Social Coordination in Unknown Price-Sensitive Populations

Transcription

1 Socal Coordnaton n Unknown Prce-Senstve Populatons Phlp N. Brown 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 knowledge 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 mere system desgn, but also effcent system utlzaton []. It s wdely known that unnfluenced socal systems often exhbt suboptmal behavor. Ths s a well-researched topc n many felds, ncludng electrc power markets [2], communcaton spectrum sharng [3], and traffc congeston [4]. Accordngly, a central desgn challenge s to nfluence socal behavor through admssble mechansms n these systems to promote effcent outcomes. To analyze the role of socal coordnaton 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 establshed by every user actng n ths way s known as a Nash equlbrum. It s wdely known 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 known as the prce of anarchy of a system, defned as the worst-possble rato between the effcency of a Nash equlbrum behavor and that of an optmal behavor [5], [6]. The prce of anarchy has been analyzed for a varety of engneered systems [7], [8], [9]. Ths leads to the central queston: how may socal behavor be nfluenced to reduce the prce of anarchy? A common methodology for socal coordnaton s known as a Pgovan tax [0]. Under a Pgovan tax, users are charged a tax equal to the negatve effect they have on other users; thus, each user s ncentvzed to choose actons that 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. maxmze the publc beneft. In a general sense, 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 [], harmful speculatve tradng n fnancal markets [2], and road congeston [3], to name a few. To llustrate a socal coordnaton settng n whch Pgovan taxes apply, we turn to the problem of road congeston. It s well-known that Nash equlbra n routng games are often neffcent [4], and a central research focus has been to nfluence routng behavor usng tolls to promote effcent routng. In the congeston settng, a Pgovan tax s known as a margnal-cost toll or congeston toll; each drver s charged a toll equvalent to the congeston caused by an addtonal unt of traffc [5], [6], [7]. An nherent challenge s that congeston s experenced n unts of tme, but tolls are charged n unts of money. If the desgner lacks perfect nformaton regardng drvers prce-senstvty, t wll not n general be possble for the desgner to levy a perfect toll. Recent research has sdestepped ths ssue by fxng all system varables and dervng a tollng strategy that optmzes a fxed nstance of the system [8]. It has been shown that under a general class of congeston games, ths type of fxed tollng can optmze Nash Equlbra [9]. Whle these results represent successes, they rely strongly on a complete characterzaton of the network structure, user capabltes, and prce senstvtes. Accordngly, t may be that small varatons n any of these parameters could lead to hghly suboptmal behavor; uncertanty may dramatcally reduce a desgner s ablty to nfluence socal behavor and could n general lead to a degradaton n effcency from the unnfluenced case. Lttle research has been done to characterze the effect of ths uncertanty on the ablty to coordnate a socal system. Wth these lmtatons n mnd, we ntate a study on the relatonshp between nformaton and a desgner s ablty to nfluence socal behavor. More concsely, we nvestgate uncertan user prce-senstvty for a routng problem n a class of congeston games: the set of network routng games such that each edge n a network possesses a congeston functon that s affne n the number of edge users. Congeston games of ths type have been extensvely studed n the exstng lterature [20], [2], [4]. For ths class of congeston games, we develop a dstrbuted tollng scheme that ncreases routng effcency for any bounded uncertanty of the user populaton s prcesenstvty. We show that under certan common utlzaton condtons, provded that the populaton s prce-senstvty s

2 range-restrcted, scaled margnal-cost tolls always ncrease effcency. Furthermore, we show that the unque margnalcost toll scale factor that mnmzes the prce of anarchy s the geometrc mean of the upper and lower bounds of the populaton s prce senstvty. That s, our toll on a road s a functon only of that road s congeston propertes and the rato of the populaton s senstvty bounds t s ndependent of the traffc flow rate and the network topology. We also provde a tght bound on the prce of anarchy for games usng our specfed tollng methodology. Ths prce of anarchy depends only on the rato of the user populaton s senstvty bounds; t s ndependent of network cost functons, network topology, and overall traffc rate. Thus, for a broad class of games, we concsely characterze the relatonshp between nformaton and effcency. A. Network Defnton II. MODEL Consder a network (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, ( where f e desgnates the flow on edge e. In ths paper, we wll focus on characterzng the effcency of flows nduced by ndvduals strategc behavor. Specfcally, we focus on the settng n whch there s a populaton of users seekng to traverse the network from a common source to a common destnaton. Consequently, each user chooses ts path from a common set of paths P = {p, 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. Network Flows For a game G, the set of feasble flows f(g = (f p, f p2,..., f pn s characterzed by F (G = {f p} p P : f p 0 and f p = r, p P where f p desgnates the flow on path p. For a gven flow f(g F (G, the latency l p (f(g of any path p P s the sum of the latences of edges n that path: l p (f(g = e p l e (f e, (2 where f e s the flow on edge e gven f(g,.e., f e = f p. (3 p P:e p Thus, the total latency on the network s the flow-weghted sum of all path latences: L(f(G = p P f p l p (f p. (4 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(g = e p l e (f e. (5 We focus n partcular on the settng n whch each user chooses the path wth the lowest cost. A routng obtaned n ths way s known as a Nash flow [4], whch corresponds to a feasble flow f ne (G F (G such that for all p, p j P > 0, j > 0 = J p (f ne (G = J pj (f ne (G, (6 > 0, j = 0 = J p (f ne (G J pj (f ne (G. (7 We defne an optmal flow on G as f opt (G = nf L(f(G. f(g F (G Then we defne the prce of anarchy of G as L(f ne (G : G PoA(G = sup G G L(f opt (G C. Tollng for Known Homogeneous Populatons The network owner, wshng to enforce optmal routng, charges a toll τ e (f e on each lnk n the network, addng to the lnk user s cost. A Pgovan toll, also known as a margnal-cost toll, s of the form τ e (f e = d e f e. In general, these margnal-cost tolls nduce optmal routng. Homogenety s a common assumpton when analyzng worst-case behavor [2]; t s a smple approach that can often be used to bound the behavor of heterogeneous populatons. The clear beneft of tollng functons of the Pgovan form s that they can be assgned n a dstrbuted manner; each edge toll s a functon only of the congeston propertes of that edge. For latency functons of the form n (, Pgovan tolls modfy (5 to gve a path cost of J p (f(g = e p (8 [2d e (f e + c e ]. (9 In [4], t s shown that cost functons of ths form always nduce optmal behavor. D. Tollng for Unknown Homogeneous Populatons In real systems, t s often true that users exhbt prcesenstvtes that are unknown or that vary from day to day. We model ths prce senstvty wth a parameter β [β L, β U ]. Intutvely, hgh β mples hgh averson to toll. Incorporatng prce senstvty, we represent the cost of a path generally as J p (f(g = e p [l e (f e + βτ e (f e ]. (0 In the case when β s known precsely, an optmal margnal-cost toll s of the form τ e (f e = d e f e /β; wth ths toll, (0 clearly smplfes to (9. 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, where µ 0 s a common factor for every edge, and scales for our estmate of the true β of the populaton. We call tolls of ths form

3 scaled margnal-cost tolls. These tolls further modfy (5 to gve the fnal form of our path cost functon J p (f(g = e p [( + βµd e f e + c e ]. ( These modfed cost functons nduce a new Nash flow whch we denote f ne (µ : G, β. For smplcty of notaton, we defne the total latency of ths Nash flow under scaled margnal-cost tolls as L ne (µ : G, β = L(f ne (µ : G, β, (2 Where L(f( s defned as n (4. Note that µ s our control parameter; G and β represent the system and populaton we are attemptng to control. Therefore, gven a game G, we seek 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 seek 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. (3 We show that under reasonable utlzaton constrants, the soluton to the optmzaton problem defned n (3 s ndependent of the underlyng game G. Defnton 3.: 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. In Secton V-A we show that ths defnton apples to a broad class of games. Theorem 3.2: Consder any game G that satsfes Defnton 3.. Let µ =. (4 βl β U Then µ satsfes L ne (µ : G, β = nf µ 0 ( sup (L ne (µ : G, β β [β L,β U] Furthermore, let G represent the class of all games G satsfyng Defnton 3.. For β [β L, β U ], 4 ( + β L /β U + β L /β U PoA (µ : G, β = 3 ( + 2. (5 β L /β U Thus, µ ensures that the worst-case total latency of a game s lmted to a mnmum for any range-restrcted β. Note that when β L / = 0, meanng that t s not possble to nfluence behavor wth tolls, we obtan the well-known prce of anarchy of 4/3 derved n [2]. On the other hand, f β L = β U, meanng that the prce-senstvty of all users s known precsely, we obtan a prce of anarchy of, mplyng that complete nformaton allows complete control. Note also 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 knowledge of the overall network topology, and guarantees a reducton n total latency from the un-tolled network. IV. PROOF OF RESULTS The proof proceeds as follows: In Lemma 4. we show that n games satsfyng Defnton 3., 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, 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 2 lead to a characterzaton of the unque margnal-cost toll scale factor that mnmzes worst-case total latency for uncertan senstvty. We then derve the prce of anarchy under our optmal margnal-cost tolls. For smplcty, we wll henceforth express (µ : G, β as merely fp ne when the dependence on µ, G, and β s clear. 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 =. (6 Intutvely, d j s the amount that affects J pj (f ne. Note that d j = d j. Lemma 4.: For any game G = (V, E, P, {l e } e E, r satsfyng Defnton 3., for any µ 0, the Nash flow on each path p P s of the form = rr + + µβ M, (7 where the coeffcents R R and M R have no dependence on µ, r, or β, and satsfy the followng equatons: R =, (8 M = 0, (9 R k d k = R k d jk, (20 M k d k + c e = M k d jk + c e, (2 e p e p j where p, p j, p k P. Proof: Frst, we the decomposton of the path flows. Then we show that due to the propertes nduced by Defnton 3., the coeffcents defned n Lemma 4. satsfy (8- (2. Combnng ( and (3, the cost of path p s J p (f ne = ( ( ( + βµd e + c e. (22 e p Defne d e (p n the followng way: k p k :e p k d e (p = { de, e p 0, e / p.

4 Now, (22 can be wrtten as J p (f ne = ( + βµ e p = ( + βµ = ( + βµ f ne p k d e (p k + de (p + e p c e c e f ne p k d k + e p c e. (23 Snce f ne s a Nash flow and every path flow s postve, by (6 all path costs are equal, so p, p j P, k (d k d jk = c e c e. (24 + βµ e pj e p Thus, each {, j} defnes an equaton of the form f p ne k a j = b j. We take any dstnct P of these = r, and we have a P -dmensonal system of lnear equatons, whch can be descrbed by the followng matrx equaton: equatons and combne them wth f ne p k Af ne = b. 2 (25 A soluton to (25 represents a Nash flow for (G, β, µ, and n [22] t s shown Nash flows always exst for congeston games of the type consdered n ths paper. Snce b s of the form b /( + βµ b =. b P /( + βµ r, (26 Any soluton f ne must be a lnear combnaton of /(+βµ and r. 3 That s, p P, can be represented as shown n (7. Note that {R } and {M } have no dependence on µ, r, or β. Clearly, snce G satsfes Defnton 3., 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, (7 s robust to ncreases n tolls. Next, we derve two mportant propertes of R and M. Substtutng (7 nto p f ne P p = r, we obtan r R + M = 0. (27 + µβ Snce (27 must hold for any arbtrarly large r and for all µ 0, we obtan the proof of (8 and (9: R =, M = 0. 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. 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. Now, substtutng (7 nto (24 and smplfyng, we obtan r R k (d k d jk + + µβ M k (d k d jk + 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 (2: R k d k = R k d j, M k d k + c e = M k d jk + c e, e p e p j Whch completes the proof. Lemma 4.2: For any game G = (V, E, P, {l e } e E, r satsfyng Defnton 3., 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 ( + µβ 2 K µ. (29 Here, we show for total latency a smlar qualty to that expressed for flows n Lemma 4.; namely, rate and tolls have ndependent effects on the total latency. Ths s a crucallymportant property of games satsfyng Defnton 3., snce t allows us to defne tollng rules wthout knowledge of r. Proof: Frst, we show the decomposton of the total latency of a Nash flow. We then show that the total latency of every Nash flow s of the specfc form shown n Lemma 4.2. Substtute (7 nto (4 and smplfy to obtan L ne (µ : G, β = L r (r + r ζ + η, (30 + µβ where L r (r = r 2 d j R R j + r c e R, (3 e p ζ = η = d j (R j M + R M j, d j ( + µβ 2 M M j + c e e p + µβ M. (32 Note that the second term n (30 depends on both r and µ. The followng argument shows that ζ = 0. By factorng and reversng sum order, we obtan ζ = M R j d j + M j R d j. Choose any p k P. By Lemma 4., p, p k P, R jd j = R jd kj, and analogously, p j, p k

5 P, p R P d j = p R P d k. Then ζ can be wrtten as ζ = R j d kj + R d k. M M j By Lemma 4., we know that each term n parentheses equals 0, so ζ = 0. Ths allows us to express the total latency as the sum of a functon dependent on r, and a functon dependent on µ and β. Thus, we see here the decouplng of the effects of rate and tolls. Consder now the /( + µβ 2 term of (32 and denote t K µ : K µ = M d j M j. (33 By Lemma 4., p, p k P, we know that d j M j = d kj M j + c e c e. e p k e p Accordngly, we can wrte K µ as M d jk M j + c e M c e. e p k e p By Lemma 4. we know that p M P = 0. Therefore K µ s smply K µ = c e M. e We can then express (30 as ( L ne (µ : G, β = L r (r + ( + µβ 2 ( + µβ = L r (r K µ µβ ( + µβ 2 K µ. (34 To complete the proof of the lemma, we show that K µ 0. Equaton (34 clearly shows that L ne (0 : G, β = L r (r. (35 That s, L r (r represents the total latency of the un-tolled system. Equatons (34 and (35 show that for any game G, f some toll decreases latency, then all tolls must decrease latency. Snce µ = /β always results n optmal routng 4 (.e., a reducton n latency, t must be true that for any G, K µ 0. Thus, scaled margnal-cost tolls always decrease total latency. Ths property of the total latency of a Nash flow lets us concsely quantfy the effect of tolls, and leads drectly to our proof of the optmal margnal-cost toll scale factor. Proof of Theorem 3.2. Consder the partal dervatve of (29 wth respect to β: L ne (µ : G, β β 4 See (9 and ( n Secton II. ( = µβ β ( + µβ 2 K µ ( µ(µβ = ( + µβ 3 K µ. (36 From (36, t can be seen that L ne (µ : G, β has a mnmum at β = /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] Consder now the partal dervatve of L ne (G, β, µ wth respect to µ: L ne ( (µ : G, β β(µβ = µ ( + µβ 3 K µ. (37 Clearly, for any β > 0, L ne (µ : G, β has a mnmum at µ = /β. Snce L ne (/β : G, β = L(f opt (G, t must be true that L ne (/β L : G, β L = L ne (/β U : G, β U. Thus, snce L ne (µ : G, β s contnuous n µ, there must exst some µ [/β U, /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 = βl β U. (38 Now we bound the neffcency of a game G under tolls as defned n (38. Snce we know 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 r (r L opt = (G L r (r 4 K µ Ths allows us to compute a bound on K µ : 4 3. (39 K µ L r (r. (40 It follows algebracally that for µ as defned n (38, β [β L, β U ], and G as defned n the statement of the theorem, 4 ( + β L /β U + β L /β U PoA(µ : G, β = 3 ( + 2. β L /β U We show that ths bound s tght n Secton V-B wth Pgou s Example. A. A Note on Full Utlzaton V. DISCUSSION To see that Defnton 3. apples to a broad class of networks, consder a smple network of n parallel lnks, wth latency functons of the form specfed n (. Due to the smplcty of the network, P = E. Let Ḡ = (V, E, P, {l e } e E, r. Suppose that every path p carres postve rate (Ḡ n a Nash flow. That s, all lnks have equal cost. Now, let G = (V, E, P, {l e } e E, r,

6 where r > r. That s, our only change from Ḡ to G s that we ncreased the traffc rate for G. By ncreasng the overall traffc rate, there must be some path p j such that j (G > f ne (Ḡ. Thus, p j l f = J pj (f ne p j (G J pj (f ne p j (Ḡ. Ths mples that for any path p p j, J p (f ne p (G J pj (f ne p j (Ḡ = J p (f ne p (Ḡ. Snce cost functons are nondecreasng, ths mples that (G (Ḡ. That s, no path flow decreased; thus, all networks of n parallel lnks such that all lnks carry postve flow satsfy Defnton 3.. There are many ways to extend ths class of networks; for example, smple arguments can show that any combnaton of parallel networks n seres wth other parallel networks also satsfes Defnton 3.. B. Pgou s Example To demonstrate these results, let us turn to a smple canoncal network routng problem known as Pgou s Example [20], llustrated n Fgure. In ths example, traffc has access to two routes: the frst, a costly constant-latency route; the second, a congeston-senstve route. Let r =. Wthout tolls, all traffc prefers route 2 snce no user can mprove her latency by usng route, and the total latency L ne (0 : G, β =. 5 However, consder the optmal routng: the traffc splts evenly between the routes so that f p = f p2 = 0.5; now half the populaton experences a latency of 0.5 and half, so the total latency s L(f opt (G = 3/4, a 25% mprovement. We know that chargng a toll equal to f 2 (correspondng to µ = /β; see (9 and ( n Secton II on route 2 wll enforce ths optmal flow. Now assume that all users share some value of β [β L, β U ]. Consder for example the case when β = β L. If our toll s f 2 /β L, the routng wll stll be optmal. However, for β > β L, a dsproportonate number of users wll dvert to route. Now, the Nash flow nduced by µ > 0 s such that f p = β/(µ + β and f p2 = µ/(µ + β. Thus the total latency s ( 2 L ne β µ (µ : G, β = (µ + β + µ + β = (µ2 + β(µ + β (µ + β 2. (4 It s easy to verfy that for µ = µ = (β L β U /2 as prescrbed by Theorem 3.2 and β = β L, the total latency s L ne (µ : G, β L = ( + β L /β U + β L /β U ( + β L /β U 2. (42 Snce L(f opt (G = 3/4, the result n (42 proves the tghtness of the prce of anarchy specfed n Theorem 3.2. l 2 f 2 = f 2 Fg.. Pgou s Example. Network used n Secton V-B to demonstrate the prce of anarchy for scaled margnal-cost tolls and uncertan user populatons. β µ τ 2 f p f p2 L ne (µ : G, β β L = f p β U = 9 9f p β L = /3 (/3f p β U = 9 /3 3f p TABLE I TOLLS AND TOTAL LATENCY IN PIGOU S EXAMPLE Table I presents some numercal examples. In partcular, f β [, 9], then for µ = µ = /3, the total latency wll never be worse than 0.825, nearly a 9% mprovement over the un-tolled case. C. Braess Topology To llustrate a network that may not satsfy Defnton 3., we turn to a second canoncal example: that of Braess s Paradox [23], [4]. The topology n Braess s Paradox has been used to show that addng lnks to a graph can ncrease the cost of a Nash flow; we use t to show that ncreasng the total rate of flow can cause an ndvdual path flow to decrease. Consder the network n Fgure 2. Note that n the canoncal Braess s Paradox network, d 2 = d 3 = 0. We consder two varants of the network: n Varant, d 2 = d 3 = 0; n Varant 2, d 2 =, d 3 = 2. Traffc routes from node A to node D, and has a choce of three paths. Defne p = {e, e 3 }, p 2 = {e, e 5, e 4 }, and p 3 = {e 2, e 4 }. The values of R (refer to Lemma 4. n the Proof sectons for each varant appear n Table II. For Varant, R 2 =. Ths means that ncreasng the total traffc rate wll actually decrease the amount of traffc usng path 2, whch s to say edge 5 (refer to Lemma 4.. Thus ths network never satsfes Defnton 3., and our optmal toll analyss s not necessarly vald. 5 The astute reader wll notce that ths example does not strctly satsfy Defnton 3.. Ths wll not adversely mpact our analyss, because we may consder the case n whch r = +ɛ for ɛ > 0. Then f p = ɛ and f p2 =, so Defnton 3. s satsfed. We take the lmt as ɛ 0, and all analyss proceeds as before.

7 .4.35 Prce of Anarchy Prce of Anarchy 4/3.05 Fg. 2. Braess s Topology. For d 2 = d 3 = 0, no game usng ths topology satsfes Defnton 3.. However, for d 2 =, d 3 = 2, games usng ths topology can satsfy Defnton 3.. Varant R R 2 R 3 2 /3 /6 /2 TABLE II VALUES OF R FOR BRAESS TOPOLOGY For Varant 2, the topology s unchanged, but the modfed latency functons have changed routng behavor dramatcally. Now R 0 for every path, so all path flows are ncreasng n r, and thus f all paths are n use, the game satsfes Defnton 3.. Thus, scaled margnal-cost tolls can never degrade network performance. D. Prce of Anarchy Our expresson for the prce of anarchy n Theorem 3.2 succnctly characterzes the relatonshp between effcency and nformaton about the user populaton. It llustrates the ntutve prncple that wth perfect nformaton, a desgner can acheve perfect effcency. Fgure 3 shows how the prce of anarchy vares wth β L /β U. Note that for any value of ths rato, the prce of anarchy s less than 4/3, that s, less than that of an un-tolled network. Though mperfect nformaton does degrade effcency, the fgure shows that hgh levels of effcency are mantaned even for a relatvely hgh level of uncertanty. Note that even when β s only known wthn a factor of 0, the effcency loss s less than 0%. Our prce of anarchy result s encouragng for two reasons. Frst, ths worst-case analyss was obtaned from assumng that the entre populaton les at one of the extremes of the range. Ths assumpton, though mportant to establsh the bound, s lkely to be unrealstc for most systems. Any β closer to our geometrc mean tollng parameter wll yeld even better performance than the prce of anarchy suggests. Ths s llustrated n Fgure 4; the fgure was generated usng values from Pgou s Example n (see Secton V-B wth r =, β L =, β U = 9, and µ = /3. The horzontal axs represents the true β of the populaton; thus t s easy to see Rato of mnmum and maxmum betas Fg. 3. Prce of anarchy for games usng our scaled margnal-cost tollng rule, as a functon of the precson wth whch we know the user populaton s prce-senstvty. Effcency Loss User Populaton s True Beta Fg. 4. Effcency Loss n Pgou s Example wth respect to true beta for µ = /3 and r =. Note that the effcency loss s equal at the endponts of the β-range, and perfect effcency s acheved at the geometrc mean of the endponts. that values close to the geometrc mean allow only very low effcency loss. Second, the analyss contans a strong assumpton of homogenety; all users are requred to have dentcal values of β. Ths s also unlkely n real systems. Consder a populaton whose β-values are dstrbuted over some range. More rgorous analyss s needed, but our smulatons have suggested that f a greater mass of users are closer to the geometrc mean than to the endponts of the range, tolls would acheve consderably mproved performance. VI. CONCLUSIONS Effectve desgn for socal coordnaton requres nformaton about a system s user populaton; we have provded ntal results on the relatonshp between the qualty of ths nformaton and the effcency of Nash equlbra resultng from a tollng scheme. In our model, mperfect nformaton

8 caused a bounded degradaton of effcency, but relatvely hgh effcency was mantaned even for hgh levels of uncertanty. Future research wll focus on extendng ths result to broader classes of games. In the model addressed n ths paper, we have shown that even under uncertanty, t s possble to acheve sgnfcant gans n the effcency of socal behavor wth a smple dstrbuted taxaton scheme. 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 the effects of heterogenety nto our models wll be a crucal step. In partcular, uncertan heterogeneous populatons may be well-suted to heterogeneous tolls or a level of prce dscrmnaton. A crucal queston s: what gans n effcency are possble f we charge dfferent tolls for dfferent users? Ths work has found the optmal toll among a specfc class of tollng functons: Pgovan margnal-cost tolls scaled by a unform factor. It remans to be shown how our tolls compare wth tolls of some other form. Are these tolls the best of any possble toll? A more comprehensve revew of general taxaton functons for uncertan populatons s requred to answer ths queston. [7] M. J. Smth, The Margnal Cost Taxaton of a Transportaton Network, Transportaton Research Part B: Methodologcal, vol. 3, no. 3, pp , 979. [8] R. Cole, Y. Dods, and T. Roughgarden, Prcng network edges for heterogeneous selfsh users, Proc. of the thrty-ffth ACM symp. on Theory of computng - STOC 03, p. 52, [9] L. Flescher, K. Jan, and M. Mahdan, Tolls for heterogeneous selfsh users n multcommodty networks and generalzed congeston games, 45th Annu. IEEE Symp. on Foundatons of Computer Scence, pp , [20] A. C. Pgou, The Economcs of Welfare [2] T. Roughgarden, How Bad Is Selfsh Routng?, Journal of the ACM JACM (2002, vol. 49, no. 2, pp , [22] J. Wardrop, Some theoretcal aspects of road traffc research, n Proc. of the Insttuton of Cvl Engneers, Part II, Vol., No. 36, pp , 952. [23] D. Braess, Uber en Paradoxon aus der Verkehrsplanung, Unternehmensforschung, pp , 968. REFERENCES [] G. E. Bolton and A. Ockenfels, Behavoral economc engneerng, Journal of Economc Psychology, vol. 33, pp , June 202. [2] C. Slva, Applcaton of mechansm desgn to electrc power markets (Republshed, IEEE Transactons on Power Systems, vol. 6, no. 4, pp , 200. [3] D. Nyato and E. Hossan, Compettve Prcng for Spectrum Sharng n Cogntve Rado Networks : Dynamc Game, Ineffcency of Nash Equlbrum, and Colluson, IEEE Journal on Selected Areas n Communcatons, vol. 26, no., pp , [4] S. Morrson, A survey of road prcng, Transportaton Research Part A: General, no. 2, pp , 986. [5] C. Papadmtrou, Algorthms, Games, and the Internet, n Proc. of the 28th Internatonal Colloquum on Automata, Languages and Programmng, 200. [6] E. Koutsoupas and C. Papadmtrou, Worst-case equlbra, Computer Scence Revew, vol. 3, pp , May [7] R. Johar and J. N. Tstskls, Effcency Loss n a Network Resource Allocaton Game, Mathematcs of Operatons Research, vol. 29, pp , Aug [8] G. Peraks and G. Roels, The Prce of Anarchy n Supply Chans: Quantfyng the Effcency of Prce-Only Contracts, Management Scence, vol. 53, pp , Aug [9] H. Youn, M. Gastner, and H. Jeong, Prce of Anarchy n Transportaton Networks: Effcency and Optmalty Control, Physcal Revew Letters, vol. 0, p. 2870, Sept [0] N. G. Mankw, Smart Taxes: An Open Invtaton to Jon the Pgou Club, Eastern Economc Journal, vol. 35, no., pp. 4 23, [] W. Baumol, On taxaton and the control of externaltes, The Amercan Economc Revew, vol. 62, no. 3, pp , 972. [2] J. Stgltz, Usng tax polcy to curb speculatve short-term tradng, Journal of Fnancal Servces Research, vol. 5, no. May, pp. 0 5, 989. [3] I. W. H. Parry, M. Walls, and W. Harrngton, Automoble externaltes and polces, SSRN Electronc Journal, [4] T. Roughgarden, Selfsh routng and the prce of anarchy. MIT Press, [5] R. Cole, Y. Dods, and T. Roughgarden, How much can taxes help selfsh routng?, Journal of Computer and System Scences, vol. 72, pp , May [6] H. Yang and H.-J. Huang, Prncple of margnal-cost prcng: how does t work n a general road network?, Transportaton Research Part A: Polcy and Practce, vol. 32, pp , Jan. 998.