Journal of Advanced Transportation

Volume 2017 (2017), Article ID 5062984, 9 pages

https://doi.org/10.1155/2017/5062984

## Numerical Bounds on the Price of Anarchy

^{1}Industrial Engineering Department, Diego Portales University, Santiago, Chile^{2}Department of Transport Engineering and Logistics, Pontificia Universidad Católica de Chile, Santiago, Chile

Correspondence should be addressed to Louis de Grange

Received 7 June 2017; Accepted 27 August 2017; Published 17 October 2017

Academic Editor: Sara Moridpour

Copyright © 2017 Louis de Grange et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Theoretical upper bounds for price of anarchy have been calculated in previous studies. We present an empirical analysis for the price of anarchy for congested transportation networks; different network sizes and demand levels are considered for each network. We obtain a maximum price of anarchy for the cases studied, which is notably lower than the theoretical bounds reported in the literature. This result should be carefully considered in the design and implementation of road pricing mechanisms for cities.

#### 1. Introduction

The “price of anarchy” [1, 2], within the context of the traffic balance of transportation networks [3], is a concept related to the percentage or relative difference of the total costs in a network obtained from a traffic assignment equilibrium (i.e., Wardrop’s first principle, [4]) with respect to the total costs in a network obtained from an optimal system assignment (i.e., Wardrop’s second principle).

In this article, we present an empirical analysis of the price of anarchy for congested transportation networks: three networks of different types and sizes and different levels of demand for each network are considered. For the three cases analyzed, we find that the maximum price of anarchy never exceeds 9%, which is significantly lower than the theoretical bounds reported in the literature. The price of anarchy tends to zero even for hypercongested networks. This result suggests that the potential benefits from an optimal road pricing scheme can be quite small within the context of a traffic assignment with fixed demand: this result should be carefully considered in the design and implementation of road pricing mechanisms for cities.

We obtain a second interesting result that the price of anarchy (which represents the relative benefit of an optimal system assignment with respect to a balanced user assignment) depends strongly on the difference between the fixed costs and the variable costs of the arcs (or routes) of the network. The price of anarchy tends to zero for cases in which the fixed cost represents nearly 100% or 0% of the cost of the routes. That is, for high and low levels of congestion, the price of anarchy tends to take on increasingly smaller values. This second result may be relevant in the design of transportation networks and their optimal use.

The analyses were performed using the Bureau of Public Roads (BPR) volume-delay functions for strategic road networks in three cities of various sizes: Santiago de Chile, Chicago, and Anaheim. The equilibrium based on Wardrop’s first and second principles was solved using the Origin-Based Assignment (OBA) traffic algorithm described by Bar-Gera [5] to ensure proper convergence of the equilibrium flows and trip times in the tests. This approach was considered also in Shi et al. [6] but including link-capacitated traffic assignment problem.

These results are important considerations for the design and implementation of road pricing mechanisms. In many cases of road pricing, a similar equilibrium is sought to that obtained based on Wardrop’s second principle (i.e., first-best pricing). Thus, if an optimal assignment of the system produces very few benefits (i.e., a low price of anarchy), it follows that first-best pricing will also generate few benefits. However, if a change in demand results (instead of only a change in the routes used by travelers within the congested network), for example, by a transfer to other transportation modes or by alternative trip schedules, the upper bounds on the price of anarchy would be higher, indicating that a first-best policy has more potential benefits.

In Section 2, we present an analytical definition of the price of anarchy, a literature review on this subject, and the central hypothesis of our research. In Section 3, we present the methodology used for and the results obtained from this research, as well as two extensions for simple networks. Finally, in Section 4, we present the main findings and discuss future extensions of this work.

#### 2. Definition of the Price of Anarchy, Literature Review, and Research Hypothesis

##### 2.1. Price of Anarchy in Transportation Networks

The concept of the “price of anarchy” was first introduced by Koutsoupias and Papadimitriou [1] and was later formally defined by Papadimitriou [2]; both of these definitions were presented within the context of computer and telecommunications networks, which become congested with use.

The concept of the price of anarchy has also been applied to transportation issues [7–12], resource allocation to broadband internet [13, 14], the design of communication networks [15, 16], supply chain management [17], and divisible goods allocation [18].

The price of anarchy in congested transportation networks [3, 7, 19] refers to the loss of efficiency, which is measured as the increase in cost for travelers who are assigned to the arcs of a congested transportation network when the travelers minimize their individual costs (i.e., a behavior criterion based on Wardrop’s first principle) with respect to the behavior that minimizes the sum of the costs for all of the individuals traveling in the network (i.e., a behavior criterion based on Wardrop’s second principle).

For a network with linear arc cost functions, Roughgarden and Tardos [7] showed that the price of anarchy has an upper bound of 4/3. That is, there is a maximum increase of 33.33% in the welfare (which is measured as the total time spent in the network) of an optimal assignment of the system traffic with respect to an optimal assignment of users. Christodoulou et al. [20] found a bound below 4/3 using coordination mechanisms for linear cost functions. Chen and Zhang [21] also obtained a bound below 4/3 for the price of anarchy using linear costs in resource allocation in telecommunications by considering incentives for agents who released information to the network.

However, the loss of efficiency (i.e., the price of anarchy) may depend on several factors such as the size and topology of the network (i.e., the number of zones, nodes, and arcs), the level of demand, the capacity of the arcs, and the type of arc cost functions. Yang et al. [22] have discussed the price of anarchy for different families of cost functions for road network arcs (i.e., linear, BPR, quadratic, or cubic functions) and have extended the analysis to the case of variable demand. The authors stated that the bounds on the price of anarchy for a variable demand depend on the parameters of the demand function; that is, the price of anarchy can grow indefinitely. For a variable demand (e.g., changes in the total number of trips, the means of transport, or the trip schedule), the additional costs for the user equilibrium with respect to the optimum balance of the system could be much higher than that for a fixed trip matrix.

Gairing et al. [23] analyzed the bounds on the price of anarchy for general polynomial functions. Cole et al. [24] showed that, for linear cost functions (and some other specific network topologies), an optimal road pricing based on marginal costs reduces costs with respect to Wardrop’s (Nash) equilibrium in the same way that eliminating some links from the network does.

Nagurney and Qiang [25] used the price of anarchy to analyze the robustness of a road transport network when the capacity of the network roads decreases over time because of the absence of maintenance. Yin and Lawphongpanich [26] applied the price of anarchy concept to the costs of polluting emissions in transportation networks. Karakostas et al. [27] applied the price of anarchy concept to a traffic assignment model in which some of the travelers did not make decisions based on the trip time but on other information, such as the minimum trip distance obtained from maps. Han and Yang [28] consider the effect of the so-called second-best tolls on the price of anarchy of the traffic equilibrium problem, where there are multiple classes of users with a discrete set of values of time. Han et al. [29] analyze the efficiency of road space rationing schemes by establishing the bounds of the reduction in the system cost associated with the restricted flow pattern at user equilibrium in comparison with the system cost at the original user equilibrium. Liu et al. [30] study the existence and efficiency of oligopoly equilibrium under simultaneous toll and capacity competition in a parallel-link network subject to congestion. They show that the bounds are demand-function-free and are only dependent upon the number of competitive roads. Xu et al. [31] analyze how ridesharing impacts traffic congestion; the model is built by combining a ridesharing market model with a classic elastic demand Wardrop traffic equilibrium model. Their results show that the ridesharing base price influences the congestion level, and the utilization of ridesharing increases as the congestion increases.

Although the price of anarchy has spawned an entire line of theoretical research, few empirical studies have been conducted. Levinson [32] reported that, in the Twin Cities, Minneapolis-Saint Paul, the additional costs from congestion generated by the traffic equilibrium for the morning rush hour (7:30–8:30 hours) were only 1.7%, despite the fact that more than 630,000 vehicles were in circulation during that period. Previously, Youn et al. [33] estimated that the price of anarchy for Boston/Cambridge, London, and New York road networks ranged between 3% and 4% on average and decreased as the number of agents (travelers) increased.

Therefore, we undertake an empirical study of the price of anarchy attained for different scenarios. As we will see later, the price of anarchy is directly related to several exogenous factors associated with the topology of the road network and the demand levels.

##### 2.2. Analytic Definition of the Network and the Price of Anarchy

A road network can be defined as a graph , where represents the set of nodes (including areas or centroids) and represents the set of arcs of the network. We also define as the set of origin-destination pairs within the network. We use the following notations: denotes the set of existing routes between the pair *, * denotes the demand (e.g., the number of trips per hour), and denotes the flow of the route, where . We characterize the arcs by defining as the cost of arc , which depends on its flow (i.e., the cost function increases monotonically with the flow). The flow is related to via , where is unity if route* p *passes through arc and is zero otherwise.

For a traffic equilibrium based on Wardrop’s first principle, which is also known as user equilibrium (UE), the flow of equilibrium on arc can be defined as , such that the cost of equilibrium of users on arc* a *is . The total cost of the system under user equilibrium can be written as follows:

Similarly, for a traffic equilibrium based on Wardrop’s second principle, which is also known as an optimum system (OS), the flow of equilibrium on arc can be defined as , and the respective cost of arc is . The total cost of the system under an optimum equilibrium of the system can be written as follows:

The ratio between the expressions given by (1) and (2) can be defined as follows:

Within the context of transportation networks, the term in (3) is known as the price of anarchy, where clearly and by construction . Let us define , where represents the percentage loss in efficiency of the optimum assignment of users with respect to the optimal system assignment. Both and are used to refer to the price of anarchy.

##### 2.3. Research Hypothesis

Considering (2) and (3), we note that (and therefore ) for the specific cases given as follows:(i)The arc cost functions are constant; that is, they do not depend on their flow ; therefore, the assignments based on Waldrop’s first and second principles are equal; that is, there is no congestion.(ii)The arc cost functions are homogeneous with degree ; that is, is satisfied. The assignments based on Waldrop’s first and second principles are also equal in this case [34, 35]. The previous result follows easily from changing variables in the objective function of Beckman et al.’s [36] classic traffic equilibrium problem, which yields the following result: Therefore, when the arc cost functions are homogeneous with degree , we can directly derive from (5) that the assignments based on Wardrop’s first and second principle are the same.

Next, let us consider the typical BPR arc cost functions:

The term in (6) represents the time or cost at free flow in arc *,* and the term , where and , represents the cost increase in arc caused by the congestion generated by the flow . The term is the fixed capacity of arc .

Considering that , , and are exogenous in (6), it is easy to see that if the parameter is the same for all of the arcs in the network, the variable part of the cost of each arc will be a homogeneous function of degree .

Thus, we present the following hypothesis for cases (i) and (ii).

*Hypothesis*. If the variable component of the trip cost is very small compared to the fixed cost, the price of anarchy approaches unity. That is,

Similarly, if the variable component of the trip cost of the arcs is very large with respect to the fixed cost, the price of anarchy approaches one. That is,

In simple terms, the hypothesis represented by (7) and (8) is that if the congestion is too low or too high (hypercongestion), the price of anarchy approaches unity; therefore, the loss of relative efficiency between the assignments based on Wardrop’s first and second principles approaches a vanishingly small value.

The hypothesis is represented graphically in Figure 1.