Phase Transitions of the Price-of-Anarchy Function in Multi-Commodity Routing Games

We consider the behavior of the price of anarchy and equilibrium flows in nonatomic multi-commodity routing games as a function of the traffic demand. We analyze their smoothness with a special attention to specific values of the demand at which the support of the Wardrop equilibrium exhibits a phase transition with an abrupt change in the set of optimal routes. Typically, when such a phase transition occurs, the price of anarchy function has a breakpoint, \ie is not differentiable. We prove that, if the demand varies proportionally across all commodities, then, at a breakpoint, the largest left or right derivatives of the price of anarchy and of the social cost at equilibrium, are associated with the smaller equilibrium support. This proves -- under the assumption of proportional demand -- a conjecture of O'Hare et al. (2016), who observed this behavior in simulations. We also provide counterexamples showing that this monotonicity of the one-sided derivatives may fail when the demand does not vary proportionally, even if it moves along a straight line not passing through the origin.


Introduction
Wardrop equilibria for nonatomic routing games provide a mathematical description of how traffic distributes over a network used by a large number of agents who do not coordinate or cooperate.Standard examples are road and telecommunication networks.This model is a special case of a nonatomic congestion game where the resources are the edges of a directed multigraph, and each edge e has an associated cost or delay c e (x e ) given by a non-decreasing and continuous function of the traffic load x e on the edge.Different types of users have different origin-destination (OD) pairs, typically called commodities.A nonnegative traffic demand is associated to each OD pair, and users can choose different paths to go from their origin to their destination.In doing so, they generate a flow over the network.A Wardrop equilibrium is a distribution of traffic which satisfies the demand on each OD pair in a such a way that all the users of any given type only use paths of minimal cost.
Traffic demand is often difficult to predict as it is eminently a dynamic phenomenon, with variations between days of the week and across different periods within any given day.Moreover, traffic demand increases on a longer time scale as a result of the growth of the population and car-ownership.As a consequence, it is relevant to analyze the sensitivity of Wardrop equilibria under different demand scenarios to understand the extent to which this equilibrium notion approximates real traffic patterns.Using this analysis, a planner can anticipate the evolution of equilibria and better design policy interventions and infrastructure modifications that the increasing traffic level requires.These observations have motivated several authors to investigate how the Wardrop equilibrium is affected by variations in the traffic demands, in terms of its continuity, smoothness, and monotonicity properties.In Section 1.2 we present a brief overview of previous works that have addressed these questions.
On the other hand, Wardrop equilibria are known to be inefficient in the sense that there may exist other traffic distributions that produce a lower total delay.Inefficiency of equilibria is usually measured via the price of anarchy (PoA), i.e., the ratio between the total equilibrium cost and the optimum total cost.This provides a benchmark for the maximal improvement that could be achieved in terms of social cost.While the initial studies on the PoA focused on finding tight bounds under worst case scenarios, recent work has turned to analyze the behavior of the PoA as a function of the traffic demand.This function is nonsmooth at demands 1, 3, 4, 6 and 27/2.Between these breakpoints the set of optimum paths remains stable and the PoA behaves smoothly.At demand 4 the set of optimum paths exhibits a contraction and the derivative jumps up.At all the other breakpoints the set of optimum paths undergoes an expansion and the derivative jumps down.
Fig. 1 shows a typical profile of the PoA as the demand varies: it is close to 1 when the demand is either very small or very large, and it oscillates in the middle range with occasional sharp kinks and smooth critical points at local minima.As reported by Youn et al. (2008), this type of profile is also observed in real networks such as Boston, London, and New York, where in addition the PoA values remain far from the worst case estimates.The fact that the PoA is close to 1 for both low and high demands was formally established under fairly general assumptions in Colini-Baldeschi et al. (2019, 2020) and Wu et al. (2021).For the intermediate range, O'Hare et al. (2016) observed that the kinks are associated with a sudden change in the set of optimum paths at equilibrium, and stated a conjecture that we analyze in this paper.A partial support for this was given by Cominetti et al. (2021): in single-commodity routing games with affine cost functions, and in demand intervals where the set of optimum paths at equilibrium does not change, the PoA is differentiable and exhibits at most one critical point, which must be a local minimum.In this paper we further explore these questions for multi-commodity networks and more general cost functions.
1.1.Our Results.We consider nonatomic routing games and analyze the behavior of the equilibrium flows, the social cost, and the PoA, as the traffic demands vary on potentially multiple OD pairs.The whole paper deals with constrained routing games in which every OD pair is allowed to use a restricted set of paths and not necessarily all the paths connecting its origin and destination.The main takeaway of our study is that when the demands vary with constant proportions across all OD pairs, the behavior of the PoA is analogue to what is observed in the single-commodity case.However, this is not what happens if the demand varies without keeping constant proportions among the different OD pairs.Specifically, we begin by generalizing to multiple OD pairs some results on the continuity and differentiability of equilibria as a function of the demands.We show that if the cost functions are C 1 with strictly positive derivative, and the demand varies along a smooth curve parameterized by a single variable t, then the equilibrium loads on each edge and the PoA are also C 1 as functions of t, as long as the demand curve stays in a region where the set of minimum-cost paths does not change.
Typically-although not always-this smoothness fails at points where the set of optimum paths changes and the equilibrium exhibits a phase transition.We call such points breakpoints.For networks with C 1 cost functions having strictly positive derivatives and proportionally varying demands, we prove a conjecture stated in O' Hare et al. (2016), showing that at a breakpoint the largest between the left and right derivatives of the PoA corresponds to the smaller set of optimum paths (see the example in Fig. 1).We also show with an example that the full conjecture, as originally stated in O' Hare et al. (2016), might fail when the demands do not vary proportionally, even if these demands move along an affine straight line in the space of demands.1.2.Related Work.The definition of equilibrium in nonatomic routing games that we adopt in this paper is due to Wardrop (1952).The characterization of Wardrop equilibria as solutions of a convex optimization problem was first described in Beckmann et al. (1956); and the first attempts to explicitly compute such equilibria were due to Tomlin (1966) for affine costs, and to Dafermos and Sparrow (1969) for general convex costs.For surveys on the topic the reader is referred to Florian and Hearn (1995) and Correa and Stier-Moses (2011).
Several authors have considered how the equilibrium flows and costs vary with the traffic demand.Hall (1978) proved that the equilibrium cost of any given OD pair is an increasing function of the amount of traffic on that OD pair.On the other hand Fisk (1979) presented an example showing that the total social cost can decrease even if the total demand increases.Further analysis of this paradoxical phenomenon can be found in Dafermos and Nagurney (1984).Concerning the smoothness of equilibrium flows, Patriksson (2004) gave necessary and sufficient conditions for the directional differentiability of equilibria, and Josefsson and Patriksson (2007) proved the directional differentiability of equilibrium costs, but observed that this might fail for the equilibrium edge loads.Concerning the set of optimum paths, Englert et al. (2010) showed examples of single OD routing games where an arbitrarily small increase in traffic demand generates a complete change of the set of paths used at equilibrium, although the change on the edge loads remains small.Moreover, for polynomial costs of degree at most d, if the demand increases by ε then the equilibrium costs increase at most by a multiplicative factor (1 + ε) d .The extension of this latter result to multiple OD pairs can be found in Takalloo and Kwon (2020).Pigou (1920) was probably the first author who studied the inefficiency of equilibria in nonatomic routing games.The formal definition of PoA to measure this inefficiency is due to Koutsoupias and Papadimitriou (1999), and acquired its name in Papadimitriou (2001).Most of the early literature on the PoA concentrated on establishing sharp bounds for the PoA for different classes of games, such as congestion games and routing games.In their landmark paper, Roughgarden and Tardos (2002) proved that in every nonatomic congestion game with affine costs the PoA is bounded above by 4/3 and showed that this bound is sharp.This result was generalized to polynomial cost functions of maximum degree d in Roughgarden (2003), showing that the PoA grows as Θ(d/ log d).Other results of this type can be found in Dumrauf and Gairing (2006), who focused on cost functions that are sums of monomials whose degrees are in a specified range, and in Roughgarden and Tardos (2004), who dealt with the class of differentiable cost functions c(x) such that x c(x) is convex.Less regular cost functions and different notions of social cost were studied in Correa et al. (2004Correa et al. ( , 2007Correa et al. ( , 2008)).
Several papers addressed the computation of the PoA in real networks.Monnot et al. (2017) analyzed data from a large sample of Singaporean students who commute to go to school, observing a PoA which is much smaller than the theoretical worst case bounds.Also, Youn et al. (2008Youn et al. ( , 2009) ) studied the PoA in the networks of Boston, London, and New York when all the OD traffic demands are scaled by the same factor, observing again values of the PoA consistently smaller than the worst case bounds.The behavior of the PoA in these three cities shows a common pattern: it is close to 1 both for small and large demands, and it oscillates in the middle range with sharp kinks at the local maxima and smooth critical points at the local minima.O'Hare et al. (2016) noted that these kinks arise when the set of optimum paths at equilibrium undergoes an expansion or a contraction, and stated the conjecture that we analyze later in this paper.
Recent efforts attempted to mathematically explain the empirical behavior of the PoA observed in the studies mentioned above.An algorithm for computing Wardrop equilibria as a function of the traffic demand was given in Klimm andWarode (2019, 2022), in the case of piecewise linear cost functions.The efficiency of equilibria when the demand is close to zero or infinity (light and heavy traffic) was analyzed in Colini-Baldeschi et al. (2019, 2020) and Wu et al. (2021).Colini-Baldeschi et al. (2019) considered the case of single OD parallel networks showing that in heavy traffic the PoA converges to 1 when the cost functions are regularly varying.These results were extended to general networks in Colini-Baldeschi et al. (2020), considering also the behavior of PoA in the light traffic regime.The case of heavy traffic was treated with different techniques by Wu et al. (2021).The study of the intermediate range of demands, when the traffic is neither light nor heavy, was the main objective of Cominetti et al. (2021), who proved that for affine cost functions the shape of the PoA function is the one observed in the empirical studies.Finally, we mention the recent paper Wu and Möhring (2022) which established the continuity of the PoA in non atomic routing games, as a function of various parameters including not only the demands, but also the cost functions.
1.3.Organization of the paper.Section 2 recalls the model to be studied, and fixes the notations used thereafter.Section 3 presents some regularity and smoothness results for the equilibrium loads, the social cost at equilibrium, the optimum social cost, and the price of anarchy.Section 4 discusses the behavior of these quantities around breakpoints, where smoothness is typically lost.The proofs of the more technical lemmas are presented in Appendix A, whereas Appendix B contains a list of the symbols used in the paper.

Network games with variable demand
This section introduces the model and notations used throughout the paper.We consider routing games with multiple OD pairs, and we study their equilibria as functions of the multivariate demand, concentrating on the case where the demand vector varies along a smooth curve in the space of demands.In particular, we study the simplest and natural case where the demands are scaled by a common factor so that they change proportionally across OD pairs.
Let G := (V, E) be a directed multigraph with vertex set V and edge set E, where each edge e ∈ E has a continuous and nondecreasing cost function c e : [0, +∞) → [0, +∞), with c e (x e ) representing the travel time of traversing the edge e when the load on that edge is x e .An origin-destination (OD) h is defined by a triple (O (h) , D (h) , P (h) ), where O (h) ∈ V is the origin vertex, D (h) ∈ V is the destination vertex, and P (h) is a subset of simple paths from O (h) to D (h) .The symbol H denotes the set of OD's.The sets P (h) are assumed nonempty but could be smaller than the sets of all paths from origin to destination in the graph G.In order to simplify our notations and formulas, we assume that these sets are pairwise disjoint so that no path is common to two different OD's.In particular, when considering the flow of a path, we do not need to specify to which OD the traffic flow belongs.We observe that our results do not depend on this simplifying assumption (see Remark 4.9).The set of all feasible routes is the disjoint union of the sets P (h) for h ∈ H, denoted by (2.1) where G is a directed multigraph, c is the vector of nondecreasing and continuous edge costs, and H is the set of OD pairs.
A traffic demand for the routing game structure (G, c, H) is given by a vector µ = (µ (h) ) h∈H ∈ R H + .Every such µ determines an associated set of feasible flows f = (f p ) p∈P , given by Each flow f ∈ F µ induces in turn a load profile x = (x e ) e∈E , where x e represents the aggregate traffic over the edge e ∈ E. The symbol X µ denotes the set of all load profiles induced by some f ∈ F µ .Explicitly, if (2.3) then the set X µ is the set of load vectors x = (x e ) e∈E such that there exist path flows f = (f p ) p∈P satisfying the linear constraints More concisely, (2.4) and (2.5) can be written in matrix form as ∆f = x, Sf = µ. (2.7) When the traffic is described by a flow vector f ∈ R P + , the travel time on a path p ∈ P is where x is the load profile induced by f .The total delay induced by a flow f is called the social cost and is denoted by 2.1.Equilibria and price of anarchy.Let (G, c, H) be a routing game structure.For each demand vector µ ∈ R H + , we obtain a classical nonatomic routing game.We recall that a Wardrop equilibrium is a flow f * ∈ F µ for which there exists λ : h) for every h ∈ H and all p ∈ P (h) with f * p = 0. (2.10) The quantity λ (h) is called the equilibrium cost associated to the OD pair h ∈ H.
From Beckmann et al. (1956) we know that for each demand vector µ ∈ R H + , the set of load profiles induced by all equilibria coincides with the set of optimum solutions of the minimization problem min An equilibrium load profile x * induces equilibrium edge costs τ e := c e (x * e ).In Fukushima (1984), the equilibrium edge costs were shown to be optimum solutions of the strictly convex dual program min (2.12)where C * e ( • ) is the Fenchel conjugate of C e ( • ), which is strictly convex.Thus, the edge costs τ e at equilibrium are uniquely defined for each demand vector µ, and are the same in every equilibrium load profile x * .This defines maps µ → τ e (µ) that assign these equilibrium edge costs to each demand vector µ.It follows that, at equilibrium, the cost c p (µ) = e∈P τ e (µ) of a path p ∈ P, as well as the equilibrium costs λ (h) (µ) = min p∈P (h) c p (µ) for each h ∈ H, are also uniquely defined for each µ and do not depend on the specific equilibrium loads x * that we might consider.Moreover, when the edge costs are strictly increasing, the equilibrium loads are also uniquely determined by x * e (µ) = c −1 e (τ e (µ)).For later reference, we introduce the concept of active regime, which associates to each µ the set of minimum-cost paths at equilibrium.Definition 2.2.Consider the routing game structure (G, c, H).
The active regime at demand µ ∈ R H + is the set All of this allows to define the equilibrium social cost with demand µ ∈ R H + as (2.14) where f * ∈ F µ is any equilibrium flow with induced equilibrium load x * ∈ X µ .Also, the optimum social cost for a demand vector µ ∈ R H + is defined as x e c e (x e ), (2.15) and the price of anarchy (2.16)

Continuity and smoothness of equilibria
Under fairly general conditions, one can prove that the PoA, as a function of the demand, is smooth at all points where the active regime is locally constant.To this end, we first proceed to establish some regularity results for the equilibrium flows, equilibrium costs, and optimum social cost.The proofs of Lemmas 3.1-3.3below are presented in Appendix A.
We begin by noting that the equilibrium costs depend continuously on the demands.This extends Cominetti et al. (2021, proposition 3.1) to the multi-OD setting, as well as Hall (1978), who considered the case of strictly increasing costs.
The next lemma establishes the regularity of the optimum social cost when the edge costs are regular.
Lemma 3.2.Let (G, c, H) be a routing game structure.If the cost functions c e are C 1 and the functions x e → x e c e (x e ) are convex, then the optimum social cost function µ → SC(µ) is convex and C 1 everywhere.
The next lemma establishes a set of sufficient conditions that guarantee the differentiability of the equilibrium loads.To this end we analyze a minimization problem without sign constraints on the variables.For this reason, we extend the edge cost functions to the whole R in such a way that the extension is continuous, non decreasing and lim x→−∞ c e (x) < 0 for all edges e ∈ E. For the second part of the lemma, we also require this extension to be differentiable with strictly positive derivative.
Given a routing game structure (G, c, H) and a fixed regime R, we consider the problem: where the feasible set X R µ comprises all (signed) load vectors x ∈ R E induced by some flow f ∈ R P with support in R, that is We also consider the optimal value function for the perturbed minimization problem s.t.Sf = µ, x = ∆f + ξ, and f p = ω p for all p / ∈ R.
Let us insist that in (P R µ ) and (P R µ,ξ,ω ) the variables have no sign constraints.
Lemma 3.3.Let (G, c, H) be a routing game structure and let R be an arbitrary fixed regime.Then, The edge costs η e := c e (x e ) are the same for every optimum x, and for each h ∈ H there exists a path cost m (h)  such that e∈p η e = m (h) for all p ∈ P (h) ∩ R.Moreover, the optimal value function V R ( • ) is everywhere finite, convex, and differentiable at (µ, 0, 0)with ∇V R (µ, 0, 0) = (m, η, ν) where If the cost functions are strictly increasing, then (P R µ ) has a unique optimum solution x R (µ).Moreover, if the costs c e are C 1 with strictly positive derivative, then µ The remaining results of this section concern the case where the demand varies on a smooth curve, parametrized by a scalar variable.For instance, a linear demand function represents a situation where the demands move proportionally across different OD, that is, µ(t) = t r, for some fixed vector r ∈ R H + .
Proposition 3.5.Let (G, c, H, µ( • )) be a nonatomic routing game with a continuously differentiable demand function µ( • ).Let the cost functions c e be C 1 with strictly positive derivative.If the active regime P(µ(t)) is constant on a neighborhood of t 0 ∈ R + , then the equilibrium load map t → x * (µ(t)) is continuously differentiable on a neighborhood of t 0 .In particular the equilibrium costs λ (h) (µ(t)) h∈H and the social cost at equilibrium SC * (µ(t)) are all of class C 1 on a neighborhood of t 0 .
Proof.A sufficient condition for a feasible pair (x, f ) to be an optimal solution of (P R µ ), is the existence of multipliers (m, η, ν) such that c e (x e ) = η e (∀e ∈ E), (3.3) For any demand µ the equilibrium load vector x * (µ) satisfies these conditions for R = P(µ), with η e = τ e = c e (x e ), m (h) = λ (h) (µ), and ν p = e∈p τ e − λ (h) (µ) for p ∈ R.This shows that the equilibrium load x * (µ) coincides with the unique optimum solution x R (µ) of (P R µ ) for R = P(µ).Now, by assumption P(µ(t)) ≡ R is constant for t near t 0 , so that x * (µ(t)) = x R (µ(t)), which is continuously differentiable as a composition of the C 1 demand function t → µ(t) and the map µ → x R (µ), which is also C 1 in view of Lemma 3.3 (b).
Observe that the previous proposition requires the active regime P(µ(t)) to be constant near t 0 ∈ R + but only along the demand curve, and not necessarily in a neighborhood of µ(t 0 ) in the ambient space

R H
+ .This covers some special situations in which the curve µ( • ) may slide along the boundary between two regions with different active regimes.
Proposition 3.5 is similar to O' Hare et al. (2016, proposition 4.4), which assumes that the edge costs are convex and twice differentiable, with strictly positive derivatives except possibly at zero.Our result does not assume convexity but requires strictly positive derivatives everywhere including at zero.The following example shows that these assumptions are close to minimal: the presence of a single edge with a non-convex cost whose derivative vanishes at a single point, may give rise to non-smooth equilibrium loads, even if the set of optimum paths is locally constant.
Example 3.6.In this example with strictly increasing C 1 cost functions, the equilibrium flow is nondifferentiable at some point which is not a breakpoint.Consider a two link parallel network where the cost functions are (3.5) These functions are C 1 everywhere and their derivatives vanish at x = 1.The equilibrium flows are: Both flows are nondifferentiable at µ = 2. Nonsmoothness can also be observed at a point where a flow vanishes.Consider the Wheatstone network in Fig. 2(a) with All three paths are active in a neighborhood of µ = 2.When the demand approaches 2 from below, the flow on the zig-zag path decreases, and it vanishes when µ = 2, whereas it increases with positive derivative after 2. A plot of the flow on the zig-zag path for µ ∈ [1, 3] is shown in Fig. 2(b).Having a smooth equilibrium load profile allows us to choose a smooth equilibrium flow, as stated in the following lemma.
Lemma 3.7.Let (G, c, H, µ( • )) be a nonatomic routing game with a differentiable demand function.Let t → x(µ(t)) be a selection of equilibrium loads defined for t in a neighborhood of t 0 , and assumed to be differentiable at t 0 .Then, x(µ(t)) can be decomposed into a flow f (µ(t)) that is also differentiable at t 0 .
Proof.Let f 0 ∈ R P be an arbitrary path decomposition of x(µ(t 0 )).Then we set where ∆ and S are as in (2.7), and the symbol L + denotes the Moore-Penrose pseudoinverse of the matrix L. Since the pseudoinverse is also a matrix, we get the conclusion.
With these preliminaries we immediately obtain the announced smoothness of the function PoA.
Theorem 3.8.Let (G, c, H, µ( • )) be a nonatomic routing game with a differentiable demand function, whose cost functions c e are C 1 with strictly positive derivatives, and let the functions x e → x e c e (x e ) be convex.If the active regime P(µ(t)) is constant in a neighborhood of t 0 ∈ R, then the function t → PoA(µ(t)) is continuously differentiable in a neighborhood of t 0 .
Remark 3.9.Invoking Remark A.1 in the Appendix A, the strict positivity of the cost derivatives in Proposition 3.5 and Theorem 3.8 can be slightly weakened by assuming that the edge costs c e ( • ) are just strictly increasing and the set of edges e with c e (x e (µ 0 )) = 0 do not contain undirected cycles.

Behavior around breakpoints
In this section we describe the behavior of the functions SC * , and PoA around breakpoints.In particular we study their left and right derivatives when the demand vector depends on a single real parameter.To formally describe this situation we make use of the following concept: Definition 4.1.We say that t ∈ R + is a P-breakpoint for (G, c, H, µ( • )) if there exist ε > 0 such that P(µ(t)) is constant as a function of t over each of the intervals [ t − ε, t) and ( t, t + ε] with P(µ( t − ε)) = P(µ( t + ε)).In this case, we use the symbols P( t− ) := P(µ( t − ε)), P( t+ ) := P(µ( t + ε)). (4.1) O'Hare et al. ( 2016), considered two regimes around a breakpoint and observed that, empirically, if one of these two regimes contains the other, then the derivatives of SC * and PoA associated to the smaller regime are larger than the derivatives in the larger regime.They formalized this idea in their conjectures 4.5, 4.6, and 4.9.Using our language and notation, we can summarize these conjectures in the following statement, where we denote g( t− ) := lim t→ t− g(t) and g( t+ ) := lim t→ t+ g(t).2016)).Let (G, c, H, µ( • )) be a routing game where the cost functions are continuous, differentiable, strictly increasing with positive second derivative, and the demand function t → µ(t) is piecewise affine and componentwise nondecreasing with t.If t ∈ R + is a P-breakpoint, then the following hold: As illustrated by the next example with affine costs and an affine demand function, this conjecture does not hold in the most general form stated above.However, we will see later that a restricted form of the conjecture is indeed valid.
Example 4.3.Consider the network in Fig. 3(a) studied in Fisk (1979), with affine cost functions and three OD pairs whose demands and routes as shown in the next table : OD Demand Feasible routes (a, b) where the map t → µ(t) is affine and only the demand of (a, c) increases with t.The following table shows the path flows and social cost at equilibrium for all t ≥ 0: We observe that at t = 11 the active regime experiences an expansion, where the route p 3 becomes optimal for the pair (a, c) at equilibrium.However, the right derivatives of SC * •µ and PoA •µ are strictly larger than the respective left derivatives.This provides a counter-example to Conjecture 4.2 (a)-(b) as originally stated in O' Hare et al. (2016).Note also that, as predicted by Lemma 3.2, both SC • µ and PoA •µ are smooth at t = 56, despite the presence of a phase transition in the social optimum.
We now proceed to prove that a restricted version of Conjecture 4.2 is true, where the demand function is assumed to be linear, so that the proportion among the demands in each OD pair is maintained constant as the total demand increases.
Proposition 4.4.Let (G, c, H, µ( • )) be a nonatomic routing game with C 1 and strictly increasing costs and linear demand function µ(t) = t r with r ∈ R H + .Let t ∈ R + be a P-breakpoint with corresponding active regimes R − := P( t− ) and R + := P( t+ ) over I − := ( t − ε, t) and I + := ( t, t + ε).Suppose that there exist continuously differentiable maps f − , f + : ( t − ε, t + ε) → R P + such that • the vector f − (t) is an equilibrium flow of demand µ(t) for all t ∈ I − • the vector f + (t) is an equilibrium flow of demand µ(t) for all t ∈ I + .
Then, the statements (a) and (b) in Conjecture 4.2 hold with weak inequalities.
Proof.For notational simplicity, for each t ∈ ( t − ε, t + ε) we let λ (h) (t) = λ (h) (µ(t)) denote the equilibrium costs and SC * (t) = SC * (µ(t)) the social cost at equilibrium with demand µ(t), so that (4.3)By Lemma 3.1, the maps t → λ (h) (t) are continuous.They are also C 1 over both I − and I + , with well defined unilateral limits (λ (h) ) ( t− ) and (λ (h) ) ( t+ ) at t. Indeed, for each h ∈ H we may fix an active path p ∈ R + so that, for all t ∈ I + , we have λ (h) (t) = c p (f + (t)), which is smooth with a well defined limit for its derivative at t+ .A similar argument using f − ( • ) can be applied for t ∈ I − .Since µ (h) (t) = r (h) t, we can express the left and the right derivatives at t of the social cost as , where for t = t we define We now derive an alternative characterization for θ( t− ) and θ( t+ ).Let us consider the latter.For t ∈ I + we have that f (t) = f + (t) is an equilibrium flow.In particular, so that differentiating we get p∈P (h) f p (t) = r (h) .Therefore for t > t we have Now, for any inactive path p ∈ R + we have f p (t) = 0, so the inner sum in (4.6) can be restricted to the active paths p ∈ R + , for which λ (h) (t) = c p (f (t)) and This allows us to express θ(t) in the equivalent form where J c ( • ) is the Jacobian matrix of the path costs function c : R n + → R n + .Letting x denote the (unique) equilibrium load for the demand µ( t), a routine computation shows that this Jacobian matrix is positive semi-definite with (4.9) We claim that θ( t+ ) coincides with the optimum value of the convex quadratic program p∈P (h)   y p = r (h) for all h ∈ H, and y p = 0 for p / ∈ R + , (4.10) with optimum solution ȳ = f ( t).Indeed, f ( t) satisfies the constraints in (4.10).On the other hand, the optimality conditions are given by the vector equation (4.11)where the m (h) are Lagrange multipliers for the constraints p∈P (h) y p = r (h) for all h ∈ H, ν p are the multipliers for the constraints y p = 0 for every p / ∈ R + , γ p is the p-th vector of the canonical basis, and 1 (h) := p∈P (h) γ p is the vector whose p-th components are 1 for p ∈ P (h) and 0 otherwise.Now, since f (t) is an equilibrium flow with P(µ(t)) = R + for all t ∈ I + , we have where h(p) is the commodity of path p.By differentiating with respect to t and letting t → t, we get Comparing (4.13) and (4.11), we see that f ( t) satisfies the optimality conditions of (4.10) with multipliers m (h) = (λ (h) ) ( t+ ) and Since the quadratic program is convex, it follows that f ( t) is an optimum solution, and then the equality (4.10) between θ( t+ ) and the optimum value of the quadratic program follows by letting t → t+ in (4.8).
Repeating the argument for t ∈ I − , this time with f (t) = f − (t), we obtain that θ( t− ) can be characterized as the optimum value of p∈P (h)   y p = r (h) for all h ∈ H, and y p = 0 for p / ∈ R − .(4.14) Since the equilibrium load vector x is unique, using (4.9) we note that the objective function in problems (4.10) and (4.14) is the same, both for θ( t− ) and θ( t+ ).Thus, the only difference between these quadratic programs are the constraints y p = 0 for all p ∈ R − or p ∈ R + : a larger regime implies fewer constraints and thus a smaller optimum value.Explicitly, if . This establishes half of Conjecture 4.2 with weak inequalities.
Concerning the derivative of PoA •µ, when it exists, it is Using the continuity of SC * at µ( t) and the fact that SC is C 1 at µ( t) (Lemmas 3.1 and 3.2), we see that the statements regarding the price of anarchy in Conjecture 4.2 also hold with weak inequalities.
As a special instance of Proposition 4.4 we obtain the following more specific result, where the existence of a differentiable path of equilibrium flows assumed in this previous proposition is derived as a consequence of Lemmas 3.3 and 3.7.Proposition 4.5.Let (G, c, H) be a routing game structure where the cost functions are C 1 with strictly positive derivatives, and consider a linear demand function µ(t) = t r with r ∈ R H + .If t ∈ R + is a P-breakpoint, then the statements (a) and (b) in Conjecture 4.2 hold with weak inequalities.
Proof.Using Lemma 3.3 we have that for every regime R one can find a C 1 solution t → x R (µ(t)) to the problem (P R µ ), defined in an open neighborhood of t, and such that x R (µ(t)) coincides with the equilibrium x(µ(t)) whenever R(µ(t)) is equal to R. Furthermore, by applying Lemma 3.7 we can find C 1 flows associated to such load profiles.If we do that in the two cases of R = P( t− ) and R = P( t+ ), we obtain differentiable flows in a neighborhood of t that are equilibria wherever the active regime coincides with R. Applying Proposition 4.4 we obtain the result.
We next show that Conjecture 4.2 holds with strict inequalities when the cost functions are affine and nondecreasing, and the demand function is linear.
Proposition 4.6.Let (G, c, H) be a routing game structure with nondecreasing affine cost functions, and let the demand µ(t) = t r be a linear function of t ∈ R + with r ∈ R H + .If t ∈ R + is a P-breakpoint, then statements (a) and (b) in Conjecture 4.2 hold.
Proof.When the cost functions are affine, to the left and to the right of t we can choose equilibrium flows in affine form ) . A proof of this in the case of a single commodity can be found in Cominetti et al. (2021, proposition 4.1).The reasoning for the multi-commodity case is completely analogous.These functions are differentiable as functions of t so that Proposition 4.4 implies that (a) and (b) in Conjecture 4.2 hold with weak inequalities.It remains to show that the inequalities are strict.By arguing as in the proof of Proposition 4.4, we need to show that the inequality between θ( t− ) and θ( t+ ) is strict, which requires to compare the values of the problems (4.10) and (4.14).As one can deduce looking at Eq. (4.9), the two problems are exactly the problems P R + µ( t) and P R − µ( t) as in Lemma 3.3 for a game on the graph G, where the cost functions are the linear functions ce (x) = c e (x e ) • x, with xe = p e f − p (µ( t)) = p e f + p (µ( t)).In these two problems, the objective function is the same, but there are more constraints in the case associated to the smaller regime between R − and R + .Suppose for instance that R − is strictly contained in R + .Then the extra constraints in (4.14) with respect to (4.10) are y p = 0 for p ∈ R + \R − , with associated Lagrange multipliers ν p given as in (4.13), that is (4.17) The path p is active when t > t, but not when t < t.This implies that there exists ε > 0 such that because p is not active, and by continuity of path costs.By assumption, the cost functions are affine; hence the path costs and equilibrium costs are also affine in the variable t ∈ ( t − ε, t).This implies that the difference ) ) (t) is a strictly negative constant on ( t − ε, t).Hence, the Lagrange multiplier ν p assumes a strictly negative value at the optimum solution.Since the multipliers are unique for the problem P R − µ( t) by Lemma 3.3, relaxing the constraint y p = 0 in (4.14) will produce a strict decrease in the optimum value of the objective function, that is, θ( t− ) > θ( t+ ), which gives in turn the strict inequalities in the statement (a) of Conjecture 4.2.
A similar argument holds for case (a) in Conjecture 4.2 when R − strictly contains R + .
Remark 4.7.Proposition 4.5 can be generalized to networks with just strictly increasing cost functions, assuming that the set of edges e where c e (x e (µ( t)) = 0 form a graph without undirected cycles.
Remark 4.8.  h) .This operation does not change the equilibrium; hence the equilibrium loads on the original edges remain unchanged.Furthermore, this transformation does not create any new undirected cycle, thus, as noted in Remark 4.7 the results are still valid.
Example 4.10.At a P-breakpoint t where neither P( t− ) ⊂ P( t+ ) nor P( t− ) ⊃ P( t+ ) are true, all inequalities are possible between the left and the right derivatives of the social cost and the price of anarchy.In order to illustrate this phenomenon, consider the single OD network in Fig. 4 where one cost function depends on a parameter > 0. For every , there is a breakpoint at μ = 2 with neither P − (μ) ⊂ P + (μ) nor P − (μ) ⊃ P + (μ), and the left and right derivatives at μ of SC * and PoA are ranked in different order depending on the value of .There are four paths and the equilibrium flow vector f is given in the following table: where we can observe that lim which implies the same relations at μ = 2 for the left and right derivatives of SC * and PoA.
As shown in Proposition 4.6, with affine cost functions the inequalities of Proposition 4.4 can never hold with equality at a breakpoint.It is an open question whether this is also the case for convex costs or Bureau of Public Roads (BPR) costs.On the other hand, the following example shows that the inequalities of Proposition 4.4 can indeed hold with equality if we allow for non-convex costs.
Acknowledgments.Valerio Dose and Marco Scarsini are members of GNAMPA-INdAM.Their work was partially supported by the GNAMPA project CUP E53C22001930001 "Limiting behavior we have W (µ + z) = v µ (z) and in particular W (µ) = v µ (0), which we consider as the primal problem.Since ϕ µ is convex, we have that z → v µ (z) = W (µ + z) is also convex, from which we deduce that µ → W (µ) is convex.Moreover, the perturbation function ϕ µ yields a corresponding dual min where ϕ * µ is the Fenchel conjugate function, that is, Since W (µ ) is finite for all µ ∈ R H + , it follows that v µ (z) = W (µ + z) is finite for z in some interval around 0, and then the convex duality theorem implies that there is no duality gap and the subgradient ∇v µ (0) at z = 0 coincides with the optimum solution set S(D µ ) of the dual problem, that is, ∇W (µ) = ∇v µ (0) = S(D µ ).
We claim that the dual problem has a unique solution, which is exactly the vector of equilibrium costs λ(µ).Indeed, fix an optimum solution f * for v µ (0) = W (µ) and recall that this is just a Wardrop equilibrium.The dual optimum solutions are precisely the λ's in R H such that This equation can be written explicitly as f q = 0, from which it follows that f = f * is an optimum solution in the latter supremum.The corresponding optimality conditions are which imply that λ (h) is the equilibrium cost of the OD pair h for the Wardrop equilibrium, that is, is not only convex but also differentiable with gradient ∇W (µ) = λ(µ).
The continuity of the equilibrium edge costs τ e = τ e (µ) is a consequence of Berge's maximum theorem (see, e.g., Aliprantis and Border, 2006, Section 17.5).Indeed, as explaind in Fukushima (1984), the equilibrium edge costs are optimum solutions for the strictly convex dual program (2.12).Hence, since the objective function is jointly continuous in (τ , µ), Berge's theorem implies that the optimum solution correspondence is upper-semicontinous.However, in this case the optimum solution is unique, so that the optimum correspondence is single-valued, and, as a consequence, the equilibrium edge costs τ e (µ) are continuous.
Proof of Lemma 3.2.The assumptions on the cost functions imply that the optimum flows are the Wardrop equilibria of the game (G, c, H), where c e (x e ) := c e (x e ) + x e c e (x e ). (A.4) For every e ∈ E, the function c e is continuous and nondecreasing.For this modified game, the minimal value of the Beckmann potential is x e c e (x e ), (A.5) which is the optimum social cost SC(µ).The result then follows by noting that the minimal value of the Beckmann potential is continuously differentiable with its gradient equal to the vector equilibrium costs, which are continuous (see Lemma 3.1).
Proof of Lemma 3. so that Φ is inf-compact, and therefore the set of minima of (P R µ,ξ,ω ) is nonempty (see for example (Rockafellar, 1997, Theorem 9.2)).This shows in particular that the value function V R ( • ) is finite everywhere.Hence, by convex duality, this function is convex and the subdifferential ∂V R (µ, 0, 0) coincides with the optimum solution set of the dual problem.We next characterize this dual and prove that it has a unique solution, from where we deduce that it is differentiable with ∇V R (µ, 0, 0) given as in the statement of part (a).
To write the dual problem we write explicitly the problem (P R µ ) as min x,f e∈E C e (x e ) s.t.
p∈P (h)   f p = µ (h) for all h ∈ H, and f p = 0 for p / ∈ R, (A.7)As mentioned above, from general convex duality, the optimum solution set of this dual coincides with the sub-differential ∂V R (µ, 0, 0) of the primal optimum value function.Since V R ( • ) is finite everywhere, this sub-differential is nonempty and the dual has optimum solutions.On the other hand, since C * e is strictly convex, the dual optimum solution (m, ν, η) is unique and therefore V R ( • ) is differentiable at (µ, 0, 0) with ∇V R (µ, 0, 0) = (m, ν, η).Writing the optimality conditions for (A.7) we obtain c e (x e ) = η e , and the characterization of the gradient follows from (A.11) and (A.12).This establishes part (a) of the lemma.Lagrange multiplier associated to edge e η edge Lagrange multiplier vector θ(t) h∈H r (h) • (λ (h) ) (t), defined in Eq. (4.4) λ (h) (µ) equilibrium cost of OD pair h with demand µ λ equilibrium cost vector µ (h)  demand at OD pair h µ µ (1) , . . ., µ (H) , demand vector ν p Lagrange multiplier associated to path p ν path Lagrange multiplier vector ξ e perturbation of load x e ξ perturbation of load vector x τ e c e (x * e ) equilibrium cost of edge e Φ(x) e∈E C e (x e ) potential ω p perturbation of flow f p ω perturbation of flow vector f 0 zero vector 1 (h)  p∈h γ p Figure1.A single-commodity example of the behavior of the PoA as a function of the demand.This function is nonsmooth at demands 1, 3, 4, 6 and 27/2.Between these breakpoints the set of optimum paths remains stable and the PoA behaves smoothly.At demand 4 the set of optimum paths exhibits a contraction and the derivative jumps up.At all the other breakpoints the set of optimum paths undergoes an expansion and the derivative jumps down.
x) := e∈E C e (x e ) with C e (x e ) := xe 0 c e (z) dz.Since the cost functions c e are continuous and nondecreasing, the objective function Φ( • ) is C 1 and convex.It follows that for every µ ∈ R H + at least one optimum solution exists. Figure2 Conjecture 4.2 (O'Hare et al. (

Figure 3 .
Figure 3.An example where Conjecture 4.2 does not hold.

Fig. 3
Fig. 3(b)  shows the plot of the PoA, which is 3. (a) In view of our extension of the costs c e ( • ) to R − , we have lim x→−∞ c e (x) < 0 and also lim x→∞ c e (x) > 0. Hence, using recession analysis, for each nonzero direction d ∈ R E \ e = p∈P δ ep f p for all e ∈ E.Introducing multipliers m = (m (h) ) h∈H , ν = (ν p ) p ∈R , and η = (η e ) e∈E , and the LagrangianL(x, f , m, ν, η) x e ) − η e x e + h∈H m (h) µ (h) + p∈R f p e∈p η e − m (h(p)) m (h(p)) − ν p , (A.8) where h(p) is the commodity of path p, the dual problem becomes sup over (x, f ) can be solved explicitly to obtain the dual in final form as sup µ (h) s.t.m (h(p)) = e∈p η e for every p ∈ R, m (h(p)) = e∈p η e − ν p for every p / ∈ R, (A.10) where C * e is the Fenchel conjugate of C e .This amounts to solving min m,η e∈E C * e (η e ) − h m (h) µ (h) s.t.e∈p η e = m (h(p)) for every p ∈ R, (A.11) and then defining ν p = e∈p η e − m (h(p)) for every p ∈ R. (A.12)

Appendix B .
List of symbols c e cost of edge e c edge cost function vector c p cost of path p C e (x e ) xe 0 c e (z) dz C * e ( • ) Fenchel conjugate of C e ( • ) D(h) To have statements (a) and (b) in Conjecture 4.2, linearity of the demand function is necessary.Indeed, as shown by Example 4.3, Proposition 4.4 can fail even if the demand function is affine (and even when restricting to affine cost functions).Hence neither Proposition 4.5 nor Proposition 4.6 can be applied.Remark 4.9.Violating the model assumption of having the sets P (h) pairwise disjoint for every h ∈ H, does not affect the results of this paper.Indeed, one can always add a dummy origin Õ(h) and attach a zero cost edge ( Õ(h) , O (h) ) to each of the original paths p ∈ P (h) f p = µ(h)for all h ∈ H , set of feasible flows, defined in (2.2) G (V, E), directed multigraph with set of vertices V and set of edges E H set of OD pairs H number of elements in H I − ( t − ε, t), defined in Proposition 4.4 I + ( t, t + ε), defined in Proposition 4.4 J c ( • ) Jacobian matrix of the path costs function c : R n + → R n p∈P