Uncertainty in Multi-Commodity Routing Networks: When does it help?

09/25/2017 ∙ by Shreyas Sekar, et al. ∙ University of Washington 0

We study the equilibrium quality under user uncertainty in a multi-commodity selfish routing game with many types of users, where each user type experiences a different level of uncertainty. We consider a new model of uncertainty where each user-type over or under-estimates their congestion costs by a multiplicative constant. We present a variety of theoretical results showing that when users under-estimate their costs, the network congestion decreases at equilibrium, whereas over-estimation of costs leads to increased equilibrium congestion. Motivated by applications in urban transportation networks, we perform simulations consisting of parking users and through traffic on synthetic and realistic network topologies. In light of the dynamic pricing policies adopted by network operators to tackle congestion, our results indicate that while users' perception of these prices can significantly impact the policy's efficacy, optimism in the face of uncertainty leads to favorable network conditions.

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I Introduction

Multi-commodity routing networks that allocate resources to self-interested users lie at the heart of many systems such as communication, transportation, and power networks (see, e.g., [1] for an overview). In all of these systems, users inherently face uncertainty and are heterogeneous. These users rarely have perfect information about the state of the system, and each have their own idiosyncratic objectives and tradeoffs between time, money, and risk [2, 3, 4]. Naturally, these private beliefs influence each user’s decision and, in turn, the total welfare of the overall system. In this paper, we provide an understanding of the effects of certain classes of uncertainties and user heterogeneity with respect to such uncertainties on network performance. In particular, we establish conditions on when they are helpful and harmful to the overall social welfare of the system.

A motivating example of a routing network that we use throughout this paper is the urban transportation network. Commuters in road networks simultaneously trade-off between diverse objectives such as total travel time, road taxes, parking costs, waiting delays, walking distance and environmental impact. At the same time, these users tend to possess varying levels of information and heterogeneous attitudes, and there is evidence [5, 6]

to suggest that the routes adopted depend not on the true costs but on how they are perceived by the users. For instance, users prefer more consistent routes over those with high variance 

[7], seek to minimize travel time over parking costs [8], and react adversely to per-mile road taxes [9].

Furthermore, the technological and economic incentives employed by planners interact with user beliefs in a complex manner [10]. For example, to limit the economic losses arising from urban congestion, cities across the world have introduced a number of solutions including road taxes, time-of-day-pricing, road-side message signs and route recommendations [11, 12, 13, 14]. However, the dynamic nature of these incentives (e.g., frequent price updates) and the limited availability of information dispersal mechanisms may add to users’ uncertainties and asymmetries in beliefs.

The effect of uncertainties on network equilibria has been examined in recent work [10, 15, 16, 17] where each user perceives the network condition to be different than the true conditions. The current results have mostly focused on simple network topologies (e.g., parallel links) or networks where a fixed percentage of the population is endowed with a specific level of uncertainty. Given the complexity of most practical networks, it is natural to ask how uncertainty (i.e. user beliefs) affects equilibria in scenarios with many types of users, whose perceptions vary according to the user type. Specifically, in this work we answer the following two questions: (i) how do equilibria depend on the type and level of uncertainty in networks with multiple types of users, and (ii) when does uncertainty improve or degrade equilibrium quality?

To address these questions, we turn to a multi-commodity selfish routing framework commonly employed by many disciplines (see, e.g., [18, 19, 20, 21]). In our model, each user seeks to route some flow from a source to a destination across a network and faces congestion costs on each link. These congestion costs are perceived differently by each user in the network, representing the uncertainties in their beliefs.

It is well-known that even in the presence of perfect information (every user knows the exact true cost), strategic behavior by the users can result in considerably worse congestion at equilibrium when compared to a centrally optimized routing solution [22]. Against this backdrop, we analyze what happens when users have imperfect knowledge of the congestion costs. A surprising outcome arises: in the presence of uncertainty, if users select routes based on perceived costs that over-estimate the true cost, the equilibrium quality is better compared to perfect information case. Conversely, if the users are not cautious and under-estimate the costs, the equilibrium quality becomes worse.

I-a Contributions

We introduce the notion of type-dependent uncertainty in multi-commodity routing networks, where the uncertainty of users belonging to type is captured by a single parameter . Specifically, for each user of type , if their true cost on edge is given by , where is the total population of users on this edge, then their perceived cost is .

For the majority of this work, we will focus on cautious behavior, where users over-estimate the costs (), for all types . Some of our results will also hold for the case where users under-estimate the costs (), for all types . The central message of this paper is that when users exhibit “caution in the face of uncertainty”, the social cost at the equilibrium is smaller compared to the case where users have perfect information (i.e. know the true congestion costs).

The following results are independent of network topology:

  1. The social cost—i.e.  where is the total population mass, summed over all user types, flowing on edge —of the equilibrium111For the multi-commodity routing game studied in this work, the equilibrium is essentially unique as all equilibria induce the same flow on the edges; see Proposition III solution where all users have the same level of uncertainty ( for all ) is always smaller than or equal to the cost of the equilibrium solution without uncertainty when and vice-versa when .

  2. The worst-case ratio of the social cost of the equilibrium to that of the socially optimal solution (i.e. the price of anarchy [22]) is , where and is the ratio of the minimum to the maximum uncertainty over user types.

Constraining network topology, we show the following:

  1. The social cost of the equilibrium where a fraction of the users exhibit an uncertainty of and the rest have no uncertainty is always smaller than or equal to the social cost of an identical system without uncertainty, as long as the network has the serially linearly independent topology (a subclass of series-parallel networks [23]).

  2. In systems having users with and without uncertainty, the routing choices adopted by the uncertain users always results in an improvement in the costs experienced by users without uncertainty, as long as the graph has a series-parallel topology [23].

At a high level, the main contribution of this work can be viewed as a nuanced characterization of the instances (i.e. network topologies, levels of uncertainty) for which uncertainty is helpful in reducing the equilibrium congestion levels. We show that these characterizations are tight by means of illustrative examples where uncertainty leads to an increase in the social cost when our characterization conditions are violated. Finally, many of our results also extend gracefully to instances having polynomial edge cost functions, which are of the form (see Section III-C).

To validate the theoretical results, we present a number of simulation results. We focus specifically on the application of parking in urban transportation networks and consider realistic urban network topologies with two types of users: through traffic and parking users. Given a parking population with uncertainties, we show that cautious behavior improves equilibrium quality while lack of caution degrades it even when uncertainty is asymmetric across user types and when the same user faces different levels of uncertainty on different parts of network.

I-B Comparison with Other Models of Uncertainty

Our work is closely related to the extensive body of work on risk-averse selfish routing [24, 25, 26] and pricing tolls in congestion networks [2, 10, 27]. The former line of research focuses on the well known mean-standard deviation

model where each self-interested user selects a path that minimizes a linear combination of their expected travel time and standard deviation. While such an objective is desirable from a central planner’s perspective, experimental studies suggest that individuals tend to employ simpler heuristics when faced with uncertainty 

[28]. Motivated by this, we adopt a multiplicative model of uncertainty similar to [29, 30, 31].

The literature on computing tolls for heterogeneous users is driven by the need to implement the optimum routing by adjusting the toll amount, which is often interpreted as the time–money tradeoff, on each edge. It is possible to draw parallels between additive uncertainty models (such as the mean–standard deviation model) and tolling; specifically, tolling can be viewed as adjusting for users’ uncertainty. While tolls can be (within reason) arbitrarily decided, the system planner has little influence over the level of uncertainty among the users. Bearing this in mind, we strive for a more subtle understanding of how equilibrium congestion depends on the level of uncertainty. Moreover, while many of the results in this paper focus on the effect of over-estimating costs, different than the existing literature, we also study the effect of cost under-estimation.

Finally, our work is thematically similar to the recent paper on the informational Braess paradox [19] whose model can be viewed as an extreme case of our model where the uncertainty parameter is so high on some edges () that users always avoid such edges. On the other hand, our model is more continuous as user attitudes are parameterized by a finite value of , which allows for a more realistic depiction of the trade-offs faced by users who must balance travel time, congestion, and uncertainty. Although both the characterizations make use of the serially linearly independent topology, it is worth pointing out that our results do not follow from [19] and require different techniques that capture a more continuous trade-off between uncertainty and social cost.

I-C Organization

The rest of the paper is structured as follows. In Section II, we formally introduce our model followed by our main results in Sections III and IV. Section V presents our simulation results on urban transportation networks with parking and routing users who face different levels of uncertainty. Finally, we conclude with discussion in Section VI.

Ii Model and Preliminaries

We consider a non-atomic, multi-commodity selfish routing game with multiple types of users. Specifically, we consider a network represented as where is the set of nodes and is the set of edges. For each edge , we define a linear cost function

(1)

where is the total population (or flow) of users on that edge and . One can interpret as the true cost or expected congestion felt by the users on this edge. However, due to uncertainty, users may perceive the cost on each edge to be different from its true cost.

To capture that users may have different perceived uncertainties, we introduce the notion of type. Specifically, we consider a finite set of user types , where each type is uniquely defined by the following tuple . The parameter denotes the total population of users belonging to type such that each of these infinitesimal users seeks to route some flow from its source node to the destination node . Moreover, the parameter captures the beliefs or uncertainties associated with users of type and affects the edge cost in the following way: users of type perceive the cost of edge to be

(2)

For illustration, consider an urban transportation network. Then may represent the constant travel time on a link (in the absence of other vehicles) and , the congestion-dependent component of the travel time. A multiplicative uncertainty of indicates that users of type adversely view costs arising due to congestion (e.g., waiting in traffic) when compared to other costs. Alternatively, could also represent a congestion-dependent toll or tax that is commonly levied in transportation infrastructure such as highways or parking, and the parameter captures the time-money trade-off [32].

A path is a sequence of edges connecting to . Define to be the set of all paths in . Let be the total flow routed by users of type on path . We use the notation for a network flow and to denote the network flow of type . Then, for each type , define the set of feasible flows to be

(3)

The action space of users of type is —that is, users of type choose a feasible flow . Further, define the joint action space —i.e. the space of feasible flows for all user types.

Path flows induce edge flows. Let be the flow on edge due to users of type . The edge and path flow for users of type are related by

Define the total flow on edge to be

Then, using this notation, we write the path cost in terms of edge flow. For any path ,

(4)

Similarly, the perceived path costs are given by

(5)

Define the game instance

The game instance captures all of the relevant information about the multi-commodity routing game including the notion of type-based uncertainty we are interested in studying.

Ii-a Nash Equilibrium Concept

We assume that the users in the system are self-interested and route their flow with the goal of minimizing their individual cost. Therefore, the solution concept of interest in such a setting is that of a Nash equilibrium, where each user type routes their flow on minimum cost paths with respect to their perceived cost functions and the actions of the other users.

[Nash Equilibrium] Given a game instance , a feasible flow is said to be a Nash equilibrium if for every , for all with positive flow, ,

(6)

For the rest of this work, we will assume that all the flows considered are feasible.

[User Beliefs] For the sake of completeness and to understand how an equilibrium is reached, we briefly comment on each user’s beliefs about the uncertainties of the other users. We assume that type-based uncertainty is not only known within the type but across all types—that is, a user of type knows the uncertainty levels of the users of all other types. While a user knowing the uncertainty level within its own type may not be unreasonable, full knowledge of the uncertainties of the other types is a strong assumption.

This being said, for the types of games we consider—games admitting a potential function, which we formally define in Proposition III—a number of myopic learning rules222By myopic learning rules, we mean rules for iterated play that require each player to have minimal-to-no knowledge of other players’ cost functions and/or strategies. converge to Nash equilibria (see, e.g., [33] and references therein). We are currently investigating layered belief structures coupled with a Bayesian Nash equilibrium concept. Some recent work has investigated such structures in a routing game context [34]; however, it is well known that equilibria in these types of games are computationally difficult to resolve.

Ii-B Social Cost and Price of Anarchy

One of the central goals in this work is to compare the quality of the equilibrium solution in the presence of uncertainty to the socially optimal flow as, e.g., computed by a centralized planner with the goal of minimizing the aggregate cost in the system. Specifically, the social cost of a flow is given by

(7)

Note that the social cost is only measured with respect to the true congestion costs and thus does not reflect users’ beliefs or uncertainties.

To capture inefficiencies, we leverage the well-studied notion of the price of anarchy which is the ratio of the social cost of the worst-case Nash equilibrium to that of the socially optimal solution. Formally, given an instance of a multi-commodity routing game belonging to some class of instances (a class refers to a set of instances that share some property) suppose that is the flow that minimizes the social cost and that is the Nash equilibrium for the given instance, then the price of anarchy is defined as follows. [Price of Anarchy] Given a class of instances , the price of anarchy for this class is

(8)

Since, we study a cost-minimizing game, the price of anarchy is always greater than or equal to one.

Iii Main Results

To support these main theoretical results, we first show that our game is a potential game. Routing games that fall into the general class of potential games have a number of nice properties in terms of existence, uniqueness, and computability [33]. General multi-commodity, selfish routing games with heterogeneous users, however, do not belong to the class of potential games unless certain assumptions on the edge cost structure are met [20].

In our case, since we have linear latencies for each type and the uncertainty parameter only appears on the term for each edge, the following proposition states that the game instances of the form we consider admit a potential function and hence, there always exists a Nash equilibrium [20]. A feasible flow is a Nash equilibrium for a given instance

of a multi-commodity routing game with uncertainty vector

if and only if it minimizes the following potential function:

(9)

Moreover, for any two minimizers , for every edge .

Note that although users perceive the multiplicative uncertainty on the term (see Equation (2)), the parameter appears in the denominator of the term in the potential function above. Conceptually, these have a similar effect: dividing Equation (6) by on both sides, one can obtain equivalent equilibrium conditions where the term is present in the denominator of the constant .

Proof:

By definition, a feasible flow is a Nash equilibrium if the following condition is satisfied for all and for all with :

Since , this is equivalent to . The remainder of proof trivially follows from standard arguments pertaining to the minimizer of a convex function. See [33] for more detail.

The second part of the proposition indicates that the equilibria are essentially unique as the cost on every edge is the same across solutions.

Iii-a Effect of Uncertainty on Equilibrium Quality

Our first main result identifies a special case of the general multi-commodity game for which uncertainty helps improve equilibrium quality—i.e. decreases the social cost—whenever users over-estimate their costs by a small factor and vice-versa when they under-estimate costs. To show this result, we need the following technical lemma. Given an instance of a multi-commodity selfish routing game with Nash equilibrium , we have that for any feasible flow ,

(10)

where .

The proof of the above lemma is provided in Appendix -A.

Given an instance of the multi-commodity routing game, we define to be the corresponding game instance with no uncertainty—that is, has the same graph, cost functions, and user types as , yet for all .

Consider any given instance of the multi-commodity routing game with Nash equilibrium and the corresponding game instance , having no uncertainty, with Nash equilibrium . Suppose for all . Then, the following hold:

  1. if .

  2. if .

Remark: What happens when the users are highly cautious, i.e., for all ? Due to the presence of a few negative examples where the social cost increases in the presence of uncertainty, we cannot conclusively state that uncertainty helps or hurts for all instances. However, these negative instances appear to be isolated—both our price of anarchy result (Theorem III-B) and our experiments (Section V) validate our claim that caution in the face of uncertainty helps the users by lowering equilibrium social costs even when —i.e., uncertainty is favorable when the users are very cautious. It is however, interesting to note that although under-estimation always leads to a worse equilibrium, over-estimation may lead to better or worse equilibria.

Proof:

Let denote the potential function for the instance and denote the potential function for where is given in (9) with . By definition of the potential function, we know that and . Expanding these two inequalities, we get that

and

where denotes the total flow on edge by users of type in the solution . Let us define , and . By summing the above inequalities, the terms cancel out giving us

where the right-hand side is less than or equal to zero. Using the fact that for all edges, we get that

(11)

Hence,

  1. When , so that .

  2. When , so that .

We finish the proof by considering two separate cases: (case 1) and (case 2) .

Let us consider (case 1) where . Applying Lemma III-A to the instance with , we obtain that

(12)

We claim that when , the right-hand side of (12) is lesser than or equal to zero. This is not particularly hard to deduce owing to the fact that in the given range and that as deduced from (11). Therefore, , which proves the claim that uncertainty with a limited amount of caution helps lower equilibrium costs.

Now, let us consider (case 2) where . Applying Lemma III-A to the instance with , and using the fact that , we have that

(13)

Once again when , we know that and from (6), we can deduce that in the given range.

The following corollary identifies a specific level of uncertainty at which the equilibrium solution is actually optimal. Given an instance of the multi-commodity routing game, let denote its Nash equilibrium and denote the socially optimal flow. If for all , then —i.e. the equilibrium is socially optimal.

Proof:

Suppose . Then, applying Lemma III-A, we have that .

Iii-B Price of Anarchy Under Uncertainty

In Theorem III-A, we showed that the equilibrium cost under uncertainty decreases (resp. increases) when users are mildly cautious (resp. not cautious) and all user types have the same level of uncertainty. This naturally raises the question of quantifying the improvement (or degradation) in equilibrium quality and whether uncertainty helps when the uncertainty parameter can differ between user types. In the following theorem, we address both of these questions by providing price of anarchy bounds as a function of the maximum uncertainty in the system and , which is the ratio between the minimum and maximum uncertainty among user types. (Price of Anarchy) For any multi-commodity routing game , the ratio between the social cost of the Nash equilibrium to that of the socially optimal solution is at most

(14)

where , and as long as .

Before proving Theorem III-B, we remark on the price of anarchy and its dependence on the level of uncertainty. The price of anarchy in (14) is plotted in Fig. 1 as a function of for three different values of . This result validates our message that uncertainty helps equilibrium quality when users over-estimate their costs and hurts equilibrium quality when users under-estimate their costs. To understand why, let us first consider the case of —i.e. the uncertainty is the same across user types. We already know that in the absence of uncertainty, the price of anarchy of multi-commodity routing games with linear costs is given by  [22]; this can also be seen by substituting , in (14). We observe that the price of anarchy is strictly smaller than for and reaches the optimum value of at thereby confirming Corollary III-A. More interestingly, even when , the price of anarchy with uncertainty is smaller than that without uncertainty, affirming our previous statement that cautious behavior helps lower congestion (at least in the worst case).

Similarly, as decreases away from one, the price of anarchy increases nearly linearly. In fact, our price of anarchy result goes one step beyond Theorem III-A as it provides guarantees even when different user types have different uncertainty levels. For example, when , and —i.e. —the price of anarchy is , which is still better than the price of anarchy without uncertainty.

The price of anarchy result reveals a surprising observation: as long as and is not too large, for any given instance of the multi-commodity routing game, either the equilibrium quality is already good or uncertainty helps lower congestion by a significant amount. Therefore, uncertainty rarely hurts the quality of the equilibrium and often helps.

Fig. 1: Price of anarchy as a function of for multi-commodity selfish routing games for three different values of . Smaller values of price of anarchy are favorable as they indicate that the cost of the equilibrium solution is comparable to the social optimum. In general, we observe that when , the price of anarchy under uncertainty is smaller than that without uncertainty and vice-versa for . Yet, too much caution or large asymmetries in the uncertainty level across user types can also lead to poor equilibrium. For a fixed value of , we observe that the price of anarchy drops initially as increases and then starts increasing once again beyond a certain threshold.

In order to prove Theorem III-B, we need the following technical lemma whose proof is provided in Appendix -B. For any two non-negative vectors of equal length, and , let and . Moreover, let be another vector of length whose entries are strictly positive. Then, if we let and , for any given function with , we have that

(15)
Proof:

Consider some instance of the multi-commodity routing game with equilibrium . Then, adopting the variational inequality conditions for a Nash equilibrium [15, 22], for any other solution , and every user type ,

(16)

must hold. Fix any edge . Let , , , and . Applying Lemma 1 with and , we have that

where

(17)

and where and .

We claim that

(18)

Recall from (III-B) that for any type , Since the price of anarchy is defined as , a worst case bound of follows from Equation (18), giving us the theorem statement.

It is worth mentioning that the proof of the above theorem carries over even when each user type has a different uncertainty parameter on each edge. For example, suppose that denotes the uncertainty parameter corresponding to users of type on edge . In this general model, a similar price of anarchy bound can be shown. Indeed, the price of anarchy in this case is

where and .

Iii-C Extensions to Polynomial Cost Functions

We now generalize our results in another direction by considering multi-commodity routing games where the edges have polynomial cost functions. Specifically, we consider games where the cost function on any edge is of the form , such that the degree , and . This class of polynomial cost functions is referred to as shifted monomials of degree [35], and has received considerable attention in the literature owing to applications in transportation [36, 37, 38]— e.g., link performance as a function of the number of cars on a road can be modeled as a shifted monomial function of degree four [36].

We now show that both of our results from this section extend gracefully to the case where the edges have shifted monomial cost functions. In fact, when the edge cost functions are super-linear, uncertainty leads to more favorable results than the linear cost case. Informally, as the degree of the monomial grows, we show that uncertainty leads to a decrease in the social cost for a larger range of the parameter . For convenience, we will assume that all edges have shifted monomial cost functions with the same degree .

We begin by a identifying the potential function that the Nash equilibrium solution minimizes for a generalization of the multi-commodity routing game where every edge has the true cost function , which users of type perceive as . Therefore, equilibrium existence is guaranteed for this more general class of functions.

(19)

The following result generalizes Theorem III-A to games with shifted monomial cost functions.

Consider any given instance of the multi-commodity routing game with shifted monomial cost functions of degree having Nash equilibrium and the corresponding game instance , having no uncertainty, with Nash equilibrium . Suppose for all . Then, the following hold:

  1. if .

  2. if .

In comparison to Theorem III-A where uncertainty results in a decrease in social cost for , we observe (surprisingly) that the range of the uncertainty parameter under which caution yields an improvement in social cost increases linearly with , i.e., uncertainty is more helpful as the cost function is more convex. On the other hand, the under-estimation of costs increases social cost regardless of the degree of the polynomial. The key idea involved in the proof of Proposition III-C is a strict generalization of Lemma III-A via a potential function argument, to obtain the following difference in costs: . Recall that in the previous inequality is the equilibrium solution for a given instance and is an arbitrary feasible flow for the same instance. The rest of the proof is analogous to that Theorem III-A.

Next, we generalize the price of anarchy bounds from Theorem III-B.

For any multi-commodity routing game with shifted monomial cost functions of degree , the ratio between the social cost of the Nash equilibrium to that of the socially optimal solution is at most

(20)

where , and as long as . Substituting in the above equation, we obtain Theorem III-B as a special case. In the absence of uncertainty, it is known that as grows large, the price of anarchy bound grows as  [39]. Assuming , for a constant , the price of anarchy bound from Equation (20) grows as . For a finite value of , the improvement in the price of anarchy is much more significant as illustrated in Fig. 2.

Fig. 2: Price of anarchy as a function of for multi-commodity selfish routing games for three different values of when edge cost functions are shifted monomials of degree . Smaller values of price of anarchy are favorable as they indicate that the cost of the equilibrium solution is comparable to the social optimum. In general, we observe that when , the price of anarchy under uncertainty is smaller than that without uncertainty and vice-versa for — this is true for all values of . Interestingly, for , uncertainty leads to near-optimal price of anarchy even for large values of (e.g., )— this is not the case when . This validates our premise that as the degree of the polynomial cost function increases, user uncertainty becomes more helpful to the system welfare.

Iv The Effect of Heterogeneity on Congestion

Now that we have a better understanding of how uncertainty affects the performance of the entire system as measured by the social cost, we move on to a more nuanced setting where different users have different levels of uncertainty. Our goal in this section is to understand the effect of this heterogeneity in uncertainty on the equilibrium congestion by studying the following two questions: (i) how do the routing choices adopted by the uncertain users impact the cost of the users without uncertainty, and (ii) when is the equilibrium social cost of a system with heterogeneous uncertainties smaller than the social cost incurred when all users have no uncertainty?

Iv-a Notation and Graph Topologies

To isolate the effect of heterogeneity on equilibrium congestion, we restrict our attention to a selfish routing game on an undirected network. There are two user types, i.e. . Users belonging to both these types seek to route their flow between source node and destination node . Finally, the uncertainty levels for the two user types are specified as and , and so, users of type are without uncertainty. We refer to this as the two-commodity game with and without uncertainty. We slightly abuse notation and use to refer to an instance of this game, and to represent the set of - paths in .

Unlike the previous sections, where we made no assumptions on the graph structure, our next characterizations will depend crucially on the network topology. Specifically, we will consider two well-studied topologies: series-parallel and linearly independent graphs. We provide the respective definitions below. (Series-Parallel [23]) An undirected graph with a single source and destination is said to be a series-parallel graph if no two - paths pass through an edge in opposite directions. There are a number of other equivalent definitions for this class of graphs; e.g., a graph is said to be series-parallel if it does not contain an embedded Wheatstone network [23]. Series-parallel graphs are an extremely well-studied topology that naturally arise in a number of applications pertaining to network routing. We refer the reader to [40, 23] for more details.

(Linearly Independent [23]) An undirected graph with a single source and destination is said to be linearly independent if every - path contains at least one edge that does not belong to any other - path. Our final definition involves a simple extension of the above topology to include linearly independent graphs connected serially. Formally, a graph is said to consist of two sub-graphs with source-destination pair and with source-destination pair connected in serial if with and .

(Serially Linearly Independent (SLI)) An undirected graph with a single source and destination is said to belong to the serially linearly independent class if (i) is linearly independent or (ii) consists of two linearly independent graphs connected in serial. This extended topology was first introduced in [19]. These three topologies are related as:

  1. Every linearly independent graph belongs to the serially linearly independent class.

  2. Every serially linearly independent graph belongs to the series-parallel class.

The first part follows from Definition IV-A. The second part is due to the fact that any linearly independent network is series-parallel (proved in [23]) and the property that when two series-parallel graphs are connected in series, the resulting graph is also series-parallel. Since series-parallel graphs happen to be the most general topology studied in this section, (without loss of generality) we focus on undirected graphs because every edge can be uniquely traversed only in one direction.

Iv-B Impact of Uncertain Users on Users without Uncertainty

We begin by studying what happens to the congestion cost faced by the users without uncertainty as the uncertainty level increases for the other users. This question is of considerable interest in a number of settings. For example, in urban transportation networks, the uncertainty about where a driver can find available street parking can often cascade into increased congestion for other drivers leading to a detrimental effect on the overall congestion cost [38, 41, 42, 43].

The following theorem shows a somewhat surprising result. As long as the network topology is series-parallel, the aggregate cost felt by users of type (users without any uncertainty) always reduces when users of are uncertain about the costs. In other words, the behavior under uncertainty by one type of users always decreases the congestion costs of other types of users who do not face any uncertainty.

Given an instance of the two-commodity game with and without uncertainty such that the graph is series-parallel, let denote a modified version of this instance with no uncertainty (i.e. ). Let and denote the Nash equilibrium for the two instances, respectively. Then,

(21)

where is the aggregate cost of users of type .

Proof:

It is well-known that [23, Lemma 3] for a series-parallel graph and any two feasible flows , , there exists a path with , such that for every edge , . Now, consider flows and . Applying the previous property, we get that, there exists a path with such that for all , .

We now bound both and in terms of the cost of the path . Specifically, note that in the solution , the path has non-zero flow on it so that

(22)

However, in the solution , we know that every user of type is using a minimum cost path with respect to the true costs and therefore, the cost of any path used by type is at least that of the path . Formally,

(23)

The final inequality follows from the monotonicity of the cost functions and the fact that for all . Therefore, we conclude that

Iv-C Characterization of Instances where Heterogeneity Helps

We now consider the impact of heterogeneity on the system performance as a whole and present a simple characterization based on the network topology and the level of uncertainty, where the presence of uncertainty (among a fraction of the user population) results in a decrease in the equilibrium social cost. Specifically, we show that for SLI networks, as long as the uncertainty level of users belonging to type is at most two (i.e., ), the social cost of the equilibrium solution is always smaller than or equal to that of the equilibrium when there is no uncertainty.

Before showing our theorem, we state the following technical lemma whose proof is deferred to Appendix -C. Given any instance of the two-commodity routing game with and without uncertainty where the graph is linearly independent, let and denote the Nash equilibria of instances and respectively. Then, it must be the case that

(24)

Informally, the above lemma states that given equilibrium flows and for any arbitrary instance and its uncertainty-free variant , the equilibrium solutions must satisfy the property that for any path , the flow on this path in the absence of uncertainty (instance ) must be greater than or equal to its magnitude due to the uncertainty-free users in .

Consider any given instance of the two-commodity game with and without uncertainty. Let denote the Nash equilibrium of this game and the corresponding game instance , having no uncertainty has Nash equilibrium . Then, as long as belongs to the serially linearly independent class and ,

Proof:

Each SLI network can be broken down into a sequence of linearly independent networks connected in series. Applying Definition IV-A recursively, we get a sequence of linearly independent sub-graphs with source-destination pairs respectively ( note that ), that are connected in series—i.e., is connected in series with such that the destination for acts as the origin for . By definition, the set of edges in these subgraphs are mutually disjoint.

Secondly, given the equilibrium flow on for instance , we can divide this flow into components such that for every , is the sub-flow of on the graph , and for every , . Finally, it is not hard to see that must be an equilibrium of the sub-instance of restricted to the graph .

Given this decomposition, we apply Lemma IV-C to each . Consider any index : since the graph is linearly independent, we can apply Lemma IV-C and get that for any - path in , .

Suppose that denotes the set of - paths in . Then, by Lemma -D provided in Appendix -D,

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so that, using the above decomposition,