1 Introduction
In multiagent systems, there are generally two paradigms of interaction. Centralized control paradigms assume that a single decision making entity is able to dictate the actions of all the agents, thus leading them to a coordinated social optimum. Decentralized control paradigms, on the other hand, assume that each agent selects its own actions, and while it is in principle possible for them to act altruistically, they are generally assumed to be selfinterested.
In this paper, we consider a routing scenario in which a subset of agents are controlled centrally (compliant agents), while the remaining are selfinterested agents. We model the system as a Stackelberg routing game [Yang, Zhang, and Meng2007] in which the decision maker for the centrally controlled agents is the leader, and the selfinterested agents are the followers. In this paper, we provide a computationally tractable methodology for 1) determining whether a given subset of centrally controlled agents are sufficient to achieve system optimum (), 2) determining the maximum number of agent that may be selfinterested such that the centrally controlled agents can be deployed in order to induce , and 3) computing the actions the leader should prescribe to a sufficient set of compliant agents in order to achieve .
It has been known for nearly a century in routing games that agents seeking to minimize their private latency need not minimize the total system’s latency [Pigou1920, Roughgarden and Tardos2002]. That is, selfinterested agents may reach a user equilibrium () that is not optimal from a system perspective. However, if all agents are assigned paths with minimum system marginal cost then the system will achieve optimal performance [Pigou1920, Beckmann, McGuire, and Winsten1956, Dietrich1969].
Therefore, from a system manager perspective, it is desirable that all agents traversing a network would strictly utilize minimal marginal cost paths, even if the path is not a minimum latency path for an individual agent. However, in many important scenarios, it will not be possible to enforce path assignment on all agents, but it may be possible to affect the behavior of a subset (the compliant agents). As a motivating example, consider an optin tolling system where drivers are given positive incentives to enroll but, in exchange, they will be subject to tolls that affect their route choice [Sharon et al.2017]. Another relevant example is virtual private network (VPN) path allocation. While each packet within the VPN might be selfinterested, a prosocial network manager might allocate virtual paths that are different from those preferred by the selfinterested packets [Fingerhut, Suri, and Turner1997, Duffield et al.1999].
However, we show that, in the general case, computing the optimal assignment of compliant agents is NPhard. Therefore, we focus on the specific scenario where the portion of compliant agents is sufficiently large to achieve . We present a novel linear program () representation for computing the maximal portion of selfinterested agents that allow the system to achieve and to determine whether a given set of compliant agents is sufficient to achieve . Furthermore, we provide a method to tractably compute the flow assignment for the compliant agents such that performance is guaranteed.
We demonstrate, using a standard traffic simulator over a wide range of road networks, that the number of compliant agents necessary to achieve system optimum is a relatively small percentage of total flow (between 13% and 53%).
2 Motivation
Recent advances in GPS based tolling technology [Numrich, Ruja, and Voß2012] open the possibility of implementing microtolling systems in which specific tolls are charged for the use of every link within a road network. Setting tolls appropriately can influence selfinterested drivers to prefer paths with minimum system marginal cost and thus, lead to improved system performance [Sharon et al.2017].
Unfortunately, political factors deter public officials from allowing such a microtolling scheme to be realized. Road pricing is known to cause a great deal of public unrest and is thus opposed by governmental institutions [Schaller2010]. To tackle this issue and avoid public unrest, it would be beneficial to have an optin microtolling system where, given some initial monetary signup incentive, drivers choose to optin to the system and be charged for each journey they take based on their chosen route. The vehicles belonging to such drivers would need to be equipped with a GPS device as well as a computerized navigation system. Given the toll values and driver’s value of time, the navigation system would suggest a minimal cost route where the cost is a function of the travel time and tolls.
While addressing the issue of political acceptance, an optin system would result in traffic that is composed of a mixture of selfinterested and compliant agents (compliant in the sense that the system manager can influence their route choice). Such a scenario raises some practical questions which are the focus of this paper, namely, what portion of selfinterested agents can the system tolerate while still reaching optimum performance? The answer to this question can help practitioners to determine both the level and the targeting of incentives in an optin system.
3 Problem definition and terminology
The terminology in this paper follows that of Roughgarden and Tardos (roughgarden2002bad). We review the relevant concepts and notation in this section.
3.1 The flow model
The flow model in this work is composed of a directed graph , and a demand function mapping a pair of vertices to a nonnegative real number representing the required amount of flow between source, , and target, .^{1}^{1}1The demand between any source and target, , can be viewed as an infinitely divisible set of agents (also known as a nonatomic flow [Rosenthal1973]). An instance of the flow model is a pair.
denotes the set of acyclic paths from to . Define as the collection of all (i.e., ). The variable represents the flow volume assigned to path . Similarly, is the flow volume assigned to link . By definition, the flow on each link () equals the summation of flows on all paths of which
is a part. Define the system flow vector as
. is said to be feasible if for all , .Each link has a latency function which, given a flow volume (), returns the latency (travel time) on . Following Roughgarden and Tardos (roughgarden2002bad) we make the following assumption:
Assumption 1.
The latency function is nonnegative, differentiable, and nondecreasing for each link .
The latency of a simple path for a given flow , is defined as . A feasible flow is defined as a user equilibrium () if for every and with it holds that (see Lemma 2.2 in [Roughgarden and Tardos2002]). In other words, at , no amount of flow can be rerouted to a path with lower latency when the rest of the flow is fixed.
Define the system cost associated with link as , the cost of a path as and the cost of a flow as . Define and . A feasible flow is defined as a system optimum () flow if for every and with it holds that (see Lemma 2.5 in [Roughgarden and Tardos2002]). In other words, at , the benefit from reducing the flow along any path is always less than or equal to the cost of by adding the same amount of flow to a parallel, alternative path. We follow Roughgarden and Tardos (roughgarden2002bad), and make the following assumption:
Assumption 2.
The cost function is convex for each link .
3.2 Problem Definition
The focus of this paper is a scenario where the demand is partitioned into selfinterested and compliant agents. We define two types of controllers that assign paths to all of the agents. These controllers are viewed as players in a Stackelberg game [Yang, Zhang, and Meng2007].

controller  Stackelberg leader, the controller aspires to assign paths to the compliant subset of agents that, taking into account the selfinterested agents’ reaction, optimizes the systems performance (i.e. minimizes total latency). controller assigned flow will be referred to as compliant flow.

controller  Stackelberg follower, considering the compliant agents’ path assignment as fixed, the controller assigns paths to the selfinterested agents, the flow, such that a state of user equilibrium (as defined above) is achieved.^{2}^{2}2The enforced by the controller applies only for the selfinterested subset of agents. That is, no selfinterested agent can benefit from unilaterally deviating from its assigned path.
The problems addressed in this paper are:

Given an instance of the flow model , what is the maximum amount of selfinterested agents that can be assigned to the controller and still permit the controller to achieve system optimum?

Given a set of compliant agents and an instance of the flow model , can the controller assign paths to them in such a way that the system achieves ?

If is achievable, how should the controller assign the compliant flow? Equivalently, what is the optimal Stackelberg equilibrium?
To the best of our knowledge, this work is the first to answer these questions in a general setting.
4 Related Work
Previous work examined mixed equilibrium scenarios where traffic is composed of: and CournotNash () controllers. A controller assigns flows to a given subset of the demand with the aim of minimizing the total travel time only for that subset. For instance, a logistic company with many trucks can be viewed as a controller.
It was shown that the equilibrium for a mixed , scenario is unique and can be computed using a convex program [Haurie and Marcotte1985, Yang and Zhang2008]. On the other hand, no tractable algorithm is known for computing the optimal Stackelberg equilibrium for scenarios that also include a controller.
Korilis et. al. (korilis1997achieving) examined mixed equilibrium scenarios that do include a controller. In their work, a technique for computing the a solution for the above questions #1 and #3 (see problem definition) was suggested for specific types of flow models. Their technique was proven to work for networks with a common source and a common target with any number of parallel links. Moreover, the latency functions were assumed to be of a very specific form (linear function with a capacity bound). As a result, their solution is not applicable when general networks with arbitrary latency functions are considered.
Other work [Roughgarden2004, Immorlica et al.2009] studied a variant of the scheduling problem where infinitesimal jobs must be assigned to a set of shared machines each of which is affiliated with a nonnegative, differentiable, and nondecreasing latency function that, given the machine load, specify the amount of time needed to complete a job. When considering a scenario where part of the jobs are assigned to machines by a controller while the rest are assigned by a controller, they show it is NPhard to compute the optimal Stackelberg equilibrium [Roughgarden2004]. Their problem can be viewed as a special case of our problem, specifically a network with a single source and target with multiple parallel links between them. Given that in this more restrictive setting computing the optimal Stackelberg equilibrium is intractable, the general question in our setting will also be computationally intractable.
5 Computing the Maximal Flow
Given that finding the optimal Stackelberg equilibrium is NPhard for an arbitrary number of compliant agents, this work focuses on scenarios where the number of compliant agents is sufficient to achieve . As we will show, finding the optimal Stackelberg equilibrium can be done in polynomial time for such cases. In this section, we will present a computationally tractable method to compute the maximal flow given an instance of a flow model , and we will provide a method to check, for a given level of compliant flow, whether is achievable.
We define as the maximal amount of demand comprised of selfinterested agents that the system can tolerate and still achieve . Additionally, we define as the amount of demand from source to target that is assigned to the controller. That is, computing is equivalent to maximizing .
We can cast the problem of maximizing as an optimization problem, specifically a linear program (). Assigning values to all variables of type must follow some constraints. Specifically, the flow from each origin to each destination must be both a subflow of some flow, and it must be an acceptable path for all selfinterested agents in the flow, given the compliant flow.
Definition 1 (Subflow of flow ).
For a directed graph and demand function , a flow is a subflow of flow if for all links , and for each pair of nodes , there exists such that
and
A path will only be acceptable for a selfinterested agent if it is the lowest latency path from the origin to the destination given the compliant flow, and in order for the path to be part of a valid flow, it must be a minimum marginal cost path. Therefore, a path , leading from vertex to vertex , will be said to be zero reduced cost if there is no other path, , leading from to with lower latency or lower marginal cost.
Definition 2 (Zero reduced cost path).
For a flow model , a zero reduced cost path with regard to flow assignment is a path such that . A link, , is defined as a zero reduced cost link with respect to source if it is part of any zero reduced cost path originating from and terminating at for some origindestination pair . We denote the set of zero reduced cost links with respect to source as
We require that the flow (flow routed by the controller) is routed solely via zero reduced cost links/paths. This is because the controller can only assign flow to minimal latency paths (otherwise selfinterested agents would deviate). However, the need to constrain flow to links/paths with minimal marginal cost () is less intuitive; this constraint will be justified later on. Note that it is sufficient to only consider whether or not a link is part of a reduced cost path from the origin to some destination (not a specific ) because either link is along a reduced cost path from , or there is no path only along links in that includes . Moreover, we can efficiently compute the set of zero reduced cost links for any origin destination pair by applying uniform cost search from to and marking all links that are part of optimal paths, once with regard to minimal total latency (, and second with regard to minimal marginal cost (.
Let the constant denote the flow vector at a solution.^{3}^{3}3A flow can be efficiently computed as a solution to a convex program [Roughgarden and Tardos2002, Dial2006]. The flow is not unique when latency functions are nondecreasing, and the maximal amount of flow permitted may, in general, depend on the specific flow. Therefore, we must efficiently search over the space of flows. This is possible due to the following lemmas.
Lemma 1.
For any two flows that achieve SO, and , .
Proof.
Given Assumption 2, a flow is the solution to a convex program [Roughgarden and Tardos2002]. The solutions to a convex program form a convex set. Suppose that there are two flows that both achieve SO, but for which . Then must be a linear function between and (to see this, note that any convex combination of and is also an solution, but if is not linear, then the total system travel time would be strictly less, a contradiction). Since is a nondecreasing function, the only way for to be linear is for to be constant between and . ∎
Lemma 2.
The set of zero reduced cost paths is identical for all solutions.
Proof.
By Lemma 1, all flows have the same latency on each link, so the solutions can differ by at most flows along a set of links with constant latency over the range of which the two flows differ on those links. Since we assume that the latency functions are differentiable, the derivatives of the latency function are zero over the range at which they are constant. Therefore, is constant over the range as well. This implies that any path that is reduced cost in one flow is also reduced cost in the other flow, since the latency functions and are constant for every link . ∎
Define the constant , i.e. is the largest flow value such that the latency on link is equal to the latency at an solution. Note that if is strictly increasing at , then . However, if is constant at , then .
Given that the zero reduced cost paths are the same for all flows (Lemma 2), and any flow has the same latency on all links (Lemma 1), it will be sufficient to only search over flows that are less than on each link .
For each vertex, , and link, , define variable denoting the amount of flow originating from source that is assigned to link . Let denote the set of links for which is the tail vertex and the set of links for which is the head vertex.
Definition 3.
For a given flow model , the linear program is:
Note that the number of variables is , and the number of constraints is also . Therefore, since the number of variables and constraints are polynomial in the flow model, the optimal solution to the linear program can be computed in polynomial time [Karmarkar1984].
Revisiting the definition of a zero reduced cost path/link (Definition 2), the need to constrain flow to minimal marginal cost paths (in addition to minimal latency paths) is explained by the fact that, at , the flow must be a subflow of . Consider the problem instance depicted in Figure 1. The latency function, flow at SO, travel time at SO, and marginal cost at are all listed above each link. The double line links have zero travel time regardless of the volume, their only purpose is to limit the possible path assignment between vertices and . Notice that, for flow traveling from vertex to , the dotted path is of minimal latency but not minimal marginal cost. Running the above on this instance would result in which is the correct value. However, if the dotted path is considered to be of zero reduced cost with respect to source 2 (despite having a non minimal marginal cost) then running the would result in .
Theorem 1.
A subflow, , defined by a feasible solution to the linear program is a subflow of a flow.
Proof.
First, note that by equations (2)–(4), the subflow subflow, , satisfies flow conservation constraints. Equation (2) states that the flow along all reduced cost paths from origin to destination must be less then total demand for . Then equations (3) and (4) state that the flow out of node must either be due to the demand generated by node or the flow into it, minus the flow that reaches as a destination. Therefore, is a subflow of a feasible flow.
What must be shown is that there must exist a flow, , such that for all . If is such that is strictly increasing at an solution, and therefore will be strictly increasing at all solutions by Lemma 1, then and constraint (5) guarantees the claim. Let be the set of links such that the latency function is constant at a flow. Therefore, it only needs to be shown that there exists a solution, , such that for , .
Suppose that there existed a set of links such that for all flows , . Let be an flow. Then there must exist an origin destination pair such that there are two sets of paths for which for all , , and for all , and all paths only differ by links in . This is because the total flow between any origindestination is larger in the flow by equation (2). Moreover, since the flow along nonconstant latency links constrains the total flow. Move units of flow from paths in set to paths in set in the flow . Denote the new flow by . The total travel time for cannot increase because the flow has only increased on constant latency links, and the new flow does not exceed on any link. The total travel time also cannot have decreased because was an flow, so is also an flow. Continue this procedure until there does not exist a link for which exceeds the transformed flow. Then we have constructed an flow, , for which for all links , , a contradiction. ∎
Lemma 3.
For a network , let be a subflow of a feasible flow . Then the flow such that is also a subflow of .
Proof.
First, , by the definition of a subflow. Now set . Then for all , , and similarly for ∎
Theorem 2.
The optimal value of the linear program for a network instance is the maximum amount of agents that the network can support and achieve .
Proof.
First, by Theorem 1, there exists an flow such that the optimal subflow, , is a subflow of the flow, and by Lemma 3, there exists a subflow of compliant agents that can achieve the solution. Moreover, by the definition of the linear program and Lemma 2, the flow is only along zero reduced cost paths. By the definition of zero reduced cost paths, all agents are willing to take the assigned paths. Therefore, the solution is achievable with the flow, and there is some volume of flow that is equal to the objective of the linear program.
Now, suppose that there was another flow assignment, , for which compliant flow could be assigned in such a way that the total system travel time was achieved and the total flow volume was larger than the value returned by the linear program. Note that this flow assignment () must be a subflow of some flow, . Moreover, by the definition of flow and the fact that all paths in a solution are minimum marginal cost paths, all paths assigned with a flow greater than zero must be a zero reduced cost path. Therefore, the flow satisfies the equations (2)(6), and since the linear program returns the optimal flow assignment, this is a contradiction. ∎
While we’ve demonstrated that we can compute the maximal flow that permits an solution given the appropriate assignment of the compliant flow, it is likely that a more common problem would be to determine, for a given set of compliant agents, whether or not it is possible to achieve with that set. Our methodology also provides an answer to this question, as the following Corollary demonstrates.
Corollary 1.
For a given network instance and given a set of compliant demand, , from each origin destination pair , there exists a compliant flow such that the network achieves if and only if there exists an for all and such that and are a solution to the linear program.
Proof.
By Theorem 1, any solution to the linear program defines a subflow of an flow. Therefore, if and is a solution, there exists an assignment of the compliant flow that achieves .
Moreover, if there exists an assignment of the complaint flow, , such that a subflow with demands achieves system optimum, then the flow is only along zero reduced cost paths by definition of flow and , and the subflow is feasible. Therefore, the decomposed flow satisfies the constraints of the linear program. ∎
6 Flow Assignment for Compliant Agents
Given that we can now determine both the maximal amount of flow that a system can tolerate and achieve system optimum and, for a given set of compliant agents, whether or not a system can achieve optimum, we are only left with assigning the compliant flow to paths. This section tackles the question of how to assign paths to a, sufficiently large, set of compliant agents such that is achieved.
The methodology from the previous section immediately suggests a solution. Given a network instance , suppose that we have compliant demand equal to for all . Then we must find a flow, , such that and permit subflows of the solution. Such a flow must exist by Theorem 1 and Corollary 1.
The first step is to compute the subflow, , given demand. From the previous section: this exists and is computationally tractable. Any feasible subflow, , with demand such that the total flow along link satisfies has latency equal to the solution, and the flow , by Lemma 1, is an solution.
We can compute with the following linear program:
subject to  
We know that a solution to the above linear program exists and it can be computed tractably.
The final step is to decompose the compliant flow, , into a per path assignment for each origindestination pair in order to assign individual agents to a path. This can be done in time using standard flow decomposition algorithms (see Section 3.5 of Ahuja, Magnanti, et. al. (Ahuja.Magnanti.ea1993) for a discussion).
7 Experimental Results
Scenario  Vertices  Links  Zones  Total Flow  TTT  TTT  % Improve  Threshold  % compliant 

Sioux Falls  24  76  24  360,600  7,480,225  7,194,256  3.82  6.19E11  13.04 
Eastern MA  74  258  74  65,576  28,181  27,323  3.04  3.04E13  19.73 
Anaheim  416  914  38  104,694  1,419,913  1,395,015  1.75  8.05E11  19.76 
Chicago S  933  2950  387  1,260,907  18,377,329  17,953,267  2.31  9.14E10  27.29 
Philadelphia  13389  40003  1525  18,503,872  335,647,106  324,268,465  3.39  4.20E09  49.59 
Chicago R  12982  39018  1790  1,360,427  33,656,964  31,942,956  5.09  4.14E07  53.34 
We are interested in the viability of optin microtolling schemes to more efficiently utilize road networks. As such, we haven undertaken an empirical study to investigate the minimal amount of compliant flow required for () in six realistic traffic scenarios over actual road networks.
7.1 Scenarios
Each traffic scenario is defined by the following attributes:

The road network, , specifying the set of vertices and links where each link is affiliated with a length, capacity and speed limit. Networks are, following standard practice, partitioned into traffic analysis zones (TAZs) and each zone contains a node belonging to called the centroid. All traffic originating and terminating within the zone is assumed to enter and leave the network at centroids.

A trip table which specifies the traffic demand between pairs of centroids. The demand function between nodes other than centroids is set to zero.
The following benchmark scenarios were chosen both for their diversity of topology and traffic volume and their widespread use within the traffic literature: Sioux Falls, Eastern Massachusetts, Anaheim, Chicago Sketch, Philadelphia, and Chicagoregional. All traffic scenarios are available at: https://github.com/bstabler/TransportationNetworks. Figure 2 depicts three representative network topologies (the three smallest networks).
7.2 The Traffic Model
A macroscopic model was used in order to evaluate traffic formation. Macroscopic models calculate the in a given scenario using algorithm B [Dial2006]. For all scenarios, the model assumed that travel times follow the Bureau of Public Roads (BPR) function [Moses and Mtoi2017] with the commonly used parameters , . The solution is computed by replacing the latency functions with and using algorithm B to obtain the equilibrium solution [Dial1999]. Since solving for the and solutions requires solving a convex program [Dial2006], we only solve them to a certain precision. To measure convergence, given an assignment of agents to paths, we define the average excess cost (AEC) as the average difference between the travel times on paths taken by the agents and their shortest alternative path. The algorithm terminates when the AEC is less than 1E12 minutes (except for Chicagoregional for which 1E10 was used due to the size of the network). Therefore, a minimum marginal cost path is only a minimum up to a threshold.
A link is defined to be zero reduced cost with respect to if it carries flow originating at in the SO solution (i.e., the link belongs to a minimum marginal cost path) and if the difference between the least latency path that include and the least latency unrestricted path, both leading from to the head vertex of , is less than a threshold .
The threshold is defined as follows. for each origin and link we calculate the least marginal cost path () leading from to the head vertex of at the solution. We do this once while restricting the path to include and once without such restriction. The difference between these two values is stored and is set to be the maximum of these difference across all the links and origins in the network.
7.3 Results
Table 1 presents the percentage of flow that must be compliant in order to guarantee an solution for six different traffic scenarios. Each scenario is affiliated with the number of vertices, links, and zones comprising the affiliated road network as well as the number of trips that make up the affiliated demand.
The columns “ TTT” and “ TTT” represent the total travel time (in minutes) over all agents for the case where 100% of the agents are controlled by the controller ( solution) and when 100% of the agents are controlled by the controller ( solution) respectively. The percentage of improvement in total travel time between TTT and TTT is also shown under “% improve”.
The percentage of required compliant flow (formally where ) as computed by the linear program (Definition 3) is presented for each scenario under “% compliant”.^{4}^{4}4Statistical analysis for Table 1 is not presented, as the macroscopic model is deterministic.
The results suggest that as the size of the network (i.e., the number of nodes and vertices) increases, a greater fraction of compliant travelers are needed to ensure the network achieves system optimum. This appears to be due to an increasing number of used paths at the solution as the network size increases. As the number of paths grow, the set of zero reduced cost paths grows more slowly, and, therefore, a higher percentage of compliant agents is required.
8 Summary
This paper discussed a scenario where a set of agents traverse a congested network, and a centralized network manager optimizes the flow (minimizes total latency) using a set of compliant agents. A methodology was presented for computing the minimal volume of traffic flow that needs to be compliant in order to reach a state of optimal traffic flow. Moreover, the methodology extends to inferring which agents should be compliant and how exactly the compliant agents should be assigned to paths. Experimental results demonstrate that the required percentage of agents that are compliant is relatively small (between 13% and 53%) for several realistic road networks.
Going forward, it would be worthwhile to explore the possibility of approximation algorithms for assigning compliant flow when the demand is too large to achieve system optimum. Given that the optimal solution to this problem is known to be NPhard, an efficient approximation algorithm would be a useful tool as optin network routing systems are implemented. Further, in order to limit the necessary optin incentives, there is work needed to develop systems that target particularly influential users to optin to these systems.
8.1 Acknowledgements
The authors would like to thank Josiah Hanna and Michael Levin for contributing useful comments and discussions in the course of this research.
A portion of this work has taken place in the Learning Agents Research Group (LARG) at the Artificial Intelligence Laboratory, The University of Texas at Austin. LARG research is supported in part by grants from the National Science Foundation (CNS1305287, IIS1637736, IIS1651089, IIS1724157), The Texas Department of Transporation, Intel, Raytheon, and Lockheed Martin. Peter Stone serves on the Board of Directors of Cogitai, Inc. The terms of this arrangement have been reviewed and approved by the University of Texas at Austin in accordance with its policy on objectivity in research.
The authors would also like to acknowledge the support of the DataSupported Transportation Operations & Planning Center and the National Science Foundation under Grant No. 1254921.
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