Network Architecture and Traffic Flows: Experiments on the Pigou-Knight-Downs and Braess Paradoxes

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1 Network Archtecture and Traffc Flows: Experments on the Pgou-Knght-Downs and Braess Paradoxes by John Morgan a, Henrk Orzen b and Martn Sefton c July 2007 Abstract Ths paper presents theory and experments to nvestgate how network archtecture nfluences route-choce behavor. We consder changes to networks that, theoretcally, exhbt the Pgou- Knght-Downs and Braess Paradoxes. We show that these paradoxes are specfc examples of more general classes of network change propertes that we term the least congestble route and sze prncples, respectvely. We fnd that techncal mprovements to networks nduce adjustments n traffc flows. In the case of network changes based on the Pgou-Knght-Downs Paradox, these adjustments undermne short-term payoff mprovements. In the case of network changes based on the Braess Paradox, these adjustments renforce the counter-ntutve, but theoretcally predcted, effect of reducng payoffs to network users. Although aggregate traffc flows are close to equlbrum levels, we see some systematc devatons from equlbrum. We show that the qualtatve features of these dscrepances can be accounted for by a smple renforcement learnng model. Acknowledgements We are grateful for the extremely helpful comments of three anonymous referees as well as the Acton Edtor. We also thank semnar partcpants at the Unversty of Lecester, the Insttute for Transport Studes, Unversty of Leeds, and at the Amsterdam 2004 Internatonal Meetng of the Economc Scence Assocaton for ther comments. a. Haas School of Busness and Department of Economcs, Unversty of Calforna, Berkeley, CA e- mal: morgan@haas.berkeley.edu b. School of Economcs, Unversty of Nottngham, Nottngham, NG7 2RD, Unted Kngdom. e-mal: henrk.orzen@nottngham.ac.uk c. School of Economcs, Unversty of Nottngham, Nottngham, NG7 2RD, Unted Kngdom. e-mal: martn.sefton@nottngham.ac.uk 1

2 1 Introducton Economc agents are often presented wth stuatons where they must decde how to get nformaton, materals, or smply themselves from pont A to pont B. In such stuatons the agents often nteract on congested networks, and an ndvdual s travel cost wll typcally depend on the decsons made by other network users. The externaltes that network users mpose on each other have mportant consequences for network performance and desgn. One consequence s that traffc flows may not be socally effcent, as each user attempts to mnmze hs own travel cost wthout takng nto account the effect of hs actons on others. A second consequence, of practcal mportance to provders and managers of networks, s that techncal mprovements to a network may have unntended consequences, as they can nduce behavoral changes that, whle sensble for each ndvdual, negatvely affect overall network performance. Ths paper nvestgates how network archtecture nfluences route-choce behavor by comparng outcomes across dfferent networks. In the theory secton, we present two prncples for planners. In what we term the least congestble route prncple we show, employng the noton of Wardrop equlbrum (Wardrop, 1952, Beckmann, McGure and Wnsten, 1956), that the planner most effectvely reduces travel tme by frst mprovng the route that s least senstve to network congeston. Ths prncple takes an extreme form when one of the routes along the network s non-congestble: Imagne a populaton of ndvduals who can choose between two routes. One route takes one hour, regardless of the number of people usng t, whle the other route s congestble, so that the travel tme ncreases wth the number of users. In equlbrum, the number of ndvduals usng the congestble route wll ensure that the travel tme s one hour. Now magne that the congestble route s mproved. Ths wll encourage people to swtch from the non-congestble to the congestble route untl, once agan, the travel tmes on both routes are equalzed the travel tme remans at one hour even after the mprovement. In ths case, the benefts from mprovng the route are completely dsspated. 1 Ths observaton s the essence of the Pgou-Knght-Downs Paradox (Downs, 1962), whch has been descrbed as beng so enshrned n transportaton plannng that t s often called the fundamental law of traffc congeston (Arnott and Small, 1994). 1 Ths wll be true so long as both optons are used n equlbrum. If the congestble route takes less than an hour when the entre populaton uses t, then further mprovements wll reduce travel tmes. 2

3 Another example of how network archtecture nfluences route choce behavor n counterntutve ways s the Braess Paradox (Braess, 1968). A verson of ths s shown n Fgure 1. Suppose 60 ndvduals want to get from O (orgn) to D (destnaton), and mnmze the tme spent dong so. The left panel shows the two possble routes and the travel tmes along each lnk where, for example, n LD represents the number of travelers usng the L-D lnk. In equlbrum, half of the populaton takes the route O-H-D, half takes the route O-L-D, and each ndvdual spends 90 mnutes travelng. If a lnk s added between H and L that takes no tme at all (rght panel), ths wll nduce all ndvduals to use the route O H L-D. As a result, travel tmes ncrease to 120 mnutes strategc effects n response to network mprovements leave all travelers worse off. H Fgure 1: The Braess Paradox H n OH 60 n OH 60 O D O 0 D 60 n LD L 60 n LD L If, however, the populaton, N, grows suffcently large (.e. N >120) the effcency of the addtonal lnk overcomes the adverse strategc effects, because, whle each user wll stll take 120 mnutes wth the lnk, her travel tme wll be 60 + N 2 mnutes wthout. Ths reversal llustrates the sze prncple the effcency gans from addng network lnks always outwegh losses from user externaltes once the number of network users s suffcently large. Whle the two prncples and ther accompanyng paradoxes reflect theoretcal results stemmng from equlbrum behavor, ther behavoral plausblty, and thus ther applcablty as plannng tools, s far from clear. Alternatves to equlbrum models, based on adaptve models of route choce behavor often mply sgnfcant dfferences between choces and equlbrum, even n the long run. For example, Horowtz (1984) studes several models of adaptve route-choce behavor on smple two-lnk networks and fnds that traffc flows sometmes converge to values 3

4 that dffer substantally from equlbrum. Moreover, Bazzan and Klügl (2003) study a network where, n theory, the Braess paradox apples, and report smulatons n whch artfcal agents employng a smple heurstc algorthm to make route choces are able to avod the adverse effects of the paradox. Examnng the behavoral relevance of equlbrum n network envronments s notorously dffcult. As Horowtz (1984) notes, the substantal dscrepances that are often found between measured traffc volumes and those computed wth equlbraton models could reflect model msspecfcaton rather than falure of equlbrum. Moreover, because of the very nterconnectedness of traffc, data, and transportaton networks, there s scant hope of fndng natural experments to cleanly and convncngly test equlbrum predctons. As a result, a number of researchers have recently turned to controlled laboratory experments to examne route-choce behavor. In these experments subjects make choces between routes and receve fnancal rewards that are lnked to the routes chosen. 2 Along these same lnes, we study the behavoral relevance of the paradoxes dentfed above by varyng the network archtecture usng controlled laboratory experments. Unlke many exstng studes whch use a full nformaton desgn, we do not nform subjects of the travel cost functons of all lnks n the network. Rather, as s commonplace n transportaton networks, our subjects get feedback about the travel cost of journeys they undertake as well as the number of other travelers sharng each lnk n the journey. We ask whether stable patterns of traffc flow emerge as ndvduals nteract repeatedly on a network, whether traffc flows adjust n response to network changes and, f so, how these relate to equlbrum and effcent traffc flows and adjustments. We fnd that, even n ths lmted nformaton settng, subjects do adjust to changes n network structure, alterng travel patterns n an attempt to reduce travel tmes. In our sessons examnng the Pgou-Knght-Downs Paradox we mprove a congestble route and observe a sgnfcant ncrease n the amount of traffc usng that route. In our sessons examnng the Braess Paradox we fnd that addng a new road results n a shft of travelers towards the congestble lnks. These results thus renforce a theoretcal pont about network desgn: When 2 We wll dscuss some of these expermental studes n the next secton. A number of other expermental studes from the transportaton lterature use a rather dfferent methodology n whch subjects respond to descrptons of hypothetcal route choce scenaros, but are not pad accordng to ther decsons. See, for example, Avner and Prashker (2006), Ida, Akyama and Uchda (1992), as well as the experments revewed n Mahmassan and Srnvasan (2004). 4

5 consderng the consequences of modfyng a network, one should pay careful attenton to nduced changes n the behavor of network users. For the Pgou-Knght-Downs networks, both equlbrum and effcency ental a shft n traffc towards the mproved congestble route. However, the magntude of actual adjustments s closer to equlbrum than to effcent route utlzaton. Improvements to the congestble route on the Pgou-Knght-Downs network lead to a small reducton n travel tmes, and an even smaller reducton when we focus on travel choces n later rounds after subjects had tme to gan experence wth the network. For the Braess networks, equlbrum entals a shft of traffc n the drecton we observe, whereas effcency entals a shft n the opposte drecton. The observed shft n route choces has the effect of ncreasng travel tmes, and the ncrease n travel tmes s even greater when we focus on choces n later rounds. Thus, the Braess Paradox s observed n our laboratory settng. Although aggregate traffc flows are close to equlbrum levels, we see some systematc devatons from equlbrum. One mportant devaton we observe s persstent varablty n route choces. In general, varablty has the effect of ncreasng average travel tmes. For the unmproved Pgou-Knght-Downs network n partcular, varablty has crtcal consequences for network performance. Whle we see the congestble road under-utlzed relatve to equlbrum, whch would result n lower travel tmes all else equal, the adverse effect of varablty n traffc flows outweghs the mean effect, and travel tmes are, n fact, even hgher than n equlbrum. Usng a smple renforcement learnng model we are able to capture the qualtatve features of these dscrepances. In partcular, the learnng model results n heterogeneous behavor among players that generates smlar patterns of varablty n route choce behavor to those observed n the experment. Furthermore, we also observe a systematc tendency toward a more even dstrbuton of traffc across routes than s consstent wth equlbrum, and ths too s an mplcaton of the learnng model. Related Expermental Lterature Selten, Chmura, Ptz, Kube and Schreckenberg (2007) conduct a route-choce experment where subjects repeatedly choose between two congestble routes on a fxed network. The nformaton structure of ther experment s smlar to ours: Subjects are not nformed of the travel cost functons assocated wth each route. They fnd that although aggregate choces are close to equlbrum proportons, traffc flows fluctuate around equlbrum rather than convergng to t. 5

6 Provdng subjects wth post-round nformaton on the travel cost of the non-chosen route dampens, but does not elmnate, traffc flow fluctuatons. Helbng (2004) replcates and extends the Selten et al. setup. He fnds that by provdng subjects wth addtonal feedback ether n the form of potental payoffs from alternatve routes or ndvdual recommendatons as to route choces further reduces the volatlty of traffc flows. Unlke our study, nether of these studes vares network archtecture. 3 As far as we are aware, the Pgou-Knght-Downs Paradox has not been prevously studed n the lab. However, the structure of the underlyng game somewhat resembles a market entry game where the profts of entrants declne wth the number of entrants. In the frst expermental study of market entry games, Kahneman (1988) found that the number of entrants was close to the number predcted by theory. Camerer (2003) provdes a useful revew of subsequent experments, and notes slght tendences toward excess entry when equlbrum predcts few entrants, and under-entry when equlbrum predcts many entrants. These results are broadly consstent wth ours; n our experment aggregate choces are close to equlbrum, and where we see devatons from equlbrum, they tend to nvolve a more even allocaton of subjects to the two routes. The Braess Paradox has recently been studed n experments by Rapoport, Kugler, Dugar and Gsches (2005, forthcomng). There are some procedural dfferences between these experments and ours. Rapoport et al. use a full nformaton desgn, larger groups, and dfferent network parameters. 4 Ther results are smlar to ours even though the nformaton condtons of subjects are qute dfferent. Lke us, they fnd that subjects route choces change n the drecton predcted by equlbrum when changes n the network occur, that the paradox s observed n the laboratory, that mean traffc flows are close to equlbrum (n partcular n the long run), and that behavor s not affected by the order n whch the baselne and the mproved networks are presented. Smlar to our fndngs, they also observe persstent fluctuatons n traffc flows and route swtchng, as well as consderable heterogenety n behavor across ndvduals. The remander of the paper proceeds as follows. In secton 2 we present the background theory, and ntroduce the least congestble route and sze prncples. Our expermental desgn and 3 Schneder and Wemann (2004) and Zegelmeyer et al. (forthcomng) also report experments on traffc networks. However, ther focus s on bottleneck models where subjects choose departure tmes. 4 Rapoport, Mak and Zwck (2006) also use a full nformaton desgn to study the effects of varaton n the number of travelers on an augmented Braess network. 6

7 procedures are descrbed n secton 3, and the results n secton 4. In secton 5 we explore alternatve explanatons of the varablty n traffc flows we fnd n the data. Secton 6 concludes. 2 Theory Consder a network where N ndvdual travelers wsh to get from some orgn, O, to a destnaton, D, to whch there are k possble routes. All travelers face the same decson and make ther choces smultaneously. Suppose further that the reward from reachng the destnaton s suffcent that all ndvduals fnd t n ther nterest to travel regardless of the actons of others. A traveler s problem, of course, s to mnmze the travel tme n gettng to D. Let the travel tme along route be gven by T = α + βn where α, β > 0 are parameters of the model and n denotes the number of travelers usng route. The free-flow parameter α should be nterpreted as the fxed tme factor to traverse a road n the absence of any congeston. The congeston parameter β s a measure of the margnal effect on travel tmes of congeston. Routes wth hgh values of β are more mpacted by an addtonal traveler than are low β roads. A standard framework for analyzng equlbrum traffc flows n network problems, and the one we adopt n ths secton, s the noton of Wardrop Equlbrum (Wardrop, 1952; Beckman et al., 1956). Such an equlbrum conssts of a vector of the number of travelers usng each of the k routes ( n, n, 1 2 K, nk ) such that the travel tmes on any used route (.e. where n > 0 ) are less than or equal to the travel tme on any other route. When the vector ( n 1, n2, K, nk ) s nteger valued, then a Wardrop equlbrum s also a Nash equlbrum. When t s not, a Wardrop equlbrum corresponds to the noton of quas-equlbrum as defned n Ellson, Fudenberg and Möbus (2004). For large N, Haure and Marcotte (1985) show that Nash equlbrum converges to Wardrop equlbrum. We wll restrct attenton to the case where all routes are used n equlbrum. Denotng the least congestble road as route 1 (.e. β1 β for all ), the followng route vablty condtons are suffcent for all routes to be used n equlbrum: α < α 1 for all 1, and N α α k 1 > = 1 β. 7

8 The frst set of condtons ensures that no route s weakly domnated by route 1 (.e. for no route s t the case that β1 β and α1 α ). The latter condton merely ensures that there are a suffcent number of travelers so that the least congestble route s used. A Wardrop equlbrum s a soluton to the system of equatons T = α + β n for all k = 1 n = N. Rewrtng the representatve equaton above for all, we obtan n = ( T α ) β. Summng over all and solvng for T then yelds the equlbrum travel tme of The equlbrum number of travelers on route s Route Improvements T n k k α 1 = + N β β = 1 = 1 T =. (1) α β. Suppose that there s fundng for a small road mprovement on one of the k routes (where an mprovement means a reducton n ether the free-flow parameter or the congeston parameter). Whch route should be mproved? A common ntuton suggests that the optmal strategy s to mprove the road that s most heavly used. Underlyng ths ntuton s the followng argument: Snce total travel tme s k nt = 1 α j or, f traffc flows do not adjust n response to changes n the network (.e. reductons n β j ) then the margnal reducton n travel tme s greatest for the most traveled route snce k k 2 ( nt 1 ) α j n = jand ( nt 1 ) β j= n = j =. Of course, mssng from ths ntuton s the dea that travelers respond (at all) to changes n the network. A slghtly more sophstcated approach mght be to mprove the road where the margnal drver s havng the greatest mpact on travel tmes. In effect, ths margnal drver s determnng the equlbrum travel tme for all drvers (snce f he or she had a better route, t would contradct the dea that the gven confguraton of route choces comprses an equlbrum). Thus, t would seem that reducng the margnal mpact of ths drver would be helpful. Much lke the (2) 8

9 frst approach, ths ntuton also fals to account for equlbrum readjustment of travel patterns followng a change n network structure. Surprsngly, ths approach s the worst possble use of the funds for road mprovement. Instead, we show that the least congestble route prncple s optmal: Least congestble route prncple: Improvements should be made on the route least senstve to congeston. To formalze the least congestble route prncple, we consder the optmal route n whch to make an nfntesmal change n the α j parameter. The equlbrum effect on travel tmes of such a change s T α j 1 = (3) k 1 β β. j = 1 Notce that the magntude of ths change s nversely related to β ; hence, the smaller s greater s the reducton n equlbrum travel tmes of the mprovement to route j. 5 j β j the When β 1 0, the Pgou-Knght-Downs Paradox emerges as a specal case of the least congestble route prncple. From equaton (1), usng L Hosptal s rule, we obtan T = α1 and t follows from equaton (3) that mprovements to routes other than to route 1 produce no reducton n travel tmes whatsoever. Improvng route 1, on the other hand, s optmal and leads to a one for one reducton n equlbrum travel tme. A modfcaton of the least congestble route prncple s requred for techncal mprovements emboded n changes to the congeston parameter β j. Dfferentatng equlbrum travel tmes wth respect to β j gves k k 1 α α j 1 2 N + 2 T β j β 1 β j β = = 1 n j = = k 2 k β j 1 1 β j β β. 1 1 = = (4) 5 Whle we have shown the mprovement prncple for networks wth lnear congeston effects, the result holds more generally snce even for networks wth nonlnear congeston, the comparatve statc property under consderaton wll represent a lnearzaton. 9

10 Note that the least congestble route prncple does not hold exactly for ths type of change because ths expresson nvolves both the volume of traffc usng a route and the congeston parameter for that route. However, for any two equally traveled routes, the better route to mprove s the one that s less congestble. More Complcated Network Structures Next, we slghtly complcate the network structure. Suppose there are k lnks, each of whch leads from the orgn O to a mdpont M, = 1, K, k. As before, the tme cost for lnk s lnear: T = α + β n. Furthermore, suppose there are another k lnks, one from each mdpont M to OM the destnaton, D. The tme cost for one of these lnks, lnk j, s gven by T = γ + δ n. MD j j j j Fnally, suppose that one of the lnks from the orgn to a mdpont and one of the lnks from a mdpont to the destnaton are non-congestble. Wthout loss of generalty, we let these be the frst among the k lnks from the orgn and the last among the k lnks to the destnaton. 6 Fgure 2 shows a representatve network wth ths structure. Fgure 2: A network wth k mdponts M 1 α 1 M 2 γ + δ n MD O α + β n OM γ + δ n MD D α + β n OM k k k γ k M k 6 Formally, we study the lmt as β, 0. 1 δ k 10

11 Notce that the orgnal network s nested as a specal case of ths network. Agan, we restrct attenton to parameter values where all routes are used. When there are no further connectons n the network, t follows from equaton (1) that the equlbrum travel tme s. 1 T α + γ = N + β + δ β + δ Now consder how the performance of the network changes when we add costless lnks connectng node M wth M j for all, j. Wth these lnks, any traveler can costlessly swtch from any node M to any other node M j. The key queston s what the addton of such lnks does to the performance of the network. A representatve network wth ths structure s shown n Fgure 3. Fgure 3: Costless swtchng between the k mdponts M 1 α 1 M 2 γ + δ n MD O α + β n OM γ + δ n MD D α + β n OM k k k γ k M k We offer the followng: The Sze Prncple: Addng costless lnks reduces travel tme f there are suffcently many travelers. The mpled possblty that the addton of costless lnks could adversely affect outcomes stems from the externalty that mdpont swtchng by one traveler mght mpose on other travelers. At the same tme, the prospect of free mdpont swtchng effectvely ncreases the 11

12 number of possble paths n the network and ths mproves the prospect of fndng a more effcent route. The sze prncple suggests that, as the number of travelers grows large, the effcency effect always domnates the externalty effect. To see the sze prncple formally, notce that the equlbrum n the network wth costless mdpont connectons essentally decomposes nto two smple network problems. In equlbrum, the travel tmes between O and all nodes M must be equal, and smlarly the travel tme between all nodes M and D must be equal. Ths mmedately mples that the equlbrum travel tme on the mproved network s T = α1 + γ k. To determne the effect of the new lnks on equlbrum travel tmes we must compare T to T. For the mprovements to actually reduce travel tme requres that N > k ( α1 α) + ( γk γ). (5) β 1 + δ = Equaton (5) mples that the effcency effect domnates the externalty effect when N s suffcently large. When N s small and the route vablty condton holds for the mproved network, the externalty effect domnates the Braess Paradox s a smple llustraton of ths. Of course, when N becomes very small, the route vablty condton for the mproved network wll no longer be satsfed. In partcular, for N suffcently small, the noncongestble lnks wll no longer be used n equlbrum n the mproved network, and the effcency effect wll agan domnate. Returnng to the example n the ntroducton, notce that, f the number of travelers was reduced from 60 to 20, then the Braess mprovement would reduce travel costs from 70 mnutes to 40 mnutes. The reason s that, when there are suffcently few travelers, the noncongestble lnks cease to be vable n the mproved network. 7 Prevous theoretcal research has also hghlghted versons of the sze prncple. Specfcally, Pas and Prncpo (1997) as well as Penchna (1997) provde versons of the prncple for the case of an ant-symmetrc network where k = 2. 8 The sze prncple s also mplct n the complete characterzaton of the k = 2 case offered by Frank (1981). 9 7 We thank the acton edtor for pontng ths out. 8 In our notaton, an ant-symmetrc network s one wth the restrcton that, for all, α = γ k + and β = δ 1 k Kameda, Altman, Kozawa and Hosokawa (2000) study Braess-lke Paradox results occurrng n a Nash equlbrum of a k = 2 network for parameter values where the Wardrop equlbrum does not yeld Braess outcomes. 12

13 3 Experment Whle the prevous secton derved propertes of varyng network archtectures usng Wardrop equlbrum and gnorng nteger constrants, mplementng networks n a laboratory settng necesstates selectng partcular parameter values and dealng wth consequent nteger ssues assocated wth the number of network users. Specfcally, n all the networks we study there are eght network users and, as we show below, we select parameter values such that the assocated Wardrop equlbrum number of travelers on each route s an nteger. 10 Pgou-Knght-Downs (PKD) Treatments Fgure 4 dsplays the frst two networks used n our experment. For the road between L and D the cost s nne tmes the number of travelers usng that road. For the other roads the costs are fxed. We wll refer to the route O-H-D as the hgh road and the route O-L-D as the low road. In the PKD Baselne network (left panel) effcency, n the sense of mnmzng average travel tme, requres that two travelers use the low road and sx use the hgh road. The average travel tme wth ths confguraton s The PKD Improved network (rght panel) has a lower cost of gettng from O to L, and allows a planner to reduce the travel tme of network users. In an effcent profle of choces three travelers use the low road and fve use the hgh road, gvng an average travel tme of Thus, f travelers use the network effcently, the mprovement wll reduce travel tmes. However, these effcent outcomes do not consttute equlbra, as n each case there s an ncentve for hgh road users to swtch to the low road. Equlbrum travel tme s determned by travel tme on the hgh road: T = 57. When four travelers take the hgh road and four the low road travel tme s equalzed across routes n the PKD Baselne network If nstead of Wardrop equlbrum, we use the noton of Nash equlbrum, then, n addton to pure strategy Nash equlbra, there are also mxed-strategy Nash equlbrum solutons for our networks. We dscuss these n detal n Secton There s an addtonal pure strategy Nash equlbrum where fve travelers take the hgh road and travel tmes on the two routes are not equalzed. In ths equlbrum, even though low road travelers enjoy faster travel tmes, hgh road travelers are ndfferent between swtchng or not. The addtonal equlbrum s an nevtable consequence of choosng parameters so that the Wardrop equlbrum where travel costs are equalzed nvolves an nteger number of users on any partcular route. Note that the addtonal equlbrum does not survve a smple refnement. If there s an (arbtrarly small) ε chance that a subject does not travel, then the unque pure strategy Nash equlbrum corresponds to the Wardrop equlbrum. 13

14 Fgure 4: PKD networks mplemented n the experment H H O D O D 21 9n LD L 3 9n LD L Turnng to the PKD Improved network, equlbrum travel tme s agan T = 57. When two travelers take the hgh road and sx take the low road travel tme s equalzed across the two routes. Ths swng n the number of travelers towards the low road drves the Pgou-Knght- Downs Paradox the benefts from the mprovement are completely dsspated by a re-allocaton of traffc from the hgh to the low road. We summarze the theoretcal propertes of the PKD networks n Table 1. Table 1: Effcent and equlbrum traffc on PKD networks Effcency Equlbrum Baselne Improved Travelers on low road 2 3 Average travel tme Travelers on low road 4 6 Average travel tme Braess Treatments Fgure 5 llustrates our mplementaton of the Braess networks. In the Braess Baselne (left panel) effcency requres that 3 travelers take the low road and 5 take the hgh road, gvng an average travel tme of In contrast, n equlbrum four travelers take the hgh road and four take the low road, gvng each traveler a travel tme of

15 Fgure 5: Braess networks mplemented n the experment H H 3n OH 45 3n OH 45 O D O D 21 9n LD L 21 9n LD L Now suppose a costless lnk s added between L and H (rght panel). By permttng traffc to be dverted between the hgh and low roads, an mproved allocaton of traffc on the network s possble and the average travel tme decreases (slghtly) when traffc flows effcently. The costless lnk n effect makes the network archtecture equvalent to two PKD networks one from O to the mdpont and one from the mdpont to D. Equlbrum on the left part of the network mples that the travel tme from O to the mdpont wll be 21 and conssts of 7 travelers leavng on the hgh road. Smlarly, the travel tme from the mdpont to D wll be 45 and conssts of only 3 travelers arrvng on the hgh road. Thus, the overall travel tme on the network ncreases to 66 per traveler. Ths s the essence of the Braess Paradox the ablty of each traveler to freely swtch between the congestble routes worsens overall congeston on the network. We summarze the theoretcal propertes of the Braess networks n Table 2. Table 2: Effcent and equlbrum traffc on Braess networks Effcency Equlbrum Baselne Improved Travelers leavng on low road 3 4 or 5 Travelers arrvng on low road 3 2 or 3 Average travel tme Travelers leavng on low road 4 1 Travelers arrvng on low road 4 5 Average travel tme

16 Procedures The experment was conducted at the Unversty of Nottngham n Sprng Subjects were recruted from a unversty-wde subject pool comprsed of undergraduates who had ndcated a wllngness to be pad volunteers n decson-makng experments. Sx sessons wth 16 subjects were conducted, wth no subject partcpatng n more than one sesson. Thus 96 subjects n total partcpated n the experment. In each sesson two groups of eght subjects were formed, and no nteracton across groups took place. Thus each group of eght can be consdered as an ndependent observaton. Throughout a sesson, no communcaton between subjects was permtted, and all choces and nformaton were transmtted va computer termnals. At the begnnng of a sesson, the subjects were seated at computer termnals and gven a set of nstructons, whch were then read aloud by the expermenter. 12 The sesson then conssted of two phases of sxty rounds. In each round subjects could earn ponts, and at the end of the sesson subjects were pad based on ther accumulated pont earnngs from all 120 rounds usng an exchange rate of 1 penny for every 5 ponts earned. Earnngs averaged for sessons lastng between 50 and 60 mnutes. 13 In each round subjects had forty seconds to complete a route on a road map by clckng on roads. If they faled to complete a route n the forty seconds they receved no ponts n that round. If they completed a route they receved a reward of 100 ponts for arrvng at ther destnaton and pad a travel cost. The actual costs were those presented above. However, subjects were only told that costs would be less than 100, would be calculated n the same way n every round of a phase, and mght depend on the number of other subjects usng the road. At the end of a round each subject was nformed of the cost ncurred from each lnk that they used, the number of subjects usng those lnks, ther own total travel cost and pont earnngs. Our ratonale for ths desgn decson was the followng. Frst, we wanted to reproduce a scenaro where network users learn about the propertes of the network through ther own experences. Specfcally, our desgn has the natural feature that, at the end of a gven journey, users dscover how long the journey took, but not how long t would have taken had they chosen an alternatve route. Second, we wanted to examne the practcal relevance of the least cost and sze prncples. One lmtaton of ths desgn decson s 12 A copy of the nstructons s ncluded as Appendx A. 13 At the tme of the experment the exchange rate as approxmately 1 = $

17 that, strctly speakng, the expermental desgn s dsconnected from a formal game-theoretc model of route choce decson makng. Thus, one must be cautous n drawng conclusons based on comparsons between the (Nash) equlbrum of such a game theoretc model and results from experments where the assumptons of the model are not met. That beng sad, to the extent one vews equlbrum analyss as offerng gudance n practcal settngs where not all of the model assumptons are lkely to be met, t seems to us sensble to take equlbrum predctons as a benchmark aganst whch to compare our expermental results. By provdng a subject wth feedback about both the cost ncurred and the volume of traffc sharng each lnk taken, a subject mght learn the cost structure of the network. Specfcally, any subject could, n prncple, calculate a lnearzaton of the cost functon for each lnk taken so long as there was any varaton n the volume of traffc on a gven lnk. Snce the cost functons for each lnk were, n fact, lnear, ths calculaton would yeld an exact estmate of the cost functon of a gven lnk. In three sessons the PKD networks were used. One group of eght subjects experenced the PKD Baselne network n phase 1 and the PKD Improved network n phase 2, and the other group of eght subjects experenced the PKD Improved network n phase 1 and the PKD Baselne network n phase 2. Smlarly, three sessons used the Braess networks, wth one group experencng the Braess Baselne network n phase 1 and the Braess Improved network n phase 2, wth the other group experencng the networks n the reverse order. 4 Results We use several dfferent performance metrcs to compare the mplcatons of Wardrop equlbrum wth two alternatve benchmarks. The frst alternatve, whch we term the reduced form heurstc, suggests that ndvduals do not respond to changes n the cost structure of the network at all. Under ths heurstc, a socal planner uses traffc flows from a gven network and changes n cost parameters to determne the benefts of a modfed network on travel tmes. The second alternatve, whch we term the effcency benchmark, postulates that ndvduals selforganze nto effcent traffc flow patterns. Throughout, we dvde the analyss nto the short-run the frst 30 perods under a gven network structure and the long run the last 30 perod under a gven network structure. 17

18 Aggregate Route Choce and Travel Tme per Route The frst performance metrc we examne s aggregate route choce. To measure ths, we compute the average number of travelers usng a gven route under a gven network structure n the short-run and the long-run. Tables 1 and 2 dsplay the equlbrum and effcency benchmarks wth respect to route choce. Whle the reduced form heurstc offers no level predcton for route choce, t does predct that route choces are unresponsve to a change n the network. Our second metrc s the average travel tme assocated wth each route. Specfcally, we compute the per perod travel tme for a gven route n a gven network. Wardrop equlbrum mples that that travel tme per route wll be equalzed across routes. In contrast, effcency requres unequal travel tmes across routes. For nstance, n the PKD networks, effcency requres systematcally shorter travel tmes along the congestble route compared to the noncongestble route. Network Performance The thrd performance metrc we examne s the average travel tme (or latency) occurrng n the network as a whole. To measure ths, we compute the average experenced travel tme for a gven network over a gven tme horzon (.e. the short-run or the long-run). As we wll see, the statstcal propertes of ths measure are of some nterest. For the PKD networks, gven some dstrbuton of route lnk choces by ndvduals, the expectaton of the travel tme experenced by a network user s k 1 E n( α βn) N + = 1 Runnng the expectatons operator through ths expresson and smplfyng yelds 1 N k ( E[ n] ( α + βe[ n] ) + β Var[ n] ). (6) = 1 Equaton (6) shows that latency depends on the varance of route choces as well as the average traffc flows over each route. 14 Moreover, equaton (6) mples that hgher varablty of route choces leads to longer experenced travel tmes. The ntuton s that every tme many travelers use a congestble road, that road s slow and many travelers are adversely affected. On the other 14 Whle the explct calculaton of equaton (6) was for the PKD networks, an analogous calculaton reveals a varance term for the Braess networks as well. 18

19 hand, every tme few travelers use a congestble road, that road s fast but only those few travelers beneft from the under-utlzaton. Tables 1 and 2 dsplay the equlbrum and effcency benchmarks wth respect to the latency metrc. The reduced form heurstc agan offers no level predcton, but does offer a drectonal predcton of the effects of changng network structure relatve to baselne behavor. 4.1 PKD Networks Aggregate Route Choces and Travel Tme per Route For the PKD networks the data consst of route choces made over 5,760 subject-rounds (48 subjects x 120 rounds). In only 7 of these (about one tenth of one percent) dd a subject fal to complete a route n the forty seconds allowed. 15 Table 3 dsplays the average number of travelers choosng the low road for each group. The frst three groups lsted experenced the Baselne network frst, whle the second three experenced the Improved network frst. Recall that both equlbrum and effcency mply that the mprovement n the network wll lead to a shft n traffc onto the mproved low road. In contrast, the reduced form heurstc predcts no shft n traffc. As Table 3 shows, the network mprovement led to a shft n traffc to the low road even n the short run n every group. Thus, there s a sgnfcant shft n the drecton mpled by equlbrum and effcency (p-value = 0.031). 16 Table 3 also shows that there are systematcally fewer travelers on the low road than n Wardrop equlbrum for both Baselne (equlbrum: 4) and Improved (equlbrum: 6) n every group n both the short-run and the long-run. Thus, there are sgnfcant dfferences between observed and equlbrum travel flows (p-value = 0.031). At the same tme, Table 3 shows that there are systematcally more travelers on the low road than mpled by effcency for both Baselne (effcent: 2) and Improved (effcent: 3) n every group n both the short-run and the long-run. Thus, there are sgnfcant dfferences between observed and effcent travel flows (pvalue = 0.031). Note, however, that n all sx groups the average number of travelers on the low road s much closer to the equlbrum number than the effcent number. 15 On the Braess networks n only 3 of 5,760 subject-rounds dd a subject fal to complete a route. 16 We found no sgnfcant dfferences between groups who experenced the baselne or mproved network frst. Thus we pool the sx ndependent groups and base statstcal tests on two-sded Wlcoxon sgned-rank tests appled to sx ndependent matched pars. Throughout the remander of the paper p-values are based on the same procedure, snce we found no sgnfcant order effects n any of our treatments. 19

20 Table 3: Traffc flow n PKD networks - Average number of travelers on the low road - PKD Baselne PKD Improved Group PKD PKD PKD PKD PKD PKD ALL A key characterstc of Wardrop equlbrum s that travel tmes across routes are equalzed. However, because the number of travelers on the congestble low route s systematcally lower than n equlbrum we can reject the travel tme equalzaton hypothess. To study ths n more detal, Table 4 presents the dfference n average travel tmes between the hgh road and the low road for each group. The congestble low route s sgnfcantly faster on both networks, whether we focus on the short-run or long-run (a two-sded Wlcoxon sgnedrank test yelds a p-value of n all cases). On the Improved network, there s a trend n the drecton of equalzaton; the dfferences are sgnfcantly smaller n the long run than short run (p = 0.031). On the Baselne network, where dfferences are generally smaller, the trend s nsgnfcant (p = 0.281). Table 4: Investgatng travel tme equalzaton n PKD networks - Travel tme on hgh road mnus travel tme on low road - Group PKD Baselne PKD Improved PKD PKD PKD PKD PKD PKD ALL

21 4.1.2 Network Performance Table 5 dsplays the experenced travel tme (or latency) for each group. Recall that, accordng to both the effcency and reduced form benchmarks, the mprovement n the network wll lead to a reducton n experenced travel tmes. In contrast, n equlbrum the mprovement should have no mpact on network performance. As Table 5 shows, experenced travel tmes were reduced after the mprovement even n the short run n every group. Thus, there s a sgnfcant shft n the drecton mpled by the effcency and reduced form benchmarks (p-value = 0.031). Group Table 5: Latency n PKD networks - Average travel tme for network users - PKD Baselne PKD Improved PKD PKD PKD PKD PKD PKD ALL However, latences systematcally devate from both the effcency and the equlbrum benchmarks. Table 5 shows that travel tmes are systematcally hgher than under effcency for both Baselne (effcent: 52.5) and Improved (effcent: ) n every group n both the shortrun and the long-run (p-value = n every case). Travel tmes are systematcally hgher than those mpled by equlbrum n the Baselne (equlbrum: 57) and then systematcally lower under the Improved network (equlbrum: 57). Havng sad ths, the observed latences are closer to the equlbrum benchmark than the effcent benchmarks n all groups n both the short run and the long run. Fnally, to examne the reduced form heurstc we need a benchmark from whch to compare. For the groups that frst encounter the Baselne and then the Improved network we take the last 30 rounds n Baselne as a benchmark. Relatve to that benchmark there s a 7% reducton n travel tme n the short run (the frst 30 rounds of the Improved network) and a 5% reducton n the long run (the second 30 rounds of the Improved network). Usng an analogous 21

22 procedure for the groups that frst encounter the Improved network, we fnd that the deleton of the network mprovement results n a 6% ncrease n travel tme n the short run and 4% ncrease n the long run. Ths s vastly less than the 15% reducton (22% ncrease), whch would have been forecast had one smply used reduced form estmates to predct changes n performance. 17 Moreover, the actual changes n travel tme appear to be systematc: Regardless of the order n whch networks were presented to subjects, travel tme s lower n the Improved than n the Baselne network n all groups. Comparng the results of Table 5 wth those n Secton reveals an apparent puzzle for the PKD Baselne network. As seen n Secton 4.1.1, low road utlzaton n Baselne s sgnfcantly less than n equlbrum, whch mples that the low road s sgnfcantly faster than n equlbrum, whereas the travel tme along the hgh road s the same as n equlbrum because t s fxed regardless of the number of travelers. However, as seen n Table 5, the average experenced travel tme exceeds the equlbrum level. The varance term n equaton (6) explans ths apparent puzzle. Despte effcency enhancng under-utlzaton of the low road, the varance n the route choces rases the experenced travel tme n the Baselne. That s, the varance effect domnates the mean effect n ths network structure. In the Improved network, the varance effect s also present but the mean effect (under-utlzaton of the low road) domnates Summary The mprovement to the PKD network leads to a sgnfcant shft n traffc flows, n clear contrast to the reduced form heurstc. Whle the shft s n the drecton consstent wth equlbrum and effcency, there are sgnfcant devatons from the effcency and equlbrum benchmarks. In addton, network users are adversely affected by traffc flow fluctuatons, whch n Baselne cause the experenced travel tmes to rse above equlbrum levels despte the underutlzaton of the congestble road. Furthermore, travel tmes are not fully equalzed, and the PKD paradox the complete dsspaton of the benefts from the road mprovement s not borne out ether n the short-run or the long-run. On the other hand, route choces and 17 For groups PKD.1,.2 and.3 the average travel tme n the Improved network would have been had they made the same choces as n the last 30 rounds of Baselne. For groups PKD.4,.5 and.6 the average travel tme n Baselne would have been had they made the same choces as n the last 30 rounds of Improved. 18 In fact, varablty s very smlar on the two networks. Averagng across groups, the standard devaton of n t s 1.15 for the Baselne and 1.13 for the Improved network. Such traffc flow fluctuatons are of course not compatble wth Wardrop equlbrum. In contrast, n a mxed-strategy Nash equlbrum one would expect varablty n traffc flows. In Secton 5 we nvestgate the explanatory power of mxed-strategy solutons n more detal. 22

23 experenced travel tmes are closer to the equlbrum than the effcency benchmark, and on average travelers could not have acheved a long-run reducton n travel tme by always choosng the faster low road. 4.2 Braess Networks Aggregate Route Choces and Travel Tme per Route Table 6 dsplays the average number of travelers choosng the low road for each group. The groups that frst encounter the Baselne network and then the Improved network are lsted frst. Note that the Improved network can be vewed as the composton of two networks n each of whch users make a bnary choce between a congestble and a non-congestble road. In Table 6 we analyze these two components separately. Recall that equlbrum requres a shft n traffc onto the congestble roads after the mprovement n the network. In contrast, effcency requres a shft n traffc onto the non-congestble road n the frst leg, and ether no shft or a shft n traffc onto the non-congestble road n the second leg. As Table 6 shows, traffc shfted n the drecton of equlbrum. There s a sgnfcant ncrease n the amount of traffc leavng the orgn on the congestble road and a sgnfcant ncrease n the amount of traffc arrvng at the destnaton on the congestble road (the p-value s n both cases and for both short and long run). Group Table 6: Traffc flow n Braess networks - Average number of travelers on the low road - Braess Baselne Braess Improved 1 st Leg Braess Improved 2 nd Leg BRS BRS BRS BRS BRS BRS ALL On the Baselne network traffc flows are not sgnfcantly dfferent from equlbrum flows of 4 travelers on each road (p-value = n both short run and long run). On the 23

24 mproved network there are systematc dfferences between equlbrum and actual traffc flows. In equlbrum, one traveler uses the low road on the frst leg, and fve use the low road on the second leg. In fact, traffc s more evenly dstrbuted across the two routes on both legs (frst leg: p = for both short and long run, second leg: p = for the short run and p = for the long run). Thus, relatve to equlbrum, the congestble roads are under-utlzed on both legs of the Improved network. Table 7 dsplays the dfference n the average travel tmes between the hgh road and the low road. Snce Baselne traffc flows do not dffer sgnfcantly from equlbrum, we cannot reject the travel tme equalzaton hypothess. For the case of the Improved network, where the congestble roads are under-utlzed relatve to equlbrum, these roads are sgnfcantly faster than the non-congestble roads, and the travel tme equalzaton hypothess s rejected. When we compare the travel tme dscrepances n the short run and long run we observe a sgnfcant trend toward equalzaton on the frst leg (p = 0.031), but not on the second leg (p = 0.438). Group Table 7: Investgatng travel tme equalzaton n Braess networks - Travel tme on hgh road mnus travel tme on low road - Braess Baselne Braess Improved 1 st Leg Braess Improved 2 nd Leg BRS BRS BRS BRS BRS BRS ALL Network Performance Table 8 dsplays latency on the Braess networks for each group. As wth the PKD networks, the externaltes road users place on one another generate neffcent travel flows. In the Baselne network, an effcent flow of traffc would generate an average travel tme of 55.5, and actual travel tmes are 6% greater than ths. Indeed, these effcency losses exceed those that would be 24

25 ncurred f the system were n equlbrum. For the Improved network average travel tmes are 62.97, better than n equlbrum (66), but stll 15% above effcent levels. Group Table 8: Latency n Braess networks - Average travel tme for network users - Braess Baselne Braess Improved BRS BRS BRS BRS BRS BRS ALL In the Baselne network, even though traffc flows do not dffer systematcally from equlbrum, travel tmes are sgnfcantly hgher than n equlbrum (p = for both the short run and long run). The reason for ths seemngly paradoxcal fndng s agan the adverse effect of varablty n traffc flows (equaton 6). On the Improved network travel tmes are sgnfcantly lower than n equlbrum (p = for both short run and long run). Although the varance effect s also present here, the overall outcome s domnated by the under-utlzaton of the congestble lnks relatve to equlbrum. Whle the addtonal lnk n the Improved network n prncple allows more effcent traffc flows than are possble n the Baselne, equlbrum entals an ncrease n travel tmes after the techncal mprovement. In the data, even though the Baselne network performs worse than n equlbrum, and the Improved network performs better than equlbrum, the table shows that all groups experence an ncrease n travel tmes after the mprovement. Thus there s a sgnfcant ncrease n travel tmes n both the short run (p = 0.031) and the long run (p = 0.031), and the Braess Paradox s observed n our data. To get an dea of the quanttatve effect, we consder the groups that frst encounter the Baselne network and then the Improved network, and agan take the last 30 rounds n Baselne 25