Competitive Comparisons of Strategic Information Provision Policies in Network Routing Games

Smart cities (i.e., cities where the sensing and communication infrastructure is developed enough to offer a central planner an informational advantage over its citizens) make it possible to envision a new kind of congestion alleviation mechanism, in which messages (whether they are they roadside signals or individual path recommendations) are actively and strategically sent to entice drivers to take socially beneficial itineraries. We compare five such information provision strategies—optimal routing, no information available, all information available, public signaling, and personal itinerary recommendation—using a competitive ratio reminiscent of the celebrated price of an anarchy metric, but adapted to stochastic networks. We also define a partial order between information provision policies by saying that a policy is maximally more efficient than another if it is more efficient on all networks and if there are instances for which the inefficiency ratio approaches the deterministic price of anarchy. With these tools, we show that a strict hierarchy exists between all policies, excluding the case of uninformed drivers. The latter is on some networks maximally better than having drivers completely informed, yet arbitrarily inefficient in other instances.


I. INTRODUCTION
R OAD congestion represents a tremendous loss of time and fuel for drivers. The additional time spent in commuting and the inability to accurately predict travel times are deleterious to business and spare time. In the US alone, road traffic congestion is responsible for an estimated annual loss of $305B in direct and indirect costs [1], which amounts to a staggering average loss of $930 per capita every year. While congestion is caused by a number of interrelated causes, one long-identified factor is the inability for drivers to coordinate their route choices. This inefficiency of selfish routing on road networks was first formalized through the use of game theory [2]- [4] and, in particular, led to Braess's paradox [5]- [9]. This revealed that: 1) increasing the capacity of a road network might decrease the welfare of drivers and 2) most probably many networks already satisfy the conditions of the paradox and would gain from the closing of some sections. More recently, the price of anarchy (PoA) has risen as a measure of the inefficiency lost in competition over cooperation in networks [10], [11]. The fact that it is bounded showed that even though drivers only mind their own itinerary, their incentives are not totally disconnected from their externalities. This has for further consequence to bound the extent of Braess's paradox.
(Advanced) Traveler Information Systems (ATIS), such as roadside signs and mobile apps, have been proposed as a way to help better synchronize drivers' decisions and fluidify traffic [12]- [14]. The expectation, in this case, is that betterinformed drivers will be more aware of the traffic situation and changes in the network and, hence, better able to modify their route choices, leading to overall congestion reduction. However, early theoretical studies of ATIS and en-route information systems [13], [15], as well as empirical observations [16], have revealed a more nuanced picture and suggested that the role of information in network routing is rather subtle and sometimes counterintuitive. For instance, more recent works such as [17] exhibited an example of "informational Braess's paradox": a network in which informed drivers (i.e., those aware of more roads than another population) are outperformed by unaware drivers in terms of individual travel time.
One reason why such information provision mechanisms may sometimes fail to achieve a socially goal is that they treat the drivers as "information takers" (in the same way that marginal pricing considers decision makers as "price takers," in contrast with mechanism design, which anticipates their self-interested reactions to announced prices) and simply relay the state of the road, instead of sending messages that actively and strategically try and modify users' decisions to align them with socially efficient outcomes.
Recently, some works, such as [18]- [20], have started adapting ideas from information design and Bayesian persuasion in economics [21] to design both public and private strategic information provision schemes for network routing and congestion alleviation. The idea at the heart of this approach is that when This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ the network is affected by uncertainty (e.g., a particular road becoming congested due to bad weather) and the planner is better informed about this vagary than the drivers, it is possible to issue a message (which can take the form of a public announcement or a personal recommendation), which is: 1) correlated with the vagary; 2) followed by the drivers; and 3) could a priori differ from "all or nothing" (and even include randomization) to specifically shape the Bayesian Wardrop equilibrium, which ultimately determines the flows that result from individual choices. In [18], Das et al. showed that the social optimum can sometimes even be implemented in the Bayesian Wardrop equilibrium using such persuasion strategies. At the conceptual level, these kinds of information provision policies would, thus, appear to offer smart cities (i.e., cities where the sensing and communication infrastructure is developed enough to offer a central planner good knowledge of the roads' conditions and to allow it to message users, personally or in aggregate) a new and more efficient way to route traffic and reduce congestion than mechanisms taking an "information-taker" view of drivers.
However, much remains open regarding the conditions under which this can be achieved or what the best achievable equilibrium may be in general. We started tackling this question in our paper [22] through the angle of competitive comparisons between the various persuasion strategies, i.e., by evaluating or bounding the worst-case ratio between the best social costs achieved by policies of a particular class, over a family of network (and network costs) of interest.
We continue and extend this work in this article by performing pairwise competitive comparisons between five different persuasion strategies: 1) no information (NI); 2) full information (FI); 3) public signaling; 4) private incentive-compatible itinerary recommendations; and 5) centrally determined itineraries. Compared to [22], our results are more general in that they pertain to general cost families (under mild hypotheses) specifying road costs, rather than merely affine costs, and we address the comparison between public signaling with personal recommendations (via the introduction of a so-called revelation mechanism), which was previously left open.
The equilibrium flows of both the uninformed drivers and the totally informed drivers can be replicated with a public signal, whereas the equilibrium flows generated by a public signal can be replicated with incentive-compatible private recommendations, and, of course, any flow could be enforced by an oracle. These facts establish a partial hierarchy; we show conversely that the respective worst-case inefficiencies amount to the original PoA. In this sense, some policies are maximally more efficient than others.
The case of uninformed drivers is special since the situation can be arbitrarily degraded compared to any other information provision. On the other hand, we show that fully informed drivers may, in some cases, route with an inefficiency nearing that of the PoA compared to uninformed drivers.
The rest of this article is structured as follows. Section II is dedicated to introducing some preliminary notions and tools and to reviewing the permanent-flow model, selfish routing, as introduced by Wardrop [3] and Beckman et al. [4], and the classical PoA introduced in [11]. Section III extends some of these notions to stochastic networks, discusses strategic information provision in that context, and presents public and private signaling strategies.
In Section IV, we derive a partial hierarchy between these strategic information provision strategies, and using a competitive ratio metrics establish that: 1) uninformed drivers can perform arbitrarily badly compared to any other routing; 2) uninformed drivers might outperform informed drivers by a factor equal to the PoA; and 3) private signaling is better than public signaling. Finally, Section V concludes this article. the conclusion summarizes the results and opens a discussion on the working hypotheses.

A. Permanent-Flow Model and Wardrop Equilibria
For the purposes of this article, we use the permanent-flow model of road networks and routing [3], [4], briefly reviewed in the following. Let G = (V, E) be a directed graph, whose vertices V represent crossroads and edges E represent road sections. We let T = V 2 denote all possible origin-destination pairs. For each trip t = (o, d) ∈ T , we denote by P t the finite set of simple paths starting from o and ending at d. We let P = t∈T P t be the set of all paths. While this definition overloads the notation "f " a bit, it should be born in mind that a flow (on paths) induces a flow on edges; hence, f will refer to a flow on paths, f P is the flow on path P , and f e will be the flow on edge e. We say that a path P is taken (for a flow f ) if f P > 0. We can now endow the graph with latencies, seen as a cost to users. Definition 2: A cost function a is a nondecreasing continuous function a : R + → R + .
If each edge e has a cost function c e , its cost under flow f is c e (f e ), and thus, path P costs As before, our notation overloads "c," but it should be clear that c is a collection of cost functions indexed by edges, while c e is the cost of edge e (function of f e , flow on edge e), and c P is the cost of path P (function of the flow on paths f ).

Definition 4:
A demand is a non-negative vector indexed by the trip set T . We say a demand d is met by a flow f if for all trips t ∈ T We will let F d be the set of such flows.
Whenever the trip set T will be a singleton, the demand will be considered scalar. A graph G with latencies c and demand d together form a road network N = (G, c, d).
Definition 5: A flow f is termed a Wardrop equilibrium if for all trip t ∈ T and path P ∈ P t which is taken, no better path P ∈ P t exists. More formally, f satisfies At a Wardrop equilibrium, all taken paths of a same trip share the same cost, and other paths of that same trip cost at least as much. This observation simplifies the social cost evaluation.
These definitions, although standard, raise some remarks. First of all, the permanent-flow hypothesis may never be exactly satisfied in reality, but is reasonable to model routing choices. Assuming that equilibrium has been achieved is also a strong hypothesis; however, the daily repetition of this game and, more recently, the availability of traffic planners bring drivers to choose their shortest path and, in turn, drive the flow to a Wardrop equilibrium. Finally, the definition of social cost is motivated by its simplicity and earlier considerations judging time as the most important resource. A more modern treatment of the question would require a more holistic perspective, including air and sound pollution and the interaction with public transports. This would, however, disconnect the incentives and externalities of drivers, falling out of the scope of this article.
The routing problem is a continuous congestion game, where drivers share roads as resources and a Wardrop equilibrium is a Nash equilibrium; therefore, it is a potential game [23], [24] of potential The following famous theorem relies heavily on this construction.
Theorem 1: Given a road network N = (G, c, d), its Wardrop equilibria are exactly the minima of Φ over F d ; hence, there exists at least one Wardrop equilibrium; furthermore, all Wardrop equilibria share the same social cost.

B. PoA and Pigou Bound
The price of anarchy (PoA) of a network refers to the inefficiency of selfish routing over an optimal routing, which could be enforced if a central entity could choose at its leisure the path of each car. We let V (N ), V * (N ) be, respectively, the social cost under selfish routing and the optimal social cost of a network N .
Definition 6: For a network N , we define the PoA as the ratio and 1 when the ratio is indeterminate. Note that the ratio is indeterminate exactly when V * (N ) = 0, as there exists a routing where no driver suffers any cost, which is thus a Wardrop equilibrium of null social cost as well. Toying with a simple two-road network, one quickly realizes that the PoA is unbounded if we let any function be an admissible cost function. This justifies restricting oneself to certain families of costs, for instance, affine costs, concave costs, or polynomial costs.
Definition 7: A cost family is a collection C of cost functions. We say that it is inhomogeneous if it contains an inhomogeneous cost, i.e., a ∈ C such that a(0) > 0, and that it is homogeneous otherwise.
The PoA of a set S of networks is the supremum of the PoA of those networks, i.e., overloading ρ We let N (C) denote the set of networks (regardless of their graph and demand), whose every edge's cost belongs to a given family C. For brevity, we let ρ(C) = ρ(N (C)) so that ρ(C) denotes the worst PoA over all networks with edge costs in C.
Pigou networks are among the simplest networks one can consider, yet are instrumental in determining the PoA. The Pigou network associated with cost function a and scalar d > 0 (see Fig. 1) consists of two parallel roads of respective costs a and constant cost a(d). Since a(d) ≥ a(x) for all possible flow x ∈ [0, d], all drivers naturally take the upper path of cost a, yielding a social cost da(d). On the other hand, the optimal social cost is we may, thus, simply require x > 0. In turn, the PoA of the class of Pigou networks associated with any (a, d) where the ratio is 1 when a(d) = 0, this is known as the Pigou bound. At first glance, the Pigou bound seems computable though rather pointless because: 1) two-road networks do not contain all possible complexity, and 2) Pigou networks do not even belong to N (C) if C does not contain constant costs. However, as expressed in the following theorem, it is a most remarkable fact that these two-road networks provide an upper bound and, for realistic cost families, actually represent worst cases. Theorem 2 (from [11]): For any family of costs C, we have In addition, if C is inhomogeneous, the converse inequality also holds and, as a result, becomes an equality. Note that this first inequality holds for all networks whose edge costs are in C, even those with multiple origin-destination pairs. We will later use this inequality as it is, granting the same generality to our results.
The Pigou bound, thus, offers a computational solution for inhomogeneous families of costs. One finds α = 4 /3 for affine costs and concave costs. On the other hand, for homogeneous polynomials, i.e., C n = {x → λx n , λ ≥ 0} for n ∈ N nonnegative integer, we find that Φ = (n + 1)Ψ; therefore, Wardrop equilibria are exactly the optimal flows and ρ(C n ) = 1 despite the fact that Computing the PoA of a homogeneous family of costs is still an open problem to the best of our knowledge. For instance, for the homogeneous family C = C 1 ∪ C 2 , the worst two-road network with unit demand is obtained by picking the cost functions x → 24x 2 and x → 25x, respectively, yielding a PoA of 729 /704. 1 This offers a lower bound on ρ(C), but it remains unclear whether this is an equality.
Our study is deeply connected to the converse inequality of Theorem 2, i.e., α(C) ≤ ρ(C) for inhomogeneous families. In fact, we will extend the proof technique used by Roughgarden [11] to stochastic networks. Before doing so, we dedicate the next section to developing a notation for some graph constructions. We also briefly review how the inequality α(C) ≤ ρ(C) for inhomogeneous C is derived in [11], because it motivates our own constructions in Section IV.

C. Graph Constructions
If a cost family C contains all constant costs, then, obviously, each Pigou network belongs to N (C) and, therefore, ρ(C) = α(C). If C is merely inhomogeneous, each Pigou network with cost a ∈ C and demand d > 0 can be replicated using cost a and an inhomogeneous cost. To this end, we consider two graph constructions.
First, we can construct an edge with cost a + b if a, b ∈ C by adding in series an edge of cost a and another of cost b, in the sense that a network with an edge a + b has the same paths, social cost, and optimal and Wardrop flows as a network with edges a and b in series (see Fig. 2). It follows that this addition 1 For a > 0, the social cost under selfish routing of the unit-demand network The PoA of this class of networks is then given as the maximum of a simple explicit optimization program. is associative and commutative. If n ∈ N * is a positive integer, we can construct na by placing the same edge with cost a in series n times.
Second, we can also place edges in parallel, which we will write a b in a manner similar to electrical circuit notations. This operation is also associative and commutative, and we may write a n = a · · · a n copies . See Fig. 3. It should be born in mind that the symbol is purely a notation and that, in particular, Parallelization takes precedence over addition, e.g., a + b c = a + (b c), and is in general not distributive over addition. With these notations, the Pigou network of Fig. 1 with parameters a and d can, thus, be written as a a(d), with demand d. Not all graphs can be represented using solely parallel and series connections (for instance, the infamous diamond network of Braess' paradox [5]); however, this will be sufficient to construct examples whose inefficiency nears the Pigou bound.
Proof of the converse inequality of Theorem 2, [11]. Now, consider a ∈ C, d > 0 with a(d) > 0 and x ∈ [0, d]. We construct a sequence of networks of N (C), whose PoA is at least .
Let b ∈ C be an inhomogeneous cost and define for all integer n large enough Consider the following sequence of networks each with demand d: The network N n has n + 1 parallel paths, path P 0 has cost q n a, and all other paths P i with i = 1, . . . , n have cost nb. Since path P 0 is never longer than the other ones, we have On the other hand, we can consider the flow Thus, the PoA of this network satisfies This holding for all a ∈ C and The worst-case instances evoked above may seem artificial in that they require an increasing number of edges to close in the upper bound; nonetheless, they show the tightness of the bound for the most general class of networks under investigation. To the best of our knowledge, characterizing the PoA for more restricted classes of network is still a widely open question.
Much like Pigou networks of this proof, our constructions will all be rather simple networks mimicking Pigou networks and yet provide asymptotically matching lower bounds. In contrast, we will directly apply the structural inequality α(C) ≥ ρ(C)valid for all networks, with possibly multiple origin-destination pairs-to extend it in the stochastic setting. Hence, all our results are valid for networks with multiple origin-destination pairs.

III. STOCHASTIC NETWORKS AND INFORMATION PROVISION POLICIES
In the now traditional setting reviewed in the previous section, the network's structure and cost functions are known to the population of decision makers, and the central planner's only way to affect routing is by directly assigning flows on each edge.
From now on, we are interested in situations where some characteristics of the network are unknown a priori and the planner has the ability to (partially) inform drivers about them. Following [18], and in keeping with the viewpoint of information design initiated in [21], we model these unknown quantities as random variables, whose realizations are privately observed by the planner. While a single stochastic network typically involves several possibly correlated uncertainties of this form, we may consider that they all rely on a single uncertain parameter, which we represent as a random variable ω, which lies in the probability space (Ω, F,μ). Henceforth, Ω will always be finite (and F = 2 Ω ), so that expected costs are cost functions (according to Definition 2), and the information provisions introduced below are well defined. For each ω ∈ Ω, we will note N ω as the deterministic network with fixed costs c[ω] e . When Ω contains only two values (i.e., when ω is Boolean) and a, b ∈ C are two cost functions, we will use the notation a b for a cost c[ω] to indicate that c[0] = a and c [1] "Information" is provided in the form of messages correlated with the uncertain variable ω, which can, in turn, be observed by decision makers and used to form a posterior belief on its value. Different kinds of messages correspond to different information provision policies, of which four are of particular interest to us: full information (FI), no information (NI), public signaling and private signaling. These policies are explained in more detail in the following. The question of whether a particular policy is more advantageous than another one is discussed in the next section, through the computation of competitive ratios.

A. No and Full Information
FI refers to the situation where drivers are communicated at the exact value of ω (in which case the message is ω itself) and, thus, know the state of the network perfectly. In this case, they simply route according to the Wardrop equilibrium of the corresponding network N ω defined by the costs c[ω] e . In expectation, the social cost of this situation is whereμ is the prior distribution over Ω, the set of values of ω. In comparison, the optimal social cost is In contrast, the NI provision method corresponds to the situation where drivers are completely ignorant of the value of ω, which can be achieved by sending no message at all or sending one that is not correlated with ω.
where, for each realization of ω, Ψ ω is the counterpart of function Ψ for network N ω , and Nμ is the network with costs c[μ] e . Accordingly, in the following, we will note Ψ μ = E μ [Ψ ω ] for any belief μ.

B. Public Signaling
As its name indicates, public signaling is a mode of information provision, in which the planner sends the same message to the entire population of decision makers. More precisely, upon observing ω, the planner sends a message m ∈ M (a finite message space) according to a stochastic policy π. Accordingly, we denote by π[ω] m the probability that message m is issued given that vagary has value ω.
The population of decision makers receives message m and updates its belief from priorμ to posterior μ [ A lemma proved in [21] shows that there exists a policy π generating the set of posteriors {μ[m]} m∈M with probabilities {τ m } m∈M (i.e., τ m is the probability that message m is issued) if and only if they satisfy Bayes plausibility, i.e., At the same time, the expected social cost may be rewritten as As a result, the best expected social cost achievable via public signaling can, hence, be computed as Furthermore, the nature of this program implies that the message space need not be greater than the event space; hence, without loss of generality, we let M = Ω [21].

C. Personal Recommendation
Another conceivable scenario would have each car equipped with a personalized itinerary recommendation system, coordinated by a smart city. If the recommendation is incentive compatible, drivers will follow it without the need to learn a best response to others' choices. The sender may, thus, choose any flow f [ω] indexed by ω and randomly recommend path P ∈ P t to drivers of trip t ∈ T , with probability f [ω] P/d t . This will be incentive compatible as long as for all t ∈ T and P, P ∈ P t whenever Eμ[f [ω] P ] > 0, where μ[P ] is the updated belief over ω after receiving recommendation P . In other words, for all recommendations (sent with positive probability), it must be weakly shorter to follow it.
Thanks to Bayes rules, this can be rewritten concisely: for all t ∈ T and P, P ∈ P t , we have ))] ≤ 0 whose optimal value defines V PRIV (N ).

IV. COMPARING INFORMATION PROVISION POLICIES
Having introduced the four kinds of information provision of interest (which we refer to collectively as "routings," much like the optimal routing), we are now in a position to compare them pairwise using a competitive ratio metric similar to the PoA. The resulting hierarchy and worst-case ratios are the main results of this article.
Definition 8: For two routings A and B and a cost family C, we define their competitive ratio as where, by convention, 0 /0 = 1 and x /0 = ∞ for x > 0, and we let N Δ (C) denote stochastic networks, where each edge has cost functions c[ω] e ∈ C.
It should be noted that this defines competitive ratios as the worst comparison over all networks, and not as mere ratios of prices of anarchy (which would be ρ A * (C) /ρ B * (C)). This definition encompasses all cases with the indeterminate fraction convention. For all γ ∈ [1, ∞) and a, b ≥ 0, one has In the following, we prove that ρ A B (C) ≥ 1; in this sense, the competitive ratio ρ A B (C) is the smallest number γ such that for all networks of N Δ (C), we have

A. Ordering Information Provisions
We begin with a lemma that reveals the simple transitive structure harbored by competitive ratios, as defined above.
Lemma 1: For all routings A, B, and C and cost family C, we have Proof: Consider a network N ∈ N Δ (C) with a single edge and unit demand; there is a unique path, and therefore, the only possible flow is f 0 = 1. As a result, the social cost of this network is equal in all situations, in particular The other inequality is trivial if either ρ A B (C) or ρ B C (C) is infinite. Otherwise, both are finite; in particular, there is no network (N ). For each network, either no cost is null, just the cost of of A is null, the cost A and B is null, or all costs are null. In all these cases, we have Therefore . We conclude that ρ B A (C) = 1 if and only if for all stochastic networks N ∈ N Δ (C), we have In this case, we say that B is more efficient than A, and note that B ≤ C A. This defines a natural partial preorder on all routings, i.e., a reflexive and transitive binary relation, since ρ A B (C) = ρ B C (C) = 1 directly implies, thanks to Lemma 1, that ρ A C (C) = 1. If A is more efficient than FI, then the inefficiency of A with respect to any other routing is upper-bounded by the deterministic PoA; this is the subject of the next proposition.
Proposition 1: Let C be a cost family and A and B be two information provisions such that A ≤ C FI; then, we have ρ A B (C) ≤ ρ(C). Proof: Indeed, for any network N ∈ N Δ (C), we first have We will then say that B is maximally more efficient than A if it is more efficient than A (that is, ρ B A (C) = 1) and ρ A B (C) = ρ(C). In this case, we will note B < C A. This defines a transitive binary relation on routings more efficient than FI, as a direct consequence of the following proposition. Proposition 2: Let C be a cost family and A, B, and C be three information provisions such that Then, C < C A.
Proof: By transitivity, C ≤ C A, and in the first case, we have whereas in the second case, we have . Armed with this strict order, we will establish a hierarchy between FI, public signals, private recommendation, and optimal routing. The case of NI stands out as it cannot be consistently compared with FI.

1) A Revelation Mechanism to Connect Public and Pri-
vate Signals: As introduced in Section III, private recommendations and public signals cannot be readily compared. Indeed, while it would seem intuitive that privately signaling each driver individually offers more flexibility than a simple public announcement, we must be mindful of the fact that the flows resulting from a private recommendation are allowed to depend directly on ω, while those induced by a public policy only depend on the message m. In addition, it may not be possible to replicate a public signal by simply sending this very signal as a private recommendation to all drivers, in the sense of Section III, since our definition requires private signals to be incentive-compatible recommendations.
However, a closer look at the definition of personal recommendation reveals that the planner could instead index flows f by any random variable of her choice, as long as the notion of incentive compatibility is computed with respect to the appropriate posterior distribution. From now on, the term private recommendation is meant to encompass those generalized policies as well.
In particular, if a flow f is indexed by a message m generated by a public policy (τ, μ), we can associate to with a private recommendation as follows: upon realization of m, flow f [m] is used to recommend itineraries, namely, path P ∈ P t is recommended to a fraction f [m] P/d t of drivers traveling on trip t. The value of this recommendation is E τ [Ψ μ [m] (f [m])], as long as it is incentive-compatible. This amounts to for all t ∈ T and P, P ∈ P t such that E τ [f [m] P ] > 0, where μ[P ] is now the belief over (m, ω) having received recommendation P . Using Bayes rule yields directly Therefore, incentive compatibility amounts to for all t ∈ T and P, P ∈ P t . The best such private recommendation is the solution of ∀t ∈ T , ∀P, P ∈ P t , whose optimal value defines V PRIV (N ).
In practice, this construction may be achieved with the help of a "revelation mechanism," which generates m given ω. The message m is then used to produce the recommendation f [m].
With this revelation mechanism and redefinition of private recommendations in hand, we are now in a position to state the following proposition.  N μ[m] ). Formally, for all t ∈ T and P, P Thus, this also holds in expectation. As a result, (τ, μ, f ) does define an incentive-compatible recommendation policy. On the other hand, the social cost is the same as that of the public signal This can also be seen as each event (m, ω) occurs with the same probability and implies the same consequences (the flow is f [m]) for both policies.
2) Structural Results: Proposition 3 shows that the equilibrium flows generated by public signals can be replicated by incentive-compatible private recommendations. This makes it possible to prove the following structural result. Proposition 4: In terms of efficiency, for all family cost C, we have * < C PRIV ≤ C PUB ≤ C FI, NI.
Proof: First of all, the optimal routing is more efficient than any routing, by definition. Furthermore, deterministic networks are instances of a stochastic network, where all information provisions induce the social cost of selfish routing; this implies * < C PRIV.
Second, both NI and FI can be replicated with a public signal. If the message m is independent of ω (for instance a constant message), drivers retain their prior belief and always route as uninformed drivers. Furthermore, if the message mirrors the state of the world, m = ω, drivers are at all time fully informed. Thence, PUB ≤ C FI, NI.
Finally, private signals can replicate public signals, as shown in Proposition 3. This construction holds for all arguments of the public program; we conclude that PRIV ≤ C PUB.
Proposition 4 is valid for any family of cost, even homogeneous ones, although we have no means to compute explicitly ρ(C) for them. The rest of this section is, thus, dedicated to: 1) understanding the relation between FI and NI and 2) presenting extreme cases to refine the ordering.

C. Need for Signaling
Without any information available, it might so happen that incentives and externalities of drivers are most of the time disconnected. It should then come as no surprise that the competitive ratio of this situation over any reasonable routing is unbounded for some families of costs.
Definition 9: A regular cost family C is an inhomogeneous cost family that contains a homogeneous cost, namely, there exists a, b ∈ C with b(0) > 0 = a(0). Proposition 5: For a regular cost family, we have Without information, both edges are equivalent; thus, we have whereas for d small enough a(d) ≤ b(0), informed drivers then all take the edge of cost a V FI (N d ) = a(d).
Then, the limit as d goes to 0 of the competitive ratio of NI over FI is infinite The rest follows directly from the hierarchy established in Proposition 4. This exhibits situations where drivers need to have access to some information. Yet, surprisingly, there are situations where the social cost is minimized when drivers are uninformed, while FI is the most nonoptimal routing (in the sense of PoA). Theorem 3: If C is an inhomogeneous cost family and 0 ∈ C, then ρ FI NI (C) = ρ(C) in particular PUB < C FI.

Proof:
We have already established that ρ FI NI (C) ≤ ρ(C); for instance, considering Proposition 1 with A = FI and B = NI, we will simply show the converse inequality. In order to accomplish this, we invoke a Pigou network with a ∈ C, d > 0, and x ∈ (0, d), and then create a situation where fully informed drivers route on edges of cost a, whereas a fraction x of uninformed drivers route on edges of cost a and another d − x route on edges of cost a(d).
Let b ∈ C with b(0) > 0 and define for all integer n large enough Consider the stochastic network N n of demand nd with n + q n branches in parallel. Each branch can be in state A or in state B (see Fig. 4). At all time, there are n branches in state A and q n branches in state B; informed drivers know which are which, and therefore, the network always appears to them as follows: (r n a + (n0) n ) n (r n 0 + (nb) n ) q n .
Since r n a(d) ≤ nb(0), they route uniformly over the branches in state A, incurring the social cost ndr n a(d).
Moreover, we let all branches be equivalent (they are each individually in state A with frequency n /n+q n ), so that uninformed drivers have a symmetric belief, route uniformly over all paths, and cause the social cost n 2 dr n n + q n a nd n + q n + nq n dn n + q n b nd n(n + q n ) .
The competitive ratio for this network N n is .
Hence, we conclude that for all a ∈ C, d > 0, and x ∈ [0, d], we have in particular ρ FI NI (C) ≥ α(C); therefore, with the help of Theorem 2, we have

D. Comparison of Public and Private Signaling
As we have already seen, private signaling is more efficient than public signaling. To complete our comparisons, we need to study the worst case of public over private signaling.  c(x). Let a, b ∈ C be homogeneous and inhomogeneous costs, respectively, and let c(d) − c(x) > > 0. We will create a sequence of networks N n with n always equivalent paths to emulate the cost 0 c(d), named P 1 , . . . , P n , and one path named P 0 to emulate c + .
For n large enough, the following integers are positive: Then, consider the binary network with demand d as We fix Ω = {0, 1} and letμ[n] = (1 − η n , η n ) with η n > 0 vanishing with n, although its exact expression will only be determined later.
Regarding public information, let μ = (1 − η, η) be the common belief that drivers share. We have Some drivers take path P 1 , and thus, we have Therefore, for all Bayesian-plausible distribution (τ, μ), noting that μ Thus, we have On the other hand, consider the following flow to be sent on P 0 conditional on ω (we let the messaging system reveal the information fully), f 0 = 0 and f 1 = x, the rest being distributed uniformly among P 1 , . . . , P n . They are incentive-compatible if and only if The first condition is ultimately true (that is for all n sufficiently large) as Likewise, the left-hand side of the second condition converges to as n → ∞; thus, the second condition is met for n large enough. The private signaling cost is, thus, ultimately r n b(x)). We can now let for n large enough so that the latter upper bound is equivalent to as n goes to infinity. As a result, we have This holds for all > 0 small enough .
In turn, ρ PUB PRIV (C) ≥ α(C) = ρ(C). The final claim is a direct consequence of Proposition 2 PRIV < C PUB ≤ C FI. Therefore, PRIV < C FI. We have already proven this fact with the more restrictive assumption that C is regular and contains 0.

V. CONCLUSION
We summarize all results in Table I, with the respective assumptions. The structural comparisons-those that do not need any assumption on C cost family-are concisely summarized by Proposition 4. Theorem 4 refines this hierarchy, stating that private signals are maximally more efficient than public ones for regular cost families. Under the same hypothesis, Proposition 5 implies the left-most column of Table I, namely that uninformed drivers may perform arbitrarily badly compared to any other routing. Furthermore, if the cost family contains the null cost, Theorem 3 completes the comparison between FI and NI routings. Overall, we obtain a total strict order for FI, public signals, private signals, and optimal routing for regular cost families containing the null cost.
Much like the original PoA, the competitive ratios have not been determined for all families of costs. In particular, the worst-case comparison between fully and uninformed drivers was only possible assuming that the null cost belongs to the family considered. Be that as it may, all results are upper bounds without any assumption. Apart from this exception, there seems to be little theoretical interest to refine these results.
We briefly note that even though our results assumed Ω finite for ease of construction, they actually pertain to any Ω, provided that information provisions and expected costs are appropriately defined. More details can be found in [25].
On top of these technical assumptions, we used multiple working hypotheses.
1) The network is in a permanent flow state: travel times depend only on flow. 2) Drivers choose their fastest route according to their belief.
3) The network state is independent of the flow. 4) Demand is constant and given. 5) A single benevolent controller witnesses the state of the roads. Within these simple hypotheses, we were able to characterize various inefficiencies between information policies, in the style of the PoA.
A fixed demand makes sense over the time scale it takes for drivers to find their shortest path. However, on a longer time scale, successfully mitigating congestion may increase demand by enticing public transportation users to drive themselves. This phenomenon is known as the Downs-Thomson paradox and can be summarized by the phrase "the travel time by car is determined by the travel time by public transports" [26], [27]. However, this phenomenon is not peculiar to the provision of information: if a network is modified with the intent of benefiting its users, it should take into account the shift of demand it will incur. This paradox adds to Braess's paradox [5], further pointing out that, over longer times scales, increasing the capacity of a network is not a guarantee of improving congestion.
In our study, the planner is a single smart city, whose information provision aims at benefiting drivers as a whole. This is quite the opposite of the present situation, where many different private path planners are in competition to gain popularity and where drivers, if considered rational, should, thus, be expected to listen to the planner providing the shortest paths in average. Against such planners, which are able to cater real-time personal information, our controller poses little challenge without any greater informational advantage; nonetheless, a smart city could possess more accurate and updated information than private parties. This competition is studied in [17] and [28].
Traffic information systems also typically collect feedback whose quality depends on the flow. For example, a small flow might not reflect the closing of a lane, while it could be a poor choice to route more drivers there, as above a threshold, a jam could form. In [29], we discuss the implication this would have on the signaling policy of a central controller. The setting is chosen so that, in terms of public policies, FI is the best choice (μ → V (N μ ) concave). The controller's best strategy to reduce congestion is to send some drivers to probe an uncertain road, even though it is more costly at first, to obtain more reliable information. Intuitively, one should expect a similar result with a planner trying to minimize the individual cost, thus sending few drivers to probe the network against their individual incentives.