موجودیت و بهره وری تعادل انحصار چند جانبه تحت تلفات و ظرفیت رقابت

This paper studies the existence and efficiency of oligopoly equilibrium under simultaneous toll and capacity competition in a parallel-link network subject to congestion. We establish sufficient conditions for the existence of a pure-strategy oligopoly equilibrium and then derive upper bounds on the efficiency loss of the oligopoly equilibrium over the social optimum under different inverse demand function assumptions, respectively. Furthermore, we show that these bounds are demand-function free and only dependent upon the number of competitive roads

With the increasing of private involvement in highway supply, several asymmetric toll roads serving for one origin–destination pair have been often seen all over the world, especially, in many developing countries (Roth, 1996, De Palma and Lindsey, 2002 and Yang and Woo, 2000). For private operators of these competitive roads, to benefit financially from their participation involves two fundamental decision variables, i.e., the toll charge and the road capacity (Verhoef and Rouwendal, 2004 and De Borger and Van Dender, 2005). The charged toll not only partly determines the total revenue that the private operator can obtain, but also to a certain extent reduces the number of road users by internalizing the congestion externality among them. The larger capacity can reduce the congestion level of a road and then increase the toll revenue due to more road usage, but it inevitably increases the construction cost of the road. As a result, the charged toll and the selected capacity both affect the private operator’s profit either positively or negatively. In this paper, we study a simultaneous toll and capacity game among multiple private road operators in the aforementioned parallel-competitive environment with elastic demand. More specifically, under the beliefs about the others’ decisions, each profit-maximizing operator is assumed to simultaneously determine its toll charge and road capacity subject to the resulting traffic flow pattern in equilibrium. Given the charged tolls and the selected capacities, which road a traveler chooses to travel is determined by the generalized travel cost, i.e., the sum of the charged toll and the experienced travel time cost. Briefly speaking, our study’s objective is to establish sufficient conditions for the existence of a pure-strategy oligopoly equilibrium, and then to provide upper bounds on the efficiency loss from the interaction between selfish users and profit-maximizing operators when competition substitutes for government regulation. This paper is related to the literature in transportation science on quantifying the inefficiency of selfish travel behavior (Roughgarden, 2003, Roughgarden and Tardos, 2002, Correa et al., 2004, Correa et al., 2008, Guo et al., 2010, Liu et al., 2009 and Yang et al., 2010). The term “price of anarchy”, first introduced by Koutsoupias and Papadimitriou (1999), has been often coined to characterize the degree of inefficiency, which is the worst-possible ratio between the total system cost under the user equilibrium (UE) flow pattern and that under the system optimum (SO) flow pattern. Readers may refer to Roughgarden, 2005 and Yang and Huang, 2005 for recent reviews. Besides that resulting from selfish travel behavior, the profit-driven efficiency loss under oligopoly competition will be also considered in this paper. There are a few studies focused on toll competition among congested markets for investigating the inefficiency resulting from both selfish routing and oligopoly competition. Considering the case of inelastic demand, Acemoglu and Ozdaglar (2007) showed that a change in the market structure from monopoly to duopoly in a two-link network typically increase the inefficiency, and a pure-strategy equilibrium always exists when travel cost functions are linear. For their simplified case, they characterized a tight bound of 6/5 on inefficiency in pure-strategy equilibria with zero latency at zero flow and a tight bound of View the MathML source1/(22-2) with nonzero latency at zero flow. In contrast, Ozdaglar, 2008, Hayrapetyan et al., 2007 and Engel et al., 2004 studied toll competition among profit-maximizing oligopolists with elastic demand. All of them discussed the existence of a pure-strategy equilibrium under the assumption that the demand function is concave. Besides, Hayrapetyan et al. (2007) also provided non-tight upper bounds on the inefficiency, in particular, arrived at 5.064 when linear cost functions and concave demand functions are considered. However, under a similar environment, Ozdaglar (2008) found that the tighter bound on the inefficiency only has 1.5. Engel et al. (2004) pointed out that the increase in competition can improve the efficiency even if demand is kept proportional to the network size. Xiao et al. (2007a) examined the inefficiency of toll competition with general inverse demand functions, but did not explicitly discuss the existence of oligopoly equilibrium. In their work, the parameterized general bounds on the inefficiency turn out to be more complicated and dependent upon the specific property of the inverse demand function. Recently, Acemoglu et al., 2009 and Xiao et al., 2007b studied the efficiency of oligopoly equilibrium in a toll and capacity competition game. In particular, Acemoglu et al. (2009) considered a two-stage game with fixed total demand and capacity constraints, where the operators first choose road capacities and then set toll charges. Comparably, Xiao et al. (2007b) indeed analyzed a one-shot game where operators compete by simultaneously choosing tolls and capacities. Xiao et al. (2007b) found that if a pure-strategy oligopoly equilibrium exist, the inefficiency of simultaneous toll and capacity competition can be upper bounded by a parameterized expression in relation to the ratio of the realized demand at oligopoly equilibrium over that at social optimum. However, they did not make a detailed analysis of the existence of the oligopoly equilibrium. In addition, their observations, that the inefficiency bound declines with the number of competitive roads, are mainly made for symmetric roads with the exponential and linear demand types. Distinguished from previous work, the main contributions in this paper focus on the following two aspects. First, a pure-strategy oligopoly equilibrium may not exist in a general one-shot game. To obtain sufficient conditions for its existence, we do not assume the concavity of inverse demand function, B(V), a restrictive assumption often made in the related literature, e.g., Engel et al., 2004 and Johari et al., 2010. In contrast, we show that if VB(V) is a concave function of the total number of users, V, and all link travel time cost functions are linear, a pure-strategy equilibrium exists. The concave VB(V) assumption captures some widely used demand functions besides the concave types, such as exponential ones ( Guo and Yang, 2009 and Tan et al., 2010). In other words, it holds for most common cumulative distribution functions representing users’ willingness-to-pay for travel (see Remark in Section 5), e.g., normal, uniform, exponential, and Pareto distributions. Many of them are convex, or concave–convex. Second, based on the concave assumption of VB(V) and considering its three special cases, i.e., that inverse demand functions are concave, log-concave or have the increasing elasticity, respectively, we derive the upper bounds on the efficiency loss of oligopoly equilibrium. We also show that these bounds are demand-function free and only dependent upon the number of competitive roads. This paper is organized as follows. Section 2 introduces some notation and assumptions used throughout this paper. In Sections 3 and 4, we present the oligopoly equilibrium and social optimum models, respectively. Section 5 bounds the efficiency loss of oligopoly equilibrium. Section 6 concludes the paper.

The contributions of this paper to the literature are summarized as follows. First, we presented an equilibrium analysis of oligopolistic competition in a congested network with parallel roads, in which operators compete for traffic by simultaneous toll and capacity choices. For assuring the existence of pure-strategy oligopoly equilibrium, we established sufficient conditions based on most common marginal benefit functions. In particular, our study extended the conventional concave function assumption to the more general situation by assuming the concavity of VB(V). Second, we provided the upper bounds on the efficiency loss of the oligopoly equilibrium compared with the social optimum under the concave VB(V) assumption and its three different subclasses about marginal benefit functions, i.e., concavity, log-concavity, and the increasing elasticity. Two conclusions can be clearly made by analyzing our bounding results. First, all the upper bounds only depend on the number of competitive toll roads, and thus they are very useful for the ex-ante estimation of the social welfare loss in a completely private road market. Second, as pointed out in a tolling game (Engel et al., 2004), the increases in simultaneous toll and capacity competition also improve the efficiency of equilibrium market. In addition, it has to be mentioned that, our bounding results in the simultaneous toll and capacity game critically rely on some simplified but very important assumptions about travel time costs and road construction costs, which might result in the outcome of identical average social cost per trip (inclusive of the investment cost) at the oligopoly equilibrium and social optimum (Tan et al., 2010). Hence, it can be expected to relax these assumptions when bounding the efficiency loss.