Monotonicity of Equilibria in Nonatomic Congestion Games

This paper studies the monotonicity of equilibrium costs and equilibrium loads in nonatomic congestion games, in response to variations of the demands. The main goal is to identify conditions under which a paradoxical non-monotone behavior can be excluded. In contrast to routing games with a single commodity, where the network topology is the sole determinant factor for monotonicity, for general congestion games with multiple commodities the structure of the strategy sets plays a crucial role. We frame our study in the general setting of congestion games, with a special focus on singleton congestion games, for which we establish the monotonicity of equilibrium loads with respect to every demand. We then provide conditions for comonotonicity of the equilibrium loads, i.e., we investigate when they jointly increase or decrease after variations of the demands. We finally extend our study from singleton congestion games to the larger class of constrained series-parallel congestion games, whose structure is reminiscent of the concept of a series-parallel network.


Introduction
Common sense suggests that a rise in traffic demands entails longer travel times.This implication is not always true.Traffic networks with a large number of vehicles are often modeled as nonatomic games, with Wardrop equilibrium as the standard solution concept: for each origin-destination (OD) pair, all the used paths have the same cost or travel time, whereas unused paths have higher costs.While Hall (1978) proved that in a single OD network the equilibrium cost is an increasing function of the traffic demand, Fisk (1979) constructed an example with three OD pairs in which the equilibrium cost of an OD pair decreases when the demand of a different OD increases.Namely, in the network depicted in Fig. 1 with three OD pairs (, ), (, ), (, ), a rise in the demand  (,) increases the cost of the link  and pushes the (, ) pair to favor the use of the direct link .This reduces the load on the link , which ultimately benefits the pair (, ) by reducing its cost.Perhaps more surprising is the fact that this phenomenon may even occur when all the demands increase by the same factor: with demands  (,) = 60,  (,) = 30,  (,) = 6 the equilibrium cost for the third OD is  (,) = 24, and when all the demands are doubled it decreases to  (,) = 18.
On the other hand, even if the equilibrium cost on single OD networks increases with the demand, the load on some edges may decrease after a surge in the demand.This can be observed in the classical Wheatstone network illustrated in Fig. 2 (see Braess, 1968, Braess et al., 2005, for the famous paradox that uses this network).
Networks in which the equilibrium loads of all the edges increase with the travel demand of every OD pair are more predictable and easier to handle for a social planner, because an edge  is never used below a certain level of demand and is always used above that level.The goal of this paper is precisely to understand when the equilibrium travel times and edge loads are monotone in the demand, so that the paradoxical phenomena observed in the two examples above cannot happen.We state our results in the general setting of congestion games, rather than routing games, with a special focus on multi-commodity singleton congestion games.
1.1.Our Results.For multi-commodity singleton congestion games, we show in Theorem 3 that, even when there exist multiple equilibrium flows, one can always select an equilibrium whose corresponding resource loads are monotone increasing with respect to each demand.
We then focus on the notion of comonotonicity, which encodes the idea that different resource loads jointly increase or decrease upon variations of the demands.We take a close look at how comonotonicity depends on the structure of equilibria, in terms of how the commodities are ranked by cost under different demand profiles, and how the resources become active or inactive as the demands vary (Theorem 5).We also provide structural results for the regions of the space of demands on which a fixed set of resources is used at equilibrium (Theorem 6).
We next extend the monotonicity results from singleton congestion games to the larger class of product-union congestion games (Theorem 10).Finally, we derive an embedding that maps congestion games into routing games (Proposition 8) and characterize the classes of congestion games with good monotonicity properties by embedding them into routing games (Theorem 11).This last result sheds light on the features that produce the non-monotonicity paradoxes, and highlights the difference between the single and multiple OD networks.When the network has a single OD pair, its topology is the sole relevant factor to guarantee the monotonicity of equilibrium loads, while for multiple ODs the structure of the set of feasible routes also plays a crucial role.
1.2.Related Work.Several authors studied the sensitivity of Wardrop equilibria in routing games with respect to changes in the demand.Hall (1978) observed that, when the costs are strictly increasing, the equilibrium loads depend continuously on the demands.Patriksson (2004) and Josefsson and Patriksson (2007) studied the directional differentiability (or lack thereof) of equilibrium costs and loads, whereas Cominetti et al. (2023) studied differentiability along a curve in the space of demands.Specific cases of differentiability, were also considered in Pradeau (2014).
As mentioned previously, Hall (1978) proved that the equilibrium cost of an OD pair increases when the demand of that OD grows.On the other hand Fisk (1979) noted that it may decrease as a consequence of a rise in the demand of a different OD.Some positive results concerning the monotonicity of equilibrium loads in series-parallel (SP) single-commodity networks can be found in Klimm and Warode (2022) for piece-wise linear costs and in Cominetti et al. (2021) for general nondecreasing costs.
Traffic equilibria in routing games exhibit a multitude of paradoxes.The most famous, due to Braess (1968), Braess et al. (2005), shows that removing an edge from a network could actually improve the equilibrium cost for all players (see Fig. 2).Also surprising is the fact observed by Fisk (1979) that an OD can reduce its cost and benefit from an increase in the demand of a different OD, even after doubling all the demands, as mentioned before.Fisk and Pallottino (1981) showed that such paradoxical phenomena could be observed in real life in the City of Winnipeg, Manitoba, Canada.Dafermos and Nagurney (1984) studied how equilibrium costs are affected by changes in the travel demand or addition of new routes under a more general non-separable cost structure.A related paradoxical phenomenon was studied by Mehr and Horowitz (2020) in a model with both regular and autonomous vehicles: despite the fact that autonomous vehicles are more efficient by allowing shorter headways and distances, replacing regular with autonomous vehicles may increase the total network delay.
A particularly simple class of congestion games is the one of singleton congestion games where each strategy comprises a single resource.Different variants of these type of games have been considered in the literature, including atomic weighted and unweighted players, with splittable or unsplittable loads, as well as nonatomic players.
For atomic splittable singleton games, Harks and Timmermans (2017) developed a polynomial time algorithm to compute a Nash equilibrium with player-specific affine costs.In a different direction, Bilò and Vinci (2017) investigated how the structure of the players' strategy sets affects the efficiency in singleton load balancing games.Atomic splittable singleton games have also been used to model the charging strategies of a population of electric vehicles (Ma et al., 2013, Deori et al., 2017, Nimalsiri et al., 2020).In a related but different direction, Castiglioni et al. (2019) studied the computational complexity of finding Stackelberg equilibria in games where one player acts as leader and the others as followers.
For atomic unsplittable singleton games, Gairing and Schoppmann (2007) provided upper and lower bounds on the price of anarchy, distinguishing between restricted and unrestricted strategy sets, weighted and unweighted players, and linear vs. polynomial costs.Fotakis et al. (2009) studied the combinatorial structure and computational complexity of Nash equilibria, including the problems of deciding the existence of pure equilibria, computing pure/mixed equilibria, and computing the social cost of a given mixed equilibrium.Harks and Klimm (2012) characterized the classes of cost functions that guarantee the existence of pure equilibria for weighted routing games and singleton congestion games.
Finally, in the nonatomic setting, which is the focus of our paper, Gonczarowski and Tennenholtz (2016) used a clever hydraulic system representation to study asymmetric singleton congestion games, presenting applications in the home internet and cellular markets, as well as in cloud computing.Another recent application of nonatomic singleton congestion games to hospital choice in healthcare systems is discussed in van de Klundert et al. (2023).In the special case of routing games, singleton games correspond to parallel networks.Despite its simple topology they are nevertheless of interest in the literature (see, e.g., Acemoglu and Ozdaglar, 2007, Wan, 2016, Harks et al., 2019)).
1.3.Organization of the paper.The paper is organized as follows.Section 2 recalls the standard model of non-atomic congestion games and reviews the basic properties of equilibria.This section includes the definition and the study of monotonic equilibrium selection (MES) and comonotonicity.Section 3 and Section 4 both deal with singleton congestion games.Section 3 contains the central monotonicity result, whereas Section 4 discusses comonotonicity and the structure of the domains associated to different sets of resources.Section 5 studies the monotonicity properties of more complex congestion games beyond the case of singleton strategies.Section 6 summarizes the results of our paper and proposes some open problems.Appendix A includes some supplementary proofs.Appendix B contains a list of the symbols used throughout the paper.

Congestion Games and Eqilibria
We begin by recalling the basic concepts and facts about nonatomic congestion games, and by fixing the notations used throughout the whole paper.The basic structural elements are: • a finite set R of resources and, for each  ∈ R, a continuous nondecreasing cost function   : R + → R + , where   (  ) represents the cost of resource  under a workload   ; and • a finite set H of commodities and, for each ℎ ∈ H, a family S ℎ ⊂ 2 R of feasible strategies, where every  ∈ S ℎ is a subset of resources  ⊂ R with which a given task in ℎ can be performed.
These elements define a congestion game structure G = (R, , S) with  (  )  ∈R the vector of cost functions and S × ℎ∈H S ℎ the set of strategy profiles.Every vector   ℎ ℎ∈H of demands  ℎ ≥ 0, determines a nonatomic congestion game (G, ) as follows.Each  ℎ splits into feasible flows  ℎ  ℎ  ∈S ℎ such that ℎ  ,  ℎ  ≥ 0, for all  ∈ S ℎ . (2.1) These flows combined induce aggregate loads over the resources, given by as well as strategy costs, defined as We let F  denote the set of feasible flows   ℎ ℎ∈H where each  ℎ satisfies Eq. (2.1), and X  the set of load profiles  = (  )  ∈R induced by all feasible flows  ∈ F  .Assuming further that for each commodity only the strategies with the smallest possible cost are selected, leads to the concept of Wardrop equilibrium: a feasible flow  ∈ F  for which there exists a nonnegative vector   ℎ ℎ∈H , such that (∀ℎ ∈ H)   ( ) =  ℎ for all  ∈ S ℎ with  ℎ  > 0,   ( ) ≥  ℎ for all  ∈ S ℎ with  ℎ  = 0. (2.4) The quantity  ℎ is called the equilibrium cost of commodity ℎ ∈ H.A strategy  ∈ S ℎ is said to be active if   ( ) =  ℎ .Similarly, a resource  ∈ R is active for commodity ℎ ∈ H if it belongs to some active strategy.
As shown by Beckmann et al. (1956), the set of load profiles induced by equilibrium flows coincides with the set of optimal solutions of the minimization problem min (2.5) where   (  ) ∫   0   () d.Since the cost functions   are continuous and nondecreasing, the above objective function is convex and differentiable.Thus, since X  is a bounded polytope, for every  there exists at least one optimal solution.
For an equilibrium load profile , we define the equilibrium resource costs     (   ).By using Fenchel's duality theory (see e.g., Remark 11 in Appendix A, or Fukushima (1984) for the special case of nonatomic routing games), we can prove that the equilibrium resource costs are optimal solutions of the strictly convex dual program min where  *  ( • ) is the Fenchel conjugate of   ( • ), which is strictly convex.Thus, for each  the equilibrium resource costs   are uniquely defined and are the same for all equilibrium loads.This implies that the strategy costs   =  ∈   and equilibrium costs  ℎ = min ∈S ℎ  ∈   depend only on  and not on the particular equilibrium flow under consideration, and hence also the active strategies and active resources only depend on .
The active regime at demand  is defined as R() ( R ℎ ()) ℎ∈H with R ℎ () the set of active resources for commodity ℎ ∈ H.We also let  ↦ → () denote the equilibrium cost map, whose basic properties are summarized in the next proposition.
Proposition 1 is a simple extension of Cominetti et al. (2021, Proposition 3.1) to the multicommodity setting.See also Hall (1978) for the case of strictly increasing costs.For the sake of completeness, we include a proof of Proposition 1 in Appendix A.
Remark 1.When the cost functions are strictly increasing, thus invertible, Proposition 1 implies that the equilibrium load vector  () is unique for every  ∈ R H + , and the map  ↦ →  () is continuous.If the costs are just nondecreasing, the equilibrium loads may be non-unique; there exists some literature about the characterization of games having the so-called uniqueness property (see, e.g., Milchtaich, 2000, Konishi, 2004, Milchtaich, 2005, Meunier and Pradeau, 2014).Another natural question for the case of multiple equilibria is whether there exists a continuous selection  ↦ →  ().
Although every equilibrium cost  ℎ is nondecreasing in the variable  ℎ , it can actually decrease strictly when the demand in some other commodity increases.It is also possible that the social cost SC() = ℎ∈H  ℎ  ℎ () decreases along a direction where the total demand ℎ∈H  ℎ increases (see Fisk, 1979).In what follows, we want to determine if a congestion game has an equilibrium selection such that the resource loads are monotone with respect to an increase in any demand.This is made precise in the following definition.
Definition 2. A congestion game structure G = (R, , S) is said to have a monotonic equilibrium selection (MES) if for each demand vector  ∈ R H + there exists an equilibrium load vector  () such that for every resource  ∈ R the map  ↦ →   () is nondecreasing with respect to each component  ℎ of the demand.
In mixed scenarios where some demands increase and other decrease, one naturally expects that the same will occur for the induced equilibrium loads.However, it is still of interest to identify groups of resources whose equilibrium loads vary in the same direction, regardless whether  and  ′ are comparable or not.In such a case, observing an increase/decrease in the load of a specific resource one can infer that all the remaining loads in the group move in the same direction.This property is captured by the notion of comonotonicity: a family of functions {  : Ω → R} ∈ is comonotonic if for all ,  ∈  we have (2.7) For singleton congestion games, we will identify subsets of resources whose equilibrium loads exhibit such comonotonic behavior in specific regions of the space of demands R H + .Informally, we will show that a group of commodities that share the same equilibrium cost behave as a single commodity, and the loads on the resources used by this group are comonotonic.

Monotonicity in singleton congestion games
In a singleton congestion game each strategy corresponds to a single resource.Thus, for every commodity ℎ ∈ H the set of feasible strategies S ℎ can be viewed as a subset R ℎ ⊂ R of the set of resources.The following result shows that the MES property holds in this case.
Theorem 3. Every singleton congestion game G = (R, , S) has a MES.
Proof.We will first prove the result for strictly increasing cost functions, and then derive the result for nondecreasing costs by a regularization argument.
Suppose first that the costs   ( • ) are strictly increasing.We will prove the existence of a MES locally by showing that for every demand vector  0 ∈ R H + and every commodity ℎ ∈ H, there exists  > 0 such that   ( 0 +   ℎ ) ≥   ( 0 ) for all  ∈ [0, ], where  ℎ is the ℎ-th vector of the canonical basis of R H .The global MES property throughout the space of demands then follows from the continuity of the map  ↦ →  () (see Remark 1).
Let R 0 be the set of resources such that   (  ( 0 )) =  ℎ ( 0 ).This set contains the active resources for commodity ℎ but may also include resources used by other commodities and that are not feasible for ℎ.By continuity of the equilibrium costs (Proposition 1), there exists  > 0 such that an increase in the demand for commodity ℎ by an amount  smaller than  can only affect the equilibrium loads of resources in R 0 , and therefore for  ∉ R 0 and  ∈ [0, ] we have   ( 0 +  ℎ ) =   ( 0 ).Let us then focus on the resources  ∈ R 0 .Fix an arbitrary  ∈ [0, ] and partition R 0 into the three subsets Suppose by contradiction that R − 0 is not empty.Since the total demand at  0 +  ℎ is larger than the total demand at  0 , any flow removed from R − 0 ∪ R = 0 must be transferred to R + 0 .This implies that there exists some commodity ℎ ′ which has feasible resources both in R − 0 ∪R = 0 and R + 0 .This contradicts the equilibrium condition for that commodity at demand  0 + ℎ because the cost of all resources in R + 0 is strictly higher than the cost of the resources in R − 0 ∪ R = 0 .This establishes the existence of a MES for the case of strictly increasing costs.
For nondecreasing costs   (  ) we perturb them as    (  )   (  ) + 2  with  > 0, to make them strictly increasing, and then consider the limit as  approaches zero.As recalled in Section 2, the equilibrium flow  (, ) for the congestion game structure is the unique solution of the Beckmann problem (2.5), which in this case has the form min with   (  ) ∫   0   () d.Tikhonov regularization (see, e.g., Attouch, 1996, section 1.1) tells us that  (, ) converges, as  approaches zero, to the minimal norm equilibrium  0 () of the original unperturbed game G. From the previous case of strictly increasing costs, for each  > 0 the map  ↦ →  (, ) is nondecreasing with respect to each demand  ℎ , and this property is inherited by  ↦ →  0 () in the limit as  ↓ 0, providing a MES as claimed.□ Remark 2. The quadratic regularizer  ∥ ∥ 2 was introduced by Tikhonov in the study of illposed inverse problems (Tikhonov, 1943, 1963, Tikhonov and Arsenin, 1977).It is also the basis of ridge regression in statistics (Hoerl, 1959, 1962, Hoerl and Kennard, 1970).In our setting this is just one choice among others, and can be replaced by a separable regularizer   =1   (  ) with  ′  ( • ) strictly increasing.Every such regularizer selects a specific optimal solution in the limit when  ↓ 0 (see Attouch (1996, theorem 2.1) and Auslender et al. (1997, proposition 2.5).Moreover, one can verify that the previous proof is still valid and yields a monotone selection of the set of Wardrop equilibria.In particular,   =1   log(  ) selects the Wardrop equilibrium of maximal entropy.A similar entropic regularization was used in Rossi et al. (1989) to select one among multiple flow decompositions of a Wardrop equilibrium (see Borchers et al., 2015, for a survey of related work).In our case we deal with multiple equilibria and the regularization is used to obtain a selection with monotonicity properties.As alternatives one may consider general penalty schemes of the form   =1  (  /), including the classical log-barrier  () = − log(), the inverse-barrier  () = 1/, the exponential penalty  () = exp(−), and more (see Cominetti, 1999).Let us also mention the multiscale regularizer  =1    2  , which yields a hierarchical selection principle: select the Wardrop equilibria that have the smallest first coordinate  2 1 , among these the ones with smallest  2 2 , and inductively with  2 3 , . . .,  2  .

Comonotonicity and Active Regimes in Singleton Congestion Games
Theorem 3 shows that the equilibrium loads in singleton congestion games respond monotonically when all the demands increase or stay the same.In mixed cases where some demands increase and others decrease, one can still identify groups of resources that behave comonotonically in specific regions of the space of demands.A trivial example is when all commodities can use every resource R ℎ ≡ R, so they can be treated as a single commodity and the equilibrium loads are just nondecreasing functions of the total demand  H = ℎ∈H  ℎ .More generally, we will show that a subset C ⊂ H of commodities that have the same equilibrium cost, behave as if they were a single-commodity on a smaller congestion game restricted to a subset R C of resources, and the equilibrium loads of these resources are nondecreasing functions of the aggregate demand  C of the group, so that they are comonotonic.
To state our result precisely, given a single commodity game structure G = (R, , S), we partition the space of demands R H + into different regions characterized by the order in which the commodities are ranked by equilibrium cost.That is, for any fixed weak order ≾ on H we consider the set of demands that rank the commodities exactly in this order, that is (see Example 1 and Fig. 3) We recall that the equivalence relation and strict order associated with ≾ are defined by The relation ∼ partitions H into equivalence classes, called cost classes: two commodities are in the same cost class if and only if ℎ ∼ ℎ ′ , that is to say, if and only if  ℎ () =  ℎ ′ () for all  ∈ Γ ≾ .To each cost class C we associate the subset R C of all the resources  ∈ R that are feasible for some commodity ℎ ∈ C, excluding those which are also feasible for higher ranked commodities ℎ ′ ≻ ℎ, that is The sub-regions delimited by horizontal and diagonal lines within a colored region, correspond to different sets of active resources as described later.
Remark 3. The regions Γ ≾ can be empty for some orders ≾ (e.g., if ℎ, ℎ ′ ∈ H are such that R ℎ ⊆ R ℎ ′ we cannot have  ℎ () <  (ℎ ′ ) ()).We stress that each commodity ℎ ∈ H belongs to a unique cost class C, whereas each resource  belongs to the cost class of the highest ranked commodity among those for which  is feasible.
Our next result describes the equilibrium within a cost class C: we show that the loads on the resources in R C coincide with those of the single-commodity game G C .In other words, in terms of equilibrium loads the commodities in C behave as if they were a single commodity.This allows in turn to analyze the comonotonicity of the equilibrium loads on R C .
Theorem 5. Let G = (R, , S) be a singleton congestion game structure, and Γ ≾ the region associated with a weak order ≾ on H.Then, for each cost class C for ≾ we have: (a) For all  ∈ Γ ≾ and every equilibrium load  of (G, ), the vector x = (  )  ∈R C is an equilibrium in the single-commodity game (G C ,  C ) with aggregate demand  C ℎ∈C  ℎ .(b) If G C has a unique equilibrium for each demand in R + , then for  ∈ Γ ≾ the equilibrium loads   () with  ∈ R C can be expressed as nondecreasing functions of the aggregate demand  C , which is equivalent to the fact that the equilibrium loads of the resources in R C are comonotonic in the region Γ ≾ .
Proof.(a) Let  be an equilibrium load vector of demand  ∈ Γ ≾ .We note that every commodity ℎ ∈ C allocates traffic only through resources in R C .Indeed, if a commodity ℎ ∈ C has a feasible resource also in , which implies that for every ,  ′ ∈ R C we have (b) By the result in (a), for each  ∈ R C and  ∈ Γ ≾ the equilibrium load   () coincides with the unique equilibrium in the single-commodity game G C with demand  C , and therefore it is a function of the aggregate demand  C .Now, according to (Cominetti et al., 2021, proposition 3.12) every single-commodity game on a series-parallel network has a nondecreasing selection of equilibria, so that   () is a nondecreasing function of  C .The equivalence with the comonotonicity of the maps  ↦ →   () for  ∈ R C throughout the region  ∈ Γ ≾ , then follows from a known result (see e.g., Dellacherie (1971) and Landsberger and Meilijson (1994)).Since we could not find a proof of this latter result in the literature, we include one in Lemma 12 in Appendix A. □ Remark 4. By Remark 1, having strictly increasing costs ensures the uniqueness of equilibria for G C , as required in Theorem 5 (b).Actually, it suffices that no two resources in R C have cost functions that are constant and equal on some (possibly different) non-degenerate intervals.Moreover, for strictly increasing costs the equilibrium loads   () for  ∈ R C and  ∈ Γ ≾ are strictly increasing with  C .Indeed, since  ∈R C   () =  C , a strict increase of  C implies that some load   () and its corresponding cost   (  ()) must strictly increase.However, across Γ ≾ the equilibrium costs of all the resources  ∈ R C remain equal, so that all their loads   () must strictly increase simultaneously.
Remark 5. Theorem 5 (b) implies that comonotonicity fails across different cost classes C ≠ C ′ : if  C increases and  C ′ decreases, the equilibrium loads of the resources R C and R C ′ will move in opposite directions.On the contrary, if both aggregate demands move in the same direction, the same holds for the corresponding equilibrium loads.
Remark 6.The comonotonicity in Theorem 5 (b) may fail when G C has multiple equilibria.Consider for instance a variant of Example 1 with costs  1 () =  3 () = 1 and  2 () = .When the demand is  = (2, 0) the equilibrium sends 1 unit of flow through  1 and  2 , and zero on  3 .Instead, at demand  = (0, 2) nothing is sent through  1 , with 1 unit of traffic on both  2 and  3 .Hence, despite the fact that at both  and  all three resources have the same equilibrium cost equal to 1, the load on resource  1 decreases when moving from  to , whereas the load on resource  3 increases, so these loads are not comonotonic.Theorem 5 (b) does not apply here because the single-commodity game G C on the three resources and aggregate demand 2 has multiple equilibria.
Although for all demands in Γ ≾ the commodities are always ranked as in ≾, the sets of active resources may vary within such a region.We may then further decompose Γ ≾ into sub-regions corresponding to different active regimes: for each tuple   ℎ ℎ∈H with  ℎ ⊂ R ℎ we define the sub-region with active regime  as The simple structure exhibited by the sub-regions actually holds more generally: even if the cost functions are nonlinear, the sub-regions are separated by hyperplanes defined by the aggregate demand of some cost class.We recall that a break point in a single commodity game is a demand μ at which the set of active resources changes, i.e., this set is not constant on any interval ( μ − , μ + ) with  > 0 (see Cominetti et al., 2021, definition 3.4).Theorem 6.Let G = (R, , S) be a singleton congestion game structure with strictly increasing costs, and ≾ a weak order on H.Then, the boundary between the sub-regions Γ ≾  coincides with the points  ∈ Γ ≾ satisfying at least one of the linear equations ∑︁ The straight lines within each region separate sub-regions corresponding to different active regimes.The regions Γ ≾ are not convex, but the boundary between sub-regions is still affine.
the hyperplanes described in Theorem 6.In the purple region where   =   with a single cost class C = {, } and R C = { 1 ,  2 ,  3 }, the equilibrium loads of all the resources are strictly increasing functions of the total demand  C =   +   , and the active regimes present break points at  C = 1 and (4.4) Similarly, in the green region where   <   with cost classes C 1 = { } and C 2 = {}, we have Remark 7. The monotonicity result in Cominetti et al. (2021, proposition 3.12) implies that the number of active regimes in a single-OD routing game on a series-parallel network is at most the number of paths.This bound does not hold for multiple commodities.In a singleton congestion game there are ℎ∈H 2 |R ℎ | − 1 potential combinations for R(), and this bound may be attained (see Example 3 below).This is not the case for single-commodity routing games: if we consider a subnetwork composed by only two paths, it is always series-parallel and only two of the three nonempty subsets of paths can actually correspond to an active regime R() for some  ∈ [0, +∞).
Example 3. Let us build a multi-commodity routing game that attains the maximal bound for the number of active regimes.Take  a positive integer and consider a routing game on a parallel network with  resources (links) R = {1, . . ., } with cost functions and  + 1 commodities where each commodity  = 1, . . .,  can only use one player-specific resource R  = {}, whereas commodity ( + 1) can use all the resources R +1 = R.
We claim that R() assumes the maximum number +1 =1 2 |R  | − 1 = 2  − 1 of possible active regimes as the demands  vary.Indeed, for each commodity  ≤  the active regime is always {}, whereas every nonempty subset  +1 ⊂ R is the active regime of the ( + 1)-th commodity for some demand .Namely, let  max = max{ ∈  +1 } and consider the demand Then, the unique equilibrium is such that commodity ( +1) allocates  max − to each resource  ∈  +1 with cost  max , whereas every resource  ∉  +1 has a cost  max +  >  max , so that the active regime for commodity ( + 1) is exactly  +1 .

Beyond Singleton Congestion Games
In this section we consider more general nonatomic congestion games, and we discuss a possible extension of the fact that for single-commodity routing games over a series-parallel network the equilibrium edge loads are nondecreasing in the traffic demand (Cominetti et al., 2021, proposition 3.12).For single-commodity routing games this is basically the most one can expect, since every network which is not series-parallel contains a Braess subnetwork for which it is well known that there exists costs that produce equilibrium loads that decrease on some ranges of the demand (see Fig. 2).
Unfortunately, for multi-commodity routing games the topology of the network alone does not provide a criterion for the monotonicity of edge loads.To better understand this general case, we start defining the class of constrained routing games.Definition 7. A constrained routing game (CRG) is a triple (, , P) where •  = (V, E) is a directed multigraph with vertex set V and edge set E, •  = (  ) ∈E is a vector of edge cost functions, • P = (P ℎ ) ℎ∈H with P ℎ a nonempty set of paths between an origin O ℎ ∈ V and a destination D ℎ ∈ V for a finite family of commodities H.
This defines a congestion game structure with resource set R = E with costs  = (  ) ∈E , and commodity set H with strategies S ℎ = P ℎ .Notice that in our setting different commodities could share the same OD pair and have some common paths.In a standard routing game each OD pair corresponds to a single commodity and P ℎ includes all the paths from O ℎ to D ℎ .All the examples in Section 4 are in fact constrained routing games, including the Braess' network in Fig. 2, which is a standard routing game.
Although restricting the paths to a subset might seem a minor detail, it is in fact a flexible feature that allows to represent any congestion game as a constrained routing game.Furthermore, we can also turn this routing game into a common-OD where all commodities have the same origin and destination, by The next proposition shows directly that every congestion game is equivalent to a common-OD routing game over an extremely simple network, and all the complexity of the game is in fact encoded into the feasible sets of paths.
Formally, two congestion games G and G are said to be equivalent if there exist one-to-one correspondences ℎ ↔ h between their commodities and  ↔ s between strategies, such that for each demand  and every feasible flow  for the first game, the flow f defined as fs =   is feasible in the second game and the strategy costs coincide c s ( f ) =   ( ).In this case the equilibria of both games are also in one-to-one correspondence.
Proposition 8. Every congestion game is equivalent to a common-OD constrained routing game over a series-parallel network.Regarding the previous result, one may naturally ask whether a given nonatomic congestion game is equivalent to an unconstrained nonatomic routing game.We are not aware of any result on this question, apart from the somewhat related result by Milchtaich (2013), who showed that every finite game can be represented as a weighted atomic routing game.
As mentioned above, for multi-OD routing games a series-parallel network topology does not suffice to guarantee the monotonicity of the equilibrium loads.Indeed, Examples 4 and 5 below show that there exist common-OD constrained routing games such that: •  is series-parallel; • every commodity uses paths P ℎ that form a series-parallel subnetwork; • the equilibrium loads  () are unique; but Example 4. Consider Fisk's network and add bypass edges  4 ,  5 with zero cost as in Fig. 5(b), producing commodities ℎ 1 , ℎ 2 , ℎ 3 where O ℎ = , D ℎ =  for every commodity ℎ, and P ℎ 1 = {( 1 ,  5 )}, P ℎ 2 = {( 4 ,  2 )}, and P ℎ 3 = {( 1 ,  2 ),  3 }.This defines an equivalent common-OD constrained routing game.As noted in the introduction, an increment in the demand of ℎ 1 pushes commodity ℎ 3 to divert more flow towards the direct path  3 , thus reducing the load on  2 (see Fisk, 1979).
Example 5. Monotonicity can also fail in a single-OD constrained routing game, even on a series-parallel graph.Indeed, the standard Braess' routing game in Fig. 2  These examples show that in addition to a series-parallel topology we need to impose further conditions on how the commodities overlap.To this end we introduce the notions of product and union of two congestion games G 1 and G 2 .Informally, the product can be seen as  a connection in series in which we consider all combinations of a commodity  ∈ H 1 followed by a commodity  ∈ H 2 to form a product commodity  ⊗ .Similarly, the union game can be seen as a parallel connection that includes all the original commodities H 1 ∪ H 2 , without introducing any new commodity that picks resources simultaneously from R 1 and R 2 .
The product and union of these games are both defined on the resource set R = R 1 ∪ R 2 with their original cost functions.Specifically: • The product game G 1 ⊗G 2 includes all the commodities  ⊗  for (, ) ∈ H 1 ×H 2 , defined by the strategy sets their original strategy sets S  1 for  ∈ H 1 and S  2 for  ∈ H 2 .• A product-union game is a congestion game which can be constructed starting from singleton congestion games and applying a finite number of product or union operations between games already constructed.
Theorem 10.Every product-union congestion game has a MES.
Proof.By induction and Theorem 3, it suffices to show that the MES property is preserved under product and union of games.To this end, let G 1 and G 2 be two congestion game structures with MES's  1 ↦ →  1 ( 1 ) and  2 ↦ →  2 ( 2 ) respectively.Then we prove the two parts: (a) The product game G 1 ⊗ G 2 has a MES.Let  = ( ⊗  ) ∈H 1 , ∈H 2 with  ⊗  the demand for the commodity  ⊗  in the product game.An equilibrium for  can be obtained by superposing  1 ( 1 ) on the resources R 1 and  2 ( 2 ) on the resources R 2 , computed respectively with demands  1 and  2 given by (5.1) Since an increase of any demand  ⊗  induces an increase in the demands   1 and   2 , the loads in  1 ( 1 ) and  2 ( 2 ) increase, so that this superposed equilibrium provides a MES for the product game.
(b) The union game G 1 ∪G 2 has a MES.For each demand  = ( 1 ,  2 ) in the union game we can directly find an equilibrium by superposing  1 ( 1 ) on the resources R 1 and  2 ( 2 ) on the resources R 2 .Since the loads in these two equilibria are monotone with respect to each individual demand, the same holds for their superposition which provides a MES for the union game.□ Remark 8.The common-OD constrained routing game of Example 4, in which Fisk's network is embedded, is not a product-union congestion game: the strategy set of ℎ 3 cannot be obtained as a strategy set of a previously present commodity when doing the union game.For a different reason, the common-OD series-parallel constrained routing game in Example 5 is not a product/union game either.Indeed, its network would be made of 5 two-link parallel network in series, each one associated to a resource, i.e., an edge of the Braess' network.The classical Braess' routing game has a single commodity with three strategies, and a strategy set with cardinality 3 cannot be obtained as the Cartesian product of the strategy sets of the 5 two-link parallel networks connected in series.A series of Pigou's has strong limitations, for example, a commodity with a number of paths divisible by a prime larger than 2 is not constructible as in Definition 9.
Whereas product-union games were defined for general congestion games, they can also be described as constrained routing games with a specific structure.The following representation is also more natural compared to the one in Proposition 8.
Theorem 11.Every product-union congestion game G = (R, , S) is equivalent to a common-OD constrained routing game (, , P) such that (i) the graph  is series-parallel, (ii) for each commodity ℎ all the paths in P ℎ visit the same vertices in the same order, (iii) for every two paths  1 ,  2 ∈ P ℎ and edges  1 ∈  1 and  2 ∈  2 connecting two subsequent vertices, the paths obtained from  1 and  2 by exchanging  1 with  2 also belong to P ℎ .
Proof.Every product-union game is constructed starting with singleton congestion games and applying a finite number of product or union operations.We start by noting that every singleton congestion game is equivalent to a constrained routing game on a parallel network with two vertices connected by links corresponding to the resources of the singleton congestion game, and any such game satisfies (i), (ii), (iii).Hence, it suffices to show that these properties are preserved under product and union operations.
The product game G 1 ⊗ G 2 is then equivalent to the constrained routing game ( , , P) where  is obtained by joining in series the graphs  1 and  2 , the costs  are the cost functions given by  1 and  2 on the corresponding edges, and the commodities are given by the sets of paths obtained choosing commodities ℎ 1 for ( 1 ,  1 , P 1 ) and ℎ 2 for ( 2 ,  2 , P 2 ), and joining every path in P ℎ 1 with every path in P ℎ 2 to construct paths in .Moreover, ( , , P) satisfies (i), (ii), (iii) because ( 1 ,  1 , P 1 ) and ( 2 ,  2 , P 2 ) do.
Similarly, the union game G 1 ∪ G 2 is equivalent to the constrained routing game ( Ḡ, c, P) where Ḡ is obtained joining in parallel  1 and  2 , the costs c are the cost functions given by  1 and  2 on the corresponding edges, and the commodities are given by the commodities of ( 1 ,  1 , P 1 ) and ( 2 ,  2 , P 2 ).Also in this case, the routing game ( Ḡ, c, P) satisfies (i), (ii), (iii) because ( 1 ,  1 , P 1 ) and ( 2 ,  2 , P 2 ) do.
This completes the proof of the first claim of the theorem.Conversely, notice that every series-parallel graph  is constructed starting with parallel networks and joining them in series or in parallel for a finite number of times.Suppose that a common-OD constrained routing game (, , P) satisfies (i), (ii), (iii).
If the graph  is obtained by joining in series two graphs  1 and  2 , we can endow them with cost functions which associate costs to edges as in .Furthermore, given a commodity ℎ for (, , P) we can define commodities ℎ 1 on  1 and ℎ 2 on  2 by determining for  = 1, 2 the set of paths P ℎ  = { path in   s.t. is part of a path in P ℎ }.Since (, , P) satisfies (iii), we have P ℎ = P ℎ 1 × P ℎ 2 , so that (, , P) is the product game of the two constrained routing game just defined on  1 and  2 .
If the graph  is obtained by joining in parallel two graphs  1 and  2 , then we can assume that the direct links from the origin and the destination of  are all contained in one of the two.We can again endow  1 and  2 with cost functions which associate costs to edges as in .Furthermore because of property (ii), for every commodity ℎ of (, , P) the paths in P ℎ all belong to one between  1 and  2 .This allows to define for each commodity of (, , P), a commodity either in  1 or  2 , so that (, , P) is the union game of the two constrained routing game just defined on  1 and  2 .□ Remark 9. Note that Braess' classical example in Fig. 2 satisfies (ii) and (iii), but does not satisfy (i).Fisk's network embedding of Example 4 satisfies (i) and (iii) but does not satisfy (ii).Finally, the constrained routing game of Example 5, obtained by embedding Braess' commodity in a series-parallel graph as in Proposition 8, satisfies (i) and (ii), but does not satisfy (iii).
Remark 10.Conditions (i), (ii), (iii) in Theorem 11 can be equivalently stated by requiring that all feasible paths for a commodity ℎ visit a specific ordered sequence of nodes; between successive nodes only a specific subset of parallel links are allowed; and P ℎ includes all possible paths in this subnetwork.Still another equivalent description is to require that for any two paths  1 ,  2 ∈ P ℎ the mixed path where we follow  1 up to an intermediate node and then continue with  2 is also in P ℎ .

Summary and open problems
This paper studied the monotonicity of equilibrium travel times and equilibrium loads in response to variations of the demands, identifying conditions under which the paradoxical phenomena of non-monotonicity cannot happen.We considered the general setting of congestion games, with a special focus on multi-commodity singleton congestion games for which we established in Theorem 3 the monotonicity of equilibrium loads with respect to the demand of every single commodity.
We next explored the notion of comonotonicity, which captures the idea that different resource loads jointly increase or decrease after variations of the demands.Theorem 5 described how comonotonicity is connected to the structure of equilibria in terms of how the commodities are ranked by cost and how the resources become active or inactive as the demands vary.Theorem 6 complemented this by a structural result on the regions of the space of demands for which the same sets of resources are used at equilibrium.
Theorem 10 extended the study of monotonicity from singleton congestion games to the larger class of congestion games having a product-union structure, reminiscent of the concept of a series-parallel network.Finally, we derived an embedding that maps congestion games into constrained routing games (see Proposition 8) and characterized the classes of congestion games with good monotonicity properties by embedding them into routing games (see Theorem 11).This last result sheds light on the features that produce the paradoxes and showcases the difference between the single and multiple OD networks.When the network has a single OD pair, its topology is the sole relevant factor to guarantee the monotonicity of equilibrium loads.In the multiple OD case the structure of the available routes for each OD pair also plays a crucial role.
A first open question not addressed in this paper, and which could be interesting to explore, is how the structural results on the regions Γ ≾ and sub-regions Γ ≾  for the different active regimes might be exploited to devise an algorithm for building a curve of equilibria along a demand curve, analog to the path-following method for piece-wise affine costs developed by Klimm and Warode (2022).A basic question here is to investigate the geometry of the regions Γ ≾ for specific classes of cost functions.For the special case of Bureau of Public Roads (BPR) costs, we conjecture that the boundaries between these regions are asymptotic to straight lines through the origin.This would imply that when the demands are scaled proportionally, the regimes will not repeat and the curve will eventually enter into a particular asymptotic region Γ ≾ and remain there forever.The latter could inspire a path following algorithm to build a curve of equilibria.
A second open problem is to find an algorithm to recognize product-union congestion games.In this regard, one could be tempted to use the equivalent game in Proposition 8 for which (i) and (ii) in Theorem 11 hold trivially, so that only (iii) would need to be checked.Unfortunately, the product-union property is not preserved under equivalence: for instance, a singleton congestion game with only one commodity is product-union by definition, but its equivalent representation in Proposition 8 is not because property (iii) fails.This suggests that recognizing product-union congestion games is not straightforward.As a possible starting point to address this question, one might try to adapt the existing algorithms for recognizing series-parallel networks (see Valdes et al., 1982, He and Yesha, 1987, Eppstein, 1992).
where  *  is the Fenchel conjugate function, that is, Since  ( ′ ) is finite for all  ′ ∈ R H + , it follows that   () =  ( + ) is finite for  in some interval around 0, and then the convex duality theorem implies that there is no duality gap and the subgradient ∇  (0) at  = 0 coincides with the optimal solution set S(D  ) of the dual problem, that is, ∇ () = ∇  (0) = S(D  ).
We claim that the dual problem has a unique solution, which is exactly the vector of equilibrium costs ().Indeed, fix an optimal solution  for   (0) =  () and recall that this is just a Wardrop equilibrium.The dual optimal solutions are precisely the 's in R H such that   (  , 0) +  *  (0, ) = 0.This equation can be written explicitly as from which it follows that  =  is an optimal solution in the latter supremum.The corresponding optimality conditions are which imply that  ℎ is the equilibrium cost of the OD pair ℎ for the Wardrop equilibrium, that is,  ℎ =  ℎ () for every ℎ ∈ H.It follows that the subgradient ∇ () = {()} so that  ↦ →  () is not only convex but also differentiable with gradient ∇ () = ().The conclusion follows by noting that every convex differentiable function is automatically of class  1 and its gradient is monotone, in the sense that ⟨∇ ( 1 ) − ∇ ( 2 ),  1 −  2 ⟩ ≥ 0 for every  1 ,  2 ∈ R H + , which in particular implies that  ℎ is nondecreasing in the variable  ℎ .The continuity of the equilibrium resource costs   =   () 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 resource costs are optimal solutions for the strictly convex dual program (2.6).Hence, since the objective function is jointly continuous in (, ), Berge's theorem implies that the optimal solution correspondence is upper-semicontinous.However, in this case the optimal solution is unique, so that the optimal correspondence is single-valued, and, as a consequence, the equilibrium resource costs   () are continuous.Finally, letting Φ()  ∈R   (  ) and   =   (   ) we clearly have  = ∇Φ( ), which is equivalent to  ∈ Φ * () where the Fenchel's conjugate is given by Φ * () =  ∈R  *  (  ).Since all the involved functions are finite and continuous, using standard subdifferential calculus rules, (A.9) is equivalent to 0 ∈ Ψ() for the convex function Ψ( ) = Φ * ()− ℎ∈H  ℎ Θ ℎ ( ) which is precisely the objective function in (2.6).
For the sake of completeness we include the following characterization of comonotonicity.This is a folk result (see e.g., Landsberger and Meilijson (1994)), but its proof is not easy to find in the literature.

Figure 2 .
Figure 2. In the Wheatstone network with three paths and a single OD pair, the equilibrium load on the vertical edge ( 1 ,  2 ) is decreasing for  ∈ [1, 2].

Figure 3 .
Figure 3. Representation of the regions Γ ≾ associated to the possible orders of the equilibrium costs in Example 1.The first commodity uses the two top links, and the second commodity uses the bottom two.The three colors represent the regions Γ ≾ for the possible orders ≾ of the equilibrium costs.
this shows that the vector x = (  )  ∈R C is a single-commodity equilibrium for G C with demand  C .It follows that  C () is in fact a function of the aggregate demand  C and so we can write it as  C ( C ).

. 3 )
Example 1. (continued).Consider again the singleton congestion game in Fig.3.In the purple region characterized by   =   , with a single cost class C = {, } and R C = { 1 ,  2 ,  3 }, there are three sub-regions depending on the value of the total demand  C =   +   .When  C ∈ (0, 1) both  and  use only the central link with active regime   =   = { 2 } and equilibrium costs   =   =  C .For  C ∈ [1, 3) we have   =   = (1 +  C )/2, with  using the central link   = { 2 } whereas  splits the flow between the top and central link with   = { 1 ,  2 }.Finally for  C ≥ 3 the active regime is   = { 1 ,  2 } and   = { 2 ,  3 } with equilibrium costs   =   = 1 +  C /3.Similarly, the green region is characterized by   <   with cost classes C 1 = { } and C 2 = {}.Throughout this green region the active regime for  is constant   = { 1 }, whereas for  we have

Figure 4 .
Figure 4.An example with quadratic costs.The first commodity uses the two top links, and the second commodity uses the bottom two.The three colors represent the regions Γ ≾ for the possible orders ≾ of the equilibrium costs.The straight lines within each region separate sub-regions corresponding to different active regimes.The regions Γ ≾ are not convex, but the boundary between sub-regions is still affine.

•
adding a super-source O connected to each O ℎ by a zero-cost edge (O, O ℎ ), • adding a super-sink D connected through zero-cost edges (D ℎ , D), and • appending the edges (O, O ℎ ) and (D ℎ , D) to each path of commodity ℎ.

Proof.
Consider a congestion game structure with resources R = { 1 , . . .,   }.Consider the series-parallel network in the figure below, where each resource is represented by two parallel of them has the original resource cost   ( • ), and the other edge provides a bypass with zero cost.Any strategy  ⊆ R can be represented as a path joining O to D that takes the top edge for each resource in , and the bypass otherwise.We can then represent the commodities of the congestion game in the routing game by prescribing that they all have the same origin O and same destination D, whereas the feasible paths correspond to their feasible strategies in the original congestion game.□ An embedding of Fisk's network in a series-parallel common-OD network.

Figure 5 .
Figure 5. Fisk's multi-commodity network can be embedded in a seriesparallel network with a common-OD, by adding two edges with zero cost.

□
Remark 11.A similar analysis where we reformulate the primal problem by including resource load variables   and considering perturbations in the flow balance equations   = ∋   +  , yields the dual problem (2.6), which characterizes the equilibrium costs   .Perhaps a more direct argument is as follows.Let us rewrite the flow balance equations  = ∋   invector form as  = ∈S     where   = (   )  ∈R denotes the indicator vector with    = 1 { ∈} .(A.6) Since  ℎ = ∈S ℎ   , by letting  ℎ  =   / ℎ for all  ∈ S ℎ we have that ∈S ℎ  ℎ  = 1 and  ℎ  ≥ 0, the latter inequality being strict only for the optimal strategies for commodity ℎ.With these notations, we can write  = ∑︁ ∈S     = ∑︁ Now, for each ℎ ∈ H the super-differential of the concave function Θ ℎ () min ∈S ℎ  ∈   is given by convex hull of the indicators of optimal strategies, namely Θ ℎ ( ) = co   :  ∈ S ℎ , ∑︁

□
Appendix B. List of symbols   cost function of resource    cost function of strategy , defined in (2.3)  vector of cost functions; it can be indexed both by elements in E or P    (  )   (  ) + 2  , perturbed cost   (  ) ∫   0   () d  *  ( • ) Fenchel conjugate of   ( • ) C subset of commodities having the same equilibrium cost D ℎ destination for OD pair ℎ (D  ) dual problem, defined in (A.4) E set of edges  ℎ ℎ-th vector of the canonical basis of R H  ℎ  flow on strategy  in commodity ℎ  ℎ  ℎ  ∈S ℎ , flow vector of commodity ℎ F  set of feasible flows for the demand vector   directed multigraph G (R, , S), congestion game structure G  (R,   , S), perturbed congestion game structure G C (R C , , S C ), defined in Definition 4 defined in (4.2) S ℎ set of feasible strategies for commodity ℎ S × ℎ∈H S ℎ , set of strategy profiles S(D  ) optimal solution set of the dual problem SC() ℎ∈H  ℎ  ℎ (), social cost   () inf    (, ), defined in (A.3)  () min ∈X   ∈R   (  ), defined in (A.1) V set of vertices   load of resource  , defined in (2.2)  (  )  ∈R , load vector X  set of load profiles induced by all feasible flows  ∈ F Γ ≾  ∈ R H + :  ℎ () ≤  ℎ ′ () ⇐⇒ ℎ ≾ ℎ ′ for all ℎ, ℎ ′ ∈ H , defined in (4.1) Γ ≾   ∈ Γ ≾ : R() =  , defined in (4.3)Fenchel conjugate of   , defined in(A.5)