A Multiagent Solution to Overcome Selfish Routing In Transportation Networks

Transcription

1 A Multiagent olution to Overome elfish Routing In Transportation Networks Mohammad Rashedul Hasan University of Nebraska-Linoln Linoln, NE 68588, UA Ana L. C. Bazzan PPGC / UFRG CP 15064, , Brazil bazzan@inf.ufrgs.br Eliyahu Friedman and Anita Raja The Cooper Union NY {friedm3,araja}@ooper.edu Abstrat It is well-known that selfish routing, where individual agents make unoordinated greedy routing deisions, does not produe a soially desirable outome in transport and ommuniation networks. In this paper, we address this general problem of the loss of soial welfare that ours due to unoordinated behavior in networks and model it as a multiagent oordination problem. peifially we study strategies to overome selfish routing in traffi networks with multiple routes where a subset of vehiles are part of a soial network that exhanges traffi related data. We investigate lassi traffi flow paradoxes that are ubiquitous in various types of networks leading to severe ongestion. We present a novel distributed traffi oordination algorithm that alleviates ongestion by harnessing the real-time information available through the driver s online soial network. We also propose a utility omputation mehanism for route hoie that generates near-optimal flows. Our extensive simulation results show that soial network based multiagent traffi route oordination ontributes to mitigate the effets of these paradoxes and signifiantly redues ongestion. I. INTROUCTION elfish routing, where individual agents make unoordinated routing deisions only in the interest of their own performane, is known to be ineffiient. It leads to ongestion problems in networked systems [7], [13], [14]. We have several motivations for the work reported in this paper. First, sine the publiation of the aforementioned (and other) works on selfish routing, the panorama has hanged. There is an inreasing prevalene in the use of soial networks where ommuniation of information among individuals using the network is real-time and seamless. This trend is giving tration to mobile apps for transport networks, suh as Waze and Roadify, designed to enable ommuters to share realtime traffi information. This work is aimed at opening a thread of inquiry to determine the underpinnings of Waze-like algorithms, whih are not aessible to the publi, with the goal of providing improvements where possible. It has also led to reent interest [9], [10], [5] in studying the potential benefits of the soial ommuniation aspets of these mobile apps and how they an be effetively applied towards minimizing traffi ongestion. Our work is motivated by suh initiatives and we seek to take advantage of these new tehnologies to use them to better distribute vehiles in the network with the simulation goals of reduing ongestion. eond, the inrease in average travel time due to selfish routing has been extensively studied with respet to two lassi traffi flow paradoxes: Pigou [11] and Braess [2]. Using a simple two route network, Pigou s paradox demonstrates the priniple that selfish behavior need not produe a soially optimal behavior. Braess paradox reinfores the sub-optimality of selfish routing by showing that network improvements (adding a zero or low ost route to redue ongestion) an result in inreased ongestion when drivers hoose routes selfishly. In real transportation networks, it has been observed [12] that this problem ould be resolved by restruturing the network (shutting down a road). Another strategy is to introdue tolls on faster routes that not only inur extra osts for drivers but also hange the ost struture of the routes. Our goal is to solve these paradoxes without restruturing the networks or by enforing tolls, both of whih ould be very expensive. Instead we use the driver s soial network data to failitate oordinated routing. Third, traffi ongestion is losely related to two fundamental onepts in traffi assignment, namely the user equilibrium (UE) and the system (or soial) optimum (O), formulated by Wardrop [16]. The former states that under equilibrium onditions, traffi arranges itself in ongested networks suh that all used routes between an origin-destination (O) pair have equal and minimum osts, while all those routes that were not used have greater or equal osts. This is Wardrop s first priniple, also known as Wardrop s user, or Nash equilibrium. Wardrop s seond priniple orresponds to the O and states that under soial equilibrium onditions, traffi should be arranged in ongested networks in suh a way that average (or total) travel ost is minimized. Further, the onept of prie of anarhy [7] measures the effet of deentralized deision-making in a system in whih its omponents at selfishly, i.e., it is defined as the ratio between the worst equilibrium (worst UE) and the optimal solution (O). Therefore, in order to ahieve a soially benefiial outome, it is important to redue the prie of anarhy by establishing optimal flows. In the event of ongestion, all drivers may stritly prefer a oordinated outome to the flow at Nash equilibrium to redue the prie of anarhy. Our approah seeks to ahieve the O in whih the Nash flow is stritly Pareto-dominated by the optimal flow. In order to reate optimal flows in transportation networks, we model this problem as a multiagent system (MA) distributed oordination problem. rivers traveling through

2 the roads at as agents that make route hoie deisions in a distributed fashion. The goal is to reate a mehanism for oordinated deision making. We envision a senario in whih a ertain fration of drivers in transportation networks are onneted via an online soial network traffi app. When a driver enounters ongestion in the route she is urrently traveling or just finished traveling, she will report the level of ongestion in the form of a post in the traffi app. Creating a post an be onstrained by the GP readings so that a driver an only post ongestion levels about a route in whih she is traveling. Based on the olletion of ongestion reports, the traffi app ontinuously omputes the utility of eah route in real-time. The utility of a route is a measure of traffi moving smoothly on that route and is omputed using the olletive reported ongestion information. Unlike onventional eletroni map apps, there is bi-diretional information flow in our app, i.e., drivers onneted via the app have aess to the route utility values whih they an hoose to use to make route hoies. We propose to augment the traffi app with a traffi route preferene funtion (TRPF) that omputes the route utility values. The TRPF is based on real-time ongestion data olleted from the drivers. rivers are able to hoose a moving time window to ompute the route utility based on data from the past n hours so that the app is able to improve its predition over the long run. Our goal is to improve the travel time for individual drivers while also reduing the average travel time of all the drivers. TRPF not only allows users to share relevant information with eah other to improve their travel experiene, but also offers valuable data that ould ultimately help transportation agenies upgrade the way they struture initiatives aimed at improving and streamlining ommutes. To gain a deeper understanding about how this traffi app ould improve the system behavior, we perform an extensive simulation study. We hoose two simple traffi networks [11], [2] for investigation that are well studied in the literature for resolving the ongestion problem. As disussed below, one of the hallenges of providing global information about the routes to drivers is that it might lead to osillation. Using simulations, we determine the optimum parameter settings based on these two simple network senarios that redue osillation and ensure a stable traffi flow aross routes. Our primary goal for this paper is to identify the fundamental aspets of TRPF to be able to implement it for larger network senarios in the future. imulation results show that the soial network based route oordination approah suessfully resolves these paradoxes while reduing both ongestion and the average travel time. To summarize, in this paper we emphasize the benefits of harnessing the online soial network platform to redue ongestion using a ouple of well-known networks as ase studies. The main ontributions of this paper are: For a given number of vehiles traveling from a soure to destination, our route-preferene approah ahieves the soially optimal (or near-optimal) distribution of drivers. M N 0.1 x (a) 2 route network M O N 0.1 x P (b) 3 route network Fig. 1. Modified Pigou s Networks x We present a MA oordination model that leverages drivers soial network to failitate establishing the optimal traffi distribution. We show that our model is able to resolve the lassi traffi flow paradoxes suh as Pigou and Braess that result from selfish routing. II. CLAIC TRAFFIC CENARIO We desribe two networks used to illustrate our approah. The simple network in Fig. 1(a) is a variant of Pigou s example [11] on the effets of selfish routing. Both M and N routes lead from a soure to a destination. Consider that a flow of 0 want to use the network. The individual osts (lateny/travel time) assoiated with eah edge are either onstant ( = ) or variable (a funtion of the flow x using an edge). The ost of M and N are 40 and + 0.1x respetively. Route N is heaper from eah individual driver s perspetive. Thus, under the assumption that eah driver aims to maximize her utility, we an expet that the entire flow uses N. Moreover, no driver has inentive to shift to M sine it also inurs a ost of 40 there. Hene, under UE only N is used. This is ertainly not the best distribution from the overall point of view: if the traffi flow ould be oordinated (instead of being a produt of selfish routing), the best distribution from the soial point of view (O), would be assigning 100 to eah route. The average ost in this ase would be 35, whih is smaller than that at UE. imilar onlusions an be reahed in the ase of three routes (Fig. 1(b)): beause MP is the heapest route from the individual point of view, the entire flow of 0 will use MP with an average ost of 60. However, the soial optimum ours when 75 use MP and 125 use MN, ausing a redued average ost of Another example vastly used in the literature on selfish routing is due to Braess [2]. An instane of it is depited here in Fig. 2. Assume flow is 4000 and = 45. The paradox ours due to the following: Initially, there were just two routes to go from to, one via v and one via w ( Fig. 2(a)). ine osts are idential, the whole flow splits equally produing an individual ost of The traffi authority then deides to onstrut a high apaity link from v to w (Fig. 2(b)). This new link has ost or lateny of 0. Obviously, now, all flow will deviate to route vw. This auses high ost in both v and w edges. The individual ost is now , whih is higher than the one experiened prior to when the additional edge was inluded.

3 v w (a) initial network Fig. 2. Braess Paradox v w 0 (b) extended network Worse, if a single driver deides to take for instane w, its travel time inreases to 85. Thus it has no inentive to do so. Like the previous examples, we ould do better if we ould route the flow to the O in the following way: 1745 take v, 1745 take w, and 510 take the route via vw. The ost in the latter group is just 45, while it is 67.5 for the others. The average ost is obviously lower than the one obtained by selfish routing (64.68 instead of 80). III. RELATE WORK The problem of routing vehiles (or pakages in a ommuniation network) has reeived great attention in the literature. For example, the aforementioned literature on selfish routing and the prie of anarhy [7], [13] aims at bounding the worst possible severity of phenomena related to selfish routing. Roughgarden [12] suggests three ways to ope with selfishness: inreasing the apaity of the network, influening traffi with edge taxes (tolls), and routing a small amount of traffi entrally. The first involves soial, environmental, and eonomi osts that lie in the area of politial deisionmaking. Regarding the seond diretion, Buriol et al. [3] proposed an approah that finds a traffi flow that is both UE and the O. However, they solve this optimization problem by finding a set of edges in whih a number of toll booths should be plaed, i.e., they solve a different problem, namely imposing additional osts to make the user equilibrium and the system optimum oinide. As for the third diretion, a solution via takelberg routing is disussed in [12]. This solution was formulated for ommuniation networks; it is unlear how a portion of the vehiular traffi an be entrally routed. In the present paper, we argue that traffi routing an be best implemented using route reommendation in onjuntion with ollaborative ongestion data olletion. In the area of guidane and reommendation, there are lassial as well as agent-based works that investigate the effets of providing information to drivers. In the interest of spae, we just mention that osillations are seen when everyone reeives the same information. A simple form of soial network is reported in Bazzan et al. [1] for a ommuting senario (restrited to a simple binary route hoie) in whih information an be provided by offie mates, by aquaintanes, by maliious agents, or by route guidane. It was found that information is benefiial. Vasserman et al. [15] investigate whether Waze an, in general, implement the soially optimal solution. It differs from ours in the sense that it does not expliitly onsider the drivers as soure of information; it assumes that all drivers use Waze; and its fous is on inentive ompatible poliies to ompute a mediated equilibria. In the VANET ommunity, there has been some early attempts [4], [8] to onsider soial relationships between vehiles or drivers. Most of them fous on reproduing the mobility models. However these models aim mainly at deteting ommunities of vehiles and they do not map the vehiles on a real topologial spae. In the same line, other works advoate for providing knowledge to vehiles and for the use of data olleted from soial networks, in order to help to improve traffi predition, as in [5], [10]. These however do not disuss the use of the methods in routing senarios. IV. OUR APPROACH In the New Cities Foundation (Conneted Commuting) study [9] on the urban ommuting problems in an Jose, the entral question posed is whether a new level of networking between ommuters ould enhane the overall ommuting experiene. One of their key observations is that ommuter omments olleted by smartphone appliations provide valuable high-quality real-time data about ommuter sentiment in relation to their ommutes. In our approah, users report the ongestion level of the route they traveled. Eah driver then alulates the Traffi Route Preferene Funtion (TRPF), whih fuses the ongestion reports into a representation of the utility of eah route. The driver an then hoose the route that maximizes the TRPF. The goal of this method is for eah vehile to use the ongestion reports to hoose a route suh that average travel time for all vehiles is minimized. A. Problem Model We model a ommuting senario. Eah edge has an assoiated ost funtion, whih represents the travel time, or lateny, a driver would experiene traveling along that edge. In our senarios, the ost funtion is linear with respet to the number of drivers on that edge. Given N drivers and k routes from whih they an hoose, we want drivers to distribute themselves in suh a way as to ahieve the optimal flow. The optimal number of drivers along a route, O i i =1, 2,...,k an be found by solving a onvex optimization problem using CPLEX (quadrati programming). Next, we introdue the three main parameters of our model: G, p, and T. To model drivers deisions, we assume that without any external stimulus, drivers will hoose the route that has previously given them the lowest travel time. This auses the driver distribution to onverge to the UE. However, with probability 1 G, a driver will stik with the same route as last time, whih is a way for us to model a person s desire to exploit their previously gained experiene rather than exploring new options. Parameter G allows us to ontrol the number of agents who hange their route eah round to determine the effet of this number on the performane of TRPF; espeially how it influenes osillation. We assume that a fration p of the drivers have aess to a Waze-like app reating a dynami soial network. This is the fration of drivers who will be using the TRPF. When drivers run into ongestion, if they are part of the soial network,

4 they will report it using one of two possible ongestion levels: Level 1: moderate ongestion (traffi moving slowly) or Level 2: heavy ongestion (standstill traffi). We define a weight metri alled the ongestion multiplier assoiated with eah ongestion level. For Level 1, a ongestion multiplier w =3 is used, and for Level 2, w =5. Eah vehile in the soial network reeives reports from the other vehiles in the soial network and alulates the TRPF. rivers may use a number of rounds, represented by the parameter T, to determine their individual utility value for eah route. B. Traffi Route Preferene Funtion (TRPF) The TRPF is a measure of whether more drivers are on a route than the optimal distribution of drivers would suggest. It fuses the traffi reports from the drivers in the soial network to result in a funtion that ahieves its maximum when fewer drivers are on a route than should be in the O. The TRPF for route i is a funtion of the traffi reports for the last T rounds. If t is the urrent round, n (t) i is the number of drivers reporting ongestion for route i at time t, and w (t) i is the ongestion multiplier for route i at time t, then the TRPF is given by: TRPF i (t) =1 P t =t P t =t T n( ) i w ( ) i T n( ) i This express the total ongestion report for route i over the last T rounds, normalized by the number of drivers reporting ongestion. The purpose of the subtration is so that the funtion is maximized when fewer drivers use route i. If fewer than O i drivers (the optimal number of drivers on route i) hoose route i, then the TRPF for route i will be 1. The TRPF will derease as the differene between the number of drivers on a route and the O of that route inreases. Our approah proeeds in the following steps: 1) Initialization: Eah driver will use the TRPF with probability p. 2) Eah driver hooses a new route. With probability 1 G, a driver will hoose the same route as last time. Of the drivers who an hange their route, those who use the TRPF will hoose the route with the maximum TRPF value and those who do not use the TRPF will hoose the route that has historially resulted in the minimum travel time. 3) rivers travel along the route and reeive the travel ost. 4) rivers who use the TRPF report the ongestion levels for their routes. 5) rivers who use the TRPF reeive the reports from other drivers and alulate the new TRPF based on the last T rounds. 6) Repeat, starting from tep 2. V. IMULATION AN REULT ANALYI We ondut simulations with the following goals: (i) investigate the performane of TRPF for reduing ongestion and average ost (travel time) (ii) determine the onditions under whih use of TRPF provides optimum performane, i.e., minimum-possible average ost with no/small osillations in the route hoies. We use the previously introdued Modified Pigou s Network with three routes (Fig. 1(b)) and Braess Network (Fig. 2) with a onstant flow of 0 and 4000 drivers respetively to study the effetiveness of our approah. In our experiments, the total number of vehiles on the ombined routes for the two networks sometimes exeed the total number of vehiles sine these are averages over a number of rounds. We vary the three parameters G (fration of drivers that hange routes in eah round), p (fration of drivers that use TRPF to hoose routes), and T (number of previous rounds used to alulate TRPF) to observe the effet of TRPF on traffi distribution (average number of drivers in eah route) and average ost. Eah simulation reports the results of 0 rounds. These results are generated by taking the average of last 100 rounds one the traffi distribution beomes stable. Table I reports the results of simulations for four senarios in Pigou s and Braess network. Rows 1-5 of Table I represent senario 1, in whih % drivers (G =0.2) may hange their routes every journey. Also, drivers having the TRPF app use data from the past one hour (T =1) to ompute TRPF values. Among the % drivers (who may hange their routes), those who do not have the app will hoose the historial lowestost (selfish) route. If they are already using the historial lowest-ost route, then they do not hange routes. However, if some (or the majority) of these drivers have aess to the TRPF app, then they swith to the route with highest TRPF value. Parameter p ontrols the fration of drivers that use the app to hange routes. When the value of p is large, a majority of the drivers use TRPF for hoosing a new route. With G and T set at fixed levels (seond and third olumns), we vary the value of p to observe how it influenes the traffi distribution and the average ost. When p = 0, the majority of the drivers hoose the historial lowest-ost route, i.e., MP in Pigou s network and vw in Braess network. As antiipated in etion II, we observe that a majority of the drivers hoose these routes, reating heavy ongestion with an inreased average ost. As the value of p inreases, the traffi distribution in the three routes approahes the O with a derease in average ost (as desribed in setion II, at O the average ost for Pigou s and Braess network would be and respetively). However, Fig. 3 shows that with p 0.5 traffi osillation gradually starts inreasing, peaking at p =0.9. The reason is that the majority of the drivers that hange their routes use the TRPF app as ompared to small values of p (apple 0.5), in whih only a small fration hanges route and the majority stik to their lowest-ost route. That is why higher values of p ( 0.5) lead to osillation. We remark however that the average ost does not derease at higher values of p. This indiates that osillation, while not preferable, does not degrade system performane (i.e., does not inrease average ost). The standard deviation of the traffi distribution inreases with the inrease of osillation for larger p values. We observe a relatively stable traffi distribution for the values

5 TABLE I MOIFIE PIGOU NETWORK WITH THREE ROUTE &BRAE NETWORK: EFFECT OF THE VARIATION OF G, T & P. Modified Pigou s Network with 3 Routes Braess Network G T p # MN drivers # MP drivers # OP drivers Cost # v drivers # w drivers # vw drivers Cost Avg td Avg td Avg td Avg td Avg td Avg td Avg td Avg td of p between in both Pigou s (Fig. 3(a) & 3()) and Braess (Fig. 3(b) & 3(d)) networks. In order to observe the effet of parameter T, senarios 2 and 3 are reated by inreasing the value of T to 3 and 5 (rows 6-15 of Table I). In other words, the drivers who have the TRPF app use soial network data (ongestion reports) from the past three and five hours to ompute their TRPF values. In rows 6-10 of Table I (senario 2), when T is set to 3, no further derease in average ost an be observed. imilarly, rows of Table I (senario 3) show that with T set to 5, there is no performane improvement when ompared with the first senario reported in rows 1-5 (T = 1). This indiates that the past one hour s ongestion report is enough to identify the less-ongested routes. Rows 16- of Table I show the results of simulations for senario 4 in whih 50% drivers (G = 0.5) hange their routes every time they begin their journey. We set T to a fixed value of 1 so that drivers having the TRPF app use data from the past one hour to ompute their TRPF values. With these two parameters set at fixed values, we vary the value of the parameter p. For Pigou s network, Fig. 3(g) shows that the amplitude of the osillation is highest at a relatively low value of p ( 0.5). In other words, if 50% of the drivers hange their routes in the beginning of their journey, then the use of TRPF app results in dereased average ost with inreased osillation. This is refleted by the inreased standard deviation values for the average number of vehiles aross all the routes for larger values of p ( ). We observe a similar situation in Braess network (Fig. 3(h)). For the above-mentioned four senarios, we performed simulations multiple times to observe whether the average number of vehiles (for the three routes) and average ost differ. However, the standard deviations were small. For spae onstraint, we do not report results from statistial analyses. VI. CONCLUION AN FUTURE WORK In this paper, we solved the problem of selfish routing in transportation networks by leveraging the driver s soial network data. We presented the Traffi Route Preferene Funtion (TRPF) that omputes utility values of the alternative routes for a given soure-destination pair. TRPF uses posts on ongestion levels (moderate or heavy ongestion) from the driver s soial network for this omputation. We emphasized two standard traffi networks, Pigou s and Braess network, that are used to investigate the phenomenon of selfish routing. Using extensive simulations we have demonstrated that TRPF is able to establish oordination among the drivers for hoosing alternative routes. Our main findings are as follows: TRPF ahieves system optimum (O) by reduing ongestion and dereasing the average travel time. TRPF is shown to perform espeially well in networks where the Nash flow is stritly pareto-dominated by the system optimum flow. Braess paradox (as well as Pigou s) an be resolved without restruturing or enforing tolls. TRPF approahes O even when less than % of the drivers use TRPF to hoose routes. Osillation beomes prominent when 50% or more drivers hange routes at the beginning of their journeys. However, osillation does not exaerbate the performane. This indiates that there is a network-speifi range for parameter p that optimizes ongestion ontrol. As future work, we intend to extend our investigation to multi-ommodity networks that have more than one origindestination pair. Also we plan to use learning algorithms so that TRPF an learn how to minimize osillations. REFERENCE [1] A. L. C. Bazzan, M. Fehler, and F. Klügl. Learning to oordinate in a network of soial drivers: The role of information. In K. Tuyls, P. J. Hoen, K. Verbeek, and. en, editors, Proeedings of the International Workshop on Learning and Adaptation in MA (LAMA 05), number 3898 in Leture Notes in Artifiial Intelligene, pages , 06. [2]. Braess. Über ein Paradoxon aus der Verkehrsplanung. Unternehmensforshung, 12:258, [3] L.. Buriol, M. J. Hirsh, P. M. Pardalos, T. Querido, M. G. Resende, and M. Ritt. A biased random-key geneti algorithm for road ongestion minimization. Optimization Letters, 4: , 10.

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