Network Topology and the Efficiency of Equilibrium *

Transcription

1 Network Topology and the Effcency of Equlbrum * IGAL MILCHTAICH Department of Economcs, Bar-Ilan Unversty, Ramat-Gan , Israel Emal: gal.mlchtach@bu.ac.l Web: Games and Economc Behavor 57 (2006), Abstract. Dfferent knds of networks, such as transportaton, communcaton, computer, and supply networks, are susceptble to smlar knds of neffcences. These arse when congeston externaltes make the cost for each user depend on the other users choce of routes. If each user chooses the least expensve (e.g., the fastest) route from the users common pont of orgn to the common destnaton, the result may be Pareto neffcent n that an alternatve choce of routes would reduce the costs for all users. Braess s paradox represents an extreme knd of neffcency, n whch the equlbrum costs may be reduced by rasng the cost curves. As ths paper shows, ths paradox occurs n an (undrected) twotermnal network f and only f t s not seres-parallel. More generally, Pareto neffcent equlbra occur n a network f and only f one of three smple networks s embedded n t. JEL Classfcaton: C72, R41. Key Words: Congeston, externaltes, topologcal effcency, nonatomc games, network topology, Braess s paradox, transportaton networks, Wardrop equlbrum * The author thanks Grant Keady, Bruce Calvert, Claude Penchna, and Hdeo Konsh for helpful dscussons. Ths research was supported by the Israel Scence Foundaton (grant no. 749/02).

2 1 Introducton In transportaton and other knds of physcal networks wth large numbers of users, congeston externaltes are a potental source of neffcency. A remarkable example of ths, known as Braess s paradox (Braess, 1968; Murchland, 1970; Arnott and Small, 1994), s shown n Fgure 1. Cars arrve at a constant rate at vertex o of the depcted network and leave t at vertex d. The network conssts of three fast roads (e 1, e 4 and e 5 ) and two slow ones (e 2 and e 3 ). The travel tme on each road s an ncreasng functon of the flow on t, or the average number of vehcles passng a fxed pont n the road per unt of tme. (Ths s a reasonable assumpton f the densty of vehcles on the road s relatvely low, so that the flow s well below the road s capacty. See Sheff, 1985, Chapter 13 and Fgure 1.8.) However, regardless of the flow, the travel tme on the route consstng of the three fast roads s shorter than on any of the alternatve routes. Therefore, at (the Wardrop) equlbrum, when the entre flow passes on the fastest routes, all vehcles use ths one. The travel tme on the network (as computed n the capton to Fgure 1) s then 21 mnutes. Suppose, however, that the transverse road, e 5, s closed, or ts physcal condton deterorates to the pont at whch the travel tme on t becomes smlar to that on each of the two slow roads. The road s new cost curve s hgher than the old one: for any flow on e 5, the travel tme s longer than before. As a result of the change n costs, the old equlbrum s replaced by a new one, n whch the transverse road s not used at all. Paradoxcally, the new travel tme s shorter than before: 20 mnutes. The longer prevous travel tme s due to the motorsts concern for ther own good only, whch results n overuse of the fast roads and consequently an neffcent equlbrum. As ponted out by Newell (1980) and Sheff (1985), traffc engneers have long known that more restrcted travel choces and reduced capacty may mprove the flow n the network as a whole. For nstance, ths s the underlyng prncple behnd many traffc control schemes, such as ramp meterng on freeway entrances (Sheff, 1985, p. 77). Braess s paradox s not lmted to transportaton networks. The potental occurrence of ths or smlar paradoxes has been demonstrated for such dverse networks as computer and telecommuncaton networks, electrcal crcuts, and mechancal systems. Remarkably, much of ths lterature (e.g., Frank, 1981; Cohen and Horowtz, 1991; Cohen and Jeffres, 1997) s concerned wth essentally the same network as n Braess s (1968) orgnal example, the Wheatstone network, shown n Fgure 1. As t turns out, there s a good reason for ths. As ths paper shows, ths s, n a sense, the only two-termnal network n whch Braess s paradox can occur. More precsely, a necessary and suffcent condton for the exstence of some cost functon exhbtng the paradox s that the network has an embedded Wheatstone network. In networks wthout ths property, called seres-parallel networks, Braess s paradox never occurs. Several alternatve characterzatons of seres-parallel networks are gven below. Braess s paradox s not the only knd of neffcency caused by congeston externaltes. Consder, for example, the seres-parallel network n Fgure 2(a), whch represents the alternatves faced by weekend vstors to a certan seasde town where the only attractons are the two nearby beaches. The two edges jonng o and v represent the alternatves of gong to the North Beach (e 1 ) or the South Beach (e 2 ) on Saturday. The two edges jonng v and d represent the same two alternatves on Sunday. The South Beach s more remote, and the cost of gettng there s two unts more than for the North Beach. On the other hand, t s 2

3 o e 1 e 2 u e 5 v e 3 e 4 d Fgure 1 Braess s paradox. A contnuum of users travels from o to d on the two-termnal Wheatstone network shown. The travel tme on each edge e s an ncreasng functon of the fracton x of the total flow from o to d that passes on e. The travel tmes, n mnutes, are gven by 1 + 6x for e 1 and e 4, and x for e 2 and e 3. If the travel tme on e 5 s also gven by 1 + 6x, then, at equlbrum, the entre flow passes on e 1, e 5 and e 4, whch consttute the fastest route from o to d. The total travel tme s then 3 ( ) = 21 mnutes. If, however, the travel tme on e 5 s longer, and gven by x, then, at equlbrum, there s no flow on that edge: half the users go though e 1 and e 3 and half though e 2 and e 4. The equlbrum travel tme s then shorter: (1 + 6 ½) + ( ½) = 20 mnutes. a longer beach, and therefore does not get crowded as fast. However, the addtonal pleasure of spendng the day on an uncrowded beach never exceeds the dfference n travel costs. Therefore, at equlbrum, all the vstors go to the North Beach, both on Saturday and on Sunday. Crowdng then costs each person four unts of pleasure. However, f people had taken turns, half of them gong to the South Beach on Saturday and the other half on Sunday, then the cost for all ndvduals would be lower, and equal to 3½. Thus, ths arrangement, whch s not an equlbrum, represents a strct Pareto mprovement over the equlbrum. The dfference between ths example and the one n Fgure 1 s that, n the case of Braess s paradox, Pareto mprovement results from rasng the costs of certan facltes (e.g., ncreasng the travel tme on the transverse road n Fgure 1), thereby creatng a new equlbrum that s better for everyone. By contrast, n the present example t s not possble to make everyone better off by rasng the costs of facltes (e.g., chargng congestondependent entry fees to beaches). Snce the networks n Fgure 2 are seres-parallel, Braess s paradox cannot occur. Hence, any Pareto mprovement necessarly nvolves non-equlbrum behavor,.e., use of certan routes for whch less costly alternatves exst. One of the man results of ths paper s that the three networks n Fgure 1 and Fgure 2 essentally represent the only knds of two-termnal network topologes n whch congeston externaltes may lead to Pareto neffcent equlbra. For example, such neffcences never arse n networks such as n Fgure 3. The crucal dfference between ths network and those prevously mentoned s that the routes n t are lnearly ndependent, n the sense that each one ncludes at least one edge that s not part of any other route. Ths s equvalent to the followng condton: none of the routes n the network has a par of edges, each of whch also belongs to some other route that does not nclude the other edge. In a smlar, but not dentcal, context, Holzman and Law-Yone (1997, 2003) show ths to be a necessary and suffcent condton for weak Pareto effcency of the equlbra for all systems of nonnegatve, 3

4 o o e 1 e 2 e 1 e 2 v e 5 e 3 e 4 e 3 e 4 d (a) d (b) Fgure 2 Another knd of neffcency caused by congeston externaltes. The cost of each edge e n network (a) s an ncreasng functon of the fracton x of the total flow from o to d that passes on e. For e 1 and e 3, the cost s gven by 2x. For e 2 and e 4, t s 2 + x. At equlbrum, only e 1 and e 3 are used, and the equlbrum cost s (2 1) + (2 1) = 4. However, ths outcome s Pareto neffcent. Splttng the flow, so that half of t goes through e 1 and e 4 and half through e 2 and e 3, would reduce the cost to (2 ½) + (2 + ½) = 3½. (Further reducton s not possble, snce t can be shown that the mean cost n ths example cannot be less than 3½.) A smlar phenomenon occurs n network (b). Indeed, snce all routes from o to d pass through the mddle edge e 5, the cost of ths edge s rrelevant. nondecreasng edge costs. Ther work dffers from ths paper manly n referrng to networks wth a fnte number of users, each of whom has a non-neglgble effect on the others. Here, by contrast, there s a contnuum of users, whch may be vewed as a mathematcal dealzaton of a very large populaton of ndvduals, each wth an almost neglgble effect on the others. As ths paper shows, the cases of fnte and nfnte populatons dffer n a number of ways. Most mportantly, n the latter but not the former case, the connecton between lnear ndependence of the routes and effcency of the equlbra also holds for heterogeneous populatons, n whch users have dfferent cost functons. Wth a fnte number of non-dentcal users, cost functons wth Pareto neffcent equlbra exst for any nontrval network,.e., one wth two or more routes. Wth a contnuum of users, an assgnment of cost functons wth a Pareto neffcent equlbrum exsts f and only f the routes n the network are not lnearly ndependent. The emphass n ths paper s on topologcal effcency: network topologes for whch cost functons gvng rse to neffcences do not exst. Most related papers put the emphass on the cost functons themselves. For example, formulas yeldng, under certan condtons, the change n cost nduced by the creaton of addtonal routes were obtaned by Stenberg and Zangwll (1983) and Dafermos and Nagurney (1984). In prncple, these formulas can be used to determne whether a form of Braess s paradox occurs n the network. They are, however, rather complcated. An nterestng (and, as shown by Calvert and Keady, 1993, rather unque) case, n whch the topology of the network s rrelevant, s that of edge costs that are 4

5 o d Fgure 3 A network wth lnearly ndependent routes. In such a network, equlbra are always Pareto effcent. gven by homogeneous functons of the same degree,.e., each of them has the form αx β, wth α, β > 0, and β s the same for all edges. In ths case, not only are the equlbra Pareto effcent but they are even socally optmal n that the mean (equvalently, aggregate) cost ncurred by the users s mnmzed. Intutvely, ths s because, n ths case, users swtchng from one route to another make proportonal changes to ther own and the socal costs (see Mlchtach, 2004). The topology of a network does not ndcate whether equlbra are socally optmal. Even for very smple networks, t s also necessary to know the functonal form of the cost functons (Mlchtach, 2004). The same s true for the so-called prce of anarchy, whch s the rato between the equlbrum cost and the mean cost ncurred by the users at a socal optmum. For general cost functons, ths rato s unbounded, but for varous natural classes of such functons, bounds do exst (Roughgarden, 2003; Roughgarden and Tardos, 2004). For example, the maxmum prce of anarchy wth lnear edge costs s 4/3 (thus, greater than the 4: 3½ rato between the equlbrum cost and the mean cost at the socal optmum n the example n Fgure 2). Under weak assumptons on the class of allowable cost functons, a network wth only two parallel edges suffces to acheve the worst possble rato. Thus, as for socal optmalty, the network topology does not play a role n determnng the prce of anarchy (Roughgarden, 2003). Ths contrasts sharply wth the stuaton for Pareto effcency, n whch, as ths paper shows, the network topology matters. Several other propertes of the equlbra n route selecton games occupy an ntermedate poston between these two extremes n that they depend on certan global propertes of the network. For example, the maxmum reducton n equlbrum cost achevable by removng one (as n the orgnal Braess s paradox) or more edges n a two-termnal network s postvely related wth the number of vertces (Roughgarden, 2004). It s also postvely related wth the number of edges removed (Ln, Roughgarden and Tardos, 2004). The rest of the paper s organzed as follows. The next secton presents n some detal the needed graph-theoretc defntons and results. Wth few exceptons (a network wth lnearly ndependent routes s a notable one), standard termnology s used (whch, however, does not always have the exact same meanng n all sources). Flows, cost functons, and related 5

6 terms are defned n Secton 3. The defnton of equlbrum wth dentcal users, and results about ts effcency, are gven n Secton 4. The frst result lnks the non-occurrence of Braess s paradox wth seres-parallel networks, and the second one lnks (the stronger property of) Pareto effcency of the equlbra wth (the smaller class of) networks wth lnearly ndependent routes. Secton 5 deals wth non-dentcal users, and shows that, n ths case, both propertes of the equlbra are lnked wth networks wth lnearly ndependent routes. In Secton 6, some of the defntons and assumptons underlyng these results are dscussed. In partcular, the advantages of dealng wth undrected rather than drected networks are explaned, and the smlartes and dfferences between the topologcal condtons for effcency and unqueness of the equlbrum (Mlchtach, 2005) are descrbed. The proofs of all the propostons and theorems n ths paper are gven n Secton 7. 2 Graph-Theoretc Prelmnares An undrected multgraph conssts of a fnte vertex set V and a fnte edge set E. Each edge e jons two dstnct vertces, u and v, whch are referred to as the end vertces of e. Thus, loops are not allowed, but more than one edge can jon two vertces. An edge e and a vertex v are sad to be ncdent wth each other f v s an end vertex of e. A path of length n (n 0) s an alternatng sequence p of vertces and edges v 0 e 1 v 1 v n 1 e n v n, begnnng and endng wth vertces, n whch each edge s ncdent wth the two vertces mmedately precedng and followng t and all the vertces (and necessarly all the edges) are dstnct. Because of the latter assumpton, each vertex and each edge n p ether precedes or follows each of the other vertces and edges. The frst and last vertces, v 0 and v n, are called the ntal and termnal vertces n p, respectvely. The path v n e n v n 1 v 1 e 1 v 0, whch ncludes the same vertces and edges as p but passes through them n reverse order, s denoted by p. If q s a path of the form v n e n+1 v n+1 v m 1 e m v m, the ntal vertex of whch s the same as the termnal vertex of p but all the other vertces and edges are not n p, then v 0 e 1 v 1 v n 1 e n v n e n+1 v n+1 v m 1 e m v m s also a path, denoted by p + q. A secton of p s any path s of the form v n1 e n1 +1v n1 +1 v n2 1e n2 v n2, wth 0 n 1 n 2 n. Each secton s unquely dentfed by ts ntal vertex u and termnal vertex v, and may therefore be denoted by p uv. If the length of s s zero,.e., t does not nclude any edges, then u and v concde. If the length s one,.e., the secton has a sngle edge e, then u and v are the two end vertces of e. In ths case, s s called an arc, and may be vewed as a specfcaton of the drecton n whch p passes through e. A two-termnal network (network, for short) s an undrected multgraph together wth a dstngushed ordered par of dstnct vertces, an orgn o and a destnaton d, such that each vertex and each edge belong to at least one path wth the ntal vertex o and termnal vertex d. Any path r wth these ntal and termnal vertces wll be called a route. The set of all routes n a network, denoted by R, s ts route set. Two networks G and G wll be sad to be somorphc f there s a one-to-one correspondence between ther vertex sets and between the edge sets such that () the ncdence relaton s preserved and () the orgn and destnaton n G are pared wth the orgn and destnaton n G, respectvely. A network G s a sub-network of a network G f the former s somorphc to a network derved from the latter by deletng a subset of ts edges 6

7 o o o o d d d e (a) (b) (c) Fgure 4 Embeddng. The left network s embedded n each of the other three, whch are obtaned from t by: (a) subdvdng an exstng edge, (b) addng a new edge, and, fnally, (c) extendng the destnaton. and vertces, whch does not nclude o or d. A network G s embedded n a network G f G s somorphc to G or to a network derved from G by applyng the followng operatons any number of tmes n any order (see Fgure 4): (a) The subdvson of an edge: ts replacement by two edges wth a sngle common end vertex. (b) The addton of a new edge jonng two exstng vertces. (c) The extenson of a termnal vertex: addton of a new edge e jonng o or d wth another, new vertex, whch becomes the new orgn or destnaton, respectvely. It can be shown that, for any network G, addton and subdvson of edges always gve a new network, wth the same orgn destnaton par. By usng both operatons, complete new paths, whch only have ther ntal and termnal vertces n G, can be added to t. Ths s done by frst jonng the two vertces by a new edge e, and then subdvdng e one or more tmes. Because each vertex and each edge n a network are n some route, ths shows that G s embedded n every network G of whch t s a sub-network. Two networks G and G wth the same orgn destnaton par, but wthout any other common vertces or edges, may be connected n parallel. The vertex and edge sets n the resultng network G are the unons of those n G and G, and the orgn and destnaton are the same as n these networks. Two networks G and G wth a sngle common vertex (and, hence, wthout common edges), whch s the destnaton n G and the orgn n G, may be connected n seres. The vertex and edge sets n the resultng network G are the unons of those n G and G, the orgn concdes wth the orgn n G, and the destnaton wth that n G. It s not dffcult to see that, when two networks G and G are connected n parallel or n seres, each of them s embedded n the resultng network G. A network s sad to be seres- parallel f two routes never pass through any edge n opposte drectons. The two networks n Fgure 2 are seres-parallel. The Wheatstone d 7

8 network n Fgure 1 s not seres-parallel, snce there are two routes passng through e 5 n opposte drectons. In fact, as the followng proposton shows, the Wheatstone network s embedded n any network that s not seres-parallel. (Theorem 1 of Duffn, 1965, makes the same asserton. However, snce that paper uses somewhat dfferent defntons, an explct proof s needed here, whch s gven n Secton 7.) Proposton 1. For a two-termnal network G, the followng condtons are equvalent: () G s seres-parallel. () For every par of dstnct vertces u and v, f u precedes v n some route r contanng both vertces, then u precedes v n all such routes. () The network n Fgure 1 s not embedded n G. As noted by Rordan and Shannon (1942), seres-parallel networks can also be defned recursvely. A network s seres-parallel f and only f t can be constructed from sngle edges by carryng out the operatons of connectng networks n seres or n parallel any number of tmes. (Hence the term seres-parallel. ) Ths can be stated as follows. Proposton 2. A two-termnal network G s seres-parallel f and only f () t has a sngle edge only, () t s the result of connectng two seres-parallel networks n parallel, or () t s the result of connectng two seres-parallel networks n seres. One corollary of Proposton 2 s that every seres-parallel network s planar and, moreover, remans so when a new edge, jonng o and d, s added to t. Equvalently (see Harary, 1969), every seres-parallel network can be embedded n the plane n such a way that o and d le on the exteror face, or boundary. Usng Proposton 2, ths corollary can easly be proved by nducton on the number of edges. Another corollary of Proposton 2 s the followng result, whch may help n verfyng that a gven network s seres-parallel. Proposton 3. A two-termnal network G s seres-parallel f and only f the vertces can be ndexed n such a way that, along each route, they have ncreasng ndces. A network wll be sad to have lnearly ndependent routes f each route has at least one edge that does not belong to any other route. (The reason for ths name s gven by Proposton 6 below.) The smplest such network s parallel network, whch conssts of one or more edges connected n parallel. Another example s shown n Fgure 3. Every network wth lnearly ndependent routes s seres-parallel but the converse s false. The two networks n Fgure 2 are seres-parallel but they do not have lnearly ndependent routes. Indeed, the next proposton mples that at least one of these two networks s embedded n any seresparallel network that does not have lnearly ndependent routes. The proposton also gves two other characterzatons of networks wth lnearly ndependent routes. The frst characterstc property s that pars of routes never merge only n the mddle: any common secton extends to ether the orgn or the destnaton. The second property s that the route set does not contan a bad confguraton (Holzman and Law-Yone, 1997, 2003), whch s 8

9 defned as a trplet of routes, the frst of whch ncludes some edge e 1 that does not belong to the second route, the second route ncludes some edge e 2 that does not belong to the frst one, and the thrd route ncludes both e 1 and e 2. Proposton 4. For a two-termnal network G, the followng condtons are equvalent: () G has lnearly ndependent routes. () For every par of routes r and r and every vertex v common to both routes, ether the secton r ov (whch conssts of v and all the vertces and edges precedng t n r) s equal to r ov, or r vd s equal to r vd. () A trplet of routes consttutng a bad confguraton does not exst. (v) None of the networks n Fgure 1 and Fgure 2 s embedded n G. The followng recursve characterzaton of networks wth lnearly ndependent routes, whch dffers from that for seres-parallel ones (Proposton 2) only n havng a more restrctve part (), reveals another facet of the dfference between these two knds of networks. Ths characterzaton s essentally a corollary of Theorem 1 of Holzman and Law- Yone (2003) (whch, however, relates to drected networks). Proposton 5. A two-termnal network G has lnearly ndependent routes f and only f () t has a sngle edge only, () t s the result of connectng two networks wth lnearly ndependent routes n parallel, or () t s the result of connectng n seres a network wth lnearly ndependent routes and one wth a sngle edge (or, equvalently, extendng the orgn or the destnaton n the frst network). Snce, n every network G, each route r has a unque set of edges, t s represented by a unque bnary vector, n whch 1 s assgned to each edge e that belongs to r and 0 to any other edge. Ths vector can be vewed as an element of the vector space F 2 E, so-called the edge space of G (Destel, 2000), where E s the number of edges n G and F 2 s the feld of the ntegers modulo 2. Thus, each collecton of routes n G corresponds to a set of vectors n the edge space. These vectors are lnearly ndependent f and only f t s not possble to wrte any one of them as the (component-wse) sum modulo 2 of some of the others. As the followng proposton shows, networks wth lnearly ndependent routes are characterzed by the property that the collecton of all routes corresponds to a lnearly ndependent set n the edge space. 1 1 Note that lnear ndependence s defned wth respect to F 2, not (the real feld) R. For example, the Wheatstone network does not have lnearly ndependent routes. Although the bnary vectors representng ts four routes are lnearly ndependent n R E, they are not so n F 2 E, snce each of them s equal to the sum modulo 2 of the other three. 9

10 Proposton 6. A two-termnal network G s a network wth lnearly ndependent routes f and only f ts route set R corresponds to a lnearly ndependent set of vectors n the edge space. 3 The Model A flow vector for a network G s a nonnegatve vector f = (f r ) r R specfyng the flow f r on each route r. The flow f p on each path p s defned as the total flow on all the routes contanng p: f p = f r. r R p s a secton of r (1) If p s a path of length zero, consstng of a sngle vertex v, then f p s a juncton flow: t gves the total flow on all the routes passng through v. In partcular, f o = f r r R (whch s clearly equal to f d ) represents the total orgn destnaton flow. If p s a path of length one, consstng of a sngle edge e and ts two end vertces, then f p s an arc flow: t gves the total flow on e n the drecton specfed by p. Each edge s assocated wth a par of arc flows, one for each drecton. In a seres-parallel network, n whch all routes pass through an edge n the same drecton, only one of these flows can be postve. In a network wth lnearly ndependent routes, n whch each route ncludes at least one edge that s not n any other route, the arc flows unquely determne the flow vector. A cost functon for a network G s a vector-valued functon c specfyng the cost c p (f) of each path p as a functon of the (entre) flow vector f. 2 The followng monotoncty condton s assumed to hold: For every path p and every par of flow vectors f and f, f f s f s and f s f s for all sectons s of p, then c p (f ) c p (f ). Ths mples, n partcular, that the cost of a path only depends on the flow on each of ts sectons and the flows n the opposte drectons. 3 A cost functon wll be sad to be ncreasng f t satsfes the followng addtonal 2 Thus, the vectors n the doman and range of c have dfferent dmensons: the former equals the number of routes n G and the latter the number of paths. Note that the costs are not assumed to be nonnegatve. However, they may well be thought of as such. Indeed, the assumpton that the costs are nonnegatve s mplct n the defnton of equlbrum (n the next secton). Ths defnton only consders routes, whch by defnton do not pass through any vertex more than once. 3 Because ths monotoncty condton nvolves a potentally long lst of premses, the restrcton t puts on the allowable cost functons s relatvely weak. Stronger condtons, e.g., a requrement that the cost of a path can ncrease only f one of the relevant arc flows ncreases, could be used nstead. However, weaker condtons here and n the next defnton are preferable snce they make for stronger results. See also the dscusson n Secton 6. Because of the dmensonalty ssue mentoned n the prevous footnote, the present monotoncty condtons are techncally ncomparable wth monotoncty and strct monotoncty as defned, e.g., by Nagurney (1999). Note, however, that the latter are wder n that they allow for crosstalk,.e., the cost of a route may be nfluenced by the flow on a parallel route. 10

11 condton: For every route r and every par of flow vectors f and f, f f s f s and f s f s for all sectons s of r, and there s at least one secton s of length one for whch f s > f s, then c r (f ) > c r (f ). A cost functon c s (addtvely) separable f the equalty c rov (f) = c rou (f) + c ruv (f) holds for every route r, every par of dstnct vertces u and v such that u precedes v n r, and every flow vector f. In other words, separablty means that the cost of each route s the sum of the costs of ts arcs. 4 Effcency of Equlbrum A flow vector f s sad to be an equlbrum f the entre flow n the network s on mnmalcost routes, that s, for all routes r wth f r > 0, c r (f ) = mn q R c q (f ). (2) For an equlbrum f, the mnmum n (2), denoted by c(f ), s the equlbrum cost. In the transportaton lterature, a flow vector satsfyng (2) s known as Wardrop, or user equlbrum. Ths condton expresses the prncple that, at equlbrum, the travel tme on all used routes s the same, and less than or equal to that of a sngle vehcle on any unused route (Wardrop, 1952). The equlbrum condton (2) can also be gven a varatonal nequalty formulaton (Nagurney, 1999, Theorem 4.5): For every flow vector f wth the same total orgn destnaton flow as f, c r (f )(f r f r ) < 0. r R If the cost functon s contnuous, then, by standard results (e.g., Nagurney, 1999, Theorem 1.4), t follows from ths formulaton that for any f o 0 there s at least one equlbrum wth a total orgn destnaton flow of f o. DEFINITION 1. Braess s paradox occurs n a network G f there are two separable cost functons c and c such that c r (f) c r (f) for all routes r and flow vectors f, but for every equlbrum4 f wth respect to c wth a total orgn destnaton flow of unty and every equlbrum 4 f wth respect to c wth a smlar total orgn destnaton flow, the equlbrum costs satsfy c (f ) < c (f ). Thus, Braess s paradox occurs f rasng the edge costs can lower the equlbrum cost. Ths s the case n the example n Fgure 1, n whch a hgher cost for the transverse edge e 5 results n a shorter equlbrum travel tme on the network. By Proposton 1, the network n Fgure 1 s embedded n every network that s not seres-parallel. Ths mples that Braess s paradox occurs n all such networks. The followng theorem shows that t occurs only n these networks. Theorem 1. Braess s paradox does not occur n a two-termnal network G f and only f G s seres-parallel. 4 Ths defnton may be changed a lttle by replacng every equlbrum wth some equlbrum. Ths change does not affect any of the results below. 11

12 Theorem 1 confrms unproven assertons made by Murchland (1970) and Calvert and Keady (1993). Murchland asserts that deleton of one or more edges from a seres-parallel network cannot be benefcal. Calvert and Keady present a theorem (Theorem 11) statng that Braess s paradox cannot occur n a seres-parallel physcal network n whch the potental dfference between the two end vertces of each edge s determned as an ncreasng functon by the quotent of the flow on the edge and an edge-specfc conductvty factor. Ths refers to a verson of Braess s paradox occurrng when the total power loss n the network can be decreased by reducng the conductvty of some edge, wth the total orgn destnaton flow held constant. Even though seres-parallel networks never exhbt Braess s paradox, they do not always have effcent equlbra. Ths s demonstrated by the example n Fgure 2, n whch the equlbrum flow can be rearranged n such a way that the costs of all used routes are below the equlbrum cost. As the next theorem shows, the reason neffcent equlbra occur n the networks n Fgure 2 s that ther routes are not lnearly ndependent. DEFINITION 2. For gven network G and cost functon c, an equlbrum f, wth equlbrum cost c(f ), s weakly Pareto effcent f, for every flow vector f wth the same total orgn destnaton flow as f, there s some route r wth f r > 0 for whch c r (f) c(f ). The equlbrum s Pareto effcent f, for every flow vector f wth the same total orgn destnaton flow as f, ether c r (f) = c(f ) for all r wth f r > 0 or there s some route r wth f r > 0 for whch c r (f) > c(f ). Theorem 2. For a two-termnal network G, the followng condtons are equvalent: () For any cost functon, all equlbra are weakly Pareto effcent. () For any ncreasng cost functon, all equlbra are Pareto effcent. () G has lnearly ndependent routes. The weak Pareto effcency of the equlbra n networks wth lnearly ndependent routes mples that, n such networks, the equlbrum cost s unquely determned by the cost functon and the total orgn destnaton flow, and can only ncrease or reman unchanged f the former or the latter ncrease. Wth separable cost functons, ths s also true for general seres-parallel networks (see Lemma 4 below). However, for a non-separable cost functon n a seres-parallel network that does not have lnearly ndependent routes, the equlbrum cost may not be unque, or may decrease rather than ncrease wth rsng costs. For example, suppose that, n Fgure 2(a), a toll s charged for usng the equlbrum route oe 1 ve 3 d. Increasng the toll from zero to 1½ decreases the (unque) equlbrum cost lnearly from 4 to 3½. (The lower equlbrum cost s also achevable by the followng turnng restrcton, whch s equvalent to nfnte toll: traffc emergng from e 1 s not allowed to turn nto e 3.) Ths example shows that Theorem 1 would not hold f, n the defnton of Braess s paradox, the assumpton of separable cost functons were dropped. Ths contrasts wth the stuaton n Theorem 2, whch does not assume separablty. Theorem 2 parallels an earler result of Holzman and Law-Yone (1997, 2003) for route selecton games n drected networks wth a fnte number of players. If the set of all 12

13 (drected) routes n a drected network does not contan a bad confguraton (as defned n Secton 2), then, for any (fnte) number of players and any nonnegatve separable cost functon, all the equlbra are weakly Pareto effcent and, moreover, strong n the sense that no group of players can make all ts members better off by changng ther route choces. Conversely, f a trplet of routes consttutng a bad confguraton exsts, then, for any number of players, there s a nonnegatve separable cost functon for whch none of the equlbra s weakly Pareto effcent. For drected networks, the absence of a bad confguraton s a stronger condton than lnear ndependence of the routes (cf. Proposton 4). For ths reason, Theorem 2, n the form gven above, does not hold for such networks. For example, the three routes n the drected Wheatstone network are lnearly ndependent n the sense that each of them has a drected edge that s not n any other route. Nevertheless, the example n Fgure 1 shows that equlbra n ths network are not always weakly Pareto effcent. 5 Non-Identcal Users The most sgnfcant dfference between the fnte route selecton games consdered by Holzman and Law-Yone (1997, 2003) and the present model s that, n the former but not n the latter, heterogenety s a potental source of neffcency. The populaton of users s heterogeneous f there are dfferences n the ntrnsc qualty they assgn to routes or the degree to whch they are affected by congeston. For example, some motorsts may be concerned prmarly wth the travel tme, and others wth the dstance traveled. In fnte populatons (Mlchtach, 1996), such dfferences may lead to neffcent equlbra. Ths s demonstrated by the smple two-person game n whch there are two parallel routes, each favored by a dfferent person. If sharng a route wth the other user s very costly, then there are two pure-strategy Nash equlbra, and the one n whch both persons use ther favorte routes strctly Pareto domnates that n whch each of them uses the other route. Ths example can easly be extended to any nontrval network (.e., one wth more than one route), whch shows that Holzman and Law-Yone s result cannot be extended to heterogeneous fnte populatons. By contrast, t s shown below that Theorem 2 can be extended. In partcular, wth a contnuum of non-dentcal users, a Nash equlbrum may be strctly Pareto domnated by another equlbrum only n a network wth an embedded network as n Fgure 1 or Fgure 2. In networks wthout ths property,.e., wth lnearly ndependent routes, the equlbra are always weakly Pareto effcent, for both heterogeneous and homogeneous populatons. Even wth a contnuum of users, however, there are dfferences between the cases of dentcal and non-dentcal users. In partcular, the result that, n a seres-parallel network, Braess s paradox cannot occur (Theorem 1) does not extend to the case of non-dentcal users. For example, n Fgure 2(a), f half the users were charged a hefty toll for usng edge e 1 and the other half for usng e 3, they would not use these edges (the former would take e 2 and e 3 and the latter e 1 and e 4 ), and consequently the equlbrum costs for all users would decrease from 4 to 3½. More precsely, ths s an example of a natural generalzaton of Braess s paradox, whch s defned below. Ths paradox does not occur n networks wth lnearly ndependent routes, n whch the equlbra are weakly Pareto effcent. As Theorem 13

14 3 below shows, wth non-dentcal users, these are, n fact, the only networks n whch Braess s paradox does not occur. Droppng the assumpton that all users of the same route ncur the same costs leads to the followng modfed verson of the model descrbed n Secton 3. The populaton of users s an nfnte set I (e.g., the unt nterval [0,1]), endowed wth a nonatomc probablty measure (e.g., Lebesgue measure), whch assgns values between zero and one to a σ-algebra of subsets of I, the measurable sets. These values are nterpreted as the set szes relatve to the total populaton. For a network G, a strategy profle s a mappng σ: I R (from users to routes) such that, for each route r, the set of all users wth σ() = r s measurable. The measure of ths set, denoted by f r (σ), s the flow on route r. Thus, for every strategy profle σ, there s a correspondng flow vector f(σ) wth a total orgn destnaton flow of unty. For each user, a cost functon c specfes the cost c p (f) of each path p n G as a functon of the flow vector f. As n the case of dentcal users, c s assumed to satsfy the condton that, for every path p and every par of flow vectors f and f, f f s f s and f s f s for all sectons s of p, then c p (f ) c p (f ). The defntons of ncreasng and separable cost functons are also smlar to those for the case of a homogeneous populaton of users. A strategy profle σ s a Nash equlbrum f each of the routes s a mnmal-cost route for ts users, that s, for each user, c σ() (f(σ)) = mn c q (f(σ)). q R (3) In ths case, the mnmum n (3), denoted by c (f(σ)), s the equlbrum cost for user. In the specal case n whch all users have the same cost functon, ths defnton essentally reduces to (2). DEFINITION 3. For gven network G and assgnment of cost functons c, a strategy profle σ s weakly Pareto effcent f, for every strategy profle τ, there s some user for whch c τ() (f(τ)) c σ() (f(σ)). A strategy profle σ s Pareto effcent f, for every strategy profle τ, ether c τ() (f(τ)) = c σ() (f(σ)) for all users or there s some for whch c τ() (f(τ)) > c σ() (f(σ)). A strategy profle σ s hyper-effcent (Mlchtach, 2004) f, for every strategy profle τ, ether c τ() (f(τ)) = c σ() (f(σ)) for all users or there s some wth τ() σ() for whch c τ() (f(τ)) > c σ() (f(σ)). (H) Braess s paradox wth non-dentcal users occurs n a network G f t s possble to assgn two separable cost functons c and c for each user such that c r (f) c r (f) for all users, routes r and flow vectors f, but for every Nash equlbrum σ wth respect to the frst assgnment and every Nash equlbrum τ wth respect to the second, the equlbrum costs for each user satsfy c (f(σ)) < c (f(τ)). Except for the noton of hyper-effcency, Defnton 3 s a straghtforward generalzaton of Defntons 1 and 2. Hyper-effcency, whch s meanngful also for dentcal uses, means that any effectve change of route choces s harmful to some of those changng routes. Clearly, any hyper-effcent strategy profle σ s both Pareto effcent and a Nash equlbrum. Indeed, 14

15 t s a strong, and even strctly strong, 5 equlbrum. Ths means that devatons are never proftable, not only for ndvduals but also for groups of users: Any devaton that makes someone n the group better off must leave someone else n t worse off. The followng theorem shows that, n a network wth lnearly ndependent routes, f all the cost functons are ncreasng, the converse s also true. That s, under these condtons, every Nash equlbrum s hyper-effcent and, hence, Pareto effcent and a strctly strong equlbrum. Clearly, a smlar result also holds n the specal case of dentcal users. Theorem 3. For a two-termnal network G, the followng condtons are equvalent: () For any assgnment of cost functons, all Nash equlbra are weakly Pareto effcent. () For any assgnment of ncreasng cost functons, all Nash equlbra are hypereffcent. () Braess s paradox wth non-dentcal users does not occur n G. (v) G has lnearly ndependent routes. 6 Dscusson Some propertes of the equlbra n route selecton games wth a contnuum of users are vrtually ndependent of the network topology, and others strongly depend on t. Socal optmalty of the equlbra and the prce of anarchy are among the former (see the ntroducton). Non-occurrence of Braess s paradox and Pareto effcency of the equlbra (ths paper) and (wth non-dentcal users) unqueness of the equlbrum (Mlchtach, 2005; and see below) are among the latter. Dependence of a property on the network topology means that t holds for all allowable cost functons f and only f the network belongs to some specfed non-trval class. As ths paper shows, wth dentcal users, the two-termnal networks n whch Braess s paradox never occurs are the seres-parallel ones (Theorem 1). Those n whch only Pareto effcent equlbra occur are the networks wth lnearly ndependent routes (Theorem 2). Wth non-dentcal users, each of these two propertes s guaranteed to hold f and only f the network has lnearly ndependent routes (Theorem 3). These condtons for topologcal effcency are dfferent from those for topologcal unqueness. A two-termnal network s sad to have the latter property f, for any assgnment of separable cost functons, the flow on each arc s the same n all Nash equlbra. (Ths refers to a heterogeneous populaton of users. Wth a homogeneous populaton, the equlbrum s always essentally unque for any network.) Two equvalent characterzatons of the class of all two-termnal networks wth the topologcal unqueness property are gven n Mlchtach (2005). Ths class s ncomparable wth the two classes consdered n ths paper: Lnear ndependence of the routes s nether a suffcent nor a necessary condton for topologcal unqueness, and the same s true for seres-parallel network. For example, multple equlbra, whch dffer even n the mean cost ncurred by the (non-dentcal) users, may exst n the network wth lnearly ndependent routes shown n Fgure 3, but not n the 5 A strategy profle σ s a strctly strong equlbrum (Voorneveld at al., 1999) f, for every strategy profle τ, c τ() (f(τ)) c σ() (f(σ)) for all users or there s some wth τ() σ() for whch c τ() (f(τ)) > c σ() (f(σ)). 15

16 Wheatstone network (Fgure 1). A parallel network (whch conssts of several edges connected n parallel) belongs to all three classes, and the network obtaned by connectng a sngle edge n parallel wth the Wheatstone network does not belong to any of them. An essental feature of the models n ths paper and Mlchtach (2005) s that networks are undrected. Ths dverges from the common practce of modelng transportaton and other knds of networks by means of drected networks (e.g., Sheff, 1985; Bell and Ida, 1997; Nagurney, 1999), so that each edge can be traversed n only one drecton. (Thus, for example, a two-way hghway s descrbed by a par of edges.) In most of the related lterature, the descrpton of a network nvolves these two knds of data: A drected graph, whch descrbes both the physcal network and the drecton of travel on each edge; and a correspondng system of edge costs, whch gves the cost of each drected edge as a functon of the flow on t. A dfferent knd of transportaton model, descrbed by Beckmann et al. (1956), assumes all roads to be two-way, wth the same travel costs n both drectons. These authors moreover assume that the costs depend only on the sum of the flows n all routes passng through the road n ether drecton. (The two knds of models concde n the specal case of seres-parallel networks. In ths case, all routes pass through an edge n the same drecton, and so the cost of passng through t n the opposte drecton and the effect of the opposte flow on the cost are rrelevant.) The model presented n ths paper subsumes both these knds of models. Here, a separable cost functon assocates wth each edge a par of edge costs one for each drecton. In each drecton, the cost s a functon of the flows on the edge n that and the opposte drecton, and possbly also the juncton flows. The frst knd of model descrbed above corresponds to the specal case n whch the cost of passng through each edge n a partcular drecton s prohbtvely hgh. The second knd corresponds to a case n whch the two costs are equal, and only depend on the sum of the arc flows n both drectons. Ths shows, n partcular, that usng an undrected network does not preclude drectonalty. It only makes t part of the cost functon rather than the network topology, and thus allows t to vary. In other words, an undrected network only precludes predetermned drectonalty. Thus, topologcal effcency essentally means that, regardless of how the edges n the network are drected and the edge costs, neffcent equlbra do not exst. Smlarly, topologcal unqueness refers to absence of multple equlbra, regardless of drectonalty and costs. Smlar propertes may also be defned for drected networks. However, there seem to be no known results lnkng the topology of drected networks wth the effcency or unqueness of the equlbra n nonatomc congeston games, other than those that can be derved from the results n ths paper and Mlchtach (2005) as specal cases. The defnton of cost functon n ths paper s rather wde. It does not assume separablty, whch means that turnng restrctons, for example, can be ncorporated smply by assgnng very hgh (effectvely, nfnte) costs to certan routes (see the example n Secton 4). It also allows route costs to be affected by the flows on ther vertces (whch may represent, for example, a crude measure of congeston at four-way stop junctons). In Mlchtach (2005), a more standard defnton s used, whch requres cost functons to be: () ncreasng, () nonnegatve, and () separable, wth (v) the cost for each user of each edge e n each drecton dependng only on the flow on e n that drecton (and not on the opposte flow or the flows on the end vertces). Adoptng the same restrctve defnton here, n ether the 16

17 homogeneous or the heterogeneous case, would not affect any of the theorems. Indeed, nspecton of the proofs of Theorems 1, 2 and 3 shows them to be also vald f the defnton of cost functon s augmented wth any subset of () (v), to whch the requrement of contnuty of the payoff functon may be added. Ths s manly because all fve propertes hold for the cost functons n the two examples gven n the ntroducton. Thus, n partcular, both the possblty of non-separable cost functons and the possble effects of juncton flows are not crucal elements of the present model. The noton of embeddng n the wde sense used n Mlchtach (2005) s also dfferent from the present noton of embeddng. The former s wder than the latter, but more complcated, and n the present context, t does not offer any advantages. However, t could be used, as both Propostons 1 and 4 can be shown to also hold wth embeddng n the wde sense replacng embeddng. In ths paper, flow s always assumed to orgnate n a sngle vertex o and termnate n a sngle vertex d. Multple orgn destnaton pars are not allowed. Ths restrcton can be partally crcumvented by connectng all sources to a sngle, fcttous, vertex, from whch all flow s assumed to orgnate, and smlarly for the snks. However, such a constructon substantally alters the network topology. Ths leaves open the queston, how does the results n ths paper change when there are more than one orgn or destnaton. 7 Proofs Ths secton gves the proofs for the results presented n ths paper. Proof of Proposton 1. () (). The network n Fgure 1 has an edge, e 5, through whch two routes pass n opposte drectons. It s easy to see that ths property s preserved under the three operatons that defne embeddng. Therefore, a network n whch the one n Fgure 1 s embedded s not seres-parallel. () (). Ths follows from the specal case n whch u and v are the two end vertces of an edge e. () (). Suppose that condton () does not hold for G: There are two routes r and r and two vertces u and v common to both routes, such that u precedes v n r but follows t n r. Suppose that these vertces are chosen n such a way that the length of the secton r uv s maxmal. Then, any vertex u common to r and r that precedes u n r must precede v n r, and any vertex v common to both routes that follows v n r must follow u n r (see Fgure 5). Let u be the last vertex before u n r that s also n r (possbly, u = o), and v the frst vertex after v n r that s also n r (possbly, v = d). All the edges n r u u, and all the vertces n ths secton of r wth the excepton of the ntal and termnal ones, do not belong to r, and the same s true for r vv. Ths mples that the network n Fgure 1 s embedded n the sub-network of G consstng of all the vertces and edges n r u u, r vv and r. Hence, t s also embedded n G. The followng lemma, whch s essentally part of Theorem 3.3 of Harary (1969), s used n the proof of Proposton 2. 17

18 o u' r r' u v v' Fgure 5 d Lemma 1. A network G can be obtaned by connectng two other networks n seres f and only f every two routes n G, dstnct or dentcal, have at least one vertex n common, other than o and d. Proof. The necessty of ths condton s clear. To prove suffcency, suppose that the condton holds, and consder the set of all trplets (p, q, v) consstng of two dstnct routes p and q and a vertex v common to both routes such that: () p ov and q ov do not have common vertces other than o and v (whch mples that v d, snce, by assumpton, routes p and q do have at least three common vertces), and () p vd = q vd (whch mples that v o, snce p q). If ths set s empty, then there s only one edge ncdent wth o, whch mples that G s the result of connectng two networks n seres, one of whch only has that sngle edge. Suppose, then, that the above set s nonempty, and choose an element (p, q, v) such that the length of p vd (= q vd ) s mnmal. CLAIM 1. The vertex v belongs to all routes n G. Suppose the contrary, that v does not belong to some route r. In that route, let v be the frst vertex that s also n p vd, and u the last vertex before v that s also n p or n q (see Fgure 6). 18

19 o p q r u v v' d Fgure 6 Wthout loss of generalty, t may be assumed that u s n p. Consder the route p = p ou + r uv + p v d. Clearly, () p ov and q ov do not have common vertces other than o and v, and () p v d = q v d. However, the secton p v d s shorter than p vd, whch contradcts the way the trplet (p, q, v) was chosen. Ths contradcton proves Clam 1. CLAIM 2. Any vertex u that precedes v n some route n G also precedes t n every other route to whch u belongs. Suppose the contrary, that there are two routes r and r such that some vertex u v common to both routes precedes v n r but follows t n r. Choosng u to be the frst such vertex n r guarantees that r ou and r ud do not have common vertces other than u (see Fgure 5). Clearly, the route r ou + r ud does not nclude v. Ths contradcts Clam 1, and thus proves Clam 2. It follows from Clams 1 and 2 that G s the result of connectng two networks n seres: the network G consstng of v (as destnaton) and all the vertces and edges that precede t n some route n G, and the network G consstng of v (as the orgn) and all the vertces and edges that follow t n some route n G. 19