^n\) such that the following conditions hold:

  1. (i)

    Every leader i can secure \(\alpha _i\) on \(S'\).

  2. (ii)

    For any \(s' \in S'\), there exists some leader i with \(s_i' \notin C_i(s',\alpha _i)\).

The game is C-secure at  \(s \in S\), if it is C-secure on \(U\cap S\) for some open set U with \(s \in U\).

Theorem 3

(Proposition 2.7 in [29]) If the game is C-secure at each \(s\in S\) that is not a PNE, then the game has a PNE.

We now turn to Stackelberg pricing games and show the existence of a PNE by using Theorem 3, i.e., we show that if a given strategy profile \(s=(z,p)\) is not a PNE, then the game is C-secure at s. To this end, we distinguish between the two cases that there are at least two leaders i with \(z_i>0\) (Lemma 6), or not (Lemma 7). Both lemmata together then imply the desired existence result. Note that the mentioned case distinction is equivalent to the case distinction that each leader i has a best response for \(s_{-i}\), or there is at least one leader i with \(\text {BR}_i(s_{-i})=\emptyset \) (see Theorem 2).

We start with the case that all best responses exist. The proof of the following lemma follows an argument in [29, p. 1647f] where McLennan et al. show that Theorem 3 implies the existence result of [30].

Lemma 6

Let \(s=(z,p)\in S\) be a strategy profile which is not a PNE. Assume that there are at least two leaders \(i\in N\) such that \(z_i>0\) holds. Then the game is C-secure at s.

Proof

We first introduce some notation used in this proof. Let \(S'\subseteq S\) be a subset of the strategy profiles and \(i\in N\). By \(S'_i\subseteq S_i\), we denote the projection of \(S'\) into \(S_i\), the set of leader i’s strategies, and \(S'_{-i}\subseteq S_{-i}\) denotes the projection of \(S'\) into \(S_{-i}=\times _{j\in N\setminus \{i\}}{S_j}\), the set of strategies of the other leaders. Note that since z has at least two positive entries \(z_i\), all strategy profiles \(s'=(z',p')\) in a sufficiently small open neighbourhood of s also have at least two entries \(z_i'>0\). In the following, whenever we speak of an open set U containing s, we implicitly require U small enough to fulfill this property. Furthermore, since it is clear that we are only interested in the elements of U which are strategy profiles, we simply write U instead of \(U\cap S\). Consequently, \(s'\in U\) denotes a strategy profile contained in U. Now we turn to the actual proof.

Since we assumed that at least two leaders have positive capacity at \(s=(z,p)\), we get that \(z_{-i}\ne 0\) holds for each leader i. Thus, by Theorem 2, each leader i has a best response to \(s_{-i}\), and the set of best responses either is a singleton, or consists of all strategies \((0,p_i)\) for \(0\le p_i\le C_i\). Since s is not a PNE, there is at least one leader j such that \(s_j=(z_j,p_j)\) is not a best response, i.e. either \(s_j\ne s_j^*\) for the unique best response \(s_j^*\), or \(z_j>0\), and all best responses \(s_j^*=(z_j^*,p_j^*)\) fulfill \(z_j^*=0\). In both cases it is clear that there is a hyperplane H which strictly separates \(s_j\) from the set of best responses to \(s_{-j}\).

Now turn to the properties in Definition 2. For each leader i, let \(s_i^*\) be a best response of leader i to \(s_{-i}\) and \(\beta _i:=\Pi _i( s_i^*,s_{-i} )\). We know that \(\Pi _i\) is continuous at \((s_i^*,s_{-i})\) for each leader i (Theorem 1). Therefore, for each \(\varepsilon >0\), there is an open set \(U(\varepsilon )\supset \{s\}\) with \(\Pi _i( s_i^*,s_{-i}' )\ge \beta _i-\varepsilon \) for each \(s'\in U(\varepsilon )\) and each leader i. That is, each leader i can secure \(\beta _i-\varepsilon \) on \(U(\varepsilon )\). Now turn to the second property of Definition 2 and consider leader j. We show that there is an \(\varepsilon >0\) and an open set \(U\subseteq U(\varepsilon )\) containing s, such that for each \(s'\in U\), the hyperplane H (which strictly separates \(s_j\) and \(B_j(s,\beta _j)\)) also strictly separates \(s_j'\) and \(B_j(s',\beta _j-\varepsilon )\), thus \(s_j' \notin C_j(s',\beta _j-\varepsilon )\) (see Fig. 1 for illustration). Since each leader i can secure \(\beta _i-\varepsilon \) on \(U\subseteq U(\varepsilon )\), both properties of Definition 2 are fulfilled, completing the proof.

Fig. 1
figure 1

Illustration of the proof construction for the case \(B_j(s,\beta _j)=\{s_j^*\}\). Note that it is not necessary that \(s_j^*\in B_j(s',\beta _j-\varepsilon )\)

To this end, choose an open set V containing \(B_j(s,\beta _j)\) such that H strictly separates \(s_j\) and V. Since \(S_j\setminus V\) is a compact set and \(\Pi _j\) is continuous at \((\widetilde{s}_j,s_{-j})\) for all \(\widetilde{s}_j\in S_j\setminus V\) (by Theorem 1), we get that \(f(s_{-j}):=\max \{\Pi _j( \widetilde{s}_j,s_{-j} ): \widetilde{s}_j \in S_j \setminus V\}\) exists. Furthermore, since \(B_j(s,\beta _j)\subset V\), we get \(f(s_{-j})<\beta _j\). Let \(0<\varepsilon <\beta _j-f(s_{-j})\), thus \(f(s_{-j})<\beta _j-\varepsilon \). Note that if we consider, for an open neighbourhood \({\bar{U}}\) of s and for fixed \(s_{-j}'\in {\bar{U}}_{-j}\), the problem of maximizing \(\Pi _j( \widetilde{s}_j,s_{-j}' )\) subject to \(\widetilde{s}_j \in S_j\setminus V\), Berge’s theorem of the maximum ([5]) yields that \(f(s_{-j}'):=\max \{\Pi _j( \widetilde{s}_j,s_{-j}' ): \widetilde{s}_j \in S_j \setminus V\}\) is a continuous function. Using the continuity of f, there is an open set \(U\subseteq U(\varepsilon )\) containing s such that \(f(s_{-j}')<\beta _j-\varepsilon \) for all \(s_{-j}' \in U_{-j}\). Additionally, let U be small enough such that H strictly separates \(U_j\) and V. Now we have the desired properties: For each \(s' \in U\) and for each \(\widetilde{s}_j\in S_j\setminus V\), we get \(\Pi _j( \widetilde{s}_j,s_{-j}' )<\beta _j-\varepsilon \), thus \(B_j(s',\beta _j-\varepsilon )\subset V\). Since \(s_j'\in U_j\) and H strictly separates \(U_j\) and V, we get that H strictly separates \(s_j'\) and \(B_j(s',\beta _j-\varepsilon )\), as desired. \(\square \)

It remains to analyze the strategy profiles \(s=(z,p)\) with at most one positive \(z_i\). Note that these profiles cannot be PNE.

Lemma 7

Let \(s=(z,p)\in S\) be a strategy profile such that \(z_i>0\) holds for at most one leader i. Then the game is C-secure at s.

Proof

We distinguish between the two cases that there is a leader with positive capacity, or all capacities are zero.

In the former case, assume that \(z_i>0\) for leader i, and \(z_j=0\) for all \(j\ne i\) hold. Choose \(\alpha _i \in (C_i-\gamma _i z_i, C_i)\) and \(0<\varepsilon <\min \{z_i,(\alpha _i+\gamma _iz_i-C_i)/\gamma _i\}\). Then, there is an open set U containing s such that leader i can secure \(\alpha _i\) on \(U\cap S=:S'\) (note that by choosing U sufficiently small, leader i can secure each profit \(<C_i\) on \(S'\)) and \(|z_i'-z_i|<\varepsilon \) holds for each \(s'=(z',p') \in S'\). For \(j\ne i\), set \(\alpha _j:=0\). It is clear that each leader \(j\ne i\) can secure \(\alpha _j=0\) on \(S'\) (by any strategy with zero capacity). In this way, property (i) of C-security is fulfilled. For property (ii), let \(s'=(z',p') \in S'\). We show that \(s_i'\notin C_i(s',\alpha _i)\) holds. To this end, note that any strategy \(s_i^*=(z_i^*,p_i^*)\in B_i(s',\alpha _i)\), i.e., with \(\Pi _i( s_i^*,s_{-i}' )\ge \alpha _i\), fulfills \(z_i^*\le z_i-\varepsilon \), since, for \(z_i^*> z_i-\varepsilon >0\), we get \( \Pi _i( s_i^*,s_{-i}' )=x_i( s_i^*,s_{-i}' )p_i^*-\gamma _iz_i^* \le C_i-\gamma _iz_i^*<C_i-\gamma _i(z_i-\varepsilon )<\alpha _i, \) where the last inequality follows from the choice of \(\varepsilon \). Clearly, any strategy in \(C_i(s',\alpha _i)\), i.e., any convex combination of strategies in \(B_i(s',\alpha _i)\), then also has this property. Since \(z_i'>z_i-\varepsilon \), we get \((z_i',p_i')\notin C_i(s',\alpha _i)\), as desired. Thus we showed that the game is C-secure at s if one leader has positive capacity.

Now turn to the case that \(z_i=0\) for all \(i\in N\). We distinguish between two further subcases, namely that there is a leader i with \(p_i<C_i\), or all prices are at their upper bounds. In the former case, let \(i\in N\) with \(p_i<C_i\), and choose \(\alpha _i\) with \(p_i<\alpha _i<C_i\). There is an open set U containing s such that leader i can secure \(\alpha _i\) on \(U\cap S=:S'\) and \(p_i'<\alpha _i\) holds for each \(s'=(z',p')\in S'\). By setting \(\alpha _j:=0\) for all \(j\ne i\), property (i) of C-security is fulfilled. For property (ii), let \(s'=(z',p') \in S'\). We show that \(s_i'\notin C_i(s',\alpha _i)\) holds. Our assumptions about \(S'\) yield \(p_i'<\alpha _i\). On the other hand, any strategy \(s_i^*=(z_i^*,p_i^*)\) with \(\Pi _i( s_i^*,s_{-i}' )\ge \alpha _i\) obviously fulfills \(p_i^*> \alpha _i\). In particular, this holds for any strategy in \(C_i(s',\alpha _i)\), thus showing that \(s_i' \notin C_i(s',\alpha _i)\). We conclude that the game is C-secure at s for the case that all \(z_i\) are zero and there is a leader i with \(p_i<C_i\).

It remains to consider the case that \((z_i,p_i)=(0,C_i)\) holds for all leaders i. For each leader i, choose \(\alpha _i\) with \((1-\frac{a_i}{2(a_i+C_i)})C_i<\alpha _i<C_i\). Note that this implies

$$\begin{aligned} 1/2<1/2+(C_i-\alpha _i)/a_i<\alpha _i/C_i. \end{aligned}$$
(1)

There is an open set U containing s such that each leader i can secure \(\alpha _i\) on \(U\cap S=:S'\) and \(z_i'<1\) holds for each \(s'=(z',p')\in S'\). Thus, property (i) of C-security is fulfilled. For property (ii), let \(s'=(z',p')\in S'\). In the following, \(s_i^*=(z_i^*,p_i^*)\) denotes a strategy of leader i with \(\Pi _i( s_i^*,s_{-i}' )\ge \alpha _i>0\). We say that \(s_i^*\) achieves a profit of at least \(\alpha _i\). Obviously \(z_i^*>0\) and \(p_i^*>\alpha _i\). Furthermore, \(\Pi _i( s_i^*,s_{-i}' )=x_i( s_i^*,s_{-i}' )p_i^*-\gamma _i z_i^*\ge \alpha _i\) implies

$$\begin{aligned} x_i( s_i^*, s'_{-i} ) \ge (\alpha _i+\gamma _iz_i^*)/p_i^*> \alpha _i/C_i>1/2, \end{aligned}$$
(2)

where the last inequality is due to (1). If \(z_i'=0\) holds for a leader i, then \(s_i'\notin C_i(s',\alpha _i)\) holds, since any strategy \((z_i^*,p_i^*)\) achieving a profit of at least \(\alpha _i>0\) fulfills \(z_i^*>0\). Thus we can assume in the following that \(z_i'> 0\) holds for all leaders i. Then, since \(n\ge 2\), there is at least one leader i with \(x_i(s')\le 1/2\). We now show that \(s_i'\notin C_i(s',\alpha _i)\) holds. If \(z_i'=0\) or \(p_i'\le \alpha _i\) holds, \(s_i'\notin C_i(s',\alpha _i)\) follows, since \(z_i^*>0\) and \(p_i^*>\alpha _i\) hold for any strategy \((z_i^*,p_i^*)\) achieving a profit of at least \(\alpha _i\). Thus we can assume in the following that \(z_i'>0\) and \(p_i'>\alpha _i\) hold. If \(x_i(s')=0\), the Wardrop equilibrium conditions yield \(K(s')\le b_i+p_i'\). Then, any strategy \(\bar{s_i}=(\bar{z_i},\bar{p_i})\) with \(\bar{p_i}\ge p_i'\) yields \(x( \bar{s_i},s_{-i}' )=x(s')\), and thus \(x_i( \bar{s_i},s_{-i}' )=0\) and \(\Pi _i( \bar{s_i},s_{-i}' )\le 0<\alpha _i\) hold. Therefore, \(p_i^*<p_i'\) holds for any strategy \((z_i^*,p_i^*)\) achieving a profit of at least \(\alpha _i\), and \(s_i'\notin C_i(s',\alpha _i)\) follows. We can thus assume in the following that \(x_i(s')>0\) holds. Summarizing, we can assume that the following inequalities are fulfilled:

$$\begin{aligned} 0<x_i( s_i',s'_{-i} )\le 1/2, \quad \alpha _i<p_i'\le C_i \quad \text {and} \quad 0<z_i'<1. \end{aligned}$$
(3)

We now show that each strategy \(s_i^*=(z_i^*,p_i^*)\) with \(\Pi _i( s_i^*, s_{-i}' )\ge \alpha _i\) fulfills \(z_i^*>z_i'\), showing that \(s_i'\notin C_i(s',\alpha _i)\) and completing the proof. Assume, by contradiction, that there is a strategy \(s_i^*=(z_i^*,p_i^*)\) which achieves a profit of at least \(\alpha _i\) and fulfills \(z_i^*\le z_i'\). For any strategy \(\tilde{s_i}\in S_i\), write \(x(\tilde{s_i}):=x( \tilde{s_i},s_{-i}' )\) and \(K(\tilde{s_i}):=K( \tilde{s_i}, s_{-i}' )\). Now consider the strategy \(\tilde{s_i}:=(z_i',\alpha _i)\). Assume, for the moment, that

$$\begin{aligned} K(\tilde{s_i})\le K(s_i^*)<K(s_i') \end{aligned}$$
(4)

holds (we prove (4) below). Using \(K(\tilde{s_i})<K(s_i')\) then implies \( a_ix_i(\tilde{s_i})/z_i'+b_i+\alpha _i<a_ix_i(s_i')/z_i' +b_i+p_i'. \) Reformulating this inequality and using (3) and (1) then yields

$$\begin{aligned} x_i(\tilde{s_i})<z_i'(p_i'-\alpha _i)/a_i+x_i(s_i')< (C_i-\alpha _i)/a_i+1/2<\alpha _i/C_i. \end{aligned}$$
(5)

The inequality \(K(\tilde{s_i})\le K(s_i^*)\) from (4) implies \(x_j(\tilde{s_i})\le x_j(s_i^*)\) for all leaders \(j \ne i\). Therefore, \(x_i(\tilde{s_i})\ge x_i(s_i^*)\) holds. Using \(x_i(s_i^*)>\alpha _i/C_i\) from (2) now leads to \(x_i(\tilde{s_i})>\alpha _i/C_i\), which contradicts (5). To complete the proof, it remains to show (4). The property \(K(s_i^*)<K(s_i')\) holds since \(K(s_i^*)\ge K(s_i')\) would imply \(x_j(s_i^*)\ge x_j(s_i')\) for all leaders \(j \ne i\), and thus \(x_i(s_i^*)\le x_i(s_i')\), but we know from (2) and (3) that \(x_i(s_i^*)>1/2\ge x_i(s_i')\). To prove the other inequality in (4), assume, by contradiction, that \(K(\tilde{s_i})> K(s_i^*)\). This implies \(x_j(\tilde{s_i})\ge x_j(s_i^*)\) for all leaders \(j \ne i\), and thus \(x_i(\tilde{s_i})\le x_i(s_i^*)\). Together with \(z_i^*\le z_i'\) and \(p_i^*>\alpha _i\), this leads to the following contradiction, and finally completes the proof:

$$\begin{aligned} K(\tilde{s_i})\le a_ix_i(\tilde{s_i})/z_i'+b_i+\alpha _i<a_ix_i(s_i^*)/z_i^*+b_i+p_i^*=K(s_i^*)<K(\tilde{s_i}). \end{aligned}$$

\(\square \)

Using Theorem 3 together with the Lemmata 6 and 7 now yields the existence of a PNE:

Theorem 4

Every Stackelberg pricing game has a pure Nash equilibrium.

Note here that in any PNE (zp), there are at least two leaders i with \(z_i>0\).

5 Uniqueness of equilibria

As we have seen in the last section, a Stackelberg pricing game always has a PNE. In this section we show that this equilbrium is essentially unique. With essentially we mean that if (zp) and \((z',p')\) are two different PNE, and \(i\in N\) is a leader such that \((z_i,p_i)\ne (z_i',p_i')\), then \(z_i=z_i'=0\) holds (and thus \(p_i\ne p_i'\)).

For a PNE \(s=(z,p)\), denote by \(N^+(z,p):=\{i\in N: z_i>0\}\) the set of leaders with positive capacity (note that \(|N^+(z,p)|\ge 2\) and \(N^+(z,p)=\{i\in N: x_i(s)>0\}\)). For \(i\in N^+(z,p)\), let \((\text {P}_i^1)(s_{-i})\) and \((\text {P}_i^2)(s_{-i})\) be the two auxiliary problems from Sect. 3.Footnote 6 By Lemma 4, the routing cost K(zp) is an optimal solution of either \((\text {P}_i^1)(s_{-i})\) or \((\text {P}_i^2)(s_{-i})\). We denote by \(N^+_1(z,p)\) the set of leaders \(i\in N^+(z,p)\) such that K(zp) is an optimal solution of \((\text {P}_i^1)(s_{-i})\), and \(N^+_2(z,p)\) contains the leaders \(i\in N^+(z,p)\) such that K(zp) is an optimal solution of \((\text {P}_i^2)(s_{-i})\). Thus \(N^+(z,p)=N^+_1(z,p)\overset{.}{\cup }N^+_2(z,p)\). Throughout this section, we use the simplified notation \(N'\setminus i\) instead of \(N' \setminus \{i\}\) for any subset \(N'\subseteq N\) of leaders and \(i\in N'\).

Note that the proofs in this section are similar to the proofs that [21] use to derive their uniqueness results. However, since our model includes price caps, some new ideas are required, in particular the decomposition of \(N^+(z,p)\) in \(N^+_1(z,p)\overset{.}{\cup }N^+_2(z,p)\).

We first derive further necessary equilibrium conditions (by using the KKT conditions) which will become useful in the following analysis.

Lemma 8

Let \(s=(z,p)\) be a PNE with \(x:=x(z,p)\) and \(K:=K(z,p)\). Let \(i \in N^+:=N^+(z,p)\). If \(p_i<C_i\) holds, then \(z_i=\sqrt{a_i/\gamma _i}x_i\), \(p_i=\frac{x_i}{\sum _{j\in N^+\setminus i}{z_j/a_j}}+\sqrt{a_i\gamma _i}\), and if \(p_i=C_i\), then \(z_i=\frac{a_ix_i}{K-b_i-C_i}\) and \(\frac{C_i}{1+a_i/z_i\cdot \sum _{j\in N^+\setminus i}{z_j/a_j}}=\frac{\gamma _iz_i^2}{a_ix_i\sum _{j\in N^+\setminus i}{z_j/a_j}}\).

Proof

Since (zp) is a PNE, \(p_j>0\) and \(x_j>0\) holds for all \(j \in N^+\), and \(x_k=0\) holds for \(k \notin N^+\). Furthermore, \((z_i,p_i)\) is a best response of leader i to \(s_{-i}\) and \(K=a_jx_j/z_j+b_j+p_j\) holds for all \(j \in N^+\). Altogether we get that \((z_i,p_i,(x_j)_{j \in N^+})\) is an optimal solution for the following optimization problem (with variables \((z_i',p_i',(x_j')_{j \in N^+})\)):

$$\begin{aligned} \max \quad x_i'p_i'-\gamma _iz_i' \quad \text { s.t.: } \quad&0 \le p_i' \le C_i, \ 0 < z_i', \ \sum _{j \in N^+}{x_j'}=1, \ x_j' \ge 0 \ \forall j \in N^+, \\&a_ix_i'/z_i'+b_i+p_i'=a_jx_j'/z_j+b_j+p_j \ \forall j \in N^+\setminus i. \end{aligned}$$

It is easy to show that the LICQ holds for \((z_i,p_i,(x_j)_{j \in N^+})=(z_i,p_i,x_i,(x_j)_{j \in N^+ \setminus i})\), thus the KKT conditions are fulfilled. We get the following equations:

$$\begin{aligned} \gamma _i-a_ix_i/z_i^2\sum _{j \in N^+\setminus i}{\lambda _j}&=0 \end{aligned}$$
(KKT1)
$$\begin{aligned} -x_i+\mu +\sum _{j \in N^+\setminus i}{\lambda _j}&=0 \end{aligned}$$
(KKT2)
$$\begin{aligned} -p_i+\lambda +a_i/z_i\sum _{j \in N^+\setminus i}{\lambda _j}&=0 \end{aligned}$$
(KKT3)
$$\begin{aligned} \lambda -\lambda _ja_j/z_j&=0 \ \forall j \in N^+\setminus i. \end{aligned}$$
(KKT4)

We now distinguish between the two cases \(p_i<C_i\) and \(p_i=C_i\).

In the first case, \(\mu =0\) holds, and (KKT2) yields \(x_i=\sum _{j \in N^+\setminus i}{\lambda _j}\). Using this, (KKT1) yields \(z_i=\sqrt{a_i/\gamma _i}x_i\). Plugging this in (KKT3) leads to \(p_i=\lambda +a_ix_i/z_i=\lambda +\sqrt{a_i\gamma _i}\). Using (KKT4), i.e., \(\lambda _j=\lambda z_j/a_j\) for all \(j \in N^+ \setminus i\), together with (KKT2) yields \(x_i=\lambda \sum _{j \in N^+\setminus i}{z_j/a_j}\), or equivalently, \(\lambda =\frac{x_i}{\sum _{j \in N^+\setminus i}{z_j/a_j}}\). This shows \(p_i=\frac{x_i}{\sum _{j \in N^+\setminus i}{z_j/a_j}}+\sqrt{a_i\gamma _i}\), as required.

The other case is \(p_i=C_i\). The formula for \(z_i\) follows from \(K=a_ix_i/z_i+b_i+C_i\). Plugging \(\lambda _j=\lambda z_j/a_j\) for all \(j \in N^+ \setminus i\) in (KKT1) and (KKT3) yields

$$\begin{aligned} \lambda= & {} \frac{\gamma _iz_i^2}{a_ix_i\sum _{j \in N^+\setminus i}{z_j/a_j}} \ \text {and} \ \lambda =C_i-a_i/z_i\cdot \lambda \cdot \sum _{j \in N^+\setminus i}{z_j/a_j} \Leftrightarrow \\ \lambda= & {} \frac{C_i}{1+a_i/z_i\sum _{j \in N^+\setminus i}{z_j/a_j}}, \end{aligned}$$

which shows the desired equality. \(\square \)

In the next lemma, we introduce two functions \(\Gamma _i^1\) and \(\Gamma _i^2\) for each leader i and derive useful properties of these functions.

Lemma 9

For each \(i \in N\), define

$$\begin{aligned} \Gamma _i^1: (\sqrt{a_i\gamma _i}+b_i,\infty ) \rightarrow {\mathbb {R}}, \ \ \Gamma _i^1(\kappa ):= & {} \frac{\sqrt{a_i\gamma _i}}{\kappa -\sqrt{a_i\gamma _i}-b_i} \quad \text {and} \\ \Gamma _i^2: (b_i+C_i,\infty ) \rightarrow {\mathbb {R}}, \ \ \Gamma _i^2(\kappa ):= & {} \frac{a_i\gamma _i/C_i}{\kappa -b_i-C_i}. \end{aligned}$$

Furthermore, let \(s=(z,p)\) be a PNE with \(x:=x(z,p)\), cost \(K:=K(z,p)\), and \(N^+:=N^+(z,p)\) with \(N^+_1:=N^+_1(z,p)\), \(N^+_2:=N^+_2(z,p)\). Then:

  1. 1.

    \(\Gamma _i^1\) and \(\Gamma _i^2\) are strictly decreasing functions.

  2. 2.

    If \(i \in N^+_1\), then \(\Gamma _i^1(K)=1-\frac{z_i/a_i}{\sum _{j\in N^+}{z_j/a_j}}<1\).

  3. 3.

    If \(i \in N^+_2\), then \(\Gamma _i^2(K)=1-\frac{z_i/a_i}{\sum _{j\in N^+}{z_j/a_j}}<1\).

  4. 4.

    \(\sum _{i \in N^+_1}{\Gamma _i^1(K)}+\sum _{i \in N^+_2}{\Gamma _i^2(K)}=|N^+|-1\).

  5. 5.

    If \(i\in N_1^+\) and there is a (different) PNE \(s'=(z',p')\) with \(i\in N^+_2(z',p')\), then \(K<K':=K(z',p')\) and \(\Gamma _i^1(K)>\Gamma _i^2(K')\).

Proof

Statement 1. is clear from the definitions of \(\Gamma _i^1\) and \(\Gamma _i^2\), so turn to statement 2. and let \(i\in N^+_1\). Lemma 4 yields \(z_i=\sqrt{a_i/\gamma _i}x_i\), \(p_i=K-\sqrt{a_i\gamma _i}-b_i\). Using Lemma 8, \(B:=\sum _{j\in N^+}{z_j/a_j}\), and \(z_i=\sqrt{a_i/\gamma _i}x_i\), we get

$$\begin{aligned} K&= \frac{x_i}{\sum _{j\in N^+\setminus i}{z_j/a_j}}+2\sqrt{a_i\gamma _i}+b_i = \frac{x_i}{B-{z_i/a_i}}+2\sqrt{a_i\gamma _i}+b_i \\&= \frac{x_i+(B-x_i/\sqrt{a_i\gamma _i})\sqrt{a_i\gamma _i}}{B-x_i/\sqrt{a_i\gamma _i}}+\sqrt{a_i\gamma _i}+b_i =\frac{B\sqrt{a_i\gamma _i}}{B-x_i/\sqrt{a_i\gamma _i}}+\sqrt{a_i\gamma _i}+b_i. \end{aligned}$$

Using this we get statement 2.:

$$\begin{aligned} \Gamma _i^1(K)=\frac{B-x_i/\sqrt{a_i\gamma _i}}{B}=1-\frac{z_i/a_i}{B}<1. \end{aligned}$$

For statement 3., let \(i \in N^+_2\). Lemmas 4 and 8 imply \(\frac{C_i}{1+a_i/z_i\cdot \sum _{j\in N^+\setminus i}{z_j/a_j}}=\frac{\gamma _iz_i^2}{a_ix_i\sum _{j\in N^+\setminus i}{z_j/a_j}}\). Rearranging and using the definition of B yields

$$\begin{aligned} \frac{a_ix_i}{z_i}= & {} \frac{\gamma _iz_i(1+a_i/z_i\cdot \sum _{j\in N^+\setminus i}{z_j/a_j})}{C_i\cdot \sum _{j\in N^+\setminus i}{z_j/a_j}}= \frac{\gamma _iz_i(1+a_i/z_i\cdot (B-z_i/a_i))}{C_i\cdot (B-z_i/a_i)}\\= & {} \frac{\gamma _ia_iB}{C_i\cdot (B-z_i/a_i)}. \end{aligned}$$

Using \(K=a_ix_i/z_i+b_i+C_i\) then yields \(K=\frac{\gamma _ia_iB}{C_i\cdot (B-z_i/a_i)}+b_i+C_i\), and statement 3. follows:

$$\begin{aligned} \Gamma _i^2(K)=\frac{B-z_i/a_i}{B}=1-\frac{z_i/a_i}{B}<1. \end{aligned}$$

Statement 4. now follows from the statements 2. and 3.:

$$\begin{aligned} \sum _{i\in N_1^+}{\Gamma _i^1(K)}+\sum _{i\in N^+_2}{\Gamma _i^2(K)}=&\sum _{i\in N_1^+}{\left( 1-\frac{z_i/a_i}{B}\right) }+\sum _{i\in N^+_2}{\left( 1-\frac{z_i/a_i}{B}\right) } = |N^+|-1 \end{aligned}$$

It remains to show statement 5. Let \(i\in N_1^+\cap N_2^+(z',p')\). Since \(i\in N_1^+\), the cost K is in particular feasible for \((\text {P}_i^1)(s_{-i})\), thus \(2\sqrt{a_i\gamma _i}+b_i \le K\le \sqrt{a_i\gamma _i}+b_i+C_i\) holds. Analogously, using \(i\in N_2^+(z',p')\), the cost \(K'\) is feasible for \((\text {P}_i^2)(s_{-i}')\), therefore \(\sqrt{a_i\gamma _i}+b_i+C_i<K'<K_i^{\max }(s'_{-i})\). Together we get \(K\le \sqrt{a_i\gamma _i}+b_i+C_i< K'\). It remains to show \(\Gamma _i^1(K)>\Gamma _i^2(K')\). By definition of \(N_2^+(z',p')\), the cost \(K'\) is an optimal solution for problem \((\text {P}_i^2)(s_{-i}')\) and in particular (see 1. and 3. of Lemma 5) yields a better objective function value than \(\sqrt{a_i\gamma _i}+b_i+C_i\) in \((\text {P}_i^1)(s_{-i}')\) (note that \(\sqrt{a_i\gamma _i}+b_i+C_i\) is feasible for \((\text {P}_i^1)(s_{-i}')\) since \(2\sqrt{a_i\gamma _i}+b_i \le K\le \sqrt{a_i\gamma _i}+b_i+C_i<K'<K_i^{\max }(s'_{-i})\)). If \(\overline{x}_i\) denotes the function occurring in the definitions of \((\text {P}_i^1)(s_{-i}')\) and \((\text {P}_i^2)(s_{-i}')\), we thus get \(\overline{x}_i(\sqrt{a_i\gamma _i}+b_i+C_i)\cdot (C_i-\sqrt{a_i\gamma _i})<\overline{x}_i(K')\cdot (C_i-a_i\gamma _i/(K'-b_i-C_i)) \le \overline{x}_i(\sqrt{a_i\gamma _i}+b_i+C_i)\cdot (C_i-a_i\gamma _i/(K'-b_i-C_i))\), where the last inequality follows from \(\sqrt{a_i\gamma _i}+b_i+C_i<K'\) and the fact that \({\overline{x}}_i\) is a decreasing function. Since \(\overline{x}_i(\sqrt{a_i\gamma _i}+b_i+C_i)>0\), we get

$$\begin{aligned} C_i-\sqrt{a_i\gamma _i}<C_i-\frac{a_i\gamma _i}{K'-b_i-C_i} \ \Leftrightarrow \ \frac{\sqrt{a_i\gamma _i}}{C_i} > \frac{a_i\gamma _i/C_i}{K'-b_i-C_i}=\Gamma _i^2(K'). \end{aligned}$$

Note that \(2\sqrt{a_i\gamma _i}+b_i\le K\le \sqrt{a_i\gamma _i}+b_i+C_i\), thus \(1/(K-\sqrt{a_i\gamma _i}-b_i)\ge 1/C_i\), and altogether

$$\begin{aligned} \Gamma _i^1(K)=\frac{\sqrt{a_i\gamma _i}}{K-\sqrt{a_i\gamma _i}-b_i}\ge \frac{\sqrt{a_i\gamma _i}}{C_i}>\Gamma _i^2(K'). \end{aligned}$$

\(\square \)

Now we turn to the uniqueness of the equilibrium, and start with the following lemma.

Lemma 10

For a fixed subset \(N^+\) of the leaders and a fixed disjoint decomposition \(N^+=N_1^+\overset{.}{\cup }N_2^+\), there is essentially at most one PNE (zp) such that \(N^+(z,p)=N^+\), \(N_1^+(z,p)=N_1^+\) and \(N_2^+(z,p)=N_2^+\).

Proof

Assume that there are two PNE (zp) and \((z',p')\) with the described properties, i.e., \(N^+(z,p)=N^+(z',p')=N^+\), \(N_1^+(z,p)=N_1^+(z',p')=N_1^+\) and \(N_2^+(z,p)=N_2^+(z',p')=N_2^+\). Let \(x:=x(z,p)\) and \(x':=x(z',p')\) with costs \(K:=K(z,p)\) and \(K':=K(z',p')\). We show that \((z_i,p_i)=(z_i',p_i')\) holds for all \(i\in N^+\), showing that (zp) and \((z',p')\) are essentially the same.

First note that \(K=K'\) holds, since \(f(\kappa ):=\sum _{i\in N_1^+}{\Gamma _i^1(\kappa )}+\sum _{i\in N_2^+}{\Gamma _i^2(\kappa )}\) is a strictly decreasing function in \(\kappa \) and \(f(K)=\sum _{i \in N^+_1}{\Gamma _i^1(K)}+\sum _{i \in N^+_2}{\Gamma _i^2(K)}=|N^+|-1=\sum _{i \in N^+_1}{\Gamma _i^1(K')}+\sum _{i \in N^+_2}{\Gamma _i^2(K')}=f(K')\) holds from 4. in Lemma 9. This implies \(p_i=p_i'\) for all \(i\in N^+\), since \(p_i=K-\sqrt{a_i\gamma _i}-b_i=K'-\sqrt{a_i\gamma _i}-b_i=p_i'\) holds for \(i\in N_1^+\), and \(p_i=C_i=p_i'\) for \(i\in N_2^+\).

If \(B:=\sum _{j \in N^+}{z_j/a_j}=\sum _{j \in N^+}{z_j'/a_j}=:B'\) holds, we also get \(z_i=z_i'\) for all \(i\in N^+\), since 2. of Lemma 9 yields \(z_i=(1-\Gamma _i^1(K))a_iB=(1-\Gamma _i^1(K'))a_iB'=z_i' \hbox { for all } i\in N_1^+\) and 3. of Lemma 9 yields \(z_i=(1-\Gamma _i^2(K))a_iB=(1-\Gamma _i^2(K'))a_iB'=z_i' \hbox { for all } i\in N_2^+\).

It remains to show \(B=B'\). First consider \(i\in N_1^+\). Using \(z_i=\sqrt{a_i/\gamma _i}\cdot x_i\) and \(z_i'=\sqrt{a_i/\gamma _i}\cdot x_i'\), as well as 2. of Lemma 9, yields \(x_i/B=(1-\Gamma _i^1(K))\sqrt{a_i\gamma _i}=(1-\Gamma _i^1(K'))\sqrt{a_i\gamma _i}=x_i'/B'\). For \(i\in N_2^+\), we use \(\frac{z_i}{a_i}=\frac{x_i}{K-b_i-C_i}\) and \(\frac{z_i'}{a_i}=\frac{x_i'}{K'-b_i-C_i}\) and 3. of Lemma 9 to achieve \(x_i/B=(1-\Gamma _i^2(K))(K-b_i-C_i)=(1-\Gamma _i^2(K'))(K'-b_i-C_i)=x_i'/B'\). Altogether we have \(x_i/B=x_i'/B'\) for all \(i\in N^+\). Using that \(\sum _{i\in N^+}{x_i}=1=\sum _{i\in N^+}{x_i'}\) yields \(B=B'\). \(\square \)

In the previous lemma, we showed that given a fixed subset \(N^+\subseteq N\) and a fixed disjoint decomposition \(N^+=N_1^+\overset{.}{\cup }N_2^+\), there is at most one PNE (zp) such that \(N^+(z,p)=N^+\), \(N_1^+(z,p)=N_1^+\) and \(N_2^+(z,p)=N_2^+\). Next, we strengthen this result by showing that for a fixed subset \(N^+\subseteq N\), there is at most one PNE (zp) with \(N^+(z,p)=N^+\) (independently of the decomposition of \(N^+\)).

Lemma 11

For a fixed subset \(N^+\) of the leaders, there is essentially at most one PNE (zp) with \(N^+(z,p)=N^+\).

Proof

Assume, by contradiction, that there are two essentially different PNE (zp) and \(({\overline{z}},{\overline{p}})\) with \(N^+(z,p)=N^+=N^+({\overline{z}}, {\overline{p}})\). Let \(N_1^+:=N_1^+(z,p)\), \(N_2^+:=N_2^+(z,p)\) and \(\overline{N}_1^+:=N_1^+({\overline{z}},{\overline{p}})\), \(\overline{N}_2^+:=N_2^+({\overline{z}},{\overline{p}})\). Further denote \(x:=x(z,p)\), \(K:=K(z,p)\) and \({\overline{x}}:=x({\overline{z}},{\overline{p}})\), \({\overline{K}}:=K({\overline{z}},{\overline{p}})\).

Lemma 10 yields that the decompositions of \(N^+\) have to be different. Without loss of generality, there is a leader \(j\in N_1^+ \setminus {\overline{N}}_1^+\). Since \(j\in \overline{N}_2^+\), statement 5. of Lemma 9 yields \(K<{\overline{K}}\). The existence of a leader \(i \in \overline{N}_1^+ \setminus N_1^+\) leads (by the same argumentation) to the contradiction \({\overline{K}}<K\), thus \(\overline{N}_1^+ \subsetneq N_1^+\) and \(N_2^+\subsetneq \overline{N}_2^+\) hold and we can write (using 4. of Lemma 9)

$$\begin{aligned} |N^+|-1&= \sum _{i \in N_1^+}{\Gamma _i^1(K)}+\sum _{i \in N_2^+}{\Gamma _i^2(K)}=\sum _{i \in N_1^+\setminus \overline{N}_1^+}{\Gamma _i^1(K)}+\sum _{i \in \overline{N}_1^+}{\Gamma _i^1(K)}\\&\quad +\sum _{i \in N_2^+}{\Gamma _i^2(K)} \ \text {and} \\ |N^+|-1&= \sum _{i \in \overline{N}_1^+}{\Gamma _i^1(\overline{K})}+\sum _{i \in \overline{N}_2^+}{\Gamma _i^2(\overline{K})}= \sum _{i \in \overline{N}_1^+}{\Gamma _i^1(\overline{K})}+\sum _{i \in \overline{N}_2^+\setminus N_2^+}{\Gamma _i^2(\overline{K})}\\&\quad +\sum _{i \in {N}_2^+}{\Gamma _i^2(\overline{K})}. \end{aligned}$$

Using \(K<\overline{K}\) and that both \(\sum _{i\in N_2^+}{\Gamma _i^2(\kappa )}\) and \(\sum _{i \in \overline{N}_1^+}{\Gamma _i^1(\kappa )}\) are decreasing in \(\kappa \) yields

$$\begin{aligned} \sum _{i \in N_2^+}{\Gamma _i^2(K)} \ge \sum _{i \in N_2^+}{\Gamma _i^2(\overline{K})} \ \ \text {and} \ \ \sum _{i \in \overline{N}_1^+}{\Gamma _i^1(K)} \ge \sum _{i \in \overline{N}_1^+}{\Gamma _i^1(\overline{K})}. \end{aligned}$$

Finally, 5. of Lemma 9 yields \(\Gamma _i^1(K)>\Gamma _i^2({\overline{K}})\) for all \(i\in N_1^+ \setminus {\overline{N}}_1^+={\overline{N}}_2^+ \setminus N_2^+\ne \emptyset \), thus

$$\begin{aligned} \sum _{i \in {N}_1^+\setminus \overline{N}_1^+}{\Gamma _i^1(K)} > \sum _{i \in \overline{N}_2^+\setminus N_2^+}{\Gamma _i^2(\overline{K})} \end{aligned}$$

holds. Altogether we get the following contradiction, completing the proof:

$$\begin{aligned} |N^+|-1=&\sum _{i \in N_1^+\setminus \overline{N}_1^+}{\Gamma _i^1(K)}+\sum _{i \in \overline{N}_1^+}{\Gamma _i^1(K)}+\sum _{i \in N_2^+}{\Gamma _i^2(K)} \\ >&\sum _{i \in \overline{N}_1^+}{\Gamma _i^1(\overline{K})}+\sum _{i \in \overline{N}_2^+\setminus N_2^+}{\Gamma _i^2(\overline{K})}+\sum _{i \in {N}_2^+}{\Gamma _i^2(\overline{K})} = |N^+|-1 \end{aligned}$$

\(\square \)

For the uniqueness of the PNE, it remains to show that there is at most one set \(N^+\) such that a PNE (zp) with \(N^+(z,p)=N^+\) exists. To this end, we first show that each leader i has a threshold \(K^*_i\) such that, for any PNE (zp), leader i has \(z_i>0\) if and only if \(K(z,p)>K^*_i\).

Lemma 12

For each \(i\in N\), define

$$\begin{aligned} K^*_i:={\left\{ \begin{array}{ll} a_i\gamma _i/C_i+b_i+C_i, &{} \text { if } \sqrt{a_i \gamma _i}>C_i, \\ 2\sqrt{a_i\gamma _i}+b_i, &{} \text {else.} \end{array}\right. } \end{aligned}$$

Then, for any PNE \(s=(z,p)\) and any leader \(i\in N\), it holds that \(z_i>0\) iff \(K(z,p)>K^*_i\).

Proof

Let \(s=(z,p)\) be a PNE with \(x:=x(z,p)\), \(K:=K(z,p)\), and \(i\in N\).

First assume that \(z_i=0\). Since (zp) is a PNE, the strategy \((z_i,p_i)=(0,p_i)\) is a best response of leader i to \(s_{-i}\). As we have seen in Theorem 2, this is equivalent to the fact that both problems \((\text {P}_i^1)(s_{-i})\) and \((\text {P}_i^2)(s_{-i})\) are infeasible. Note that \(K=K_i^{\max }(s_{-i})\) holds due to the definition of \(K_i^{\max }(s_{-i})\) (cf. Sect. 3.2 and \(x_j=(K-b_j-p_j)z_j/a_j\) for all \(j\in \{j'\in N: x_{j'}>0\}=\{j' \in N\setminus i: z_{j'}>0, b_{j'}+p_{j'}{<}K\}\). To show \(K\le K^*_i\), we have to distinguish between the two cases \(\sqrt{a_i \gamma _i}>C_i\) and \(\sqrt{a_i \gamma _i}\le C_i\). First consider \(\sqrt{a_i \gamma _i}>C_i\), thus \(K^*_i=a_i\gamma _i/C_i+b_i+C_i\). Since \((\text {P}_i^2)(s_{-i})\) is infeasible and \(\sqrt{a_i\gamma _i}+b_i+C_i<a_i\gamma _i/C_i+b_i+C_i\), we get the desired inequality \(K=K_i^{\max }(s_{-i})\le a_i\gamma _i/C_i+b_i+C_i=K^*_i\). Now consider \(\sqrt{a_i \gamma _i}\le C_i\), i.e. \(K^*_i=2\sqrt{a_i\gamma _i}+b_i\). Since \((\text {P}_i^1)(s_{-i})\) is infeasible and \(2\sqrt{a_i\gamma _i}+b_i\le \sqrt{a_i\gamma _i}+b_i+C_i\), we get \(K=K_i^{\max }(s_{-i})\le 2\sqrt{a_i\gamma _i}+b_i=K^*_i\), as desired. We have seen that \(z_i=0\) implies \(K\le K^*_i\), or, equivalently, \(K>K^*_i\) implies \(z_i>0\).

It remains to show the other direction, i.e., \(z_i>0\) implies \(K> K^*_i\). We consider the cases \(\sqrt{a_i \gamma _i}>C_i\) and \(\sqrt{a_i \gamma _i}\le C_i\) and use our results from Lemma 4 and Theorem 2. If \(\sqrt{a_i \gamma _i}>C_i\), thus \(K^*_i=a_i\gamma _i/C_i+b_i+C_i\), the cost K is an optimal solution for problem \((\text {P}_i^2)(s_{-i})\) with positive objective function value (note that \((\text {P}_i^1)(s_{-i})\) is infeasible), therefore \(K^*_i=a_i\gamma _i/C_i+b_i+C_i<K\). In the second case, i.e., \(\sqrt{a_i \gamma _i}\le C_i\) and \(K^*_i=2\sqrt{a_i\gamma _i}+b_i\), the cost K either is optimal for \((\text {P}_i^1)(s_{-i})\), or optimal for \((\text {P}_i^2)(s_{-i})\), and has positive objective function value in both cases. We get the desired property, since \(K^*_i=2\sqrt{a_i\gamma _i}+b_i<K\) holds for the first case and \(K^*_i=2\sqrt{a_i\gamma _i}+b_i\le \sqrt{a_i\gamma _i}+b_i+C_i<K\) holds for the second, completing the proof. \(\square \)

We can now show the remaining result for the desired uniqueness of PNE.

Lemma 13

There is at most one \(N^+\subseteq N\) such that a PNE (zp) with \(N^+(z,p)=N^+\) exists.

Proof

Assume, by contradiction, that there are different subsets \(N^+\) und \(\overline{N}^+\) with PNE (zp) and \(({\overline{z}},{\overline{p}})\), such that \(N^+(z,p)=N^+\) and \(N^+({\overline{z}}, \overline{p})=\overline{N}^+\). Let \(N_1^+\overset{.}{\cup } N_2^+\) and \(\overline{N}_1^+\overset{.}{\cup }\overline{N}_2^+\) be the decompositions of \(N^+\) and \(\overline{N}^+\), i.e., \(N^+_1(z,p)=N^+_1\), \(N^+_2(z,p)=N^+_2\), \(N^+_1({\overline{z}},\overline{p})={\overline{N}}^+_1\) and \(N^+_2({\overline{z}},{\overline{p}})=\overline{N}^+_2\). Finally denote \(x:=x(z,p)\), \(K:=K(z,p)\) and \(\overline{x}:=x({\overline{z}},{\overline{p}})\), \({\overline{K}}:=K(\overline{z},{\overline{p}})\).

Using Lemma 12, we can assume w.l.o.g. that \(K< {\overline{K}}\) and \(N^+\subsetneq \overline{N}^+\). Then, \(N_2^+\subseteq \overline{N}_2^+\) holds, since the existence of a leader \(i \in N_2^+ \setminus {\overline{N}}_2^+\), i.e. \(i \in N_2^+ \cap {\overline{N}}_1^+\), leads to the contradiction \({\overline{K}}<K\) by statement 5. of Lemma 9. Furthermore, if there is a leader \(i\in N_1^+ \setminus {\overline{N}}_1^+\), i.e. \(i \in N_1^+ \cap {\overline{N}}_2^+\), statement 5. of Lemma 9 yields \(\Gamma _i^1(K) >\Gamma _i^2({\overline{K}})\). Finally, \(\Gamma _i^1({\overline{K}})<1\) holds for all \(i \in {\overline{N}}_1^+\), and \(\Gamma _i^2(\overline{K})<1\) holds for all \(i\in {\overline{N}}_2^+\) (see 2. and 3. of Lemma 9). Altogether, this leads to the following contradiction, and completes the proof (where we additionally use \(K <{\overline{K}}\), and the statements 1. and 4. of Lemma 9):

$$\begin{aligned} |{\overline{N}}^+|-1 =&\sum _{i \in {\overline{N}}_1^+}{\Gamma _i^1({\overline{K}})}+\sum _{i \in {\overline{N}}_2^+}{\Gamma _i^2({\overline{K}})} \\ =&\sum _{i \in {\overline{N}}_1^+ \cap N_1^+}{\Gamma _i^1({\overline{K}})}+\sum _{i \in {\overline{N}}_1^+ \setminus N^+}{\Gamma _i^1({\overline{K}})}+\sum _{i \in {\overline{N}}_2^+ \cap N_1^+}{\Gamma _i^2({\overline{K}})}+\sum _{i \in N_2^+}{\Gamma _i^2({\overline{K}})}\\&\quad +\sum _{i \in {\overline{N}}_2^+ \setminus N^+}{\Gamma _i^2({\overline{K}})}\\ <&\sum _{i \in N_1^+ \cap {\overline{N}}_1^+}{\Gamma _i^1(K)}+\sum _{i \in N_1^+\cap {\overline{N}}_2^+ }{\Gamma _i^1(K)}+\sum _{i \in N_2^+}{\Gamma _i^2(K)}+|{\overline{N}}^+|-|N^+|\\ =&\sum _{i \in N_1^+}{\Gamma _i^1(K)}+\sum _{i \in N_2^+}{\Gamma _i^2(K)}+|{\overline{N}}^+|-|N^+| =|{\overline{N}}^+|-1. \end{aligned}$$

\(\square \)

Together with the existence result in Theorem 4, the preceding Lemmata 11 and 13 show:

Theorem 5

Every Stackelberg pricing game has an essentially unique PNE, i.e., if (zp) and \((z',p')\) are different PNE and \(i\in N\) with \((z_i,p_i)\ne (z_i',p_i')\), then \(z_i=z_i'=0\) holds.

6 Quality of equilibria

In the last section, we showed that a Stackelberg pricing game has an (essentially) unique PNE. Now we analyze the quality of the PNE compared to a social optimum. Define the social cost C(zp) of a strategy profile \(s=(z,p)\) as

$$\begin{aligned} C(z,p)={\left\{ \begin{array}{ll} \sum _{i\in \{1,\ldots ,n\}: z_i>0}{\left( \ell _i(x_i(s),z_i)x_i(s)+\gamma _iz_i\right) }, &{} \text { if } \sum _{i=1}^{n}{z_i}>0, \\ \infty , &{} \text { else.} \end{array}\right. } \end{aligned}$$

The function C(zp) measures utilitarian social welfare over leaders and followers (the price component cancels out). Common notions to measure the quality of equilibria are the Price of Anarchy (PoA) and the Price of Stability (PoS), which are defined as the worst case ratios of the cost of a worst, respectively best, PNE, and a social optimum. Note that for Stackelberg pricing games, PoA and PoS are the same: Since a Stackelberg pricing game G has an essentially unique PNE, all PNE of G have the same social cost. If we denote this cost by \(C(\text {PNE}(G))\), and the minimum social cost in G (compared to all possible strategy profiles) by \(\text {OPT}(G)\), we thus get that \(\text {PoA}=\text {PoS}=\sup _G C(\text {PNE}(G))/\text {OPT}(G)\).

In Sect. 6.1, we show that the PoA for Stackelberg pricing games is unbounded. For the proof, we use a family of instances with heterogeneous leaders, that is, the leaders have different parameters. In Sect. 6.2, we then turn to the homogeneous case, where all leaders have the same parameters. We derive a closed-form expression of the ratio \(C(\text {PNE}(G))/\text {OPT}(G)\). In particular, this expression implies that the PoA is unbounded also for homogeneous leaders. Finally, in Sect. 6.3, we briefly discuss other definitions for the social cost of a strategy profile, and consequences for the quality of equilibria.

6.1 Unboundedness of the PoA

The following theorem shows that PoA and PoS are unbounded for Stackelberg pricing games.

Theorem 6

\(\text {PoA}=\text {PoS}=\infty \). The bound is attained even for games with only two leaders.

Proof

Consider the Stackelberg pricing game \(G_M\) with \(n=2, \ a_1=\gamma _1=C_1=1, b_1=0\), and \(a_2=\gamma _2=C_2=M, b_2=0, \text { where } M\ge 1\).

By \(z_1=1,p_1=z_2=p_2=0\), we get a profile with social cost 2, thus OPT\((G_M)\le 2\). We now show \(C(\text {PNE}(G_M))>M\), which implies \(\text {PoA}=\text {PoS}\ge C(\text {PNE}(G_M))/\text {OPT}(G_M)>M/2\). By \(M\rightarrow \infty \), this yields the desired result. It remains to show \(C(\text {PNE}(G_M))>M\). For fixed \(M\ge 1\), let \(s=(z,p)\) be a PNE of \(G_M\) with induced Wardrop flow \(x:=x(s)\) and cost \(K:=K(s)\). Note that \(z_i>0\) holds for \(i\in \{1,2\}\), since any PNE has at least two positive capacities. Lemma 4 together with Theorem 2 yields that, for each leader \(i\in \{1,2\}\), the cost K either is optimal for \((\text {P}_i^1)(s_{-i})\), or optimal for \((\text {P}_i^2)(s_{-i})\), and has positive objective function value in both cases. Since for each leader \(i\in \{1,2\}\), the only candidate for a feasible solution of \((\text {P}_i^1)(s_{-i})\) is \(2\sqrt{a_i\gamma _i}+b_i=\sqrt{a_i\gamma _i}+b_i+C_i\) and this yields an objective function value of 0, we get that K is an optimal solution of \((\text {P}_i^2)(s_{-i})\). In particular this yields, by considering leader 2, that \(2M=\sqrt{a_2\gamma _2}+b_2+C_2<K\). Furthermore we get \(z_1=\frac{x_1}{K-1}\) and \(z_2=\frac{M(1-x_1)}{K-M}\) for the capacities. Altogether, the desired inequality for \(C(s)=C(\text {PNE}(G_M))\) follows: \(C(s) = \frac{1}{z_1}x_1^2+\frac{M}{z_2}(1-x_1)^2+z_1+Mz_2> (K-1)x_1+(K-M)(1-x_1) \ge K-M> M\). \(\square \)

Note that for \(M>1\), the two leaders in the proof of Theorem 6 do not have the same parameters. This raises the question whether the PoA is bounded if we restrict ourselves to homogeneous leaders. This question is analyzed in the next subsection.

6.2 Homogeneous leaders

In this subsection, we analyze the quality of equilibria for the case that the leaders are homogeneous, that is, there exist \(a>0,b\ge 0,C>0\) and \(\gamma >0\) such that \(a_i=a, b_i=b\), \(C_i=C\) and \(\gamma _i=\gamma \) for all \(i\in N\). We derive the following.

Theorem 7

Assume that G is a Stackelberg pricing game with n homogeneous leaders, i.e., there exist \(a>0,b\ge 0,C>0\) and \(\gamma >0\) such that \(a_i=a, b_i=b, C_i=C\) and \(\gamma _i=\gamma \) for all \(i\in N\). Then, the ratio between the equilibrium cost \(C(\text {PNE})\) and the optimal cost \(\text {OPT}\) of G is

$$\begin{aligned} \frac{C(\text {PNE})}{\text {OPT}}= {\left\{ \begin{array}{ll} 1, &{} \text { if } \frac{n}{n-1}\sqrt{a\gamma }<C, \\ \frac{\frac{n}{n-1}\cdot \frac{a\gamma }{C}+b+\frac{n-1}{n}C}{2\sqrt{a\gamma }+b}, &{} \text { if } \frac{n}{n-1}\sqrt{a\gamma }\ge C. \end{array}\right. } \end{aligned}$$

Note that this shows that even for homogeneous leaders, the PoA for Stackelberg pricing games is unbounded (consider \(C\rightarrow 0\)). On the other hand, if the caps are large enough, one can ensure an optimal equilibrium.

Proof

First, we analyze the social cost \(\text {OPT}\) of an optimal profile, and show \(\text {OPT}=2\sqrt{a\gamma }+b\). Note that given z with \(\sum _{i\in N}{z_i}>0\), the flow \(x^*\) defined by \(x_i^*:=z_i/\sum _{j\in N}{z_j}\) for all \(i\in N^+:=\{i\in N: z_i>0\}\), minimizes the total congestion cost \(\sum _{i\in N^+}{\ell _i(x_i,z_i)x_i}\): Consider

$$\begin{aligned} \min \quad \sum _{i\in N^+}{\left( ax_i/z_i+b\right) x_i}=a\cdot \sum _{i\in N^+}{x_i^2/z_i+b} \quad \text { s.t. } \quad \sum _{i\in N^+}{x_i}=1, \ x_i\ge 0 \ \forall i\in N^+. \end{aligned}$$

For this problem, the KKT conditions are necessary and sufficient for the unique optimal solution. Therefore, \(x^*\) is optimal if and only if there exist \(\lambda \in {\mathbb {R}}\) and \(\mu _i \in {\mathbb {R}}\) for all \(i\in N^+\), such that \(2ax_i^*/z_i+\lambda -\mu _i=0, \mu _ix_i^*=0, \text { and } \mu _i\ge 0\) hold for all \(i\in N^+\). Setting \(\lambda :=-2a/\sum _{j\in N}{z_j}\) and \(\mu _i:=0\) for all \(i\in N^+\), these conditions are fulfilled, and we conclude that \(x^*\) is optimal. Furthermore, we can induce \(x^*\) as Wardrop-flow, for example by \(p_i:=0\) for all \(i\in N\), since then, \(ax_i^*/z_i+b+p_i=a/\sum _{j\in N}{z_j}+b\) for all \(i\in N^+\). This shows that \(\text {OPT}\) equals the optimal objective function value of the problem

$$\begin{aligned} \min&\quad a/\sum _{j\in N}{z_j}+b+\gamma \cdot \sum _{j\in N}{z_j} \quad \text {s.t.} \quad \sum _{j\in N}{z_j}>0, \ z_j\ge 0 \ \forall j\in N. \end{aligned}$$

Since the objective function only depends on the sum of the capacities, we can also minimize \(a/\zeta +b+\gamma \zeta \) for \(\zeta >0\). The optimal solution of the latter problem is \(\zeta ^*=\sqrt{a/\gamma }\), and the optimal objective function value is \(2\sqrt{a\gamma }+b\). We have thus shown that \(\text {OPT}=2\sqrt{a\gamma }+b\).

We now analyze the cost \(C(\text {PNE})\). Note that if (zp) is a PNE, \(z_i>0\) holds for each player \(i\in N\) (this follows from the threshold Lemma 12). This shows that in case of homogeneous leaders, there is a unique PNE, and we denote this PNE with \((z^E,p^E)\). The uniqueness implies that \((z^E,p^E)\) is symmetric, that is, there exist \(\zeta >0,\rho \in (0,C]\) such that \(z_i^E=\zeta \) and \(p_i^E=\rho \) for all \(i\in N\). This immediately implies that \(x_i^E=1/n\) holds for all \(i\in N\), where \(x^E\) denotes the Wardrop flow induced by the PNE. Next, we exploit Lemma 8 to derive explicit formulas for \(\zeta \) and \(\rho \), leading to

$$\begin{aligned} (\zeta ,\rho )={\left\{ \begin{array}{ll} (\sqrt{a/\gamma }\cdot \frac{1}{n},\frac{n}{n-1}\cdot \sqrt{a\gamma }), &{} \text { if } \rho <C, \\ (\frac{n-1}{n^2}\cdot \frac{C}{\gamma },C), &{} \text { if } \rho =C. \end{array}\right. } \end{aligned}$$
(6)

We show below that \(\rho <C\) holds if and only if \(n/(n-1)\cdot \sqrt{a\gamma }< C\). Using this, we get

$$\begin{aligned} (\zeta ,\rho )&={\left\{ \begin{array}{ll} (\sqrt{a/\gamma }\cdot \frac{1}{n},\frac{n}{n-1}\cdot \sqrt{a\gamma }), &{} \text { if } \frac{n}{n-1}\cdot \sqrt{a\gamma }< C, \\ (\frac{n-1}{n^2}\cdot \frac{C}{\gamma },C), &{} \text { if } \frac{n}{n-1}\cdot \sqrt{a\gamma }\ge C,\end{array}\right. } \qquad \text {and} \\ \nonumber C(z^E,p^E)&=\frac{a}{\zeta n}+b+n\gamma \zeta ={\left\{ \begin{array}{ll} 2\sqrt{a\gamma }+b, &{} \text { if } \frac{n}{n-1}\cdot \sqrt{a\gamma }< C, \\ \frac{n}{n-1}\cdot \frac{a\gamma }{C}+b+\frac{n-1}{n}C, &{} \text { if } \frac{n}{n-1}\cdot \sqrt{a\gamma }\ge C. \end{array}\right. } \end{aligned}$$
(7)

Together with \(\text {OPT}=2\sqrt{a\gamma }+b\), this implies the theorem.

It remains to show that \(\rho<C \Leftrightarrow n/(n-1) \sqrt{a\gamma }< C\). Using (6), \(\rho <C\) clearly implies \(n/(n-1)\sqrt{a\gamma }< C\). We now assume that \(n/(n-1) \sqrt{a\gamma }< C\). Assume, by contradiction, that \(\rho =C\), and thus \(\zeta =\frac{n-1}{n^2}\cdot \frac{C}{\gamma }\) by (6). If \(K^E\) denotes the routing cost induced by \(x^E\), we get \(K^E=a/(\zeta n)+b+C=an\gamma /((n-1)C)+b+C<\sqrt{a\gamma }+b+C\). By our characterization of best responses, this shows that \(K^E\) is the optimal solution of \((\text {P}_i^1)(z_{-i},p_{-i})\), where \((z_j,p_j)=(\zeta ,\rho )\) holds for each \(j\in N \setminus i\), and thus \(\rho =K^E-\sqrt{a\gamma }-b<C\); contradiction. \(\square \)

At the end of this subsection, we briefly address the natural question if one can achieve a closed-form expression of the ratio between equilibrium and optimal cost, as in Theorem 7, also for heterogeneous leaders, where the parameters do not have to be equal. We found that at least for two leaders, this is possible. But the resulting expression is rather complicated and hardly insightful, thus we did not include it in this paper.

6.3 Different social cost functions

To complete our analysis of the quality of PNE, we now briefly discuss other definitions of the social cost of a strategy profile. Note that the function C(zp) defined at the beginning of this section measures utilitarian social welfare over followers and leaders. Alternatively, one could consider only the followers, or only the leaders. But as we show below, the worst-case quality of the equilibrium is also unbounded for these cases.

For a strategy profile \(s=(z,p)\) with \(\sum _{i\in N}{z_i}>0\), the total cost of the followers equals K(s), the cost of the induced Wardrop flow. (If \(\sum _{i\in N}{z_i}=0\), i.e., there is no resource to choose for the followers, we define, as before, that the cost is \(\infty \).) But for this alternative definition of the cost of a strategy profile, the optimal cost is not attained, and may even tend to zero: Obviously, \(K(s)> b_{\min }:=\min \{b_i: i\in N\}\) holds for each profile s. On the other hand, the cost of the profile \(s_{M}\) defined by \(z_i=M, p_i=0\) for some leader i with \(b_i=b_{\min }\), and \(z_j=p_j=0\) else, tends to \(b_{\min }\) for \(M \rightarrow \infty \). In other words, in a “near-optimal” profile, the congestion effects are extinguished by very high capacities. But naturally, such capacities also induce high investment costs. To impose an upper bound on the total investment cost (and thus on the capacities) for the optimal profile does not only seem natural, but would also imply that the optimal cost is attained, and is strictly positive.Footnote 7 Assume, e.g., that there is a budget parameter \(B>0\) such that \(\sum _{i\in N}{\gamma _iz_i}\le B\) needs to hold for an optimal profile. With this restriction, we now analyze the equilibrium quality, and again get that PoA and PoS are unbounded: To see this, consider the family of instances \((G_M)_{M\ge 1}\) given in the proof of Theorem 6, and assume that \(B=1\). It follows from the proof of Theorem 6 that the equilibrium cost is larger than 2M, whereas the optimal cost is at most 1, leading to a lower bound of 2M for the PoA. By \(M\rightarrow \infty \), we get that PoA and PoS are unbounded.Footnote 8

Alternatively, one could measure the quality of a strategy profile via the total profits of the leaders. Here, it is again the case that an optimal profile (maximizing the sum of the leaders’ profits) does not exist: The total profit is always strictly smaller than \(C_{\max }:=\max \{C_i: i\in N\}\), but for \(z_i=\varepsilon , p_i=C_i\) for some leader i with \(C_i=C_{\max }\), and \(z_j=p_j=0\) else, we get a total profit arbitrarily near to \(C_{\max }\) if \(\varepsilon \) tends to 0. Irrespective of whether we define OPT as \(C_{\max }\), or restrict the optimal profile for example by some lower bound \(\beta \) on the sum of capacities \(\sum _{i\in N}{z_i}\ge \beta \), the worst-case quality of the PNE is unbounded: To see this, consider (7), where we state the equilibrium strategies for the case that the leaders are homogeneous. Assume that we have a homogeneous instance with \(\sqrt{a\gamma }\ge C\). The total profit of the PNE then is \(C-(n-1)/n\cdot C=C/n\). As n gets large, this clearly tends to zero. On the other hand, the optimal total profit (C or \(C-\beta \gamma \)) is independent of n, and strictly positive (for \(\beta < C/\gamma \)). Altogether, this shows that if we consider the total profits of the leaders, the quality of the PNE may also get arbitrarily bad.