Selfishness need not be bad: a general proof

Transcription

1 Selfihne need not be bad: a general proof arxiv: v1 [c.gt] 20 May 2018 Zijun Wu zijunwu1984a@163.com, Center of Traffic and Big Data CTB), Sino-German Intitute for Applied Optimization SG-IAO), Department of Computer Science and Technology DCST), Hefei Univerity HU) Rolf H. Möhring rolf.moehring@me.com, Beijing Intitute for Scientific and Engineering Computing BISEC), Beijing Univerity of Technology BJUT) Dachuan Xu xudc@bjut.edu.cn, Beijing Intitute for Scientific and Engineering Computing BISEC), Beijing Univerity of Technology BJUT) May 22, 2018 Abtract Thi article tudie the uer behavior in non-atomic congetion game. We conider non-atomic congetion game with continuou and non-decreaing function and invetigate the it of the price of anarchy when the total uer volume approache infinity. We deepen the nowledge on aymptotically well deigned game [28], it game [28], calability [28] and gaugeability [7] that were recently ued in the it analye of the price of anarchy for non-atomic congetion game. We develop a unified framewor and derive new technique that allow a general it analyi of the price of anarchy. With thee new technique, we are able to prove a global convergence on the price of anarchy for non-atomic congetion game with arbitrary polynomial 1

2 price function and arbitrary uer volume vector equence, ee Theorem 2. Thi mean that non-atomic congetion game with polynomial price function are aymptotically well deigned. Moreover, we how that thee new technique are very flexible and robut and apply alo to non-atomic congetion game with price function of other type. In particular, we prove that non-atomic congetion game with regularly varying price function are alo aymptotically well deigned, provided that the price function are lightly retricted, ee Theorem 3 and Theorem 4. Our proof are direct and very elementary without uing any heavy machinery. They only ue baic propertie of Nah equilibrium and ytem optimum profile, imple fact about the aymptotic notation O ), Ω ), etc, and induction. Our reult greatly generalize recent reult from [8], [6], [7] and [28]. In particular, our reult further upport the view of [28] with a general proof that elfihne need not be bad for non-atomic congetion game. 1 Introduction 1.1 Motivation Nowaday, traffic congetion ha almot become a daily annoyance to every citizen in large citie of China. According to the newet data from AMap [1] in 2017, more than 26% of citie in China experienced traffic congetion in ruh hour, 55% of citie experienced low peed, and only 19% of citie did not uffer from traffic congetion. Traffic congetion doe not only coniderably enlarge travel latency, but alo caue eriou economic lo. We tae the capital city of China, Beijing, a an example. The average economic lo caued by congetion in 2017 wa about 4,013 RMB per peron, ee [2], which account for 3.1% of the annual GDP of Beijing in that year. Note that the annual GDP growth of Beijing wa only about 6.8% in Thi mean that traffic congetion ha almot detroyed one third of the potential economic growth of Beijing. To alleviate problem caued by traffic congetion, the government of China ha actively implemented a erie of traffic management meaure in ome large citie in recent year, including the even and odd licene plate number rule, licene plate lotterie, encouraging public tranportation and other. Thee meaure definitely prevent further deterioration of traffic, but not yet completely cure congetion. Road traffic condition are a direct reult from imultaneou travel of cit- 2

3 izen in a particular area. Given road condition, the routing behavior of traveler almot determine how the traffic develop. Thu, to comprehenively cure congetion, a preinary tep i to well undertand the routing behavior of traveler. In particular, we need to find out the extent to which the autonomou routing behavior of traveler contribute to congetion. Thi motivate the preent article. 1.2 The tatic model To that end, one need to model road traffic appropriately. A popular tatic model for road traffic i the o-called non-atomic congetion game NCG), ee [18] or [13]. NCG are non-cooperative game of perfect information. In an NCG, uer player) are collected into K different group according to ome meaurement on their imilaritie, for a fixed integer K N +. Aociated with each group K := {1,..., K} i a finite non-empty et S containing all trategie only available to uer from group. Every uer engaged in the game chooe a trategy S := K S that he will follow, and every choen trategy S conume ra, ) unit of reource a for each a A. Here, A i a finite non-empty et containing all available reource, and ra, ) i a fixed non-negative contant denoting the conumed or demanded) volume of reource a by trategy for each a A and each S. The eventual price of a reource a A depend only on it conumed volume. Given a vector d = d ) K of uer volume, a feaible trategy profile f i an aignment that aign to each of the d uer from group K a feaible trategy S for each group K. See [28] or Section 2 for detail. Obviouly, NCG model road traffic on an macrocopic level. Then, reource a A will be arc treet) of the underlying road networ, a group K will be a travel origin-detination OD) pair, and a trategy S will be a path from the -th origin to the -th detination, for each K. The contant ra, ) i jut an indicator function of the memberhip relation a for each arc a A and each path S. A feaible trategy profile f i then a feaible traffic path) aignment [10] for all the T d) := K d traveler. A our reult hold on a more general level, we will not tic to thee terminologie of road traffic in the equel, but till be able to undertand the routing behavior of traveler. In an NCG, the price of a reource a A i often expreed a an nonnegative, non-decreaing and continuou function τ a ) of it demanded volume, ee, e.g., [19], [20], [22], [13], [23]. Popular price function are polynomial. For intance, latency function τ a ) in road traffic are conventionally 3

4 aumed to be Bureau of Public Road BPR) function [14], which are polynomial of degree 4. In our tudy, we will follow thi fahion, and emphaize on polynomial price function τ a ) and other that are related to polynomial, e.g., regularly varying function [3]. However, the polynomial we will conider are general, i.e., they are allowed to have different degree. We do not conider trategie that are completely free, i.e., a A ra, ) τ ax) 0 for all x 0, for ome group K and ome trategy S. Thi i rather reaonable in congetion game, ince uer with choice of free trategie are actually outide the underlying game! 1.3 Selfih uer behavior NCG are non-cooperative, and o uer are coniderd to be elfih. They would lie to ue trategie minimizing their own cot. For intance, traveler would lie to follow a quicet path, o a to reduce their travel latency. In general, the cot of a uer i jut the cot of the trategy adopted by that uer. Given a vector d = d ) K of uer volume and a feaible profile f, the cot of a trategy S equal a A ra, ) τ af a ), where f a denote the total conumed volume of reource a w.r.t. profile f, for each a A. Obviouly, for road traffic, the cot of a trategy S i jut the total travel time latency) along path. The elfih behavior of uer will eventually lead the underlying game into a o-called Wardrop equalibrium WE) [27], in which every uer follow a cheapet trategy he could follow given the choice of all other uer, ee Section 2 for detail. Under our aumption of continuou and non-decreaing price function τ a ), a WE i actually a pure Nah equilibrium NE) [18], in which all uer will loyally adhere to the choice they have done, ince no unilateral change in trategy can introduce any extra profit. Therefore, the game will enter a teady tate if no external force interfere. An NE equivalently, a WE) i thu a macro model of uer elfih) behavior. The average cot of uer in an NE profile therefore reflect the cot uer need to pay in practice. Note that under our etting of continuou and non-decreaing price function τ a ), all NE profile will have the ame cot, ee, e.g., [26]. A quetion of great interet in NCG i whether uer elfih) behavior i harmful. Thi actually concern whether the elfih behavior of uer will demage ocial welfare, i.e., increae the average cot of uer engaged in the game. If thi i the cae, then we may need to employ ome external force or meaure to brea up the equilibrium induced by the elfih behavior, o a to get cloer to ocial welfare. For road traffic, poible external meaure 4

5 could be ome road guidance policie lie congetion pricing, ee, e.g., [4], [11], [5], [17] and [12]. The price of anarchy PoA), a concept temming from [16], i a popular meaure for the defficiency of elfih uer behavior. Given a vector d = d ) K of uer volume, a feaible trategy profile f i aid to be at ytem optimum SO) if it minimize the average cot of uer engaged in the game. The value of the PoA for non-decreaing price function τ a ) equal the ratio of the average cot of uer in an NE profile over that in an SO profile. Obviouly, the larger the value of PoA, the more elfih uer behavior demage the ocial welfare. The value of the PoA doe not only reflect the extent to which the elfih uer behavior ruin ocial welfare, but alo the potential benifit we could get if all uer were appropriately guided. For road traffic, the value of the PoA thu how the extent to which the average latency can be reduced in principle, if all traveler ue the right path. Therefore, for our purpoe, we need a cloe inpection of the value of the PoA. 1.4 The tate of the art Traditionally, elfih uer behavior i conidered to be harmful, ee, e.g., the tudie in [19], [20], [21], [24], [22], [9], [25], and [23]. Thee tudie invetigated an upper bound of the PoA for everal clae of price function. They demontrated that the wort-cae upper bound can be very large. A famou example motivating thee tudie i Pigou game, ee, e.g., [13] or Figure 1. In Pigou game, there i only one uer group with two trategie x β o 1 t Figure 1: Pigou game of price function x β and 1, repectively, for ome contant β 0. The PoA of thi game equal T/ T β + 1) 1/β + β + 1) 1), where T 1 i the volume of uer engaged in the game. Obviouly, conidering all poible β, 5

6 the wort-cae upper bound of the PoA i infinity, ince the PoA tend to a β, if T = 1. Wort-cae upper bound of the PoA are actually not a valid evidence to how that elfih uer behavior i bad, epecially not for a large total volume of uer. For intance, the PoA of Pigou game actually tend to 1, a T approache infinity, for each fixed β 0. Thi mean that elfih uer behavior may well guarantee ocial welfare if the total uer volume i large. Generally, the total travel demand in ruh hour i uually very large. Therefore, to comprehenively undertand elfih routing) behavior, a cloer inpection of the value of the PoA i till needed, epecially for the cae of heavy traffic, i.e., the cae when the total travel demand T d) = K d i large. Recently, everal tudie have been done in thi direction, ee [8], [6], [7], and [28]. Colini et al. [8] conidered two pecial cae: i) game with a ingle OD pair, and ii) game with a ingle OD pair with parallel feaible path. For cae i), they proved that if one of the feaible path ha a latency function that i bounded by a contant from above, then the PoA will converge to 1 a the total demand T d) tend to infinity. For cae ii), they proved that if the latency function τ a ) are regularly varying [3], then the PoA will converge to 1 a the total demand T d) tend to infinity. Colini et al. [7] continued the tudy of [8]. They invetigated more general cae, i.e., game with multiple OD pair. Uing the notion of regular variation [3], they propoed the concept of gaugeable game that are defined only for particular uer volume vector equence {d n) } n N fulfilling the condition that K tight d n) T d n) ), ee [7] or Subection 3.1 for detail. With the technique of regular variation [3], they proved that the PoA of gaugeable game converge to 1 a the total demand T d) = K d tend to infinity. However, thi convergence reult only hold for particular uer volume vector equence, due to the equence-pecific nature of gaugeable game. See Subection 3.1 for detail about the convergence reult of gaugeable game. Colini et al. [6] further continued the tudy of [7]. They applied the technique of gaugeability [7] to NCG with polynomial price function. Due to the equence-pecific nature of gaugeability [7], they aumed a uer volume d n) T d n) ) vector equence {d n) } n N with > 0 for each group K. They proved that if all τ a ) are polynomial, then PoAd n) ) = 1, provided that T d n) ) =. Moreover, Colini et al. [6] further extended the tudy of [7] to light traffic, i.e., T d) 0. All the reult of [8], [6] and [7] are derived with the technique of regular 6

7 variation [3]. A different tudy wa done by Wu et al. [28]. They aumed arbitrary uer volume vector equence {d n) } n N and aimed to explore propertie of aymptotic well deigned game, i.e., game in which the PoA tend to 1, a the total volume T d) approache infinity. They propoed the concept of calable game, and proved that all calable game are aymptotically well deigned. They alo proved that gaugeable game are pecial cae of calable game w.r.t. the particular uer volume vector equence aumed by gaugeability. Moreover, they provided example that are calable, but not gaugeable. In addition, Wu et al. [28] made a detailed tudy of the cae that all τ a ) are BPR-function. They proved for thi particular cae that an SO profile i an ɛ-approximate NE profile [19] for a mall ɛ > 0, and the PoA equal 1 + O T d) γ), where γ i the common degree of the BPR-function. Thi prove a conjecture propoed by O Hare et al. [15]. They alo proved for thi particular cae that the cot of both, an NE-profile and an SO-profile, can be aymptotically approximated by Ld) T d) γ, where Ld) i a computable contant that only depend on ditribution d := d /T d) ) of uer among K group, when the total volume T d) i large enough. However, Wu et al. [28] till failed to how that NCG with general polynomial are aymptotically well deigned. In ummary, [8], [6], [7] and [28] definitely how that elfih behavior need not be bad for the cae of a large T d) under certain condition. However, one important quetion ha remained open, namely, whether NCG with arbitrary polynomial price function τ a ) are aymptotically well deigned. Although [6] and [7] have preinary reult toward thi quetion, their reult only partially anwer thi quetion due to the equence-pecific nature of their tudy. Polynomial function are o popular becaue they uually erve a a firt prototype to undertand quantitative relation between variable. Price function τ a ) are ey component of an NCG and model the quantitative relation between reource price and demanded volume. NCG with polynomial price function τ a ) are thu of great importance in practice. The open quetion about the PoA for polynomial thu concern propertie of the elfih uer behavior in uch game for a large uer volume T d) and i thu of great interet to our purpoe of undertanding heavy traffic. 7

8 1.5 Our main reult We will continue the tudy of [28]. However, to better undertand the tate of the art, we will firt dicu the notion of calability [28] and gaugeability [7], and give a detailed decription of the nown reult from [28], [6] and [7], ee Subection 3.1. We then apply the concept of calability and it game temming from [28]. We firt prove that if the it game exit, then an NCG i aymptotically well deigned if and only if it i calable, ee Theorem 1. Thi deepen the nowledge about calability and aymptotically well deigned game. For an even deeper undertanding, we adapt ome algebraic idea to our analyi and conider decompoition of NCG. We prove that the cla of aymptotically well deigned game i actually cloed under direct um, ee Corollary 2. Thi demontrate in a certain ene the extent of the notion of aymptotically deigned game. To obtain a general proof for the convergence of the PoA, we develop a new technique called aymptotic decompoition. Thi technique generalize the idea of direct um, and i deigned for handling general NCG in the it analyi. We are able to demontrate it power by applying it to NCG with arbitrary polynomial price function and NCG with regularly varying price function, ee Theorem 2, Theorem 3 and Theorem 4. With the aymptotic decompoition, we are able to prove that NCG with arbitrary polynomial price function are aymptotically well deigned, ee Theorem 2. Thi completely olve the convergence of the PoA of NCG with polynomial price function, and thu the aforementioned open quetion. Thi reult greatly extend the finding of [28], [8], [7] and [6] for road traffic, and deepen the undertanding that elfihne need not be bad, and might be the bet choice in a bad environment. Moreover, thi reult alo indicate that elfih routing i actually not the main caue of congetion, when the total travel demand T d) i large. In particular, if the total travel demand tay high, then we cannot ignificantly reduce the average travel latency by any road guidance policie. Theorem 2 alo bring ome inight into free maret economic. In maret economic, reource correpond to factor of production, group correpond to et of upplier manufacturing a particular type of product, and reource price τ a ) are the purchaing price of thoe production factor. In a free maret ytem, the price τ a ) of production factor are completely determined by the demanded volume, and are often aumed to be polynomial function. Theorem 2 then how that given the demand of each product 8

9 type, the free maret will autonomouly minimize the average manufacturing cot, when the total number of upplier i large. Aymptotic decompoition alo applie to NCG with price function of other type. To demontrate thi, we alo applied thi technique to NCG with regularly varying price function. The reult how that thee NCG are in general alo aymptotically well deigned, ee Theorem 3. In particular, with thi technique, we are able to remove the equence itation for gaugeable game and generalize the main reult Theorem 4.4 in [7] for gaugeability, ee Theorem 4. Theorem 2, Theorem 3 and Theorem 4 definitely demontrate the power of aymptotic decompoition. They aume an arbitrary uer volume vector equence, and thu the reult hold globally. In particular, together they contitute a general proof that elfihne need not be bad for NCG. Their proof are direct and very elementary without uing any heavy machinery, and only ue ome baic propertie of Nah equilibrium and ytem optimum profile, imple fact about aymptotic notation O ), Ω ), etc, and a imple induction over the group et K. 1.6 The tructure of the paper The remainder of the article i arranged a follow: Section 2 give a detailed decription of the NCG model that we will tudy. Section 3 give a detailed decription on nown reult and then report our reult. Section 4 give a brief ummary of the whole article. To improve readability, we move the elementary but long proof of our reult to the Appendix. 2 The Model In our tudy, we follow the formulation of NCG in [28]. Thi formulation i lightly different from the traditional model commonly ued in the literature, ee, e.g., [13]. Traditionally, a trategy i aumed to be a ubet of reource. In our tudy, we employ a contant ra, ) 0 to refelct the relation between a reource a A and a trategy S. Thi lightly generalize our reult. where: An NCG i repreented by a tuple Γ = K, A, S, ra, )) a A, S, τ a ) a A, d ), 9

10 K i a finite non-empty et of group. We aume, w.l.o.g., that K = {1,..., K}, i.e., there are K group of uer. A i a finite non-empty et of reource that will be demanded by uer engaged in the game. S = K S i a finite non-empty et of available trategie. Herein, each S contain all trategie available to uer in group for each K. We aume that S S =, provided that, for each, K. ra, ) 0 denote the demanded volume of reource a by a uer adopting trategy, for each a A and each S. τ a : [0, + ) [0, + ) denote the price function of reource a for each a A. We aume that each τ a x) i a continuou function that i non-negative and non-decreaing for all x 0, for all a A. d = d ) K i a non-negative uer volume vector, where each component d 0 repreent the volume of uer belonging to group K. In our tudy, we aume further that for each group K and each trategy S, a S ra, ) τ a x) 0. 1) Note that 1) i a reaonable aumption. Otherwie, there would be a group K, whoe uer can conume reource without paying any price. Thi actually conflict with the pirit of congetion game in practice. In an NCG, uer uually want to adopt trategie that minimize their own cot. However, the cot of a uer doe not only depend on hi/her choice, but alo on the choice of other uer, i.e., the cot of a uer i eventually determined by the trategy profile formed by all uer engaged in the game. Herein, a feaible trategy profile f can be repreented by a vector f = f ) S, where: p1) Each component f 0 repreent the total volume of uer adopting trategy, for each trategy S. p2) S f = d, for each group K, which indicate that every uer mut chooe a trategy to follow. 10

11 Conider a feaible trategy profile f = f ) S. The demanded volume or conumed volume) of each reource a A w.r.t profile f, denoted by f a, can be computed a f a = ra, ) f. S Thu, the price of a reource a A w.r.t. profile f equal τ a f a ). Therefore, the cot of a trategy S w.r.t. profile f, denoted by τ f), equal τ f) = a A ra, ) τ a f a ). Then, the average cot of uer w.r.t. profile f equal Cf) := 1 T d) f τ f) = 1 T d) f a τ a f a ), S where T d) = K d denote the total volume of uer in the game. The elfihne of uer will autonomouly lead their choice to eventually form a feaible profile f = f atifying that ) S K, S f > 0 = τ f) τ f) ), 2) i.e., every uer chooe a cheapet trategy he/he could follow w.r.t. profile f. Such profile are called Wardrop equilibria WE, [27]). Under our aumption on the price function τ a ), Wardrop equilibria coincide with the pure Nah equilibria NE). A feaible trategy profile f = f ) S i aid to be at NE, if a A K, S f > 0 = ɛf > ɛ > 0) = τ f) τ f 1+ɛ )) ), 3) where f 1+ɛ = f 1+ɛ ) S i a feaible trategy profile that move ɛ uer from trategy to trategy, i.e., for each trategy S, f if / {, }, f 1+ɛ = f ɛ if =, f + ɛ if =. In the equel, we hall alway put a tilde above a trategy profile, if the trategy profile i an NE profile or, equivalently a Wardrop equilibrium). 11

12 An NE profile f i a macro model for the elfih behavior of uer in practice. Under our aumption on price function τ a ), all NE profile f have the ame average cot, ee e.g. [26] for a proof. Obviouly, thee profile are uer optimal 2), and table 3) to ome extent. Beide NE profile, ytem optimum SO) profile are alo of great interet, for the ae of achieving ocial welfare. Formally, a feaible trategy profile f = f ) S i an SO profile if it olve the following program: min.t. Cf) S f = d, K, f 0, S. In the equel, we hall alway ue a tar in the upercript of a feaible profile to indicate that it i an SO profile. In general, an NE profile need not be a olution to the program 4). The PoA i a popular index to how the extent to which the elfih uer behavior detroy ocial welfare in practice. It i a concept temming from [16], and can be defined a follow PoA := C f) Cf ) = S f τ f) where f i an NE profile, and f i an SO profile. 4) S f τ f ), 5) A mentioned, we will invetigate the it of the PoA when the total volume T d) = K d approache infinity. Therefore, to avoid ambiguity, we hall denote by PoAd) the correponding PoA for uer volume vector d = d ) K in the equel. 3 Selfihne need not be bad: a general dicuion In thi Section, we conider the it of the PoA under our aumption of continuou, non-decreaing and non-negative price function τ a ). In particular, we will emphaize on the polynomial function and regularly varying function that have been recently tudied in [28], [8], [6] and [7]. To better undertand our reult, we firt introduce ome relevant concept and reult from [28] and [7]. 12

13 3.1 Scalability and gaugeability NCG are tatic model for deciion-maing behavior of elfih uer player) in ytem with ited reource. Deigning an NCG uch that the elfih choice of uer autonomouly optimize ocial welfare i in general not eay, ee e.g. [20]. However, uch game exit, ee [28]. Definition 1 See [28]). An NCG Γ i aid to be a well deigned game WDG), if PoAd) = 1 for each given uer volume vector d = d ) K with total uer volume T d) > 0. Obviouly, elfih behavior of uer in a WDG hould be trongly favored, a it lead the underlying ytem into a teady tate with minimum average cot. Reader may refer to [28] for example of WDG. A mentioned, WDG are often too retrictive for deigning them in practice. Neverthele, an important goal in NCG concern how to effectively allocate ited reource to a large volume of uer. Therefore, an alternative choice to deigning a WDG i to deign an NCG that will approximate a WDG when the total uer volume T d) become large. Thi inpire the concept of an aymptotically well deigned game AWDG) [28]. Definition 2 ee [28]). An NCG Γ i aid to be an AWDG, if the PoAd) converge to 1 a T d) approache infinity. For later ue, we alo denote the cla of all AWDG a AWDG. Scalable game introduced by Wu et al. [28] are example of AWDG. Thee game require the exitence of a well deigned it game for each equence {d n) } n N of uer volume vector with T dn) ) = d n) =, K where d n) i the -th component of d n) = d n) ) volume in group for each K and each n N. K and denote the uer Definition 3. Given a equence { d n) = d n) ) K of uer volume vector n N with T d n) ) = K dn) =, an NCG Γ = K, A, S, ra, )) a A, S, τa ) a A, d ) i called a it of the NCG ) Γ = K, A, S, ra, )) a A, S, τ a ) a A, d 13 }

14 w.r.t. the uer volume vector equence {d n) } n N, if there exit an infinite ubequence {n i } i N uch that: L1) For each K, i d n i) T d n i) ) = d, where d [0, 1] i the it volume of group. L2) There exit a equence {g i } i N of poitive caling factor, uch that τ a T d n i ) )x ) = τa x) i g i for all x > 0, where τ a ) i the it price of reource a, for each a A. L3) Each it price function τ a ) i either a continuou and non-decreaing function, or τ a x) for all x > 0, for each a A. And for each group K, L3.1) either group i negligible w.r.t. caling factor {g i } i N, i.e., i S f n i) τ f n i) ) T d n i) ) g i = 0, where each f n i) i an arbitrary feaible trategy profile of Γ w.r.t. uer volume vector d n i) for each i N, L3.2) or there exit a trategy S that i tight w.r.t. caling factor {g i } i N, i.e., τ a x) for x > 0, for each reource a A with ra, ) > 0. L4) Put S := { S : i tight w.r.t. {g i } i N }, K := { K : i not negligible, or S S }. The cot of NE profile of the it game Γ i poitive w.r.t. the it uer volume vector d ) K. Definition 4 See alo [28]). An NCG Γ i called a calable game if, for each uer volume equence {d n) } n N with total volume T d n) ) a n, there i a well deigned game Γ that i a it of Γ w.r.t. the uer volume equence {d n) } n N. 14

15 Condition L3) and L4) of Definition 3 are impoed to guarantee that the cot of NE profile in the it game Γ will be neither unbounded, nor vanihing w.r.t. the caling factor {g i } i N. Therefore, the caling factor {g i } i N hould be carefully choen with reference to the equence {T d n) )} n N of uer volume vector and price function τ a ), o a to fulfill thee condition. Note that it game only conider tight trategie S and nonnegligible group K. Thi i actually reaonable, ince uer will aymptotically adopt only tight trategie w.r.t. both, NE profile and SO profile, ee the proof of Lemma 1 and Theorem 1 in the Appendix. Condition L2) of Definition 3 can be further relaxed. Note that for each reource a A, only thoe x > 0 mae ene that are poible conumed volume of reource a w.r.t. the it Γ. We can therefore require only that the it price function τa x) exit for x I a 0, ), where I a i a non-empty et containing all the poible conumed volume of reource a w.r.t. the it game Γ, for each a A. See [28] for detail. Note that it i poible that there are everal it game for a given uer volume equence {d n) } n N, ee the following example. 2 x+1 O 1 3 x+1 t 1 4 x x 2 +1 O 2 t 2 Figure 2: An NCG with double it Example 1. Conider the NCG Γ hown in Figure 2. The game ha two group OD pair), each of which ha two trategie two parallel path). The price function are lited above the correponding path. Let {d n) = d n) 1, d n) 2 )} n N be a uer volume vector equence uch that d n) 1 = { 0 if n i odd, n if n i even, d n) 2 = { n if n i odd, 0 if n i even, where d n) 1 denote uer volume of the upper group, and d n) 2 denote uer volume of the lower group, for each n N. Obviouly, w.r.t. ubequence 15

16 {2i} i N and caling factor equence {g i = i} i N, the lower group i negligible ince d 2i) 2 0 for all i N. Moreover, it price function 2 2i x) + 1 i 2i = 2x and i 3 2i x) + 1 2i exit and are continuou and non-decreaing for all x > 0. = 3x Furthermore, the NCG game Γ 1 coniting of thee two it price function and the upper group ha poitive average cot for NE profile w.r.t. the it uer volume vector d 1 ), where d 1 = 1. Thu, Γ 1 i a it game of Γ w.r.t. the given uer volume vector equence {d n) } n N. Similarly, conidering the ubequence {2i + 1} i N and caling factor equence {g i = 2i + 1) 2 } i N, we can define another it game Γ 2 coniting of the lower group and two price function 4x, 5x, repectively. Obviouly, thee two it game Γ 1 and Γ 2 are different. However, both of them are it of the given NCG Γ w.r.t. the uer volume equence {d n) } n N. Wu et al. [28] proved that for an NCG Γ and a given uer volume vector {d n) } n N with total volume T d n) ) a n, if Γ i the it of Γ for an infinite ubequence {n i } i N and a caling factor equence {g i } i N, then the average cot of NE profile normalized by the caling factor equence {g i } i N converge to the total cot of NE profile of Γ. Lemma 1. Conider an NCG ) Γ = K, A, S, ra, )) a A, S, τ a ) a A, d, in which all price function τ a ) are non-negative, non-decreaing and continuou. Let {d n) } n N be an arbitrary equence of uer volume vector uch that the total volume T d n) ) = K dn) a n, where each d n) = d n) ) K for each n N. Let f n) = f n) ) be an NE profile of Γ S for the uer volume vector d n), for each n N. If Γ ha a it Γ = K, A, S, ra, )) a A, S, τa ) a A, d ) for the uer volume vector equence {d n) } n N, then there exit an infinite ubequence {n i } i N and a equence {g i } i N of poitive number uch that C f ni) ) i g i = f τ f ), S where f = f ) S i ome NE profile of the it game Γ. 16

17 Proof. See the proof of Theorem 3.2 in [28], or the appendix for an alternative proof. Uing Lemma 1, Wu et al. [28] then proved that all calable game are aymptotically well deigned. In that proof, the condition that the it game i well deigned play a pivotal role, which actually implie that the average cot of NE profile i aymptotically not larger than the average cot of SO profile. We ummarize thi in Lemma 2. Lemma 2. Every calable game i an AWDG. Proof. See the proof of Theorem 3.2 in [28] for detail. Theorem 1 below continue the tudy of Wu et al. [28]. It tate that calable game and AWDG actually coincide when they have a it game. Moreover, we howed in the proof of Theorem 1 that uer will aymptotically adopt only tight trategie w.r.t. SO profile, and Lemma 1 alo applie to SO profile. Therefore, it i reaonable to conider only tight trategie in it game. Theorem 1. Conider an NCG Γ and a uer volume vector {d n) } n N with the total volume T d n) ) a n. If Γ ha a it game Γ w.r.t. the given uer volume vector, and the it game Γ i not well deigned, then Γ i not an AWDG. Proof. See the Appendix. Specific for polynomial price function, Theorem 3.2 from Wu et al. [28] directly yield that NCG with polynomial price function τ a ) of the ame degree are aymptotically well deigned. In that cae, we can tae caling factor g n = T d n) ) γ for each given uer volume vector equence {d n) } n N with T d n) ) a n, where γ 0 i the common degree of all polynomial. Then, the correponding it game i well deigned. To how thi for general polynomial price function τ a ) with different degree, we will ue ome helpful notation from [6]. For each reource a A, let ρ a ) be the degree of polynomial τ a ). Put ρ = max{ρ a : ra, ) > 0, a A} and ρ = min{ρ : S } for each S and K. If ρ = ρ l for all, l K, then the above argument how that the underlying NCG i calable, and therefore aymptotically well deigned. We ummarize thi in Corollary 1. 17

18 Corollary 1. Conider an NCG Γ with polynomial price function τ a ). If ρ = ρ l for all, l K, then Γ i aymptotically well deigned. Proof. Let {d n) } n N be an arbitrary uer volume vector uch that T d n) ) a n. Let ρ = ρ for ome K. Put g n = T d n) ) ρ for each n N. By aumption 1), one can then eaily how that the it of Γ w.r.t. equence {n} n N and the caling factor equence {g n } n N i well deigned. The reult of Wu et al. [28] doe not directly apply if ρ ρ l for ome l, K. The reaon i that, in thi cae, there need not exit a unified it game for all group for ome uer volume vector equence {d n) } n N. We thu need additional argument in thi cae and leave the proof of thi cae to Subection 3.3. To better undertand the current tate of the art, we now introduce ome relevant reult from [6] and [7]. They employed an alternative technical path to prove the convergence of the PoA. They introduced the o-called gaugeable game, which conider only particular equence of uer volume vector. Gaugeability i a concept baed on the notion of regular variation [3]. A non-negative function g ) i aid to be regularly varying, if the it exit for all x > 0. gt x) t gt) = qx) 0, ) 6) Definition 5 See alo [7]). An NCG Γ i aid to be gaugeable for a uer volume vector {d n) }, if there exit a regularly varying function g ) uch that: G1) The it exit for all reource a A. τ a x) gx) =: q a [0, ] G2) For each group K, there exit a trategy S uch that q a <, for all reource a A with ra, ) > 0. G3) The lower it K tight n) d T d n) ) > 0, 7) 18

19 where K tight i the et of all tight group, and a group i aid to be tight in uch a cae if 0 < min S max{q a : ra, ) > 0, a A} <. With the technique of regular variation [3], Colini et al. [7] proved that for each NCG Γ, if Γ i gaugeable w.r.t. a uer volume vector {d n) } n N, then PoAd n) ) 1 a n, ee Theorem 4.4 in [7]. Here, we recall that PoAd n) ) denote the price of anarchy w.r.t. uer volume vector d n), for each n N. In fact, if Γ i gaugeable w.r.t. a uer volume vector {d n) } n N, then Γ ha a well deigned it w.r.t. that uer volume vector equence. Let g ) be the required regularly varying function in Definition 5. By G1), the it price function ) τa τ a T d n) )x ) x = = q a x ρ g n exit, where the caling factor g n = g T d n) ) ), and ρ 0 i the regular variation index of g in Karamata Characterization Theorem [3]. Moreover, by G2) and G3), one can eaily chec that the it game coniting of group K = K and price function τ a i well deigned. See Wu et al. [28] for detail. Therefore, gaugeable game are pecial cae of calable game. To explicitly how thi, we define the equential counterpart of calable game in a natural way. Definition 6. Conider an NCG Γ and ome uer volume vector {d n) } n N with T d n) ) a n. Γ i aid to be calable w.r.t. the equence {d n) } n N, if for each infinite ubequence {n i } i N, Γ ha a well deigned it w.r.t. the ubequence {d n i) } i N. Obviouly, an NCG Γ i calable if and only if Γ i calable w.r.t. each uer volume equence {d n) } n N with T d n) ) =. However, if Γ i only calable w.r.t. ome uer volume equence {d n) } n N, then Γ need not to be globally calable. Neverthele, Lemma 3 below tate that thi retricted notion of calability already generalize the gaugeability of [7]. Lemma 3. Conider an NCG Γ and ome uer volume vector {d n) } n N with T d n) ) a n. If Γ i gaugeable w.r.t. the equence {d n) } n N, then Γ i alo calable w.r.t. that uer volume vector equence. Proof. See the proof of Corollary 3.1 in Wu et al. [28] for detail. 19

20 The difference between calable game and gaugeable game are therefore obviou. Scalable game conider arbitrary uer volume vector equence, while gaugeable game conider particular uer volume vector equence atifying 7). Thu, the convergence reult of the PoA for calable game i global, while that for gaugeable game hold only locally. Actually, calablity i more general than gaugeability even for a pecific uer volume vector equence {d n) } n N. Gaugeability require that there exit a uniform regularly varying function g ) for the whole equence that fulfill condition G1)-G3). A hown above, thi reult in the ame well deigned it game for every ubequence of {d n) } n N. However, calability allow different ubequence of {d n) } n N to have different well deigned it game. The NCG Γ in Example 1 ha two it game w.r.t. the given uer volume vector equence, and both of them are well deigned, ee Wu et al. [28] for detail. We ummarize thi in Lemma 4. Lemma 4. There exit an NCG Γ and a uer volume vector equence {d n) } n N with T d n) ) a n, uch that Γ i calable w.r.t. {d n) } n N, but not gaugeable w.r.t. {d n) } n N. Proof. See the proof of Theorem 3.4 in Wu et al. [28], or Example 1. Now, let u return to the dicuion of NCG with arbitrary polynomial price function τ a ). By conidering a particular uer volume vector equence {d n) } n N, Colini et al. [6] proved that PoAd n) ) 1, a n, ee alo Corollary 4.7 in [7]. They aumed that for each K, d n) T d n) ) > 0. 8) Obviouly, 8) fulfill 7). Let gx) = T d n) ) ρ, where ρ = max{ρ : K}. They actually proved that if {d n) } n N atifie 8), then the underlying NCG i gaugeable w.r.t. {d n) } n N and regularly varying function g ). We ummarize thi in Lemma 5. Lemma 5. Conider an NCG Γ with polynomial price function τ a ), and a uer volume vector equence {d n) } n N uch that T d n) ) a n, and atifie 8). Then, Γ i calable w.r.t. {d n) } n N, i.e., PoAd n) ) 1 a n. Proof. The proof follow immediately from Lemma 3. 20

21 Although Lemma 5 how an inpiring reult for general polynomial price function, we can not directly conclude that NCG with polynomial price function are aymptotically well deigned, due to the equence-pecific nature of Lemma 5. To how that NCG with polynomial price function are aymptotically well deigned, we till need a more ophiticated analyi. Motivated by Lemma 3, Lemma 4, and Example 1, we will ue the idea of calability for a general proof. 3.2 AWDG i cloed under direct um A direct application of calability doe not lead to a general proof. To ee thi, conider again Example 1, but now with uer volume vector equence d n) = d n) 1 = n 2, d n) 2 = n). In thi cae, the two group are mutually nonnegligible, and we cannot find a uitable caling factor equence that reult in a well deigned it game. However, a cloer inpection how that either of the two group ha it own well deigned it game w.r.t. the given uer volume equence. Thi inpire u to conider the two group eparately. Definition 7. An NCG Γ i aid to have mutually dijoint group MDG), if 1 {x:x>0} ra, ) ) 1, 9) K for each a A, for each K-dimenional trategy vector 1,..., K ) K S, where 1 {x:x>0} ) i the indicator function of et {x : x > 0}. Condition 9) in Definition 7 implie that uer from different group cannot hare reource. Therefore, uer from different group of an NCG with MDG will not affect each other when they determine trategie to follow. Thi mean that each group in an NCG with MDG form an independent ubgame. Let ) Γ = K, A, S, ra, )) a A, S, τ a ) a A, d be an NCG with MDG. For each group K, we denote by ) Γ = {}, A, S, ra, )) a A, S, τ a ) a A, d ) the -marginal of Γ, and denote by f = f ) S the -marginal of a feaible trategy f = f ) S. 21

22 Lemma 6. Conider an NCG Γ with MDG. Then, f = f ) S i an SO profile of Γ if and only if the -th marginal profile f = ) f i an SO S profile of the -marginal game Γ, for each K. Thi hold imilarly for NE profile. Proof. Trivial. With Lemma 6 and by applying calability to each of the mutually independent marginal, we can eaily derive that NCG with mutually dijoint and calable marginal are aymptotically well deigned. Lemma 7. Conider an NCG Γ with MDG. If all the marginal are calable, then Γ i aymptotically well deigned. Proof. Thi i eay by oberving the trivial fact that C f) PoAd) = Cf ) = K S f τ f) S f τ f ), K and that each marginal i calable, and thu the marginal PoA S f τ f) S f τ f ), tend to 1, a d, for each K. Herein we recall aumption 1) that there are no free trategie, and that if S, for each K. τ f) = τ f ) and τ f ) = τ f ), Combining Corollary 1 and Lemma 7, we obtain immediately that NCG with MDG and polynomial price function τ a ) are aymptoticall well deigned. A direct extenion of Lemma 7 conider the direct um of aymptotically well deigned game. Let ) Γ l = K l, A l, S l, ra, )) a Al, S l, τ a ) a Al, dl) be an NCG, for l = 1,..., m, uch that A 1,..., A m are mutually dijoint. Then, we call the game m m m K l, A l, S l, ra, )) a m l=1 A l, m l=1 S, τ l a) a m l=1 A, m ) dl) l l=1 l=1 l=1 22 l=1

23 the direct um of Γ 1,..., Γ m, denoted by m l=1 Γ l, where m l=1 dl) mean the concatenation of vector d1),..., dm). Obviouly, direct um of aymptotically well deigned game are again aymptotically well deigned. Corollary 2. The cla AWDG i cloed under direct um. Proof. Trivial. Corollary 2 ugget a poible approach to chec whether an NCG i aymptotically well deigned. One can try to decompoe the underlying NCG into a direct um of everal independent marginal, and then chec the calability of each marginal. Here, we allow compound marginal, i.e., each marginal can contain more than one group. The independence between them then mean that uer from different marginal do not affect each other when they determine trategie to follow. However, in general, it could be difficult to find uch a decompoition, ince different group might compete for the ame reource. The above dicuion ha already hown that if colliion are heavy or light, then it i eay to chec whether the underlying NCG i aymptotically well deigned. In fact, if all group heavily collide on reource, i.e., every reource can be ued by all group, then the game i not decompoable and Lemma 2 applie, ee e.g. Corollary 1. On the other hand, if group do not collide on reource, i.e., if they can be partitioned into everal mutually dijoint clae w.r.t. the ue of reource, then the game i decompoable and Corollary 2 may apply. The above two cae are, omehow, regular. Below, we conider the cae of irregular colliion, which might be the general cae in practice. 3.3 Aymptotic decompoition: a general proof for polynomial price function In general, it may be difficult to directly apply the idea of calability, ince there need not exit a unified well deigned game for all group for ome uer volume vector equence. Moreover, it may alo be difficult to directly decompoe the game, due to irregular colliion of group on reource. If thi i the cae, then one may conider an aymptotic decompoition. The idea i imilar to direct um. However, one need to additionally deal with the problem caued by the irregular colliion. An aymptotic decompoition i baed on a uitable partition of the group et K. Conider an arbitrary equence {d n) } n N of uer volume vector uch 23

24 that T d n) ) a n. The decompoition aim to eventually partition K into K 0,..., K t, for ome integer t 0, uch that K = tm=0 K m, and n) f τ f n) ) m u=0 Ku S m u=0 Ku S f n) τ f n) ) = 1, 10) for each m = 0,..., t, where f n), f n) are SO and NE profile w.r.t. d n), repectively, for each n N. Obviouly, if uch partition can indeed be contructed, then the underlying NCG i well deigned, due to the arbitrary choice of {d n) } n N. One can try to contruct the partition inductively. In the beginning of each inductive tep l = 0,..., t, we aume that we have already contructed clae K 0,..., K l 1, uch that K 0 K,..., K l 1 K, and 10) hold for 0 tep m = l 1, where we employ the convention that K 1 =, = 1, 0 and K 0, K 1 mean. The objective of tep l then i to contuct cla K l K\ l 1 u=0 K u uch that 10) hold again for m = l. To contruct K l, one can inpect the remaining group more cloely, and pic thoe group K\ l 1 u=0 K u that have a non-vanihing it proportion in the remaining total uer volume T l d n) ) := T d n) ) l 1 u=0 S dn), ince thee group are mot ignificant in the it. To how 10) for m = l, one need to argue that thee group are either aymptotically independent of the group that have been conidered before, or negligible compared to them. To that end, one need to uitably etimate the cot of uer w.r.t. NE profile f n) and SO profile f n), repectively. By comparing the cot of uer from group K l with thoe from group l 1 u=0 K u, one can then learn whether group K l are negligible. If they are negligible, then 10) hold trivially for m = l. Otherwie, group K l will be aymptotically independent of group l 1 u=0 K u, ince the cot of uer from group l 1 u=0 K u will be negligible compared with the cot of uer from group K l. If thi i the cae, one can then chec the calability of group from K l under the condition that uer from group l 1 u=0 K u adopt trategie that they ued in NE profile f n) and SO profile f n), repectively. Moreover, if thee group are calable, then 10) follow immediately for m = l. Thi procedure can tactically avoid the impact of poible irregular colliion in the it analyi by comparing the cot of uer from different clae K l. If the above partition can be contructed, then the underlying game decompoe into everal aymptotically independent ubgame correponding to the partition K 0,..., K t. Although thee ubgame hare re- 24

25 ource et A, they are aymptotically independent, ince the choice of uer from one ubgame will be aymptotically independent of thoe from other ubgame. Aymptotic decompoition can be uccefully applied to NCG with arbitrary polynomial price function, which directly implie that NCG with arbitrary polynomial price function are aymptotically well deigned. Theorem 2 ummarize thi reult. We move the detailed decompoition procedure to the Appendix. Theorem 2. Conider an NCG Γ with polynomial price function τ a ) uch that each τ a x) i non-negative and non-decreaing for all x 0 for a A. Then, Γ i aymptotically decompoable, and thu aymptotically well deigned. Proof. See the appendix. Different from the direct um in Subection 3.2, an SO profile need not be locally a ytem optimum w.r.t. ome marginal correponding to the partition in the aymptotic decompoition. Thi introduce extra difficultie in the application of calability. The proof of Theorem 2 overcome them by conidering SO profile a NE profile w.r.t. price function c a x) := xτ ax) + τ a x), where each τ a ) i the derivative function of τ a ). Note that NE profile are till at pure) Nah equilibrium w.r.t. each marginal under the condition that uer from other marginal adhere to the trategie they ued in correponding NE profile. The proof of Theorem 2 i baed on three baic propertie of polynomial function. The firt i that polynomial function can be aymptotically orted according to their degree, which form a bae for the cot comparion and the contruction of caling factor at each inductive tep. The econd i that the price function c a x) = xτ ax) + τ a x) are comparable with the price function τ a x), i.e., x c ax) τ ax) = q a for ome contant q a 0, ), when all τ a ) are polynomial. Thi play a pivotal role when we chec calability for marginal in the inductive tep. The lat properly i the relatively clear tructure of polynomial function, from which we can obtain uitable caling factor g l) n at each inductive tep l. The proof of Theorem 2 i very elementary. It doe not involve any advanced technique, but only mathematical induction, baic calculu, and a very crude raning of non-negative function. Therefore, it hould be widely readable. 25

26 Theorem 2 fully ettle the convergence of the PoA for arbitrary polynomial price function. Thi greatly extend the partial reult from [6], [7] and [28] for polynomial price function. Due to the popularity of polynomial function, Theorem 2 may apply to a more general context other than road traffic, e.g. the enario of free maret mentioned in the Introduction. 3.4 A further extenion: a general proof for regularly varying price function The idea of aymptotic decompoition may apply alo to NCG with price function of other type. Polynomial function are pecial cae of regularly varying function. Thi ubection aim to apply the aymptotic decompoition to thi more general notion. By Karamata Characterization Theorem [3], a regularly varying function τ ) can be written a τx) = x ρ Qx), where ρ R i called the regular variation index of τ ) and Qx) i a lowly varying function, i.e., for each x > 0, Qtx) t Qt) = 1. The cla of regularly varying function i very extenive and include many popular analytic function, e.g., all affine function, polynomial, logarithm, and other. Although regularly varying function are more extenive than polynomial function, they actually have imilar propertie. Lemma 8 ummarize thee propertie. Lemma 8. Conider a regularly varying function τ ) with index ρ 0. Then, the following tatement hold. a) For each ɛ > 0, x τx) = 0, and xρ+ɛ x b) For each non-negative function g ), if gx) x τx) = q 0, ) τx) =. xρ ɛ for ome contant q, then g ) i alo regularly varying with index ρ. 26

27 c) For each regularly varying function g ) with index ρ, the qoutient gx) τx) i again regularly varying, but with index ρ ρ. Therefore, the qoutient of two lowly varying non-zero function i again lowly varying. Proof. See the Appendix, or [3]. Lemma 8 a) and Lemma 8 c) indicate a partial ordering on the cla of regularly varying function, i.e., regularly varying function of different indice can be orted according to their indice. However, two regularly varying function of the ame index are generally incomparable. Therefore, we cannot directly reue the imple ordering of polynomial function when we apply the aymptotic decompoition to NCG with regularly varying function. Lemma 8 b) will be implicitly ued in our dicuion. It guarantee that the auxiliary price function c a x) = xτ ax) + τ a x) are again regularly varying and have the ame indice a the price function τ a x). Due to the generality of regularly varying function, we cannot have a uniform argument. To implify the dicuion, we hall conider regularly varying function with particular propertie in thi Subection, e.g., convex and differentiable regularly varying function. Lemma 9. Conider a regularly varying function τ ) that i non-decreaing, non-negative, convex and differentiable on [0, ). Then, the regular variation index of ρ i non-negative, and Proof. See the Appendix. x τ x) x τx) = ρ 0. 11) Lemma 8 b) and Lemma 9 yield immediately that the auxiliary price function c a x) = xτ ax) + τ a x) are again regularly varying, provided that the price function τ a x) are regularly varying and convex. Moreover, the marginal game in an aymptotic decompoition will be aymptotically well deigned in thi cae, ince 11) hold. We ummarize thi reult in Theorem 3. 27