# Value of Information Systems in Routing Games

We study a routing game in an environment with multiple heterogeneous information systems and an uncertain state that affects edge costs of a congested network. Each information system sends a noisy signal about the state to its subscribed traveler population. Travelers make route choices based on their private beliefs about the state and other populations' signals. The question then arises, "How does the presence of asymmetric and incomplete information affect the travelers' equilibrium route choices and costs?" We develop a systematic approach to characterize the equilibrium structure, and determine the effect of population sizes on the relative value of information (i.e. difference in expected traveler costs) between any two populations. This effect can be evaluated using a population-specific size threshold. One population enjoys a strictly positive value of information in comparison to the other if its size is below the corresponding threshold, but not otherwise. We also consider the situation when travelers may choose an information system based on its value, and characterize the set of equilibrium adoption rates delineating the sizes of subscribed traveler populations. The resulting routing strategies are such that all travelers face an identical expected cost, and no traveler has the incentive to change her subscription.

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## Authors

• 8 publications
• 16 publications
• 2 publications
• ### Learning an Unknown Network State in Routing Games

We study learning dynamics induced by myopic travelers who repeatedly pl...
05/11/2019 ∙ by Manxi Wu, et al. ∙ 0

• ### Bayesian Learning in Dynamic Non-atomic Routing Games

We consider a discrete-time nonatomic routing game with variable demand ...
09/24/2020 ∙ by Emilien Macault, et al. ∙ 0

• ### The route to chaos in routing games: Population increase drives period-doubling instability, chaos & inefficiency with Price of Anarchy equal to one

We study a learning dynamic model of routing (congestion) games to explo...
06/06/2019 ∙ by Thiparat Chotibut, et al. ∙ 0

• ### Access to Population-Level Signaling as a Source of Inequality

We identify and explore differential access to population-level signalin...
09/12/2018 ∙ by Nicole Immorlica, et al. ∙ 0

• ### Inferring the prior in routing games using public signalling

This paper considers Bayesian persuasion for routing games where informa...
09/13/2021 ∙ by Jasper Verbree, et al. ∙ 0

• ### A Semidefinite Approach to Information Design in Non-atomic Routing Games

We consider a routing game among non-atomic agents where link latency fu...
05/06/2020 ∙ by Yixian Zhu, et al. ∙ 0

• ### Equilibrium in Wright-Fisher models of population genetics

For multivariant Wright-Fisher models in population genetics, we introdu...
06/28/2020 ∙ by D. Koroliouk, et al. ∙ 0

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## 1 Introduction

Travelers are increasingly relying on traffic navigation services to make their route choice decisions. In the past decade, numerous services have come to the forefront, including Waze/Google maps, Apple maps, INRIX, etc. These Traffic Information Systems (TISs) provide their subscribers with costless information about the uncertain network condition (state), which is typically influenced by several exogenous factors such as weather, incidents, and road conditions. The information provided by TIS can be especially useful in making travel decisions when a change in state corresponds to changes in travel times of multiple edges of the network. Experiential evidence suggests that the accuracy levels of TISs are less than perfect, and exhibit heterogeneities due to the inherent technological differences in data collection and analysis approaches. Moreover, travelers may use different TISs or choose not to use them at all, depending on factors such as marketing, usability, and availability. Therefore, we can reasonably expect that travelers face an environment of asymmetric and incomplete information about the network state.

Importantly, information heterogeneity can directly influence the travelers’ route choice decisions, and the resulting congestion externalities. For example, travelers who are informed by their TIS that a certain route has an incident and take a detour may not only reduce their own travel time, but also benefit the uninformed travelers by shifting traffic away from the affected route. However, if too many travelers take the detour, then this alternate route will also start getting congested, limiting the benefits of information. Thus, the question arises as to how information heterogeneity impacts the travelers’ route choice and costs.

In this article, we develop a game-theoretic approach to study this question. We consider a routing game in which the travelers are privately informed about the network state by their respective TIS, and choose strategies based on their beliefs about the state and other travelers’ behavior. Our approach enables a systematic study of equilibrium structure and travel costs for this routing game. The game is played on a general network by multiple traveler populations that are heterogeneous in the access and accuracy of state information. Specifically, we provide a complete characterization of how the population size vector impacts the equilibrium structure and the relative value of information faced by travelers subscribed to one TIS in comparison to the ones subscribed to another TIS.

Furthermore, we study the situation in which travelers can choose their TIS subscription based on the relative values of different available TISs. In general, the adoption rate of one TIS (i.e. the fraction of travelers subscribing to it) depends on a variety of factors such as TIS accuracies, uncertainty in network states, and cost parameters. Thus, determining the equilibrium adoption rates is a non-trivial yet practically-relevant problem. We provide a rather straightforward characterization of the set of equilibrium adoption rates. Particularly, we show that this set is comprised of the population size vectors for which all travelers face an identical expected cost in equilibrium (i.e. social fairness is achieved), and no traveler has an incentive to change her TIS subscription (i.e. stability is achieved).

### 1.1 Our Model and Contributions

We model the traffic routing problem in an asymmetric and incomplete information environment as a Bayesian routing game. We consider a general traffic network with a single origin-destination pair, and an uncertain state that is realized from a finite set according to a prior probability distribution. The cost function (travel time) of each edge in the network is increasing in the aggregate traffic load on that edge. Moreover, the edge costs are

state-dependent

in that the state can affect them in various ways. All travelers have identical preferences, and the total demand is inelastic. There are multiple heterogeneous TISs, each sending a noisy signal of the state to its subscribed traveler population. The signal sent by each TIS is privately known to only its subscribers, but the joint distribution of the state and all signals is known by all travelers (common knowledge). This joint distribution is the common prior of the game, and each population’s private belief of the state and other populations’ private signals is derived from it. Our information environment is general in that we do not impose any structural assumption on the common prior. The signals of different TISs can be correlated or independent, conditional on the state. Also, one TIS can be more accurate than another in some states, but less in the other states. Therefore, we do not assume that the TISs are ordered according to the accuracy of their signals.

We use Bayesian Wardrop Equilibrium (BWE) as the solution concept of our game. In a BWE, all populations assign demand on routes with the smallest expected cost based on their private beliefs. In fact, our game is a weighted potential game, i.e. the set of BWE is the optimal solution set of a convex optimization problem that minimizes the weighted potential function over the set of feasible routing strategies of traveler populations (Theorem 1). This property establishes the essential uniqueness of BWE, i.e. the equilibrium edge load vector is unique. However, the strategy-based optimization problem is not directly useful for analyzing how the set of equilibrium strategies and population costs depend on the population size vector. To address this issue, we provide a characterization of the set of feasible route flows (i.e. route flows induced by feasible strategy profiles) as a convex polytope (Proposition 1), and show that this set is the optimal solution set of another convex optimization problem (Proposition 2). The optimal value of this flow-based formulation is the value of weighted potential function in equilibrium.

The flow-based formulation enables us to analyze the sensitivity of equilibrium structure with respect to perturbations in the size vector. The constraints in this formulation include: basic route feasibility constraints, a set of size-independent equality constraints, and a set of size-dependent inequality constraints. The equality constraints ensure that any shift in route flows resulting from a change of signal received by one population is independent of the signals received by other populations. Each inequality constraint corresponds to a single population, and ensures that the maximum extent to which the received signals impact its equilibrium routing behavior is limited by the population’s size. Hence, we refer to these inequality constraints as information impact constraints (IICs). Consequently, the effects of perturbation in the size vector on the equilibrium structure can be studied by evaluating the tightness of the IICs corresponding to the perturbed populations at the optimum of the flow-based formulation.

In particular, Theorem 2

describes how the qualitative properties of equilibrium route flows change under perturbations in the sizes of any two populations, with sizes of all other populations being fixed (i.e. directional perturbations of the size vector). Among the two perturbed populations, we say that a population is the “minor population” if its size is smaller than a certain (population-specific) threshold. The corresponding IIC is tight in equilibrium, i.e. the impact of information on the minor population is fully attained. These population-specific thresholds depend on the common prior distribution as well as sizes of all other populations, and each threshold can be computed by solving a linear program. Based on the two thresholds, we can distinguish three qualitatively distinct equilibrium regimes: In the two side regimes exactly one population assumes the minority role, on the other hand, in the middle regime neither population is minor.

We can apply Theorem 2 to analyze the sensitivity of the equilibrium value of the weighted potential function under directional perturbation of population sizes (Proposition 3). In the middle regime, perturbing the relative sizes does not change the equilibrium value of the potential function. On the other hand, in the two side regimes the value of the potential function monotonically decreases as the size of the minor population increases. Thanks to the essential uniqueness of BWE, the equilibrium edge load vector does not change with this directional perturbation if and only if the size vector falls in the middle regime.

These results allow us to compare the expected cost in equilibrium faced by travelers in any pair of populations. In particular, we can evaluate how this cost difference — which we call the relative value of information — changes with pairwise size perturbations. By using the results on sensitivity analysis of general convex optimization problems (Fiacco (1984), Rockafellar (1984)), we show that the relative value of information is proportional to the derivative of the equilibrium value of potential function in the direction of perturbation. Combining this observation with Proposition 3, we obtain that the minor population faces a lower cost relative to the other population in the two side regimes; whereas both populations face identical costs in the middle regime. Importantly, the relative value of information is non-increasing in the size of its subscribed population (Theorem 3). This result is based on the intuition that an individual traveler faces higher congestion externality when more travelers have access to the same information, and hence make their route choices according to the same strategy. Thus, an increase in the size of minor population decreases the relative imbalance in congestion externality, thereby reducing the advantage enjoyed by its travelers over the other population.

Our results can be easily specialized to a simpler information environment in which one of the populations does not have an access to TIS (uninformed population). In this case, we obtain that the equilibrium cost of the uninformed population is no less than that of any other population regardless of the size vector. That is, having access to a TIS always leads to a non-negative relative value of information in comparison to being uninformed (Proposition 4).

We also extend our approach of pairwise comparison of populations to study how the equilibrium outcome depends on population sizes in general. In particular, we characterize a non-empty set of size vectors, where all the IICs can be dropped from the flow-based optimization problem without changing its optimal value. The equilibrium edge load vector is size-independent in this set (Proposition 5). Furthermore, Theorem 4 shows that this set is comprised of all size vectors such that, in equilibrium, all travelers face identical costs (socially fair) and no travelers have incentive to change TIS subscription (stable). We say that such an outcome exhibits the stable social fairness property.

Theorem 4 is useful for evaluating the equilibrium adoption rate of each TIS when travelers have the flexibility to choose their TIS subscription. Particularly, we consider a two-stage game, in which travelers first choose their TIS, and then play the Bayesian routing game for the size vector induced from the first stage. A subgame perfect equilibrium (SPE) of this game requires that travelers have no incentive to change their respective TIS subscriptions, and that the routing strategies are chosen according to BWE of the subgame. Hence, any TIS with a positive adoption rate must incur the lowest expected equilibrium cost among all TIS. We conclude that the set of equilibrium adoption rates in SPE is indeed the set of size vectors for which the induced equilibrium outcome satisfies the stable social fairness property.

Finally, we also identify a sufficient condition on the edge cost functions, under which there is no inefficiency caused by selfish routing. When this condition is satisfied, we obtain that equilibrium efficiency can be achieved in the stable socially fair set of size vectors (Proposition 6).

### 1.2 Related Work

Congestion games. Well-known results in classical congestion games include their equivalence with potential games (Rosenthal (1973), Monderer and Shapley (1996), Sandholm (2001), and Sorin and Wan (2016)), analysis of network formation games as congestion games (Gopalakrishnan et al. (2014), and Tardos and Wexler (2007)), and equilibrium inefficiency (Roughgarden and Tardos (2004), Koutsoupias and Papadimitriou (1999), Correa et al. (2007), Acemoglu and Ozdaglar (2007), and Nikolova and Stier-Moses (2014)). Some models of congestion games in asymmetric information environments have been also reported. For example, Heumen et al. (1996) and Facchini et al. (1997) showed that for Bayesian congestion games with atomic players, a pure Nash equilibrium exists when the game has a common prior, but may not exist otherwise. Milchtaich (1996), and Mavronicolas et al. (2007) have studied congestion games with player-specific cost functions. These games can model both heterogeneous private information and heterogeneous preferences. However, the existence of a potential function or even a pure Nash equilibrium is not guaranteed. Since our game has non-atomic traveler populations, the existence of a pure equilibrium is guaranteed. Furthermore, in our game, the heterogeneity in the expected costs among populations arises only due to heterogeneous private beliefs, which are derived from a common prior. This feature makes our game a weighted potential game Sandholm (2001).

Effect of Traffic Information Systems. Prior work has studied the effects of TIS on travelers’ departure time choices (Arnott et al. (1991), and Khan and Amin (2018)), and on their route choices (Ben-Akiva et al. (1991), and Ben-Akiva et al. (1996)). In particular, Mahmassani and Jayakrishnan (1991) conducted a simulation to study the effect of real-time information on the performance of a congested traffic corridor, and concluded that as more travelers receive information, the informed travelers gradually start facing higher costs and their relative value of being informed diminishes. Our analysis, when applied to the game with one uninformed population and other more informed ones, also leads to similar conclusions. Our results are more general because they are applicable to routing games with multiple heterogeneously informed populations with arbitrary TIS accuracies.

Another related work is by Acemoglu et al. (2018), which studies a congestion game where travelers have different information sets about the available edges (routes). The authors identify a sufficient and necessary condition on the network topology under which receiving additional information does not increase the traveler costs. While their work focuses on heterogeneous information about the network structure, we use a Bayesian approach to model the information heterogeneity resulting from the differences in TIS access and accuracy.

Value of Information. In a classical paper, Blackwell et al. (1951) showed that for a single decision maker, more informative signal always results in higher expected utility. The Blackwell’s criterion for the comparison of information has been refined by Athey and Levin (2017) and Persico (2000) for a class of decision making problems. In game-theoretic settings, it is generally difficult to determine whether the value of information in equilibrium for individual players and/or society is positive, zero or negative (see Hirshleifer (1971), and Haenfler (2002)). However, the value of information is guaranteed to be positive when certain conditions are satisfied; see for example Neyman (1991), Bassan et al. (2003), Gossner and Mertens (2001), and Lehrer and Rosenberg (2006). Since travelers are non-atomic players in our game, the relative value of information between any two TISs is equivalent to the value of information for an individual traveler when her subscription changes unilaterally. We give precise conditions on the population sizes under which the value of information in our Bayesian routing game is positive, zero, or negative.

The paper is organized as follows: In Section 2 we motivate our analysis using a simple routing game. Section 3 introduces our Bayesian routing game model; and in Section 4 we show that this game is a weighted potential game; this property leads to characterizations of the set of equilibrium strategy profiles and the set of route flows. In Section 5, we analyze how the equilibrium structure and the relative value of information between any two populations change with their sizes. In Section 6, we characterize equilibrium TIS adoption rates for the case when travelers have the flexibility to choose or switch their subscription. Concluding remarks are drawn in Section 7.

The complete proofs, along with supplementary results, are provided in the appendix.

## 2 Motivating Example

In this section, we motivate our analysis using a simple game of two asymmetrically informed traveler populations routing over a network of two parallel routes, denoted and . The network state belongs to the set , where the state represents an incident condition on , and the state represents the nominal condition. The state occurs with probability . The network faces a unit size of demand (), which is comprised of two traveler populations: population 1 with size and population 2 with size . Each population receives a signal of the state from its TIS. Thus, the signal space of population is . Assume for simplicity that population 1 receives the correct state with probability 1 (i.e. complete information), and population 2 receives signal or with probability , independent of the state (i.e. no information). This information structure is common knowledge.

Let denote the traffic demand assigned to route by population when receiving signal ; the remaining demand is assigned to route . Since the signal is independent of the state, we have . A feasible demand assignment must satisfy the constraints: and . For this example, we can represent a routing strategy profile as . We denote the aggregate route flow on as . Again, for simplicity, consider that the cost of each route is an affine function of the route flow, and both routes have identical free-flow travel time. That is, the cost function of is in state , and in state ; the cost function of is . Furthermore, we assume that . Since population 1 has complete information, its travelers know the exact cost function in both states. However, since population 2 travelers are uninformed, they make their route choices based on the expected cost of each route, evaluated according to the prior distribution of states.

This routing game with heterogeneously informed traveler populations admits a Bayesian Wardrop equilibrium, as discussed in Section 3. Let denote an equilibrium strategy profile. Each population, given the signal it receives, can either assign all its demand on one of the two routes, or splits on both routes. Thus, there are possible cases. Our results can be used to study how the equilibrium strategies and route flows change as the size of a population varies from 0 to 1. Detailed analysis for this simple routing game and some interesting variants are available in Wu et al. (2017). Specifically, we find that there exists a threshold size of population 1, , such that the qualitative structure of equilibrium routing strategies is different based on whether or .

In the first regime, i.e. when , the game admits a unique equilibrium: , , where . This equilibrium regime corresponds to the following outcome: in state (resp. state ), population 1 assigns all its demand on route (resp. route ), and population 2 splits its demand on both routes. The induced equilibrium flow on is given by if , and if .

On the other hand, in the second regime, i.e. when , the equilibrium set may not be singleton, and can be represented as follows: , , where . Thus, the equilibrium set is a one-dimensional interval for , and a singleton set for or . In this regime, both populations face identical expected route costs in equilibrium. Consequently, each population splits its demand on both routes. Moreover, the equilibrium route flow on each route is unique and independent of : if , and if .

Notice that when , we have , i.e. population 1 shifts all its demand to when receiving the signal about the incident on . However, if , we have , i.e. the change in the received signal only influences a part of travelers in population 1. One can say that the information impacts the entire demand of population 1 in the first regime, but not in the second regime.

For any feasible , we can calculate the equilibrium population costs, denoted . If , since population 1 takes in state and in state , we can write . Population 2 uses both routes, and thus . It is easy to check that , i.e. when the state information is only available to a small fraction of travelers, the informed travelers have an advantage over the uninformed ones. On the other hand, if the size of informed population exceeds the threshold , then . In this case, all travelers face identical cost in equilibrium, no uninformed traveler has the incentive to become informed (and vice versa).

Finally, the equilibrium average cost is simply the average expected cost in equilibrium, i.e. . We can check that monotonically decreases with in the first regime, and attains a constant (minimum) value in the second regime. Thus, increasing the size of informed population decreases the equilibrium average cost but only when it is below .

We illustrate the aforementioned results in Fig. 1 using the following parameters: , , , , and . The costs are normalized by the socially optimal cost, denoted , which is the minimum cost achievable by a social planner with complete information of the state.

## 3 Model

### 3.1 Environment

To generalize the simple routing game in Section 2, we consider a transportation network modeled as a directed graph with a single origin-destination pair. Let denote the set of edges and denote the set of routes. The finite set of network states, denoted , represents the set of possible network conditions, such as incidents, weather, etc. The network state, denoted , is randomly drawn by a fictitious player “Nature” from according to a distribution , which determines the prior probability of each state. For any edge and state , the state-dependent edge cost function is a positive, increasing, and differentiable function of the load through the edge . Note that the state can impact the edge costs in various ways.

The network serves a set of non-atomic travelers with a fixed total demand . We assume that each traveler is subscribed exclusively to one of the TIS in the set . We refer to the set of travelers subscribed to the TIS as population . All travelers within a population receive an identical signal from their TIS. Let denote the ratio of population ’s size and the total demand . We also consider degenerate situations when the sizes of one or more populations approach 0. Thus, a vector of population sizes is feasible if it satisfies the following constraints:

 ∑i∈Iλi =1, (1a) λi ≥0,∀i∈I. (1b)

The size vector is considered as given in our analysis of equilibrium structure and costs (Sections 4 and 5). In Section 6 we consider a more general situation where results from the travelers’ TIS subscription choices.

Each TIS sends a noisy signal of the state to population . The signal received by each population determines its type (private information). We assume that the type space of population is a finite set, denoted as . Note that the type spaces and the state space need not be of the same size. Let denote a type profile, i.e. vector of signals received by the traveler populations; thus, . The joint probability distribution of the state and the vector of signals is denoted , and it is the common prior of the game. The marginal distribution of on states is consistent with the common prior, i.e. for all . The conditional probability of type profiles on the state is given by , i.e. the joint distribution of signals received by the populations when the network state is . In our modeling environment, the signals of different TIS can be correlated, conditional on the state. Each population generates a belief about the state and the other populations’ types based on the signal received from the information system . We denote the population ’s belief as .

The routing strategy of each population is a function of its type, denoted as . One way to describe the generation of routing strategies is that each TIS sends a noisy signal of the state to its subscribed population, and the individual route choices of non-atomic travelers results in an aggregate routing strategy . An alternative viewpoint is that is a direct result of strategy route recommendations sent by each TIS to its subscribed population. That is, each TIS routes travelers in population according to the function . For our purpose, these two viewpoints are equivalent in that given any population , and any type , the demand of travelers on route is .

We say that a routing strategy profile is feasible if it satisfies the following constraints:

 ∑r∈Rqir(ti) =λiD,∀ti∈Ti,∀i∈I, (2a) qir(ti) (2b)

For a given size vector , let denote the set of all feasible strategies of population . From (2a)-(2b), we know that the set of feasible strategy profiles is a convex polytope.

### 3.2 Bayesian Routing Game

The Bayesian routing game for a fixed size vector can be defined as , where

• : Set of populations,

• : Set of states with prior distribution

• : Set of population type profiles with element

• : Set of feasible strategy profiles for a given size vector , with element

• : Set of state-dependent edge cost functions

• : is the population ’s belief on state and other populations’ types

All parameters including the common prior are common knowledge, except that populations privately receive signals about the network state from their respective TIS. The game is played as shown in Fig. 2.

For any and , the interim belief of population is derived from the common prior:

 βi(s,t−i|ti)=π(s,ti,t−i)Pr(ti),∀s∈S,∀t−i∈T−i, (3)

where . For a strategy profile , the induced route flow is denoted , where is the aggregate flow assigned to the route by populations with type profile , i.e.

 fr(t)=∑i∈Iqir(ti),∀r∈R,∀t∈T. (4)

Note that the dependence of on is implicit and is dropped for notational convenience.

Again, for the strategy profile , we denote the induced edge load as , where is the aggregate load on the edge assigned by populations with type profile :

 we(t)=∑r∋e∑i∈Iqir(ti)∑r∋efr(t),∀e∈E,∀t∈T. (5)

The corresponding cost of edge in state is . Then, the cost of route in state can be obtained as: . Finally, the expected cost of route for population can be expressed as follows:

 E[cr(q)|ti]∑s∈S∑t−i∈T−i∑e∈rβi(s,t−i|ti)cse(we(ti,t−i)) ∑s∈S∑t−i∈T−i∑e∈rπ(s,ti,t−i)Pr(ti)cse(we(ti,t−i)),∀r∈R,∀ti∈Ti,∀i∈I, (6)

where is given by (5).

The equilibrium concept for our game is Bayesian Wardrop equilibrium (BWE). A strategy profile is a BWE if for any and any :

 ∀r∈R,qi∗r(ti)>0⇒E[cr(q∗)|ti]≤E[cr′(q∗)|ti],∀r′∈R. (7)

That is, in a BWE, each population with type assigns its demand only on routes that have the smallest expected cost based on its interim belief .

We define the equilibrium population cost, denoted , as the expected cost incurred by a traveler of a given population across all types and network states in equilibrium: . In fact, from (7), we can write:

 Ci∗(λ)1λiD∑ti∈TiPr(ti)(∑r∈Rqi∗r(ti))minr∈RE[cr(q∗)|ti]∑ti∈TiPr(ti)minr∈RE[cr(q∗)|ti]. (8)

Note that is a degenerate case for population as its size approaches 0. In this case, the cost can be viewed as the expected cost faced by an individual (non-atomic) traveler who subscribes to the TIS .

Finally, the equilibrium average cost, denoted , is the average cost incurred by a traveler of any population across all network states in equilibrium:

 C∗(λ)Δ=∑i∈IλiCi∗(λ)=1D∑i∈I∑ti∈TiPr(ti)∑r∈RE[cr(q∗)|ti]qi∗r(ti). (9)

## 4 Equilibrium Characterization

In this section, we show that the game is a weighted potential game. This property enables us to express the sets of equilibrium strategy profiles and route flows as optimal solution sets of certain convex optimization problems.

### 4.1 Equilibrium Strategy Profiles

Following Sandholm (2001), the game is a weighted potential game if there exists a continuously differentiable function and a set of positive, type-specific weights such that:

 ∂Φ(q(t))∂qir(ti)=γ(ti)E[cr(q)|ti],∀r∈R,∀ti∈Ti,∀i∈I. (10)

Indeed, the game is a weighted potential game with the function defined as follows:

 Φ(q)Δ=∑s∈S∑e∈E∑t∈Tπ(s,t)∫∑r∋e∑i∈Iqir(ti)0cse(z)dz, (11)

and the positive type-specific weights are for any and , see Lemma A.1.

Using (4) and (5), can be equivalently expressed as a function of the route flow or the edge load induced by a strategy profile :

 ˆΦ(f) Δ=∑s∈S∑e∈E∑t∈Tπ(s,t)∫∑r∋efr(t)0cse(z)dz (12) Δ=∑s∈S∑e∈E∑t∈Tπ(s,t)∫we(t)0cse(z)dz. (13)

Thus, for any feasible strategy profile , we can write , where and are the route flow and edge loads induced by the strategy profile . Moreover, is twice continuously differentiable and strictly convex in , see Lemma A.2.

Our first result provides a characterization of the set of equilibrium strategy profiles:

###### Theorem 1.

A strategy profile is a BWE if and only if it is an optimal solution of the following convex optimization problem:

 minΦ(q)s.t.q∈Q(λ), (OPT-Q)

where is the set of feasible strategy profiles. The equilibrium edge load vector is unique.

The existence of BWE follows directly from Theorem 1. For any size vector , we denote the set of BWE for the game as . Importantly, since the equilibrium edge load is unique, the equilibrium population cost for each population in (8) and the equilibrium average cost in (9) must also be unique for any . Thus, the equilibria of can be viewed as essentially unique. We denote the optimal value of (OPT-), i.e. the value of the weighted potential function in equilibrium as .

The Lagrangian of (OPT-) that we use in proving Theorem 1 is given as follows:

 L(q,μ,ν,λ)=Φ(q)+∑i∈I∑ti∈Tiμti(λiD−∑r∈Rqir(ti))−∑r∈R∑i∈I∑ti∈Tiνtirqir(ti), (14)

where and are Lagrange multipliers associated with the constraints (2a) and (2b), respectively. In fact, we show in Lemma A.4 that for any BWE , the optimal Lagrange multipliers and in (14) associated with are unique, and can be written as follows:

 μti∗ =minr∈RPr(ti)E[cr(q∗)|ti],∀ti∈Ti,∀i∈I (15a) νti∗r =Pr(ti)E[cr(q∗)|ti]−μti∗,∀r∈R,∀ti∈Ti,∀i∈I. (15b)

This result follows from the fact that (OPT-) satisfies the Linear Independence Constraint Qualification (LICQ) condition (Wachsmuth (2013)), which ensures the uniqueness of Lagrange multipliers at the optimum of (OPT-); see Lemma A.3.

Equations (15a) - (15b) connect the expected route costs for each type in equilibrium with the Lagrange multipliers and at the optimum of (OPT-), and will be used in Section 5 for studying the relative ordering of equilibrium population costs.

### 4.2 Equilibrium Route Flows

Our main question of interest is how the set of BWE , i.e. optimal solution set of (OPT-), and more importantly, the equilibrium edge load , change with the perturbations in the size vector . However, characterizing the effect of directly from (OPT-) is not so straightforward. Recall that in the simple routing game in Section 2, the effects of perturbations in on the equilibrium route flow are relatively easier to describe in comparison to the effects on the set of equilibrium strategy profiles, because the equilibrium route flow remains fixed in a certain range of , whereas the set of equilibrium strategy profiles do not. Thus, our approach involves first studying how effects the set of equilibrium route flows. We show two results in this regard: (i) The set of feasible route flows and the set of feasible strategy profiles that induces a particular route flow can be both expressed as polytopes (Proposition 1); (ii) The set of equilibrium route flows is the optimal solution set of a convex optimization problem (Proposition 2). These results enable us to evaluate how the equilibrium edge load and population costs change with perturbations in .

Let us start by introducing the following set of route flows:

 F(λ)Δ={f∈R|R|×|T|∣∣f satisfies (???)-(???)}, (16)

where the constraints are given by:

 fr(ti,t−i)−fr(~ti,t−i) =fr(ti,~t−i)−fr(~ti,~t−i),∀r∈R, ∀ti,~ti∈Ti,and ∀t−i,~t−i∈T−i,∀i∈I, (17a) ∑r∈Rfr(t) =D,∀t∈T, (17b) fr(t) ≥0,∀r∈R,∀t∈T, (17c) D−∑r∈Rminti∈Tifr(ti,t−i) ≤λiD,∀t−i∈T−i,∀i∈I. (17d)

The constraints (17a)-(17c) do not depend on the size vector and can be understood as follows: (17a) captures the fact that the change in the flow through any route resulting from change in the type of population does not depend on the particular types of the remaining populations; (17b) ensures that all the demand is routed through the network; and (17c) guarantees that the demand assigned to any route is nonnegative.

On the other hand, the constraints in (17d) depend on the size vector , wherein the size of each population , , appears linearly in the constraint corresponding to that population. To further interpret (17d), we define an “impact of information” metric for any given population as the maximum extent to which the signal received from its TIS can influence the routing behavior of travelers within the population. Specifically, for any strategy profile and population , we define the impact of information on population as follows:

 Ji(q)Δ=λiD−∑r∈Rminti∈Tiqir(ti). (18)

Using (2a), we can re-write (18) as: , where is an arbitrary type in . That is, for each population , is the summation over all routes of the maximum difference between the demands assigned to each route by the type and any other type . Thus, we can say that the metric evaluates the impact of information on the population ’s strategy. We can alternatively express this metric in terms of the flow induced by :

 ˆJi(f)Δ=Ji(q)∑r∈Rmaxti∈Ti(fr(ˆti,ˆt−i)−fr(ti,ˆt−i))D−∑r∈Rminti∈Tifr(ti,ˆt−i), (19)

where is any type profile in .

Now the constraints (17d) can be equivalently stated as follows:

 ˆJi(f) ≤λiD,∀i∈I. (IIC)

These constraints ensure that the impact of signals on any population’s strategy is bounded by its size. We will refer to them as information impact constraints (IIC). We use (IIC) and (17d) interchangeably, and refer the constraint in (IIC) corresponding to population as (IIC). Also, it is easy to see that for each , (IIC) can be written as a set of affine inequalities:

 D−∑r∈Rfr(tir,ˆt−i) ≤λiD,∀ti1∈Ti,…,∀ti|R|∈Ti. (21)

Thus, , as defined in (16), is a convex polytope. The following proposition relates the set of feasible strategy profiles and the induced route flows; see Fig. 3.

###### Proposition 1.

The set of feasible route flows is the convex polytope . Furthermore, for a given route flow , any feasible strategy profile that induces can be expressed as:

 qir(ti) =fr(ti,ˆt−i)−fr(ˆti,ˆt−i)+χir,∀r∈R,∀ti∈Ti,∀i∈I, (22)

where is any type profile in , and is an -dimensional vector satisfying the following constraints:

 ∑r∈Rχir =λiD,∀i∈I, (23a) ∑i∈Iχir =fr(ˆt),∀r∈R, (23b) χir ≥maxti∈Ti(fr(ˆti,ˆt−i)−fr(ti,ˆt−i)),∀r∈R,∀i∈I. (23c)

The next proposition provides a characterization of the set of equilibrium route flows, and is analogous to Theorem 1 which characterizes the set of equilibrium strategy profiles.

###### Proposition 2.

A feasible route flow is an equilibrium route flow if and only if is an optimal solution of the following convex optimization problem:

 minˆΦ(f)s.t.f∈F(λ), (OPT-F)

where is given by (12), and is the set of feasible route flow vectors, as defined by (16).

We denote the set of equilibrium route flows in the game as . From Theorem 1 and (11)-(13), we know that for any size vector , and any , , we have

 (24)

The Propositions 1 and 2 form the basis of our analysis of how the perturbations of size vector effects the equilibrium structure and population costs.

## 5 Pairwise Comparison of Populations

In this section, we first analyze the effects of perturbations in the relative sizes of any two populations on the equilibrium structure. Next, we study how the cost difference between any two populations depends on the population sizes.

### 5.1 Equilibrium Regimes

To study the effects of perturbations in the relative sizes of any two populations, we employ the notion of directional perturbation of size vector . In particular, for any two populations and , we consider the -dimensional direction vector:

 zijΔ=(…0…,1|i−th,…0…,−1|j−th,…0…).

When is perturbed in the direction of , the size of population (resp. population ) increases (resp. decreases), and the sizes of the remaining populations do not change.

For any size vector and any two populations and , let the vector of the remaining populations’ sizes be denoted . The total size of the remaining populations is . To avoid triviality in pairwise comparison, we only consider the case when the sizes of both populations are strictly positive so that satisfies the constraint , and the range of the perturbations in the population ’s size is . We denote the set of admissible as .

Now consider an optimization problem that is similar to (OPT-), except that the two constraints in the (IIC) set corresponding to the populations and are replaced by a single constraint:

 minˆΦ(f)s.t.(???), (???)% , (???), (???)∖{i,j}, (???),

where the constraints (IIC) indicate that all but (IIC) and (IIC) from the original set (IIC) are included, and the constraint (IIC) is defined as follows:

 ˆJi(f)+ˆJj(f)≤(1−|λ−ij|)D. (IICij)

The constraint (IIC) ensures that the total impact of information on population and does not exceed their total demand. We denote the set of optimal solutions for (5.1) as . Analogously to Theorem 1, we can show that any induces a unique edge load , which can be obtained by (5); see Lemma B.1. Then, the optimal solution set of (5.1) can be written as the following polytope:

 Fij,†={f∣∣∣[]lf satisfies (???), (???), (???), (???)∖{i,j}, and (???),∑r∋efr(t)=wij,†e(t),∀e∈E,∀t∈T}. (25)

Note that both and depend on