1 Introduction
We consider a model for oligopolistic capacity and price competition in service markets exhibiting congestion effects. There is a set of firms offering a service to customers and the overall customer utility depends on the congestion level experienced and the price charged. In addition to setting a price, a firm can invest in installing capacity to reduce congestion. Firms compete against each other in the market by simultaneously choosing capacity and price levels while customers react subsequently choosing the most attractive firms (in terms of congestion and price levels). This model captures many aspects of realistic oligopolistic markets such as road pricing in traffic networks (see Xiao, Yang and Han [30] and Zhang and Han [32]) and competition among WIFIproviders or cloud computing platforms (see Acemoglu and Ozdaglar [2] and Anselmi et al. [4]).
In a landmark paper, Johari, Weintraub and Van Roy [21] (JWVR for short) studied the fundamental question of existence, uniqueness and worstcase quality of pure Nash equilibria for this model. They derived several existence and uniqueness results under fairly general assumptions on the functional form of the congestion functions. Specifically, they showed that for models with elastic demand, a unique pure Nash equilibrium exists. The existence result is based on the Kakutani fixedpoint theorem (see [22] and the generalizations of Fan and Glicksberg [17, 19]) and crucially exploits the assumption that demand is elastic leading to nonemptyness of the bestresponse correspondences of firms. For the case of inelastic demand
, that is, there is a fixed volume of customers requesting service, much less is known in terms of existence and uniqueness of equilibria. Many works in the transportation science and algorithmic game theory community (see, e.g.,
[7, 12, 13, 14, 31, 18] and [28, 8, 3, 16], respectively) assume inelastic demand and this case is usually considered as fundamental base case. As we show in this paper, in terms of equilibrium existence, the case of inelastic demand is much more complicated compared to the seemingly more general case of elastic demands: the bestresponse correspondence may be empty in the “inelastic case” putting standard approaches out of reach. Besides this theoretical aspect, the case of inelastic demand is also interesting from a practical point of view as it applies to realistic situations (such as road pricing games), where customers have already made investments on a longer timescale (like buying a car) so that they are not willing to opt out of the game.1.1 Related Work
JWVR [21] derived several existence and uniqueness results for pure Nash equilibria assuming elastic demand of customers. The elasticity is modeled by a differentiable strictly decreasing inverse demand function. Depending on the generality of the allowed congestion cost functions, further concavity assumptions on the inverse demand function are imposed by JWVR. The existence proof is based on Kakutani’s fixedpoint theorem which crucially exploits (together with further convexity and compactness properties) that bestresponse correspondences of the firms are nonempty.
For the model with inelastic demand – that we consider in this paper – JWVR derive an existence result assuming homogeneous firms, that is, the cost function of every firm is equal. As shown by JWVR, homogeneity implies that there is only one symmetric equilibrium candidate profile. For this specific symmetric strategy profile, they directly prove stability using concavity arguments of the bestreply correspondences. This proof technique is clearly not applicable in the general nonhomogeneous case. As already mentioned, a further difficulty stems from the fact that one cannot directly apply standard fixedpoint arguments as even a best response of a firm might not exist.
Acemoglu, Bimpikis and Ozdaglar [1] study a capacity and price competition game assuming that the capacities represent “hard” capacities bounding the admissible customer volume for a firm. They observe that pure Nash equilibria do not exist and subsequently study a twostage model, where in the first stage firms determine capacities and in a second stage they set prices. For this model, they investigate the existence and worstcase efficiency of equilibria (see also Kreps and Scheinkman [23] for earlier work on the twostage model). Further models in which capacities are determined centrally in order to reduce the travel cost of the resulting Wardrop equilibria have been considered in the (bilevel) optimization community for some time already, see [11, 24]. In a very recent work [29], Schmand, Skopalik and Schröder study existence and inefficiency of equilibria in a network investment game in which the firms invest in edges of a seriesparallel graph (but do not directly set prices).
1.2 Our Results and Proof Techniques
We study in this paper oligopolistic capacity and price competition assuming that there is a fixed population of customers requesting service. This assumption applies to road pricing games, where customers usually own a car and are not willing to opt out of the game given that prices range in realistically bounded domains. Consequently, our model allows for the possibility of a priori upper bounds with respect to the prices set by the firms. This situation appears naturally, if there are legislative regulations imposing a hard price cap in the market (see Correa et al. [9] and Harks et al. [20] and further references therein). As reported in Correa et al. [9], even different price caps for different firms is current practice in the highway market of Santiago de Chile, where 12 different operators set tolls on different urban highways each with a unique price cap. While firmspecific price caps generalize the model of JWVR, we impose, on the other hand, more restrictive assumptions on the congestion functions, that is, they are assumed to be linear with respect to the volume of customers and inverse linear with respect to the installed capacity.
As our first result, we completely characterize the structure of bestresponse correspondences of firms; including the possibility of nonexistence of bestreplies (Theorem 3.6). Our second main result then establishes the existence of equilibria (Theorem 4.6). For the proof we use the concept of security introduced by McLennan, Monteiro and Tourky [25], which in turn resembles ideas of Reny [27]. A strategy profile is called secure, 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) neighbourhood. 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 main result of McLennan, Monteiro and Tourky [25] states that a game admits an equilibrium, if every nonequilibrium profile is secure. It is important to note that the concept of security does not rely on quasiconcavity of utility functions nor on their continuity. With our characterization of bestresponse correspondences at hand, we show that the capacity and price game with inelastic demand, linear congestion functions and price caps is secure and, thus, admits pure Nash equilibria. As our third main result, we show that there is a unique equilibrium (Theorem 5.7). The proof technique follows the general approach as in JWVR [21], however, since our model allows for price caps, some new ideas are required. We finally study the worst case efficiency of the unique equilibrium compared to a natural benchmark, in which we relax the equilibrium conditions of the firms, but not the equilibrium conditions of the customers. We show that the unique equilibrium might be arbitrarily inefficient (Theorem 6.1).
2 Model
There is a set , , of firms offering a service to customers. Customers are represented by the continuum (each consumer is assumed to be infinitesimally small and represented by a number in ) and we denote by the standard simplex of assignments of customers to firms.^{1}^{1}1All results hold for arbitrary intervals by a standard scaling argument. The effective quality of the service of firm offered to a customer depends on two key factors: the level of congestion and the price charged by the firm. The congestion function depends on the volume of customers and the service capacity installed. Clearly grows with the volume of customers to be served but decreases with the service capacity . If no capacity is installed, i.e. , we assume infinite congestion, and for the case that , we assume that congestion depends linearly on the volume of customers and inverselinearly on the installed capacity, that is,
where and are given parameters for . Note that the case can be interpreted as if firm is just opting out of the market and does not offer the service at all. Each firm additionally decides on a price which is charged for offering service to its customers, where
is a given price cap. For a capacity vector
with , i.e. the service is offered by some firms, and a price vector , customers choose rationally the most attractive service in terms of the effective costs, that is, congestion and price experienced. This is expressed by the Wardrop equilibrium conditions:holds for all with . Note that for given capacities and prices , there is exactly one satisfying the Wardrop equilibrium conditions (see, e.g., [10]). Call this flow the Wardrop flow induced by . In particular, there exists a constant such that holds for each with , and holds for each with . For a Wardrop flow , we call the corresponding constant the (routing) cost of .
If we denote the perunit cost of installed capacity by , the profit function of a firm can now be represented as
We assume that each firm seeks to maximize her own profit. For each firm , let be the strategy set. A vector consisting of strategies is called a strategy profile, and denotes the set of strategy profiles. We will usually write a strategy profile in the form , where denotes the vector consisting of all capacities for , and is the vector of prices for . The profit of firm for a strategy profile is then defined as . Furthermore, we write for the Wardropflow induced by and for the routing cost of . For firm , denote by the vector consisting of strategies . We then write for the strategy profile where firm chooses , and the other firms choose . Moreover, we use the simplified notation and . A strategy profile is a pure Nash equilibrium (PNE) of the thus defined capacity and price competition game, if for each firm :
For given strategies of the other firms, the bestresponse correspondence of firm is defined as
If is clear from the context, we just write .
We conclude this section with the following fundamental result about the continuity of the profit functions. We will use Theorem 2.1, which completely characterizes the strategy profiles having the property that all profit functions , , are continuous in , several times during the rest of the paper.
Theorem 2.1.
Let . Then: All profit functions , , are continuous in if and only if or .
Proof.
We start with the strategy profile and show that for each firm , her profit function is continuous in . Let . For , define and let with . If , then . Otherwise,
holds, showing that is continuous in .
Now consider with . We again need to show that all profit functions , , are continuous in . Since , we get that . Furthermore, for sufficiently small, holds for all with . Write , where denotes the strategies of the firms in , and denotes the strategies of the firms in .
Now let . We need to show that is continuous in . For all with , firm ’s profit is . Thus it is sufficient to show that is continuous in . The idea of the proof is to show that, for a slightly perturbed strategy profile , the difference between and , as well as the difference between and , is small.
In the following, let with . It is well known ([5]) that is the unique optimal solution of the following optimization problem :
Furthermore, for . Therefore, the values are the unique optimal solution of
which is equivalent to
By Berge’s theorem of the maximum [6], for all there is such that for all with . That is, is continuous in if we only allow changes in , but not in . Furthermore, if denotes the optimal objective function value of , and if only changes in are allowed, also is continuous in , i.e. for all there is such that for all with .
We now distinguish between the two cases that or .
First consider the case , that is, , and let . Note that , thus we need to find such that for all with . To this end, define
and let with . In particular, . Furthermore, holds since . If , we immediately get . Thus assume and assume, by contradiction, that . Then, by definition of , we get the following contradiction:
Therefore, holds, showing that is continuous in if .
Now consider the case , i.e. . For , we need to find such that for all with . To this end, define
and let with . In particular, , thus . Furthermore, since , we get for all . If for all , we get and thus , as desired. Otherwise, there is with . In particular, . We now use a result about the sensitivity of Wardrop equilibria due to Englert et al. [15, Theorem 2]. They show that if firm is deleted from the game, the resulting change in the Wardrop flow can be bounded by the flow that received. More formally, if is the Wardrop flow for the game with firms , and is the Wardrop flow if firm is deleted from the game, then for all . Obviously, changing firm ’s capacity from to has the same effect on the Wardrop flow as deleting firm . Therefore, if we change, one after another, the capacities of all firms having to , we get (note that the flow values for can always be always upperbounded by ):
Using this, we now get the desired inequality:
Altogether we showed that all profit functions are continuous at if .
To complete the proof, it remains to show that if all profit functions , , are continuous in , then or holds. We show this by contraposititon, thus assume that fulfills and . We need to show that there is a firm such that is not continuous in . To this end, let with . Define the sequence of strategy profiles by for all , and . Obviously, for . But for the profits, we get
Since , this shows that is not continuous in . ∎
3 Characterization of Best Responses
The aim of this section is to derive a complete characterization of the bestresponse correspondences of the firms. We will make use of this characterization in all our main results, i.e. existence, uniqueness and quality of PNE.
Given a firm and fixed strategies for the other firms, we characterize the set of best responses of firm to . To this end, we distinguish between the two cases that (Subsection 3.1) and (Subsection 3.2). Subsection 3.3 then contains the derived complete characterization. In Subsection 3.4, we discuss how our results about the best responses influence the applicability of Kakutani’s fixed point theorem.
3.1 The Case
In this subsection, assume that the strategies of the other firms fulfill . Under this assumption, firm does not have a best response to :
Lemma 3.1.
.
Proof.
Whenever firm chooses a strategy with , then holds for the induced Wardropflow , thus firm ’s profit is . On the other hand, any strategy with yields a profit of 0. Thus, firm ’s profit depends solely on her own strategy , and can be stated as follows:
Obviously, holds for each , i.e. for and . On the other hand, by for , firm gets a profit of arbitrarily near to , that is, . This shows . ∎
3.2 The Case
In this subsection, assume that the strategies of the other firms fulfill . For a strategy of firm , write for firm ’s profit function, for the Wardropflow induced by and for the corresponding routing cost.
For , firm ’s profit is . It is clear that each strategy with yields , and thus . On the other hand, each strategy with yields negative profit since
Therefore, each best response fulfills since it yields nonnegative profit. Thus, can be described as the set of optimal solutions of the following optimization problem :
has an optimal solution, since the feasible set of is compact and is continuous in (see Theorem 2.1), and thus the theorem of Weierstrass can be applied. Since can be described as the set of optimal solutions of , we get .
Note that is a bilevel optimization problem (since can be described as the optimal solution of a minimization problem ([5])), and these problems are known to be notoriously hard to solve.
The characterization of that we derive here has the advantage that it only uses ordinary optimization problems, namely the following two (1dimensional) optimization problems in the variable ,
where
with and .
Note that is a continuous function which is equal to 1 for , and strictly decreasing for with . Therefore, there is a unique constant with the property . Obviously, if and only if .
Furthermore, the function is closely related to Wardropflows, as described in the following lemma:
Lemma 3.2.

[1.]

If with , then for .

If with , and fulfills and , then and .
Proof.
We start with 1, so let such that . By definition of , we get for all with , and for all with . Since
we get that . Therefore, for each , we get , which is equivalent to . Using yields
as desired. Now we show 2, so let with and let be a strategy with and . Consider the vector defined by
It is clear that holds for all with , and . Furthermore, the definition of yields . We now show that fulfills the Wardrop equilibrium conditions. The uniqueness of the Wardropflow then implies ; and
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