An Improved Tax Scheme for Selfish Routing

LIPICS - Leibniz International Proceedings in Informatics, Dec 2016

We study the problem of routing traffic for independent selfish users in a congested network to minimize the total latency. The inefficiency of selfish routing motivates regulating the flow of the system to lower the total latency of the Nash Equilibrium by economic incentives or penalties. When applying tax to the routes, we follow the definition of [Christodoulou et al, Algorithmica, 2014] to define ePoA as the Nash total cost including tax in the taxed network over the optimal cost in the original network. We propose a simple tax scheme consisting of step functions imposed on the links. The tax scheme can be applied to routing games with parallel links, affine cost functions and single-commodity networks to lower the ePoA to at most 4/3 - epsilon, where epsilon only depends on the discrepancy between the links. We show that there exists a tax scheme in the two link case with an ePoA upperbound less than 1.192 which is almost tight. Moreover, we design another tax scheme that lowers ePoA down to 1.281 for routing games with groups of links such that links in the same group are similar to each other and groups are sufficiently different.

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An Improved Tax Scheme for Selfish Routing

I S A A C An Improved Tax Scheme for Selfish Routing? Te-Li Wang 0 Chih-Kuan Yeh 0 Ho-Lin Chen 0 0 Department of Electrical Engineering, National Taiwan University , Taipei , Taiwan Department of Electrical Engineering, National Taiwan University , Taipei , Taiwan Department of Electrical Engineering, National Taiwan University , Taipei , Taiwan We study the problem of routing traffic for independent selfish users in a congested network to minimize the total latency. The inefficiency of selfish routing motivates regulating the flow of the system to lower the total latency of the Nash Equilibrium by economic incentives or penalties. When applying tax to the routes, we follow the definition of [8] to define ePoA as the Nash total cost including tax in the taxed network over the optimal cost in the original network. We propose a simple tax scheme consisting of step functions imposed on the links. The tax scheme can be applied to routing games with parallel links, affine cost functions and single-commodity networks to lower the ePoA to at most 34 ? , where only depends on the discrepancy between the links. We show that there exists a tax scheme in the two link case with an ePoA upperbound less than 1.192 which is almost tight. Moreover, we design another tax scheme that lowers ePoA down to 1.281 for routing games with groups of links such that links in the same group are similar to each other and groups are sufficiently different. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems and phrases selfish routing; price of anarchy; tax Introduction We study the problem of routing traffic for independent selfish users in a congested network to minimize the total cost (latency). In many settings, it is very expensive or impossible to regulate the traffic precisely. In the absence of regulation, users usually only focus on minimizing his own cost measured by the total time needed to traverse his chosen route. Many works focus on the degradation in network performance measured by comparing the cost of the Nash equilibrium flow and the cost of the optimal setting. The ratio of total cost of Nash Equilibria to the minimum possible cost is defined to be the Price of Anarchy (PoA). Therefore one could consider PoA as an index of the inefficiency of the lack of regulation in a network of selfish behavior. In [25], it is proven that the PoA is ? 34 for ? Research supported by MOST grant number 104-2221-E-002-045-MY3 and MOST grant number 104-2815-C-002-086-E. ? The first two authors contributed equally to the paper and are listed in alphabetical order. ? Corresponding author. affine latency functions, and the upper bound 43 is tight in a well-known example called the Pigue?s example [20]. There are many well-known works on the selfish routing game, such as [22, 23, 24, 21, 6, 18]. The inefficiency of selfish routing motivates regulating the flow of the system to lower the total latency of the Nash Equilibria by economic incentives or penalties. Marginal cost pricing is an ancient idea proposed in [20]. Marginal cost taxes may induce the minimum-latency flow as a flow at a Nash equilibrium, assuming all network users choose the routes to minimize the sum of latency and tax [2]. One major research is to lower price of anarchy to 1 for users having different sensitivity to tax in a single-commodity network [10], with an upper bound of tax with complexity O(n3). Several further researches improved the result above, such as generalizing the result for single commodity to multi-commodity [12, 15] and generalizing the result for giving an tax upperbound with complexity O(n) [11]. In [4], optimal tax with constraints can be derived in certain circumstances. Another similar concept is the coordination mechanisms introduced in [7]. Coordination Mechanisms have been used to improve the PoA in scheduling problems for parallel and related machines [7, 14, 17] as well as for unrelated machines [1, 5]. In the above researches, the system is efficient only if the tax is returned to the users, otherwise dis-utility for users due to large tax may exist. In [16], an PoA upperbound of 2 is given if tax is included as a part of the cost. The bound becomes 5/4 particularly for affine latency case. On the other hand, it has been proven [9] that marginal tax could not help reduce total cost if tax is considered as a part of the cost for affine cost functions. It is also proven [8] that continuous tax functions yield no improvement to the total latency. In the above modelings, the total flow r is specified as a part of the game. However, there are situations that the total flow is unknown beforehand, thus finding a good tax scheme becomes more difficult. Christodoulou et al. [8] studied this type of problem for single-commodity routing games with affine cost functions. They designed a tax scheme such that the PoA is at most 43 ? over all possible amount of flow, where is a constant that approaches 0 when the number of links go to infinity. In this work, arbitrary tax function is allowed as along as the sum of tax and the original cost (latency) function is monotone increasing. In our work, we focus on step-function congestion tolls. This type of tax scheme has been studied by transport economists to model the effects of the traffic lights on traffic regulations [13, 19]. Compared to arbitrary tax schemes, the step-function congestion tolls is more feasible in transportation regulations. This motivates us to investigate the possibility of improving ePoA using only step-function congestion tolls in settings similar to [8]. Our Result: We provide a simple tax scheme consisting of step functions imposed on the links. The tax scheme is applied to routing games with parallel links, affine cost function, single-commodity networks to lower the ePoA below 43 ? , where depends on the discrepancy between the links but not the number of links. Moreover, we consider a special case in which all links can be clustered into several groups. The latency function is similar among links in the same group and are sufficiently different between links in different groups. Each group could be seen as different transportation methods. For example, all freeway may belong to one group, and all local roads and railroad may each belong to another group. In this case, we propose a tax scheme which reduces ePoA to 1.281. The rest of the paper is organized as follows. In Section 2 we describe the basic routing game model and the type of tax scheme that we will use. In Section 3 we define the parameters in the tax scheme more formally and prove some essential results on the relationship between the ePoA and the imposed tax. In Section 4 we show that step function tolls perform equally well to previous optimal (arbitrary) tax scheme for networks with two parallel links. In Section 5 we propose a tax scheme for 43 ? ePoA. In Section 6 we give an 1.281 upperbound of ePoA for networks with groups of similar links. 2 Model We consider single-commodity congestion games on networks, defined by a directed traffic network G = (V, E, l), with vertex set V , edge set E, and cost function (or latency function such as [25]) set l. l is the set of cost function le for each link e ? E. There is only a start node and an end node in V , while each link e ? E connects the start node directly to the end node, and we denote all links E = {e1, e2, . . . , em}. r is defined to be the rate of traffic or the total flow, which is independent of the network G. Unlike some previous works, r is not part of the game. We aim to lower the ePoA for any value of r, instead of choosing a different tax scheme for different value at r. A flow is a function that maps every link e ? E to a non-negative real number. Given G and r, we call a flow feasible if ?e?Efe = r. le is the cost function of link e ? E, which is non-decreasing, non-negative and affine. Therefore we order the links by an increasing order of the constant of the latency of the links. Without loss of generality, we let lei (f ) = ai ? f + bi and bi ? bj for any i, j > 0 such that i < j. The concept of User Equlibrium [3] is adopted as Nash Equilibrium in this work. Formally, a flow f feasible for traffic network G and total rate r, is at User Equilibrium if and only if for every e1, e2 ? E with fe1 > 0, le1 (f ) ? l?im0 le2 (f + 1e2 ? 1e1). It has been proven that for the case where all discontinuity is lower semicontinuous, the User Equilibrium exists as a theorem in [3]. The definition follows an equivalent definition in [25] when the latency function is continuous. We call the flow at Nash Equilibrium, or User equilibrium simply the Nash flow in the rest of the paper. The cost of flow f in traffic network G is C(f, G) = ?e?Efe ? le(fe). We use Copt(r, G) to denote the minimum cost of any flow feasible at rate r, or the cost of the optimal flow. Therefore the optimal flow is the flow that minimizes the cost of flow for given (G, r), which would be referred to as OPT. Moreover, we say that a flow uses j links when there are j links with non-zero flow-value. We use CN (r, G) to denote the cost of the Nash flow at rate r, while the uniqueness of the Nash equilibrium is guaranteed in theorem in [3]. When the context is clear, we may omit G, using C(f ) for C(f, G), Copt(r) for Copt(r, G), CN (r) for CN (r, G). The Price of Anarchy is defined as P oA(r) = CCoNpt((rr)) , and P oA = maxr>0P oA(r). It should be noted that the P oA defined here is not a function of r as in most previous works. The P oA in our work is the worst case of P oA(r) among any r-value for a particular network G. In the remainder of the paper, we focus on single commodity, parallel-link networks G = (V, E, l), where E consists of m links {e1, ? ? ? , em}, and cost function of link ei is of the form lei (f ) = ai ? f + bi. 2.1 Tax On each edge, the original cost function before imposing the tax is lei (f ) = ai ? f + bi, the tax-modified cost function becomes ?lei (f ) = a?i ? f + ?bi, and a? = a. The tax scheme used in our work adds tax ?bj ? bj to the cost for users using link j, where ?bj is a function of total flow r. b?j = bj + X(bi ? bi?1) ? hi ? u(r ? wi), i>j ( 1 ) where hi < 1 and wi are constants to be chosen, and u is the unit step function. Note that to guarantee the existence of User Equilibrium, the unit step function is defined to be lower-semicontinuity. One point to be noted is that under this form of tax, the Nash flow accounting tax on any link is non-decreasing while total rate r increases. This is a desired property, which makes taxing feasible and efficient, since rerouting existing traffic when total traffic increases may be very costly if at all possible. For the taxed network, we consider adding tax to be a modification to the original network. Therefore, we call G? the tax-modified network obtained by imposing tax on G. All notations for the taxed network G? is denoted with a hat, such as the expression ?bj defined above. We specify that the C?N (r) is the total cost of the Nash equilibrium flow of the tax modified network at rate r, where the cost of each edge and the Nash flow are both affected by the tax. We formally define ePoA = maxr>0ePoA(r) = maxr>0Po?A(r) = maxr>0 CC?oNpt((rr)) . 3 Useful Inequalities on the PoA and Tax Before proving the main results, we need to prove some lemmas on the cost of the Nash equilibrium and OPT. I Definition 1. We follow notations in previous works. Given a traffic network G, let ?j = 1/aj, ?j = bj/aj, ?j = ?i=1?i, ?j = ?i=1?j and rj = ?ji=?11(bi+1 ? bi)?i. We also j j define uj = rj/rj?1 and vj = ?j/?j?1. Intuitively, rj is the amount of flow at which the (j + 1)-th edge starts to have non-zero Nash flow. PoA is locally maximized at each rj. The tax schemes we design also seeks to reduce PoA near these values. Cost of the Nash flow and the OPT on this type of traffic network has been well studied, and closed-form expressions were given [8]. We restate some essential results in Lemma 2. I Lemma 2 ([8]). The Nash flow uses link j for r > rj and the OPT uses link j for r > rj/2. If the OPT uses exactly j links at rate r then 1 If the Nash flow uses exactly j links at rate r then If s < r and OPT uses exactly j links at s and r then 1 If s < r and the Nash flow uses exactly j links at s and r then CN (r) = CN (s) + 1 ((r ? s)2 + (?j + 2s)(r ? s)). j h X X(bh ? bi)2?h?i /(4?j). h=1 i=1 Directly from Lemma 2, we know that both the OPT and the Nash flow start to use links with the same b-value simultaneously because ri = rj if bi = bj. I Lemma 3. Given a traffic network G, if there exists an index i such that bi = bi+1, we can find a network G0 having one less link than G such that CN (r, G) = CN (r, G0), Copt(r, G) = Copt(r, G0) for all r. Using Lemma 3, given a traffic network G, we can replace all links with the same b values by one link and let the cost of the Nash and the OPT remain the same. Furthermore, if we apply tax in the new game, we can apply the same tax on every corresponding links in the old game, as a result, we only consider traffic networks such that bi 6= bj, ?i 6= j in the rest of the paper. Informally, the tax scheme we design works in the following way. For every flow value ri which corresponds to a local maximum in the PoA-r curve, we add a set of step functions which reduces the tax in the flow range [?ri, ?ri] if the original PoA at flow ri is greater than a certain threshold. This set of step functions has no effect on PoA when the total flow is less than ?ri but increases PoA marginally when the total flow is greater than ?ri. A tax scheme can be described by a set of parameters (T, A, B), where T is the threshold, A = {?1, ? ? ? , ?m}, B = {?1, ? ? ? , ?m} describes the range of flow in which PoA is supressed. When the tax is imposed on a flow value ri, a step function are added onto the original cost functions for the first i links, where the heights and positions of those step functions are chosen such that the Nash flow on these i links stop increasing when the total flow r is between [?ri, ?ri], causing the Nash flow to use new links. The detailed definition of the tax scheme being used is the following: I Definition 4. Given a traffic network G, let Gj be an identical network of G with links e1 to ej?1 removed. Let fj be the Nash flow on a given a network Gj and rate r, let CNj(r) be the cost of fj on Gj. I Definition 5. Given a traffic network G, constants T , ?i and ?i such that ?i < 1, ?i > 1 for 1 ? i ? m, Let A = {?1, ? ? ? , ?m}, B = {?1, ? ? ? , ?m}, S(T ) be the set of all index i such that PoA(ri) > T and G? be the network obtained from applying tax(T, A, B) to G. The parameters hj and wj in equation ( 1 ) (Section 2.1), which correspond to the heights and the locations of the step functions are chosen as following, ( CNj((?j??j)?rj) (?j??j)?rj ? C?N?(j??jr?jrj) /(bj ? bj?1), if j ? S(T ) otherwise. hj = We also set two parameters, hmax = maxihi, vmin = mini?S(T )vi. Follow the definition, we can describe the cost of the Nash flow on G? with Lemma 6. I Lemma 6. Given a traffic network G, constants T , ?i and ?i such that ?i < 1, ?i > 1 for 1 ? i ? m, and tax(T, A, B) imposed on G, C?N (r) = C?N (?j ? rj) + CNj(r ? ?j ? rj) for r ? [?j ? rj, ?j ? rj] and j ? S(T ). (r2 + r ? ??j(r)) for r ?/ (?j ? rj, ?j ? rj)?j ? S(T ), where ??j(r) = ?ij=1?bi(r)/a?i = ?ij=1?bi(r)/ai. If the Nash flow uses exactly n links on Gj at rate r, CNj(r) = 1 ?j+n?1 ? ?j?1 (r2 + (?j+n?1 ? ?j?1) ? r). Proof. Equations can be derive directly from Lemma 2. J In this paper, all tax schemes are designed in a way that after the tax is being applied, the ePoA is determined by the Nash/OPT costs at total flow ?iri or ?iri for some i. In order to have a good estimate of the ePoA, we first derive Theorem 7 which gives us a good estimate of the original PoA at total flow ?iri and ?iri. In this theorem, the first inequality gives a good upper bound on PoA at ?iri and the second inequality gives a good upper bound on the PoA at ?iri. All upper bounds are described using parameters ?j since these values play an important role in determining the PoA [8]. I Theorem 7. If the Nash flow uses exactly j links and the OPT uses exactly h links at rate r then PoA(r) ? maxn 4r r2?j?1 + r ? rj(?j??11 ? ?j?1) 4r ? rj?1 , r2?j??11 ? ?ih=j(r ? ri/2)2 ? (?i??11 ? ?i?1) If the Nash flow uses exactly j-1 links and the OPT uses exactly h links at rate r then PoA(r) ? maxn 4r 4r ? rj?1 , r2?j??11 ? ?ih=j(r ? ri/2)2 ? (?i??11 ? ?i?1) The proof is omitted due to space constraints. In the PoA-r curve, local maximum only exists at r = rj. The following lemma gives an upper bound on PoA which will be used to show that PoA in the region [?iri, ?i+1ri+1] is bounded by the PoA of this region?s two endpoints. I Lemma 8. Given a traffic network G, if the Nash flow uses exactly j links at rate s and t for s < t, then PoA(r) ? max{PoA(s), PoA(t)}, ?r ? [s, t] . Most of our proof relies on Theorem 7 and Lemma 8, first with Lemma 8 to bound the PoA for total flow far away from the peak values ri, then with Theorem 7 to provide a good bound for total flow close to these peak values. As previously mentioned, the step functions that decrease PoA near ri will increase PoA when the total flow is greater than ?iri. Lemma 9 shows that our tax will only increase the total cost by a constant factor. I Lemma 9. Given a traffic network G, constants T , ?i and ?i such that ?i < 1, ?i > 1 for 1 ? i ? m, let G? be the traffic network obtained by imposing tax(T, A, B) on G, then CC?NN ((rr)) ? 1 + hvmmainx , for r such that r ?/ (?j ? rj, ?j ? rj) for all j ? S(T ). Proof. For total flow r ? [rj?1, rj] and r ?/ (? ? ri, ? ? ri) for all i ? S(T ), let k be the largest i ? S(T ) such that i < j, from Lemma 2, C?N (r) CN (r) ? r + ??j?1(r) = 1 + r + ?j?1 Pij=?11(?bi(r) ? bi)?i r + ?j?1 . The largest possible tax added to a link when r ? [rj?1, rj] is hmax ? bk, and only link 1 to k-1 have non-zero tax added rate r, ? 1 + hmax ? bk?k?1 = 1 + hmax ? bk?k?1 . rj?1 + ?j?1 bj?1?j?1 Since k < j, bk ? bj?1 and ?k ? ?j?1, C?N (r) hmax CN (r) ? 1 + ?k/?k?1 = 1 + hmax ? 1 + hmax . vk vmin 4 The ePoA for Two-Link Networks J In this section, we study the networks with two parallel links. In this special case, we give an upperbound of ePoA for the step function tolls which is 1.192. This result shows that applying step function tolls is as powerful as arbitrary tax scheme proposed in [8]. In fact, when the total flow is between 0 and ?r1, our step function tax is exactly identical to the tax scheme in [8]. When the total flow is greater than ?r1, the previous tax scheme remove the previously added step-function tax and does not impose tax on any link. In this paper, removing the step functions is not allowed. We prove that even though these step functions only increase PoA when the total flow is greater than ?r1, the influence is marginal and the maximum value always happen at total flow ?r1. The proof is omitted due to space constraints. I Theorem 10. Given a two link traffic network G, there always exist a pair of ? ? ( 21 , 1 ), ? ? (1, ?) such that if tax(T = 1.192, {?}, {?}) is imposed, then ePoA ? 1.192. 5 Upperbound of the ePoA for Multiple Parallel-Link Networks In this section we consider parallel-link networks. Given a traffic netowrk G, we consider that ratio between two adjacent peak values rir?i 1 . Let = min(mini>1ui, 2) ? 1 = min(mini>1 rir?i 1 , 2) ? 1. We prove that the ePoA has an upper bound less than 43 ? 3 ( 3 )3. 1 Notice that in this case, only depends on the discrepancy between the links and is independent of the number of links in the network. The main result of this section is the following theorem. I Theorem 11. Given a traffic network G, and tax(T = 43 ? ( 3 )3, {?1 = ? ? ? = ?m = 1 ? 2( 3 )3}, {?1 = ? ? ? = ?m = 1 + 3( 3 )3}) is imposed. Then ePoA < 43 ? 31 ( 3 )3. incrNeaosteicse, atnhdatthrjru?js1 isisalegsosotdhainndbijc?a1tor of the discrepancy between the links. bj and increases when the difference between aj and aj?1 In order to prove Theorem 11, we need the following lemmas. Intuitively, we first use Lemma 14 and 15 to prove that the PoA of the original network is at most T when the total flow is ?rj of ?rj. Combining with Lemma 8 and 9, we know that ePoA? T (1 + hvmmainx ) for all r ?/ (?rj, ?rj). Lemma 16 shows that when the total flow is between ?rj and ?rj, the ePoA is also bounded. Plug in the value of hmax and vmin from Lemma 12 and 13 to finish the proof. For the constants T , ?i, ?i chosen, we can bound all related parameters needed in Theorem 11 with some straightforward calculations. I Lemma 12. Given a traffic network G, a constant T such that T > 43++44 . vmin = mini?S(T )vi = mini?S(T ) ??i?i 1 > (24 ?? 32T)T Proof. By definition of set S, PoA(ri) > T for all i ? S(T ). From Theorem 7, PoA(rj) ? maxn From condition of T, 4 J I Lemma 14. Given a traffic network G, constants T and ? such that ? ? T +(T 22?T ) 21 , 4??rj 4??rj?rj?1 ? T and 1 < T < 34 , then PoA(? ? rj) ? T . I Lemma 15. Given a traffic network G, constants T and ? such that 2 > ? ? 4(TT?1) and 1 < T < 43 , then PoA(? ? rj) ? T . I Lemma 16. Given a traffic network G, constants T , ?i and ?i such that ?i < 1, ?i > 1 for 1 ? i ? m, and tax(T, A, B) imposed on G, ePoA(r) ? maxnePoA(?j ? rj), (?j ? ?j) ?j ???jj?1 + (1 ? ?j + ?j)o for r ? [?j ? rj, ?j ? rj] and j ? S(T ). The Proof of Lemma 13 to 16 are omitted due to space constraints. Proof of Theorem 11. Let ? = ?1 = ? ? ? = ?m = 1 ? 2( 3 )3, ? = ?1 = ? ? ? = ?m = 1 + 3( 3 )3. First consider the case when total flow r ?/ (??rj, ? ?rj) ?j ? S(T ). Since ? ?rj < ??rj+1 ?j, we can apply the result of Lemma 8, PoA(r) ? maxnmaxi?/S(T )PoA(ri), maxi?S(T )PoA(? ? ri), maxi?S(T )PoA(? ? ri)o. From Lemma 14, 15 and the definition of S(T ), all terms above are bounded by the threshold T, PoA(r) ? T for r ?/ (? ? rj, ? ? rj) ?j ? S(T ). ePoA(r) is bounded by PoA(r) times the ratio between cost of the Nash flow on G? and G, From Lemma 9, ePoA(r) = PoA(r) ? CC?NN ((rr)) ? T (1 + hvmmainx ) for r ?/ (? ? rj, ? ? rj) ?j ? S(T ). (2) We then consider ePoA(r) when total flow r ? [? ? rj, ? ? rj], and j ? S(T ). From Lemma 16, ePoA(r) ? maxnePoA(? ? rj), (? ? ?) For the second term above, since j ? S(T ), the ratio of ?j and ?j?1 is bounded, from Lemma 12, ?j (? ? ?) + (1 ? ? + ?) From previous case, we know that ePoA(? ? rj) ? T (1 + hvmmainx ), therefore ePoA(r) ? T (1 + hmax ) for r ? [? ? rj, ? ? rj] if j ? S(T ). vmin Combine (2) and (3), we have an upperbound of ePoA(r) for all r > 0, ePoA ? T (1 + hmax ). vmin From Lemma 12 and 13, 4 3 ePoA < 3 ? 27 1 ? ( 3 2(2 ? 2)( 34 ? 27 ) + 3 3/9 3 (2 ? 2)( 34 ? 27 ) ? 3/9 ) + 2 ? 9(2 ? ) In previous sections, we have given an upperbound of ePoA when it is strictly less than 43 . In this section, we study a special case in which the links can be classified int many groups. Links in the same group all have similar ri and thus similar cost functions. This special case is closely related to the case in which there are many types of transportation methods, or just many types of roads (such as freeways and local roads). We give an upper bound of ePoA for a specific case of groups of similar link defined below. I Definition 17. A traffic network Gc is a network with clustered latencies if and only if there exists N intervals [L1, R1], . . . , [LN , RN ], and RLii <= 1.05 for i ? [1, 2, . . . , N ], and Li+1 Ri ? 20 and any rj for j ? 2 is in one of the intervals [Li, Ri]. The main result of this section is ePoA? 1.281 for a traffic network Gc with clustered latencies. Before proving the main result, we introduce the following transformation, and several lemmas. I Definition 18. Given any traffic network Gc with clustered latencies, we define the aggregated network of Gc, Ga as the following. For all ri in Gc, inside a certain interval [Lk, Rk], we re-label the index i to be k1, k2, . . . , knk so that Lk ? rk1 ? rk2 ? ? ? ? ? rknk ? Rk. An intermediate network Gtemp is obtained by increasing the constant of the cost functions bki to b?ki = bknk for all i < nk. Now all links ei with ri in the same interval in Gc has the same b-value, which is bknk . Thus, by Lemma 3, these links can be merged through a transformation of graph without changing either the Nash flow or the OPT. After the merge, the resulting network is Ga. The transformation Tr is the combination of increasing the constants of links in Gc to get Gtemp, and merging edges of Gtemp to get Ga. (3) J I Lemma 19. In a traffic networks G1 with rj?1 and rj where , bj?1 is increased to bj , and the two links are merged to index inew, as stated in Definition 18 then the position of rinew is betweeen (rj?1, rj ). Proof. By the basic equation in Definition 1, rj = ?ij=?11(bi+1 ? bi)?i. rj ? rinew = (bj ? bj?1)(?j?1 ? ?j?2) > 0 rinew ? rj?1 = (bj ? bj?1)(?j?1) > 0 Thus, rj?1 ? rinew ? rj . Following Definition 18, with Lemma 19 used recursively, we see that the resulting rk after merging all links in section k lies in [Lk, Rk]. Therefore, after the transformation, the resulting traffic network Ga has min rjr+j1 ? 20, which is directly from the fact that LRi+i1 ? 20. The ratio of the optimal cost between the network after the transformation and before the transformation is less than the ratio of the largest bb?ii ? 1.05. The formal lemma and proof are below. I Lemma 20. For any traffic network Gc with clustered latencies and its corresponding aggregated network Ga, CCoopptt((rr,,GGac)) ? 1.05. The following lemma is similar to Lemma 9, for a slightly different situation. I Lemma 21. In a traffic network G with rate r and ri+1 ri ? 20, where constants T , ?i < 1, ?i > 1. When tax(T, A, B) is imposed on the network G. We have C?N (r) hmax CN (r) ? 1 + 20 ? s for any s satisfying ?j ? rj ? r ? s ? rj , and s ? rj ? ?j?1rj?1, s ? 1. Similarly, we have for ?j+1 ? rj+1 ? r ? ?j ? rj . We now introduce the tax scheme and upper bound the corresponding ePoA for an aggregated network. The tax scheme chooses different values of ?i, ?i, with different regions of vi. I Lemma 22. For any traffic network Gc with clustered latencies and its corresponding aggregated network Ga, there exists a tax scheme Ga such that ePoA of Ga ? 1.22 when the tax is applied to Ga. Proof. The tax scheme tax(1.198, A, B), where ?j , ?j are decided according to the value of v in Table 1 satisfies the requirement. The proof is omitted due to space constraints. J With Lemma 20 and 22 we prove the main theorem in this section by simply multiplying 1.22 and 1.05. I Theorem 23. The ePoA is at most 1.281 for a traffic network with clustered latencies. J (4) (5) Proof. For any traffic network Gc with clustered latencies and its corresponding aggregated network Ga, with Lemma 20 and 22 we know that ePoA of Ga ? 1.22, and CCoopptt((rr,,GGac)) ? 1.05. We view the transformation Tr on Gc as tax T1, and the tax imposed on Ga as tax T2. The final tax scheme imposed on Gc is T1 + T2. While the tax scheme imposed on Ga is T2. Now we prove the theorem eP oA = C?N (r, Gc) Copt(r, Gc) = C?N (r, Ga) Copt(r, Ga) Copt(r, Ga) ? Copt(r, Gc) ? 1.22 ? 1.05 = 1.281 J A point to be noted is that the lower bound of PoA is proved in [8] to be 1.191 for two edge network, therefore that proving ePoA ? 1.22 is clearly close to optimal since additional tax is further accounted while in [8] the tax could be retrieved and that rj+1 is ?, where in rj Lemma 22 the restriction is much stricter, while only increasing the ePoA by less than 3 percent. 7 Open Problems The goal of this work is to design a taxing scheme with unit step function which is able to be applied to general networks. In the case of parallel links in our study, we have demonstrated different possible approaches to bound the ePoA. 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Te-Li Wang, Chih-Kuan Yeh, Ho-Lin Chen. An Improved Tax Scheme for Selfish Routing, LIPICS - Leibniz International Proceedings in Informatics, 2016, 61:1-61:12, DOI: 10.4230/LIPIcs.ISAAC.2016.61