© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In this study, we describe a new coordination mechanism for non-atomic congestion games that leads to a (selfish) social cost which is arbitrarily close to the non-selfish optimal. This mechanism incurs no additional cost, in contrast to tolls that typically differ from the social cost as expressed in terms of delays.Peer ReviewedPostprint (author's final draft

Díaz Cort, Josep

Giotis, Ioannis

Kirousis, Lefteris

Mourtos, Ioannis

Serna Iglesias, María José

ICT Express

© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In this study, we describe a new coordination mechanism for non-atomic congestion games that leads to a (selfish) social cost which is arbitrarily close to the non-selfish optimal. This mechanism incurs no additional cost, in contrast to tolls that typically differ from the social cost as expressed in terms of delays.Peer Reviewe

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The social cost of congestion games by imposing variable delays

In this study, we describe a new coordination mechanism for non-atomic congestion games that leads to a (selfish) social cost which is arbitrarily close to the non-selfish optimal. This mechanism incurs no additional cost, in contrast to tolls that typically differ from the social cost as expressed in terms of delays

Josep Díaz

Ioannis Giotis

Lefteris Kirousis

Ioannis Mourtos

Maria Serna

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The Social Cost of Congestion Games by Imposing Variable DelaysJosep Dı´aza,1, Ioannis Giotisa,b,1,2, Lefteris Kirousisc,2,∗, Ioannis Mourtosb,2, Maria Sernaa,1aDepartament de Cie`ncies de la Computacio´, Universitat Polite`cnica de Catalunya, BarcelonabDepartment of Management Science and Technology, Athens University of Economics and Business, GreececDepartment of Mathematics, National and Kapodistrian University of Athens, Greeceand Computer Technology Institute Press “Diophantus”, Patras, GreeceAbstractWe describe a new coordination mechanism for non-atomic congestion games that leads to a (selfish) social cost which isarbitrarily close to the non-selfish optimal. This mechanism incurs no additional cost, in contrast to tolls that typicallydiffer from the social cost as expressed in terms of delays.Keywords: Congestion games, Price of anarchy, Coordination mechanisms.1. IntroductionSelfish behavior is one of the primary reasons manysystems with multiple agents deviate from desirable out-comes. Allowing players to prioritize solely their own ben-efit can lead to social inefficiency, even in outcomes whereno one is better off compared to an optimal solution.This type of behavior has been analyzed in various con-texts and has often been verified in practice. A key sucharea is transportation and network routing where selfishselection among possible routes can lead to congestion withaccompanying economical and environmental issues.Various approaches have been proposed to steer theselfishly constructed outcome towards optimal social wel-fare. The main idea is usually to incentivize the usersto alter their selections to ones that lead to socially bet-ter outcomes, typically through the use of tolls or similarmeasures.We propose an alternative approach that alters the wayusers experience latency and can offer significant improve-ments on social cost, however drivers still get to pick theirown route In more detail, instead of all users experiencing∗Corresponding authorEmail addresses: diaz@cs.upc.edu (Josep Dı´az),igiotis@cs.upc.edu (Ioannis Giotis), lkirousis@math.uoa.gr(Lefteris Kirousis), mourtos@aueb.gr (Ioannis Mourtos),mjserna@cs.upc.edu (Maria Serna)1Supported by funds from the Spanish Ministry of Economyand Competitiveness (MINECO) and the European Union (FEDERfunds) under grant TIN2013-46181-C2-1-R (COMMAS) and grantSGR 2014 1137 (ALBCOM) of the Catalan government.2This research has been co-financed by the European Union (Eu-ropean Social Fund ESF) and Greek national funds through theOperational Program “Education and Lifelong Learning” of the Na-tional Strategic Reference Framework (NSRF) - Research FundingProgram: Thales. Investing in knowledge society through the Euro-pean Social Fund.the same latency, we propose to implement variable laten-cies through a prioritization scheme. That is, we allowfor some users to experience smaller latencies than before,while others to experience longer ones. We employ knownresults to show that our system, if users behave selfishlyas expected, achieves the optimal social welfare. To makesuch a system practical we present a discretization of thetheoretical continuous functions that approximates the op-timal social welfare.We also wish to emphasize the distributed and decen-tralized nature of our system. As explained in the next sec-tions, each resource (road or highway in the transportationsetting) implements the desired changes individually andindependently. It is important to note that our system’saverage latency on each road, as experienced by the users,is at least equal, and actually closely matches, the road’saverage latency without the system in place, hence our sys-tem falls under the notion of coordination mechanisms, i.e.no “cheating” in the form of network improvements, whichtypically carry significant cost, is introduced. Also, thereis no imposing of tolls (transfer of social cost to a differenttype); we simply distribute the resource differently. Thisholds on any instance, not just in equilibrium settings,which means that even in non-stable situations we do notget worse performance. Furthermore, we do not need toknow the demand in advance, i.e. our system delivers closeto the social optimum for all possible total amounts of traf-fic. Our only requirement is that the latency induced oneach road is a non-negative, non-decreasing, continuouslydifferentiable and convex function of the traffic.We believe that our system has a strong applicabil-ity potential. For example, some countries have alreadyimplemented metered highway entrance ramps which canvary the latency of incoming drivers. Traffic lights mayalso be used in an urban environment to implement thisPreprint submitted to Elsevier July 8, 2017aspect of our mechanism. We deliberately leave the pri-oritization scheme generic to allow for different such ap-proaches with our only requirement being that users choos-ing to alter their current selection are forced to experiencemaximal latency in their new selection, a reasonable re-quirement as typically someone that alters her selection ina running system ends up at the end of the queue.We examine our system in the generic scheme of con-gestion games, to emphasize that it admits applicationsbeyond traffic routing. One interesting application couldbe in the context of job scheduling on computing resources.Again, in a typical model, each user choosing to use a par-ticular resource experiences the same latency, for examplecomputing jobs running in parallel on a computer. By pri-oritizing jobs according to our proposed mechanism, suchthat some jobs complete faster and some slower than be-fore, we can achieve optimal average job completion timesunder selfish behavior. We note that this can easily be im-plemented by an administrator (human or computerized)using system priorities.2. Related workThe fact that selfish behavior can lead to inefficiencyhas long been studied in the context of transportation the-ory [1, 2]. More recently, Koutsoupias and Papadimitriouintroduced the Price of Anarchy as a measure of this inef-ficiency [3, 4]. Exploration of this metric in the context ofselfish routing was then greatly progressed by Roughgar-den and Tardos [5, 6] who bounded the price of anarchyfor different classes of latency functions.Naturally, ways to improve inefficient outcomes havebeen investigated, with a prime example being the im-posement of tolls [7, 8, 9]. While this approach achievesoptimal social welfare regarding latencies, it introduces acost separation to the players as the tolls’ cost is affectingbehavior but is not accounted for in the objective function.Coordination mechanisms have recently been introducedby Christodoulou et al. [10] as a way to “shape” latencyfunctions to steer the selfishly dictated outcome towardsgreater social welfare. There are two main restrictionsin the type of coordination mechanisms defined in [10],namely that the latency per resource is not decreased andthat the benchmark optimal social welfare, against whichthe mechanism is measured, is still the original one withoutany additional latencies possibly imposed by the mecha-nism. It has recently been shown that indeed such mech-anisms can positively affect social welfare [11]. Our ap-proach sustains the non-decreasing latency on average butnot on every user, as a prerequisite for achieving, through‘coordination’, a significantly lower price of anarchy thanthe mechanism of [10]. In fact, the average latency peruser within our system can be made arbitrarily close tothe unique latency per user without the system in place.The approach of differentiating the latency per resourceis also explored from an algorithmic perspective by Harkset al. [12] but not with the same scheme. The resultsof Farzad et al. [13] are closer to our work. However, inthe later, it is the strategic equilibrium without the mech-anism of [13] in place that matches the optimal underthat mechanism; i.e., the strategic equilibrium under thatmechanism may in general differ substantially from thenon-selfish optimal.3. ModelWe define a congestion game (E, l,S, P, d) in the genericsense but using network routing (or alternatively, flow) ter-minology for convenience. First, a set E of edges with anassociated non-negative, non-decreasing, continuously dif-ferentiable and convex le() latency function for each edge.We note that these assumptions are typical for latencyfunctions.In P , we have n player types 1, 2, . . . , n. For each playertype i we have a source-sink pair (si, ti). We also have afinite set Si of finite sequences of elements of E, called thestrategy set of player type i. A particular element S ∈ Siis a single strategy of player type i, also referred to asa path from si to ti. Finally, we have a flow (or traffic)demand di for each player type i.We assume that each player type corresponds to a con-tinuum of nonatomic players, each with negligible flow, i.e.they can be arbitrarily divided into the various paths. Aninfinitesimal part of the flow or traffic will often referred toas a user. Let xSi denote a nonnegative real representingthe part of demand corresponding to player type i thatuses strategy (path) S and xi the vector for the strategyset Si, i.e. xi = (xSi )S∈Si . The vector x for all xi’s is calleda flow if for all player types i,∑S∈SixSi = di. We definethe part of the demand of a player type i that uses edge easxie =∑{S:S∈Si,e∈S}xSi ,and the total flow through an edge e asxe =∑i=1...nxie.In related literature, the cost induced to each player typei by a flow x is defined to be ci(x) =∑e∈E le(xe) ·xie. Thecost of the total flow through an edge e is defined to bece(xe) = le(xe) · xe,whereas the social cost is defined to beC(x) =∑e∈Ele(xe) · xe.For reference, we now give the notion of Wardrop equilib-rium in our setting.Definition 1. We say that the flow vector x is in Wardropequilibrium if for all player types i and for any pairs ofstrategies (paths) S1, S2 ∈ Si, if xS1i > 0 then the follow-ing holds: ∑e∈S1le(xe) ≤∑e∈S2le(xe). (1)24. Variable delay mechanismGiven a congestion game (E, l,S, d) with non-negative,non-decreasing, continuously differentiable and convex la-tency functions, we propose a coordination mechanismwhich differentiates the latency experienced by differentusers as follows:Let N = (Ne)e∈E be a sequence of positive integersindexed by the set of elements (edges) E to be called abatch system. A positive integer b ≤ Ne is referred to as abatch index (or just batch) at edge e. At each edge e, thetotal flow xe through e is split into Ne batches of equal sizexe/Ne each. Each batch induces a different latency costto its corresponding flow, with batches of a larger indexgetting larger latencies, as formally defined below. Notethat this means that different parts of the flow of someplayer type could receive different latencies. The way thatthis split is implemented does not affect our results, i.e.,the assignment of flow to batches can be performed by anydesired policy (e.g., randomly or first-come-first-served orthrough priority lists).Now consider the following functions, known as marginal–cost latency functions:lˆe(xe) = c′e(xe) = le(xe) + l′e(xe) · xe, (2)where c′e(), l′e() are the derivatives of ce(), le(), respec-tively. The latency induced is not going to be equal amongusers at an edge e. Instead, the flow of any player type andthrough any path S at batch b receives latency lˆe((b/Ne)xe)per unit. Users are interested in minimizing their own la-tency. We refer to the model of applying equal latency toall users as the uniform latency or classical model.Since each batch b receives latency lˆe((b/Ne)xe) perunit, we define the cost with respect to the batch systemat an edge with flow xe to be:cˆe(xe) = (xe/Ne)Ne∑b=1lˆe((b/Ne)xe)and the social cost with respect to the batch systemCˆ(x) =∑ecˆe(xe).Note that cˆe(xe) ≥∫ xe0lˆe(z)dz = le(xe)xe = ce(xe), there-fore we do not decrease the cost as per the coordinationmechanisms’ doctrine and as we shall see later, any costincrease can be made arbitrarily small.Although we state above that all batches of the flowat a particular edge are of equal size xe/Ne, let us notethat this is not essential for our proofs, i.e., all technicalarguments go through for arbitrary batch sizes. We as-sume batches of equal size to avoid cumbersome notationand thus improve clarity. In addition, when the numberof batches tends to infinity, all batch sizes approach zero;that asymptotic case is important because, then, the Priceof Anarchy under our model approaches 1 while the costapproaches the cost under the uniform latency model.Given a path S, a sequence of batch indices be, e ∈ Sis called a batch assignment for S.Definition 2. We say that the flow vector x is in equilib-rium with respect to the batch system if for all player typess i and for any pairs of strategies (paths) S1, S2 ∈ Si, ifxS1i > 0, then for every batch assignment be, e ∈ S1, thefollowing holds:∑e∈S1lˆe((be/Ne)xe) ≤∑e∈S2lˆe(xe). (3)Intuitively, if a user (infinitesimal part of flow) changespath, then we assume that it gets to the last batch of ev-ery edge of the new path. Indeed this is so because in theright hand side of the above equation the cost of the lastbatch appears for all edges; whereas on the left we have thecost of an arbitrary batch sequence be along S1. What theabove equation expresses is that under this assumption,there is no strict gain in cost a user experiences if it uni-laterally implements a change of path (users are assumedto be anonymous, so the batch at an edge for a particularuser is not well defined; this is the reason arbitrary batchsequences within the various edges of a path are taken onthe left side).We now have the following:Lemma 3. The flow vector x is in equilibrium with re-spect to the batch system iff it is in Wardrop equilibriumwith respect to the marginal–cost latency functions lˆe(xe) =le(xe) + l′e(xe) · xe, i.e. iff for all players i and for anypairs of strategies (paths) S1, S2 ∈ Si, if xS1i > 0, then∑e∈S1lˆe(xe) ≤∑e∈S2lˆe(xe).Proof. For the sufficiency of Wardrop equilibrium withrespect to the marginal-cost latency notice that since le()are convex, lˆe are non-decreasing. For the necessity noticethat because the inequality in Definition 2 holds for anyselection of batch indices, and therefore also for be = Ne.We now state the following well known theorems derivedfrom the literature [14, 6].Theorem 4. When the latencies are non-negative, con-tinuous and non–decreasing, there always exists at leastone Wardrop equilibrium.Theorem 5. If x and x′ are flow vectors in Wardrop equi-librium then le(xe)xe = le(x′e)x′e for all edges e. This alsoshows a unique social cost for all Wardrop equilibria.Theorem 6. A flow vector x in Wardrop equilibrium withrespect to the marginal–cost latencies le(xe) + l′e(xe) · xehas optimal social cost C(x) with respect to the latencyfunctions le.3b bl2(x) = xl1(x) = 1(a) Original Pigou’s networkb blˆ2(x) = 2xlˆ1(x) = 1121latencytraffic batches(b) Pigou’s network with variable delaysFigure 1: An example on Pigou’s networkWe will now transfer these results in our variable delaybatch setting.Theorem 7. Under the variable delay mechanism, anybatch system has a unique equilibrium (as defined in Def-inition 2). Moreover, there is always a suitable batch sys-tem whose cost, with respect to the batch system, whenin equilibrium (in the sense of Definition 2), is arbitrar-ily close to its optimal social cost of the uniform latencymodel.Proof. Indeed by the preceding Theorems 4–6, and byLemma 3, it suffices to prove that if a flow x is in equilib-rium with respect to a suitable batch system, then its costCˆ(x) with respect to the batch system is arbitrarily closeto the social cost C(x) of the uniform latency model withrespect to latencies le. This however is immediate to seesincecˆe(xe) = (xe/Ne)Ne∑b=1lˆe((b/Ne)xe)can be made arbitrarily close to∫ xe0lˆe(z)dz = le(xe)xe = ce(xe)by choosing for each e a large enoughNe. Note that at thispoint one needs an upper bound of the demand to pick alarge enough Ne but this is not required if the approxima-bility bound is not needed. 5. DiscussionInterestingly, even a small number of batches can offersignificant improvements in certain situations. We illus-trate the classic Pigou’s network as an example with atotal traffic of 1. In the original network (see Figure 1a),it is well known that the Price of Anarchy is 4/3 sinceall users would pick the lower edge inducing latency of 1to everyone while the optimal solution would be for theusers to be split evenly among the edges with an averagelatency of 3/4. By using 3 batches with variable delays(Figure 1b), in equilibrium, half of the traffic would pickthe lower edge and be distributed uniformly among the 3batches with an induced latency of 1/3, 2/3 and 1 respec-tively. The average latency would be0.5 · 1 + 0.5 · 1/3 · (1/3 + 2/3 + 1) = 5/6.The resulting Price of Anarchy compared to the optimalsolution in the original network would be 10/9, a signifi-cant improvement from 4/3. Even with just two batches,the resulting Price of Anarchy would be 7/66. AcknowledgementsWe would like to thank George Christodoulou for thevery helpful discussions. Also we thank an anonymousreferee of a previous draft of this work for pointing to usthe results by Farzad et al. [13] and Harks et al. [12]References[1] A. C. Pigou, The Economics of Welfare, Macmillan, 1920.[2] J. G. Wardrop, Some theoretical aspects of road traffic research,in: ICE Proceedings: Engineering Divisions, Vol. 1, ThomasTelford, 1952, pp. 325–362.[3] E. Koutsoupias, C. H. Papadimitriou, Worst-case equilibria,Computer Science Review 3 (2) (2009) 65–69.[4] C. H. Papadimitriou, Algorithms, games, and the internet, in:J. S. Vitter, P. G. Spirakis, M. Yannakakis (Eds.), STOC, ACM,2001, pp. 749–753.[5] T. Roughgarden, E´. Tardos, How bad is selfish routing?, J. ACM49 (2) (2002) 236–259.[6] T. Roughgarden, Selfish routing and the price of anarchy, MITPress, 2005.[7] R. Cole, Y. Dodis, T. Roughgarden, Pricing network edges forheterogeneous selfish users, in: L. L. Larmore, M. X. Goemans(Eds.), STOC, ACM, 2003, pp. 521–530.[8] L. Fleischer, K. Jain, M. Mahdian, Tolls for heterogeneous self-ish users in multicommodity networks and generalized conges-tion games, in: Foundations of Computer Science, 2004. Pro-ceedings. 45th Annual IEEE Symposium on, IEEE, 2004, pp.277–285.[9] G. Karakostas, S. G. Kolliopoulos, Edge pricing of multicom-modity networks for heterogeneous selfish users, in: Proceedingsof the 45th Annual IEEE Symposium on Foundations of Com-puter Science, IEEE Computer Society, 2004, pp. 268–276.[10] G. Christodoulou, E. Koutsoupias, A. Nanavati, Coordinationmechanisms, Theor. Comput. Sci. 410 (36) (2009) 3327–3336.[11] G. Christodoulou, K. Mehlhorn, E. Pyrga, Improving the priceof anarchy for selfish routing via coordination mechanisms, in:Proceedings of the 19th European conference on Algorithms,Springer-Verlag, 2011, pp. 119–130.[12] T. Harks, S. Heinz, M. E. 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The social cost of congestion games by imposing variable delays

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