Social Coordination in Unknown Price-Sensitive Populations

Transcription

1 Socal Coordnaton n Unnown Prce-Senstve Populatons Phlp N. Brown 1 and Jason R. Marden 2 Abstract In ths paper, we nvestgate the relatonshp between uncertanty and a desgner s ablty to nfluence socal behavor. Pgovan taxes are a common approach to socal coordnaton. However, guaranteeng effcent behavor typcally requres that the system desgner has complete nowledge of the user populaton s senstvty to taxaton. In ths paper, we explore the effect of relaxng ths requrement n the context of congeston games wth affne costs. Focusng on the class of scaled Pgovan taxes, we derve the optmal tollng scheme that mnmzes the worst-case effcency loss under uncertanty n user senstvty. Furthermore, we derve explct bounds whch hghlght how the level of uncertanty n senstvty degrades performance. I. INTRODUCTION As engneered systems and ther users become ncreasngly nterconnected, engneers must consder not only system desgn, but also effcent system utlzaton [1]. It s wdely nown that unnfluenced socal systems often exhbt suboptmal behavor. Ths s a well-researched topc n many felds, ncludng electrc power marets [2], communcaton spectrum sharng [3], and traffc congeston [4]. Accordngly, a central desgn challenge s to nfluence socal behavor through admssble mechansms to promote effcency. To analyze the role of socal nfluence n engneered systems, we model the behavor of a system s users as a game between the users, n whch each user chooses hs actons to maxmze hs personal beneft. Equlbrum behavor n whch every user acts n ths way s nown as a Nash equlbrum. It s wdely nown that Nash equlbra need not be effcent n any global sense. It s common to quantfy the effcency losses nherent n Nash equlbra by a metrc nown as the prce of anarchy of a system, defned as the worst-possble rato between the effcency of Nash equlbrum behavor and that of optmal behavor [5], [6]. The prce of anarchy has been analyzed for a varety of engneered systems [7], [8]. For example, the prce of anarchy for congeston on certan road networs has been shown to be 4/3 [9]. Ths leads to the central queston: how may we nfluence socal behavor to reduce the prce of anarchy? A common methodology for ncreasng the effcency of socal systems s nown as a Pgovan tax [10]. Under a Pgovan tax, users are charged a tax equal to the negatve Ths research was supported by AFOSR grants #FA and #FA and by ONR grant #N P. N. Brown s a graduate research assstant wth the Department of Electrcal, Computer, and Energy Engneerng, Unversty of Colorado, Boulder, CO 80309, phlp.brown@colorado.edu. J. R. Marden s wth the Department of Electrcal, Computer, and Energy Engneerng, Unversty of Colorado, Boulder, CO 80309, jason.marden@colorado.edu. Correspondng author. effect they have on other users; thus, each user s ncentvzed to choose actons that maxmze the publc beneft [11]. Generally, Pgovan taxes are attractve snce they guarantee that Nash equlbrum behavor exactly equals system optmal behavor. Ths type of taxaton strategy has been suggested as a soluton to ndustral polluton [12], harmful speculatve tradng n fnancal marets [13], and road congeston [14], to name a few. We turn to the problem of road congeston to llustrate a settng n whch Pgovan taxes apply. In routng games, an equlbrum s nown as a Nash flow; t s well-nown that Nash flows are often neffcent [15]. A central research focus has been to charge tolls that nfluence drvers to mae more effcent routng choces. In the congeston settng, a Pgovan tax s nown as a margnal-cost toll; each drver s charged a toll equvalent to the congeston caused by an addtonal unt of traffc [16], [17]. Ths s a fully-dstrbuted tollng system: a road s toll depends only on the road s congeston propertes. An nherent challenge s that congeston s experenced n unts of tme, but tolls are charged n unts of money. If the desgner lacs perfect nformaton regardng drvers prce-senstvty, t may not be possble to levy an accurate toll. For example, f drver prce-senstvtes are unnown, a Nash flow may be as much as 33% more congested than an optmal flow [9]. On the other hand, f the prce-senstvty of all drvers s nown perfectly, Pgovan tolls cause all Nash flows to be optmal [17], [18]. Recent research has sdestepped ths prce-senstvty ssue by fxng all system varables and dervng a tollng strategy that optmzes a fxed nstance of the system [19]. It has been shown that under a general class of congeston games, ths type of fxed tollng can optmze Nash equlbra [20]. Whle these results represent successes, they rely strongly on a complete characterzaton of the networ structure, user capabltes, and prce senstvtes. It s unnown how varatons n these parameters degrade a desgner s ablty to mtgate effcency losses through tolls. Furthermore, fxed tolls typcally rely on global nformaton and do not provde a dstrbuted soluton. Wth these lmtatons n mnd, we ntate a study on the relatonshp between nowledge of prce senstvty and a desgner s ablty to nfluence socal behavor wth tolls. Specfcally, we nvestgate the set of networ routng games such that each edge n a networ possesses a congeston functon that s affne n the number of edge users. We analyze worst-case Nash flow effcency n these games over a range of user prce-senstvtes. Affne-cost congeston games wth nown senstvtes are common n the lterature [18], [9],

2 [15]; however, to our nowledge, ths paper s the frst prce senstvty analyss of ts nd for congeston games. For ths class of games, under mld assumptons, we show n Theorem 3.2 that the unque margnal-cost-toll scale factor that mnmzes worst-case effcency losses s the geometrc mean of the bounds of the user populaton s prce-senstvty. Ths s a dstrbuted tollng rule: a road s toll depends only on ts congeston propertes and the amount of traffc on the road; t s ndependent of the overall networ topology. Furthermore, we show that ths tollng rule always ncreases effcency wth respect to the un-tolled networ, and we provde a tght bound on the prce of anarchy for games usng our tollng methodology. Ths prce of anarchy depends only on the rato of the user populaton s senstvty bounds; t s ndependent of cost functons, networ topology, and overall traffc rate. For example, f the prce senstvty s only nown wthn an order of magntude, our result gves a prce of anarchy of Ths s a mared mprovement from the the nown prce of anarchy of 4/3 for un-tolled affne-cost congeston games [9]. A. Networ Defnton II. MODEL Consder a networ (V, E, whch conssts of a vertex set V and edge set E (V V. Each edge e E s assocated wth a latency functon of the affne form l e (f e = d e f e + c e, (1 where f e 0 desgnates the flow on edge e, and d e 0 and c e 0 are edge-specfc constants. In ths paper, we focus on the effcency of routng nduced by ndvduals selfsh behavor. Specfcally, we focus on the settng n whch there s a populaton of users seeng to traverse the networ from a common source to a common destnaton. Each user chooses hs path from a common set of paths P = {p 1, p 2,..., p n } where each path p P conssts of a set of edges,.e., P 2 E. We defne the total mass of the populaton to be r > 0. Usng these defntons, let a game G be defned by the tuple G = (V, E, P, {l e } e E, r, and let G represent the class of all such games G. B. Networ Flows For a game G, we call a flow f = (f p1, f p2,..., f pn feasble f p P, f p 0 and p P f p = r, where f p denotes the flow on path p. We denote the set of feasble flows for game G as F(G. For a gven flow f, the latency l p (f of any path p P s the sum of the latences of edges n that path: l p (f = l e (f e, (2 e p where f e s the flow on edge e gven f,.e., f e = p P:e p f p. Thus, the total latency on the networ s the flow-weghted sum of all path latences: L(f = p P f p l p (f p. (3 Users are concerned wth mnmzng ther latences, so the cost J p to a user of any path p P s the path latency: J p (f = e p l e (f e. (4 We focus n partcular on the settng n whch each user chooses the path wth the lowest cost gven the choces of others. A routng obtaned n ths way s nown as a Nash flow [15], whch corresponds to a feasble flow f ne F(G such that for all p, p j P > 0, j > 0 = J p (f ne = J pj (f ne, (5 > 0, j = 0 = J p (f ne J pj (f ne. (6 We defne an optmal flow on G as f opt = nf L(f. f F(G Then we defne the prce of anarchy of G as L(f ne : G PoA(G = sup G G L(f opt : G, (7 Where (f ne : G and (f opt : G denote Nash and optmal flows for G, respectvely. C. Tollng for Known Homogeneous Populatons The networ owner charges a toll τ e (f e on each ln n the networ n an attempt to nfluence drvers to mae effcent routng choces, leadng to a modfed path cost of J τ p (f = e p [l e (f e + τ e (f e ]. (8 A margnal-cost toll for latency functons of the form n (1 s of the form τ e (f e = d e f e. (9 In [15], t s shown that tolls of the form n (9 always nduce optmal behavor, dong so n a dstrbuted manner: each edge toll s a functon only of the flow and congeston propertes of that edge. D. Tollng for Unnown Homogeneous Populatons In real systems, t s often true that users exhbt prcesenstvtes that are unnown or that vary from day to day. We model ths prce senstvty wth a parameter β 0. Intutvely, hgh β mples hgh averson to toll. Incorporatng prce senstvty, we represent the cost of a path generally as J τ p (f = e p [l e (f e + βτ e (f e ]. (10 In the case when β s nown precsely, an optmal margnal-cost toll s of the form τ e (f e = d e f e /β, snce then (10 resolves to (8. However, suppose the senstvty s some unnown β [β L, β U ]. We desre to explot the dstrbuted nature of margnal-cost tolls, so we nvestgate tollng functons of the form τ e (f e = µd e f e, (11 where our control parameter µ 0 s a common factor for every edge that scales wth our estmate of the true β of the

3 populaton. We call tolls of ths form scaled margnal-cost tolls. These tolls further modfy (4 to gve the fnal form of our path cost functon J τ p (f = e p [(1 + βµd e f e + c e ]. (12 These modfed cost functons combned wth (5 and (6 nduce a new Nash flow f ne. For smplcty of notaton, we denote the total latency of a flow f ne gven µ, G, and β as L ne (µ; G, β. Therefore, gven a game G, we see to characterze the scaled margnal-cost tolls whch mnmze the worst-case total latency for G gven a range of user prce senstvtes. That s, we see to characterze solutons µ to the optmzaton problem ( L ne (µ ; G, β = nf µ 0 sup (L ne (µ; G, β β [β L,β U] III. MAIN RESULT. (13 We frst mae the followng assumpton on the utlzaton of the networ. Defnton 3.1: A game G = (V, E, P, {l e } e E, r s fully-utlzed f r r, p P, (µ = 0; G, β > 0. That s, all paths are n use n an un-tolled Nash flow, and ncreasng r wll not cause any path flow to decrease to zero. Smple arguments can show that ths defnton apples ntutvely to a large class of networs, ncludng all networs of parallel lns. In Secton V-A we dscuss some mplcatons of ths defnton. Theorem 3.2: Consder any game G that satsfes Defnton 3.1. Let µ 1 =. (14 βl β U Then µ unquely satsfes ( L ne (µ ; G, β = nf µ 0 sup (L ne (µ; G, β β [β L,β U] and L(µ ; G, β < L(0; G, β. Furthermore, let G represent the class of all games G satsfyng Defnton 3.1. For β [β L, β U ], PoA (µ ; G, β = 4 (1 + β L /β U + β L /β U 3 ( (15 β L /β U Thus, µ ensures that the worst-case total latency of a game s mnmzed for any range-restrcted β. Note that µ depends only on β L and β U ; t s ndependent of G. Ths allows us to defne a dstrbuted rule for our edge tolls so that we can set each edge s toll wthout nowledge of the networ topology or total traffc rate, and guarantees a reducton n total latency from the un-tolled networ. Our expresson for the prce of anarchy n (15 succnctly characterzes the relatonshp between effcency and uncertanty wth respect to prce senstvty. Fgure 1 shows how the prce of anarchy vares wth β L /β U. Note that when β L /β U = 0, meanng that t may not be possble to nfluence, Fg. 1. Prce of anarchy for games usng our scaled margnal-cost tollng rule, as a functon of the precson wth whch we now the user populaton s prce-senstvty. behavor wth tolls, (15 resolves to the well-nown prce of anarchy of 4/3 derved n [9]. On the other hand, f β L = β U > 0, meanng that the prce-senstvty of all users s nown precsely, we obtan a prce of anarchy of 1. Though mperfect nformaton does degrade effcency, the fgure shows that hgh levels of effcency are mantaned even for a relatvely hgh level of uncertanty. For example, even when β s only nown wthn an order of magntude, the prce of anarchy s IV. PROOF OF RESULTS The proof proceeds as follows: 1 In Lemma 4.1 we show that n games satsfyng Defnton 3.1, the flow on any path can be decomposed as the sum of a term dependng on r and a term dependng on µ. 2 In Lemma 4.2 we show that for such games, Lemma 4.1 mples that scaled margnal-cost tolls always decrease latency by a quantty that s ndependent of the traffc rate r. 3 Lastly, we show that the propertes proved n Lemma 4.2 lead to a characterzaton of the unque margnal-cost toll scale factor that mnmzes worstcase total latency for uncertan senstvty. We then derve the prce of anarchy under our optmal margnalcost tolls. Our proof utlzes the followng defnton: for any p, p j P, { e p d j = p j d e, p p j 0, p p j =. (16 Intutvely, d j s the amount that affects Jp τ j (f ne. Note that d j = d j. Lemma 4.1: For any game G satsfyng Defnton 3.1, for any µ 0, the Nash flow on each path p P s of the form 1 = rr µβ M, (17

4 where the coeffcents R R and M R have no dependence on µ, r, or β, and satsfy the followng equatons: R = 1, (18 M = 0, (19 R d = R d j, (20 M d + c e = M d j + c e, (21 e p e p j where p, p j, p P. Proof: Frst, we prove the decomposton of the path flows. Then we show that due to the propertes nduced by Defnton 3.1, the coeffcents R and M satsfy (18-(21. Combnng (12 and (II-B, the cost of path p s Jp τ (f ne = ( ( (1 + βµd e + c e. (22 e p Defne d e (p n the followng way: Now, (22 can be wrtten as p :e p d e (p = { de, e p 0, e / p. Jp τ (f ne = (1 + βµ e p = (1 + βµ = (1 + βµ f ne p d e (p + de (p + e p c e c e f ne p d + e p c e. (23 Snce f ne s a Nash flow and every path flow s postve, by (5 all path costs are equal, so p, p j P, 1 (d d j = c e c e. ( βµ e pj e p Thus, each {, j} defnes an equaton of the form f p ne a j = b j. We tae any dstnct 1 P 1 of these = r to obtan a P -dmensonal system of lnear equatons, whch can be descrbed by the followng matrx equaton: 2 equatons and combne them wth f ne p Af ne = b. (25 A soluton to (25 represents a Nash flow for (G, β, µ, and n [21] t s shown Nash flows always exst for congeston games of the type consdered n ths paper. Snce b s of the form b 1 /(1 + βµ b =. b P 1 /(1 + βµ r, (26 1 Note that {, j} s not dstnct from {j, }. 2 Wth abuse of notaton, here we represent the tuple of path flows as a column vector. any soluton f ne must be a lnear combnaton 3 of 1/(1+βµ and r. That s, p P, can be represented as shown n (17. Note that {R } and {M } have no dependence on µ, r, or β. Clearly, snce G satsfes Defnton 3.1, R must be nonnegatve for every path p P. Note also that µ > 0, (µ; G, β > 0 snce (0; G, β > 0. That s, (17 s robust to ncreases n tolls. Next, we derve two mportant propertes of R and M. Substtutng (17 nto p f ne P p = r, we obtan r R M = 0. ( µβ Snce (27 must hold for any r and for all µ 0, we obtan the proof of (18 and (19. Now, substtutng (17 nto (24 and smplfyng, we obtan r R (d d j µβ M (d d j + c e c e = 0. e p j e p (28 Smlarly to above, snce (28 must hold for any arbtrarly large r and for all µ 0, we obtan the proof of (20 and (21, whch completes the proof. Next, we show that rate and tolls have ndependent effects on the total latency. Ths wll allow us to defne tollng rules wthout nowledge of r. Lemma 4.2: For any game G satsfyng Defnton 3.1, there exsts a functon L r (r dependng only on r, and a constant K µ 0, such that L ne µβ (µ; G, β = L r (r (1 + µβ 2 K µ. (29 Proof: Frst, we show the decomposton of the total latency of a Nash flow. Substtute (17 nto (3 and smplfy to obtan L ne (µ; G, β = L r (r + r ζ + η, ( µβ where L r (r = r 2 d j R R j + r c e R, (31 e p ζ = M R j d j + M j R d j. (32 η = d j (1 + µβ 2 M M j + e p c e µβ M. (33 3 To see ths, perform Gaussan elmnaton on the augmented matrx [A b] to solve for f. Note that matrx row operatons only nvolve sums of elements of b, never products.

5 Note that ζ depends on both r and µ. Here we show that ζ = 0. Choose any p P. By Lemma 4.1, p, p P, p R j P jd j = p R j P jd j, and analogously, p j, p P, p R P d j = p R P d. Thus, (32 can be wrtten ζ = R j d j + R d. M M j By Lemma 4.1, we now that each term n parentheses equals 0, so ζ = 0. Thus, the total latency s the sum of a functon dependent on r and a functon dependent on µ and β, thus decouplng the effects of rate and tolls. Consder now the 1/(1 + µβ 2 term of (33 and denote t K µ : K µ = M d j M j. (34 By (21 n Lemma 4.1, p, p P, we now that d j M j = d j M j + c e c e. e p e p Accordngly, we can wrte K µ as M d j M j + c e c e M. e p e p By Lemma 4.1 we now that p M P = 0. Therefore K µ s smply K µ = c e M. e Factorng out K µ, we can then express (30 as L ne (µ; G, β = L r (r µβ (1 + µβ 2 K µ. (35 To complete the proof of the lemma, we show that K µ 0. Equaton (35 clearly shows that L ne (0; G, β = L r (r. (36 That s, L r (r represents the total latency of the untolled system. Equatons (35 and (36 show that for a gven G, f some toll decreases latency, then all tolls must decrease latency. Snce µ = 1/β always results n optmal routng 4 (.e., a reducton n latency, t must be true that K µ 0. Thus, scaled margnal-cost tolls always decrease total latency. Completon of Proof of Theorem 3.2. Consder the partal dervatve of (29 wth respect to β: L ne (µ; G, β β = ( µ(µβ 1 (1 + µβ 3 K µ. (37 From (37, t s clear that L ne (µ; G, β has a mnmum at β = 1/µ, so the end-ponts of the β-range gve the worst possble latences: sup {L ne (µ; G,β}=max{L ne (µ; G,β L,L ne (µ; G,β U }. β [β L,β U] 4 See Secton II-D. Consder now the partal dervatve of L ne (µ; G, β wth respect to µ: L ne ( (µ; G, β β(µβ 1 = µ (1 + µβ 3 K µ. (38 Clearly, for any β > 0, L ne (µ; G, β has a mnmum at µ = 1/β. Furthermore, snce L ne (1/β; G, β = L(f opt, t must be true that L ne (1/β L ; G, β L = L ne (1/β U ; G, β U. Thus, snce L ne (µ; G, β s contnuous n µ, there must exst some µ [1/β U, 1/β L ] such that L ne (µ ; G, β L = L ne (µ ; G, β U. It can easly be verfed from (29 that µ s as defned n (14 n the theorem statement. Now we bound the neffcency of a game G under tolls as defned n (14. Snce we now that an un-tolled latency can never be more than 4/3 tmes an optmal latency, from (29 we can wrte L ne (0; G, β L(f opt L r (r = L r (r 1 4 K 4 µ 3. (39 Ths allows us to compute a bound on K µ : K µ L r (r. (40 It follows algebracally that for µ as defned n (14, β [β L, β U ], and G as defned n the statement of the theorem, the Prce of Anarchy s gven by the expresson n (15. We show that ths bound s tght n Secton V-B wth Pgou s Example. A. A Note on Full Utlzaton V. DISCUSSION It remans an open queston as to the mportance of Defnton 3.1 for the above results. It s straghtforward to show that many networs ntutvely satsfy the defnton, such as all networs that are composed entrely of smple combnatons of parallel and seres subnetwors. However, there do exst networs for whch Defnton 3.1 never apples, notably the networ used n Braess s Paradox [22]. Ths does not mply that t s mpossble to acheve effcency gans for Braess s networ wth a dstrbuted tollng rule; t smply means that the tollng rule specfed n ths paper s not guaranteed to be the soluton to the optmzaton problem (13. Defnton 3.1 s a suffcent condton for Theorem 3.2, but t should not be nterpreted as necessary. B. Pgou s Example Consder a networ n whch drvers have access to two routes: the frst, a congeston-senstve route; the second, a costly constant-latency route; ths s the canoncal routng problem nown as Pgou s Example [18], llustrated n Fgure 2. Let r = 1. Wthout tolls, all traffc 5 prefers route 1 snce no drver can do better by usng route 2, and the total latency L ne (0; G, β = 1. 5 Note that ths example does not strctly satsfy Defnton 3.1 snce f p2 = 0. Ths wll not adversely mpact our analyss, because we may consder the case n whch r = 1 + ɛ for ɛ > 0. Then f p2 = ɛ and f p1 = 1, so Defnton 3.1 s satsfed. We tae the lmt as ɛ 0, and all analyss proceeds as before.

6 Fg. 2. Pgou s Example. Networ used n Secton V-B to demonstrate the prce of anarchy for scaled margnal-cost tolls and uncertan user populatons. On the other hand, the flow whch mnmzes the total congeston s f p1 = f p2 = 0.5; now half the populaton experences a latency of 0.5 and half 1, so the total latency s L(f opt = 3/4; the prce of anarchy for ths nstance s thus 4/3. Assume that all users share some value of β [β L, β U ]. Charge a toll on the frst ln of τ 1 (f = f 1 µ, where µ 0. The Nash flow nduced by ths toll s such that f p1 = µ/(µ+ β and f p2 = β/(µ + β. Thus the total latency s ( 2 µ L ne β (µ; G, β = + (41 µ + β (µ + β It s easy to verfy that µ = (β L β U 1/2 as prescrbed by Theorem 3.2 mnmzes the worst-case total latency for any β [β L, β U ] and that ths worst-case total latency s L ne (µ ; G, β L = (1 + β L /β U + β L /β U (1 + β L /β U 2. (42 Snce L(f opt = 3/4, (42 proves the tghtness of the prce of anarchy specfed n Theorem 3.2. VI. CONCLUSIONS Effectve desgn for socal coordnaton requres nformaton about a system s user populaton. If ths nformaton s naccurate, an attempt by the desgner to nfluence user behavor may result n users mang neffcent choces. It s essental to quantfy the effect of these naccuraces n order to ensure that attempts to mtgate effcency losses do not nflame the problem. Toward ths end we have demonstrated a postve result: we have developed a dstrbuted tollng strategy for certan road networ settngs that always reduces Nash flow road congeston, even f the user populaton s prce senstvty s not precsely nown. In our model, uncertantes n prce senstvty dd reduce effcency, but relatvely hgh effcency was mantaned even for hgh levels of uncertanty. Ths area has many open questons. In the congeston settng, t remans to be shown whether these results extend to more general congeston games. For example, future research wll focus on elmnatng the requrement that games satsfy Defnton 3.1. It s not nown whether the effcency guarantees proved n ths paper wll stll apply after the removal of ths requrement. Another mportant ssue s that of heterogenety. Our model strongly assumes homogenety of the populaton s prce senstvtes; as research progresses, we hope to relax ths assumpton. Homogenety s rare n practce, and ncorporatng ts effects nto our models wll be an mportant step. Ths style of analyss, n whch we optmze our ncentve scheme over a range of user prce senstvtes, could be appled to a wde varety of problems, both engneerng and economc n nature. From electrc power marets to energy effcency tax credts, prce senstvty plays an ntegral role n many engneerng and socal systems. Ths paper represents an ntal foray nto a broader analyss of the mpact of prce senstvty uncertanty on desgners ablty to promote effcency through fnancal ncentves. REFERENCES [1] G. E. Bolton and A. Ocenfels, Behavoral Economc Engneerng, Journal of Economc Psychology, vol. 33, pp , June [2] C. Slva, Applcaton of Mechansm Desgn to Electrc Power Marets (Republshed, IEEE Transactons on Power Systems, vol. 16, no. 4, pp , [3] D. Nyato and E. Hossan, Compettve Prcng for Spectrum Sharng n Cogntve Rado Networs: Dynamc Game, Ineffcency of Nash Equlbrum, and Colluson, IEEE Journal on Selected Areas n Communcatons, vol. 26, no. 1, pp , [4] S. Morrson, A Survey of Road Prcng, Transportaton Research Part A: General, no. 2, pp , [5] C. Papadmtrou, Algorthms, Games, and the Internet, n Proc. of the 28th Internatonal Colloquum on Automata, Languages and Programmng, [6] E. Koutsoupas and C. Papadmtrou, Worst-Case Equlbra, Computer Scence Revew, vol. 3, pp , May [7] R. Johar and J. N. Tstsls, Effcency Loss n a Networ Resource Allocaton Game, Mathematcs of Operatons Research, vol. 29, pp , Aug [8] H. Youn, M. Gastner, and H. Jeong, Prce of Anarchy n Transportaton Networs: Effcency and Optmalty Control, Physcal Revew Letters, vol. 101, pp to 4, Sept [9] T. Roughgarden, How Bad Is Selfsh Routng?, Journal of the ACM JACM (2002, vol. 49, no. 2, pp , [10] N. G. Manw, Smart Taxes: An Open Invtaton to Jon the Pgou Club, Eastern Economc Journal, vol. 35, no. 1, pp , [11] W. Sandholm, Negatve Externaltes and Evolutonary Implementaton, The Revew of Economc Studes, vol. 72, pp , July [12] W. Baumol, On Taxaton and the Control of Externaltes, The Amercan Economc Revew, vol. 62, no. 3, pp , [13] J. Stgltz, Usng Tax Polcy to Curb Speculatve Short-Term Tradng, Journal of Fnancal Servces Research, vol. 3, no. 2-3, pp , [14] I. W. H. Parry, M. Walls, and W. Harrngton, Automoble Externaltes and Polces, SSRN Electronc Journal, [15] T. Roughgarden, Selfsh Routng and the Prce of Anarchy. MIT Press, [16] R. Cole, Y. Dods, and T. Roughgarden, How Much Can Taxes Help Selfsh Routng?, Journal of Computer and System Scences, vol. 72, pp , May [17] W. Sandholm, Evolutonary Implementaton and Congeston Prcng, The Revew of Economc Studes, vol. 69, no. 3, pp , [18] A. C. Pgou, The Economcs of Welfare [19] R. Cole, Y. Dods, and T. Roughgarden, Prcng Networ Edges for Heterogeneous Selfsh Users, Proc. of the thrty-ffth ACM symp. on Theory of computng - STOC 03, p. 521, [20] L. Flescher, K. Jan, and M. Mahdan, Tolls for Heterogeneous Selfsh Users n Multcommodty Networs and Generalzed Congeston Games, 45th Annu. IEEE Symp. on Foundatons of Computer Scence, pp , [21] J. Wardrop, Some Theoretcal Aspects of Road Traffc Research, n Proc. of the Insttuton of Cvl Engneers, Part II, Vol. 1, No. 36, pp , [22] D. Braess, A. Nagurney, and T. Waolbnger, On a Paradox of Traffc Plannng, Transportaton Scence, vol. 39, pp , Nov