Braess' paradox in the age of traffic information
aa r X i v : . [ phy s i c s . s o c - ph ] S e p Braess’ paradox in the age of traffic information
S Bittihn and A Schadschneider
Institut f¨ur Theoretische Physik, Universit ¨at zu K¨oln, 50937 K¨oln, GermanyE-mail: [email protected] , [email protected] Abstract.
The Braess paradox describes the counterintuitive situation that theaddition of new roads to road networks can lead to higher travel times for all networkusers. Recently we could show that user optima leading to the paradox exist innetworks of microscopic transport models. We derived phase diagrams for two kindsof route choice strategies that were externally tuned and applied by all network users.Here we address the question whether these user optima are still realized if intelligentroute choice decisions are made based upon two kinds of traffic information. We findthat the paradox still can occur if the drivers 1) make informed decisions based ontheir own past experiences or 2) use traffic information similar to that provided bymodern navigation apps. This indicates that modern traffic information systems arenot able to resolve Braess’ paradox.
Keywords : traffic, network, Braess’ paradox, exclusion process
Submitted to:
J. Stat. Mech.
ONTENTS Contents1 Introduction 22 Background information 5
Everyday experience shows that we spend a lot of time in traffic jams [1–3]. This timecan add up to more than 100h per year in certain parts of the world [4]. Two potentialsolutions to the problem of congestions come to mind: 1) building more roads and 2)providing traffic information for the drivers to make better decisions. However, it isknown for some time that new roads are not necessarily a solution. The
FundamentalLaw of Road Congestion [4] states that more roads will lead to more road users whichwill lead to congestions on the new roads as well.
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Braess’ paradox . In compact form it can be formulated in the following way:In road networks of selfish users additional roads can lead to higher travel timesfor all users.Although it is formulated using terminology from traffic engineering, the paradox hasbeen shown to occur in a variety of other systems as well, ranging from general transportnetworks [7–9] to mechanical and electrical systems [10, 11], pedestrian dynamics [12],microfluidic networks [13], oscillator networks and power grids [14, 15]. A review ofsome examples from mechanical systems, biological networks, to power grids can befound in [16].Braess has exemplified the paradox for a simple network that has just five edges.One of the essential ingredients for the occurrence of the paradox is that one has todistinguish two different kinds of optima in the system, the user optimum (uo, also calledNash equilibrium), and the system optimum (so) [17]. The uo is realized if network usersdistribute themselves onto the routes such that the travel times of all used routes areequal and lower than those of any unused routes. It reflects the perspective of individualdrivers who can not improve their travel times by choosing a different route and is thuswidely considered to be the stable state of a traffic network used by selfish users. Theso corresponds to a global perspective, e.g. of an engineer or politician. In the so thedrivers are distributed onto the routes such that a global parameter is minimal. Severaldifferent global parameters could be considered. Prominent examples are the weightedaverage of all travel times [17] or the sum of the travel times of all drivers [18]. Braessconsidered yet another definition of the system optimum: in his work [5, 6] as well asin our previous works on the Braess paradox [19, 20] and also in the present articlethe system optimum is defined as the state which minimizes the maximum travel timeof all used roads. Indeed, the uo and the so can be different in certain situations, i.e.correspond to a different distribution of drivers on the available routes.Braess’ example [5, 6] consists of simple networks with 4 and 5 links (where linksrepresent roads), respectively. In the network with 4 links, the uo and so coincide. Theaddition of a new road (i.e. the fifth link) changes the network such that the uo andso become different, or – more specifically – in the network that includes the new road,the individual travel times in the uo are larger than those in the so. The travel timefunctions are chosen in such a way, that the travel times in the uo of the network withthe fifth link are higher than those in the network without the fifth link.Although the paradox considers a simplified scenario, it has been observed in severalreal world situations. Newly built roads can worsen the traffic situation, or inversely,the closing of roads can improve traffic [21–23]. A concept related to Braess’ paradox isthe price of anarchy . It measures the reduced efficiency of a system due to the selfishbehavior of agents [24].
ONTENTS if those potentially accessible user optimumstates are also realized in more realistic situations with real (imperfectly deciding) humannetwork users who base their decisions on real (imperfect) traffic information.
In fact it has been shown that often travel time minimization is not the only factordetermining route choice decisions [28–30] and that even if it is, drivers do not decideperfectly rational on this basis [31, 32]. Therefore variations of the above mentioneddefinition of the user optimum were introduced [33, 34]. While perfect traffic informationis also not present in road networks, with the introduction of smartphone routing appsand personal navigational systems, more accurate information is available [35]. It hasrecently been shown, that this might actually lead to the realisation of user optima insome cases of road networks [36, 37].Here we examine whether Braess’ paradox is realized in a microscopic transportmodel if users decide intelligently based upon information similar to that availablefor real modern day road networks. This is meant in the following sense: are theuser optima in the systems with and without the new road which are accessible byexternally tuning the users’ decisions reached if users intelligently base their decisionson information similar to that available in present day real world networks. Two typesof traffic information are considered: information based on the drivers own memory orexperience (as e.g. in a commuter scenario) and information similar to that providedby smartphone apps. It is shown that both types drive the system into its user optima,realizing Braess’ paradox. This is a strong indication for answering the question if theparadox still occurs in present day road networks or if it is potentially resolved due tomodern traffic information: it seems that the paradox is indeed still of great importance!It is furthermore shown that user optima of different phases of the system (that do not
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2. Background information
Before presenting the results of our study we define some terminology and provide somemore details about Braess’ paradox and some background to traffic information androute choice processes. Then we define the model investigated here and give a shortsummary of previous results.
Since this paper is concerned with traffic networks we will use the terms “edge” and“road” interchangeably. A “route” is a connection between an origin and a destinationin a traffic network. A route can be comprised of multiple roads and also of so-called“junction sites” which connect roads. The “travel time” of a road refers to the time ittakes to traverse the road. The travel time on a route refers to the time needed to traversethe route, i.e. to get from the origin to the destination on that route. Furthermore,the terms “car” and “particle” are used interchangeably as well as “driver”, “user” and“agent”.A “strategy” of an agent refers to its route choice. Two specific types of strategiesthat were used in our previous research [19, 20] are “pure strategies” and “mixedstrategies”. For pure strategies the driver chooses exactly one specific route, whereas“mixed strategy” refers to the case in which one route is chosen out of several routeswith a certain probability. In real road networks, if a network user has to perform routechoices repeatedly, e.g. mixtures of these two strategy-types can be at play. A “state”of the network is given by the distribution of the cars onto the routes, i.e. the set of thestrategies of all drivers.The “user optimum” state is often considered to be the stable state of trafficnetworks with “selfish users”, i.e. agents who choose their routes non-altruistically onlyaccording to their own intentions. The user optimum is reached if the travel times of allcars are such that they are equal on all used routes and, at the same time, lower thanthose of any unused routes [17]. A “pure user optimum” (puo) is reached if all agentsfollow pure strategies. In this case the numbers of cars using each route are integernumbers (or zero). This corresponds to a “pure Nash equilibrium” in game theory [38].A “mixed user optimum” (muo) is reached if all agents follow mixed strategies and the mean values of the travel times of all used routes are equal and lower than those of anyunused routes. In this case the average numbers of cars following routes can be positivenon-integer values or zero. It corresponds to a “mixed Nash equilibrium” [38].
ONTENTS The network proposed by Braess in his original work [5, 6] isshown in Fig. 1. In his scenario all agents move from the same origin to the samedestination. Road 5 is the road which is added to the system. Thus in the networkwithout road 5, which we call “4link network” from now on, there are two routes fromorigin to destination: route 14 and route 23. In the network with road 5, called “5linknetwork” in the following, there is the additional route 153 to the destination ‡ . In the Figure 1.
Braess’ network as presented in his original work [5, 6]. All agents movefrom the same origin to the same destination. There are three routes available, namedafter the edges they are comprised of: routes 14, 23 and 153. Edge 5 is supposed tobe newly added to the network. Route 153 is only available after this addition. Thenetworks without and with edge 5 are called 4link and 5link networks, respectively. original example, traffic flow was characterized only by travel time functions T i linearin the number of cars n using road i : T = T = 10 n, (1) T = T = 50 + n, (2) T = 10 + n. (3)The 4link network is thus symmetric.Braess showed that for a total number of N = 6 cars, the pure user optimumof the 4link network is given by half the cars using route 14 and the other half route23, respectively: n (4)14 , puo = n (4)23 , puo = 3. This results in equal travel times of bothroutes: T (4)puo = T (4)14 , puo = T (4)23 , puo = 83. In the 5link network the pure user optimumis given by n (5)14 , puo = n (5)23 , puo = n (5)153 , puo = 2 with equal travel times on all routes, T (5)puo = T (5)14 , puo = T (5)23 , puo = T (5)153 , puo = 92. Thus, T (5)puo > T (4)puo , i.e. the new road leads tolarger pure user optimum travel times. ‡ Routes are named according to the (ordered!) edges they are comprised of. From here on, variablescorresponding to the 4link and 5link networks are marked with superscripts (4) and (5), respectively.
ONTENTS § . For N = 6, in the4link network the mixed user optimum state is found if routes 14 and 23 are chosen withequal probabilities: p (4)14 , muo = p (4)23 , muo = 1 /
2. This leads to equal travel time expectationvalues of h T (4)muo i = h T (4)14 , muo i = h T (4)23 , muo i = 88 .
5. In the 5link system the mixed useroptimum is given for p (5)14 , muo = p (5)23 , muo = 5 /
13 and p (5)153 , muo = 3 /
13 with travel timeexpectation values h T (5)muo i = h T (5)14 , muo i = h T (5)23 , muo i = h T (5)153 , muo i = 93 . h T (5)muo i > h T (4)muo i . Results of further research.
Since the initial description of the paradox by Braess alot of efforts were made to understand the phenomenon in more detail in the contextof mathematical models of traffic networks. It was shown that in the original modelof Braess, the paradox occurs for several amounts of total users, and not only forBraess’ specific example of N = 6 [9]. The paradox also occurs for different choicesof (linear) travel time functions in Braess’ original network [25] and also in differentnetwork topologies [18]. A general framework for predicting the occurrence of theparadox in networks of uncorrelated links was established [7, 26]. Mathematical modelsincluding correlations between the roads were studied in [8]. It was furthermore shownfor arbitrary networks and models with monotonically increasing travel time functions,that if the paradox occurs at a certain density, the new road will be ignored completelyfor densities higher than a certain threshold [27]. In an attempt to get an understanding of the paradox in a more realistic context,in two recent articles [19, 20] we have shown that the Braess paradox can also beobserved in networks of stochastic, microscopic traffic models, i.e. in networks of totallyasymmetric exclusion processes (TASEPs). The description of traffic flow in Braess’original example, as summarized above, was rather basic, being more of a proof ofprinciple instead of a realistic model. Braess used only linear travel time functionswhich is not realistic. In addition, microscopic interactions and the stochastic nature oftraffic were omitted. Furthermore, correlations between the roads were not taken intoaccount.Modelling traffic flow in the network by coupled TASEP segments is a first step toa more realistic traffic description by including these aspects while, at the same time,keeping the system simple enough to be analysed. There is a vast amount of researchdedicated to the many variants of TASEPs and their properties. For some condensedinformation the reader is referred to e.g. [39–41].The Braess network of TASEPs for two different route assignment types is shownin Figure 2. The network has the same structure as Braess’ original network (Fig. 1).The edges E i are now made up of TASEPs of lengths L i joined through junction sites j k . Furthermore we use periodic boundary conditions via the additional link E . This § For a more detailed discussion of the mixed user optima, see Appendix A.
ONTENTS Figure 2.
Braess’ network of TASEPs with periodic boundary conditions for twodifferent types of externally tuned strategies. The structure of the network correspondsto that used by Braess in his original article, as shown in Figure 1. Here, edges E to E are made up of TASEPs, coupled through junction sites j to j . The added edge E realizes the periodic boundary conditions. Part (a) shows the system with fixedstrategies, in which fixed numbers of cars N i use routes i as studied in [20]. In part(b) the system with turning probabilities, as studied in [19], is shown. Particles sittingon junctions j or j turn left with probabilities γ and δ , respectively. has the advantage that the total number of particles M is conserved which allows tocompare the travel times of the 4link and 5link systems in their respective user optimafor the same number of particles. To reduce the number of parameters, in the followingwe will present results only for the following edge lengths: L = L = 100 , L = L = 500 , L = 1 . (4)The length of E will be varied, subject to the condition that the length of the newroute is smaller or equal to that of the two old routes (which are of equal length).Figure 2 (a) shows the network with fixed personal strategies as analysed in [20]. Inthis case, each particle keeps its personal pure strategy of always choosing one specificroute. Numbers of N , N , and N particles choose routes 14, 23 and 153 respectively,with N + N + N = M . The three numbers N , N , and N can also be expressedthrough the two quantities n ( j )l = 1 − N M , (5) n ( j )l = N N + N , (6)which describe the fraction of particles turning ’left’ at junctions j and j . User optimacan be found by varying the N , N , N . The user optima obtained in this scenarioare pure user optima. ONTENTS j jumps to the left (i.e. onto E ) with probability γ and to theright (i.e. onto E ) with probability 1 − γ . In the 5link network, particles on junction j jump left (i.e. onto E ) and right (i.e. onto E ) with probabilities δ and 1 − δ ,respectively. User optima in this network are found by varying the γ and δ . They aremixed user optima.These two types of route assignment are from here on called “externally tunedstrategies”. This is meant in the sense that all decisions are set at the beginning of eachsimulation run externally , i.e. not intelligently by the particles themselves. These routechoices will later on be distinguished from route choice decisions made by ‘intelligent’agents following our route choice algorithm.The phase diagrams for both networks are shown in Fig. 3. The phase of the systemdepends on the ratio ˆ L / ˆ L of the lengths of the new route 153, ˆ L , and the twoold routes, ˆ L = ˆ L ; and the global density. Since the phase of the system describeshow the travel times of the 4link and 5link systems’ user optima are related, the globaldensities of both systems ρ (4)global = M/ (4 + P i =0 L i ) and ρ (5)global = M/ (4 + P i =0 L i ) areshown. Figure 3.
Phase diagrams of Braess’ network of TASEPs (Fig. 2) for (a) fixed routechoices and (b) turning probabilities. The phase of the system depends on the routelength ratio ˆ L / ˆ L between the new and old routes and the global densities ρ (4 / in 4link/5link systems. In both phase diagrams four states ⋆ ⋆ In both systems at low densities the user optima in the 5link networks have lowertravel times than those of the corresponding 4link networks, thus the system is not in a
ONTENTS N , N , N ) and ( γ, δ ). By varying these parameters, user optimawere found and the travel times of the 4link and corresponding 5link systems werecompared to construct the phase diagrams. In the present article we address the questionif these user optima are actually realized if the particles choose their routes intelligently,based on different kinds of information. As in the original example presented by Braessin his mathematical model, the pure and mixed user optima for one combination ofthe ( ˆ L / ˆ L , ρ (4 / ) do not necessarily coincide, i.e. the ( n ( j )l , n ( j )l ) and ( γ, δ ) whichrealize a pure and mixed user optimum respectively do not always have equal values.In the following, we summarize some results of previous research on trafficinformation and route choice processes and then present our route choice algorithm.This algorithm is then applied to the four states marked in Figure 3. The travel timesof the user optima of these states can be found in Appendix B. Information about the state of traffic networks is called traffic information . It canconsist of information on various aspects of the traffic network, such as the positionsof all vehicles, average speeds, traffic light phases and many more. Here we focusspecifically on information about travel times on roads and routes in the network.Information available to network users can be grouped into two main categories.(i) “Public Information” is in principle accessible to all network users.(ii) “Personal Information” is only known to individual network users. It is usuallybased on the user’s personal experience and/or specifically designed for a specificuser, e.g. based on his/her current position, destination etc.These two main categories can contain information from three different sub-categories [42].a) “Historical Information” describes travel times measured in the network in previoustime periods.b) “Current Information” refers to the most up-to-date information available. It can begiven in the form of providing network users with the current state of the network,e.g. providing the current traffic densities or the currently measured (average)speeds on certain routes as e.g. in [43].If one sticks strictly to this definition, in real traffic networks travel time informationcannot be current information. This is due to the following problem: if e.g. a
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In present dayroad networks various kinds of information are available. Individual network users mighthave personal historical information about travel times on specific routes based on theirown experiences if they used these routes before (personal historical information, in thiscase also called “experiential information” [35]). This is often the case in commuterscenarios. Furthermore, individual users might have some insights from friends or otherpersonal sources. Public information of various kinds is available from numerous sources.Public historical information can be found in internet databases and is also available invarious smartphone routing apps and personal navigational systems. Current publicinformation is available from radio traffic forecasting and various advanced trafficinformation systems (ATIS) [45] such as variable road signs.Personal navigational systems and smartphone routing apps also provide predictivepublic information. This type of information is considered public since these devices arein principle accessible for everyone. In contrast to the former mentioned information,these tools also provide prescriptive route choice information. Among many alternatives,Google Maps was the most popular routing app in the US in 2018 [46]. The maindifference from more traditional types of traffic information is that such apps rely oncrowdsourcing [47]. This means that all users of the app send their location data toGoogle where this data is in turn used to get an accurate picture of the current trafficsituation of the network (given there are enough Google Maps users in the network atthat moment). This current information is combined with large quantities of historicalinformation to provide fairly accurate public predictive (prescriptive) information (abouttravel times). Details about how the Google Maps algorithm works are not known to
ONTENTS whole traffic system. Such information would be given indifferent forms than just information about (predictive) travel times on available routes.It could be designed e.g. to drive traffic networks into their system optima and couldthus lead to a reduction of traffic congestion, as shown e.g. in [49]. Even though thenecessary data for such information is in principle available, such systems are (to ourknowledge) currently not in use, which is why they are not considered here.
The question how road users choose their routes given certain types of traffic informationand consequently the question if user optima are realized in networks of selfish usershas been analysed in various scientific disciplines. The research approaches can besubdivided into three groups: analyses of real world data, mathematical models andsimulation studies, and laboratory experiments.
Since traffic networks are generally highly complexstructures with numerous users that all decide individually it is difficult to gainobjective knowledge about what drives the decision making processes in these networks.Next to much anecdotal knowledge about route choices, some large scale real worldobservations and experiments were performed. A study from 2001 [29] hints at traveltime minimization not being the only factor driving route choice decisions. Furthermore,travel time seems to be systematically misperceived by many drivers [28]. In a studyin which vehicles of a large number of network users were equipped with GPS units itwas shown that only approximately one third of traffic network users chooses the fastestavailable path [30]. For a nice recap of results of previous research, the reader is alsoreferred to [30].The introduction of smartphone routing apps lead to a different usage of roadnetworks in many parts of the world. Data suggests that minimal travel time becomesmore important when using those apps and hints at the realization of user optima dueto the heavy usage of these tools [37]. There are also many negative side-effects of suchtools, such as increasing use of smaller side roads to avoid congested main roads, leadingto complaints by residents [50].An effect that has already been predicted in mathematical traffic models [42, 51]and queueing models [52] and was observed in simulations [44] has also been observedas a consequence of routing apps: routing apps provide predictive information about
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In earlier days, research on routechoices was performed using mathematical models [42, 51, 53] which already predictedmany effects that were later confirmed e.g. by simulations. In [54, 55] the Braess paradoxwas considered in a discrete time macroscopic mathematical model. It was shown thatin the given model, the user optima leading to Braess’ paradox can be avoided with aspecial kind of personal historical information.For route choice research on the basis of microscopic simulations it has proven to beuseful to implement so-called “multi-agent techniques” [44, 56]. The traffic flow itself,forming the “tactical layer”, is modelled by a stochastic microscopic traffic model. Theacquisition of information and the route choice decisions, the so-called “strategic layer”,are modelled by an algorithm. This multi-agent approach is also used in the presentpaper.Multi-agent models with the tactical layer being described by the Nagel-Schreckenberg model [57] have been studied for certain types of traffic information:It was shown that the availability of public historical information in a symmetrictwo-route network with open boundary conditions leads to oscillations around theuser optimum [44]. In [44] the latest experienced travel times on both routes weremade available to all network users. This leads to overreactions since this travel timeinformation is based on the network states previous to when the information is madeavailable. The user optimum is reached when agents choose each route with equalprobability. Instead oscillations between periods of all cars using just one route andtimes of all cars using just the other route are realized. This observation lead to theproposition of many other types of information with the aim of realizing the user optimain this specific network (see e.g. [58–60]). A good review of these information types andhow they perform is found in [61]. There it is also pointed out that most of these realizeuser optima only in the specific symmetric two route scenario. It was also shown thatthe paradox can be observed in the Braess network, if traffic flow is modelled by theNagel-Schreckenberg model [55].In [62] a two route model with periodic boundary conditions and dynamics similarto TASEP is studied with users with personal historical information, i.e. users that havememories of certain lengths that decide based on their own experiences. It is shown thatthis type of information realizes user optima in the network. This is similar to some ofthe results to be presented later in the present paper.Large-scale simulations of systems with information similar to that provided bysmartphones suggest that user optima are realized in these systems [37]. To ourknowledge no models based on simulations in connection with personal historicalinformation or public predictive information like that provided by smartphones,implemented in the way given in the present paper, have been studied in controlled
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Next to research based on mathematical modelsand simulations which was mainly conducted in the traffic science and trafficengineering community, in the social sciences and behavioral economics many laboratoryexperiments on route choice processes were performed with the aim of understandinghow humans decide. Typically, a road network is implemented and the traffic flow isdescribed by a mathematical model. Human subjects are then asked to perform routechoices repeatedly given various types of information. Usually real money is paid outas an incentive to perform well in the task of travel time minimization.A nice review of route choice experiments is found in [63]. Here we focus onexperiments that either directly address the Braess paradox or are closely related. In [32]and [35] scenarios with two and three unconnected links from origin to destination werestudied, respectively. In both studies, when participants relied on personal historicalinformation, i.e. their own experience of travel times of previous rounds, user optimawere reached on average with some persisting fluctuations. In [32] also the situationwith public historical information, in which participants had knowledge of travel timesalso on routes not taken, was studied. The user optimum was also reached on average.In [64] route choice decisions in the 5link version of the Braess network were studiedin a virtual experiment. Participants had to chose routes daily in the app “WeChat”.Subject to public historical information the user optimum was reached.In [31] the Braess paradox was tested directly in the laboratory. Participantsperformed route choices first in Braess’ 4link and then in the 5link network. Subjectto public historical information user optima were reached in both cases (in the 4linkon average, while fluctuations decreased in the 5link) and the paradox was realized.In a further network of different topology, Braess’ paradox was realized as well [31].Furthermore, in [65] another network exhibiting Braess-behaviour was studied. Theparadox was also realized here.
The research performed in thedifferent scientific disciplines employing the various approaches mentioned above all addto the understanding of route choice processes while all of them have their advantagesand disadvantages. In observations of real world data, typically the system cannotbe controlled as well as in the toy systems studied in simulations and laboratoryexperiments. Here, there is always a larger underlying network. The objective of the‘participants’ is not clear either. Nevertheless, on one hand, this lead to importantdiscoveries such as the minimization of travel time not necessarily being the sole goal ofnetwork users. On the other hand, these conclusions cannot be proven rigorously, sincethe objectives and motivations for the route choice of the agents are not known.The two more controlled approaches also differ in important ways: in all the workscited in the paragraph on laboratory experiments, the traffic description is limited todeterministic, macroscopic travel time functions of the individual roads in the network.
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3. Route choice algorithms
To find out whether user optima are realized in the 4link and 5link versions of a Braessnetwork of TASEPs (Fig. 2), when used by intelligently deciding agents, we implementedthe following route choice algorithm. We examined variants where the agents have accessto personal historical and public predictive information. With this we want to combinea more realistic tactical layer (i.e. a microscopic traffic modelled by coupled TASEPs)with a more realistic strategic layer with decisions based on realistic traffic information.Due to the periodic boundary conditions all M particles stay in the system andthus decisions based on personal experiences, i.e. on the memory of the particles, canbe implemented.The system is always initialized by placing M particles on the routes randomlyand assigning an initial pure strategy (to choose either route 14, 23 or 153) randomlyto each agent. The system then undergoes a relaxation procedure which is differentdepending on the type of information used. The relaxation procedures are explained inthe following paragraphs which describe the different types of implemented information.Once the system is relaxed, all agents have information about the (expected) travel timesof all routes. The information for all three routes 14, 23 and 153 is from now on called T , info , T , info and T , info , respectively. If an agent went once from j to E (i.e. afterjumping out of j ), this agent has finished one “round”.After relaxation is complete, route choice decisions can occur at three points: beforestarting a new round (when ‘sitting’ on E ) and during a round when sitting on j or j and not being able to jump to the desired target site. Before a new round, whensitting on E before jumping to j , each particle generally chooses the route i with thelowest T i, info . To make such decisions more realistic, the two following parameters areintroduced to the algorithm.(i) With probability p info the particle bases its decision on the available T i, info . Withprobability 1 − p info the particle chooses one of the two or three routes (depending ONTENTS p info as described in (i), an information-based decision willbe taken, the difference between the expected travel times on the routes ∆ T = | T , info − T , info | + | T , info − T , info | + | T , info − T , info | is calculated k . If thisdifference is below the threshold of ∆ T thres , the agent stays on the route of theprevious round. If ∆ T ≥ ∆ T thres , the agent switches to the route with the lowest T info . Thus the agents act “boundedly rational” [34].Additionally to these decisions before any new round, agents can make route choicedecisions during the rounds. These decisions work as follows. Consider an agent in the4link network sitting on j who chose to take route 23 before the round began. If thisagent cannot jump to its target site (first site of E ) since this site is occupied, (s)hemay re-decide for another route. If T , info ≥ T , info (agent chose route 23 based on arandom decision before the round), (s)he will then immediately switch to route 14. If T , info < T , info , the particle will keep trying to jump onto E for κ j , thres times theto-be-expected saved time on route 23, i.e. for κ j , thres · ( T , info − T , info ) time steps. Ifafter this waiting time a jump to its target site is not possible (s)he will switch to theother route.This algorithm is slightly more complicated in the 5link system but works in thesame sense: if an agent, due to a random decision, chose a route which does nothave the lowest expected travel time and this route is blocked, (s)he will immediatelydecide for another route. If the chosen route does have the lowest expected traveltime and the routes’ entrance is blocked, on j the agent will wait κ j , thres times theto-be-expected saved travel time before switching. In the 5link network, an analogousalgorithm operates at junction j where the parameter κ j , thres is introduced. For moredetails, see [66] where the algorithms for decision making are shown in pseudo code.In the following the different types of information that are used for the algorithm areexplained. Furthermore the relaxation procedures used in the simulations are explained. Public historical information is implemented as follows: each time any user finishes oneround (jumps out of j ), the experienced travel time is recorded. This travel time isthen made available to all agents as their T info . This information is historical since, asexplained in Sec. 2.4, the traffic state might have changed during the round. For this typeof information a short relaxation process is needed: at the beginning all agents followrandomly assigned routes. As soon as each route has been used at least once, the systemis considered to be relaxed. It has already been shown that this type of informationdoes not lead to user optima in various two route scenarios but rather to very strong k In the 4link system this expression reduces to ∆ T = | T , info − T , info | ONTENTS
Public predictive information is provided on the basis of the current positions of allagents in the network. It is implemented as an approximation of the traffic informationprovided by smartphone apps in real road networks. To provide estimates of traveltimes for all edges, the densities ρ i are determined from the current number of particleson each edge E i : ρ i = n i /L i where n i is the number of particles on edge E i . From thisdensity a travel time prediction T i, pred is calculated employing the formula T i, pred = L i − ρ i . (7)This equation is the exact stationary state expression for the travel time of a particlein a TASEP of length L i with periodic boundary conditions and density ρ i [19]. In ourcase it is only an approximation for the travel time on an edge: the edges neither haveperiodic boundary conditions nor are they (necessarily) in stationary states. It will showto produce reasonably accurate approximations (at least at low global densities) in theBraess network. From the approximated travel times of all edges the expected routetravel times, which are used as the traffic information for all agents in the route choicealgorithm, are calculated as T , info = T , pred + T , pred , (8) T , info = T , pred + T , pred , (9) T , info = T , pred + T , pred + T , pred . (10)Here further approximations are introduced: potential waiting times at the junctionsites are not considered and the system is only described in a mean field fashion sincecorrelations are neglected.At each decision point the current T i, info is calculated based on the current positionsof all particles. This means that if an agent bases its decision on the T i, info beforestarting a round and then re-decides on one of the junctions, the T i, info might alreadyhave changed at the time of the second decision.For this type of information no relaxation process is required, the agents are justplaced on random positions in the network. Personal historical information is information based on the agents’ experiences fromprevious rounds. Each agent is assigned a memory capacity of c mem rounds. For thelast c mem rounds, each agent remembers which routes it took and their corresponding ONTENTS T i, info are calculated: they are the mean values ofthe travel times of each route as experienced in the last c mem rounds. Additionally, eachagent remembers its last experienced travel times on all three routes. Like this, even ife.g. route 23 was not used in the last c mem rounds, the agent will still remember thetravel time of that route from the last usage (that lies more than c mem rounds in thepast).For this kind of information there is a two-fold relaxation process. First, each agentis placed on a random position with a random strategy. Then it tries to gather one traveltime value for each route. Once these values are obtained, the system keeps evolvinguntil each agent has experienced c mem rounds. Once each agent used each route at leastonce and has a filled its memory of capacity of c mem rounds, the system is consideredrelaxed.
4. Simulation results
We applied the algorithms described in the previous section to determine the effects ofthe route choice behavior in the Braess network with TASEP-based traffic dynamics.The edge lengths L , . . . , L are given by Eq. (4). We focussed on four differentcombinations of L , the length of the added edge, and M , the number of agents. Theseare indicated by ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ p info = 0 . , (11) ONTENTS T thres = 10 , (12) κ j , thres = κ j , thres = 0 . , (13) c mem = 30 , (14)where c mem is only needed for the case of personal historic information. As we considerthe continuous route choices in the Braess network to mirror a commuter scenario, itis reasonable to assume that in the majority of times, drivers choose their routes withthe aim of minimizing their travel times, i.e. p info = 0 .
9. Furthermore it appears to berealistic that a commuter remembers approximately one month of her last experiencedtravel times ( c mem . = 30). Assuming that drivers may not switch from their preferredroute if the expected saved time is really low, a value of ∆ T thres = 10 seems reasonable.Figs. 4–11 show the results for all four states ⋆ ⋆
4, each with public predictiveand personal historical information. All figures have parts (a) and (b). Parts (a) showhow the mean values of the travel times ¯ T i of the routes i develop with the system timewhere all times are measured in numbers of performed Monte Carlo sweeps. For detailson the simulation process the reader is referred to [66]. The values of both routes inthe 4link system and of the three routes in the corresponding 5link systems are shownfor comparison. One can thus see if the new road leads to higher or lower travel times.Additionally the travel times that are expected from the pure and mixed user optima ofthe 4link and 5link systems with externally tuned strategies are shown for comparisonby the dotted grey lines whose values are given by the τ i on the second y -axis on theright.Parts (b) of Figs. 4–11 show the two variables m ( j )l = 1 − M /M and m ( j )l = M / ( M + M ) ¶ against the system time + . Here, the M , M and M are thenumbers of cars which follow routes 14, 23 and 153 at that system time. The values ofthe two variables m ( j )l and m ( j )l capture the fractions of agents using the three differentroutes. Similar to the variables n ( j )l and n ( j )l , which we introduced in Eqs. (5),(6) inthe context of externally tuned fixed personal strategies, they describe the fraction ofparticles turning ‘left’ of junctions j and j , but in the present moment . They representthe strategies that the particles choose as a result of the route choice algorithm. Dueto the algorithm their values can change before and during the rounds.The m ( j )l and m ( j )l can be compared to the ( n ( j )l , n ( j )l ) that realize the pure useroptima for externally tuned fixed personal strategies and the ( γ, δ ) that realize the mixeduser optima for externally tuned turning probabilities. Those strategies realizing theuser optima by externally tuning the route choices are also shown by the dotted greylines whose values are given by the σ i on the second y -axis on the right of parts (b)of Figs. 4–11. This allows to determine whether the algorithm drives the system intoa user optimum. This gives an indication on how close it is to an expected pure ormixed optimum. To see if real pure user optima or real mixed user optima are realized,further analysis is needed: in a real pure user optimum no individual users would switch ¶ In the 4link system only m ( j )l is needed which reduces to m ( j )l = M /M + In [66] these two variables are called implicit turning probabilities γ imp . and δ imp . . ONTENTS m ( j )l and m ( j )l , since they only show thesums of particles following specific routes). Situations without any individual particlesswitching routes can not be obtained in our algorithm since p info <
1. Still, if thenumber of switches is low one can presume that the algorithm brings the system closeto a pure user optimum. In a mixed user optimum one would expect a higher numberof individual route switches. To test if indeed a real mixed user optimum is realized afurther statistical analysis of the behaviour of all individual particles is needed. Suchan analysis is not shown here. The interested reader is referred to [66], where it isperformed (in parts) for the state ⋆ The results for the algorithm with public predictive information are shown in Figs. 4–7for the four different states ⋆ ⋆
4. Fig. 4 shows results for state ⋆
1. One can see thatthe user optima of both the 4link and the 5link networks are realized in a stable manner.In the 4link network approximately half the agents choose route 14 and the other halfroute 23 (Fig. 4 (b)). Their mean travel times (Fig. 4 (a)) equalize at the value expectedfrom the user optima obtained in networks with externally tuned strategies. In the 5linknetwork, apart from small fluctuations, almost all agents choose route 153 (Fig. 4 (b))and this route has a lower travel time than the other two (almost unused) routes in the5link and also lower than those in the 4link system (Fig. 4 (a)). The “ E optimal, all153” state that is expected is thus realized.The results for state ⋆ M/ on average . Thenumbers of cars on the three routes (as represented through the m ( j )(5)l and m ( j )(5)l inFig. 5 (b)) oscillate around the values expected from the ( n ( j )(5)l , puo , n ( j )(5)l , puo ) and ( γ (5)muo , δ (5)muo )from the pure and mixed user optima for externally tuned strategies. Due to thesefluctuations the mean travel times of the three routes are close to each other but notequal (Fig. 5 (a)). Opposed to the expectation, the mean travel times of routes 14 and23 are actually higher than those in the 4link system. Thus, even if the user optimum ofthe 5link system is realized on average, the algorithm with public predictive informationdrives the system into a state which is more of ‘Braess nature’ in the sense that the 5linktravel times are higher than the 4link’s, opposed to the expected “ E optimal” state.Fig. 6 shows results for the case of public predictive information in state ⋆
3. As inthe two previous states, the user optimum of the 4link system is realized. As seen in
ONTENTS m e a n t r a v e l t i m e s (a) ¯ T (4)14 ¯ T (4)23 ¯ T (5)14 ¯ T (5)23 ¯ T (5)153 τ τ ≈ τ p a r t i c l e s t r a t e g i e s (b) m ( j m ( j m ( j σ σ σ Figure 4.
Results in state ⋆ T (4)max , puo = τ , T (4)max , muo = τ , T (5)max , puo = T (5)max , muo = τ . Part (b) shows that inthe 4link both routes are, as expected, used by equal amounts of agents, and inthe 5link almost all cars use route 153. The strategies realizing the user optima byexternally tuning the strategies are given for comparison with n ( j )(4)l , puo = γ (4)muo = σ , (cid:16) n ( j )(5)l , puo , n ( j )(5)l , puo (cid:17) = (cid:16) γ (5)muo , δ (5)muo (cid:17) = ( σ , σ ). The “ E optimal, all 153” state isrealized. Figure 6 (b) half of the particles use routes 14 and 23. As can be seen in Figure 6 (a) thetravel times of both routes equalize at the expected value. As detailed in Appendix B,the 5link system of state ⋆ n ( j )(5)l , puo(i / ii) , n ( j )(5)l , puo(i / ii) ) of neither of thetwo pure optima puo(i) and puo(ii) coincides with the values of the ( γ (5)muo , δ (5)muo ) of themixed user optimum muo. In the 5link system the route choice algorithm producesstrong fluctuations around the pure user optimum puo(ii) (Fig. 6 (b)), a state in whichroute 23 is not used and half the agents choose route 14 and the other half route 153.Due to the fluctuations around the user optimum the travel times of the two used routes(route 14 and 153) are close to each other but not exactly equal (Fig. 6 (a)). They are allhigher than those of the two routes in the 4link system. Thus a Braess state is realized.The almost unused route 23 has an even higher travel time, as expected.Fig. 7 shows results for state ⋆
4. In the 4link system the user optimum is realized.
ONTENTS m e a n t r a v e l t i m e s (a) τ τ ¯ T (4)14 ¯ T (4)23 ¯ T (5)14 ¯ T (5)23 ¯ T (5)153 τ ≈ τ p a r t i c l e s t r a t e g i e s (b) m ( j m ( j m ( j σ σ σ Figure 5.
Results for state ⋆ T (4)max , puo = τ , T (4)max , muo = τ , T (5)max , puo = τ , T (5)max , muo = τ . Part (b) shows that inthe 4link both routes are, as expected, used by equal amounts of agents. In the 5linksystem fluctuations around the expected user optima are observed. The strategiesrealizing the user optima by externally tuning the strategies are given for comparisonwith n ( j )(4)l , puo = γ (4)muo = σ , (cid:16) n ( j )(5)l , puo , n ( j )(5)l , puo (cid:17) = (cid:16) γ (5)muo , δ (5)muo (cid:17) = ( σ , σ ). Judging fromthe mean travel time values, a Braess state instead of the expected “ E optimal” stateis observed. Even if fluctuations around the 4link user optimum are also small in state ⋆
4, the traveltimes of both routes equalize at a slightly higher value than expected (Fig. 7 (a)). This isa consequence of jamming effects in front of j which play a larger role at higher densities.In the 5link network with externally tuned particles only a pure user optimum exists(see Appendix B). In the system with externally tuned turning probabilities fluctuatingdomain walls are observed at such high densities and thus no (short term) user optimumexists [19]. The route choice algorithm drives the system close to the pure user optimum.The resulting m ( j )(5)l and m ( j )(5)l are slightly different from the expected ( n ( j )(5)l , puo , n ( j )(5)l , puo )in the pure user optimum. Furthermore they fluctuate (Fig. 7 (b)). Thus the travel timesof the three routes are not equal. Routes 23 and 153 have higher travel times than theroutes of the 4link system (Fig. 7 (b)), which can be interpreted as a kind of Braess ONTENTS m e a n t r a v e l t i m e s (a) ¯ T (4)14 ¯ T (4)23 ¯ T (5)14 ¯ T (5)23 ¯ T (5)153 τ ≈ τ τ τ p a r t i c l e s t r a t e g i e s (b) m ( j m ( j m ( j σ σ σ σ σ Figure 6.
Results for state ⋆ T (4)max , puo = τ , T (4)max , muo = τ , T (5)max , puo(i) = T (5)max , puo(ii) = τ , T (5)max , muo = τ . Part (b) shows that in the 4link both routes are, as expected, usedby equal amounts of particles. In the 5link system strong fluctuations around oneof the three theoretically accessible user optima, i.e. around the pure user optimumpuo(ii), are observed. Route 23 is not used by many particles. The strategies realizingthe user optima by externally tuning the strategies are given for comparison with n ( j )(4)l , puo = γ (4)muo = σ , (cid:16) n ( j )(5)l , puo(i) , n ( j )(5)l , puo(i) (cid:17) = ( σ , σ ), (cid:16) n ( j )(5)l , puo(ii) , n ( j )(5)l , puo(ii) (cid:17) = ( σ , σ ), (cid:16) γ (5)muo , δ (4)muo (cid:17) = ( σ , σ ). The “Braess 1” state is realized on average. behaviour.One can conclude that the route choice based on public predictive informationtypically drives the system into user optima in the 4link systems. This is not surprisingif one remembers how the predicted travel times are calculated (see Eqs. (8)–(10)): sincethe 4link system is symmetric and the pure user optima are always given for an equaldistribution onto both routes, the algorithm which counts the numbers of particles forits travel time predictions will always realize such user optima.In the 5link networks user optima are not always realized since road 5 breaks thesymmetry of the 4link network. In the 5link network, the public predictive informationrealizes user optima at low global densities. This is the case since in this density regime ONTENTS m e a n t r a v e l t i m e s (a) τ τ ¯ T (4)14 ¯ T (4)23 ¯ T (5)14 ¯ T (5)23 ¯ T (5)153 τ x ≈ τ x p a r t i c l e s t r a t e g i e s (b) m ( j m ( j m ( j σ σ σ Figure 7.
Results for state ⋆ T (4)max , puo = τ , T (5)max , puo = τ . Part (b)shows that in the 4link both routes are, as expected, used by equal amounts of agents.In the 5link system strong fluctuations around the theoretically accessible pure useroptima are observed. The strategies realizing the user optima by externally tuning thestrategies are given for comparison with n ( j )(4)l , puo = σ , (cid:16) n ( j )(5)l , puo , n ( j )(5)l , puo (cid:17) = ( σ , σ ). the correlations between the roads do not influence the route travel times strongly.At higher densities the correlations become more important leading to traffic jamsnear junction sites. Here the travel time predictions become less accurate, leadingto fluctuations around the user optima. This can be seen in Appendix C, where theaccuracy of the predicted travel times is shown. Results for the case of personal historical information are shown in Figs. 8–11 for thefour different states ⋆ ⋆
4. In all parts (a) and (b) of these figures four vertical linesare shown. The two lines in brighter and darker grey correspond to the two relaxationtimes of the 4link and 5link systems, respectively. The line further to the left indicatesthe system time at which all agents have gathered at least one travel time experience foreach route. The line further to the right indicates the system time at which the memory
ONTENTS ⋆
1. The user optimum in the 4link system is reachedwith small remaining fluctuations around the user optimum (Fig. 8 (b)). The traveltimes of both routes in the 4link equalize at the expected values (Fig. 8 (a)). In the m e a n t r a v e l t i m e s (a) ¯ T (4)14 ¯ T (4)23 ¯ T (5)14 ¯ T (5)23 ¯ T (5)153 τ τ ≈ τ p a r t i c l e s t r a t e g i e s (b) m ( j m ( j m ( j σ σ σ Figure 8.
Results for state ⋆ T (4)max , puo = τ , T (4)max , muo = τ , T (5)max , puo = T (5)max , muo = τ . Part (b) shows that in the 4link and5link networks, the strategies develop as expected in the theoretically accessible useroptima. In the 5link, route 153 is used almost exclusively. The strategies realizingthe user optima by externally tuning the strategies are given for comparison with n ( j )(4)l , puo = γ (4)muo = σ , (cid:16) n ( j )(5)l , puo , n ( j )(5)l , puo (cid:17) = (cid:16) γ (5)muo , δ (5)muo (cid:17) = ( σ , σ ). The “ E optimal,all 153” state is realized. ⋆ ⋆ ONTENTS m e a n t r a v e l t i m e s (a) τ τ ¯ T (4)14 ¯ T (4)23 ¯ T (5)14 ¯ T (5)23 ¯ T (5)153 τ ≈ τ p a r t i c l e s t r a t e g i e s (b) m ( j m ( j m ( j σ σ σ Figure 9.
Results for state ⋆ T (4)max , puo = τ , T (4)max , muo = τ , T (5)max , puo = τ , T (5)max , muo = τ . Part (b) shows that in the 4link and 5linknetworks, the strategies develop as expected in the theoretically accessible user optima.The strategies realizing the user optima by externally tuning the strategies are given forcomparison with n ( j )(4)l , puo = γ (4)muo = σ , (cid:16) n ( j )(5)l , puo , n ( j )(5)l , puo (cid:17) = (cid:16) γ (5)muo , δ (5)muo (cid:17) = ( σ , σ ).The “ E optimal” state is realized. than the 4link’s and the system thus showed Braess behaviour (Fig. 6 (b)), here theexpected “ E optimal” state is realized.Fig. 10 shows results for state ⋆
3. As in the previous states ⋆ ⋆ M/ ⋆
4. Onecan see that in this case a long relaxation process is needed. This is due to the highglobal density: as all agents want to gather travel time experiences for all three routesin the beginning, this leads to routes getting blocked. The blockages are not permanentgridlocks since agents will re-decide their route choices if they have to wait very long
ONTENTS m e a n t r a v e l t i m e s (a) ¯ T (4)14 ¯ T (4)23 ¯ T (5)14 ¯ T (5)23 ¯ T (5)153 τ ≈ τ τ τ p a r t i c l e s t r a t e g i e s (b) m ( j m ( j m ( j σ σ σ σ σ Figure 10.
Results for state ⋆ T (4)max , puo = τ , T (4)max , muo = τ , T (5)max , puo(i) = T (5)max , puo(ii) = τ , T (5)max , muo = τ . Part (b) shows that inthe 4link the strategies develop as expected in the theoretically accessible user optima.In the 5link system, the pure user optimum puo(ii) is approached with some smallfluctuations. In the 5link system, routes 14 and 153 are used by approximately halfthe agents each, while route 23 is almost not used at all. The strategies realizingthe user optima by externally tuning the strategies are given for comparison with n ( j )(4)l , puo = γ (4)muo = σ , (cid:16) n ( j )(5)l , puo(i) , n ( j )(5)l , puo(i) (cid:17) = ( σ , σ ), (cid:16) n ( j )(5)l , puo(ii) , n ( j )(5)l , puo(ii) (cid:17) = ( σ , σ ), (cid:16) γ (5)muo , δ (4)muo (cid:17) = ( σ , σ ). The “Braess 1” state is realized. at junctions j or j . Nevertheless it takes quite long until the whole system is relaxed.Once it is relaxed it stabilises quickly. As can be seen in Fig. 11 (b), in the 4link systemthe user optimum is reached with larger fluctuations around the expected state thanfor states ⋆ ⋆
3. The effect of these larger fluctuations can also be seen in the meantravel times of the two routes in the 4link: Fig. 11 (a) shows that the travel times’ meanvalues only equalize after a relatively long time. The 5link systems’ user optimum is alsorealized on average. Fluctuations around the pure user optimum persist (Fig. 11 (b)).The mean travel times of the three routes are almost equal to the expected value of thepure user optimum, above the travel times of the 4link system. Thus, here also a Braessstate is realized.
ONTENTS m e a n t r a v e l t i m e s (a) τ τ ¯ T (4)14 ¯ T (4)23 ¯ T (5)14 ¯ T (5)23 ¯ T (5)153 τ x ≈ τ x p a r t i c l e s t r a t e g i e s (b) m ( j m ( j m ( j σ σ σ Figure 11.
Results for state ⋆ ⋆ ⋆ T (4)max , puo = τ , T (5)max , puo = τ . Part (b) shows that in the 4link and 5linkthe strategies develop as expected in the theoretically accessible user optima. Thestrategies realizing the user optima by externally tuning the strategies are given forcomparison with n ( j )(4)l , puo = σ , (cid:16) n ( j )(5)l , puo , n ( j )(5)l , puo (cid:17) = ( σ , σ ). The “Braess 1” state,which for externally tuned route choices only exists with fixed route choices, is realized. One can conclude that user optima of both the 4link and the 5link systems arerealized in all four states by the algorithm with personal historical information.
5. Conclusions
For drivers using public predictive information, user optima are realized at low globaldensities. At higher global densities the user optima are still reached on average. Thekind of predictive information that we implemented depends on the current positionsof all agents in the system. It employs an approximation formula for the travel timesbased on these positions. This kind of information is similar to that used in modernsmartphone apps which rely on crowdsourced data. The fact that the expected useroptima are realized, also in the Braess states, is a strong hint at the importance of theparadox in modern traffic networks. As already proposed by previous observations of
ONTENTS
Acknowledgements
Financial support by Deutsche Forschungsgemeinschaft (DFG), Germany under grantSCHA 636/8-2 is gratefully acknowledged. Also support by the Bonn-Cologne GraduateSchool of Physics and Astronomy (BCGS) is acknowledged. Monte Carlo simulationswere carried out on the CHEOPS (Cologne High Efficiency Operating Platform forScience) cluster of the RRZK (University of Cologne).
Appendix A. Mixed user optima in Braess’ original example
As described in Sec. 2.2, the Braess paradox is also observed in Braess’ original modelif users choose their routes according to mixed strategies. Let p , p and p bethe probabilities with which all users choose routes 14, 23 and 153, respectively. Theprobabilities are subject to p + p = 1 or p + p + p = 1 for the 4link and 5linksystems, respectively.In the 4link system, for mixed strategies (ms) the expectation values, denoted by h T (4) i, ms i , of the travel times on the routes 14 and 23 are h T (4)14 , ms i = 50 + (1 + p · ( N − ·
11 (A.1) h T (4)23 , ms i = 50 + (1 + p · ( N − ·
11 (A.2)for each car.For N = 6 a mixed user optimum state (muo) is found for p = p = 1 / h T (4)14 , muo i = h T (4)23 , muo i = 88 . h T (5)14 , ms i = (1 + ( p + p )( N − ·
10 + 50 + 1 + p ( N −
1) (A.3) h T (5)23 , ms i = (1 + ( p + p )( N − ·
10 + 50 + 1 + p ( N −
1) (A.4)
ONTENTS h T (5)153 , ms i = (2 + ( p + p + 2 p )( N − ·
10 + 10 + 1 + p ( N − . (A.5)Here a mixed user optimum is given for p = p = 5 /
13 and p = 3 /
13 with traveltime values h T (5)14 , muo i = h T (5)23 , muo i = h T (5)153 , muo i = 93 . N = 6given by distributing the users equally on the three routes. The mixed equilibrium isnot achieved by all users choosing the routes with equal probability! Appendix B. The test states
The parameter sets for which the algorithm was tested are marked in Figs. 3 (a) and (b).The corresponding travel time values that are expected in the existing pure and mixeduser optima (as found previously in [19] and [20]) are given in the following. See [19, 20]as well for the naming scheme for the different states.All these states are boundedly rational user optima [34]: in [19, 20] we found theuser optima of the systems by tuning the decisions of the particles externally (eitherthe N , N , N for fixed personal strategies or the γ, δ for turning probabilities) andstates for which ∆ T = | T − T | + | T − T | + | T − T | <
100 were considered tobe user optima. This means that the travel times on the three routes are not necessarilyexactly equal but are reasonably close to each other and we thus consider the states tobe user optima. Since the travel times are not necessarily exactly equal, we give herethe maximum travel time observed in those states as the reference time, which is usedas a comparison for the system with intelligently deciding particles. state ⋆ : has the parameters L = 97 and M = 156 which correspond to ˆ L / ˆ L =0 . ρ (4)global ≈ . ρ (5)global = 0 .
12. For externally tuned particles this is an “ E optimal,all 153” state both for fixed personal strategies and for turning probabilities.In the 4link system • a pure user optimum is given for N (4)14 , puo = N (4)23 , puo = 78 which corresponds to n ( j )(4)l , puo = 0 . T (4)max , puo ≈
692 and ∆ T (4)puo → • a mixed user optimum is given for γ (4)muo = 0 . T (4)max , muo ≈
693 and ∆ T (4)muo → • a pure user optimum is given for N (5)14 , puo = N (5)23 , puo = 0 and N (5)153 , puo = 156 whichcorresponds to n ( j )(5)l , puo = 1 . n ( j )(5)l , puo = 0 . T (5)max , puo ≈
615 and ∆ T (5)puo = 0. • a mixed user optimum is given for γ (5)muo = 1 . δ (5)muo = 0 . T (5)max , muo ≈ T (5)muo = 0. ONTENTS state ⋆ : has the parameters L = 339 and M = 154 which correspond to ˆ L / ˆ L =0 . ρ (4)global ≈ . ρ (5)global = 0 .
1. For externally tuned particles this is an “ E optimal”state both for fixed personal strategies and for turning probabilities.In the 4link system • a pure user optimum is given for N (4)14 , puo = N (4)23 , puo = 77 which corresponds to n ( j )(4)l , puo = 0 . T (4)max , puo ≈
691 and ∆ T (4)puo → • a mixed user optimum is given for γ (4)muo = 0 . T (4)max , muo ≈
692 and ∆ T (4)muo → • a pure user optimum is given for N (5)14 , puo = 36, N (5)23 , puo = 30 and N (5)153 , puo = 88 whichcorresponds to n ( j )(5)l , puo ≈ . n ( j )(5)l , puo ≈ . T (5)max , puo ≈
670 and ∆ T (5)puo ≈ • a mixed user optimum is given for γ (5)muo = 0 . δ (5)muo = 0 . T (5)max , muo ≈ T (5)muo ≈ state ⋆ : has the parameters L = 37 and M = 248 which correspond to ˆ L / ˆ L =0 . ρ (4)global ≈ . ρ (5)global = 0 .
2. For externally tuned particles this is a “Braess 1” stateboth for fixed personal strategies and for turning probabilities.In the 4link system • a pure user optimum is given for N (4)14 , puo = N (4)23 , puo = 124 which corresponds to n ( j )(4)l , puo = 0 . T (4)max , puo ≈
764 and ∆ T (4)puo → • a mixed user optimum is given for γ (4)muo = 0 . T (4)max , muo ≈
763 and ∆ T (4)muo → • two pure user optimum exist for(i) N (5)14 , puo(i) = 0, N (5)23 , puo(i) = 124 and N (5)153 , puo(i) = 124 which corresponds to n ( j )(5)l , puo(i) = 0 . n ( j )(5)l , puo(i) = 0 . T (5)max , puo(i) ≈
978 and ∆ T (5)puo(i) = 10.(ii) N (5)14 , puo(ii) = 124, N (5)23 , puo(ii) = 0 and N (5)153 , puo(ii) = 124 which corresponds to n ( j )(5)l , puo(ii) = 1 . n ( j )(5)l , puo(ii) = 0 . T (5)max , puo(ii) ≈
978 and ∆ T (5)puo(ii) = 11. • a mixed user optimum is given for γ (5)muo = 0 .
87 and δ (5)muo = 0 . T (5)max , muo ≈ T (5)muo ≈ state ⋆ : has the parameters L = 218 and M = 712 which correspond to ˆ L / ˆ L =0 . ρ (4)global ≈ . ρ (5)global = 0 .
5. For externally tuned particles this is an “Braess 1” statefor fixed personal strategies. For turning probabilities no user optima could be foundfor these parameters.In the 4link system • a pure user optimum is given for N (4)14 , puo = N (4)23 , puo = 356 which corresponds to n ( j )(4)l , puo = 0 . T (4)max , puo ≈ T (4)puo → • there is no short term mixed user optimum due to fluctuating domain walls ONTENTS • a pure user optimum is given for N (5)14 , puo = 357, N (5)23 , puo = 178 and N (5)153 , puo = 177which corresponds to n ( j )(5)l , puo = 0 .
75 and n ( j )(5)l , puo ≈ .
67 with T (5)max , puo ≈ T (5)muo ≈ • there is no mixed user optimum due to fluctuating domain walls. Appendix C. Accuracy of the travel time predictions used for publicpredictive information
In this section of the Appendix we discuss the accuracy of the travel time predictions thatare provided in the case of public predictive information, i.e. we discuss how accuratethe T i, info as given by Eqs. (8)–(10) are. As can be seen in Figs. 4 to 7 and as discussedin Section 4, only in state ⋆ ⋆ ⋆ T i, info compared to the actually measured T i are given: the T i, info ofall routes i , predicted before starting a new round are saved for each particle. Once anagent finishes a round, the travel time that the agent actually experienced on its chosenroute i ( T i ) is measured. From this the relative error is computed as ( T i, info − T i ) /T i . r e l a t i v ee rr o r o f t h e T i , i n f o (a) T , info − T T T , info − T T r e l a t i v ee rr o r o f t h e T i , i n f o (b) T , info − T T T , info − T T T , info − T T Figure C1.
Accuracy of the public predictive travel time predictions T i, info for state ⋆ In Fig. C1 it can be seen that in state ⋆ ONTENTS − . . . r e l a t i v ee rr o r o f t h e T i , i n f o (a) T , info − T T T , info − T T − . . . r e l a t i v ee rr o r o f t h e T i , i n f o (b) T , info − T T T , info − T T T , info − T T Figure C2.
Accuracy of the public predictive travel time predictions T i, info for state ⋆ As can be seen in Fig. 5, in state ⋆ ⋆
1. In the 5link system the user optimum is only reached on average(Fig. 5). In Fig. C2 (b) it can be seen that the predicted travel times are off for up toalmost ± ⋆
3, the predictions in the 4link system are still fairly accurate, as seen inFig. C3 (a). Accordingly also the user optimum in the 4link is realized well (Fig. 6). Inthe 5link system we can see, that the predicted travel times are of by a large degree,especially for route 153, they are at times more than 100% off. In Fig. C3 (b) one cansee, that at times when the travel time on route 153 is predicted too high, the traveltime on route 14 is predicted too low and vice versa. This could be the cause for thefluctuations around the user optimum that can be seen in Fig. 6.In state ⋆ ρ (4)global ≈ .
59 and ρ (5)global = 0 . ⋆ ONTENTS r e l a t i v ee rr o r o f t h e T i , i n f o (a) T , info − T T T , info − T T − r e l a t i v ee rr o r o f t h e T i , i n f o (b) T , info − T T T , info − T T T , info − T T Figure C3.
Accuracy of the public predictive travel time predictions T i, info for state ⋆ r e l a t i v ee rr o r o f t h e T i , i n f o (a) T , info − T T T , info − T T r e l a t i v ee rr o r o f t h e T i , i n f o (b) T , info − T T T , info − T T T , info − T T Figure C4.
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