Optimal Taxes in Atomic Congestion Games

Optimal T axes in Atomic Congestion Games D ARIO P A CCA GNAN, Imperial College London, UK RAH UL CHAND AN, University of California at Santa Barbara, USA BRY CE L. FERGUSON, University of California at Santa Barbara, USA JASON R. MARDEN, University of California at Santa Barbara, USA First version: November 2019 . This version: March 2021. Abstract. How can we design mechanisms to promote ecient use of shared resources? Here, we answer this question in relation to the well-studied class of atomic congestion games, used to mo del a variety of problems, including trac routing. Within this context, a metho dology for designing tolling me chanisms that minimize the system ineciency (price of anarchy) exploiting solely local information is so far miss- ing in spite of the scientic interest. In this manuscript we resolve this problem through a tractable linear programming formulation that applies to and beyond polynomial congestion games. When specializing our approach to the polynomial case, we obtain tight values for the optimal price of anarchy and corr esponding tolls, uncovering an unexpected link with load balancing games. W e also derive optimal tolling mechanisms that are constant with the congestion le vel, generalizing the results of [ 8 ] to polynomial congestion games and beyond. Finally , we apply our techniques to compute the eciency of the marginal cost me chanism. Surprisingly , optimal tolling me chanism using only local information p erform closely to existing mecha- nism that utilize global information [ 6 ], while the marginal cost me chanism, known to be optimal in the continuous-ow model, has low er eciency than that encountered le vying no toll. All results ar e tight for pure Nash equilibria, and extend to coarse correlated equilibria. CCS Concepts: • Theory of computation → Algorithmic game the ory ; Quality of equilibria ; Additional K ey W ords and Phrases: Congestion games, optimal taxation mechanisms, price of anarchy , Nash equilibrium, coarse correlated equilibrium, selsh routing, approximation algorithms. 1 INTRODUCTION Modern society is based on large-scale systems often at the ser vice of end-users, e.g., transporta- tion and communication networks. A s the performance of such systems heavily depends on the interaction between the users’ individual behaviour and the underlying infrastructure (e.g., drivers on a road-trac network), the operation of such systems requires interdisciplinary considerations at the conuence between economics, engineering, and computer science. A common issue arising in these settings is the performance degradation incurred when the users’ individual objectives are not aligned to the “greater good” . A prime e xample of how users’ behaviour degrades the performance is provided by road-trac routing: when drivers cho ose routes that minimize their individual travel time, the aggregate congestion could be much higher compared to that of a centrally-impose d routing. A fruitful paradigm to tackle this issue is to This manuscript has been deposite d to the arXiv on 22 Nov 2019. A preliminary version appeared in “Proce edings of the 14th W orkshop on the Economics of Networks, Systems and Computation” (NetEcon ’19), se e [ 9 ]. This research was carried out when the rst author was aliated with the University of California at Santa Barbara. This work was supported by the Swiss National Science Foundation Grant P2EZP2_181618, ONR Grant #N00014-20-1-2359, AFOSR Grant #F A9550-20-1-0054. A uthors’ addresses: Dario Paccagnan, Imperial College London, Department of Computing, London, UK, d.paccagnan@ imperial.ac.uk; Rahul Chandan, University of California at Santa Barbara, Center for Control, Dynamical Systems and Computation, Santa Barbara, CA, 93105, USA, r chandan@ucsb.edu; Bryce L. Ferguson, University of California at Santa Barbara, Center for Control, Dynamical Systems and Computation, Santa Barbara, CA, 93105, USA, blf@ucsb.edu; Jason R. Marden, University of California at Santa Barbara, Center for Control, Dynamical Systems and Computation, Santa Barbara, CA, 93105, USA, jrmarden@ucsb.edu. A CM Transactions on Economics and Computation (to appear) 2 Paccagnan, et al. employ appropriately designed tolling mechanisms, as widely acknowledged in the economic and computer science literature [5, 22, 26]. Pursuing a similar line of research, this paper focuses on the design of tolling mechanisms that maximize the system eciency associated with self-interested decision making in atomic congestion games , i.e., that minimize the resulting price of anarchy [ 21 ]. Within this context, we develop a methodology to compute the most ecient local tolling mechanism, i.e., the most ecient mechanism whose tolls levied on each resource depend only on the local properties of that r esource. W e do so for both the congestion-aware and congestion-independent settings, whereby tolls have or do not have access to the curr ent congestion levels. A summar y of our results, including a comparison with the literature, can be found in T able 1. Perhaps surprisingly , optimal local tolls deliver a price of anarchy close to that of existing tolls designed using global information [6]. 1.1 Congestion games, local and global mechanisms Congestion games were introduced in 1973 by Rosenthal [ 31 ], and since then have found applications in diverse elds, such as energy markets [ 29 ], machine sche duling [ 35 ], wireless data networks [ 36 ], sensor allocation [ 23 ], network design [ 2 ], and many more. While our results can be applied to a variety of problems, we consider trac routing as our prime application. In a congestion game, we are giv en a set of users 𝑁 = { 1 , . . . , 𝑛 } , and a set of resources E . Each user can choose a subset of the set of r esources which she intends to use . W e list all feasible choices for user 𝑖 ∈ 𝑁 in the set A 𝑖 ⊆ 2 E . The cost for using each resource 𝑒 ∈ E depends only on the total number of users concurrently selecting that resource, and is denoted with ℓ 𝑒 : N → R ≥ 0 . Once all users have made a choice 𝑎 𝑖 ∈ A 𝑖 , each user incurs a cost obtaine d by summing the costs of all resources she selected. Finally , the system cost represents the cost incurred by all users SC ( 𝑎 ) =  𝑖 ∈ 𝑁  𝑒 ∈ 𝑎 𝑖 ℓ 𝑒 ( | 𝑎 | 𝑒 ) , (1) where | 𝑎 | 𝑒 is the number of users selecting resource 𝑒 in allocation 𝑎 = ( 𝑎 1 , . . . , 𝑎 𝑛 ) . W e denote with G the set containing all congestion games with a maximum of 𝑛 agents, and where all resource costs { ℓ 𝑒 } 𝑒 ∈ E belong to a common set of functions L . Local and global tolling mechanisms. W e assume users to b e self-interested, and observe that self- interested decision making often results in a highly sub optimal system cost [ 30 ]. Consequently , there has been considerable interest in the application of tolling mechanisms to inuence the resulting outcome [ 5 , 6 , 8 , 14 , 18 , 26 ]. Formally , a tolling me chanism 𝑇 : 𝐺 × 𝑒 ↦→ 𝜏 𝑒 is a map that associates an instance 𝐺 ∈ G and a resource 𝑒 ∈ E to the corresponding toll 𝜏 𝑒 . Here 𝜏 𝑒 : { 1 , . . . , 𝑛 } → R is a congestion-dependent toll, i.e., 𝜏 𝑒 maps the number of users in resource 𝑒 to the corresponding toll. As a result, user 𝑖 ∈ 𝑁 incurs a cost factoring both the cost of the resources and the tolls, i.e., C 𝑖 ( 𝑎 ) =  𝑒 ∈ 𝑎 𝑖 [ ℓ 𝑒 ( | 𝑎 | 𝑒 ) + 𝜏 𝑒 ( | 𝑎 | 𝑒 ) ] . (2) Designing tolling mechanisms that utilize global information ( such as knowledge of all resour ce costs, or knowledge of the feasible sets { A 𝑖 } 𝑛 𝑖 = 1 ) is often dicult, as that might require the central planner to access private users information, in addition to the associated computational bur den. Within the context of trac routing ( see Example 1 below), a global mechanism might produce a toll on edge 𝑒 that depends on the structure of the network, on the travel time o ver all edges, as well as on the exact origin and destination of ev ery user . On the contrary , lo cal tolling mechanisms require much less information, are scalable, accommodate resources that are dynamically added or removed, and are r obust against a number of variations, e.g., the common scenario where drivers modify their destination. Therefore , a signicant p ortion of the literature has focused on designing A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 3 tolls that exploit only local information. Formally , we say that a tolling mechanism 𝑇 is local if the toll on each resource only uses information on the cost function ℓ 𝑒 of the same resource 𝑒 , and no other information. If this is the case, we write 𝜏 𝑒 = 𝑇 ( ℓ 𝑒 ) with slight abuse of notation. On the contrary , if the mechanism utilizes additional information on the instance, we say it is global . Example 1 (traffic routing). Within the context of trac routing, E represents the set of e dges dening the underlying road network over which users wish to travel. For this p urp ose, each user 𝑖 ∈ 𝑁 can select any path connecting her origin to her destination node, thus producing a list of feasible paths A 𝑖 . The travel time incurred by a user transiting on edge 𝑒 ∈ E is captured by the function ℓ 𝑒 , and depends only on the number of users traveling through that very edge. In this context, the function ℓ 𝑒 takes into consideration ge ometric properties of the e dge, such as its length, the number of lanes and speed limit [ 37 ]. The system cost in (1) represents the time spent on the network by all users, whereas tolls are imp osed on the edges to incentivize users in sele cting paths that minimize the total travel time (1) . 1.2 Performance metrics The performance of a tolling mechanism is typically measured by the ratio between the system cost incurred at the worst-performing emergent allo cation and the minimum system cost. As users are assume d to b e self-interested, the emergent allo cation is describe d by any of the follo wing equilibrium notions: pure or mixe d Nash equilibrium, correlated or coarse correlated equilibrium – each being a superset of the previous [ 32 ]. 1 When considering pure Nash e quilibria, the performance of a mechanism 𝑇 , referred to as the price of anarchy [21], is dened as Po A ( 𝑇 ) = sup 𝐺 ∈ G NECost ( 𝐺 , 𝑇 ) MinCost ( 𝐺 ) , (3) where MinCost ( 𝐺 ) is the minimum social cost for instance 𝐺 as dened in (1) , and NECost ( 𝐺 , 𝑇 ) denotes the highest social cost at a Nash equilibrium obtained when employing the mechanism 𝑇 on the game 𝐺 . Similarly , it is possible to dene the price of anarchy for mixed Nash, corr elated and coarse corr elated equilibria. While these dier ent metrics need not be equal in general, they do coincide within the setting of interest to this manuscript, as we will later clarify . Therefore, in the fol- lowing, we will use Po A ( 𝑇 ) to r efer to the eciency values of any and all these equilibrium classes. 1.3 Related work The interest in the design of tolls dates back to the early 1900s [ 30 ]. Since then, a large body of literature in the areas of transportation, e conomics, and computer science has investigated this approach [ 5 , 14 , 18 , 26 , 33 ]. Designing tolling mechanisms that optimize the eciency is particularly challenging in the context of ( atomic) congestion games, as observed, e.g., by [ 16 ], in part due to the multiplicity of the e quilibria. While most of the research [ 1 , 3 , 13 , 32 ] has focused on providing eciency bounds for given schemes or in the un-tolled case, much less is known regarding the design question. Owing to the technical diculties, only partial results are available for global tolling mechanisms [ 6 , 8 , 15 ], while r esults for local mechanisms ar e limite d to ane congestion games [8]. Reference [ 8 ] initiated the study of tolling mechanisms in the context of congestion games, restricting their attention to ane resource costs. Relative to this setting, they show how to compute a congestion-indep endent global toll yielding a (tight) price of anarchy of 2 for mixed Nash equilibria, in addition to a congestion-independent local toll yielding a (tight) price of anarchy of 1 + 2 / √ 3 ≈ 2 . 155 for pure Nash e quilibria. Our result pertaining to the design of optimal 1 For a congestion game, existence of pur e Nash equilibria (and thus of all other equilibrium notions mentioned above) is guaranteed even in the presence of tolls, due to the fact that the resulting game is potential. A CM Transactions on Economics and Computation (to appear) 4 Paccagnan, et al. congestion-independent local tolls generalizes this nding to any polynomial (and non polynomial) congestion game and holds tightly for both pure Nash and coarse correlated equilibria. More recently [ 6 ] studies congestion-dependent global tolls for pure Nash and coarse correlated equilibria, in addition to one-round walks from the empty state. Relativ e to unweighted congestion games, they derive tolling mechanisms yielding a price of anarchy for coarse correlated equilibria equal to 2 for ane resource costs; and similarly for higher order p olynomials. While the latter work provides a numb er of interesting insights (e.g., some close d form price of anarchy expressions), all the derived tolling schemes require global information such as network and user kno wledge - an often impractical scenario. In contrast, our results on optimal local tolls fo cuses on the design of optimal mechanisms that exploit local information only , and thus are more widely applicable. Even if utilizing much less information, optimal local mechanisms are still competitive. For example, congestion-dependent optimal lo cal tolls yield a price of anarchy of 2 . 012 for coarse correlated equilibria and ane congestion games (to be compared with a value of 2 mentioned ab ove). Related works have also explored modications of the setup considered here: [ 8 ] also studies singleton congestion games, [ 15 ] focuses on symmetric network congestion games, [ 25 ] on altruistic congestion games, while [ 17 , 19 , 20 ] consider tolling a subset of the resources. Preprint [ 34 ] focuses on the computation of approximate Nash equilibria in atomic congestion games. Therein, the authors design modied resource costs leveraging a methodology similar to that developed in [9, 12, 27]. Finally , we note that the design of tolling mechanisms is far better understood when the original congestion game is replaced by its continuous-ow appro ximation, as uniqueness of the Nash equilibrium is guaranteed. Limited to this setting, marginal cost tolls produce an e quilibrium which is always optimal [ 4 ]. Within the atomic setting, our work demonstrates that marginal cost tolls do not improve - and instead signicantly deteriorate - the r esulting system eciency . 1.4 Preview of our contributions The core of our work is centred on designing optimal local tolling mechanisms and on deriving their corresponding prices of anarchy for various classes of congestion games. Our work develops on parallel directions, and the contributions include the following: i) The design of optimal local tolling mechanisms; ii) The design of optimal congestion-independent local tolling mechanisms; iii) The study of marginal cost tolls and their ineciency . T able 1 highlights the impact of our results on congestion games with polynomial cost functions of varying degree, though our methodology extends further . The following paragraphs describe our contributions in further details, while supporting Python and Matlab code can be found in [10]. Optimal local tolls. In The orems 1 and 2 we pro vide a methodology for computing optimal local tolling mechanisms for congestion games. The resulting price of anarchy values for the case of p olynomial congestion games are presented in T able 1 (fourth column), where we provide a comparison with those derived in [ 6 , 8 ] (third column), which instead make use of global information, for example, by letting the tolling function on r esource 𝑒 depend on all the other resource costs. Perhaps unexpe ctedly , the eciency of optimal tolls designed using only lo cal information is almost identical to that of existing tolls designed using global information [ 6 , 8 ]. In addition to providing similar p erformances by means of less information, local tolls ar e robust against uncertain scenarios (e .g., modications in the origin/destination pairs) and can be compute d eciently . Extensive w ork has focused on quantifying the price of anarchy for load balancing games, that is congestion games where all action sets are singletons, i.e., A 𝑖 ⊆ E . Within this setting, and when all r esource costs ar e ane and identical, the price of anarchy is ≈ 2 . 012 [ 7 , 35 ]. Our results connect with this line of work, demonstrating that optimally tolled ane congestion games have a price of A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 5 𝑑 No toll Global toll Optimal local toll Optimal constant local toll Marginal cost toll [1] from [6, 8] (this work) (this work) (this work) 1 2 . 50 2 2 . 012 2 . 15 3 . 00 2 9 . 58 5 5 . 101 5 . 33 13 . 00 3 41 . 54 15 15 . 551 18 . 36 57 . 36 4 267 . 64 52 55 . 452 89 . 41 391 . 00 5 1513 . 57 203 220 . 401 469 . 74 2124 . 21 6 12 345 . 20 877 967 . 533 3325 . 58 21 337 . 00 T able 1. Price of anarchy values for congestion games with resource costs of degree at most 𝑑 . All results are tight for pure Nash and also hold for coarse corr elated equilibria. The columns feature the price of anarchy with no tolls, with global tolls from [ 6 , 8 ], with optimal lo cal tolls, with optimal constant (i.e. congestion- independent) local tolls, and with marginal cost tolls, respectively . Columns four , fiv e, and six, are composed of entirely novel results, e xcept for the case of constant tolls with 𝑑 = 1 , which recovers [ 8 ]. Note that i) optimal tolls r elying only on local information perform closely to optimal tolls designed using global information, with a dierence in performance below 1% for 𝑑 = 1 ; ii) congestion-independent tolls result in a price of anarchy that is comparable to that obtained using congestion-aware local tolls for polynomials of low degree. The code used to generate this table can be downloaded from [10]. anarchy matching this value, and are tight already within this class. Stated dierently , the price of anarchy of un-tolled and optimally tolled ane load balancing games with identical resources is the same. W e believe such statement holds true more generally . Optimal congestion-independent lo cal tolls. Our methodology can also be exploited to derive optimal local mechanisms under more stringent structural constraints. One such constraint, studied in numerous settings, consists in the use of congestion-indep endent mechanisms, which are attractive b ecause of their simplicity . A linear program to compute optimal congestion-independent local mechanisms is presented in Theorem 3, while the corr esponding optimal prices of anarchy for the case of polynomial congestion games are displayed in T able 1 (fth column) and derived in Corollary 1 as well as Se ction 5. All the results are novel, except for the case of 𝑑 = 1 , which recovers [ 8 ]. W e observe that the performance of congestion-indep endent mechanisms is comparable with that of congestion-aware mechanisms for polynomials of low degree ( 𝑑 ≤ 3 ), and still a go od improvement over the un-tolled setup for high degr ee polynomials. In these cases, congestion- independent mechanisms are not only robust and simple to implement, but also relatively ecient. Marginal cost tolls are worse than no tolls. In non-atomic congestion games, any Nash equilibrium resulting fr om the application of the marginal contribution mechanism is optimal, i.e., it has a price of anar chy equal to one . Corollary 2 sho ws how to utilize our approach to compute the eciency of the marginal cost mechanism in the atomic setup . The resulting values of the price of anarchy are presented in the last column of T able 1 for polynomial congestion games. While the marginal cost mechanism ensures that a Nash equilibrium is optimal (i.e., its price of stability is one), utilizing the marginal cost mechanism on the atomic model yields a price of anarchy that is worse than that experienced levying no toll at all (compar e the second and last column in Table 1). In other wor ds, the design principle derived from the continuous-ow model does not carry over to the original atomic setup. This phenomenon manifests itself already in very simple settings, as we demonstrate in Fig. 3. W e conclude by observing that our result diers signicantly from that in [ 24 ], where the authors show that, as the number of agents grow large , marginal cost tolls become optimal. This dierence stems from the fact that, in [ 24 ], the network structure is xed as the number of agents grows, whereas the w orst case instance in our setting is a function of the number of agents. A CM Transactions on Economics and Computation (to appear) 6 Paccagnan, et al. 1.5 T echniques and high-level ideas Underlying the developments presented above are a number of technical results, which stem from the observation that, in the majority of the existing literature, the set L contains all resource costs of the form ℓ ( 𝑥 ) = Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑏 𝑗 ( 𝑥 ) with 𝛼 𝑗 ≥ 0 , and given basis functions { 𝑏 1 , . . . , 𝑏 𝑚 } . This describ es the fact that each resour ce cost featured in the corr esponding game belongs to a known class of functions. For example, in polynomial congestion games of maximum degree 𝑑 , each resource is associated to a cost of the form 𝛼 1 + 𝛼 2 𝑥 + · · · + 𝛼 𝑑 + 1 𝑥 𝑑 , corresponding to the choice of basis functions { 1 , . . . , 𝑥 𝑑 } . More generally , the decomp osition ℓ ( 𝑥 ) = Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑏 𝑗 ( 𝑥 ) allows us to leverage a common framework to study dierent classes of pr oblems not limited to polynomial congestion games. In this context, we rst show in Theorem 1 that an optimal local tolling mechanisms is a linear map from the set of resource costs to the set of tolls. More precisely , we show that, for every resource cost ℓ ( 𝑥 ) = Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑏 𝑗 ( 𝑥 ) , there exists an optimal local mechanism satisfying 𝑇 opt ( ℓ ) = 𝑇 opt ( Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑏 𝑗 ) = Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑇 opt ( 𝑏 𝑗 ) , where the me chanism is obtained as a linear combination of 𝑇 opt ( 𝑏 𝑗 ) , with the same coecients 𝛼 𝑗 used to dene ℓ . 2 It is worth noting that this rst result allows for a decoupling argument, whereby an optimal tolling function 𝑇 opt ( 𝑏 𝑗 ) can be separately computed for each of the basis 𝑏 𝑗 . The key idea underpinning the result on linearity of optimal tolls lies in observing that any congestion game utilizing resource cost functions with coecients 𝛼 𝑗 ∈ R > 0 and a possibly non-linear tolling mechanism 𝑇 , can be mapped to a corr esponding congestion game where i) all coecients 𝛼 𝑗 are identical to one, ii) only the linear part of the tolling mechanism 𝑇 is used, and iii) the price of anarchy is identical to that of the original game (as the number of resources grows). Complementary to this, our second result in Theorem 1 reduces the problem of designing optimal basis tolls { 𝑇 opt ( 𝑏 𝑗 ) } 𝑚 𝑗 = 1 to a polynomially solvable linear program that also returns the tight value of the optimal price of anar chy . W e do so by building upon the r esults in [ 11 , 28 ], which allow us to determine the p erformance of a tolling mechanism through the solution of a linear program (see Eq. (11)), in a similar spirit to [ 6 ], although with a provably tight characterization for any number of agents and tolling function. W e exploit this result and construct a polynomially-size d linear program that, for a given basis 𝑏 𝑗 , searches over all linear tolls to nd 𝑇 opt ( 𝑏 𝑗 ) . When the basis functions are conve x and increasing ( e.g., in the well-studied polynomial case), we are able to explicitly solv e the linear program and provide an analytic expression for the optimal tolling function, as well as a semi-analytic expr ession for optimal price of anarchy (Theorem 2). The fundamental idea consists in showing that the set of active constraints at the solution gives rise to a telescopic recursion, whereby the optimal toll to be levied when 𝑢 + 1 agents are selecting a resource can be written as a function of the optimal toll to be le vied when only 𝑢 agents are present. This is the most technical part of the manuscript, and the expression of the optimal price of anarchy re veals an unexpected conne ction with that for un-tolled load balancing games on identical machines [ 7 , 35 ]. While the sizes of the linear programs appearing in Theorems 1 and 2 grow (polynomially) with the number of agents 𝑛 , in Section 4 we show how to design optimal tolling mechanisms that apply to any 𝑛 (possibly innite). Our approach consists in two steps: we rst solve a linear program of nite dimension, and then extend its solution to arbitrary 𝑛 . Congestion-independent optimal local me chanisms as well as the eciency of the marginal cost mechanism can also b e computed through linear programs (Theorem 3 and Corollary 2). In the rst case, admissible tolls basis { 𝑇 opt ( 𝑏 𝑗 ) } 𝑚 𝑗 = 1 are constrained to be constant with the congestion, while in the second case the expr ession for marginal cost tolls is substituted in the program. W e provide analytical solutions to these two pr ograms for the case of polynomial congestion games. 2 As word of caution, we r emark that linearity of the optimal mechanism in the sense claried ab ove does not mean that the corresponding tolls are linear in the congestion level, i.e ., does not mean that 𝜏 𝑒 ( 𝑥 ) = 𝑎 𝑒 𝑥 + 𝑏 𝑒 for some 𝑎 𝑒 , 𝑏 𝑒 . A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 7 1.6 Organization In Section 2 we derive linear programs to compute optimal local tolling mechanisms. W e also provide optimal price of anarchy values for polynomial congestion games. In Section 3 we obtain an explicit solution to these programs that applies when resource costs ar e convex and increasing. Section 4 generalizes the pre vious results to arbitrarily large number of agents. In Section 5 and Section 6 we derive congestion independent tolling mechanism and evaluate the eciency of the marginal cost mechanism. In these sections we also specialize the results to the polynomial case. 2 OPTIMAL TOLLING MECHANISMS In this section we de velop a methodology to compute optimal local tolling me chanisms thr ough the solution of tractable linear programs. T o ease the notation, we introduce the set of integer triplets I = { ( 𝑥 , 𝑦 , 𝑧 ) ∈ Z 3 ≥ 0 s.t. 1 ≤ 𝑥 + 𝑦 + 𝑧 ≤ 𝑛 and either 𝑥𝑦𝑧 = 0 or 𝑥 + 𝑦 + 𝑧 = 𝑛 } , for given 𝑛 ∈ N . Theorem 1. A local mechanism minimizing the price of anarchy over congestion games with 𝑛 agents, resource costs ℓ ( 𝑥 ) = Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑏 𝑗 ( 𝑥 ) , 𝛼 𝑗 ≥ 0 , and basis functions { 𝑏 1 , . . . , 𝑏 𝑚 } is given by 𝑇 opt ( ℓ ) = 𝑚  𝑗 = 1 𝛼 𝑗 · 𝜏 opt 𝑗 , where 𝜏 opt 𝑗 : { 1 , . . . , 𝑛 } → R , 𝜏 opt 𝑗 ( 𝑥 ) = 𝑓 opt 𝑗 ( 𝑥 ) − 𝑏 𝑗 ( 𝑥 ) (4) and 𝜌 opt 𝑗 ∈ R , 𝑓 opt 𝑗 : { 1 , . . . , 𝑛 } → R solve the following linear programs (one per each 𝑏 𝑗 ) max 𝑓 ∈ R 𝑛 , 𝜌 ∈ R 𝜌 s . t . 𝑏 𝑗 ( 𝑥 + 𝑧 ) ( 𝑥 + 𝑧 ) − 𝜌 𝑏 𝑗 ( 𝑥 + 𝑦 ) ( 𝑥 + 𝑦 ) + 𝑓 ( 𝑥 + 𝑦 ) 𝑦 − 𝑓 ( 𝑥 + 𝑦 + 1 ) 𝑧 ≥ 0 ∀ ( 𝑥 , 𝑦 , 𝑧 ) ∈ I , (5) where we dene 𝑏 𝑗 ( 0 ) = 𝑓 ( 0 ) = 𝑓 ( 𝑛 + 1 ) = 0 . Correspondingly , Po A ( 𝑇 opt ) = max 𝑗 { 1 / 𝜌 opt 𝑗 } . 3 These results are tight for pure Nash equilibria, and extend to coarse correlated e quilibria. Optimal T axes in Atomic Congestion Games 7 Congestion-indep endent optimal lo cal me chanisms as w ell as the e  ciency of the marginal cost me chanism can also b e compute d thr ough linear pr ograms (The or em 3 and Cor ollar y 2). In the  rst case , admissible tolls basis { ) opt ( 1 9 )} < 9 = 1 ar e constraine d to b e constant with the congestion, while in the se cond case the e xpr ession for marginal cost tolls is substitute d in the pr ogram. W e pr o vide analytical solutions to these tw o pr ograms for the case of p olynomial congestion games. 1.6 Organization In Se ction 2 w e deriv e linear pr ograms to compute optimal lo cal tolling me chanisms. W e also pr o vide optimal price of anar chy values for p olynomial congestion games. In Se ction 3 w e obtain an e xplicit solution to these pr ograms that applies when r esour ce costs ar e conv e x and incr easing. Se ction 4 generalizes the pr e vious r esults to arbitrarily large numb er of agents. In Se ction 5 and Se ction 6 w e deriv e congestion indep endent tolling me chanism and e valuate the e  ciency of the marginal cost me chanism. In these se ctions w e also sp e cialize the r esults to the p olynomial case . 2 OPTIMAL T OLLING MECHANISMS In this se ction w e de v elop a metho dology to compute optimal lo cal tolling me chanisms thr ough the solution of tractable linear pr ograms. T o ease the n otation, w e intr o duce the set of integer triplets I = {( G , ~ , I ) 2 Z 3  0 s.t. 1  G + ~ + I  = and either G~ I = 0 or G + ~ + I = = } , for giv en = 2 N . T  1. A lo cal me chanism minimizing the price of anar chy o v er congestion games with = agents, r esour ce costs ✓ ( G ) = Õ < 9 = 1 U 9 1 9 ( G ) , U 9  0 , and basis functions { 1 1 ,. . . , 1 < } is giv en by ) opt ( ✓ ) = < ’ 9 = 1 U 9 · g opt 9 , wher e g opt 9 : { 1 ,. . . , = } ! R , g opt 9 ( G ) = 5 opt 9 ( G )  1 9 ( G ) (4) and d opt 9 2 R , 5 opt 9 : { 1 ,. . . , = } ! R solv e the follo wing linear pr ograms ( one p er each 1 9 ) max 5 2 R = , d 2 R d s.t. 1 9 ( G + I )( G + I )  d1 9 ( G + ~ )( G + ~ )+ 5 ( G + ~ ) ~  5 ( G + ~ + 1 ) I  0 8 ( G , ~ , I ) 2 I , (5) wher e w e de � ne 1 9 ( 0 ) = 5 ( 0 ) = 5 ( = + 1 ) = 0 . Corr esp ondingly , Po A ( ) opt ) = max 9 { 1 / d opt 9 } . 3 These r esults ar e tight for pur e Nash e quilibria, and e xtend to coarse corr elate d e quilibria. Fig. 1. Graphical r epr esentation of the main r esult on the design of optimal tolls. The input consists of a giv en latency ✓ ( G ) e xpr esse d as a combination of basis 1 9 ( G ) with co e � icients U 9 . For each basis, w e compute the asso ciate d optimal toll g opt 9 ( G ) = 5 opt 9 ( G )  1 9 ( G ) by solving the linear pr ogram (LP) app earing in (5) . The r esulting optimal toll is obtaine d as the linear combination of g opt 9 ( G ) with the same co e � icients U 9 . The quantities g opt 9 ( G ) can b e pr e compute d and stor e d in a librar y , o � loading the solution of the linear pr ograms. U 1 U < ✓ = Õ < 9 = 1 U 9 1 9 1 1 1 < . . . ⇥ ⇥ + g opt 1 g opt < ) opt ( ✓ ) Solve LP with 1 1 Solve LP with 1 < 3 If w e r e quir e tolls to b e non-negativ e , an optimal me chanism is as in (4) , wher e w e set g opt 9 ( G ) = 5 opt 9 ( G )· Po A opt  1 9 ( G ) . A CM T ransactions on Economics and Computation, V ol. 1, No . 1, Article . Publication date: Mar ch 2021. Fig. 1. Graphical representation of the main result on the design of optimal tolls. The input consists of a resource cost ℓ ( 𝑥 ) expressed as a combination of basis 𝑏 𝑗 ( 𝑥 ) with coeicients 𝛼 𝑗 . For each basis, we compute the associated optimal toll 𝜏 opt 𝑗 ( 𝑥 ) = 𝑓 opt 𝑗 ( 𝑥 ) − 𝑏 𝑗 ( 𝑥 ) by solving the linear program (LP) appearing in (5) . The resulting optimal toll is obtained as the linear combination of 𝜏 opt 𝑗 ( 𝑥 ) with the same coeicients 𝛼 𝑗 . The quantities 𝜏 opt 𝑗 ( 𝑥 ) can be precomputed and stored in a librar y , o loading the solution of the linear programs. The above statement contains tw o fundamental results. The rst part of the statement shows that an optimal tolling mechanism applied to the function ℓ ( 𝑥 ) = Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑏 𝑗 ( 𝑥 ) can be obtained as the linear combination of 𝜏 opt 𝑗 ( 𝑥 ) , with the same co ecients 𝛼 𝑗 used to dene ℓ . Complementary to this, the second part of the statement pr ovides a practical technique to compute 𝜏 opt 𝑗 ( 𝑥 ) for each 3 If we r equire tolls to be non-negative , an optimal mechanism is as in (4) , where w e set 𝜏 opt 𝑗 ( 𝑥 ) = 𝑓 opt 𝑗 ( 𝑥 ) · PoA opt − 𝑏 𝑗 ( 𝑥 ) . A CM Transactions on Economics and Computation (to appear) 8 Paccagnan, et al. of the basis 𝑏 𝑗 ( 𝑥 ) as the solution of a tractable linear program. A graphical repr esentation of this process is included in Fig. 1, while Python/Matlab code to design optimal tolls can b e found in [ 10 ]. W e solved the latter linear programs for 𝑛 = 100 and polynomials of maximum degree 1 ≤ 𝑑 ≤ 6 . The corresponding results are display ed in T able 1, while Section 4 shows that these r esults hold identically for arbitrarily large 𝑛 . In the case of 𝑑 = 1 , the optimal price of anarchy is approximately 2 . 012 , matching that of un-tolled load balancing games on identical machines [ 7 , 35 ]. W e observe that, in this restricted setting, the price of anarchy cannot b e improved at all through local tolling mechanisms. In fact, no matter what non-negative tolling me chanism we are given, we can always construct a load balancing game on identical machines with a price of anarchy no lo wer than 2 . 012 . 4 W e conclude obser ving that the decomposition of resource costs as linear combination of basis functions is, strictly sp eaking, not required for Theorem 1 to hold. Nevertheless, pursuing this approach would r equire to solve a linear program for each function in L , a task that becomes daunt- ing when L contains innitely many functions, e.g., in the case of polynomial congestion games. In this case, Theorem 1 allows to compute optimal tolls by solving nitely many linear programs. Proof. W e divide the proof in two parts to ease the exposition. Part 1. W e show that any local me chanism minimizing the price of anarchy over all linear local mechanisms, does so also over all linear and non-linear local mechanisms. W e let 𝑇 opt be a mecha- nism that minimizes the price of anarchy over all linear local mechanisms, i.e ., over all 𝑇 satisfying 𝑇 𝑚  𝑗 = 1 𝛼 𝑗 𝑏 𝑗 ! = 𝑚  𝑗 = 1 𝛼 𝑗 𝑇 ( 𝑏 𝑗 ) , for all 𝛼 𝑗 ≥ 0 . W e intend to show that Po A ( 𝑇 opt ) ≤ Po A ( 𝑇 ) for any possible 𝑇 (linear or non-linear). T owards this goal, assume, for a contradiction, that ther e exists a tolling mechanism ˆ 𝑇 such that Po A ( 𝑇 opt ) > Po A ( ˆ 𝑇 ) . (6) Let G 𝑏 be the class of games in which any resource 𝑒 can only utilize a resource cost ℓ 𝑒 ∈ { 𝑏 1 , . . . , 𝑏 𝑚 } . Since G 𝑏 ⊂ G , we have Po A ( ˆ 𝑇 ) ≥ sup 𝐺 ∈ G 𝑏 NECost ( 𝐺 , ˆ 𝑇 ) MinCost ( 𝐺 ) . (7) Additionally , let G ( Z ≥ 0 ) ⊂ G be the class of games with 𝛼 𝑗 ∈ Z ≥ 0 for all 𝑗 ∈ { 1 , . . . , 𝑚 } , for all resources in E . Construct the mechanism ¯ 𝑇 by “linearizing” the mechanism ˆ 𝑇 , i.e., as ¯ 𝑇 ( ℓ ) = ¯ 𝑇 𝑚  𝑗 = 1 𝛼 𝑗 𝑏 𝑗 ! = 𝑚  𝑗 = 1 𝛼 𝑗 ˆ 𝑇 ( 𝑏 𝑗 ) . W e obser ve that the eciency of any instance 𝐺 ∈ G 𝑏 to which the tolling mechanism ˆ 𝑇 is applied, coincides with that of an instance 𝐺 ∈ G ( Z ≥ 0 ) to which ¯ 𝑇 is applied, and vice-versa. Thus, sup 𝐺 ∈ G 𝑏 NECost ( 𝐺 , ˆ 𝑇 ) MinCost ( 𝐺 ) = sup 𝐺 ∈ G ( Z ≥ 0 ) NECost  𝐺 , ¯ 𝑇  MinCost ( 𝐺 ) = Po A ( ¯ 𝑇 ) , (8) where the last equality holds due to Lemma 1 in App endix A.1. Putting together Eqs. (6) to (8) gives Po A ( 𝑇 opt ) > Po A ( ¯ 𝑇 ) . (9) 4 T o do so, it is sucient to utilize the instance in [ 35 , Thm 3.4], where the the resource cost 𝑥 used therein is replaced with 𝑥 + 𝜏 ( 𝑥 ) . The Nash equilibrium and the optimal allocation will remain unchanged, yielding the same price of anar chy value. A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 9 Since 𝑇 opt minimizes the price of anarchy over all linear me chanisms, and since ¯ 𝑇 is linear by construction, it must be Po A ( 𝑇 opt ) ≤ Po A ( ¯ 𝑇 ) , a contradiction of (9) . Thus, 𝑇 opt minimizes the price of anarchy over any mechanism. Part 2. W e will derive a linear program to design optimal linear mechanisms. Putting this together with the claim in Part 1 will conclude the proof. T owards this goal, we will prove that any mechanism of the form 𝑇 ( ℓ ) = 𝑚  𝑗 = 1 𝛼 𝑗 𝜏 opt 𝑗 with 𝜏 opt 𝑗 ( 𝑥 ) = 𝜆 · 𝑓 opt 𝑗 ( 𝑥 ) − 𝑏 𝑗 ( 𝑥 ) (10) is optimal, regardless of the value of 𝜆 ∈ R > 0 . While this is slightly mor e general than ne eded, setting 𝜆 = 1 will give the rst claim. Additionally , setting 𝜆 = Po A opt will give the second claim as this choice will ensure non-negativity of the tolls. Before turning to the pr oof, we recall a result from [ 11 ] that allows us to compute the price of anarchy for given linear tolling mechanism 𝑇 ( ℓ ) = Í 𝑚 𝑗 = 1 𝛼 𝑗 𝜏 𝑗 . Upon dening 𝑓 𝑗 ( 𝑥 ) = 𝑏 𝑗 ( 𝑥 ) + 𝜏 𝑗 ( 𝑥 ) for all 1 ≤ 𝑥 ≤ 𝑛 and 𝑗 ∈ { 1 , . . . , 𝑚 } , the authors show that the price of anarchy of 𝑇 computed over congestion games G is identical for pure Nash and coarse corr elated equilibria and is given by Po A ( 𝑇 ) = 1 / 𝜌 opt , where 𝜌 opt is the value of the following program max 𝜌 ∈ R ,𝜈 ∈ R ≥ 0 𝜌 s . t . 𝑏 𝑗 ( 𝑥 + 𝑧 ) ( 𝑥 + 𝑧 ) − 𝜌 𝑏 𝑗 ( 𝑥 + 𝑦 ) ( 𝑥 + 𝑦 ) + 𝜈 [ 𝑓 𝑗 ( 𝑥 + 𝑦 ) 𝑦 − 𝑓 𝑗 ( 𝑥 + 𝑦 + 1 ) 𝑧 ] ≥ 0 ∀ ( 𝑥 , 𝑦, 𝑧 ) ∈ I , ∀ 𝑗 ∈ { 1 , . . . , 𝑚 } , (11) W e also remark that, when all functions { 𝑓 𝑗 } 𝑚 𝑗 = 1 are non-decreasing, it is sucient to only consider a reduced set of constraints, following a similar argument to that in [ 28 , Cor . 1]. In this case, the linear program simplies to max 𝜌 ∈ R ,𝜈 ∈ R ≥ 0 𝜌 s . t . 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑓 𝑗 ( 𝑢 ) 𝑢 − 𝑓 𝑗 ( 𝑢 + 1 ) 𝑣 ] ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 ≤ 𝑛, ∀ 𝑗 ∈ { 1 , . . . , 𝑚 } , 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑓 𝑗 ( 𝑢 ) ( 𝑛 − 𝑣 ) − 𝑓 𝑗 ( 𝑢 + 1 ) ( 𝑛 − 𝑢 ) ] ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 > 𝑛, ∀ 𝑗 ∈ { 1 , . . . , 𝑚 } . (12) W e now leverage (11) to prove that any mechanism in (10) is optimal, as required. T owards this goal, we begin by observing that the optimal price of anarchy obtained when the resour ce costs are generated using all the basis functions { 𝑏 1 , . . . , 𝑏 𝑚 } is no smaller than the optimal price of anarchy obtained when the resource costs ar e generated using a single basis function { 𝑏 𝑗 } at a time (and therefore is no smaller than the highest of these optimal price of anarchy values). This follows readily since the former class of games is a superset of the latter . Additionally , observe that a set of tolls minimizing the price of anarchy over the games generated using a single basis function { 𝑏 𝑗 } is precisely that in (10) . This is because minimizing the price of anarchy amounts to designing 𝑓 𝑗 to maximize 𝜌 in (11), i.e., to solving the following program max 𝑓 ∈ R 𝑛 max 𝜌 ∈ R ,𝜈 ∈ R ≥ 0 𝜌 s . t . 𝑏 𝑗 ( 𝑥 + 𝑧 ) ( 𝑥 + 𝑧 ) − 𝜌 𝑏 𝑗 ( 𝑥 + 𝑦 ) ( 𝑥 + 𝑦 ) + 𝜈 [ 𝑓 ( 𝑥 + 𝑦 ) 𝑦 − 𝑓 ( 𝑥 + 𝑦 + 1 ) 𝑧 ] ≥ 0 ∀ ( 𝑥 , 𝑦 , 𝑧 ) ∈ I , A CM Transactions on Economics and Computation (to appear) 10 Paccagnan, et al. which can be equivalently written as max ˜ 𝑓 ∈ R 𝑛 , 𝜌 ∈ R 𝜌 s . t . 𝑏 𝑗 ( 𝑥 + 𝑧 ) ( 𝑥 + 𝑧 ) − 𝜌 𝑏 𝑗 ( 𝑥 + 𝑦 ) ( 𝑥 + 𝑦 ) + ˜ 𝑓 ( 𝑥 + 𝑦 ) 𝑦 − ˜ 𝑓 ( 𝑥 + 𝑦 + 1 ) 𝑧 ≥ 0 ∀ ( 𝑥 , 𝑦 , 𝑧 ) ∈ I , where we dened ˜ 𝑓 = 𝜈 · 𝑓 . While 𝑓 opt 𝑗 is dened in (5) precisely as the solution of this last program, resulting in a price of anarchy of 1 / 𝜌 opt 𝑗 , note that 𝜆 · 𝑓 opt 𝑗 is also a solution since its price of anarchy matches 1 / 𝜌 opt 𝑗 (in fact, it can be computed using (11) for which ( 𝜌 , 𝜈 ) = ( 𝜌 opt 𝑗 , 1 / 𝜆 ) ar e feasible). The above reasoning shows that the optimal price of anarchy for a game with resource costs generated by { 𝑏 1 , . . . , 𝑏 𝑚 } must b e no smaller than max 𝑗 { 1 / 𝜌 opt 𝑗 } . W e now show that this holds with equality . T owards this goal, we note, thanks to (11) , that utilizing tolls as in (6) for a game generated by { 𝑏 1 , . . . , 𝑏 𝑚 } results in a price of anarchy of pr ecisely max 𝑗 { 1 / 𝜌 opt 𝑗 } . This follows as ( min 𝑗 { 𝜌 opt 𝑗 } , 1 / 𝜆 ) is feasible for this program for any choice of 𝜆 > 0 . This proves, as requested, that any tolling mechanism dened in (10) is optimal. W e now verify that the choice 𝜆 = Po A opt = max 𝑗 { 1 / 𝜌 opt 𝑗 } ensures positivity of the tolls, which is equivalent to 𝑓 opt 𝑗 ( 𝑥 ) − 𝑏 𝑗 ( 𝑥 ) / 𝜆 ≥ 0 for all 𝑥 ∈ { 1 , . . . , 𝑛 } . This follows readily , as setting 𝑥 = 𝑧 = 0 in (5) results in the constraint 𝑓 ( 𝑦 ) − 𝜌 𝑏 𝑗 ( 𝑦 ) ≥ 0 for all 𝑦 ∈ { 1 , . . . , 𝑛 } . Since 𝑓 opt 𝑗 and 𝜌 opt 𝑗 must be feasible for this constraint, we have 𝑓 opt 𝑗 ( 𝑦 ) − 𝜌 opt 𝑗 𝑏 𝑗 ( 𝑦 ) ≥ 0 . One concludes obser ving that 𝑓 opt 𝑗 ( 𝑦 ) − 𝑏 𝑗 ( 𝑦 ) / 𝜆 ≥ 𝑓 opt 𝑗 ( 𝑦 ) − 𝜌 opt 𝑗 𝑏 𝑗 ( 𝑦 ) ≥ 0 , since 𝜆 ≥ 1 / 𝜌 opt 𝑗 . W e conclude remarking that all results hold for both Nash and coarse correlated equilibria, as they were derived from (11). □ 3 EXPLICIT SOLU TION AND SIMPLIFIED LINEAR PROGRAM In this section we derive a simplied linear program as well as an analytical solution to the pr oblem of designing optimal tolling mechanisms. W e do so under the assumption that all basis functions are positive, incr easing, and convex in the discrete sense . 5 Theorem 2. Consider congestion games with 𝑛 agents, where resource costs take the form ℓ ( 𝑥 ) = Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑏 𝑗 ( 𝑥 ) , 𝛼 𝑗 ≥ 0 , and basis 𝑏 𝑗 : { 1 , . . . , 𝑛 } → R are positive, convex, strictly increasing. 6 i) A tolling mechanism minimizing the price of anarchy is as in (4) , where each 𝑓 opt 𝑗 : { 1 , . . . , 𝑛 } → R solves the following simplied linear program 𝜌 opt 𝑗 = max 𝑓 ∈ R 𝑛 , 𝜌 ∈ R 𝜌 s . t . 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝑓 ( 𝑢 ) 𝑢 − 𝑓 ( 𝑢 + 1 ) 𝑣 ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 ≤ 𝑛, 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝑓 ( 𝑢 ) ( 𝑛 − 𝑢 ) − 𝑓 ( 𝑢 + 1 ) ( 𝑛 − 𝑢 ) ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 > 𝑛, (13) with 𝑓 ( 0 ) = 𝑓 ( 𝑛 + 1 ) = 0 . The corresponding optimal price of anarchy is max 𝑗 { 1 / 𝜌 opt 𝑗 } . ii) A n explicit expression for each 𝑓 opt 𝑗 is given by the following recursion, where 𝑓 opt 𝑗 ( 1 ) = 𝑏 𝑗 ( 1 ) , 𝑓 opt 𝑗 ( 𝑢 + 1 ) = min 𝑣 ∈ { 1 ,. . .,𝑛 } 𝛽 ( 𝑢, 𝑣 ) 𝑓 opt 𝑗 ( 𝑢 ) + 𝛾 ( 𝑢, 𝑣 ) − 𝛿 ( 𝑢 , 𝑣 ) 𝜌 opt 𝑗 , 𝛽 ( 𝑢, 𝑣 ) = min { 𝑢, 𝑛 − 𝑣 } min { 𝑣 , 𝑛 − 𝑢 } , 𝛾 ( 𝑢 , 𝑣 ) = 𝑏 ( 𝑣 ) 𝑣 min { 𝑣 , 𝑛 − 𝑢 } , 𝛿 ( 𝑢, 𝑣 ) = 𝑏 ( 𝑢 ) 𝑢 min { 𝑣 , 𝑛 − 𝑢 } , (14) 5 W e say that a function 𝑓 : { 1 , . . . , 𝑛 } → R is convex if 𝑓 ( 𝑥 + 1 ) − 𝑓 ( 𝑥 ) is non-decreasing in its domain. 6 The result also holds if convexity and strict increasingness of 𝑏 𝑗 ( 𝑥 ) are weakened to strict convexity of 𝑏 𝑗 ( 𝑥 ) 𝑥 and 𝑏 𝑗 ( 𝑛 ) > 𝑏 𝑗 ( 𝑛 − 1 ) . One such example is that of 𝑏 𝑗 ( 𝑥 ) = √ 𝑥 . A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 11 𝜌 opt 𝑗 = min ( 𝑣 1 ,. . .,𝑣 𝑛 ) ∈ { 1 ,. . .,𝑛 } 𝑛 − 1 × { 0 ,. . .,𝑛 } ( 𝑛 − 𝑣 𝑛 )  Î 𝑛 − 1 𝑢 = 1 𝛽 𝑢 𝑏 𝑗 ( 1 ) + Í 𝑛 − 2 𝑢 = 1  Î 𝑛 − 1 𝑖 = 𝑢 + 1 𝛽 𝑖  𝛾 𝑢 + 𝛾 𝑛 − 1  + 𝑏 ( 𝑣 𝑛 ) 𝑣 𝑛 ( 𝑛 − 𝑣 𝑛 )  Í 𝑛 − 2 𝑢 = 1  Î 𝑛 − 1 𝑖 = 𝑢 + 1 𝛽 𝑖  𝛿 𝑢 + 𝛿 𝑛 − 1  + 𝑏 ( 𝑛 ) 𝑛 , (15) where we use the short-hand notation 𝛽 𝑢 instead of 𝛽 ( 𝑢 , 𝑣 𝑢 ) , and similarly for 𝛾 𝑢 and 𝛿 𝑢 . Before delving into the proof, we observe that the key diculty in designing optimal tolls resides in the expressions of 𝜌 opt 𝑗 arising from (15) . Nevertheless, for any possible choice of ¯ 𝜌 𝑗 that approximates 𝜌 opt 𝑗 from below , i.e., ¯ 𝜌 𝑗 ≤ 𝜌 opt 𝑗 , one can directly utilize the recursion in (14) to design a valid tolling me chanism. The resulting price of anarchy would then amount to max 𝑗 { 1 / ¯ 𝜌 𝑗 } > max 𝑗 { 1 / 𝜌 opt 𝑗 } . This follows from the ensuing proof. Proof. As shown in Theorem 1, computing an optimal tolling mechanism amounts to utilizing (4) , where each 𝜏 opt 𝑗 has been designed through the solution of the program in (5) . In light of this, we prove the theorem as follows: rst, we consider a simplie d linear program, where only a subset of the constraints enforced in (5) are considered. Second, we show that a solution of this simplie d program is given by ( 𝜌 opt 𝑗 , 𝑓 opt 𝑗 ) as dened ab ove . Third, we show that 𝑓 opt 𝑗 is non-decreasing, thus ensuring that ( 𝜌 opt 𝑗 , 𝑓 opt 𝑗 ) is also feasible for the original over constrained program in (5) . From this we conclude that ( 𝜌 opt 𝑗 , 𝑓 opt 𝑗 ) must also be a solution of (5) , i.e., the second claim in the Theorem. W e conclude with some cosmetics, and transform the simplied linear program whose solution is given by ( 𝜌 opt 𝑗 , 𝑓 opt 𝑗 ) in (13) , thus obtaining the rst claim. Throughout the proof, we drop the index 𝑗 from 𝑏 𝑗 as the proof can be repeated for each basis separately . Simplified linear program. W e begin rewriting the program in (5) , where instead of the indices ( 𝑥 , 𝑦 , 𝑧 ) , we use the corr esponding indices ( 𝑢, 𝑣 , 𝑥 ) dened as 𝑢 = 𝑥 + 𝑦 , 𝑣 = 𝑥 + 𝑧 . The constraint indexed by ( 𝑢, 𝑣 , 𝑥 ) reads as 𝑏 ( 𝑣 ) 𝑣 − 𝜌𝑏 ( 𝑢 ) 𝑢 + 𝑓 ( 𝑢 ) ( 𝑢 − 𝑥 ) − 𝑓 ( 𝑢 + 1 ) ( 𝑣 − 𝑥 ) ≥ 0 . W e now consider only the constraints where 𝑥 is set to 𝑥 = min { 0 , 𝑢 + 𝑣 − 𝑛 } , and 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } , Such constraints read as 𝑏 ( 𝑣 ) 𝑣 − 𝜌𝑏 ( 𝑢 ) 𝑢 + min { 𝑢, 𝑛 − 𝑣 } 𝑓 ( 𝑢 ) − min { 𝑣 , 𝑛 − 𝑢 } 𝑓 ( 𝑢 + 1 ) ≥ 0 . 7 Finally , we exclude the constraints with 𝑣 = 0 , 𝑢 ∈ { 1 , . . . , 𝑛 − 1 } and obtain the following simplied linear program max 𝑓 ∈ R 𝑛 , 𝜌 ∈ R 𝜌 s . t . 𝑏 ( 𝑣 ) 𝑣 − 𝜌𝑏 ( 𝑢 ) 𝑢 + min { 𝑢, 𝑛 − 𝑣 } 𝑓 ( 𝑢 ) − min { 𝑣 , 𝑛 − 𝑢 } 𝑓 ( 𝑢 + 1 ) ≥ 0 ∀ ( 𝑢, 𝑣 ) ∈ { 0 , . . . , 𝑛 } × { 1 , . . . , 𝑛 } ∪ ( 𝑛, 0 ) (16) Proof that ( 𝜌 opt , 𝑓 opt ) solve (16) . T owards the stated goal, we begin by observing that ( 𝜌 opt , 𝑓 opt ) is feasible by construction. For 𝑢 = 0 this follows as the tightest constraints in (16) read as 𝑓 opt ( 1 ) ≥ 𝑏 ( 1 ) and we selected 𝑓 opt ( 1 ) = 𝑏 ( 1 ) . Feasibility is immediate to verify for 𝑢 ∈ { 1 , . . . , 𝑛 − 1 } , 𝑣 ∈ { 1 , . . . , 𝑛 } as applying its denition gives 𝑓 opt ( 𝑢 + 1 ) ≤ 𝛽 ( 𝑢, 𝑣 ) 𝑓 opt ( 𝑢 ) + 𝛾 ( 𝑢, 𝑣 ) − 𝛿 ( 𝑢 , 𝑣 ) 𝜌 opt . Using the expressions of 𝛽 , 𝛾 , 𝛿 , and rearranging giv es exactly the constraint ( 𝑢, 𝑣 ) in (16) . The only element of diculty consists in showing that also the constraints with 𝑢 = 𝑛 , 𝑣 ∈ { 0 , . . . , 𝑛 } are satised. T owards this goal, we observe that utilizing the recursive denition of 𝑓 opt we obtain an expression for 𝑓 opt ( 𝑛 ) as a function of 𝜌 opt with a nested succession of minimizations, which can be jointly extracted as follows 𝑓 opt ( 𝑛 ) = min 𝑣 𝑛 − 1  · · · + min 𝑣 𝑛 − 2  · · · + min 𝑣 1 { . . . }   = min 𝑣 𝑛 − 1 min 𝑣 𝑛 − 2 . . . min 𝑣 1 { . . . } . 7 Note that considering all these constraints with 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } results precisely in (13) . T o see this, simply distinguish the cases based on whether 𝑢 + 𝑣 ≤ 𝑛 or 𝑢 + 𝑣 > 𝑛 . A CM Transactions on Economics and Computation (to appear) 12 Paccagnan, et al. This holds as 𝑓 opt ( 𝑢 + 1 ) = min 𝑣 𝑢  𝛽 𝑢 min 𝑣 𝑢 − 1 ( 𝛽 𝑢 − 1 𝑓 opt ( 𝑢 − 1 ) − 𝛿 𝑢 − 1 𝜌 opt + 𝛾 𝑢 − 1 ) − 𝛿 𝑢 𝜌 opt + 𝛾 𝑢  , and since 𝛽 𝑢 ≥ 0 , the latter simplies to 𝑓 opt ( 𝑢 + 1 ) = min 𝑣 𝑢 min 𝑣 𝑢 − 1 𝛽 𝑢 𝛽 𝑢 − 1 𝑓 opt ( 𝑢 − 1 ) − ( 𝛽 𝑢 𝛿 𝑢 − 1 + 𝛿 𝑢 ) 𝜌 opt + 𝛽 𝑢 𝛾 𝑢 − 1 + 𝛾 𝑢 . Repeating the argument recursively gives the desired expr ession. Hence, 𝑓 opt ( 𝑛 ) = min ( 𝑣 1 ,. . .,𝑣 𝑛 − 1 ) ∈ { 1 ,.. .,𝑛 } 𝑛 − 1 𝑛 − 1 Ö 𝑢 = 1 𝛽 𝑢 𝑏 𝑗 ( 1 ) + 𝑛 − 2  𝑢 = 1 𝑛 − 1 Ö 𝑖 = 𝑢 + 1 𝛽 𝑖 ! ( 𝛾 𝑢 − 𝛿 𝑢 𝜌 opt ) + ( 𝛾 𝑛 − 1 − 𝛿 𝑛 − 1 𝜌 opt )  min ( 𝑣 1 ,. . .,𝑣 𝑛 − 1 ) ∈ { 1 ,. . .,𝑛 } 𝑛 − 1 𝑞 ( 𝑣 1 , . . . , 𝑣 𝑛 − 1 ; 𝜌 opt ) , where we implicitly dened 𝑞 ( 𝑣 1 , . . . , 𝑣 𝑛 − 1 ; 𝜌 opt ) . The constraints we intend to verify read as 𝑏 ( 𝑣 ) 𝑣 − 𝜌𝑏 ( 𝑛 ) 𝑛 + ( 𝑛 − 𝑣 ) 𝑓 opt ( 𝑛 ) ≥ 0 for all 𝑣 ∈ { 0 , . . . , 𝑛 } , and can be equivalently written as min 𝑣 𝑛 ∈ { 0 ,. . ., 𝑛 } [ 𝑏 ( 𝑣 𝑛 ) 𝑣 𝑛 − 𝜌𝑏 ( 𝑛 ) 𝑛 + ( 𝑛 − 𝑣 𝑛 ) 𝑓 opt ( 𝑛 ) ] ≥ 0 . W e substitute the resulting expres- sion of 𝑓 opt ( 𝑛 ) , extract the minimization over 𝑣 𝑛 as in the above, and are therefore left with min ( 𝑣 1 ,. . .,𝑣 𝑛 − 1 ,𝑣 𝑛 ) ∈ { 1 ,. . .,𝑛 } 𝑛 − 1 × { 0 ,. . .,𝑛 } [ 𝑏 ( 𝑣 𝑛 ) 𝑣 𝑛 − 𝜌 𝑏 ( 𝑛 ) 𝑛 + ( 𝑛 − 𝑣 ) 𝑞 ( 𝑣 1 , . . . , 𝑣 𝑛 − 1 ; 𝜌 opt ) ] ≥ 0 , which holds if and only if 𝑏 ( 𝑣 𝑛 ) 𝑣 𝑛 − 𝜌 𝑏 ( 𝑛 ) 𝑛 + ( 𝑛 − 𝑣 𝑛 ) 𝑞 ( 𝑣 1 , . . . , 𝑣 𝑛 − 1 ; 𝜌 opt ) ≥ 0 for all possible tuples ( 𝑣 1 , . . . , 𝑣 𝑛 ) . Rearranging these constraints and solving for 𝜌 opt will result in a set of inequalities on 𝜌 opt (one inequality for each tuple). Our choice of 𝜌 opt in (15) is pr ecisely obtained by turning the most binding of these into an equality . This ensures that ( 𝜌 opt , 𝑓 opt ) are feasible also when 𝑢 = 𝑛 . W e now prove, by contradiction, that ( 𝜌 opt , 𝑓 opt ) is optimal. T o do so, we assume that there e xists ˆ 𝑓 , that is feasible and achieves a higher value ˆ 𝜌 > 𝜌 opt . Since ( ˆ 𝑓 , ˆ 𝜌 ) is feasible, using the constraint with 𝑢 = 0 , 𝑣 = 1 , we have ˆ 𝑓 ( 1 ) ≤ 𝑏 ( 1 ) = 𝑓 opt ( 1 ) . Observing that min { 𝑣 , 𝑛 − 𝑢 } > 0 due to 𝑣 > 0 , 𝑢 < 𝑛 and leveraging the constraints with 𝑢 = 1 as well as the corresponding sp ecic choice of 𝑣 = 𝑣 ∗ 1 (for given 𝑢 ∈ { 1 , . . . , 𝑛 − 1 } , we let 𝑣 ∗ 𝑢 be an index 𝑣 ∈ { 1 , . . . , 𝑛 } where the minimum in (14) is attained), it must be that ˆ 𝑓 ( 2 ) satises ˆ 𝑓 ( 2 ) ≤ 𝑏 ( 𝑣 ∗ 1 ) 𝑣 ∗ 1 − ˆ 𝜌𝑏 ( 1 ) + min { 1 , 𝑛 − 𝑣 ∗ 1 } ˆ 𝑓 ( 1 ) min { 𝑣 ∗ 1 , 𝑛 − 1 } < 𝑏 ( 𝑣 ∗ 1 ) 𝑣 ∗ 1 − 𝜌 opt 𝑏 ( 1 ) + min { 1 , 𝑛 − 𝑣 ∗ 1 } 𝑓 opt ( 1 ) min { 𝑣 ∗ 1 , 𝑛 − 1 } = 𝑓 opt ( 2 ) . Here the rst inequality follows by feasibility of ˆ 𝑓 , the se cond is due to ˆ 𝜌 > 𝜌 opt and ˆ 𝑓 ( 1 ) ≤ 𝑓 opt ( 1 ) . The nal equality follows due to the denition of 𝑓 opt ( 2 ) . Hence we have shown that ˆ 𝑓 ( 2 ) < 𝑓 opt ( 2 ) . Noting that the only information we used to move from level 𝑢 to 𝑢 + 1 is that ˆ 𝜌 > 𝜌 opt and ˆ 𝑓 ( 𝑢 ) ≤ 𝑓 opt ( 𝑢 ) , one can apply this argument recursively up until 𝑢 = 𝑛 − 1 , and thus obtain ˆ 𝑓 ( 𝑛 ) < 𝑓 opt ( 𝑛 ) . Nevertheless, le veraging the constraints with 𝑢 = 𝑛 and 𝑣 = 𝑣 ∗ 𝑛 gives 𝑏 ( 𝑣 ∗ 𝑛 ) 𝑣 ∗ 𝑛 − ˆ 𝜌𝑏 ( 𝑛 ) 𝑛 + ( 𝑛 − 𝑣 ∗ 𝑛 ) ˆ 𝑓 ( 𝑛 ) ≥ 0 , or equivalently ˆ 𝜌 ≤ ( 𝑏 ( 𝑣 ∗ 𝑛 ) 𝑣 ∗ 𝑛 + ( 𝑛 − 𝑣 ∗ 𝑛 ) ˆ 𝑓 ( 𝑛 ) ) / ( 𝑏 ( 𝑛 ) 𝑛 ) . Thus ˆ 𝜌 ≤ 𝑏 ( 𝑣 ∗ 𝑛 ) 𝑣 ∗ 𝑛 + ( 𝑛 − 𝑣 ∗ 𝑛 ) ˆ 𝑓 ( 𝑛 ) 𝑏 ( 𝑛 ) 𝑛 ≤ 𝑏 ( 𝑣 ∗ 𝑛 ) 𝑣 ∗ 𝑛 + ( 𝑛 − 𝑣 ∗ 𝑛 ) 𝑓 opt ( 𝑛 ) 𝑏 ( 𝑛 ) 𝑛 = 𝜌 opt , where we use d the fact that 𝑛 − 𝑣 ∗ 𝑛 ≥ 0 and ˆ 𝑓 ( 𝑛 ) < 𝑓 opt ( 𝑛 ) . Note that ˆ 𝜌 ≤ 𝜌 opt contradicts the assumption ˆ 𝜌 > 𝜌 opt , thus concluding this part of the proof. Proof that 𝑓 opt is non-decreasing. By contradiction, let us assume 𝑓 opt is decreasing at some index. Lemma 2 in the Appendix shows that, if this is the case, then 𝑓 opt continues to decrease, so that 𝑓 opt ( 𝑛 ) ≤ 𝑓 opt ( 𝑛 − 1 ) . Note that it must be 𝑓 opt ( 𝑛 ) > 0 , as if it were 𝑓 opt ( 𝑛 ) ≤ 0 , then by denition of 𝜌 opt we would have 𝜌 opt = min 𝑣 ∈ { 0 ,. . .,𝑛 } 𝑏 ( 𝑣 ) 𝑣 + ( 𝑛 − 𝑣 ) 𝑓 opt ( 𝑛 ) 𝑛𝑏 ( 𝑛 ) = 0 + 𝑓 opt ( 𝑛 ) 𝑏 ( 𝑛 ) ≤ 0 , since the minimum is attained at the lowest feasible 𝑣 due to 𝑏 ( 𝑣 ) 𝑣 and − 𝑣 𝑓 opt ( 𝑛 ) non-decreasing and increasing, respectively . This is a contradiction as the price of anar chy is bounded already in the A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 13 un-tolled setup. 8 It must therefore be that the price of anarchy is bounded also when optimal tolls are used. Additionally , as we have removed a number of constraints from the linear pr ogram, the corresponding price of anarchy will be e ven low er . Therefore it must b e that 1 / 𝜌 opt is non-negative and bounded, so that 𝜌 opt > 0 contradicting the last e quation. Thus, in the following we proceed with the case of 𝑓 opt ( 𝑛 ) > 0 . It must b e that 𝜌 opt = min 𝑣 ∈ { 0 ,. . .,𝑛 } 𝑏 ( 𝑣 ) 𝑣 + ( 𝑛 − 𝑣 ) 𝑓 opt ( 𝑛 ) 𝑛𝑏 ( 𝑛 ) ≤ min 𝑣 ∈ { 1 ,. . .,𝑛 } 𝑏 ( 𝑣 ) 𝑣 + ( 𝑛 − 𝑣 ) 𝑓 opt ( 𝑛 ) 𝑛𝑏 ( 𝑛 ) ≤ min 𝑣 ∈ { 1 ,. . .,𝑛 } 𝑏 ( 𝑣 ) 𝑣 + ( 𝑛 − 𝑣 ) 𝑓 opt ( 𝑛 − 1 ) 𝑛𝑏 ( 𝑛 ) , where the rst inequality holds as we are restricting the domain of minimization, the second because 𝑓 opt ( 𝑛 ) ≤ 𝑓 opt ( 𝑛 − 1 ) and 𝑛 − 𝑣 ≥ 0 . Let us observe that 𝑓 opt ( 𝑛 ) is dened as 𝑓 ( 𝑛 ) = min 𝑣 ∈ { 1 ,. . .,𝑛 } [ 𝑏 ( 𝑣 ) 𝑣 + ( 𝑛 − 𝑣 ) 𝑓 opt ( 𝑛 − 1 ) ] − 𝜌 opt ( 𝑛 − 1 ) 𝑏 ( 𝑛 − 1 ) . Substituting min 𝑣 ∈ { 1 ,. . .,𝑛 } [ 𝑏 ( 𝑣 ) 𝑣 + ( 𝑛 − 𝑣 ) 𝑓 opt ( 𝑛 − 1 ) ] = 𝑓 opt ( 𝑛 ) + 𝜌 opt ( 𝑛 − 1 ) 𝑏 ( 𝑛 − 1 ) in the former bound on 𝜌 opt , we get 𝜌 opt ≤ 𝑓 opt ( 𝑛 ) + 𝜌 opt ( 𝑛 − 1 ) 𝑏 ( 𝑛 − 1 ) 𝑛𝑏 ( 𝑛 ) = ⇒ 𝜌 opt ≤ 𝑓 opt ( 𝑛 ) 𝑛𝑏 ( 𝑛 ) − ( 𝑛 − 1 ) 𝑏 ( 𝑛 − 1 ) . W e want to prove that this gives rise to a contradiction. T o do so, we will show that 𝑓 opt ( 𝑛 ) 𝑛𝑏 ( 𝑛 ) − ( 𝑛 − 1 ) 𝑏 ( 𝑛 − 1 ) < min 𝑣 ∈ { 0 ,. . .,𝑛 } 𝑏 ( 𝑣 ) 𝑣 + ( 𝑛 − 𝑣 ) 𝑓 opt ( 𝑛 ) 𝑛𝑏 ( 𝑛 ) . (17) As a matter of fact, if the latter inequality holds true, the proof is immediately concluded as 𝜌 opt ≤ 𝑓 opt ( 𝑛 ) 𝑛𝑏 ( 𝑛 ) − ( 𝑛 − 1 ) 𝑏 ( 𝑛 − 1 ) < min 𝑣 ∈ { 0 ,. . .,𝑛 } 𝑏 ( 𝑣 ) 𝑣 + ( 𝑛 − 𝑣 ) 𝑓 opt ( 𝑛 ) 𝑛𝑏 ( 𝑛 ) = 𝜌 opt = ⇒ 𝜌 opt < 𝜌 opt , where the rst inequality has b een shown above, the second is what remains to b e proved, and the latter equality is by denition. Therefore, we are left to show (17) , which holds if we can show that ∀ 𝑣 ∈ { 0 , . . . , 𝑛 } it is 𝑔 ( 𝑣 )  ℎ ( 𝑣 ) + ( 𝑛 − 𝑣 ) 𝑓 opt ( 𝑛 ) ℎ ( 𝑛 ) − 𝑓 opt ( 𝑛 ) ℎ ( 𝑛 ) − ℎ ( 𝑛 − 1 ) > 0 , where ℎ : R → R ≥ 0 is a function such that ℎ ( 𝑣 ) = 𝑏 ( 𝑣 ) 𝑣 for 𝑣 ∈ { 0 , . . . , 𝑛 } . W e choose ℎ to be continuously dierentiable, strictly incr easing, and strictly conve x; one such function always exists. 9 W e rst consider the p oint 𝑣 = 0 . Obser ve that 𝑔 ( 0 ) > 0 when 𝑛 > 1 as 𝑔 ( 0 ) = 𝑓 opt ( 𝑛 ) 𝑏 ( 𝑛 ) − 𝑓 opt ( 𝑛 ) 𝑛𝑏 ( 𝑛 ) − ( 𝑛 − 1 ) 𝑏 ( 𝑛 − 1 ) > 0 ⇐ ⇒ 𝑓 opt ( 𝑛 ) [ ( 𝑛 − 1 ) 𝑏 ( 𝑛 ) − ( 𝑛 − 1 ) 𝑏 ( 𝑛 − 1 ) ] > 0 , which holds as 𝑓 opt ( 𝑛 ) > 0 , 𝑛 > 1 , and 𝑏 ( 𝑛 ) > 𝑏 ( 𝑛 − 1 ) strictly . If 𝑔 ′ ( 𝑣 ) ≥ 0 at 𝑣 = 0 , the proof is complete as 𝑔 is convex and due to 𝑔 ′ ( 0 ) ≥ 0 it is non-decreasing for any 𝑣 ≥ 0 so that the constraint will be satised for all 𝑣 ≥ 0 . If this is not the case, then 𝑔 ′ ( 0 ) < 0 , which we consider now . Note that, at the point 𝑣 = 𝑛 − 1 , the derivative 𝑔 ′ ( 𝑛 − 1 ) = [ ℎ ′ ( 𝑛 − 1 ) − 𝑓 opt ( 𝑛 ) ] / ℎ ( 𝑛 ) satises ℎ ( 𝑛 ) 𝑔 ′ ( 𝑛 − 1 ) = ℎ ′ ( 𝑛 − 1 ) − 𝑓 opt ( 𝑛 ) ≥ ℎ ′ ( 𝑛 − 1 ) − ( ℎ ( 𝑛 − 1 ) − ℎ ( 𝑛 − 2 ) ) ≥ 0 8 T o see this, consider the linear program used to determine the price of anarchy in the un-tolled case, i.e., (12) where we set 𝑓 𝑗 ( 𝑥 ) = 𝑏 𝑗 ( 𝑥 ) . When 𝜈 = 1 , it is always possible to nd 𝜌 > 0 , so that the corresponding price of anarchy is bounded. 9 Observe that the function 𝑏 ( 𝑣 ) 𝑣 is positive, strictly increasing, and strictly convex in the discrete sense in its domain due to the assumptions. A CM Transactions on Economics and Computation (to appear) 14 Paccagnan, et al. where the last inequality is due to convexity , while the rst inequality holds as 𝑓 opt ( 𝑛 ) ≤ ℎ ( 𝑛 − 1 ) − ℎ ( 𝑛 − 2 ) thanks to Lemma 2 and 𝑛 ≥ 2 . 10 Therefore since 𝑔 ′ ( 0 ) < 0 , 𝑔 ′ ( 𝑛 − 1 ) ≥ 0 and 𝑔 convex, there must exist an unconstrained minimizer 𝑣 ∗ ∈ ( 0 , 𝑛 − 1 ] . W e will guarantee that 𝑔 ( 𝑣 ∗ ) > 0 so that for any (real and thus integer) 𝑣 ∈ [ 0 , 𝑛 ] it is 𝑔 ( 𝑣 ) > 0 . The unconstraine d minimizer satises 𝑓 opt ( 𝑛 ) = ℎ ′ ( 𝑣 ∗ ) , which we substitute, and are thus left with proving the nal inequality ℎ ( 𝑣 ∗ ) + ( 𝑛 − 𝑣 ) ℎ ′ ( 𝑣 ∗ ) ℎ ( 𝑛 ) − ℎ ′ ( 𝑣 ∗ ) ℎ ( 𝑛 ) − ℎ ( 𝑛 − 1 ) > 0 , which is equivalent to [ ℎ ( 𝑛 ) − ℎ ( 𝑛 − 1 ) ] ℎ ( 𝑣 ∗ ) > ℎ ′ ( 𝑣 ∗ ) [ ( 𝑛 − 𝑣 ∗ ) ( ℎ ( 𝑛 − 1 ) − ℎ ( 𝑛 ) ) + ℎ ( 𝑛 ) ] , where we recall 0 < 𝑣 ∗ ≤ 𝑛 − 1 . As the left hand side is p ositive due to ℎ increasing and 𝑣 ∗ > 0 , the inequality holds trivially if the right hand side is less or equal to zero, i.e., if ℎ ( 𝑛 ) ≤ ( 𝑛 − 𝑣 ∗ ) ( ℎ ( 𝑛 ) − ℎ ( 𝑛 − 1 ) ) . In the other case, when ( 𝑛 − 𝑣 ∗ ) ( ℎ ( 𝑛 − 1 ) − ℎ ( 𝑛 ) ) + ℎ ( 𝑛 ) > 0 , we lev erage the fact that ℎ ′ ( 𝑣 ∗ ) < ( ℎ ( 𝑛 ) − ℎ ( 𝑣 ∗ ) ) / ( 𝑛 − 𝑣 ∗ ) by strict convexity of ℎ ( 𝑥 ) in 𝑥 = 𝑣 ∗ > 0 , so that ℎ ′ ( 𝑣 ∗ ) [ ( 𝑛 − 𝑣 ∗ ) ( ℎ ( 𝑛 − 1 ) − ℎ ( 𝑛 ) ) + ℎ ( 𝑛 ) ] < ℎ ( 𝑛 ) − ℎ ( 𝑣 ∗ ) 𝑛 − 𝑣 ∗ [ ( 𝑛 − 𝑣 ∗ ) ( ℎ ( 𝑛 − 1 ) − ℎ ( 𝑛 ) ) + ℎ ( 𝑛 ) ] = ℎ ( 𝑛 ) 𝑛 − 𝑣 ∗ [ ℎ ( 𝑛 ) − ℎ ( 𝑣 ∗ ) ] + [ ℎ ( 𝑛 ) − ℎ ( 𝑛 − 1 ) ] [ ℎ ( 𝑣 ∗ ) − ℎ ( 𝑛 ) ] ≤ [ ℎ ( 𝑛 ) − ℎ ( 𝑛 − 1 ) ] ℎ ( 𝑣 ∗ ) , where the last inequality follows since [ ℎ ( 𝑛 ) − ℎ ( 𝑛 − 1 ) ] [ ℎ ( 𝑣 ∗ ) − ℎ ( 𝑛 ) ] ≤ 0 and from ℎ ( 𝑛 ) − ℎ ( 𝑣 ∗ ) 𝑛 − 𝑣 ∗ ≤ ℎ ( 𝑛 ) − ℎ ( 𝑛 − 1 ) , which holds for 0 < 𝑣 ∗ ≤ 𝑛 − 1 by convexity . This concludes this part of the proof. Proof that ( 𝜌 opt , 𝑓 opt ) is feasible also for (5) and final cosmetics. Recall fr om the rst part of the proof that the constraints in (5) can b e e quivalently written as 𝑏 ( 𝑣 ) 𝑣 − 𝜌𝑏 ( 𝑢 ) 𝑢 + 𝑓 ( 𝑢 ) ( 𝑢 − 𝑥 ) − 𝑓 ( 𝑢 + 1 ) ( 𝑣 − 𝑥 ) ≥ 0 . Since 𝑓 opt is non-decreasing, following the argument in [ 28 , Cor . 1] one veries that the tightest constraints are obtained when 𝑥 = min { 0 , 𝑢 + 𝑣 − 𝑛 } . These constraints are already included in our simplied program of (16) , with the exception of those with 𝑣 = 0 and 𝑢 ∈ { 0 , . . . , 𝑛 − 1 } which we have remov ed. T o show that also these hold, we note that the constraint with 𝑣 = 0 reads as 𝑢 𝑓 opt ( 𝑢 ) ≥ 𝜌 opt 𝑢𝑏 ( 𝑢 ) , and is trivially satised for 𝑢 = 0 . W e now show that also the constraints with 𝑣 = 0 , 𝑢 > 0 hold. T o do so, w e consider the constraint corresponding to 𝑣 = 1 𝑏 ( 1 ) 1 − 𝜌 𝑏 ( 𝑢 ) 𝑢 + 𝑢 𝑓 opt ( 𝑢 ) − 𝑓 opt ( 𝑢 + 1 ) ≥ 0 . Since 𝑓 opt is non-decreasing as shown in previous point then 𝑓 opt ( 𝑢 + 1 ) ≥ 𝑓 opt ( 1 ) = 𝑏 ( 1 ) . Hence, 0 ≤ 𝑏 ( 1 ) 1 − 𝜌𝑏 ( 𝑢 ) 𝑢 + 𝑢 𝑓 opt ( 𝑢 ) − 𝑓 opt ( 𝑢 + 1 ) ≤ 𝑏 ( 1 ) 1 − 𝜌 opt 𝑏 ( 𝑢 ) 𝑢 + 𝑢 𝑓 opt ( 𝑢 ) − 𝑏 ( 1 ) . Thus, from the left and right hand sides we obtain the desired result 𝑢 𝑓 opt ( 𝑢 ) ≥ 𝜌 opt 𝑢𝑏 ( 𝑢 ) . W e conclude with some cosmetics: the simplied linear program in (16) is almost identical to that in (13) , except for the constraints with 𝑣 = 0 and 𝑢 ∈ { 0 , . . . , 𝑛 − 1 } , which we hav e remov ed in (16) . Nevertheless, we have just veried that an optimal solution does satisfy these constraints too. Hence, we simply add them back to obtain (13). □ 10 In fact, either 𝑛 is the rst index starting from which 𝑓 opt decreases (i.e. 𝑓 opt ( 𝑛 ) < 𝑓 opt ( 𝑛 − 1 ) ) in which case 𝑓 opt ( 𝑛 ) ≤ 𝜌 opt [ 𝑏 ( 𝑛 − 1 ) ( 𝑛 − 1 ) − 𝑏 ( 𝑛 − 2 ) ( 𝑛 − 2 ) ] ≤ 𝑏 ( 𝑛 − 1 ) ( 𝑛 − 1 ) − 𝑏 ( 𝑛 − 2 ) ( 𝑛 − 2 ) due to 𝜌 opt ≤ 1 , or the function starts decreasing at a 𝑢 + 1 < 𝑛 in which case Lemma 2 also shows that 𝑓 opt ( 𝑛 ) ≤ · · · ≤ 𝑓 opt ( 𝑢 + 1 ) ≤ 𝜌 opt [ 𝑏 ( 𝑢 ) 𝑢 − 𝑏 ( 𝑢 − 1 ) ( 𝑢 − 1 ) ] ≤ 𝑏 ( 𝑢 ) 𝑢 − 𝑏 ( 𝑢 − 1 ) ( 𝑢 − 1 ) ≤ 𝑏 ( 𝑛 − 1 ) ( 𝑛 − 1 ) − 𝑏 ( 𝑛 − 2 ) ( 𝑛 − 2 ) , where the inequalities hold due to 𝜌 opt ≤ 1 and the convexity of 𝑏 ( 𝑢 ) 𝑢 . A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 15 4 OPTIMAL TOLLING MECHANISMS FOR ARBITRARY N UMBER OF AGENTS While the linear programming formulations introduced in (5) and (13) provide an optimal tolling mechanism and the corresponding optimal price of anarchy when the number of agents is upper- bounded by 𝑛 (nite), in this section we show how to design optimal tolling mechanisms for polynomial congestion games that apply to any 𝑛 (possibly innite), by solving a linear program of xed size. The resulting values of the price of anarchy ar e those already displayed in T able 1. For ease of exposition, we consider congestion games where the set of resource costs is produced by non-negative combinations of a single monomial 𝑥 𝑑 , 𝑑 ≥ 1 at a time. This is without loss of generality , as one can derive optimal tolling mechanisms for p olynomial congestion games with maximum degree 𝑑 , i.e., generated by { 1 , 𝑥 , . . . , 𝑥 𝑑 } , simply repeating the ensuing reasoning separately for all polynomials of degree higher than one and low er-equal to 𝑑 . No toll need to be applied to polynomials of order zero as the corresponding price of anarchy is one. The idea w e leverage is as follo ws: rst, we solve a linear program of xe d size ¯ 𝑛 , from which we obtain a set of tolls that are then e xtended analytically to any number of agents. This produces a mechanism for which we are able to quantify the corresponding price of anarchy over games with possibly innitely many agents. Such price of anarchy value is an upper bound on the true optimal price of anarchy over games with possibly innitely many agents, as the mechanism we design is not necessarily optimal. At the same time, we solve the linear program in (13) , and thus obtain the optimal price of anarchy for games with a maximum of ¯ 𝑛 agents. The latter is a low er bound for the optimal price of anarchy over games with possibly innitely many agents. Letting ¯ 𝑛 grow , the upper bound matches the lower bound already for small values of ¯ 𝑛 , as showcased in T able 2. While the construction of the low er b ound follows r eadily by solving the linear program in (13) with ¯ 𝑛 agents, in the following we describe the procedure to derive the upp er bound. More specically , we clarify i) what program of dimension ¯ 𝑛 we solve; ii) how we extend its solution from ¯ 𝑛 to innity; and iii) how we compute the resulting price of anarchy over games with possibly innitely many agents. In the remainder of this section, we will always select ¯ 𝑛 nite and even. As for the rst point, we consider the following linear pr ogram max 𝑓 ∈ R ¯ 𝑛 , 𝜌 ∈ R 𝜌 s . t . 𝑣 𝑑 + 1 − 𝜌𝑢 𝑑 + 1 + 𝑓 ( 𝑢 ) 𝑢 − 𝑓 ( 𝑢 + 1 ) 𝑣 ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , ¯ 𝑛 } 𝑢 + 𝑣 ≤ ¯ 𝑛, 𝑣 𝑑 + 1 − 𝜌𝑢 𝑑 + 1 + 𝑓 ( 𝑢 ) ( ¯ 𝑛 − 𝑣 ) − 𝑓 ( 𝑢 + 1 ) ( ¯ 𝑛 − 𝑢 ) ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , ¯ 𝑛 } 𝑢 + 𝑣 > ¯ 𝑛, 𝑓 ( 𝑢 ) ≤ 𝑢 𝑑 ∀ 𝑢 ∈ { 1 , . . . , ¯ 𝑛 } 𝑓 ( 𝑢 ) ≥ 𝑓 ( 𝑢 − 1 ) ∀ 𝑢 ∈ { 2 , . . . , ¯ 𝑛 } (18) with the usual convention that 𝑓 ( 0 ) = 𝑓 ( ¯ 𝑛 + 1 ) = 0 . Note that the previous program is identical to that in (13) with 𝑏 ( 𝑥 ) = 𝑥 𝑑 , except that we have included two additional sets of constraints. W e let ( 𝑓 opt , 𝜌 opt ) b e a solution of this program and utilize it to dene 𝑓 ∞ : N → R as follows 𝑓 ∞ ( 𝑥 ) = ( 𝑓 opt ( 𝑥 ) for 𝑥 ≤ ¯ 𝑛 / 2 𝛽 · 𝑥 𝑑 for 𝑥 > ¯ 𝑛 / 2 , where 𝛽 = 𝑓 opt ( ¯ 𝑛 / 2 ) ( ¯ 𝑛 / 2 ) 𝑑 . (19) Informally , the idea is to extend 𝜏 ∞ ( 𝑥 ) = 𝑓 ∞ ( 𝑥 ) − 𝑏 ( 𝑥 ) from ¯ 𝑛 / 2 to innity with a polynomial of the same order of the original 𝑥 𝑑 . Note that 𝛽 ≥ 0 is chosen so that the two e xpressions dening 𝑓 ∞ match for 𝑥 = ¯ 𝑛 / 2 . 11 While the expression of 𝑓 ∞ and all forthcoming quantities depends on the 11 Observe that 𝑓 ∞ ( 1 ) ≥ 0 since having 𝑓 ∞ ( 1 ) < 0 would always result in a lo wer performance, as shown in [ 28 ]. Therefor e 𝑓 ∞ ( ¯ 𝑛 / 2 ) ≥ 0 as it is feasible for (18), which includes the constraint 𝑓 ( 𝑥 + 1 ) ≥ 𝑓 ( 𝑥 ) . Hence, 𝛽 ≥ 0 . A CM Transactions on Economics and Computation (to appear) 16 Paccagnan, et al. ¯ 𝑛 𝑑 = 1 𝑑 = 2 𝑑 = 3 LB UB LB UB LB UB 10 2 . 011 825 2 . 038 237 5 . 097 187 5 . 316 382 15 . 530 175 17 . 138 429 20 2 . 012 067 2 . 019 844 5 . 100 974 5 . 147 543 15 . 550 847 15 . 751 993 30 2 . 012 067 2 . 014 335 5 . 100 974 5 . 119 149 15 . 550 852 15 . 684 195 40 2 . 012 067 2 . 012 067 5 . 100 974 5 . 100 974 15 . 550 852 15 . 550 859 T able 2. Lower and upper b ounds (LB and UB) on the values of the optimal price of anarchy for polynomial congestion games with arbitrarily large number of agents and 𝑑 = 1 , 2 , 3 . The LB vs UB shows how the tolls derived from 𝑓 ∞ defined in (19) are approximately optimal up the fih decimal digit when we select ¯ 𝑛 = 40 . choice of ¯ 𝑛 , we do not make this explicit to ease the notation. Lemma 3 in the Appendix ensures that the price of anarchy of 𝑓 ∞ is identical for pure Nash and coarse correlated equilibria, and is upper bounded over games with possibly innitely many agents by 1 / 𝜌 ∞ , where 𝜌 ∞ is given by 𝜌 ∞ = min ( 𝜌 opt , 𝛽 − 𝑑  1 + 2 ¯ 𝑛  𝑑 + 1  𝛽 𝑑 + 1  1 + 1 𝑑 ) . As claried above this represents an upper b ound on the optimal price of anarchy . The upper and lower bounds displayed in T able 2 have been computed according to the procedure just described, and demonstrate that, for a relatively small ¯ 𝑛 = 40 , the mechanism obtained from 𝑓 ∞ is approximately optimal up to the fth decimal digit for polynomial congestion games with 𝑑 = 1 , 2 , 3 . Finally , we obser ve that the tolling mechanism 𝑇 ∞ ( 𝛼 ℓ ) = 𝛼𝑇 ∞ ( ℓ ) = 𝛼𝜏 ∞ , where 𝜏 ∞ ( 𝑥 ) = 𝑓 ∞ ( 𝑥 ) − 𝑏 ( 𝑥 ) might not satisfy 𝜏 ∞ ( 𝑥 ) ≥ 0 for all 𝑥 ∈ N (i.e., they might be monetar y incentives and not tolls). Nevertheless, multiplying 𝑓 ∞ with a factor 𝛾 > 0 produces tolls 𝛾 𝑓 ∞ ( 𝑥 ) − 𝑏 ( 𝑥 ) with identical price of anar chy (the proof of Lemma 3 will hold with 𝜈 = 1 / 𝛾 in place of 𝜈 = 1 ). Ther efore, one simply needs to consider tolls of the form 𝛾 𝑓 ∞ ( 𝑥 ) − 𝑏 ( 𝑥 ) , where 𝛾 is chosen suciently large to ensure that 𝛾 𝑓 ∞ ( 𝑥 ) − 𝑏 ( 𝑥 ) ≥ 0 for all 𝑥 ∈ N ; one such 𝛾 always exists. When multiple basis are present, we select 𝛾 as a common scaling factor to ensure non-negativity of all tolls basis. 5 CONGESTION-INDEPENDENT TOLLING MECHANISMS In this section we pr ovide a general methodology to compute optimal congestion-indep endent local tolling mechanisms for games generated by { 𝑏 1 , . . . , 𝑏 𝑚 } . W e also specialize the result to polynomial congestion games providing explicit expressions for the tolls and the corresponding price of anarchy . In this section we consider basis functions that are convex in the discr ete sense (see Footnote 5). Theorem 3. A lo cal congestion-indep endent mechanism minimizing the price of anarchy over congestion games with 𝑛 agents, and resource costs ℓ ( 𝑥 ) = Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑏 𝑗 ( 𝑥 ) , 𝛼 𝑗 ≥ 0 , with convex p ositive non-decreasing basis functions { 𝑏 1 , . . . , 𝑏 𝑚 } is given by 𝑇 opt ( ℓ ) = 𝑚  𝑗 = 1 𝛼 𝑗 · 𝜏 opt 𝑗 , where 𝜏 opt 𝑗 ∈ R , 𝜏 opt 𝑗 =  1 𝜈 opt − 1  𝑏 𝑗 ( 1 ) (20) and 𝜌 opt ∈ R , 𝜈 opt ∈ R ≥ 0 solve the linear program max 𝜌 ∈ R ,𝜈 ∈ [ 0 , 1 ] 𝜌 s . t . 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑏 𝑗 ( 𝑢 ) 𝑢 − 𝑏 𝑗 ( 𝑢 + 1 ) 𝑣 ] + 𝑏 𝑗 ( 1 ) ( 1 − 𝜈 ) ( 𝑢 − 𝑣 ) ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 ≤ 𝑛 𝑢 ≥ 𝑣 , ∀ 𝑗 ∈ { 1 , . . . , 𝑚 } , 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑏 𝑗 ( 𝑢 ) ( 𝑛 − 𝑣 ) − 𝑏 𝑗 ( 𝑢 + 1 ) ( 𝑛 − 𝑢 ) ] + 𝑏 𝑗 ( 1 ) ( 1 − 𝜈 ) ( 𝑢 − 𝑣 ) ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 > 𝑛 𝑢 ≥ 𝑣 , ∀ 𝑗 ∈ { 1 , . . . , 𝑚 } . (21) A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 17 where we dene 𝑏 𝑗 ( 0 ) = 𝑏 𝑗 ( 𝑛 + 1 ) = 0 . Correspondingly , Po A ( 𝑇 opt ) = 1 / 𝜌 opt , and the optimal tolls are non-negative. 12 The result is tight for pure Nash equilibria and extends to coarse correlated equilibria. The optimal price of anarchy arising from the solution of (21) for polynomials of order at most 𝑑 = 1 , 2 , . . . , 6 and 𝑛 = 100 ar e shown in the fth column of T able 1. Before pr oceeding with pr oving the theorem, we specialize its result to polynomial congestion games with 𝑑 ≥ 2 and arbitrarily large 𝑛 . This allows us to deriv e explicit expressions matching the values featured in T able 1 and holding for arbitrarily large 𝑛 . W e do not study the case of 𝑑 = 1 as this has been analyzed in [ 8 ], resulting in an optimal price of anarchy of 1 + 2 / √ 3 ≈ 2 . 15 , which we also recover through the solution of the linear program above. Corollary 1. Consider polynomial congestion games of maximum degree 𝑑 = 2 and arbitrarily large number of agents, i.e., congestion game where the cost on resource 𝑒 is ℓ 𝑒 ( 𝑥 ) = 𝛼 𝑒 𝑥 2 + 𝛽 𝑒 𝑥 + 𝛾 𝑒 , with non-negative 𝛼 𝑒 , 𝛽 𝑒 , 𝛾 𝑒 . A n optimal congestion-independent me chanism satises 𝑇 opt ( ℓ 𝑒 ) = 3 𝛼 𝑒 , Po A ( 𝑇 opt ) = 16 3 ≈ 5 . 33 . (22) The result is tight for pure Nash equilibria and extends to coarse correlated equilibria. Following a similar line of reasoning to that of Cor ollary 1 (see the next page for its proof ), it is possible to derive an expression for the optimal price of anarchy with constant tolls also in the case of 3 ≤ 𝑑 ≤ 6 , i.e., Po A ( 𝑇 opt ) = ¯ 𝑢 ( ¯ 𝑢 + 1 ) 𝑑 + 1 − ¯ 𝑢 𝑑 + 1 [ ¯ 𝑢 + ( ¯ 𝑢 + 2 ) 𝑑 ] + ( ¯ 𝑢 + 1 ) 2 𝑑 + 1 − ( ¯ 𝑢 + 1 ) 𝑑 + 1 ¯ 𝑢 ( ¯ 𝑢 + 1 ) ( ( ¯ 𝑢 + 1 ) 𝑑 − ¯ 𝑢 𝑑 ) + ( ¯ 𝑢 + 1 ) 𝑑 + 1 − ¯ 𝑢 ( ¯ 𝑢 + 2 ) 𝑑 − 1 , (23) where ¯ 𝑢 is the oor of the unique real positive solution to 𝑢 𝑑 + 1 + 1 = ( 𝑢 + 1 ) 𝑑 + 𝑢 . For example, 𝑑 = 3 = ⇒ ¯ 𝑢 = 2 , Po A ( 𝑇 opt ) = 2 · 3 4 − 2 4 · ( 2 + 4 3 ) + 3 7 − 3 4 2 · 3 · ( 3 3 − 2 3 ) + 3 4 − 2 · 4 3 − 1 = 1212 66 ≈ 18 . 36 . Similarly , with 𝑑 = 4 , . . . , 6 , it is, respectively , Po A ( 𝑇 opt ) = 111588 / 1248 ≈ 89 . 41 , Po A ( 𝑇 opt ) = 1922184 / 4092 ≈ 469 . 74 , Po A ( 𝑇 opt ) = 32963196 / 9912 ≈ 3325 . 58 , matching the values in T able 1. While we do not formally prove the expression (23) in the interest of conciseness, the key idea consists in observing that the two most binding constraints appearing in the linear program of Theorem 3 ar e those obtained with ( 𝑢, 𝑣 ) = ( ¯ 𝑢 , 1 ) and ( 𝑢, 𝑣 ) = ( ¯ 𝑢 + 1 , 1 ) . T urning the corresponding inequalities into equalities and solving for 𝜌 and 𝜈 gives the result in (23). W e now turn focus on proving Theorem 3, followed by Corollary 1. Proof of Theorem 3. The fact that optimal lo cal congestion-independent tolls are linear in the sense that 𝑇 opt ( ℓ ) = 𝑇 opt ( Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑏 𝑗 ) = Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑇 opt ( 𝑏 𝑗 ) can be proven following the same steps of Theorem 1. Therefore it suces to determine the b est linear local congestion-independent toll. T owards this goal, we observe that the price of anarchy of a given linear lo cal constant toll 𝑇 ( Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑏 𝑗 ) = Í 𝑚 𝑗 = 1 𝛼 𝑗 𝜏 𝑗 , 𝜏 𝑗 ∈ R ≥ 0 can be determined as the solution of the following program, which applies, thanks to (12), since 𝑓 𝑗 ( 𝑥 ) = 𝑏 𝑗 ( 𝑥 ) + 𝜏 𝑗 is non-decreasing max 𝜌 ∈ R ,𝜈 ∈ R ≥ 0 𝜌 s . t . 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ ( 𝑏 𝑗 ( 𝑢 ) + 𝜏 𝑗 ) 𝑢 − ( 𝑏 𝑗 ( 𝑢 + 1 ) + 𝜏 𝑗 ) 𝑣 ] ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 ≤ 𝑛, ∀ 𝑗 ∈ { 1 , . . . , 𝑚 } , 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ ( 𝑏 𝑗 ( 𝑢 ) + 𝜏 𝑗 ) ( 𝑛 − 𝑣 ) − ( 𝑏 𝑗 ( 𝑢 + 1 ) + 𝜏 𝑗 ) ( 𝑛 − 𝑢 ) ] ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 > 𝑛, ∀ 𝑗 ∈ { 1 , . . . , 𝑚 } . (24) 12 The result also holds if convexity of 𝑏 𝑗 ( 𝑥 ) is weakened to convexity of 𝑏 𝑗 ( 𝑥 ) 𝑥 . One example is that of 𝑏 𝑗 ( 𝑥 ) = √ 𝑥 . A CM Transactions on Economics and Computation (to appear) 18 Paccagnan, et al. W e also recall that (24) tightly characterizes the price of anarchy for pure Nash equilibria, and the corresponding bound extends to coarse correlated equilibria. Determining the best non-negative toll amounts to letting ( 𝜏 1 , . . . , 𝜏 𝑚 ) ∈ R 𝑚 ≥ 0 be decision variables, over which we need to maximize. While this would result in a bi-linear program, we dene 𝜎 = ( 𝜎 1 , . . . , 𝜎 𝑚 ) ∈ R 𝑚 ≥ 0 with 𝜎 𝑗 = 𝜈 𝜏 𝑗 , and consider the following linear program max 𝜌 ∈ R , 𝜈 ∈ R ≥ 0 , 𝜎 ∈ R 𝑚 ≥ 0 𝜌 s . t . 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑏 𝑗 ( 𝑢 ) 𝑢 − 𝑏 𝑗 ( 𝑢 + 1 ) 𝑣 ] + 𝜎 𝑗 ( 𝑢 − 𝑣 ) ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 ≤ 𝑛, ∀ 𝑗 ∈ { 1 , . . . , 𝑚 } , 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑏 𝑗 ( 𝑢 ) ( 𝑛 − 𝑣 ) − 𝑏 𝑗 ( 𝑢 + 1 ) ( 𝑛 − 𝑢 ) ] + 𝜎 𝑗 ( 𝑢 − 𝑣 ) ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 > 𝑛, ∀ 𝑗 ∈ { 1 , . . . , 𝑚 } . (25) which is an exact reformulation of (24) , except for the fact that we are not including the (non-linear ) constraint r equiring 𝜎 𝑗 = 0 whene ver 𝜈 = 0 . W e will rectify this at the end by showing that 𝜈 opt > 0 . Lemma 5 in the Appendix leverages the fact that the basis functions are conv ex, positive, non- decreasing by assumption, so that only the constraints with 𝑢 ≥ 𝑣 , 𝑢 ≥ 1 and ( 𝑢, 𝑣 ) = ( 0 , 1 ) need to be accounted for in (25) . Due to the fact that 𝑢 − 𝑣 ≥ 0 for 𝑢 ≥ 1 , in order to maximize 𝜌 , we choose 𝜎 𝑗 as large as possible. Observing that the only upper bound on 𝜎 𝑗 arises from the choice of ( 𝑢, 𝑣 ) = ( 0 , 1 ) and reads as 𝜎 𝑗 ≤ ( 1 − 𝜈 ) 𝑏 𝑗 ( 1 ) , we set 𝜎 𝑗 = ( 1 − 𝜈 ) 𝑏 𝑗 ( 1 ) , and translate the constraint 𝜎 𝑗 ≥ 0 into 𝜈 ≤ 1 , thus obtaining (21) . T o conclude we are left to show that 𝜈 opt solving (21) is non- zero. T o do so, note that solving the program for xed 𝜈 = 0 results in 𝜌 = 𝑏 ( 1 ) / 𝑏 ( 𝑛 ) (the tightest constraint is ( 𝑢, 𝑣 ) = ( 𝑛, 0 ) ), while an arbitrarily small but positive 𝜈 would give a strictly higher 𝜌 . Once 𝜈 opt is determined, the optimal tolls can be derived from 𝜈 opt 𝜏 opt 𝑗 = ( 1 − 𝜈 opt ) 𝑏 𝑗 ( 1 ) , r ecalling that 𝜈 opt > 0 , thus yielding (20) . Non-negativity of the tolls follow from the fact that we impose 𝜈 ≤ 1 so that 𝜏 opt 𝑗 =  1 / 𝜈 opt − 1  𝑏 𝑗 ( 1 ) ≥ 0 □ W e now focus on Corollary 1, and prove the result following a dierent approach other than directly applying Theorem 3, with the hop e of providing the reader with an independent persp ective. Proof of Corollary 1. W e prove the claim in two steps. First, we show that the price of anarchy for any constant toll and pure Nash equilibria is lower-bounded by 16 / 3 . Second, we show that the price of anarchy of 𝑇 opt is upper-bounded by 16 / 3 for b oth Nash and coarse correlated equilibria. For the low er bound, it suces to consider resour ce costs of the form ℓ 𝑒 ( 𝑥 ) = 𝛼 𝑒 𝑥 2 , whereby any constant linear tolling mechanisms takes the form 𝑇 ( ℓ 𝑒 ) = 𝛼 𝑒 𝜏 , for some scalar 𝜏 ≥ 0 . For any 𝜏 ≥ 3 we consider the following problem instance: there are 8 agents each with two actions 𝑎 ne 𝑖 and 𝑎 opt 𝑖 . In action 𝑎 ne 𝑖 , user 𝑖 selects 6 of the available 8 resources, which are associated to 𝑐 1 𝑥 2 / 8 ; in 𝑎 opt 𝑖 user 𝑖 selects the remaining two r esources with costs 𝑐 1 𝑥 2 / 8 , as well as one r esource with cost 𝑐 2 𝑥 2 / 8 (we will x 𝑐 1 and 𝑐 2 at a later stage). Each player has a similar pair of actions, but each subsequent agent is oset by one from the prior user , as depicted in Fig. 2. In this game, the system and user costs can be computed as in the following SC ( 𝑎 ne ) = Í 𝑒 | 𝑎 | 𝑒 ℓ 𝑒 ( | 𝑎 | 𝑒 ) = 8 · 6 ℓ 𝑒 ( 6 ) = 8 · 6 𝑐 1 𝑏 ( 6 ) / 8 , 𝐶 𝑖 ( 𝑎 ne ) = Í 𝑒 ∈ 𝑎 𝑖 [ ℓ 𝑒 ( | 𝑎 | 𝑒 ) + 𝛼 𝑒 𝜏 ] = 6 · ( 𝑐 1 𝑏 ( 6 ) / 8 + 𝑐 1 𝜏 / 8 ) = 6 𝑐 1 ( 𝑏 ( 6 ) + 𝜏 ) / 8 , and similarly SC ( 𝑎 ne ) = 6 ( 𝑐 1 𝑏 ( 6 ) ) = 216 𝑐 1 , SC ( 𝑎 opt ) = 2 𝑐 1 𝑏 ( 2 ) + 𝑐 2 𝑏 ( 1 ) = 8 𝑐 1 + 𝑐 2 , 𝐶 𝑖 ( 𝑎 ne ) = 6 8 𝑐 1 ( 𝑏 ( 6 ) + 𝜏 ) = 1 8 ( 216 𝑐 1 + 6 𝑐 1 𝜏 ) , 𝐶 𝑖 ( 𝑎 opt 𝑖 , 𝑎 ne − 𝑖 ) = 2 8 𝑐 1 ( 49 + 𝜏 ) + 1 8 𝑐 2 ( 1 + 𝜏 ) . (26) W e normalize the costs in the game setting SC ( 𝑎 ne ) = 1 , which results in 𝑐 1 = 1 / 216 from (26) . T o ensure that the joint action 𝑎 ne is a Nash e quilibrium (at least weakly), we impose that A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 19 𝐶 𝑖 ( 𝑎 ne ) = 𝐶 𝑖 ( 𝑎 opt 𝑖 , 𝑎 ne − 𝑖 ) for any play er 𝑖 . This condition is satised when 𝑐 2 = ( 2 𝜏 + 59 ) / ( 108 ( 1 + 𝜏 ) ) . Hence, the price of anarchy in this game is lower-bounded by SC ( 𝑎 ne ) / SC ( 𝑎 opt ) = 1 / SC ( 𝑎 opt ) = 1 / ( 8 𝑐 1 + 𝑐 2 ) . This expr ession is no smaller than 16 / 3 for any choice of 𝜏 ≥ 3 (in particular , equal when we set 𝜏 = 3 ), where we have utilized the values of 𝑐 2 from above and 𝑐 1 = 1 / 216 . For 𝜏 < 3 we construct a game with similar features. There are 3 users each with actions 𝑎 ne 𝑖 and 𝑎 opt 𝑖 . In action 𝑎 ne 𝑖 , user 𝑖 selects 2 of the 3 available resources featuring a cost 𝑐 1 𝑥 2 / 3 ; in 𝑎 opt 𝑖 they select the remaining resource with cost 𝑐 1 𝑥 2 / 3 , as well as one resource with cost 𝑐 2 𝑥 3 / 3 . Each user has a similar pair of actions, but each subsequent agent is oset by one from the prior (see Fig. 2). In this game, we obtain the following system and user costs: SC ( 𝑎 ne ) = 2 ( 𝑐 1 𝑏 ( 2 ) ) = 8 𝑐 1 , SC ( 𝑎 opt ) = 𝑐 1 𝑏 ( 1 ) + 𝑐 2 𝑏 ( 1 ) = 𝑐 1 + 𝑐 2 𝐶 𝑖 ( 𝑎 ne ) = 2 3 𝑐 1 ( 𝑏 ( 2 ) + 𝜏 ) = 1 3 ( 8 𝑐 1 + 2 𝑐 1 𝜏 ) , 𝐶 𝑖 ( 𝑎 opt 𝑖 , 𝑎 ne − 𝑖 ) = 𝑐 1 3 ( 9 + 𝜏 ) + 𝑐 2 3 ( 1 + 𝜏 ) . As in the previous example, we set SC ( 𝑎 ne ) = 1 implying 𝑐 1 = 1 / 8 . T o ensure that the joint action 𝑎 ne is a Nash equilibrium, w e impose that 𝐶 𝑖 ( 𝑎 ne ) = 𝐶 𝑖 ( 𝑎 opt 𝑖 , 𝑎 ne − 𝑖 ) for any player 𝑖 , resulting in 𝑐 2 = ( 𝜏 − 1 ) / ( 8 ( 1 + 𝜏 ) ) . The resulting price of anarchy is lower-bounded by 1 / SC ( 𝑎 opt ) = 1 / ( 𝑐 1 + 𝑐 2 ) . This quantity is no smaller than 16 / 3 for any choice of 𝜏 < 3 . Finally , for the xed toll 𝜏 = 3 , we upper-bound the price of anarchy at 16 / 3 . For ease of presentation, we rst consider the case where the cost on resource 𝑒 is ℓ 𝑒 ( 𝑥 ) = 𝛼 𝑒 𝑥 2 for some 𝛼 𝑒 ≥ 0 . W e will show at the end how to extend this result to the case of ℓ 𝑒 ( 𝑥 ) = 𝛼 𝑒 𝑥 2 + 𝛽 𝑒 𝑥 + 𝛾 𝑒 . T owards this goal, let 𝑎 ne (resp. 𝑎 opt ) be an equilibrium (resp. optimum) allocation in a congestion game 𝐺 , with 𝑛 users, resources in 𝑒 ∈ E . The cost at equilibrium satises SC ( 𝑎 ne ) ≤ 𝑛  𝑖 = 1 𝐶 𝑖 ( 𝑎 opt 𝑖 , 𝑎 ne − 𝑖 ) − 𝑛  𝑖 = 1 𝐶 𝑖 ( 𝑎 ne ) + SC ( 𝑎 ne ) (27) =  𝑒 ∈ E 𝛼 𝑒  𝑧 𝑒 ( ( 𝑥 𝑒 + 𝑦 𝑒 + 1 ) 2 + 𝜏 ) − 𝑦 𝑒 ( ( 𝑥 𝑒 + 𝑦 𝑒 ) 2 + 𝜏 ) + ( 𝑥 𝑒 + 𝑦 𝑒 ) 3  (28) ≤  𝑒 ∈ E 𝛼 𝑒 [ ( 𝑥 𝑒 + 𝑧 𝑒 ) ( ( 𝑥 𝑒 + 𝑦 𝑒 + 1 ) 2 + 3 ) − ( 𝑥 𝑒 + 𝑦 𝑒 ) ( ( 𝑥 𝑒 + 𝑦 𝑒 ) 2 + 3 ) + ( 𝑥 𝑒 + 𝑦 𝑒 ) 3 ] (29) 𝑐 1 / 8 𝑐 1 / 8 𝑐 1 / 8 𝑐 1 / 8 𝑐 1 / 8 𝑐 1 / 8 𝑐 1 / 8 𝑐 1 / 8 𝑎 ne 1 𝑎 opt 1 𝑎 ne 2 𝑎 opt 2 𝑐 2 / 8 𝑐 2 / 8 𝑐 2 / 8 𝑐 2 / 8 𝑐 2 / 8 𝑐 2 / 8 𝑐 2 / 8 𝑐 2 / 8 𝑎 opt 1 𝑎 opt 2 𝑐 1 / 3 𝑐 1 / 3 𝑐 1 / 3 𝑎 ne 1 𝑎 opt 1 𝑎 ne 2 𝑎 opt 2 𝑐 2 / 3 𝑐 2 / 3 𝑐 2 / 3 𝑎 opt 1 𝑎 opt 2 Fig. 2. Game construction used to lower bound the price of anarchy for quadratic congestion games with fixed tolls 𝜏 ≥ 3 (le) and 𝜏 < 3 (right). On the le (resp. right), the available actions of two of the eight (resp. three) agents are shown. The solid red shape contains the resources utilized by the first user in the action 𝑎 ne 1 , while the solid blue shape contains the resources utilized by the first user in the action 𝑎 opt 1 . User 2 has similar actions but rotated clockwise on each circle by one resource. Each of the remaining agents’ actions are defined similarly by rotating about the apparent “ring” . A CM Transactions on Economics and Computation (to appear) 20 Paccagnan, et al. =  𝑒 ∈ E 𝛼 𝑒  3 ( ( 𝑥 𝑒 + 𝑧 𝑒 ) − ( 𝑥 𝑒 + 𝑦 𝑒 ) ) + ( 𝑥 𝑒 + 𝑧 𝑒 ) ( 𝑥 𝑒 + 𝑦 𝑒 + 1 ) 2  ≤  𝑒 ∈ E 𝛼 𝑒  4 ( 𝑥 𝑒 + 𝑧 𝑒 ) 3 + 1 4 ( 𝑥 𝑒 + 𝑦 𝑒 ) 3  = 4SC ( 𝑎 opt ) + 1 4 SC ( 𝑎 ne ) , (30) where 𝑦 𝑒 = | 𝑎 ne | 𝑒 − 𝑥 𝑒 , 𝑧 𝑒 = | 𝑎 opt | 𝑒 − 𝑥 𝑒 , and 𝑥 𝑒 = | { 𝑖 ∈ 𝑁 s.t. 𝑒 ∈ 𝑎 ne 𝑖 ∩ 𝑎 opt 𝑖 } | . Obser ve that (27) holds from the denition of Nash equilibrium, while (28) follows from the parameterization intr oduced in [ 28 ], and substituting 𝑏 𝑒 ( 𝑥 ) = 𝑥 2 . Equation (29) follows by replacing 𝜏 = 3 and by 𝑥 𝑒 ≥ 0 . T o see that (30) holds for all integers 𝑥 𝑒 , 𝑦 𝑒 ≥ 0 , we dene 𝑢 = 𝑥 𝑒 + 𝑦 𝑒 ≥ 0 , 𝑣 = 𝑥 𝑒 + 𝑧 𝑒 ≥ 0 , and divide the argument in two parts depending on whether the integer tuple ( 𝑢, 𝑣 ) ∈ { 𝑢 ≥ 22 or 𝑣 ≥ 8 } or not. For the case of ( 𝑢, 𝑣 ) ∈ { 𝑢 ≥ 22 or 𝑣 ≥ 8 } , we observe that 4 𝑣 3 + 1 4 𝑢 3 − 3 𝑣 + 3 𝑢 − 𝑣 ( 𝑢 + 1 ) 2 ≥ 4 𝑣 3 + 1 4 𝑢 3 − 𝑣 ( 𝑢 2 + 2 𝑢 + 4 ) ≥ 4 𝑣 3 + 1 4 𝑢 3 − 𝑣 ( 𝑢 + 2 ) 2 , and therefore we are left to pr ove 4 𝑣 3 + 1 4 𝑢 3 − 𝑣 ( 𝑢 + 2 ) 2 ≥ 0 . (31) For every xed 𝑢 ≥ 0 , dierentiating with r espect to 𝑣 shows that the left hand side of (31) has a unique global minimum in the positive orthant at 𝑣 = ( 𝑢 + 2 ) / √ 12 . For any 𝑢 > 22 , this minimum satises (31) , thus for any 𝑣 ≥ 0 and 𝑢 > 22 (31) is satised. Additionally observe that, when 𝑣 = 8 , (31) is satise d for each 𝑢 ∈ { 0 , . . . , 22 } . Further , for xe d 0 ≤ 𝑢 ≤ 22 , the left hand side of (31) is increasing in 𝑣 for 𝑣 ≥ 8 . This implies that (31) holds for every 𝑣 ≥ 8 as well. Therefore (31) (and consequently (30) ) is satised for all ( 𝑢, 𝑣 ) ∈ { 𝑢 ≥ 22 or 𝑣 ≥ 8 } . One can enumerate the nitely-many non-negative integers ( 𝑢, 𝑣 ) with 𝑢 < 22 , 𝑣 < 8 and v erify that (30) holds. The inequality in (30) implies that the price of anarchy is upper bounded by 4 1 − 1 / 4 = 16 / 3 when resource costs take the form ℓ 𝑒 ( 𝑥 ) = 𝛼 𝑒 𝑥 2 for some 𝛼 𝑒 ≥ 0 . Obser ve that this b ound holds for arbitrarily large 𝑛 and matches the solution of the linear program, stated in T able 1. W e now generalize this result to ℓ 𝑒 ( 𝑥 ) = 𝛼 𝑒 𝑥 2 + 𝛽 𝑒 𝑥 + 𝛾 𝑒 , where 𝛼 𝑒 , 𝛽 𝑒 and 𝛾 𝑒 are non-negative. T o do so, we start from (27) , and note that (28) now contains the sum of three contributions: contributions relative to 𝛼 𝑒 , contributions r elative to 𝛽 𝑒 and contributions r elative to 𝛾 𝑒 . Hence, it suces to prove the following two additional inequalities to complete the reasoning, that is  𝑒 ∈ E 𝛽 𝑒  𝑧 𝑒 ( 𝑥 𝑒 + 𝑦 𝑒 + 1 ) − 𝑦 𝑒 ( 𝑥 𝑒 + 𝑦 𝑒 ) + ( 𝑥 𝑒 + 𝑦 𝑒 ) 2  ≤  𝑒 ∈ E 𝛽 𝑒  4 ( 𝑥 𝑒 + 𝑧 𝑒 ) 2 + 1 4 ( 𝑥 𝑒 + 𝑦 𝑒 ) 2  , (32)  𝑒 ∈ E 𝛾 𝑒 [ 𝑧 𝑒 − 𝑦 𝑒 + 𝑥 𝑒 + 𝑦 𝑒 ] ≤  𝑒 ∈ E 𝛾 𝑒  4 ( 𝑥 𝑒 + 𝑧 𝑒 ) + 1 4 ( 𝑥 𝑒 + 𝑦 𝑒 )  , (33) where we recall that no toll is associated to the presence of 𝛽 𝑒 or 𝛾 𝑒 in (22) . Summing these two inequalities with the inequality from (28) and (30) will, in fact, yield the desired claim. While the proof of (33) is immediate, the argument used to show (32) is similar to that following (28) . In fact, since 𝑥 𝑒 ≥ 0 , w e have 𝑧 𝑒 ( 𝑥 𝑒 + 𝑦 𝑒 + 1 ) − 𝑦 𝑒 ( 𝑥 𝑒 + 𝑦 𝑒 ) + ( 𝑥 𝑒 + 𝑦 𝑒 ) 2 ≤ ( 𝑥 𝑒 + 𝑧 𝑒 ) ( 𝑥 𝑒 + 𝑦 𝑒 + 1 ) − ( 𝑥 𝑒 + 𝑦 𝑒 ) 2 + ( 𝑥 𝑒 + 𝑦 𝑒 ) 2 . Thus, we are left to show that for every non-negative integer 𝑢 and 𝑣 it is 4 𝑣 2 + 1 4 𝑢 2 − 𝑣 − 𝑢 𝑣 ≥ 0 , where we make use of the same coordinates introduced earlier . This inequality is satised by all non-negative integer points since 4 𝑣 2 + 1 4 𝑢 2 − 𝑣 − 𝑢 𝑣 ≥ 0 describes the region outside an ellipse locate d in the ( 𝑢, 𝑣 ) plane entirely on the left of the line 𝑢 = 1 and entir ely south of the line 𝑣 = 1 , where the inequality is trivially satised for 𝑢 = 𝑣 = 0 . Finally , we observe that the technique use d to bound the price of anarchy extends to coarse correlated equilibria due to linearity of the expectation. □ A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 21 6 (IN)EFFICIENCY OF THE MARGINAL COST MECHANISM In this section we study the eciency of the marginal cost mechanism, whereby the toll imposed to each user corresponds to her marginal contribution to the system cost. In the atomic setup, the marginal cost me chanism takes the form 𝑇 p ( ℓ ) = 𝜏 p , and the corresponding tolls read 𝜏 p ( 𝑥 ) = ( 𝑥 − 1 ) ( ℓ ( 𝑥 ) − ℓ ( 𝑥 − 1 ) ) , where we set ℓ ( 0 ) = 0 . W e recall that 𝑇 p ensures that the best performing equilibrium is optimal, i.e., its price of stability is one [ 23 ]. The following Corollary shows how to compute Po A ( 𝑇 p ) through the solution of a linear program when bases are discrete conv ex (see Footnote 5). W e also provide the analytical expression of Po A ( 𝑇 p ) for polynomial congestion games with 𝑑 = 1 , and note that a similar argument carries over when 𝑑 ≥ 1 . Corollary 2. The price of anarchy of the marginal cost mechanism 𝑇 p ( ℓ ) = 𝜏 p , with 𝜏 p ( 𝑥 ) = ( 𝑥 − 1 ) ( ℓ ( 𝑥 ) − ℓ ( 𝑥 − 1 ) ) over congestion games with 𝑛 agents, and resource costs generated by a non-negative linear combination of conv ex basis functions { 𝑏 1 , . . . , 𝑏 𝑚 } equals 1 / 𝜌 opt , where 𝜌 opt solves the following linear program max 𝜌 ∈ R ,𝜈 ∈ R ≥ 0 𝜌 s . t . 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ ( 𝑢 2 − 𝑢 𝑣 ) 𝑏 𝑗 ( 𝑢 ) − 𝑢 ( 𝑢 − 1 ) 𝑏 𝑗 ( 𝑢 − 1 ) − 𝑣 ( 𝑢 + 1 ) 𝑏 𝑗 ( 𝑢 + 1 ) ] ≥ 0 ∀ 𝑢, 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 ≤ 𝑛, ∀ 𝑗 ∈ { 1 , . . . , 𝑚 } , 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑢𝑏 𝑗 ( 𝑢 ) ( 2 𝑛 − 𝑢 − 𝑣 ) + ( 𝑢 − 1 ) 𝑏 𝑗 ( 𝑢 − 1 ) ( 𝑣 − 𝑛 ) + ( 𝑢 + 1 ) 𝑏 𝑗 ( 𝑢 + 1 ) ( 𝑢 − 𝑛 ) ] ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 > 𝑛, ∀ 𝑗 ∈ { 1 , . . . , 𝑚 } . where we set 𝑏 𝑗 ( − 1 ) = 𝑏 𝑗 ( 0 ) = 𝑏 𝑗 ( 𝑛 + 1 ) = 0 . 13 For ane congestion games with arbitrarily large number of agents, we have Po A ( 𝑇 p ) = 3 . Both results are tight for pure Nash equilibria, and also hold for coarse correlated equilibria. Proof. W e b egin with the rst claim, and observe that the marginal cost mechanism is linear , in the sense that 𝑇 p ( Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑏 𝑗 ) = Í 𝑚 𝑗 = 1 𝛼 𝑗 𝑇 p ( 𝑏 𝑗 ) for all non-negative 𝛼 𝑗 and for all basis functions. Additionally , the functions 𝑓 𝑗 ( 𝑥 ) = 𝑏 𝑗 ( 𝑥 ) + ( 𝑥 − 1 ) ( 𝑏 𝑗 ( 𝑥 ) − 𝑏 𝑗 ( 𝑥 − 1 ) ) ar e non-decreasing in their domain. This is because 𝑓 𝑗 ( 𝑥 + 1 ) − 𝑓 𝑗 ( 𝑥 ) = 𝑏 𝑗 ( 𝑥 + 1 ) + 𝑥 ( 𝑏 𝑗 ( 𝑥 + 1 ) − 𝑏 𝑗 ( 𝑥 ) ) − 𝑏 𝑗 ( 𝑥 ) − ( 𝑥 − 1 ) ( 𝑏 𝑗 ( 𝑥 ) − 𝑏 𝑗 ( 𝑥 − 1 ) ) = ( 𝑥 + 1 ) 𝑏 𝑗 ( 𝑥 + 1 ) − 𝑥 𝑏 𝑗 ( 𝑥 ) − ( 𝑥 𝑏 𝑗 ( 𝑥 ) − ( 𝑥 − 1 ) 𝑏 𝑗 ( 𝑥 − 1 ) ) ≥ 0 , for all 𝑥 ∈ { 1 , . . . , 𝑛 − 1 } , where the inequality holds as each function 𝑏 𝑗 ( 𝑥 ) 𝑥 is convex (since each 𝑏 𝑗 ( 𝑥 ) is so), and thus its discrete derivative is non-decreasing. It follows that the price of anarchy can be computed using the linear program in (12) which provides tight results for pure Nash equilibria that extend to coarse correlated equilibria. Substituting 𝑓 𝑗 ( 𝑥 ) = 𝑏 𝑗 ( 𝑥 ) + ( 𝑥 − 1 ) ( 𝑏 𝑗 ( 𝑥 ) − 𝑏 𝑗 ( 𝑥 − 1 ) ) in (12) we obtain the desired result. W e now focus on ane congestion games, and pr ove that Po A ( 𝑇 p ) = 3 . T owards this goal, we observe that Fig. 3 is an example of an ane congestion game using the marginal cost mechanism. Thus, we conclude that Po A ( 𝑇 p ) ≥ 3 for ane congestion games and pur e Nash equilibria. W e now show that Po A ( 𝑇 p ) ≤ 3 whenever each resour ce 𝑒 is asso ciated to a cost ℓ 𝑒 ( 𝑥 ) = 𝛼 𝑒 𝑥 + 𝛽 𝑒 . In this case, we have 𝑇 p ( ℓ 𝑒 ) = 𝜏 p , where 𝜏 p ( 𝑥 ) = 𝛼 𝑒 ( 𝑥 − 1 ) is indep endent of 𝛽 𝑒 , thanks to its 13 The result also holds under the weaker requirement that only 𝑏 𝑗 ( 𝑥 ) 𝑥 are convex. A CM Transactions on Economics and Computation (to appear) 22 Paccagnan, et al. O 1 O 2 D 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 Nash routing, 𝜏 𝑒 ( 𝑥 ) = 0 (A ) System Cost: 2 O 1 O 2 D 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 Nash routing, 𝜏 p 𝑒 ( 𝑥 ) = 𝑥 − 1 (B) System cost: 6 Fig. 3. Instance used to demonstrates that the price of anarchy asso ciated to the marginal cost toll mechanism is at least 3 in aine congestion games. T wo users ar e willing to travel from O 1 / O 2 to D , where each edge features a cost ℓ 𝑒 ( 𝑥 ) = 𝑥 . In the un-tolled case ( A) the system cost at the worst Nash-equilibrium is 2 . The situation worsens when using marginal cost tolls (B), as the worst Nash e quilibrium gives a system cost of 6 . denition. For an equilibrium allocation 𝑎 ne ∈ A and 𝑎 opt ∈ A and optimal allocation, we have SC ( 𝑎 ne ) ≤ 𝑛  𝑖 = 1 C 𝑖 ( 𝑎 opt 𝑖 , 𝑎 ne − 𝑖 ) − 𝑛  𝑖 = 1 C 𝑖 ( 𝑎 ne ) + SC ( 𝑎 ne ) =  𝑒 ∈ E 𝛼 𝑒  𝑧 𝑒 ( 2 𝑥 𝑒 + 2 𝑦 𝑒 + 1 ) − 𝑦 𝑒 ( 2 𝑥 𝑒 + 2 𝑦 𝑒 − 1 ) + ( 𝑥 𝑒 + 𝑦 𝑒 ) 2  + 𝛽 𝑒 [ 𝑧 𝑒 − 𝑦 𝑒 + ( 𝑥 𝑒 + 𝑦 𝑒 ) ] ≤  𝑒 ∈ E 𝛼 𝑒  ( 𝑥 𝑒 + 𝑦 𝑒 ) − ( 𝑥 𝑒 + 𝑦 𝑒 ) 2 + 2 ( 𝑥 𝑒 + 𝑦 𝑒 ) ( 𝑥 𝑒 + 𝑧 𝑒 ) + ( 𝑥 𝑒 + 𝑧 𝑒 )  + 𝛽 𝑒 [ 𝑥 𝑒 + 𝑧 𝑒 ] ≤  𝑒 ∈ E 𝛼 𝑒  3 ( 𝑥 𝑒 + 𝑧 𝑒 ) 2  + 𝛽 𝑒 [ 3 ( 𝑥 𝑒 + 𝑧 𝑒 ) ] = 3 · SC ( 𝑎 opt ) , where we utilize the notation 𝑦 𝑒 = | 𝑎 ne | 𝑒 − 𝑥 𝑒 , 𝑧 𝑒 = | 𝑎 opt | 𝑒 − 𝑥 𝑒 , and 𝑥 𝑒 = | { 𝑖 ∈ 𝑁 s.t. 𝑒 ∈ 𝑎 ne 𝑖 ∩ 𝑎 opt 𝑖 } | . The rst inequality holds by denition of Nash equilibrium, and the second holds due to non- negativity of 𝑥 𝑒 and 𝛼 𝑒 . One veries that the last inequality holds, using 𝑢 = 𝑥 𝑒 + 𝑦 𝑒 ≥ 0 , 𝑣 = 𝑥 𝑒 + 𝑧 𝑒 ≥ 0 , and obser ving that the region 3 𝑣 2 + 𝑢 2 − 𝑢 − 2 𝑢 𝑣 − 𝑣 ≥ 0 is the exterior of an ellipse containing all ( 𝑢, 𝑣 ) ∈ Z 2 ≥ 0 . W e remark that the latter reasoning extends identical to coarse correlated equilibria exploiting linearity of the expectation. Rearranging the above inequality , we get Po A ( 𝑇 p ) ≤ SC ( 𝑎 ne ) / SC ( 𝑎 opt ) ≤ 3 for pur e Nash as well as coarse correlated equilibria. □ 7 CONCLUSIONS AND OPEN PROBLEMS This w ork derives optimal local tolling mechanisms and corr esponding prices of anarchy for atomic congestion games. W e do so for b oth the setup where tolls are congestion-awar e and congestion- independent. Finally , we derive price of anarchy values for the marginal cost mechanism. Our results generalize those of [ 8 ], and show that the eciency of optimal tolls utilizing solely local information is comparable to that of existing tolls using global information [ 6 ]. Further , we show that utilizing the marginal cost mechanism on the atomic setup is worse than levying no toll. Open questions. Our work leaves a number of open questions, two of which are discussed next. - While we observed that the price of anarchy for optimally tolled ane congestion games matches that of ane load balancing games on identical machines, we conjecture such r esult holds more generally , at least for polynomial congestion games. - In this manuscript we focused on the worst-case eciency metric both with respect to the game instance, and the resulting equilibrium. It is currently unclear if, and to what extent, optimizing the price of anarchy impacts other more optimistic performance metrics. A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 23 A APPENDIX A.1 Results used in Section 2 Lemma 1. Consider the class of congestion games G . For any linear tolling me chanism 𝑇 , it is Po A ( 𝑇 ) = sup 𝐺 ∈ G ( Z ≥ 0 ) NECost ( 𝐺 , 𝑇 ) MinCost ( 𝐺 ) , where G ( Z ≥ 0 ) ⊂ G is the subclass of games with 𝛼 𝑗 ∈ Z ≥ 0 for all 𝑗 ∈ { 1 , . . . , 𝑚 } , for all resources in E . Proof. W e divide the proof in two steps. First, we show that Po A ( 𝑇 ) = sup 𝐺 ∈ G ( Q ≥ 0 ) NECost ( 𝐺 , 𝑇 ) MinCost ( 𝐺 ) , (34) where G ( Q ≥ 0 ) ⊂ G is the subclass of games with 𝛼 𝑗 ∈ Q ≥ 0 for all 𝑗 ∈ { 1 , . . . , 𝑚 } , for all resources in E . T owards this goal, obser ve that (34) holds trivially with ≥ in place of the e quality sign, as R ≥ 0 ⊃ Q ≥ 0 . T o show that the converse inequality also holds, obser ve that the price of anarchy of a given linear mechanisms 𝑇 (computed over all meaningful instances where NECost ( 𝐺 , 𝑇 ) > 0 ) can be computed utilizing the linear program reported in (11) . By strong duality , we have Po A ( 𝑇 ) = 1 / 𝐶 ∗ , where 𝐶 ∗ is the value of the dual program of (11), i.e., 𝐶 ∗ = min 𝜃 ( 𝑥 , 𝑦 ,𝑧, 𝑗 )  𝑥 , 𝑦 ,𝑧, 𝑗 𝑏 𝑗 ( 𝑥 + 𝑧 ) ( 𝑥 + 𝑧 ) 𝜃 ( 𝑥 , 𝑦, 𝑧 , 𝑗 ) (35a) s . t .  𝑥 , 𝑦 ,𝑧, 𝑗  𝑓 𝑗 ( 𝑥 + 𝑦 ) 𝑦 − 𝑓 𝑗 ( 𝑥 + 𝑦 + 1 ) 𝑧  𝜃 ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) ≤ 0 (35b)  𝑥 , 𝑦 ,𝑧, 𝑗 𝑏 𝑗 ( 𝑥 + 𝑦 ) ( 𝑥 + 𝑦 ) 𝜃 ( 𝑥 , 𝑦, 𝑧 , 𝑗 ) = 1 (35c) 𝜃 ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) ≥ 0 ∀ ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) ∈ I (35d) where we dene 𝑏 𝑗 ( 0 ) = 𝑓 𝑗 ( 0 ) = 𝑓 𝑗 ( 𝑛 + 1 ) = 0 for convenience, I = { ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) ∈ Z 4 ≥ 0 s.t. 1 ≤ 𝑥 + 𝑦 + 𝑧 ≤ 𝑛, 1 ≤ 𝑗 ≤ 𝑚 } , and the minimum is intended over the entire tuple { 𝜃 ( 𝑥 , 𝑦, 𝑧 , 𝑗 ) } ( 𝑥 ,𝑦 ,𝑧, 𝑗 ) ∈ I . Let { 𝜃 ∗ ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) } ( 𝑥 ,𝑦 ,𝑧, 𝑗 ) ∈ I denote an optimal solution (which exists, due to the non-emptiness and boundedness of the constraint set, which can be proven using the same argument in [ 28 , Thm. 2]). If all 𝜃 ∗ ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) are rational, then consider the game 𝐺 dened as follows. For e very 𝑖 ∈ { 1 , . . . , 𝑛 } and for every ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) ∈ I , we create a resource identied with 𝑒 ( 𝑥 , 𝑦 , 𝑧 , 𝑗 , 𝑖 ) , and assign to it the resource cost 𝛼 𝑗 𝑏 𝑗 , where 𝛼 𝑗 = 𝜃 ∗ ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) / 𝑛 . The game 𝐺 features 𝑛 players, where player 𝑝 ∈ { 1 , . . . , 𝑛 } can either select the resources in the allocation 𝑎 opt 𝑝 or in 𝑎 ne 𝑝 , dened by 𝑎 opt 𝑝 = ∪ 𝑛 𝑖 = 1 ∪ 𝑚 𝑗 = 1 { 𝑒 ( 𝑥 , 𝑦 , 𝑧, 𝑗 , 𝑖 ) : 𝑥 + 𝑦 ≥ 1 + ( ( 𝑖 − 𝑝 ) mo d 𝑛 ) } , 𝑎 ne 𝑝 = ∪ 𝑛 𝑖 = 1 ∪ 𝑚 𝑗 = 1 { 𝑒 ( 𝑥 , 𝑦 , 𝑧, 𝑗 , 𝑖 ) : 𝑥 + 𝑧 ≥ 1 + ( ( 𝑖 − 𝑝 + 𝑧 ) mod 𝑛 ) } . Note that the above construction is an extension of that appearing in [ 28 ] to the case of multiple basis functions. Since 𝐺 has NECost ( 𝐺 , 𝑇 ) =  𝑥 , 𝑦 ,𝑧, 𝑗 𝑏 𝑗 ( 𝑥 + 𝑦 ) ( 𝑥 + 𝑦 ) 𝜃 ∗ ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) = 1 , MinCost ( 𝐺 ) ≤  𝑥 , 𝑦 ,𝑧, 𝑗 𝑏 𝑗 ( 𝑥 + 𝑧 ) ( 𝑥 + 𝑧 ) 𝜃 ∗ ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) = 𝐶 ∗ , (see [ 28 , Thm. 2] for this), its price of anarchy is no smaller than 1 / 𝐶 ∗ . Observe that 𝐺 features only non-negative rational resource costs’ coecients (i.e., 𝐺 ∈ G ( Q ≥ 0 ) ), therefore (34) follows readily . A CM Transactions on Economics and Computation (to appear) 24 Paccagnan, et al. If at least one entry in the tuple { 𝜃 ∗ ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) } ( 𝑥 ,𝑦 ,𝑧, 𝑗 ) ∈ I is not rational, we will pro ve the existence of a sequence of games 𝐺 𝑘 ∈ G ( Q ≥ 0 ) whose worst-case eciency converges to Po A ( 𝑇 ) as 𝑘 → ∞ . This would imply that (34) holds with ≤ in place of the equality sign, concluding the proof. T o do so, let us consider the set 𝑆 = { { 𝜃 ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) } ( 𝑥 ,𝑦 ,𝑧, 𝑗 ) ∈ I s.t. (35b), and (35d) hold } . Observe that 𝑆 is non-empty , and that for any tuple belonging to 𝑆 , we can nd a sequence of non-negative rational tuples { { 𝜃 𝑘 ( 𝑎, 𝑥 , 𝑏 , 𝑗 ) } ( 𝑥 ,𝑦 ,𝑧, 𝑗 ) ∈ I } ∞ 𝑘 = 1 (i.e., 𝜃 𝑘 ( 𝑎, 𝑥 , 𝑏 , 𝑗 ) ∈ Q ≥ 0 for all 𝑎, 𝑥 , 𝑏 , 𝑗 and 𝑘 ), that conv erges to it. Let { { 𝜃 𝑘 ( 𝑎, 𝑥 , 𝑏 , 𝑗 ) } ( 𝑥 ,𝑦 ,𝑧, 𝑗 ) ∈ I } ∞ 𝑘 = 1 be a sequence of tuples converging to { 𝜃 ∗ ( 𝑎, 𝑥 , 𝑏 , 𝑗 ) } ( 𝑥 ,𝑦 ,𝑧, 𝑗 ) ∈ I , which b elongs to 𝑆 . For each tuple { 𝜃 𝑘 ( 𝑎, 𝑥 , 𝑏 , 𝑗 ) } ( 𝑥 ,𝑦 ,𝑧, 𝑗 ) ∈ I in the sequence, dene the game 𝐺 𝑘 following the same construction intr oduced above with 𝜃 𝑘 ( 𝑎, 𝑥 , 𝑏 , 𝑗 ) in place of 𝜃 ∗ ( 𝑎, 𝑥 , 𝑏 , 𝑗 ) . Fol- lowing the same reasoning as above, it is NECost ( 𝐺 𝑘 , 𝑇 ) = Í 𝑥 , 𝑦 ,𝑧, 𝑗 𝑏 𝑗 ( 𝑥 + 𝑦 ) ( 𝑥 + 𝑦 ) 𝜃 𝑘 ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) , and MinCost ( 𝐺 𝑘 ) ≤ Í 𝑥 , 𝑦 ,𝑧, 𝑗 𝑏 𝑗 ( 𝑥 + 𝑧 ) ( 𝑥 + 𝑧 ) 𝜃 𝑘 ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) . Therefore Po A 𝑘 = NECost ( 𝐺 𝑘 , 𝑇 ) MinCost ( 𝐺 𝑘 ) ≥ Í 𝑥 , 𝑦 ,𝑧, 𝑗 𝑏 𝑗 ( 𝑥 + 𝑦 ) ( 𝑥 + 𝑦 ) 𝜃 𝑘 ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) Í 𝑥 , 𝑦 ,𝑧, 𝑗 𝑏 𝑗 ( 𝑥 + 𝑧 ) ( 𝑥 + 𝑧 ) 𝜃 𝑘 ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) , from which we conclude that lim 𝑘 →∞ Po A 𝑘 ≥ lim 𝑘 →∞ Í 𝑥 , 𝑦 ,𝑧, 𝑗 𝑏 𝑗 ( 𝑥 + 𝑦 ) ( 𝑥 + 𝑦 ) 𝜃 𝑘 ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) Í 𝑥 , 𝑦 ,𝑧, 𝑗 𝑏 𝑗 ( 𝑥 + 𝑧 ) ( 𝑥 + 𝑧 ) 𝜃 𝑘 ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) = Í 𝑥 , 𝑦 ,𝑧, 𝑗 𝑏 𝑗 ( 𝑥 + 𝑦 ) ( 𝑥 + 𝑦 ) 𝜃 ∗ ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) Í 𝑥 , 𝑦 ,𝑧, 𝑗 𝑏 𝑗 ( 𝑥 + 𝑧 ) ( 𝑥 + 𝑧 ) 𝜃 ∗ ( 𝑥 , 𝑦 , 𝑧, 𝑗 ) = 1 𝐶 ∗ , as 𝜃 𝑘 ( 𝑎, 𝑥 , 𝑏 , 𝑗 ) → 𝜃 ∗ ( 𝑎, 𝑥 , 𝑏 , 𝑗 ) for 𝑘 → ∞ . This completes the rst step. The second and nal step consist in showing that sup 𝐺 ∈ G ( Q ≥ 0 ) NECost ( 𝐺 , 𝑇 ) MinCost ( 𝐺 ) = sup 𝐺 ∈ G ( Z ≥ 0 ) NECost ( 𝐺 , 𝑇 ) MinCost ( 𝐺 ) . T owards this goal, for any given game from the ab ove-dened sequence 𝐺 𝑘 ∈ G ( Q ≥ 0 ) , let 𝑑 𝐺 𝑘 denote the lowest common denominator among the resource cost coecients 𝛼 𝑗 , across all the resources of the game. Dene ˆ 𝛼 𝑗 = 𝛼 𝑗 · 𝑑 𝐺 𝑘 ∈ Z ≥ 0 for all 𝑗 ∈ { 1 , . . . , 𝑚 } , for all r esources in E . Since the tolling mechanisms 𝑇 is linear by assumption, the equilibrium conditions are independent to uniform scaling of the resource costs and tolls by the co ecient 𝑑 𝐺 𝑘 . Therefore any game in the sequence 𝐺 𝑘 with tolling mechanism 𝑇 and resource cost coecients { 𝛼 𝑗 } 𝑚 𝑗 = 1 has the same worst-case equilibrium eciency as a game ˆ 𝐺 𝑘 which is identical to 𝐺 𝑘 except that it has resource cost coecients { ˆ 𝛼 𝑗 } 𝑚 𝑗 = 1 . Observing that ˆ 𝐺 𝑘 belongs to G ( Z ≥ 0 ) concludes the proof. □ A.2 Results used in Section 3 Lemma 2. Let 𝑏 : N → R ≥ 0 be a nondecreasing, convex function, and let 0 < 𝜌 ≤ 1 b e a given parameter . Further , dene the function 𝑓 : { 1 , . . . , 𝑛 } → R such that 𝑓 ( 1 ) = 𝑏 ( 1 ) and 𝑓 ( 𝑢 + 1 )  min 𝑣 𝑢 ∈ { 1 ,. . ., 𝑛 } min { 𝑢, 𝑛 − 𝑣 𝑢 } · 𝑓 ( 𝑢 ) − 𝑏 ( 𝑢 ) 𝑢 · 𝜌 + 𝑏 ( 𝑣 𝑢 ) 𝑣 𝑢 min { 𝑣 𝑢 , 𝑛 − 𝑢 } , (36) for all 𝑢 ∈ { 1 , . . . , 𝑛 − 1 } . Then, for the lowest value 1 ≤ ˆ 𝑢 ≤ 𝑛 − 1 such that 𝑓 ( ˆ 𝑢 + 1 ) < 𝑓 ( ˆ 𝑢 ) , it must hold that 𝑓 ( 𝑢 + 1 ) < 𝑓 ( 𝑢 ) for all 𝑢 ∈ { ˆ 𝑢 , . . . , 𝑛 − 1 } . Proof. The proof is presented in two parts as follows: in Part 1, we identify inequalities given that 𝑓 ( ˆ 𝑢 + 1 ) < 𝑓 ( ˆ 𝑢 ) , for 1 ≤ ˆ 𝑢 ≤ 𝑛 − 1 as dened in the claim; and, in Part 2, we use a recursive argument to prove that 𝑓 ( 𝑢 + 1 ) < 𝑓 ( 𝑢 ) holds for all ˆ 𝑢 + 1 ≤ 𝑢 ≤ 𝑛 − 1 , using the ine qualities derived in Part 1. A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 25 Part 1. W e dene 𝑣 ∗ 𝑢 as one of the arguments that minimize the right-hand side of (36) for each 𝑢 ∈ { 1 , . . . , 𝑛 − 1 } . By assumption, it must hold that 𝑓 ( ˆ 𝑢 + 1 ) < 𝑓 ( ˆ 𝑢 ) , which implies that 𝑓 ( ˆ 𝑢 ) > min 𝑣 ∈ { 1 ,. . .,𝑛 } min { ˆ 𝑢 , 𝑛 − 𝑣 } min { 𝑣 , 𝑛 − ˆ 𝑢 } 𝑓 ( ˆ 𝑢 ) − 𝑏 ( ˆ 𝑢 ) ˆ 𝑢 min { 𝑣 , 𝑛 − ˆ 𝑢 } 𝜌 + 𝑏 ( 𝑣 ) 𝑣 min { 𝑣 , 𝑛 − ˆ 𝑢 } = min 𝑣 ∈ { 1 ,. . .,𝑛 } min { ˆ 𝑢 − 1 , 𝑛 − 𝑣 } min { 𝑣 , 𝑛 − ˆ 𝑢 + 1 } 𝑓 ( ˆ 𝑢 − 1 ) − 𝑏 ( ˆ 𝑢 − 1 ) ( ˆ 𝑢 − 1 ) min { 𝑣 , 𝑛 − ˆ 𝑢 + 1 } 𝜌 + 𝑏 ( 𝑣 ) 𝑣 min { 𝑣 , 𝑛 − ˆ 𝑢 + 1 } + min { ˆ 𝑢 , 𝑛 − 𝑣 } min { 𝑣 , 𝑛 − ˆ 𝑢 } 𝑓 ( ˆ 𝑢 ) − min { ˆ 𝑢 − 1 , 𝑛 − 𝑣 } min { 𝑣 , 𝑛 − ˆ 𝑢 + 1 } 𝑓 ( ˆ 𝑢 − 1 ) − 𝑏 ( ˆ 𝑢 ) ˆ 𝑢 min { 𝑣 , 𝑛 − ˆ 𝑢 } 𝜌 + 𝑏 ( ˆ 𝑢 − 1 ) ( ˆ 𝑢 − 1 ) min { 𝑣 , 𝑛 − ˆ 𝑢 + 1 } 𝜌 + 𝑏 ( 𝑣 ) 𝑣 min { 𝑣 , 𝑛 − ˆ 𝑢 } − 𝑏 ( 𝑣 ) 𝑣 min { 𝑣 , 𝑛 − ˆ 𝑢 + 1 } , where the strict inequality holds by the denition of 𝑓 ( ˆ 𝑢 + 1 ) . Recall that 𝑓 ( ˆ 𝑢 )  min 𝑣 ∈ { 1 ,. . .,𝑛 } min { ˆ 𝑢 − 1 , 𝑛 − 𝑣 } min { 𝑣 , 𝑛 − ˆ 𝑢 + 1 } 𝑓 ( ˆ 𝑢 − 1 ) − 𝑏 ( ˆ 𝑢 − 1 ) ( ˆ 𝑢 − 1 ) min { 𝑣 , 𝑛 − ˆ 𝑢 + 1 } 𝜌 + 𝑏 ( 𝑣 ) 𝑣 min { 𝑣 , 𝑛 − ˆ 𝑢 + 1 } . Thus, if 𝑣 ∗ ˆ 𝑢 ≤ 𝑛 − ˆ 𝑢 , the above strict inequality with 𝑓 ( ˆ 𝑢 ) can only b e satised if 𝑓 ( ˆ 𝑢 + 1 ) < 𝑓 ( ˆ 𝑢 ) ≤ ˆ 𝑢 𝑓 ( ˆ 𝑢 ) − ( ˆ 𝑢 − 1 ) 𝑓 ( ˆ 𝑢 − 1 ) < [ 𝑏 ( ˆ 𝑢 ) ˆ 𝑢 − 𝑏 ( ˆ 𝑢 − 1 ) ( ˆ 𝑢 − 1 ) ] · 𝜌 . Similarly , if 𝑣 ∗ ˆ 𝑢 ≥ 𝑛 − ˆ 𝑢 + 1 , then it must hold that ( 𝑛 − 𝑣 ∗ ˆ 𝑢 )  𝑓 ( ˆ 𝑢 ) 𝑛 − ˆ 𝑢 − 𝑓 ( ˆ 𝑢 − 1 ) 𝑛 − ˆ 𝑢 + 1  +  1 𝑛 − ˆ 𝑢 − 1 𝑛 − ˆ 𝑢 + 1  𝑏 ( 𝑣 ∗ ˆ 𝑢 ) 𝑣 ∗ ˆ 𝑢 <  𝑏 ( ˆ 𝑢 ) ˆ 𝑢 𝑛 − ˆ 𝑢 − 𝑏 ( ˆ 𝑢 − 1 ) ( ˆ 𝑢 − 1 ) 𝑛 − ˆ 𝑢 + 1  · 𝜌 = ⇒  1 𝑛 − ˆ 𝑢 − 1 𝑛 − ˆ 𝑢 + 1  [ ( 𝑛 − 𝑣 ∗ ˆ 𝑢 ) 𝑓 ( ˆ 𝑢 ) + 𝑏 ( 𝑣 ∗ ˆ 𝑢 ) 𝑣 ∗ ˆ 𝑢 ] <  𝑏 ( ˆ 𝑢 ) ˆ 𝑢 𝑛 − ˆ 𝑢 − 𝑏 ( ˆ 𝑢 − 1 ) ( ˆ 𝑢 − 1 ) 𝑛 − ˆ 𝑢 + 1  · 𝜌 ⇐ ⇒ 𝑓 ( ˆ 𝑢 + 1 ) < [ 𝑏 ( ˆ 𝑢 ) ˆ 𝑢 − 𝑏 ( ˆ 𝑢 − 1 ) ( ˆ 𝑢 − 1 ) ] 𝜌 , where the rst line implies the second line because 𝑓 ( ˆ 𝑢 ) ≥ 𝑓 ( ˆ 𝑢 − 1 ) , by the denition of ˆ 𝑢 in the claim, and the second line is e quivalent to the thir d by the denitions of 𝑓 ( ˆ 𝑢 + 1 ) and 𝑣 ∗ ˆ 𝑢 . This concludes Part 1 of the proof. Part 2. In this part of the proof, we show by recursion that if 𝑓 ( ˆ 𝑢 + 1 ) < 𝑓 ( ˆ 𝑢 ) , then 𝑓 ( 𝑢 + 1 ) < 𝑓 ( 𝑢 ) for all 𝑢 ∈ { ˆ 𝑢 + 1 , . . ., 𝑛 − 1 } . W e do so by showing that, if 𝑓 ( 𝑢 ) < 𝑓 ( 𝑢 − 1 ) < · · · < 𝑓 ( ˆ 𝑢 + 1 ) for any 𝑢 ∈ { ˆ 𝑢 + 1 , . . . , 𝑛 − 1 } , then it must hold that 𝑓 ( 𝑢 + 1 ) < 𝑓 ( 𝑢 ) . Thus, in the following reasoning, we assume that 𝑢 ∈ { ˆ 𝑢 + 1 , . . . , 𝑛 − 1 } , and that 𝑓 ( 𝑢 ) < 𝑓 ( 𝑢 − 1 ) < · · · < 𝑓 ( ˆ 𝑢 + 1 ) . W e b egin with the case of 𝑣 ∗ 𝑢 − 1 < 𝑛 − 𝑢 + 1 , which gives us that 𝑣 ∗ 𝑢 − 1 ≤ 𝑛 − 𝑢 . Observe that 𝑓 ( 𝑢 + 1 )  min 𝑣 𝑢 ∈ { 1 ,. . ., 𝑛 } min { 𝑢, 𝑛 − 𝑣 𝑢 } min { 𝑣 𝑢 , 𝑛 − 𝑢 } 𝑓 ( 𝑢 + 1 ) − 𝑏 ( 𝑢 ) 𝑢 min { 𝑣 𝑢 , 𝑛 − 𝑢 } 𝜌 + 𝑏 ( 𝑣 𝑢 ) 𝑣 𝑢 min { 𝑣 𝑢 , 𝑛 − 𝑢 } = min 𝑣 𝑢 ∈ { 1 ,. . ., 𝑛 } min { 𝑢 − 1 , 𝑛 − 𝑣 𝑢 } min { 𝑣 𝑢 , 𝑛 − 𝑢 + 1 } 𝑓 ( 𝑢 − 1 ) − 𝑏 ( 𝑢 − 1 ) ( 𝑢 − 1 ) min { 𝑣 𝑢 , 𝑛 − 𝑢 + 1 } 𝜌 + 𝑏 ( 𝑣 𝑢 ) 𝑣 𝑢 min { 𝑣 𝑢 , 𝑛 − 𝑢 + 1 } + min { 𝑢, 𝑛 − 𝑣 𝑢 } min { 𝑣 𝑢 , 𝑛 − 𝑢 } 𝑓 ( 𝑢 + 1 ) − min { 𝑢 − 1 , 𝑛 − 𝑣 𝑢 } min { 𝑣 𝑢 , 𝑛 − 𝑢 + 1 } 𝑓 ( 𝑢 − 1 ) − 𝑏 ( 𝑢 ) 𝑢 min { 𝑣 𝑢 , 𝑛 − 𝑢 } 𝜌 + 𝑏 ( 𝑢 − 1 ) ( 𝑢 − 1 ) min { 𝑣 𝑢 , 𝑛 − 𝑢 + 1 } 𝜌 + 𝑏 ( 𝑣 𝑢 ) 𝑣 𝑢 min { 𝑣 𝑢 , 𝑛 − 𝑢 } − 𝑏 ( 𝑣 𝑢 ) 𝑣 𝑢 min { 𝑣 𝑢 , 𝑛 − 𝑢 + 1 } ≤ 𝑓 ( 𝑢 ) + 𝑢 𝑣 ∗ 𝑢 − 1 𝑓 ( 𝑢 ) − 𝑢 − 1 𝑣 ∗ 𝑢 − 1 𝑓 ( 𝑢 − 1 ) − 𝑏 ( 𝑢 ) 𝑢 − 𝑏 ( 𝑢 − 1 ) ( 𝑢 − 1 ) 𝑣 ∗ 𝑢 − 1 𝜌 A CM Transactions on Economics and Computation (to appear) 26 Paccagnan, et al. < 𝑓 ( 𝑢 ) + 1 𝑣 ∗ 𝑢 − 1 𝑓 ( ˆ 𝑢 + 1 ) − 1 𝑣 ∗ 𝑢 − 1 [ 𝑏 ( 𝑢 ) 𝑢 − 𝑏 ( 𝑢 − 1 ) ( 𝑢 − 1 ) ] 𝜌 < 𝑓 ( 𝑢 ) , where the rst inequality holds by evaluating the minimization at 𝑣 𝑢 = 𝑣 ∗ 𝑢 − 1 , the second ine quality holds because 𝑓 ( 𝑢 ) < 𝑓 ( 𝑢 − 1 ) and 𝑓 ( 𝑢 ) ≤ 𝑓 ( ˆ 𝑢 + 1 ) , by assumption, and the nal inequality holds by the result showed in Part 1 and because 𝑏 ( ·) is nondecr easing and convex. Next, we consider the scenario in which 𝑣 ∗ 𝑢 − 1 > 𝑛 − 𝑢 + 1 . Obser ve that 𝑓 ( 𝑢 + 1 ) ≤ 𝑓 ( 𝑢 ) + ( 𝑛 − 𝑣 ∗ 𝑢 − 1 )  𝑓 ( 𝑢 ) 𝑛 − 𝑢 − 𝑓 ( 𝑢 − 1 ) 𝑛 − 𝑢 + 1  +  1 𝑛 − 𝑢 − 1 𝑛 − 𝑢 + 1  𝑏 ( 𝑣 ∗ 𝑢 − 1 ) 𝑣 ∗ 𝑢 − 1 − 𝑏 ( 𝑢 ) 𝑢 𝑛 − 𝑢 𝜌 + 𝑏 ( 𝑢 − 1 ) ( 𝑢 − 1 ) 𝑛 − 𝑢 + 1 𝜌 < 𝑓 ( 𝑢 ) +  1 𝑛 − 𝑢 − 1 𝑛 − 𝑢 + 1  [ ( 𝑛 − 𝑣 ∗ 𝑢 − 1 ) 𝑓 ( 𝑢 − 1 ) + 𝑏 ( 𝑣 ∗ 𝑢 − 1 ) 𝑣 ∗ 𝑢 − 1 ] − 𝑏 ( 𝑢 ) 𝑢 𝑛 − 𝑢 𝜌 + 𝑏 ( 𝑢 − 1 ) ( 𝑢 − 1 ) 𝑛 − 𝑢 + 1 𝜌 = 𝑓 ( 𝑢 ) +  1 𝑛 − 𝑢 − 1 𝑛 − 𝑢 + 1  [ ( 𝑛 − 𝑢 + 1 ) 𝑓 ( 𝑢 ) + 𝑏 ( 𝑢 − 1 ) ( 𝑢 − 1 ) 𝜌 ] − 𝑏 ( 𝑢 ) 𝑢 𝑛 − 𝑢 𝜌 + 𝑏 ( 𝑢 − 1 ) ( 𝑢 − 1 ) 𝑛 − 𝑢 + 1 𝜌 ≤ 𝑓 ( 𝑢 ) + 1 𝑛 − 𝑢 𝑓 ( ˆ 𝑢 + 1 ) − 1 𝑛 − 𝑢 [ 𝑏 ( 𝑢 ) 𝑢 − 𝑏 ( 𝑢 − 1 ) ( 𝑢 − 1 ) ] 𝜌 < 𝑓 ( 𝑢 ) , where the rst inequality holds by evaluating the minimization at 𝑣 𝑢 = 𝑣 ∗ 𝑢 − 1 , the second ine quality holds because 𝑓 ( 𝑢 ) < 𝑓 ( 𝑢 − 1 ) , by assumption, the equality holds by the denitions of 𝑓 ( 𝑢 ) and 𝑣 ∗ 𝑢 − 1 , the third inequality holds because 𝑓 ( 𝑢 ) ≤ 𝑓 ( ˆ 𝑢 + 1 ) , by assumption, and the nal inequality holds by the identity we showed in Part 1 and because 𝑏 is nondecreasing and conve x. Finally , we consider the scenario in which 𝑣 ∗ 𝑢 − 1 = 𝑛 − 𝑢 + 1 . Observe that 𝑓 ( 𝑢 + 1 ) ≤ 𝑓 ( 𝑢 ) + 𝑢 − 1 𝑛 − 𝑢 𝑓 ( 𝑢 ) − 𝑢 − 1 𝑛 − 𝑢 + 1 𝑓 ( 𝑢 − 1 ) − 𝑏 ( 𝑢 ) 𝑢 𝑛 − 𝑢 𝜌 + 𝑏 ( 𝑢 − 1 ) ( 𝑢 − 1 ) 𝑛 − 𝑢 + 1 𝜌 +  1 𝑛 − 𝑢 − 1 𝑛 − 𝑢 + 1  𝑏 ( 𝑛 − 𝑢 + 1 ) ( 𝑛 − 𝑢 + 1 ) < 𝑓 ( 𝑢 ) +  1 𝑛 − 𝑢 − 1 𝑛 − 𝑢 + 1  [ ( 𝑛 − 𝑣 ∗ 𝑢 − 1 ) 𝑓 ( 𝑢 − 1 ) + 𝑏 ( 𝑣 ∗ 𝑢 − 1 ) 𝑣 ∗ 𝑢 − 1 ] − 𝑏 ( 𝑢 ) 𝑢 𝑛 − 𝑢 𝜌 + 𝑏 ( 𝑢 − 1 ) ( 𝑢 − 1 ) 𝑛 − 𝑢 + 1 𝜌 < 𝑓 ( 𝑢 ) , where the rst inequality holds by evaluating the minimization at 𝑣 𝑢 = 𝑣 ∗ 𝑢 − 1 , the second ine quality holds because 𝑓 ( 𝑢 ) < 𝑓 ( 𝑢 − 1 ) , by assumption, and the nal inequality holds by the same reasoning as for 𝑣 ∗ 𝑢 − 1 > 𝑛 − 𝑢 + 1 . □ A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 27 A.3 Results used in Section 4 Lemma 3. For given 𝑑 ≥ 1 , consider the linear program dened in (18) , and let ( 𝑓 opt , 𝜌 opt ) be a solution. Further let 𝑓 ∞ be dene d in (19) utilizing ( 𝑓 opt , 𝜌 opt ) . The price of anarchy of 𝑓 ∞ over congestion games of degree 𝑑 with possibly innitely many agents is upper bounded by 1 / 𝜌 ∞ , where 𝜌 ∞ = min ( 𝜌 opt , 𝛽 − 𝑑  1 + 2 ¯ 𝑛  𝑑 + 1  𝛽 𝑑 + 1  1 + 1 𝑑 ) . This result is tight for pure Nash equilibria and holds for coarse correlated equilibria too. Proof. Since 𝑓 ∞ is non-decreasing by construction, we characterize its performance over games with a maximum of 𝑛 agents through the linear program in (12) with 𝑏 ( 𝑥 ) = 𝑥 𝑑 , that is max 𝜌 ∈ R ,𝜈 ∈ R ≥ 0 𝜌 s . t . 𝑏 ( 𝑣 ) 𝑣 − 𝜌 𝑏 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑓 ∞ ( 𝑢 ) 𝑢 − 𝑓 ∞ ( 𝑢 + 1 ) 𝑣 ] ≥ 0 ∀ 𝑢, 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 ≤ 𝑛, 𝑏 ( 𝑣 ) 𝑣 − 𝜌𝑏 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑓 ∞ ( 𝑢 ) ( 𝑛 − 𝑣 ) − 𝑓 ∞ ( 𝑢 + 1 ) ( 𝑛 − 𝑢 ) ] ≥ 0 ∀ 𝑢 , 𝑣 ∈ { 0 , . . . , 𝑛 } 𝑢 + 𝑣 > 𝑛 . (37) Upper bounding the price of anarchy of 𝑓 ∞ amounts to nding a feasible solution to this linear program; the challenging task is that we intend to do so for 𝑛 arbitrary large. W e claim that ( 𝜌 , 𝜈 ) = ( 𝜌 ∞ , 1 ) is feasible for any (p ossibly innite) 𝑛 , so that the claime d upper bound on the price of anarchy follo ws. T o prov e this, we divide the discussion in two parts as the expr essions are dierent for 𝑢 + 𝑣 ≤ 𝑛 and 𝑢 + 𝑣 > 𝑛 . Before doing so, we study the degenerate case of 𝑢 = 0 , for which the constraints reduce to 𝑓 ∞ ( 1 ) ≤ 1 , which holds as 𝑓 ∞ is feasible for the linear program in (18) which already includes this constraint. – Case of 𝑢 + 𝑣 ≤ 𝑛 , 𝑢 ≥ 1 . The constraints read as 𝑣 𝑑 + 1 − 𝜌𝑢 𝑑 + 1 + 𝜈 [ 𝑢 𝑓 ∞ ( 𝑢 ) − 𝑣 𝑓 ∞ ( 𝑢 + 1 ) ] ≥ 0 (38) which we want to hold for any integers 𝑢 ≥ 1 , 𝑣 ≥ 0 (the bound on the indices 𝑢 + 𝑣 ≤ 𝑛 can be dropped as we are interested in the case of arbitrary 𝑛 ). 𝑢 𝑣 𝑢 = 𝑣 Region A Region B Region C 1 2 ¯ 𝑛 / 2 . . . 0 1 . . . Fig. 4. Regions A, B, and C utilized in proof for the case 𝑢 + 𝑣 ≤ 𝑛 . • In the region where 0 ≤ 𝑣 ≤ 𝑢 , 1 ≤ 𝑢 < ¯ 𝑛 / 2 (region A in Fig. 4) these constraints certainly hold with 𝜌 ≤ 𝜌 opt , 𝜈 = 1 . This follows because 𝑓 ∞ ( 𝑢 ) = 𝑓 opt ( 𝑢 ) when 𝑢 ≤ ¯ 𝑛 / 2 , and 𝑓 opt is feasible for the program in (18) which includes these constraints. • In the region where 𝑣 > 𝑢 , 1 ≤ 𝑢 < ¯ 𝑛 / 2 (region B in Fig. 4) these constraints also hold with 𝜌 ≤ 𝜌 opt , 𝜈 = 1 thanks to Lemma 4 part a). A CM Transactions on Economics and Computation (to appear) 28 Paccagnan, et al. • W e are left with the region where 𝑢 ≥ ¯ 𝑛 / 2 , 𝑣 ≥ 0 (region C in Fig. 4). In this case, 𝑓 ∞ ( 𝑢 ) = 𝛽 · 𝑢 𝑑 by denition. With the choice of 𝜈 = 1 , the constraints in (38) read 𝑣 𝑑 + 1 − 𝜌 𝑢 𝑑 + 1 + 𝛽 𝑢 𝑑 + 1 − 𝛽 𝑣 ( 𝑢 + 1 ) 𝑑 ≥ 0 . (39) Observe that, for a xe d choice of integer 𝑢 ≥ ¯ 𝑛 / 2 , the left hand side of (39) is a convex function of 𝑣 for 𝑣 ≥ 0 . Its corresponding minimum value over the non-negativ e reals is  𝛽 𝑑 + 1  1 + 1 𝑑 ( 𝑢 + 1 ) 𝑑 + 1 − 𝛽  𝛽 𝑑 + 1  1 𝑑 ( 𝑢 + 1 ) 𝑑 + 1 + ( 𝛽 − 𝜌 ) 𝑢 𝑑 + 1 . Hence, (39) is satised for a xed choice of 𝑢 ≥ ¯ 𝑛 / 2 and all integers 𝑣 ≥ 0 if the latter expression is non-negative . Simple algebra shows that this is the case if 𝜌 ≤ 𝛽 − 𝑑  1 + 1 𝑢  𝑑 + 1  𝛽 𝑑 + 1  1 + 1 𝑑 . (40) Since we would like (39) to hold for all 𝑢 ≥ ¯ 𝑛 / 2 , and since the right-hand side in (40) is increasing in 𝑛 , it suces to ask for (40) to hold at the lowest admissible 𝑢 , i.e. 𝑢 = ¯ 𝑛 / 2 . Therefore , in order for (39) to be satised for any 𝑢 ≥ ¯ 𝑛 / 2 , it suces to select 𝜌 ≤ 𝛽 − 𝑑  1 + 2 ¯ 𝑛  𝑑 + 1  𝛽 𝑑 + 1  1 + 1 𝑑 . (41) – Case of 𝑢 + 𝑣 > 𝑛 , 𝑢 ≥ 1 . Lemma 4 part b) shows that the constraints corresponding to 𝑢 + 𝑣 > 𝑛 are satised for arbitrary 𝑛 with the choice of 𝜈 = 1 , and 𝜌 as in (41) thanks to the fact that 𝑓 ∞ : N → R is non-de creasing. In conclusion, w e veried that ( 𝜌 , 𝜈 ) = ( 𝜌 ∞ , 1 ) is feasible for the program in (37) . It follo ws that the price of anarchy of 𝑓 ∞ over games with arbitrarily large 𝑛 is upper bounde d as in the claim. □ Lemma 4. a) Let 𝑓 : { 1 , . . . , 𝑛 } → R , 𝜌 ≥ 0 , and 𝑓 ( 𝑥 ) ≤ 𝑥 𝑑 for all 1 ≤ 𝑥 ≤ 𝑛 and 𝑑 ≥ 1 . The constraints 𝑣 𝑑 + 1 − 𝜌𝑢 𝑑 + 1 + 𝑢 𝑓 ( 𝑢 ) − 𝑣 𝑓 ( 𝑢 + 1 ) ≥ 0 obtained for any 𝑣 ∈ N , 𝑣 ≥ 𝑢 , 𝑢 ∈ { 1 , . . . , 𝑛 − 1 } are satised if the same inequality holds for all 𝑣 = 𝑢 ∈ { 1 , . . . , 𝑛 − 1 } . b) Let 𝑓 : N → R be non-decreasing, 𝑑 ≥ 1 , 𝜌 ≥ 0 . If the constraints 𝑣 𝑑 + 1 − 𝜌𝑢 𝑑 + 1 + 𝑢 𝑓 ( 𝑢 ) − 𝑣 𝑓 ( 𝑢 + 1 ) ≥ 0 hold for all non-negative integers 𝑢 , 𝑣 , then 𝑣 𝑑 + 1 − 𝜌𝑢 𝑑 + 1 + ( 𝑛 − 𝑣 ) 𝑓 ( 𝑢 ) − ( 𝑛 − 𝑢 ) 𝑓 ( 𝑢 + 1 ) ≥ 0 hold for all non-negative integers 𝑢 , 𝑣 with 𝑢 + 𝑣 > 𝑛 , for any choice of 𝑛 ≥ 1 integer . Proof. W e prove the tw o claims separately . First claim. For 𝑣 = 𝑢 the constraints of interest reduces to 𝜌𝑢 𝑑 + 1 ≤ 𝑢 𝑑 + 1 + 𝑢 ( 𝑓 ( 𝑢 ) − 𝑓 ( 𝑢 + 1 ) ) , while for 𝑣 = 𝑢 + 𝑝 (with 𝑝 ≥ 1 ) the constraint reads as 𝜌𝑢 𝑑 + 1 ≤ ( 𝑢 + 𝑝 ) 𝑑 + 1 + 𝑢 𝑓 ( 𝑢 ) − ( 𝑢 + 𝑝 ) 𝑓 ( 𝑢 + 1 ) . Proving the claim amounts to showing 𝑢 𝑑 + 1 + 𝑢 ( 𝑓 ( 𝑢 ) − 𝑓 ( 𝑢 + 1 ) ) ≤ ( 𝑢 + 𝑝 ) 𝑑 + 1 + 𝑢 𝑓 ( 𝑢 ) − ( 𝑢 + 𝑝 ) 𝑓 ( 𝑢 + 1 ) ⇐ ⇒ 𝑓 ( 𝑢 + 1 ) ≤ ( 𝑢 + 𝑝 ) 𝑑 + 1 − 𝑢 𝑑 + 1 𝑝 for 𝑝 ≥ 1 . The right hand side is minimized at 𝑝 = 1 (it describes the slope of the se cant to the function 𝑢 𝑑 + 1 at abscissas 𝑢 and 𝑢 + 𝑝 ) due to the convexity of 𝑢 𝑑 + 1 . Therefore, it suces to ensure that 𝑓 ( 𝑢 + 1 ) ≤ ( 𝑢 + 1 ) 𝑑 + 1 − 𝑢 𝑑 + 1 for any choice of 𝑢 ∈ { 1 , . . . , 𝑛 − 1 } . By assumption, 𝑓 ( 𝑢 + 1 ) ≤ ( 𝑢 + 1 ) 𝑑 , so that we can equivalently pro ve ( 𝑢 + 1 ) 𝑑 ≤ ( 𝑢 + 1 ) 𝑑 + 1 − 𝑢 𝑑 + 1 . The latter holds, as required, for all 𝑢 ∈ { 1 , . . . , 𝑛 − 1 } since ( 𝑢 + 1 ) 𝑑 ≤ ( 𝑢 + 1 ) 𝑑 + 1 − 𝑢 𝑑 + 1 ⇐ ⇒ ( 𝑢 + 1 ) 𝑑 ≤ ( 𝑢 + 1 ) 𝑑 ( 𝑢 + 1 ) − 𝑢 𝑑 + 1 ⇐ ⇒ 𝑢 𝑑 ≤ ( 𝑢 + 1 ) 𝑑 . A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 29 Second claim. By assumption, for all non-negative integers 𝑢 , 𝑣 it is 𝜌𝑢 𝑑 + 1 ≤ 𝑣 𝑑 + 1 + 𝑢 𝑓 ( 𝑢 ) − 𝑣 𝑓 ( 𝑢 + 1 ) , while we intend to show that for all non-negative integers 𝑢 , 𝑣 with 𝑢 + 𝑣 > 𝑛 it is 𝜌𝑢 𝑑 + 1 ≤ 𝑣 𝑑 + 1 + ( 𝑛 − 𝑣 ) 𝑓 ( 𝑢 ) − ( 𝑛 − 𝑢 ) 𝑓 ( 𝑢 + 1 ) , regardless for the choice of 𝑛 ≥ 1 integer . Proving this is equivalent to showing 𝑢 𝑓 ( 𝑢 ) − 𝑣 𝑓 ( 𝑢 + 1 ) ≤ ( 𝑛 − 𝑣 ) 𝑓 ( 𝑢 ) − ( 𝑛 − 𝑢 ) 𝑓 ( 𝑢 + 1 ) ⇐ ⇒ ( 𝑢 + 𝑣 − 𝑛 ) ( 𝑓 ( 𝑢 ) − 𝑓 ( 𝑢 + 1 ) ) ≤ 0 , which holds, as required, due to the fact that 𝑢 + 𝑣 − 𝑛 > 0 and 𝑓 ( 𝑢 ) − 𝑓 ( 𝑢 + 1 ) ≤ 0 due to 𝑓 being non-decreasing. □ A.4 Results used in Section 5 Lemma 5. Let 𝑛 ∈ N , and assume that the basis functions { 𝑏 1 , . . . , 𝑏 𝑚 } , 𝑏 𝑗 : N → R are convex (in the discrete sense), positive, and non-decreasing, for all 𝑗 = 1 , . . . , 𝑚 . Then, the constraints app earing in (25) with 𝑢 = 0 , 𝑣 = { 1 , . . . , 𝑛 } are satised if the constraint with ( 𝑢, 𝑣 ) = ( 0 , 1 ) holds. Similarly , the constraints with 𝑢 ∈ { 1 , . . . , 𝑛 } , 𝑣 ∈ { 0 , . . . , 𝑛 } and 𝑢 < 𝑣 are satised if the constraints with 𝑢 ∈ { 1 , . . . , 𝑛 } , 𝑣 ∈ { 0 , . . . , 𝑛 } and 𝑢 ≥ 𝑣 hold. Proof. W e b egin with the constraints obtained for 𝑢 = 0 , which read as 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜈 𝑏 𝑗 ( 1 ) 𝑣 − 𝜎 𝑗 𝑣 ≥ 0 . The constraint with 𝑣 = 0 holds trivially , while the tightest constraint for 𝑣 > 0 is obtained with 𝑣 = 1 due to the fact that 𝑏 𝑗 ( 𝑣 ) 𝑣 is increasing for 𝑣 > 0 . This shows that it is sucient to consider the constraint with ( 𝑢 , 𝑣 ) = ( 0 , 1 ) . W e now consider the constraints obtained for 𝑢 ≥ 1 and divide the pr oof in three parts, according to the regions in Fig. 5. 𝑢 𝑣 𝑣 = 𝑢 A B C 0 1 𝑛 . . . 0 1 . . . 𝑛 Fig. 5. Regions A, B, and C utilized in proof of Lemma 5. • In the region where 𝑣 > 𝑢 and 𝑢 + 𝑣 ≤ 𝑛 (Region A in Fig. 5), we show that if the constraint obtained with 𝑣 = 𝑢 holds, then the constraints with 𝑣 > 𝑢 also hold. Note that feasible values of 𝜈 are upp er bounded by 𝜈 ≤ 1 . This is because the constraint with ( 𝑢, 𝑣 ) = ( 0 , 1 ) reads as ( 1 − 𝜈 ) 𝑏 𝑗 ( 1 ) ≥ 𝜎 𝑗 , and since 𝜎 𝑗 ≥ 0 , 𝑏 𝑗 ( 1 ) > 0 , every feasible 𝜈 must satisfy 1 − 𝜈 ≥ 0 . The constraint with 𝑣 = 𝑢 + 𝑝 , 𝑝 ≥ 1 read as 𝑏 𝑗 ( 𝑢 + 𝑝 ) ( 𝑢 + 𝑝 ) − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑏 𝑗 ( 𝑢 ) 𝑢 − 𝑏 𝑗 ( 𝑢 + 1 ) ( 𝑢 + 𝑝 ) ] − 𝜎 𝑗 𝑝 ≥ 0 , the tightest of which is obtaine d for the largest feasible value of 𝜎 𝑗 , that is 𝜎 𝑗 = ( 1 − 𝜈 ) 𝑏 𝑗 ( 1 ) . The constraint with 𝑣 = 𝑢 reads as 𝑢𝑏 𝑗 ( 𝑢 ) − 𝜌 𝑢𝑏 𝑗 ( 𝑢 ) + 𝜈𝑢 ( 𝑏 𝑗 ( 𝑢 ) − 𝑏 𝑗 ( 𝑢 + 1 ) ) ≥ 0 . Therefore, we intend to show that for ev ery 𝜌 and 0 ≤ 𝜈 ≤ 1 satisfying 𝑢𝑏 𝑗 ( 𝑢 ) − 𝜌 𝑢𝑏 𝑗 ( 𝑢 ) + 𝜈𝑢 ( 𝑏 𝑗 ( 𝑢 ) − 𝑏 𝑗 ( 𝑢 + 1 ) ) ≥ 0 , it also holds for all 𝑝 ≥ 1 that 𝑏 𝑗 ( 𝑢 + 𝑝 ) ( 𝑢 + 𝑝 ) − 𝜌 𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑏 𝑗 ( 𝑢 ) 𝑢 − 𝑏 𝑗 ( 𝑢 + 1 ) ( 𝑢 + 𝑝 ) ] − ( 1 − 𝜈 ) 𝑏 𝑗 ( 1 ) 𝑝 ≥ 0 . Since both constraints describ e straight lines in the plane ( 𝜈 , 𝜌 ) , it suces to verify that this is true at the extreme points 𝜈 = 0 and 𝜈 = 1 . A CM Transactions on Economics and Computation (to appear) 30 Paccagnan, et al. When 𝜈 = 0 the constraint with 𝑣 = 𝑢 and 𝑣 = 𝑢 + 𝑝 read as 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 ≤ 𝑏 𝑗 ( 𝑢 ) 𝑢 , and 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 ≤ 𝑏 𝑗 ( 𝑢 + 𝑝 ) ( 𝑢 + 𝑝 ) − 𝑏 𝑗 ( 1 ) 𝑝 . W e therefore intend to show that 𝑏 𝑗 ( 𝑢 ) 𝑢 ≤ 𝑏 𝑗 ( 𝑢 + 𝑝 ) ( 𝑢 + 𝑝 ) − 𝑏 𝑗 ( 1 ) 𝑝 ⇐ ⇒ 𝑏 𝑗 ( 𝑢 + 𝑝 ) ( 𝑢 + 𝑝 ) − 𝑢𝑏 𝑗 ( 𝑢 ) 𝑝 ≥ 𝑏 𝑗 ( 1 ) , (42) which holds since 𝑏 𝑗 ( 𝑢 ) 𝑢 is convex and 𝑏 𝑗 ( 𝑢 ) is non-decreasing so that 𝑏 𝑗 ( 𝑢 + 𝑝 ) ( 𝑢 + 𝑝 ) − 𝑢𝑏 𝑗 ( 𝑢 ) 𝑝 ≥ 𝑏 𝑗 ( 𝑢 + 1 ) ( 𝑢 + 1 ) − 𝑢𝑏 𝑗 ( 𝑢 ) ≥ 𝑏 𝑗 ( 𝑢 + 1 ) ≥ 𝑏 𝑗 ( 1 ) . When 𝜈 = 1 , following a similar reasoning, we are left to show that 𝑢𝑏 𝑗 ( 𝑢 ) ≤ 𝑏 𝑗 ( 𝑢 + 𝑝 ) ( 𝑢 + 𝑝 ) − 𝑝 𝑏 𝑗 ( 𝑢 + 1 ) , which holds thanks to the non-decreasingness of 𝑏 𝑗 ( 𝑢 ) , indeed 𝑏 𝑗 ( 𝑢 + 𝑝 ) ( 𝑢 + 𝑝 ) − 𝑝 𝑏 𝑗 ( 𝑢 + 1 ) ≥ 𝑏 𝑗 ( 𝑢 + 𝑝 ) 𝑢 ≥ 𝑏 𝑗 ( 𝑢 ) 𝑢 . • In the region where 𝑣 > 𝑢 , 𝑢 + 𝑣 ≥ 𝑛 and 𝑢 ≤ 𝑛 / 2 (Region B in Fig. 5), we intend to show that the following constraints hold 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑏 𝑗 ( 𝑢 ) ( 𝑛 − 𝑣 ) − 𝑏 𝑗 ( 𝑢 + 1 ) ( 𝑛 − 𝑢 ) ] + 𝜎 𝑗 ( 𝑢 − 𝑣 ) ≥ 0 . W e do so by observing that the proof of the previous point did not r equire at all that 𝑢 + 𝑣 ≤ 𝑛 . Therefore , exploiting the same proof, w e have 𝑏 𝑗 ( 𝑣 ) 𝑣 − 𝜌𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑏 𝑗 ( 𝑢 ) 𝑢 − 𝑏 𝑗 ( 𝑢 + 1 ) 𝑣 ] + 𝜎 𝑗 ( 𝑢 − 𝑣 ) ≥ 0 also for 𝑢 + 𝑣 ≥ 𝑛 , i.e., in the region B of inter est. W e exploit this to conclude, and, in particular , we show that the satisfaction of latter constraint implies the desired result. T owards this goal we need to show that, for 𝑢 + 𝑣 > 𝑛 it is 𝑏 𝑗 ( 𝑢 ) ( 𝑛 − 𝑣 ) − 𝑏 𝑗 ( 𝑢 + 1 ) ( 𝑛 − 𝑢 ) ≥ 𝑏 𝑗 ( 𝑢 ) 𝑢 − 𝑏 𝑗 ( 𝑢 + 1 ) 𝑣 ⇔ ( 𝑏 𝑗 ( 𝑢 ) − 𝑏 𝑗 ( 𝑢 + 1 ) ) ( 𝑛 − 𝑢 − 𝑣 ) ≥ 0 , which holds due to the non-decreasingness of 𝑏 𝑗 and to 𝑢 + 𝑣 > 𝑛 . • In the region where 𝑣 > 𝑢 , 𝑢 + 𝑣 ≥ 𝑛 and 𝑢 > 𝑛 / 2 (Region C in Fig. 5), we use the same approach of that in the rst point. In particular , we intend to show that the when the constraints with 𝑣 = 𝑢 hold, also the constraints with 𝑣 > 𝑢 do so. Following a similar reasoning as in the above, this amount to showing for e very 𝜌 and 0 ≤ 𝜈 ≤ 1 satisfying 𝑏 𝑗 ( 𝑢 ) 𝑢 − 𝜌 𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 ( 𝑛 − 𝑢 ) ( 𝑏 𝑗 ( 𝑢 ) − 𝑏 𝑗 ( 𝑢 + 1 ) ) ≥ 0 , it also holds for all 𝑝 ≥ 1 that 𝑏 𝑗 ( 𝑢 + 𝑝 ) ( 𝑢 + 𝑝 ) − 𝜌 𝑏 𝑗 ( 𝑢 ) 𝑢 + 𝜈 [ 𝑏 𝑗 ( 𝑢 ) ( 𝑛 − 𝑢 − 𝑝 ) − 𝑏 𝑗 ( 𝑢 + 1 ) ( 𝑛 − 𝑢 ) ] − ( 1 − 𝜈 ) 𝑏 𝑗 ( 1 ) 𝑝 ≥ 0 . Since b oth constraints describe straight lines in the plane ( 𝜈 , 𝜌 ) , it suces to verify that this is true at the extreme points 𝜈 = 0 and 𝜈 = 1 . When 𝜈 = 0 we are left with 𝑏 𝑗 ( 𝑢 ) 𝑢 ≤ 𝑏 𝑗 ( 𝑢 + 𝑝 ) ( 𝑢 + 𝑝 ) − 𝑏 𝑗 ( 1 ) 𝑝 , which we already prov ed in (42) . When 𝜈 = 1 , we need to show that 𝑏 𝑗 ( 𝑢 + 𝑝 ) ( 𝑢 + 𝑝 ) − 𝑝 𝑏 𝑗 ( 𝑢 ) ≥ 𝑢𝑏 𝑗 ( 𝑢 ) , which holds by non-decreasingness of 𝑏 𝑗 . □ A CM Transactions on Economics and Computation (to appear) Optimal T axes in Atomic Congestion Games 31 REFERENCES [1] Sebastian Aland, Dominic Dumrauf, Martin Gairing, Burkhard Monien, and F lorian Schoppmann. 2011. 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