When is selfish routing bad? The price of anarchy in light and heavy traffic
Riccardo Colini-Baldeschi, Roberto Cominetti, Panayotis Mertikopoulos, Marco Scarsini
WWHEN IS SELFISH ROUTING BAD?THE PRICE OF ANARCHY IN LIGHT AND HEAVY TRAFFIC
RICCARDO COLINI-BALDESCHI ∗ , ROBERTO COMINETTI ‡ ,PANAYOTIS MERTIKOPOULOS § , AND MARCO SCARSINI ¶ Abstract.
This paper examines the behavior of the price of anarchy as afunction of the traffic inflow in nonatomic congestion games with multipleorigin-destination (O/D) pairs. Empirical studies in real-world networks showthat the price of anarchy is close to in both light and heavy traffic, thusraising the question: can these observations be justified theoretically? Wefirst show that this is not always the case: the price of anarchy may remain apositive distance away from for all values of the traffic inflow, even in simplethree-link networks with a single O/D pair and smooth, convex costs. On theother hand, for a large class of cost functions (including all polynomials), theprice of anarchy does converge to in both heavy and light traffic, irrespectiveof the network topology and the number of O/D pairs in the network. Wealso examine the rate of convergence of the price of anarchy, and we showthat it follows a power law whose degree can be computed explicitly when thenetwork’s cost functions are polynomials. “ Traffic congestion is caused by vehicles, not by people in themselves. ”— Jane Jacobs,
The Death and Life of Great American Cities ∗ Core Data Science Group, Facebook Inc., 1 Rathbone Place, London, W1T 1FB,UK. ‡ Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile. § Univ. Grenoble Alpes, CNRS, Inria, LIG, F-38000 Grenoble, France. ¶ Dipartimento di Economia e Finanza, LUISS, Viale Romania 32, 00197 Roma,Italy.
E-mail addresses : [email protected], [email protected],[email protected], [email protected] .2010 Mathematics Subject Classification.
Primary 91A13; secondary 91A43, 91A80.
Key words and phrases.
Nonatomic congestion games; price of anarchy; light traffic; heavytraffic; regular variation.We thank an anonymous reviewer of an earlier conference version of this paper for suggestingone of the examples with variable inflows in Section 6.This research benefited from the support of the FMJH Program PGMO under grantHEAVY.NET and from the support of EDF, Thales, and Orange. R. Colini-Baldeschi andM. Scarsini are members of GNAMPA-INdAM. R. Cominetti and P. Mertikopoulos gratefullyacknowledge the support and hospitality of LUISS during a visit in which this research was ini-tiated. R. Cominetti’s research is also supported by FONDECYT 1130564 and Núcleo MilenioICM/FIC RC130003 “
Información y Coordinación en Redes .” P. Mertikopoulos was partiallysupported by the ECOS/CONICYT Grant C15E03 and the Huawei HIRP Flagship project UL-TRON. P. Mertikopoulos and M. Scarsini also gratefully acknowledge the support and hospitalityof FONDECYT 1130564 and Núcleo Milenio “
Información y Coordinación en Redes .” a r X i v : . [ c s . G T ] A p r R. COLINI-BALDESCHI, R. COMINETTI, P. MERTIKOPOULOS, AND M. SCARSINI
1. Introduction
Almost every commuter in a major metropolitan area has experienced the frus-tration of being stuck in traffic. At best, this might mean being late for dinner; atworst, it means more accidents and altercations, not to mention the vastly increaseddamage to the environment caused by huge numbers of idling engines.To name but an infamous example, China’s G110 traffic jam in August 2010brought to a standstill thousands of vehicles for 100 kilometers between Hebei andInner Mongolia. The snarl-up lasted twelve days and resulted in drivers beingunable to move for more than 1 kilometer per day, reportedly spending up to fivedays trapped in the jam. Not caused by weather or a natural disaster, this massive -day tie-up was instead laid at the feet of a bevy of trucks swarming on theshortest route to Beijing, thus clogging the highway to a halt (while ironicallycarrying supplies for construction work to ease congestion). This, therefore, raisesthe question: how much better would things have been if all traffic had been routed bya social planner who could calculate ( and enforce ) the optimum traffic assignment? In game-theoretic terms, this question boils down to the inefficiency of Nashequilibria that are not Pareto optimal. The most widely used quantitative measureof this inefficiency is the so-called price of anarchy (PoA): introduced by Koutsou-pias and Papadimitriou (1999) and so dubbed by Papadimitriou (2001), the price ofanarchy is simply the ratio of the social cost of the least efficient Nash equilibriumdivided by the minimum achievable social cost. By virtue of this straightforwarddefinition, deriving worst-case bounds for the price of anarchy has given rise to avigorous literature at the interface of operations research, economics and computerscience, often leading to surprising and counter-intuitive results.In the context of network congestion, Pigou (1920) was probably the first tonote the inefficiency of selfish routing, and his elementary two-road example witha PoA of / is one of the two prototypical examples thereof. The other exampleis due to Braess (1968), and consists of a four-road network where the additionof a zero-cost segment makes things just as bad as in the Pigou case. These twoexamples were the starting point for Roughgarden and Tardos (2002) who showedthat the price of anarchy in (nonatomic) routing games with affine costs may notexceed / . On the other hand, if the network’s cost functions are polynomials ofdegree at most d , the price of anarchy may become as high as Θ( d/ log d ) , implyingthat selfish routing can become arbitrarily bad in networks with polynomial costs(Roughgarden, 2003).By this token, and given the typically nonlinear relation between traffic loads andtravel times, the intervention of a central planner seems necessary in order to regainsome degree of efficiency. At the same time however, these worst-case instances aretypically realized in networks with delicately tuned traffic loads and costs: if anetwork operates beyond this regime, it is not clear whether the price of anarchyremains high. In view of this, our aim in this paper is to examine the asymptoticbehavior of the price of anarchy at both ends of the congestion spectrum: light andheavy traffic .Using both analytical and numerical methods, a very recent study by O’Hareet al. (2016) suggests that the price of anarchy is usually close to for both highand low traffic, and it fluctuates in the intermediate regime (typically exhibitingmultiple local maxima). In a similar setting, Monnot et al. (2017) used a hugedataset on commuting students in Singapore to estimate the so-called “stress of HEN IS SELFISH ROUTING BAD? 3
O Dc ( x ) = [1 + 1 / x )] x c ( x ) = x c ( x ) = [1 + 1 / x )] x ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ �� - � ��� � �� ����������������������� ����� ( � ) � � � ����� �� ������� �� � �������� �� ������ ����� Figure 1.
A network where selfish routing remains inefficient for bothlight and heavy traffic. catastrophe”: this majorant of the ordinary price of anarchy was estimated toa value between . and . , suggesting that the actual value of the price ofanarchy in Singapore is lower (and definitely below the / worst-case bound of thePigou/Braess examples).All this leads to the following natural questions: a ) Under what conditions does the price of anarchy converge to in light/heavytraffic? b ) Do these conditions depend on the network topology, its cost functions, orboth? c ) Can general results be obtained for networks with multiple origin-destination(O/D) pairs? d ) When these conditions are satisfied, how fast is this convergence?1.1. Our contributions.
Our first result is a cautionary tale: we show that the priceof anarchy may oscillate between two bounds strictly greater than for all valuesof the traffic inflow, even in simple parallel-link networks with a single O/D pair(cf. Fig. 1). The cost functions in our example are convex and differentiable, soneither convexity nor smoothness seem to play a major role in the efficiency ofselfish routing. Moreover, our construction only involves a three-link network, sosuch phenomena may arise in any network containing such a three-link component.Heuristically, the reason for this irregular – and, perhaps, counter-intuitive –behavior is that the growth rate of the network’s cost functions exhibits higher-order oscillations which persist at any scale, in both light and heavy traffic. Todispense with such pathologies, we focus on networks whose cost functions c e ( x ) are asymptotically comparable to a benchmark function c ( x ) which is itself assumedto be regularly varying (cf. Definition 4.1). In so doing, we obtain a classificationof the network’s edges, paths, and O/D pairs as fast , slow or tight relative to thechosen benchmark. Then, thanks to this classification, we obtain the followinggeneral result: If the routing cost of the “most costly” O/D pair in the networkbehaves like the benchmark, the network’s price of anarchy converges to in bothlight and heavy traffic. Polynomial cost functions satisfy all of the above requirements, leading to thecomprehensive asymptotic principle:
In networks with polynomial costs,the price of anarchy becomes under both light and heavy traffic. R. COLINI-BALDESCHI, R. COMINETTI, P. MERTIKOPOULOS, AND M. SCARSINI
In other words, a benevolent social planner with full control of traffic assignmentwould not do any better than selfish agents in conditions of high or low congestion.In particular, only if the traffic falls in an intermediate regime can there be asubstantial gap between optimum and equilibrium states.To assess how wide this intermediate regime might be in practice, we also ex-amine the speed at which the price of anarchy converges to as a function of thetraffic inflow. Specializing to networks with polynomial costs, we find that in bothregimes the convergence follows a power law with respect to the total traffic inflow,and we derive explicit sharp estimates for the corresponding rates.1.2. Related work.
Establishing worst-case bounds for the price of anarchy underdifferent conditions has been a staple of the literature on congestion games eversince the seminal result of Roughgarden and Tardos (2002). In words, this resultstates that in networks with affine costs the price of anarchy is no higher than / , independently of the network topology and/or the number of O/D pairs in thenetwork. Furthermore, this bound is sharp in that, for every M > , there existsa network with traffic inflow M and affine costs such that the price of anarchyis exactly equal to / . Importantly, our analysis shows that the order of thequantifiers in the above statement cannot be exchanged: in any network with affinecosts, the price of anarchy gets arbitrarily close to if the traffic inflow is sufficientlylarge or small.Worst-case bounds for the price of anarchy have been obtained for larger classesof cost functions. For polynomial costs with degree at most d , Roughgarden (2003)showed that the worst possible instance grows as Θ( d/ log d ) while Dumrauf andGairing (2006) provided sharper bounds for monomials of maximum degree d andminimum degree q . Extending the above results, Roughgarden and Tardos (2004)provided a unifying result for costs that are differentiable with xc ( x ) convex, whileCorrea et al. (2004, 2008) considered less regular classes of cost functions. Correaet al. (2007) also studied the price of anarchy when the goal is to minimize themaximum – rather than the average – latency in the network. For a survey, thereader is referred to Roughgarden (2007).In a more practical setting, Youn et al. (2008) studied the difference betweenoptimal and actual system performance in real transportation networks, focusingin particular on Boston’s road network. They observed that the price of anarchydepends crucially on the total traffic inflow: it starts at , it then grows withsome oscillations, and ultimately returns to as the flow increases. González Vayáet al. (2015) studied optimal scheduling for the electricity demand of a fleet ofplug-in electric vehicles: without using the term, they showed that the price ofanarchy goes to as the number of vehicles grows. Cole and Tao (2016) showedthat in large Walrasian auctions and in large Fisher markets the price of anarchygoes to one as the market size increases. Finally, Feldman et al. (2016) took adifferent asymptotic approach and considered atomic games where the number ofplayers grows to infinity. Applying the notion of ( λ, µ ) -smoothness to the resultingsequence of atomic games, they showed that the price of anarchy converges to thecorresponding nonatomic limit.From an analytic standpoint, the closest antecedent to our paper is the recentwork of Colini-Baldeschi et al. (2016) who studied the heavy traffic limit of the priceof anarchy in parallel networks with a single O/D pair. Their analysis identifiedthat regular variation plays an important role in this setting; however, it offered HEN IS SELFISH ROUTING BAD? 5 no insights into non-parallel networks with multiple O/D pairs or the light trafficregime. Our paper provides an in-depth answer to these questions: we show that( a ) regular variation yields asymptotic efficiency under both light and heavy trafficconditions; ( b ) the topology of the network doesn’t matter; and ( c ) the existenceof several O/D pairs doesn’t matter as long as they admit a common benchmark(which is always the case if the network’s cost functions are polynomial).Building on a previous unpublished version of the present paper, Wu et al. (2017)introduced a class of congestion games, called scalable , whose price of anarchyconverges to as the total demand diverges. They also computed the rate ofconvergence of the price of anarchy for the special case of BPR cost functions ofthe same degree. Stidham (2014) also studied the behavior of the price of anarchyfor queueing networks in heavy traffic under various assumptions on the structureof the network and on the stochastic properties of the queues. His results can beused for the analysis of network routing models with capacity constraints.Our work should also be compared to that of Monnot et al. (2017) who per-formed an empirical study of the price of anarchy based on data from thousands ofcommuting students in Singapore. Focusing on the network’s “stress of catastrophe”(an empirical majorant of the network’s price of anarchy), they showed that routingchoices are near-optimal and the incurred price of anarchy is much lower than whattraditional worst-case bounds suggest. Interestingly, the study of Monnot et al.(2017) also suggests that the Singapore road network is often lightly congested: assuch, their results can be seen as a practical validation of the light traffic resultspresented here (and, conversely, our results provide a theoretical justification fortheir empirical observations).1.3. Outline of the paper.
The paper is organized as follows. In Section 2, weintroduce the basic model and concepts that will be used in the rest of the paper.Section 3 provides two motivating examples for the analysis to follow. In Section 4,we treat networks with a single O/D pair, whereas Section 5 examines networkswith multiple O/D pairs. The more complicated case of variable relative inflowsis treated in Section 6. Finally, in Section 7, we study the rate of convergence ofthe price of anarchy in light and heavy traffic. To streamline our presentation, theproofs of our main results have been relegated to a series of appendices at the endof the paper.
2. Model and preliminaries
Network model.
Following Beckmann et al. (1956) and Roughgarden and Tar-dos (2002), the basic component of our model will be a directed multi-graph
G ≡ G ( V , E ) with vertex set V and edge set E (both finite). We further assumethat there is a finite set of origin-destination (O/D) pairs indexed by i ∈ I , eachwith an individual traffic demand m i ≥ that is to be routed from the pair’s originnode O i ∈ V to its destination D i ∈ V . To route this traffic, the i -th O/D pairemploys a set P i of paths joining O i to D i , with each path p ∈ P i comprisinga sequence of edges that meet head-to-tail in the usual way; specifically, we donot assume that P i is necessarily the set of all paths joining O i to D i , but onlysome subset thereof. [This distinction is particularly relevant for packet-switchednetworks (such as the Internet) where only paths with a low hop count are typi-cally employed.] For bookkeeping reasons, we will also make the following standingassumptions throughout our paper: R. COLINI-BALDESCHI, R. COMINETTI, P. MERTIKOPOULOS, AND M. SCARSINI a ) The total inflow rate M = (cid:80) i ∈I m i is positive (so there is a nonzero amountof traffic in the network). b ) The path sets P i are disjoint (which in particular holds trivially if all pairs ( O i , D i ) are distinct).Now, writing P ≡ (cid:83) i ∈I P i for the union of all such paths, the set of feasible routing flows f = ( f p ) p ∈P in the network is defined as F = (cid:110) f ∈ R P + : (cid:80) p ∈P i f p = m i for all i ∈ I (cid:111) . (2.1)In turn, a routing flow f ∈ F induces a load on each edge e ∈ E as x e = (cid:88) p (cid:51) e f p , (2.2)and we write x = ( x e ) e ∈E for the corresponding load profile on the network.Given all this, the delay (or latency) experienced by an infinitesimal traffic ele-ment traversing edge e is determined by a nondecreasing continuous cost function c e : [0 , ∞ ) → [0 , ∞ ) : more precisely, if x = ( x e ) e ∈E is the load profile induced bya feasible routing flow f = ( f p ) p ∈P , the incurred delay on edge e ∈ E is c e ( x e ) .Hence, with a slight abuse of notation, the associated cost of path p ∈ P will begiven by c p ( f ) = (cid:88) e ∈ p c e ( x e ) . (2.3)Putting together all of the above, the tuple Γ = ( G , I , { m i } i ∈I , {P i } i ∈I , { c e } e ∈E ) will be referred to as a ( nonatomic ) routing game . When we want to explicitly keeptrack of the total inflow rate M = (cid:80) i m i , we write Γ M instead of Γ ; also, whenthere is a single O/D pair, we will drop all reference to i and I altogether.2.2. Equilibrium, optimality, and the price of anarchy.
In routing games, the no-tion of Nash equilibrium is captured by Wardrop’s first principle: at equilibrium,the delays along utilized paths are equal and no higher than those that would be expe-rienced by an infinitesimal traffic element going through an unused route (Wardrop,1952).Formally, a routing flow f ∗ is said to be a Wardrop equilibrium (WE) of Γ if, forall i ∈ I , we have c p ( f ∗ ) ≤ c p (cid:48) ( f ∗ ) for all p, p (cid:48) ∈ P i such that f ∗ p > . (2.4)From the work of Beckmann et al. (1956), it is known that Wardrop equilibriacoincide with the solutions of the (convex) minimization problemminimize (cid:88) e ∈E C e ( x e ) , subject to x e = (cid:88) p (cid:51) e f p , f ∈ F , (WE)where C e ( x e ) = (cid:82) x e c e ( w ) dw denotes the primitive of c e . Analogously, sociallyoptimum (SO) flows are defined as solutions of the total cost minimization problemminimize (cid:88) p ∈P f p c p ( f p ) , subject to f ∈ F . (SO) HEN IS SELFISH ROUTING BAD? 7
By a simple rearrangement of terms, the objective function of (SO) can berewritten as L ( x ) = (cid:80) e ∈E x e c e ( x e ) , so the value of the above problem can beexpressed equivalently (and somewhat more concisely) as Opt(Γ) = min x ∈X L ( x ) , (2.5)where X = { x ∈ R E + : x e = (cid:80) p (cid:51) e f p , f ∈ F} denotes the set of all load profiles ofthe form (2.2). Thus, to quantify the gap between solutions to (WE) and (SO), let Eq(Γ) = L ( x ∗ ) (2.6)where x ∗ is the load profile induced by a Wardrop equilibrium f ∗ of Γ (by a standardresult of Beckmann et al. (1956), all such flows incur the same total cost). Thegame’s price of anarchy (PoA) is then defined as PoA(Γ) = Eq(Γ)Opt(Γ) . (2.7)For this ratio to be well-defined, we must have Opt(Γ) > ; otherwise, if this is notthe case, we will vacuously set PoA(Γ) = 1 . To avoid such technicalities, we willtacitly assume that
Opt(Γ) > throughout.Of course, PoA(Γ) ≥ with equality if and only if Wardrop equilibria are alsosocially efficient. Our main objective in what follows will be to study the asymptoticbehavior of this ratio when M → or M → ∞ .
3. First results
Sioux Falls: a representative case study.
To motivate our analysis, we beginby examining the behavior of the price of anarchy in the road network of SiouxFalls, a standard case study in the transportation literature. For concreteness, thenetwork’s (two-way) arterial roads are shown in Fig. 2(a) and their delay functionsare taken to be of the BPR (Bureau of Public Roads) type c e ( x ) = a e + b e x d e (3.1)with coefficients a e , b e and degrees d e (typically d e = 4 ) taken from the standardreference work of LeBlanc et al. (1975, Table 1). To analyze the network’s price ofanarchy as a function of the total traffic inflow, we considered all O/D pairswith nonzero inflow, and for each O/D pair i ∈ I , we restricted P i to containonly the five shortest paths in terms of free-flow travel time (i.e., the time taken totraverse a path when empty). We then scaled up or down these inflows preservingthe ratios between different O/D pairs and we plotted the network’s price of anarchyfor various values of the total inflow M .As can be seen from Fig. 2(b), the network’s price of anarchy is identically equalto when the total inflow is small enough (approximately up to . × trips perhour); for intermediate values of M , the price of anarchy becomes strictly greaterthan , and, ultimately, it decreases monotonically to in the heavy traffic limit.Interestingly, LeBlanc et al. (1975) report a value of M avg ≈ . × trips/hourfor the network’s median traffic inflow; this value is well within the range wherethe price of anarchy decreases monotonically to and, indeed, the observed value isapproximately equal to . , indicating a . difference between socially optimumand equilibrium flows under median traffic conditions.Similar conclusions have been drawn in the literature from empirical studies inLondon, New York and Boston (Youn et al., 2008), as well as Sioux Falls with R. COLINI-BALDESCHI, R. COMINETTI, P. MERTIKOPOULOS, AND M. SCARSINI �� ���� ��� ������ �������� ���� �� �������� �� (a) The Sioux Falls road network. ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ � � � � � � ������������������������������� ����� ������ [ �� � ����� / ���� ] � � � ����� �� ������� �� � �������� �� ������� ������ (b) Anarchy and efficiency in Sioux Falls. Figure 2.
The price of anarchy in the Sioux Falls metropolitan area asa function of the traffic inflow. different subsets of O/D pairs and connecting paths per pair (O’Hare et al., 2016).In particular, in all cases, it was observed that there is an initial interval of valuesof M for which the price of anarchy is identically equal to ; our first result showsthat this is not mere happenstance: Proposition 3.1.
If the network’s cost functions are of the form (3.1) with a e , b e > and d e ≡ d for all e ∈ E , we have PoA(Γ M ) = 1 for all sufficiently small M . In fact, as the following result shows, this behavior arises whenever each O/Dpair admits a single “best” path under zero inflow:
Proposition 3.2.
Let P i min = arg min p ∈P i c p (0) denote the set of minimal cost pathsof the i -th O/D pair under zero inflow. If P i min is a singleton for all i ∈ I , we have PoA(Γ M ) = 1 for all sufficiently small M . The above results (both proven in Appendix A) provide a reasonable theoreticaljustification for the light traffic behavior of the price of anarchy that is observedin Fig. 2 (the heavy traffic limit is discussed in detail in the next sections). Atthe same time however, the BPR and “unique best path” assumptions in Proposi-tions 3.1 and 3.2 respectively suggest that there is a finer mechanism at play whichbecomes apparent when the total cost at low traffic depends more delicately on thedistribution of traffic in the network. We make this precise in the following sectionwhere we provide an example of a three-link network where the price of anarchyoscillates between two values strictly greater than , for all values of the trafficinflow.3.2. A network where selfish routing is always inefficient.
To construct an exampleof an “always inefficient” network, our approach will be to take a network witha certain degree of periodicity, obtain an explicit handle for its price of anarchyover a compact interval, and then tessellate this behavior over the entire trafficspectrum (0 , ∞ ) . To carry this out, let Γ M be a nonatomic routing game consistingof a single O/D pair with traffic inflow M . This traffic is to be routed over thethree-link parallel graph of Fig. 1 with cost functions c ( x ) = x d (cid:2) sin(log x ) (cid:3) , (3.2a) c ( x ) = x d , (3.2b) HEN IS SELFISH ROUTING BAD? 9 c ( x ) = x d (cid:2) cos(log x ) (cid:3) , (3.2c)where d is a positive integer. It is easy to see that these cost functions are increasing,strictly convex and smooth on [0 , ∞ ) for all d ≥ , and they all grow as Θ( x d ) atboth traffic limits ( x → and x → ∞ ). Furthermore, the functions x e c e ( x e ) are strictly convex, so the optimum traffic allocation problem (SO) admits a uniquesolution. Hence, the only way for the game’s price of anarchy to be equal to iswhen the game’s (also unique) Wardrop equilibrium coincides with the network’ssocially optimum flow.As we show in Appendix A, the equations determining the network’s equilibriumand optimum flows never admit a common solution, so the price of anarchy is strictlygreater than over any compact interval (showing in this way that the conclusion ofPropositions 3.1 and 3.2 already fails for this example). Moreover, the trigonometricterms in (3.2) imply that these equations are periodic in a logarithmic scale (i.e.,in log M ). Hence, combining these two properties, we obtain: Proposition 3.3.
In the three-link parallel network defined above,
PoA(Γ M ) is peri-odic in log M and oscillates between two values strictly greater than . Colini-Baldeschi et al. (2016) already provided examples of networks where lim sup M →∞ PoA(Γ M ) > but the cost functions involved were fairly irregularand the lim inf of the price of anarchy was still (i.e., selfish routing was still ef-ficient infinitely often). By contrast, in the above example, the price of anarchyis bounded away from for all possible demands, and this despite the fact thatthe network’s cost functions are smooth, strictly convex and grow as Θ( x d ) at bothends of the congestion spectrum. This is a considerable sharpening of the exampleof Colini-Baldeschi et al. (2016) as it shows that there are cases where efficiency is never achieved at equilibrium – not even asymptotically.
4. Networks with a single O/D pair
Despite the highly smooth and convex structure of the example network of Propo-sition 3.3, closer inspection reveals that the growth rate of its cost functions exhibitspersistent oscillations at both and ∞ . This naturally leads to the following ques-tion: Does selfish routing remain bad for “reasonable” cost functions that do notbehave irregularly in the limit?
To quantify – and discard – such irregularities, we will employ the seminal notionof regular variation (recalled below). For clarity and concision, we will focus fornow on networks with a single O/D pair; the case of multiple O/D pairs will bediscussed in detail later, in Sections 5 and 6.4.1.
Regular variation and edge classification.
To present a unified perspective, wewill tackle both ends of the congestion spectrum simultaneously by introducing the traffic limit indicator ω ∈ { , ∞} : letting M → ω gives the light traffic limit for ω = 0 and the heavy traffic limit for ω = ∞ . Regular variation at either limit isthen defined as follows: Definition 4.1.
A function g : (0 , ∞ ) → (0 , ∞ ) is said to be regularly varying at ω if lim t → ω g ( tx ) g ( t ) is finite and nonzero for all x > . (4.1) In words, regular variation means that g ( t ) grows at the same rate when viewed atdifferent scales (determined here by x ). The concept itself dates back to Karamata(1930, 1933) and has been used extensively in functional analysis, probability, andlarge deviations theory (see e.g., de Haan and Ferreira, 2006; Jessen and Mikosch,2006; Resnick, 2007); for a comprehensive survey we refer the reader to Binghamet al. (1989).Standard examples of regularly varying functions include all affine functions,polynomials, logarithms, and, more generally, all analytic functions (barring thosewith an essential singularity at ω = ∞ ). [Recall here that a function g ( x ) is analyticon a domain U if it is equal to its Taylor series on U .] On the other hand, despitebeing bounded from above and below as Θ( x d ) , the oscillatory cost functions (3.2)used in the counterexample of Section 3 are not regularly varying. Indeed, at either ω = 0 or ω = ∞ , the limit lim t → ω c ( tx ) c ( t ) = lim t → ω sin(log t + log x )1 + sin(log t ) x d (4.2)does not exist in (0 , ∞ ) unless log x = kπ for some k ∈ Z (and likewise for c ). Inthis way, regular variation provides a much finer view than polynomial growth.With all this at hand, we will dispose of growth irregularities like the above bypositing that the network’s cost functions can be compared asymptotically to someregularly varying function c ( x ) . Specifically, given an ensemble of cost functions C = { c e } e ∈E , we will say that a regularly varying function c : (0 , ∞ ) → (0 , ∞ ) is a benchmark for C at ω if the following (possibly infinite) limit exists for all e ∈ E α e = lim x → ω c e ( x ) c ( x ) . (4.3)This limit will be called the index of edge e at ω , and e will be called fast , slow ,or tight (relative to c at ω ) if α e is respectively , ∞ , or in-between. In particular,when e is tight, c e ( x ) is also regularly varying and exhibits the same asymptoticbehavior as the benchmark function c ( x ) at ω ; if e is fast, then c e ( x ) = o ( c ( x )) ; and,finally, if e is slow, then c ( x ) = o ( c e ( x )) . As such, a benchmark function groupsthe network’s edges into three equivalence classes that exhibit the same qualitativebehavior with respect to c ( x ) .Of course, this partition depends on the chosen benchmark and the traffic limit(light or heavy): for instance, x is fast with respect to x at , but it is slow at ∞ . For concision, we will not keep track of this dependence explicitly and insteadrely on the context to resolve any ambiguities. However, it will be important tokeep in mind that the classification of fast and slow edges could be flipped whentransitioning from heavy to light traffic and vice versa.Now, since bottlenecks along a path are caused by its slowest edges, we alsodefine the index of a path p ∈ P as α p = max e ∈ p α e , (4.4)and we say that p is fast , slow , or tight based on whether α p is , ∞ , or in-between.Finally, given that traffic will tend to be routed along the fastest paths in thenetwork, we define the index of the network as α = min p ∈P α p , (4.5) HEN IS SELFISH ROUTING BAD? 11 and we say that the network is itself tight if < α < ∞ . In words, a path is fast(resp. tight/slow) if its slowest edge is fast (resp. tight/slow), and a network istight if its fastest path is tight.Defined this way, tightness guarantees that the network admits a path whosecost behaves asymptotically as a (positive) multiple of the benchmark function c ( x ) .The importance of this requirement is again illustrated by the cost model (3.2) ofthe previous section: if we only assumed that the network admits a path whosecost behaves as Θ( c ( x )) , then we would not be able to rule out the pathologicaloscillations of the example in Section 3.4.2. The light traffic limit.
Thanks to the above legwork, we are in a position tostate our main result for lightly congested networks with a single O/D pair:
Theorem 4.2.
Let Γ M be a nonatomic routing game with a single O/D pair. If thenetwork is tight under light traffic ( ω = 0 ) , then lim M → PoA(Γ M ) = 1 . (4.6)In words, Theorem 4.2 simply states that if the cost of the network’s fastestpath is regularly varying at , selfish routing becomes efficient in light traffic. Tostreamline our presentation, Theorem 4.2 is proved in Appendix B as a special caseof a much more general statement. Here, we focus on some immediate corollariesthereof: Corollary 4.3.
Suppose that, for every edge e ∈ E , the limit lim x → c e ( x ) /x q e isfinite and nonzero for some q e ≥ . Then, PoA(Γ M ) → as M → .Proof. Referring to q e as the order of e , define the order of a path p ∈ P as q p =min e ∈ p q e and that of the network as q = max p ∈P q p . Clearly, lim x → c e ( x ) /x q = 0 ifand only if q e > q ; lim x → c e ( x ) /x q = ∞ if and only if q e < q ; and lim x → c e ( x ) /x q ∈ (0 , ∞ ) if and only if q e = q . This shows that the network is tight with respect to c ( x ) = x q at , so Theorem 4.2 applies. (cid:3) Corollary 4.4.
In a single O/D-network with analytic costs we have
PoA(Γ M ) → as M → .Proof. If c e ( x ) = (cid:80) ∞ k =0 c k,e x k for small enough x , we have lim x →∞ c e ( x ) /x q e ∈ (0 , ∞ ) for q e = min { k ∈ N : c k,e (cid:54) = 0 } . Our claim then follows from Corollary 4.3. (cid:3) Corollary 4.5.
In a single O/D-network with polynomial costs we have
PoA(Γ M ) → as M → . Of the above results, Corollaries 4.4 and 4.5 are of special practical interest be-cause most latency models that have been proposed in the literature are polynomialor analytic at . In urban networks, the golden standard is the Bureau of PublicRoads (BPR) quartic model c e ( x ) = a e + b e x , while basically all of the establishedqueueing models used in the theory of packet-switched networks ( M/M/ , M/G/k , M/M/c , etc.) are analytic at (Bertsekas and Gallager, 1992).Despite appearances, the very wide applicability of Theorem 4.2 and its corollar-ies is fairly surprising. Indeed, at first sight, one would expect that when M → ,traffic is so light that it doesn’t really matter how it is routed. This is indeed thecase if, for instance, all paths in the network exhibit different positive costs when M = 0 (cf. Proposition 3.2). However, if the cost of an empty path is zero, this is no longer the case: the optimum and equilibrium traffic assignments could be fairlydifferent (even when the network is lightly congested), so there is no a priori reasonfor the price of anarchy to converge to as M → (the example of Section 3 clearlyillustrates this phenomenon). Theorem 4.2 shows that all that is needed for this tooccur is for the network’s cost functions to be faithfully represented by a commonbenchmark function: when this condition is met, optimum and equilibrium costsno longer fluctuate but, instead, they converge to the same value.4.3. The heavy traffic limit.
Our main result for highly congested networks with asingle O/D pair is as follows:
Theorem 4.6.
Let Γ M be a nonatomic routing game with a single O/D pair. If thenetwork is tight under heavy traffic ( ω = ∞ ) , then lim M →∞ PoA(Γ M ) = 1 . (4.7)In words, Theorem 4.6 simply states that if the cost of the network’s fastestpath is regularly varying at ∞ , selfish routing becomes efficient in heavy traffic.To compare and contrast the light and heavy traffic regimes, we relegate the proofof Theorem 4.6 to Appendix B and only focus here on some immediate corollariesthereof: Corollary 4.7.
Suppose there exists a path p ∈ P with bounded costs, that is, lim x →∞ c e ( x ) < ∞ for all e ∈ p . Then, PoA(Γ M ) → as M → ∞ .Proof. Taking c ( x ) = 1 , we get α e = lim x →∞ c e ( x ) ∈ (0 , ∞ ] for all e ∈ E . Byassumption, there exists a path such that < α p < ∞ , so Theorem 4.6 applies. (cid:3) Corollary 4.8.
Suppose that the limit lim x →∞ c e ( x ) /x q e is finite and nonzero forsome q e ≥ and all e ∈ E . Then, PoA(Γ M ) → as M → ∞ .Proof. Shadowing the proof of Corollary 4.3, let q p = max e ∈ p q e and q = min p ∈P q p (but note the reversal of the max and min operators). Clearly, lim x →∞ c e ( x ) /x q =0 if and only if q e < q ; lim x →∞ c e ( x ) /x q = ∞ if and only if q e > q ; finally, lim x →∞ c e ( x ) /x q ∈ (0 , ∞ ) if and only if q e = q . This shows that the network istight with respect to c ( x ) = x q at ∞ , so Theorem 4.6 applies. (cid:3) Corollary 4.9.
In a single O/D-network with polynomial costs we have
PoA(Γ M ) → as M → ∞ . In a certain, precise sense, Theorems 4.2 and 4.6 show that the high and lowcongestion regimes can be seen as different sides of the same coin. By excludingpathological oscillations at either end of the congestion spectrum, regular variationensures asymptotic regularity and guarantees that selfish routing becomes efficientin the limit: specifically, tightness at guarantees efficiency in light traffic whiletightness at ∞ guarantees efficiency in heavy traffic. By this token, taking Corol-laries 4.5 and 4.9 in tandem implies that selfish routing becomes efficient underboth light and heavy traffic in networks with polynomial costs and a single O/Dpair.That being said, there are still important, quantitative differences between thelight and heavy traffic limits. For instance, even though Corollaries 4.8 and 4.9are direct analogues of their light traffic counterparts, the conclusion of Corol-lary 4.7 is false in light traffic (the three-link network of Section 3 serves again asa counterexample). In fact, even in the case of polynomial costs (Corollary 4.5 vs. HEN IS SELFISH ROUTING BAD? 13
Corollary 4.9), there is an important reversal of roles that takes place between fastand slow edges. Specifically, edges that are fast in light traffic typically becomeslow under heavy traffic and vice versa (importantly however, tight edges are notre-classified under this regime change). Nevertheless, despite this reversal, the priceof anarchy still goes to in both cases.
5. Networks with multiple O/D pairs
We now extend our analysis to networks with multiple O/D pairs. In this case,if the inflow rate of the i -th O/D pair is m i , the total traffic inflow in the networkis given by M = (cid:88) i ∈I m i , (5.1)and we write λ i = m i M (5.2)for the relative inflow rate of the i -th O/D pair – i.e., the fraction of the total trafficgenerated by the pair in question. In the rest of this section, we will assume thatthe relative inflow of every O/D pair i ∈ I is a fixed positive constant that doesnot depend on M ; the case of variable inflow rates will be discussed in detail inSection 6.The key difference with the single-pair setting is that routing costs for differentO/D pairs may exhibit completely different asymptotic behaviors in the limit. As aresult, in the presence of multiple O/D pairs, the definition of the network’s index(and the related notion of tightness) must be re-examined. To do so, given thatthe traffic generated by an O/D pair will tend to be routed along the pair’s fastestpath, we first define the index of an O/D pair i ∈ I as α i = min p ∈P i α p . (5.3)Just like edges and paths, this index can be used to classify O/D pairs as fast , slow or tight depending on whether α i is respectively , ∞ , or in-between. The index ofthe network is then defined as α = max i ∈I α i , (5.4)and we say that the network is tight if < α < ∞ . Heuristically, this definitionsimply captures the fact that the leading contribution to congestion is due to the“costliest” O/D pairs in the network; obviously, if there is but a single O/D pair,this last point is moot and (5.4) reduces to (4.5). Example . To illustrate the above concepts, consider a Wheatstone network withtwo O/D pairs and cost functions as in Fig. 3. Focusing first on the heavy trafficlimit, the benchmark c ( x ) = x would classify edge as tight, edges and as slow,and edges and as fast. Accordingly, the first O/D pair would be classified astight while the second O/D pair would be classified as fast; since no pair is slowand at least one pair is not fast, the network is itself tight.In the light traffic limit, the same benchmark would classify edges and astight, edges and as slow, and edge as fast. Under this classification, the firstO/D pair would again be tight, but the second O/D pair would now be classified asslow (because all its paths contain a slow edge), so the network would no longer betight. A moment’s reflection shows that the reason for this is that the benchmark O D O D c ( x ) = xc ( x ) = x c ( x ) = log(1 + x ) c ( x ) = 1 + √ xc ( x ) = e x Figure 3.
A Wheatstone network with two O/D pairs (cf. Example 5.1below). In heavy traffic, the network is tight relative to the benchmarkfunction c ( x ) = x ; in light traffic, the network is tight relative to thebenchmark c ( x ) = 1 . function c ( x ) = x is not well-suited for the second O/D pair. Instead, if we takethe benchmark c ( x ) = 1 , the first O/D pair would be classified as fast (because ithas a fast path, namely edge ) and the second pair would be classified as tight,so the network would now be tight. For a systematized version of this benchmarkselection procedure, see the proof of Corollary 5.2 below.With all this at hand, our next result states that if the costliest O/D pair in thenetwork admits a tight path, selfish routing becomes asymptotically efficient in thelimit: Theorem 5.1.
Let Γ M be a nonatomic routing game. If the network is tight in thelimit as M → ω , then lim M → ω PoA(Γ M ) = 1 . (5.5)In words, if ( a ) every O/D pair has a path which is not slow, and ( b ) the fastestpath of the slowest O/D pair has a regularly varying cost, selfish routing becomesefficient in the limit. Motivated by the strong connection between Theorems 4.2and 4.6, Theorem 5.1 has been stated in a way that does not discriminate betweenthe light and heavy traffic regimes. The reason for this is to highlight the role of thetightness assumption: tightness at guarantees efficiency in light traffic ( ω = 0 )while tightness at ∞ guarantees efficiency in heavy traffic ( ω = ∞ ).Of course, both Theorems 4.2 and 4.6 follow as corollaries of Theorem 5.1 bytaking respectively ω = 0 or ∞ and specializing to a single O/D pair (in whichcase Eqs. (4.5) and (5.4) coincide). Other than that, however, the same caveatsapply regarding the passage from light to heavy traffic: the classification of fastand slow edges could be reversed, the equilibrium/socially optimum flows could bedrastically different in the two regimes, etc. To illustrate all this, we proceed withsome further corollaries of Theorem 5.1 (which we prove in Appendix B): Corollary 5.2.
If the network’s costs are regularly varying at ω and the ( possiblyinfinite ) limit α e,e (cid:48) = lim x → ω c e ( x ) /c e (cid:48) ( x ) (5.6) exists for all e, e (cid:48) ∈ E , then PoA(Γ M ) → as M → ω .Proof. Define a total preorder among the network’s edges by setting e (cid:52) e (cid:48) if andonly if α e,e (cid:48) ≤ . For each path p ∈ P , choose a maximal element e p of p , i.e., an HEN IS SELFISH ROUTING BAD? 15 edge e p ∈ p such that e (cid:52) e p for all e ∈ p . Then, for each O/D pair i ∈ I , choose apath p i for which e p i is minimal, i.e., e p i (cid:52) e p (cid:48) for all p (cid:48) ∈ P i . Finally, pick an O/Dpair i ∈ I such that e p i is maximal, i.e., e p j (cid:52) e p i for all j ∈ I . Setting e ∗ = e p i and taking c ( x ) ≡ c e ∗ ( x ) as a benchmark, it is easy to verify that the network istight at ω , so Theorem 5.1 applies. (cid:3) The proof of Corollary 5.2 shows that the “comparison index” α e,e (cid:48) induces apreference relation which refines the coarser classification of the network’s edgesinto fast, slow and tight. Of course, this ordering could be reversed when passingfrom light to heavy traffic, but the existence thereof (along with regular variation)guarantees that the price of anarchy is asymptotically equal to in both cases.In addition to the above, Corollary 5.2 also gives an alternative way to prove thefollowing analogue of Corollaries 4.3 and 4.8: Corollary 5.3.
Suppose that the limit lim x → ω c e ( x ) /x q e is finite and nonzero forsome q e ≥ and all e ∈ E . Then, PoA(Γ M ) → as M → ω .Proof. Observe that c e ( x ) c e (cid:48) ( x ) = c e ( x ) x q e x q e x q e (cid:48) x q e (cid:48) c e (cid:48) ( x ) , (5.7)so lim x → ω c e ( x ) /c e (cid:48) ( x ) exists for all e ∈ E . Our claim then follows from Corol-lary 5.2. (cid:3) We thus obtain the following important corollary for polynomial cost functions:
Corollary 5.4.
In networks with polynomial costs,
PoA(Γ M ) → as M → ω . In words, Corollary 5.4 yields the general principle that we stated in the intro-duction:
In networks with polynomial costs,the price of anarchy becomes under both light and heavy traffic. We find this principle particularly appealing as it indicates that the price of anarchymay attain high values only in an intermediate regime (where the traffic is neitherlight nor heavy).
6. Networks with variable inflow rates
An important assumption in the analysis of the previous section is that therelative inflow rate λ i = m i /M of each O/D pair i ∈ I does not fluctuate in thelimit – i.e., all pairs are assumed to generate a constant fraction of the overall traffic.In general however, this assumption need not hold: for instance, in an urban roadnetwork, central O/D pairs generate disproportionately more traffic during rushhour than peripheral, suburban destinations, so it is not reasonable to assume thattraffic scales up maintaining a constant traffic ratio between different O/D pairs.To understand the impact of this variability, consider two independent links, e and e , with corresponding cost functions c ( x ) = x and c ( x ) = x . Supposefurther that these links are joining two uncoupled O/D pairs with inflow rates m and m and total inflow M = m + m . If both inflows scale in the limit as Θ( M ) , the cost of the first pair will scale as Θ( M ) while that of the second pairwill scale as Θ( M ) . As such, the leading contribution to congestion will be thatof the first O/D pair in light traffic, and that of the second pair in heavy traffic. If, however, the inflow of the first pair scales as Θ( M ) but that of the second pairscales as Θ( M / ) , the induced costs will scale respectively as Θ( M ) and Θ( M / ) ;consequently, the most costly O/D pair will now be the second one in light trafficand the first one in heavy traffic.This reversal of roles shows that the asymptotic behavior of the relative inflowrates λ i = m i /M could end up painting a completely different picture in the limit.In particular, if these relative rates oscillate wildly in the limit, the price of anarchymay exhibit a likewise irregular asymptotic behavior, even if the underlying networkis well-behaved (for instance, even if it is tight; cf. Example 6.2 below). As a result,special care must be taken to define and study the asymptotic regime in networkswith variable traffic demands.To that end, let Γ n be a sequence of nonatomic routing games with total inflow M n = (cid:80) i ∈I m in induced by a sequence of inflow rates m in for each O/D pair i ∈ I .The light and heavy traffic limits are obviously recovered depending on whetherthe total inflow M n converges respectively to ω = 0 or ω = ∞ as n → ∞ . However,the relative inflow rates λ in = m in /M n could now exhibit very different behaviors as n → ∞ : in particular, as discussed above, the relative inflow of an O/D pair couldoscillate – or even vanish – in the limit. To capture such phenomena, we introducebelow the notion of salience : Definition 6.1.
Let Γ n be a sequence of nonatomic routing games with relativeinflow rates λ in , i ∈ I . We say that a subset I (cid:48) ⊆ I of O/D pairs is salient if lim inf n →∞ (cid:88) i ∈I (cid:48) λ in > , (6.1)i.e., if the total fraction of the traffic generated by the O/D pairs in I (cid:48) is non-negligible in the limit.Obviously, if the sequence of relative inflow vectors λ n = ( λ in ) i ∈I converges toa well-defined limit, I (cid:48) will be salient if and only if some O/D pair of I (cid:48) is itselfsalient – i.e., if and only if lim inf n →∞ λ in > for some i ∈ I (cid:48) . However, if this is notthe case, a set of O/D pairs may be salient even if none of its constituent pairs issalient: for instance, if there are two O/D pairs, “ + ” and “ − ”, with relative inflows λ ± n = (1 ± ( − n ) / , neither pair is salient but their union is (since λ + n + λ − n = 1 for all n ). Thus, the notion of salience does not rule out fluctuations in the relativeinflows of individual O/D pairs; it only posits that the set of O/D pairs in questioncarries enough traffic in the limit.Bearing all this in mind, our main result for networks with variable inflow ratesis as follows:
Theorem 6.2.
Let Γ n be a sequence of nonatomic routing games with inflow rates m in and total inflow M n = (cid:80) i ∈I m in . Suppose further that: ( a ) Traffic is either light or heavy in the limit, i.e., lim n →∞ M n = ω ∈ { , ∞} . ( b ) Every O/D pair has a path which is not slow, i.e., α i < ∞ for all i ∈ I . ( c ) The set of tight O/D pairs is salient, i.e., lim inf n →∞ (cid:80) i : α i > λ in > .Then, PoA(Γ n ) → as n → ∞ . Heuristically, Condition ( a ) above simply indicates the traffic regime under study(light or heavy), whereas Condition ( b ) guarantees that the network’s benchmarkfunction correctly classifies the paths that are not too costly in each O/D pair. HEN IS SELFISH ROUTING BAD? 17 O D O D c ( x ) = 1 c ( x ) = xc ( x ) = 0 Figure 4.
A simple network with two uncoupled O/D pairs.
Finally, Condition ( c ) guarantees that tight O/D pairs are indeed relevant in termsof traffic, i.e., they account for a non-negligible fraction of the total inflow.In view of all this, Theorem 6.2 can be seen as a direct extension of our “fixed-rate” analysis in Sections 4 and 5: indeed, in the case of constant (positive) relativeinflows, salience boils down to asking that the network admits at least one tightO/D pair, so Theorem 5.1 can be obtained as a special case of Theorem 6.2. Below,we provide some further corollaries of Theorem 6.2 along these lines: Corollary 6.3.
If every O/D pair in the network is tight, then
PoA(Γ n ) → . Corollary 6.4.
If the network is tight and every O/D pair is salient, then
PoA(Γ n ) → . On the other hand, it is worth noting that if salience fails, we can draw nodefinitive conclusions for the price of anarchy. We illustrate the main reasons forthis via two examples below:
Example . Consider the simple network of Fig. 4,where two “uncoupled” O/D pairs respectively encounter a standard Pigou networkand an independent link with zero cost. In heavy traffic, the benchmark function c ( x ) = 1 classifies the O/D pair i = 1 as tight and the pair i = 2 as fast, so thenetwork is itself tight. Note also that the second O/D pair does not affect thenetwork’s price of anarchy because it has a single routing option and its cost isidentically equal to ; however, it still affects the definition of relative inflows.If we take the inflow sequence m n = √ n and m n = n , we get M n = n + √ n → ∞ and λ n → as n → ∞ , so the first O/D pair is not salient. Since the second O/Dpair is not tight, Condition ( c ) fails; nevertheless, if we apply Theorem 4.6 to thefirst O/D pair by itself, we obtain lim M →∞ PoA(Γ M ) = 1 (recall here that thesecond pair does not affect the network’s PoA). In other words, Condition ( c ) isnot necessary for selfish routing to be asymptotically efficient. Example . Consider the same network as abovebut take the periodically oscillating inflow sequence m n = (cid:40) for n odd , n for n even , and m n = (cid:40) n for n odd , for n even . (6.2)We then have M n = 2 n + 1 → ∞ and lim inf n →∞ λ n = 0 so, again, Condition ( c )fails (but in a different way). This time, whenever n is odd, the network’s price ofanarchy is equal to that of a Pigou network with inflow (because the second O/Dpair is costless). Thus, for n odd, we get PoA(Γ n ) = 4 / , which is the worst-casevalue for networks with affine costs; as such, the conclusion of Theorem 6.2 doesnot hold in general if we just drop the salience condition. O Dc ( x ) = x d c ( x ) = x d Figure 5.
A two-link Pigou network with monomial costs.
The above examples suggest that there is a finer mechanism at work which is notcaptured by the intersection of tightness and salience. At a high level, the crucialcomponent of this mechanism seems to be that asymptotic efficiency is guaranteedif the network remains tight after suitably modifying the network’s cost functionsto take into account the scaling of the inflow of each O/D pair. However, gettingan exact statement at this level of generality is fairly cumbersome, so we do notattempt it here.
7. Rate of convergence
The analysis of the previous sections provides a wide range of sufficient conditionsguaranteeing that selfish routing becomes efficient in the limit; however, it does notprovide any indication for the rate at which the network’s price of anarchy convergesto . In this section, we derive such rates (including subleading terms) for networkswith polynomial costs of the form c e ( x ) = d e (cid:88) k = q e c e,k x k , (7.1)where all coefficients are assumed nonnegative ( c e,k ≥ ) and q e and d e respectivelydenote the smallest and largest powers present (so c e,q e , c e,d e > ); by convention,we also take q e = ∞ and d e = 0 when c e ( x ) ≡ . [This follows the standard –if somewhat surprising at first – convention that sup ∅ = −∞ , inf ∅ = ∞ .] Thismodel covers in particular the BPR “constant plus monomial” model (3.1) but alsoextends more easily to networks with more intricate cost functions.In contrast to our qualitative analysis, the two traffic limits (light and heavy)exhibit different quantitative behaviors, so we treat them separately.7.1. The light traffic case.
We begin with the light traffic limit ( ω = 0 ). To motivateour analysis, we start with a simple example of a Pigou network with monomialcosts as shown in Fig. 5: for d , d > , the zero-flow travel time of both links iszero, so Proposition 3.2 does not apply. Instead, as we show below, the network’sprice of anarchy decays to following a power law: Proposition 7.1.
Consider a two-link parallel network with cost functions c ( x ) = x d and c ( x ) = x d , < d < d , and a single O/D pair with inflow M . Then PoA(Γ M ) = 1 + bM a + o ( M a ) (7.2) where a = d /d − , (7.3) HEN IS SELFISH ROUTING BAD? 19 ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ ○ � � = �� � � = � □ � � = �� � � = � ◇ � � = �� � � = � △ � � = �� � � = ��� - � �� - � ����� ����� ����� ��� - �� �� - � �� - � �� - � ����� ������ ( � ) � � � - � ����� �� ������� �� � �������� �� ������ ( ����� ������� ) ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ ○ � � = �� � � = � □ � � = �� � � = � ◇ � � = �� � � = � △ � � = �� � � = ��� - � �� - � ����� ����� ����� �������������� ����� ������ ( � ) � - � ( � � � - � ) ���������� ����� �� ������� �� � �������� �� ������ ( ����� ������� ) Figure 6.
The rate of convergence of the price of anarchy in a lightlycongested Pigou network as in Fig. 5. The figure on the left shows theprice of anarchy as a function of the total traffic inflow M for differentvalues of the exponents d and d . In the figure to the right, the priceof anarchy has been rescaled by M − a with a = d /d − , showingthat PoA(Γ M ) ∼ bM a for some b > ; the horizontal asymptotescorrespond precisely to the expression (7.4) for b . and b = d (cid:18) d d (cid:19) /d − d > . (7.4)In words, Proposition 7.1 shows that the rate of convergence of the price ofanarchy is controlled by the ratio d /d : the largest the ratio of degrees, the fastestthe decay of the price of anarchy (for a numerical illustration, see Fig. 6). Thisbehavior is consistent with Proposition 3.2 which predicts that PoA(Γ M ) ≡ if M is small enough and d = 0 . Indeed, taking d → in (7.2) shows that PoA(Γ M ) =1 + O ( M a ) for any a > , suggesting in turn that the rate of decay of PoA(Γ M ) isqualitatively different in this case.Another case worth noting is when d = d , i.e., when both links are equivalentin terms of performance. In this case, PoA(Γ M ) is identically equal to for allvalues of M (Proposition 3.1 already guarantees as much when M is not large).However, (7.2) would seem to suggest that the price of anarchy can remain large as M → (since a = 1 − when d = d ). The solution of this apparent paradoxis provided by looking at the multiplicative constant b : when d = d , we also have b = 0 , so the resulting contribution to the price of anarchy is – not Ω(1) .The above highlights the importance of the relative rate of decay of the network’sedge costs as a function of the inflow. Since monomials with lower exponents aremore costly in the low traffic limit, this rate is dominated by the smallest power in(7.1). Thus, motivated by the index machinery of Sections 4 and 5, we respectivelydefine the order of an edge e ∈ E , that of a path p ∈ P , of an O/D pair i ∈ I , andthat of the network itself as q e = min { k : c e,k > } (7.5a) q p = min e ∈ p q e , (7.5b) q i = max p ∈P i q p , (7.5c) q = min i ∈I q i . (7.5d)In view of the above, the network is tight with respect to the benchmark function c ( x ) = x q , and an edge e ∈ E is fast when q e > q , tight when q e = q , and slow if q e < q . We then denote the set of the network’s slow edges as E slow = { e ∈ E : q e < q } , (7.6)and we write q slow = max e ∈E slow q e (7.7)for the order of the fastest edge in E slow (again employing the standard conventionthat max ∅ = −∞ , so q slow = −∞ when there are no slow edges).Building on the intuition gained from Proposition 7.1, our main quantitativeresult for low traffic is that the network’s PoA decays to following a power lawthat depends only on the ratio between the order of the network ( q ) and that of itsfastest slow edge ( q slow ): Theorem 7.2.
Let Γ M be a nonatomic routing game with polynomial costs, totalinflow M , and fixed relative inflows. Then, there exist non-negative constants K ≥ and K a ≥ such that PoA(Γ M ) ≤ K M + K a M a , (7.8) where a = q/q slow − and K a = 0 whenever E slow = ∅ . Theorem 7.2 was stated for networks with fixed relative inflows for simplicityonly: in Appendix C, we state and prove a more general result for networks withvariable relative inflows as in Section 5. In terms of intuition, we only note herethat Theorem 7.2 complements the insights gained from Propositions 3.1 and 3.2in an important way: when there is no single “best path” under zero inflow, thenetwork’s price of anarchy is no longer identically equal to for small inflows butinstead behaves as a power law.7.2. The heavy traffic case.
We now turn to the heavy traffic limit ( ω = ∞ ). Asin the light traffic case, we start with a simple – but illuminating – example of atwo-link Pigou network where precise results can be obtained: Proposition 7.3.
Consider a two-link parallel network with cost functions c ( x ) = x d and c ( x ) = x d , < d < d , and a single O/D pair with inflow M . Then PoA(Γ M ) = 1 + bM − a + o ( M − a ) (7.9) where a = 1 − d /d (7.10) and b = d (cid:18) d d (cid:19) /d − d > . (7.11)For illustration purposes, we plotted in Fig. 7 the asymptotic behavior of thenetwork’s price of anarchy for different values of d and d . In the same vein as inthe light traffic limit, two special cases that are of interest here are when d = ∞ and when d = d . In the former ( d = ∞ ), Eq. (7.10) gives a = 1 , indicating aconvergence rate of the order of O (1 /M ) : since a < for all finite d , this is the HEN IS SELFISH ROUTING BAD? 21 ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ ○ � � = �� � � = � □ � � = �� � � = � ◇ � � = �� � � = � △ � � = �� � � = ���� �� � �� � �� - � �� - � �� - � �� - � ��������������� ����� ������ ( � ) � � � - � ����� �� ������� �� � �������� �� ������ ( ����� ������� ) ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○□ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ ○ � � = �� � � = � □ � � = �� � � = � ◇ � � = �� � � = � △ � � = �� � � = ���� �� � �� � ������������������������ ����� ������ ( � ) � � ( � � � - � ) ���������� ����� �� ������� �� � �������� �� ������ ( ����� ������� ) Figure 7.
The rate of convergence of the price of anarchy in a heavilycongested Pigou network as in Fig. 5. The figure on the left shows theprice of anarchy as a function of the total traffic inflow M for differ-ent values of the exponents d and d . In the figure to the right, theprice of anarchy has been rescaled by M a with a = 1 − d /d , showingthat PoA(Γ M ) ∼ b/M a for some b > ; the horizontal asymptotescorrespond precisely to the expression (7.11) for b . best possible rate that can be achieved in the heavy traffic limit. For the latter( d = d ), Eq. (7.10) gives a = 0 , suggesting that the price of anarchy can remainlarge as M → ∞ . This seems to be inconsistent with the fact that the network’sprice of anarchy is identically equal to when d = d but a closer look revealsthat the multiplicative constant b of (7.11) is also when d = d , thus reconcilingthe two results.Now, even though the above result does not apply to more general networkswith polynomial costs, it still highlights the main mechanism at play. Specifically,for large edge loads x , the dominant term in (7.1) is the one with highest degree d e . As we’ve discussed before, this indicates a complete reversal of roles betweenlight and heavy traffic: for d < d , x d is slower than x d when x → , but faster when x → ∞ . Thus, with an obvious adaptation of what we did for light traffic,we define the order of an edge e ∈ E , that of a path p ∈ P , of an O/D pair i ∈ I ,and of the network itself as d e = max { k : c e,k > } (7.12a) d p = max e ∈ p d e , (7.12b) d i = min p ∈P i d p , (7.12c) d = max i ∈I d i . (7.12d)With all this at hand, we see that the network is tight with respect to the benchmark c ( x ) = x d , so an edge e ∈ E is fast when d e < d , tight when d e = d , and slow if d e > d . The set of the network’s slow edges is then denoted as E slow = { e ∈ E : d e > d } , (7.13)and we write d slow = min e ∈E slow d e (7.14) for the order of the fastest edge in E slow (again employing the standard conventionthat min ∅ = ∞ , so d slow = ∞ when there are no slow edges).Mutatis mutandis, this definition is the same as in light traffic except for areversal of the max/min operators. Our main result for heavily congested networksconfirms this intuition: Theorem 7.4.
Let Γ M be a nonatomic routing game with polynomial costs, totalinflow M , and fixed relative inflows. Then, there exist non-negative constants K ≥ and K a ≥ such that PoA(Γ M ) ≤ K M + K a M a , (7.15) where a = 1 − d/d slow and K a = 0 whenever E slow = ∅ .Remark . As in the light traffic case, if the costs are monomials of the samedegree, then K = K a = 0 and PoA(Γ M ) ≡ for all M > .In words, given that a < when there is at least one slow edge in the network( a = 1 and K a = 0 otherwise), Theorem 7.4 states the network’s price of anarchyconverges to as O (1 /M a ) with an O (1 /M ) subleading term. In particular, inthe presence of a single slow edge e with d e > d , the convergence exponent a in(7.15) can become as small as / ( d + 1) , thus pointing to a slower convergence ratein networks with routing costs of high degree and a small gap between the degreeof tight and slow edges. On the other hand, if there are no slow edges we have E slow = ∅ and we get an O (1 /M ) rate of convergence.On this issue, Wu et al. (2017) recently showed that if all the network’s costfunctions are of the BPR type c e ( x ) = a e + b e x d with the same degree d , then PoA(Γ M ) = 1 + O ( M − d ) as M → ∞ . In this special case, the rate of convergenceis faster than the prediction of Theorem 7.4, a gap which points to a sharp discon-tinuity that occurs when all costs have the same degree. To see this in a concreteexample, consider again the two-link Pigou network of Fig. 5. By symmetry, if d = d , the fraction of traffic routed on edge at optimum and at equilibriumcoincide ˜ y = y ∗ = 12 for all M > , (7.16)implying in turn that PoA(Γ M ) is identically equal to . On the other hand, when d < d , both fractions ˜ y and y ∗ converge to as M → ∞ . Proposition 7.3 showsthat the rate of convergence of the price of anarchy in this case is exactly of order Θ(1 /M a ) and cannot be improved.Put differently, Proposition 7.3 shows that the slightest difference in edge degreescauses the rate of convergence of the price of anarchy to drop abruptly to Θ( M − a ) ;in fact, Eq. (7.11) even provides an explicit expression for the proportionality con-stant in the high congestion rate Θ( M − a ) . By this token, the bounds providedby Theorem 7.4 are tight and cannot be improved in general, even in the class oftwo-link parallel networks with monomial costs.
8. Discussion
Most of the literature on the price of anarchy – for congestion games and notonly – has traditionally focused on worst-case upper bounds for different classesof networks, cost functions, and/or types of players. Several of these results have
HEN IS SELFISH ROUTING BAD? 23 become milestones in the field and have had a significant impact in practical consid-erations for traffic networks. However, real-world situations involve a fixed networkand traffic flows that are not necessarily close to these worst-case scenarios. Thus,in addition to determining how bad can selfish routing become, it is also importantto determine when these cases are relevant.Our goal in this paper was to provide an answer to this question by examining thebehavior of the price of anarchy at each end of the congestion spectrum. Under fairlymild assumptions (that always include networks with polynomial costs), we foundthat the PoA goes to in both cases, independently of the network’s topology, andeven when there are multiple O/D pairs. What we find appealing about this resultis that it is essentially independent of the underlying graph and/or the distributionof O/D pairs in the network. Especially in the heavy traffic limit, this means thatselfish routing is not the real cause of increased delays: from a social planner’spoint of view, sophisticated tolling/rerouting schemes that target the optimumtraffic assignment will not yield considerable gains over a “laissez-faire” approachwhere each traffic element takes the fastest available path. Appendix A. Proofs of the results of Section 3
We start this appendix with the proofs of Propositions 3.1 and 3.2. To that end,we first establish the following result:
Lemma A.1.
For sufficiently small M , equilibrium and optimum traffic allocationsonly employ paths in P i min = arg min p ∈P i c p (0) .Proof. In a slight abuse of notation, let c p ( M ) = (cid:80) e ∈ p c e ( M ) denote the cost ofthe path p if all its edges carry load equal to the total inflow M . Clearly, if M is small enough, we have c p (cid:48) (0) > c p ( M ) for all p ∈ P i min and all p (cid:48) ∈ P i \ P i min .Hence, for an equilibrium flow f ∗ , we have c p (cid:48) ( f ∗ ) ≥ c p (cid:48) (0) > c p ( M ) ≥ c p ( f ∗ ) , (A.1)implying in turn that f ∗ p (cid:48) = 0 . Likewise, since an optimal flow ˜ f is an equilibriumfor the marginal costs ˜ c e ( x ) = c e ( x ) + x c (cid:48) e ( x ) and ˜ c p (cid:48) (0) > ˜ c p ( M ) , similar consid-erations show that an optimum flow profile cannot route any traffic along a path p (cid:48) ∈ P i \ P i min . (cid:3) To proceed, it is more convenient to start with Proposition 3.2:
Proof of Proposition 3.2.
By Lemma A.1, when M is small enough, both the equi-librium and the optimum must route the total inflow m i along the unique path in P i min . Hence the equilibrium and optimal flows coincide and therefore the price ofanarchy is equal to . (cid:3) With this result at hand, we have:
Proof of Proposition 3.1. If P i min is a singleton for all i ∈ I , our claim follows fromProposition 3.2. Otherwise, by Lemma A.1, if M is small enough, for every i ∈ I ,only paths in P i min are used in equilibrium. Moreover, if p, p (cid:48) ∈ P i min , then, for M small enough, we have (cid:88) e ∈ p a e + b e ( x ∗ e ) d = (cid:88) e ∈ p (cid:48) a e + b e ( x ∗ e ) d , (A.2) and hence (cid:88) e ∈ p b e ( x ∗ e ) d = (cid:88) e ∈ p (cid:48) b e ( x ∗ e ) d . (A.3)Again, by Lemma A.1, if M is small enough, for every i ∈ I , only paths in P i min are used at the optimum. If p, p (cid:48) ∈ P i min , then, for M small enough, we have (cid:88) e ∈ p a e + ( d + 1) b e ˜ x de = (cid:88) e ∈ p (cid:48) a e + ( d + 1) b e ˜ x de , (A.4)that is, (cid:88) e ∈ p ( d + 1) b e ˜ x de = (cid:88) e ∈ p (cid:48) = ( d + 1) b e ˜ x de . (A.5)Comparing (A.3) and (A.5), we see that the two equations are satisfied by the sameloads. Therefore, for M small enough, there exist an equilibrium and an optimumhaving the same flows. Uniqueness of the equilibrium and optimum costs providesthe result. (cid:3) We now present the proof of the counterexample with an oscillating PoA ofSection 3.2:
Proof of Proposition 3.3.
Since an unused edge has a cost of zero under (3.2), allthree edges must be used at equilibrium. Hence, for a given value of the total inflow M = x + x + x , the load profile x = ( x , x , x ) is a Wardrop equilibrium if andonly if c ( x ) = c ( x ) = c ( x ) . In that case, the normalized profile z = x/M satisfies z d (cid:2) sin(log M z ) (cid:3) = z d = z d (cid:2) cos(log M z ) (cid:3) . (A.6)Likewise, after differentiating and rearranging, the conditions for the network’ssocially optimum flow are z d (cid:104) sin(log M z ) + d +1) cos(log M z ) (cid:105) = z d = z d (cid:104) cos(log M z ) − d +1) sin(log M z ) (cid:105) . (A.7)We now show that Eqs. (A.6) and (A.7) never admit a common solution. Indeed,this can occur if and only if cos(log M z ) = 0 = sin(log M z ) , (A.8)i.e., if and only if there exist integers k , k ∈ Z such that log M z = k π + π , log M z = k π. (A.9)This implies that sin(log M z ) = ± and cos(log M z ) = ± , leading to the follow-ing cases:Case 1: sin(log M z ) = 1 , cos(log M z ) = 1 . Substituting in (A.6) we get z d = z d so (A.9) gives k π + π k π. (A.10)This gives k − k = 1 / , which cannot hold for integer values of k and k . HEN IS SELFISH ROUTING BAD? 25
Case 2: sin(log
M z ) = 1 , cos(log M z ) = − . As above, from (A.6) we get z d = z d , so (A.9) gives d log 3 + k π + π k π. (A.11)This yields log 3 π = d ( k − k − ) , which again cannot hold for k , k , d ∈ Z .The remaining two cases lead to a contradiction in the same way, implying thatthe game’s Wardrop equilibrium and socially optimum flows cannot coincide forany value of M . Since Eqs. (A.6) and (A.7) are periodic in log M , it follows thatthe game’s price of anarchy oscillates periodically at a logarithmic scale. Thus,focusing on the period ≤ M ≤ e π , we conclude that inf M> PoA(Γ M ) = min ≤ M ≤ e π PoA(Γ M ) > , (A.12)i.e., the Wardrop equilibria of the network in Fig. 1 remain strictly inefficient underboth light and heavy traffic. (cid:3) Appendix B. Convergence of the price of anarchy
We now prove Theorem 6.2; Theorems 4.2, 4.6 and 5.1 will then follow as specialcases of this more general result. To that end, we begin with two auxiliary lemmasconcerning the asymptotic behavior of regularly varying functions:
Lemma B.1 (Karamata, 1933) . If g is regularly varying at ω , there exists some ρ ∈ R such that lim t → ω g ( tx ) g ( t ) = x ρ for all x > . (B.1)Lemma B.1 is a classical result in the theory of regularly varying functions andgives rise to the term “ ρ -regularly varying” for functions satisfying (B.1); for aproof, see, e.g., Bingham et al. (1989).The second lemma is a more technical asymptotic comparison result allowing usto replace a ρ -regularly varying function by a monomial of degree ρ in the limit: Lemma B.2.
Let ω ∈ { , ∞} and consider two functions f, g : (0 , ∞ ) → (0 , ∞ ) suchthat: (1) f is nondecreasing. (2) g is ρ -regularly varying at ω for some ρ > . (3) lim x → ω f ( x ) /g ( x ) = α ∈ [0 , ∞ ) .If M n → ω and z n → z ∈ [0 , ∞ ) , then lim n →∞ f ( M n z n ) g ( M n ) = αz ρ . (B.2) Proof.
We first consider the case ω = ∞ . If z > , the sequence x n = M n z n diverges to infinity, so our claim follows from Theorem 1.5.2 in Bingham et al.(1989) by writing f ( M n z n ) g ( M n ) = f ( x n ) g ( x n ) · g ( M n z n ) g ( M n ) → αz ρ . (B.3)If z = 0 , then, for all ε > , we have z n ≤ ε if n is sufficiently large. Then, usingthe monotonicity of f and the previous argument, we get ≤ lim sup n →∞ f ( M n z n ) g ( M n ) ≤ lim sup n →∞ f ( M n ε ) g ( M n ) = αε ρ . (B.4) Taking ε → , we conclude that f ( M n z n ) /g ( M n ) → αz ρ , as claimed.The case ω = 0 is even simpler. Indeed, we now have that x n = M n z n tends to0, so that the result follows using (B.3) and invoking Theorem 1.5.2 in Binghamet al. (1989). (cid:3) Now, to proceed with the proof of Theorem 6.2, we will require some additionalnotation. First, fix some inflow vector m = ( m i ) i ∈I with total inflow M = (cid:80) i ∈I m i and relative inflows λ = ( λ i ) i ∈I . Instead of working directly with the flow variables f ∈ F , it will be more convenient to introduce the normalized traffic allocation variables y i = ( y ip ) p ∈P i defined as y ip = f p /m i for all p ∈ P i , i ∈ I . (B.5)We clearly have (cid:80) p ∈P i y ip = 1 for all i ∈ I ; we will also write Y i = ∆( P i ) forthe simplex of traffic allocations of i ∈ I and Y = × i ∈I Y i for the product thereof.Moreover, descending to the edge level, we define the normalized load induced bythe i -th O/D pair on e ∈ E as z ie ( y ) = (cid:88) p ∈P i ,p (cid:51) e y ip (B.6)and we denote respectively the normalized and total load on edge e ∈ E as ζ e ( y, λ ) = (cid:88) i ∈I λ i z ie ( y ) and x e ( y, m ) = M ζ e ( y, λ ) = (cid:88) i ∈I m i z ie ( y ) . (B.7)Finally, based on the index framework of Sections 4 and 5, we will respectivelydenote the set of the network’s fast, tight and slow edges as E fast = { e ∈ E : α e = 0 } , (B.8a) E tight = { e ∈ E : 0 < α e < ∞} , (B.8b) E slow = { e ∈ E : α e = ∞} , (B.8c)and, in obvious notation, we will write e.g., P slow for the set of the network’s slowpaths, I tight for the set of tight O/D pairs, etc.The following asymptotic approximation result provides the heavy lifting for theproof of Theorem 6.2: Lemma B.3.
Consider a network with nondecreasing cost functions g e , with g e (0) =0 for e ∈ E , and suppose that it admits a benchmark function g at ω , which is ρ -regularly varying with ρ > . Consider also a sequence of inflow vectors m n = M n λ n such that:a ) M n → ω and the vector of relative inflows λ n converges to some λ ∈ ∆( I ) .b ) Every O/D pair has a path which is not slow ( relative to g ) .c ) There exists an O/D pair i ∈ I which is tight ( relative to g ) and has λ i > .Then, the optimal allocation problem G n = min y ∈Y (cid:88) e ∈E g e ( x e ( y, m n )) (B.9) satisfies lim n →∞ G n g ( M n ) = V ρ ( λ ) , (B.10) HEN IS SELFISH ROUTING BAD? 27 where V ρ ( λ ) ∈ (0 , ∞ ) is the solution value of the problem V ρ ( λ ) = min y ∈Y (cid:88) e ∈E α e ζ e ( y, λ ) ρ (B.11) and, by convention, we have set α e z ρe = 0 if α e = ∞ and z e = 0 . Moreover, if ˆ y n is a sequence of optimal solutions of G n , every limit point of ˆ y n solves V ρ ( λ ) .Proof. The arguments in the proof are similar to the line of reasoning in epi-convergence arguments as in Attouch (1984). To streamline the presentation, webreak up the proof in five steps as follows:
Step 1: V ρ ( λ ) < ∞ . By Condition ( b ), each O/D pair admits a path that isnot slow; therefore, routing all traffic through said path gives a finite value for theobjective of (B.11), implying in turn that V ρ ( λ ) < ∞ . More precisely, for every i ∈ I , take a traffic allocation y i ∈ Y i that assigns zero weight to the slow paths P i slow of i . Then, for every slow edge e ∈ E slow , we have z ie ( y ) = 0 and, a fortiori , ζ e ( y, λ ) = 0 ; hence V ρ ( λ ) ≤ (cid:88) e : α e < ∞ α e ζ e ( y, λ ) ρ < ∞ . (B.12) Step 2: V ρ ( λ ) > . By Condition ( c ), there exists a tight O/D pair i ∈ I tight such that λ i > . For every y ∈ Y we have (cid:80) p ∈P i y p = 1 , so there exists someroute p ∈ P i with y ip ≥ / |P i | . This gives z ie ( y ) ≥ / |P i | for all e ∈ p , and hence (cid:88) e ∈E α e ζ e ( y, λ ) ρ ≥ (cid:88) e ∈E α e (cid:0) λ i z ie ( y ) (cid:1) ρ ≥ (cid:88) e ∈ p α e (cid:0) λ i / |P i | (cid:1) ρ ≥ α i (cid:0) λ i / |P i | (cid:1) ρ > . (B.13)Minimizing over y ∈ Y then yields V ρ ( λ ) > , as claimed. Step 3: lim sup n →∞ G n /g ( M n ) ≤ V ρ ( λ ) . Fix an optimal solution ˆ y ∈ Y of(B.11). By the finiteness of V ρ ( λ ) , we have ζ e (ˆ y, λ ) = 0 for every slow edge e ∈ E slow (i.e., when α e = ∞ ). If λ i > , this implies that z ie (ˆ y ) = 0 . Otherwise, if λ i = 0 ,the objective function of (B.11) does not depend on y i , so every y i with z ie ( y ) = 0 is also optimal. Hence we can choose the solution ˆ y of (B.11) so that all traffic isrouted along edges that are not slow.Now, from optimality we have G n g ( M n ) ≤ g ( M n ) (cid:88) e ∈E g e ( M n ζ e (ˆ y, λ n )) . (B.14)Using Lemma B.2, for every non-slow edge e ∈ E \ E slow (i.e., α e < ∞ ), we get lim n →∞ g e ( M n ζ e (ˆ y, λ n )) g ( M n ) = α e ζ e (ˆ y, λ ) ρ . (B.15)Otherwise, if e ∈ E slow is slow (i.e., α e = ∞ ), we have ζ e (ˆ y, λ n ) = 0 ; thus, since g e (0) = 0 , we get lim n →∞ g e ( M n ζ e (ˆ y, λ n )) g ( M n ) = lim n →∞ g e (0) g ( M n ) = 0 = α e ζ e (ˆ y, λ ) ρ . (B.16)Combining the previous three displayed equations, we obtain lim sup n →∞ G n g ( M n ) ≤ (cid:88) e ∈E lim n →∞ g e ( M n ζ e (ˆ y, λ n )) g ( M n ) = (cid:88) e ∈E α e ζ e (ˆ y, λ ) ρ = V ρ ( λ ) . (B.17) Step 4: lim inf n →∞ G n /g ( M n ) ≥ V ρ ( λ ) . Passing to a subsequence if necessary,we may assume that lim inf n →∞ G n /g ( M n ) is attained as a limit. Thus, letting ˆ y n be a sequence of solutions of G n , and taking a further subsequence if necessary, wemay assume that ˆ y n converges to some ˆ y ∈ Y . Then, ignoring the network’s slowedges, we have G n g ( M n ) ≥ (cid:88) e : α e < ∞ g e ( M n ζ e (ˆ y n , λ n )) g ( M n ) , (B.18)and hence, by Lemma B.2, we obtain lim inf n →∞ G n g ( M n ) ≥ (cid:88) e : α e < ∞ α e ζ e (ˆ y, λ ) ρ . (B.19)To proceed, we will show that ζ e (ˆ y, λ ) = 0 for every slow edge. Indeed, if thiswere not the case, we could find some ε > such that ζ e (ˆ y n , λ n ) > ε for allsufficiently large n . With g e nondecreasing, we then get G n g ( M n ) ≥ g e ( M n ζ e (ˆ y n , λ n )) g ( M n ) ≥ g e ( M n ε ) g ( M n ) = g e ( M n ε ) g ( M n ε ) g ( M n ε ) g ( M n ) → α e ε ρ = ∞ , (B.20)in contradiction to Steps 1 and 3 above. From all this, it follows that lim inf n →∞ G n g ( M n ) ≥ (cid:88) e ∈E α e ζ e (ˆ y, λ ) ρ ≥ V ρ ( λ ) . (B.21) Step 5: Optimality of limit points.
As above, let ˆ y n be a sequence of optimalsolutions of (B.9) and, by descending to a subsequence if necessary, assume that itconverges to some ˆ y ∈ Y . From the previous steps we have G n /g ( M n ) → V ρ ( λ ) so,proceeding as in Step 4, we get V ρ ( λ ) = lim n →∞ G n g ( M n ) ≥ (cid:88) e ∈E α e ζ e (ˆ y, λ ) ρ ≥ V ρ ( λ ) , (B.22)showing that ˆ y solves (B.11). (cid:3) Armed with Lemma B.3, we are finally in a position to prove Theorem 6.2:
Proof of Theorem 6.2.
To begin with, express the objective function of (SO) interms of the normalized flow variables y as L n ( y ) = (cid:88) e ∈E x e ( y, m n ) c e ( x e ( y, m n )) . (B.23)Now, let y ∗ n , ˜ y n be the normalized traffic allocation profiles of a Wardrop equilibriumand a socially optimum flow, respectively. Then, the network’s price of anarchy maybe expressed as PoA(Γ n ) = Eq(Γ n )Opt(Γ n ) = L n ( y ∗ n ) L n (˜ y n ) . (B.24)Notice that Opt(Γ n ) > thanks to Assumptions (b) and (c).In order to prove that PoA(Γ n ) → it suffices to take a subsequence Γ n k realizingthe lim sup n →∞ PoA(Γ n ) as a limit and to prove that PoA(Γ n k ) → . Relabelingindices and extracting a further subsequence if necessary, we may assume withoutloss of generality that: ( a ) the limit lim n →∞ PoA(Γ n ) exists; ( b ) the sequence λ n of relative inflows converges to some λ ∈ ∆( I ) ; and ( c ) the sequences y ∗ n and ˜ y n HEN IS SELFISH ROUTING BAD? 29 converge to some y ∗ , ˜ y ∈ Y respectively. With all this, we will use Lemma B.3 toderive the asymptotic behavior of Opt(Γ n ) and Eq(Γ n ) .First, for Opt(Γ n ) , combining Lemma B.1 with Proposition 1.5.1 of Binghamet al. (1989) and the fact that the network’s cost functions are nondecreasing, weimmediately see that the network’s benchmark function c is β -regularly varying forsome β ≥ . Then, letting g e ( x ) = xc e ( x ) and g ( x ) = xc ( x ) , we also get that g is ρ -regularly varying with ρ = 1+ β > and lim x → ω g e ( x ) /g ( x ) = lim x → ω c e ( x ) /c ( x ) = α e . This means that the hypotheses of Lemma B.3 are all satisfied, implying inturn that L n (˜ y n ) = Opt(Γ n ) ∼ V ρ ( λ ) g ( M n ) as n → ∞ , (B.25)with the notation “ f n ∼ g n ” meaning here that lim n →∞ f n /g n = 1 .In view of this, and since < V ρ ( λ ) < ∞ , it remains to show that L n ( y ∗ n ) ∼ V ρ ( λ ) g ( M n ) . To that end, we first analyze the asymptotic behavior of the convexminimization problem W (Γ n ) = min y ∈Y (cid:88) e ∈E C e ( x e ( y, m n )) (B.26)by applying Lemma B.3 to the primitives C e and C of c e and c respectively. Bya standard result (Bingham et al., 1989, Theorem 1.5.11), C is ρ -regularly varyingwith ρ = 1 + β ; moreover, by L’Hôpital’s rule we also have lim x → ω C e ( x ) /C ( x ) =lim x → ω c e ( x ) /c ( x ) = α e . By Lemma B.3, it follows that W (Γ n ) /C ( M n ) → V ρ ( λ ) .In addition, since the Wardrop equilibrium traffic allocations y ∗ n are solutions of W (Γ n ) , the limit y ∗ of y ∗ n is optimal for V ρ ( λ ) by Lemma B.3.Noting that x e ( y ∗ n , m n ) = M n ζ e ( y ∗ n , λ n ) , we obtain L n ( y ∗ n ) g ( M n ) = (cid:88) e ∈E g e (cid:0) M n ζ e ( y ∗ n , λ n ) (cid:1) g ( M n ) . (B.27)By Lemma B.2, we also have the following limit for every non-slow edge e ∈ E \E slow : g e ( M n ζ e ( y ∗ n , λ n )) g ( M n ) → α e ζ e ( y ∗ , λ ) ρ . (B.28)To establish a similar limiting result when e is slow, we first claim that there existsa constant B ≥ such that g e ( M n ζ e ( y ∗ n , λ n )) ≤ B ζ e ( y ∗ n , λ n ) g ( M n ) (B.29)This is trivially satisfied when ζ e ( y ∗ n , λ n ) = 0 , so it suffices to consider the case ζ e ( y ∗ n , λ n ) > . The above inequality is then equivalent to asking that c e ( M n ζ e ( y ∗ n , λ n )) ≤ B c ( M n ) (B.30)Now, ζ e ( y ∗ n , λ n ) > implies that the edge e receives some equilibrium traffic fromat least one O/D pair i ∈ I , so it must belong to a path p ∈ P i with minimal cost.Thus, if we consider an alternative path p (cid:48) ∈ P i all of whose edges are tight or fast,we have c e ( M n ζ e ( y ∗ n , λ n )) ≤ (cid:88) e (cid:48) ∈ p c e (cid:48) ( M n ζ e (cid:48) ( y ∗ n , λ n )) ≤ (cid:88) e (cid:48) ∈ p (cid:48) c e (cid:48) ( M n ζ e (cid:48) ( y ∗ n , λ n )) . (B.31)Using the trivial bound M n ζ e (cid:48) ( y ∗ n , λ n ) ≤ M n , we further get c e ( M n ζ e ( y ∗ n , λ n )) ≤ (cid:88) e (cid:48) ∈ p (cid:48) c e (cid:48) ( M n ) ≤ (cid:88) e (cid:48) : α e (cid:48) < ∞ c e (cid:48) ( M n ) . (B.32) However, for every non-slow edge e (cid:48) ∈ E \ E slow , the sequence c e (cid:48) ( M n ) /c ( M n ) con-verges to α e (cid:48) so we can find a constant B e (cid:48) such that c e (cid:48) ( M n ) /c ( M n ) ≤ B e (cid:48) for all n ∈ N ; consequently, (B.30) follows by taking B = (cid:80) e (cid:48) : α e (cid:48) < ∞ B e (cid:48) . Thus, given that y ∗ is optimal for V ρ ( λ ) , we get ζ e ( y ∗ n , λ n ) → ζ e ( y ∗ , λ ) = 0 , and hence g e ( M n ζ e ( y ∗ n , λ n )) g ( M n ) ≤ B ζ e ( y ∗ n , λ n ) → α e ζ e ( y ∗ , λ ) ρ . (B.33)Combining (B.28), (B.33) and (B.27), we then get L n ( y ∗ n ) /g ( M n ) → (cid:88) e ∈E α e ζ e ( y ∗ , λ ) ρ = V ρ ( λ ) , (B.34)as was to be shown. (cid:3) Appendix C. Speed of convergence
In this appendix, we provide the proofs of the results presented in Section 7.C.1.
Rates in the light traffic regime.
First we present the proof of Proposition 7.1on the light traffic rates in the case of a Pigou network.
Proof of Proposition 7.1.
Let x denote the flow on edge e . At equilibrium, thecosts on both edges must be equal so that ( x ∗ ) d = ( M − x ∗ ) d , which is equivalentto x ∗ + ( x ∗ ) d /d = M . Since M tends to 0 it follows that x ∗ will be small andsince d < d the term ( x ∗ ) d /d dominates the right hand side. Therefore x ∗ = M d /d (1 + o (1)) (C.1)so the equilibrium cost Eq(Γ M ) = M · c ( x ∗ ) = M · c ( M − x ∗ ) scales as Eq(Γ M ) = M · (cid:104) M − M d /d (1 + o (1)) (cid:105) d = M d +1 − d M d + d /d + o ( M d + d /d ) . (C.2)Similarly, if ˜ x is the optimal flow on edge e , both edges have the same marginalcost (1 + d )˜ x d = (1 + d )( M − ˜ x ) d . (C.3)Therefore, if we let ρ = (cid:18) d d (cid:19) /d , (C.4)we get ρ ˜ x + ˜ x d /d = ρM as before, and hence ˜ x = ( ρM ) d /d (1 + o (1)) . (C.5)It follows that the optimal cost scales as Opt(Γ M ) = ˜ x · c (˜ x ) + ( M − ˜ x ) · c ( M − ˜ x )= (cid:104) ( ρM ) d /d (1 + o (1)) (cid:105) d +1 + (cid:104) M − ( ρM ) d /d (1 + o (1)) (cid:105) d +1 = ( ρM ) d + d /d + M d +1 − ( d + 1) M d ( ρM ) d /d + o ( M d + d /d )= M d +1 − ρ d /d (cid:2) ( d + 1) − ρ d (cid:3) M d + d /d + o ( M d + d /d )= M d +1 − ( b + d ) M d + d /d + o ( M d + d /d ) (C.6)where the last equality follows from the identity ρ d /d (cid:2) ( d + 1) − ρ d (cid:3) = b + d . HEN IS SELFISH ROUTING BAD? 31
Combining the previous expressions we get
PoA(Γ M ) = Eq(Γ M )Opt(Γ M ) = Opt(Γ M ) + bM d + d /d + o ( M d + d /d )Opt(Γ M )= 1 + bM d + d /d + o ( M d + d /d ) M d +1 + o ( M d +1 )= 1 + bM a + o ( M a ) . (C.7)To complete our proof, it remains to show that b > . After a small rearrange-ment, this is equivalent to establishing the inequality (cid:18) d d (cid:19) d = (cid:18) d − d d (cid:19) d > (cid:18) d − d d (cid:19) d = (cid:18) d d (cid:19) d , (C.8)which itself follows from the fact that the function (1 + g/x ) x is increasing in x whenever g is positive. (cid:3) Now we prove the following more general version of Theorem 7.2.
Theorem C.1.
Let Γ n be a sequence of nonatomic routing games satisfying theassumptions of Theorem 6.2, with M n → . Suppose further that the edge costs arepolynomials as in (7.1) , and let a = q/q slow − with q and q slow given by (7.5d) and (7.7) respectively. Then, there exist non-negative constants K ≥ and K a ≥ such that PoA(Γ n ) ≤ K M n + K a M an (C.9) with K a = 0 whenever E slow = ∅ .Proof. Let y ∗ n , ˜ y n ∈ Y be an equilibrium and an optimum flow for Γ n with inducededge flows x ∗ e,n = M n ζ e ( y ∗ n , λ n ) and ˜ x e,n = M n ζ e (˜ y n , λ n ) . The social cost of y ∗ n can be estimated as L n ( y ∗ n ) = (cid:88) e ∈E x ∗ e,n c e ( x ∗ e,n ) = (cid:88) e ∈E d e (cid:88) k = q e c e,k · ( x ∗ e,n ) k +1 = (cid:88) e ∈E d e (cid:88) k = q e (cid:20) q + 1 k + 1 + k − qk + 1 (cid:21) c e,k · ( x ∗ e,n ) k +1 = ( q + 1) (cid:88) e ∈E C e ( x ∗ e,n ) + (cid:88) e ∈E d e (cid:88) k = q e k − qk + 1 c e,k · ( x ∗ e,n ) k +1 ≤ ( q + 1) (cid:88) e ∈E C e (˜ x e,n ) + (cid:88) e ∈E d e (cid:88) k = q +1 k − qk + 1 c e,k · ( x ∗ e,n ) k +1 (C.10)where C e is the primitive of c e and for the last inequality we used the fact that x ∗ e,n minimizes the first sum, and in the double sum we dropped the negative termswith k ≤ q . Now, the first sum in (C.10) can be further bounded as ( q + 1) (cid:88) e ∈E C e (˜ x e,n ) = (cid:88) e ∈E d e (cid:88) k = q e q + 1 k + 1 c e,k · (˜ x e,n ) k +1 = (cid:88) e ∈E d e (cid:88) k = q e c e,k · (˜ x e,n ) k +1 + (cid:88) e ∈E d e (cid:88) k = q e q − kk + 1 c e,k · (˜ x e,n ) k +1 ≤ Opt(Γ n ) + (cid:88) e ∈E slow q − (cid:88) k = q e q − kk + 1 c e,k · (˜ x e,n ) k +1 where we used the optimality of ˜ x n in the first double sum, and we dropped thenegative terms in the second. Note that in the latter only the slow edges with q e < q are relevant. Combining these estimates we get L n ( y ∗ n ) ≤ Opt(Γ n )+ (cid:88) e ∈E slow q − (cid:88) k = q e q − kk + 1 c e,k · (˜ x e,n ) k +1 + (cid:88) e ∈E d e (cid:88) k = q +1 k − qk + 1 c e,k · ( x ∗ e,n ) k +1 (C.11)Let us call L I n the first double sum in (C.11) and L II n the second. In order tobound L II n we assume that n is large enough so that M n ≤ . This assumption isdone for convenience and it only affects the value of the constants K and K a : byredefining them appropriately, the bound (C.9) will hold for all n . Then by setting G = (cid:88) e ∈E d e (cid:88) k = q +1 k − qk + 1 c e,k , (C.12)and noting that x ∗ e,n ≤ M n , we get L II n ≤ (cid:88) e ∈E d e (cid:88) k = q +1 k − qk + 1 c e,k · M k +1 n ≤ GM q +2 n . (C.13)In order to bound L I n we note that this term vanishes when E slow is empty.Otherwise, consider any edge e ∈ E slow that contributes to the sum with ˜ x e,n > .We note that the optimum flow ˜ y n is an equilibrium for the marginal cost functions ˜ c e ( x ) = c e ( x ) + x c (cid:48) e ( x ) = d e (cid:88) k = q e ( k + 1) c e,k x k . (C.14)The edge e must therefore belong to an optimal path p ∈ P i (w.r.t. the costs ˜ c e ( x ) )for some i ∈ I . Hence, taking any alternative path p (cid:48) ∈ P i which is not slow, i.e.,with q e (cid:48) ≥ q for all e (cid:48) ∈ p (cid:48) , and denoting B = (cid:88) e (cid:48) (cid:54)∈E slow d e (cid:48) (cid:88) k = q e (cid:48) ( k + 1) c e (cid:48) ,k , (C.15)we get the bound (recall that M n ≤ ) ˜ c e (˜ x e,n ) ≤ (cid:88) e (cid:48) ∈ p ˜ c e (cid:48) (˜ x e,n ) ≤ (cid:88) e (cid:48) ∈ p (cid:48) ˜ c e (cid:48) (˜ x e,n ) ≤ (cid:88) e (cid:48) ∈ p (cid:48) ˜ c e (cid:48) ( M n ) ≤ BM qn . (C.16)In particular, letting ˜ c = min e ∈E slow ( q e + 1) c e,q e we have ˜ c · (˜ x e,n ) q e ≤ ( q e + 1) c e,q e · (˜ x e,n ) q e ≤ ˜ c e (˜ x e,n ) ≤ BM qn . (C.17)Now, for n large we have BM qn / ˜ c ≤ and since q e ≤ q slow we obtain ˜ x e,n ≤ ( BM qn / ˜ c ) /q slow . Combining this latter bound with (C.16), and denoting D = B ( B/ ˜ c ) /q slow |E slow | , we deduce L I n = (cid:88) e ∈E slow q − (cid:88) k = q e q − kk + 1 c e,k · (˜ x e,n ) k +1 HEN IS SELFISH ROUTING BAD? 33 ≤ (cid:88) e ∈E slow q − (cid:88) k = q e ( k + 1) c e,k · (˜ x e,n ) k +1 ≤ (cid:88) e ∈E slow ˜ x e,n ˜ c e (˜ x e,n ) ≤ (cid:88) e ∈E slow ( BM qn / ˜ c ) /q slow BM qn ≤ DM q + q/q slow n . (C.18)Plugging (C.18) and (C.13) into (C.11) we get PoA(Γ n ) = L n ( y ∗ n )Opt(Γ n ) ≤ Opt(Γ n ) + GM q +2 n + DM q + q/q slow n Opt(Γ n ) . (C.19)Now, if we set H = min e ∈E c e,q e , we have the following lower bound for theoptimal cost Opt(Γ n ) = (cid:88) e ∈E ˜ x e,n · c e (˜ x e,n ) ≥ H (cid:88) e ∈E (˜ x e,n ) q e +1 . (C.20)We claim that the latter is of order at least O ( M q +1 n ) . Indeed, let us take ε > with (cid:80) i ∈I tight λ in ≥ ε for sufficiently large n . For each n ∈ N we may find i ∈ I tight such that λ in ≥ ε/ |I tight | and, similarly, there exists a path p ∈ P i with ˜ y p,n ≥ / |P i | ≥ / |P| . Then, setting κ = 1 / ( |I tight | × |P| ) we have ζ e (˜ y n , λ n ) ≥ κε andtherefore ˜ x e,n ≥ M n κε for all e ∈ p . For n large we may assume that M n κε ≤ and,since the path p contains at least one edge e ∈ p with q e ≤ q , setting ¯ H = H ( κε ) q +1 we get Opt(Γ n ) ≥ H ( M n κε ) q e +1 ≥ ¯ HM q +1 n . (C.21)This lower bound, combined with (C.19), yields (C.9) with K = G/ ¯ H and K a = D/ ¯ H . We conclude by noting that when E slow = ∅ we have L I n = 0 andtherefore we may take K a = 0 . (cid:3) C.2.
Rates in the heavy traffic regime.
We proceed with the proof of Proposi-tion 7.3 on the heavy traffic rates in the case of a Pigou network.
Proof of Proposition 7.3.
Let x denote the flow on edge e . At equilibrium, thecosts on both edges must be equal so ( x ∗ ) d = ( M − x ∗ ) d , which is equivalent to x ∗ + ( x ∗ ) d /d = M . It thus follows that x ∗ = M d /d (1 + o (1)) , (C.22)implying in turn that the equilibrium cost Eq(Γ M ) = M · c ( x ∗ ) = M · c ( M − x ∗ ) scales as Eq(Γ M ) = M · (cid:104) M − M d /d (1 + o (1)) (cid:105) d = M d − d M d + d /d + o ( M d + d /d ) . (C.23)Similarly, if ˜ x is the optimal flow on edge e , both edges have the same marginalcost, namely (1 + d )˜ x d = (1 + d )( M − ˜ x ) d , (C.24)and hence ˜ x = ( θM ) d /d (1 + o (1)) , (C.25) where we set θ = (cid:18) d d (cid:19) /d . (C.26)Therefore, the optimal social cost scales as Opt(Γ M ) = ( M − ˜ x ) · c ( M − ˜ x ) + ˜ x · c (˜ x )= (cid:2) M − ( θM ) d /d (1 + o (1)) (cid:3) d +1 + (cid:2) ( θM ) d /d (1 + o (1)) (cid:3) d +1 = M d +1 − ( d + 1) M d ( θM ) d /d + ( θM ) d + d /d + o ( M d + d /d )= M d +1 − θ d /d [( d + 1) − θ d ] M d + d /d + o ( M d + d /d ) , (C.27) = M d +1 − ( b + d ) M d + d /d + o ( M d + d /d ) , (C.28)where b is defined as in (7.11).Combining the previous expressions, we then get PoA(Γ M ) = Opt(Γ M ) + bM d + d /d + o ( M d + d /d )Opt(Γ M ) (C.29) = 1 + bM d + d /d + o ( M d + d /d ) M d +1 + o ( M d +1 )= 1 + bM − a + o (cid:0) M − a (cid:1) , (C.30)which establishes the first part of Proposition 7.3. Finally, the positivity of b > ,follows again from the fact that (1 − g/x ) x increased with x when g > . (cid:3) The following more general result subsumes Theorem 7.4.
Theorem C.2.
Let Γ n be a sequence of nonatomic routing games satisfying theassumptions of Theorem 6.2, with M n → ∞ . Suppose further that the edge costsare polynomials as in (7.1) , and let a = 1 − d/d slow with d and d slow given by (7.12d) and (7.14) . Then there exist non-negative constants K ≥ and K a ≥ such that PoA(Γ n ) ≤ K M n + K a M an (C.31) with K a = 0 whenever E slow = ∅ .Proof. The proof follows a similar pattern as the one of Theorem C.1. Let again y ∗ n , ˜ y n ∈ Y be an equilibrium and an optimum for Γ n , respectively. Denote x ∗ e,n = M n ζ e ( y ∗ n , λ n ) and ˜ x e,n = M n ζ e (˜ y n , λ n ) the corresponding induced edge flows. Asbefore, the social cost of y ∗ n can be estimated as L n ( y ∗ n ) = (cid:88) e ∈E x ∗ e,n c e ( x ∗ e,n ) = (cid:88) e ∈E d e (cid:88) k = q e c e,k · ( x ∗ e,n ) k +1 = (cid:88) e ∈E d e (cid:88) k = q e (cid:20) d + 1 k + 1 + k − dk + 1 (cid:21) c e,k · ( x ∗ e,n ) k +1 = ( d + 1) (cid:88) e ∈E C e ( x ∗ e,n ) + (cid:88) e ∈E d e (cid:88) k = q e k − dk + 1 c e,k · ( x ∗ e,n ) k +1 ≤ ( d + 1) (cid:88) e ∈E C e (˜ x e,n ) + (cid:88) e ∈E slow x ∗ e,n · c e ( x ∗ e,n ) , (C.32) HEN IS SELFISH ROUTING BAD? 35 where in the last inequality we used the fact that x ∗ e,n minimizes the first sum,while in the double sum we dropped the edges e (cid:54)∈ E slow since ( k − d ) / ( k + 1) ≤ for all k ≤ d e ≤ d , and we used the inequality ( k − d ) / ( k + 1) ≤ to bound theremaining terms e ∈ E slow by factoring out x ∗ e,n and using the expression (7.1) for c e ( x ) . Now, the first sum in (C.32) can be further bounded as ( d + 1) (cid:88) e ∈E C e (˜ x e,n ) = (cid:88) e ∈E d e (cid:88) k = q e d + 1 k + 1 c e,k · (˜ x e,n ) k +1 = (cid:88) e ∈E d e (cid:88) k = q e c e,k · (˜ x e,n ) k +1 + (cid:88) e ∈E d e (cid:88) k = q e d − kk + 1 c e,k · (˜ x e,n ) k +1 ≤ Opt(Γ n ) + (cid:88) e ∈E d − (cid:88) k = q e d − kk + 1 c e,k · (˜ x e,n ) k +1 , where in the inequality we used the optimality of ˜ x n for the first sum and wedropped the negative terms in the second sum. Putting all this together we obtainthe bound L n ( y ∗ n ) ≤ Opt(Γ n ) + (cid:88) e ∈E d − (cid:88) k = q e d − kk + 1 c e,k · (˜ x e,n ) k +1 + (cid:88) e ∈E slow x ∗ e,n · c e ( x ∗ e,n ) . (C.33)Now, call L I n the first double sum and L II n the last sum in (C.33). In order tobound L I n we assume that n is large enough so that M n ≥ . Then, denoting G = (cid:88) e ∈E d − (cid:88) k = q e d − kk + 1 c e,k (C.34)and using the fact that ˜ x e,n ≤ M n , we can bound L I n as L I n ≤ (cid:88) e ∈E d − (cid:88) k = q e d − kk + 1 c e,k · M k +1 n ≤ GM dn . (C.35)In order to bound L II n we note that this term vanishes whenever E slow is empty.Otherwise, consider any edge e ∈ E slow that contributes to the sum with x ∗ e,n > .Since y ∗ n is an equilibrium, the edge e must belong to a path p ∈ P i with minimalcost for some i ∈ I . Hence, taking any alternative path p (cid:48) ∈ P i which is not slow,and denoting B = (cid:88) e (cid:48) (cid:54)∈E slow d e (cid:48) (cid:88) k = q (cid:48) e c e (cid:48) ,k , (C.36)we get the bound c e ( x ∗ e,n ) ≤ (cid:88) e (cid:48) ∈ p c e (cid:48) ( x ∗ e,n ) ≤ (cid:88) e (cid:48) ∈ p (cid:48) c e (cid:48) ( x ∗ e,n ) ≤ (cid:88) e (cid:48) ∈ p (cid:48) c e (cid:48) ( M n ) ≤ BM dn . (C.37)In particular, letting c = min e ∈E slow c e,d e we have c · (cid:0) x ∗ e,n (cid:1) d e ≤ c e,d e · (cid:0) x ∗ e,n (cid:1) d e ≤ c e ( x ∗ e,n ) ≤ BM dn . (C.38)Now, for n large we have BM dn /c ≥ and since d e ≥ d slow we get x ∗ e,n ≤ (cid:0) BM dn /c (cid:1) /d slow . Combining this latter bound with (C.37), and denoting D = B ( B/c ) /d slow |E slow | , we deduce L II n = (cid:88) e ∈E slow x ∗ e,n · c e ( x ∗ e,n ) ≤ (cid:88) e ∈E slow (cid:0) BM dn /c (cid:1) /d slow BM dn ≤ DM d + d/d slow n . (C.39)Plugging (C.35) and (C.39) into (C.33) we get PoA(Γ n ) = L n ( y ∗ n )Opt(Γ n ) ≤ Opt(Γ n ) + GM dn + DM d + d/d slow n Opt(Γ n ) . (C.40)Now, if we set H = min e ∈E c e,d e , we have the following lower bound for theoptimal cost Opt(Γ n ) = (cid:88) e ∈E ˜ x e,n · c e (˜ x e,n ) ≥ H (cid:88) e ∈E (˜ x e,n ) d e +1 . (C.41)We claim that the latter is of order at least O ( M d +1 n ) . Indeed, let us take ε > with (cid:80) i ∈I tight λ in ≥ ε for sufficiently large n . For each n ∈ N we may find i ∈ I tight such that λ in ≥ ε/ |I tight | and, similarly, there exists a path p ∈ P i with ˜ y p,n ≥ / |P i | ≥ / |P| . Then, setting κ = 1 / ( |I tight | × |P| ) we have ζ e (˜ y n , λ n ) ≥ κε andtherefore ˜ x e,n ≥ M n κε for all e ∈ p . For n large we may assume that M n κε ≥ and,since the path p contains at least one edge e ∈ p with d e ≥ d , setting ¯ H = H ( κε ) d +1 we get Opt(Γ n ) ≥ H ( M n κε ) d e +1 ≥ ¯ HM d +1 n . (C.42)This lower bound, combined with (C.40), yields (C.31) with K = G/ ¯ H and K a = D/ ¯ H . We conclude by noting that when E slow = ∅ we have L II n = 0 andtherefore we may take K a = 0 . (cid:3) References
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