The Effectiveness of Subsidies and Tolls in Congestion Games
11 The Effectiveness of Subsidies and Tolls inCongestion Games
Bryce L. Ferguson, Philip N. Brown, and Jason R. Marden
Abstract —Are rewards or penalties more effective in influ-encing user behavior? This work compares the effectiveness ofsubsidies and tolls in incentivizing user behavior in congestiongames. The predominantly studied method of influencing userbehavior in network routing problems is to institute taxes whichalter users’ observed costs in a manner that causes their self-interested choices to more closely align with a system-levelobjective. Another conceivable method to accomplish the samegoal is to subsidize the users’ actions that are preferable froma system-level perspective. We show that, when users behavesimilarly and predictably, subsidies offer superior performanceguarantees to tolls under similar budgetary constraints; however,in the presence of unknown player heterogeneity, subsidies failto offer the same robustness as tolls.
I. I
NTRODUCTION
In systems governed by a collective of multiple decisionmaking users, system performance is often dictated by thechoices those users make. Though each user may make deci-sions rationally, the emergent behavior observed in the systemneed not align with the objective of the system designer. Thisphenomenon appears in many engineering settings includingdistributed control [2], resource allocation problems [3], elec-tric power grids [4], and transportation networks [5], as wellas many logistical problem settings such as marketing [6] andsupply-chain management [7]. A prominent metric to quantifythis emergent inefficiency is the price of anarchy , defined asthe worst-case ratio between the social welfare experiencedwhen users make self interested decisions and the optimalsocial welfare [8], [9].A promising method of mitigating this inefficiency is byintroducing incentives to the system’s users, influencing theirdecisions to more closely align with the system optimal [10].One example of such incentives is to levy taxes , elicitingmonetary fees from users will affect their preferences overthe available actions (e.g., tolls in transportation) [11]–[13].Such taxes have been shown to be effective in reducing systeminefficiency as measured by the price of anarchy ratio [14]–[17]. Another method to influence user behavior is to subsidize the actions that are preferable from a system level perspective.Subsidies have been studied as a tool to influence users intransportation [18], supply chains [19], congestion [20], and
This research was supported by ONR grant { blferguson,jrmarden } @ece.ucsb.edu .P. N. Brown is with the Department of Computer Science, University ofColorado at Colorado Springs, [email protected] The conference version of this paper appeared in [1] emissions [21]. Though subsidies require the system operatorto pay its users, it is possible that the savings obtained fromefficient use of the infrastructure outweigh the cost incurredfrom the implemented incentives [22], [23]; additionally, onecould consider implementing subsidies as rebates to a fixed,opt-in fee, to prevent a loss of revenue for the system operator.Though the use of subsidies is feasible in theory and inimplementation, this method has been studied significantly lessthan the tax equivalent; the relative performance of each is thusunknown.In this paper, we seek to understand the relative performanceof subsidies and taxes in influencing user behavior in socio-technical systems. Specifically, we consider a network routingproblem in which users must traverse a network with con-gestible edges with delays that grow as a function of the localmass of users. Finding a route for each user that minimizesthe total latency in the system is straightforward if the systemdesigner has full control in directing the users. However, whenusers select their own routes, the resulting network flow neednot be optimal [24]. Modeling the selfish routing problem as acongestion game, we adopt the
Nash flow as a solution conceptof the emergent behavior in the system. From the users’ selfishrouting, the price of anarchy may be large [25]. To alleviatethis emergent inefficiency, we introduce incentives to the users’which alter their observed costs and preferences. The objectiveof such incentives is to shape the users’ preferences so theperformance of the resulting Nash flow will improve.A well studied method of incentivizing users in congestiongames is to tax the users, i.e., introducing tolls to links inthe network [11]–[17], [26], [27]. In each of these referencedworks, the price of anarchy is used to measure the effectivenessof a tolling scheme. Indeed in the most elementary settings,tolls exist that influence users to self route in line with the sys-tem optimum [24]. However, when more nuance is introducedin the form of player heterogeneity (i.e., players differingin their response to incentives), the task of designing tollsbecomes more involved. When the toll designer possesses suf-ficient knowledge of the network structure and user population,they may still compute and implement tolls which incentivizeoptimal routing [17]. However, in the case where the systemdesigner has some uncertainty in the network parameters orbehavior of the user population, it may not be possible todesign tolls that give optimal system performance; thus, tollsare often designed to minimize inefficiency measured by theprice of anarchy ratio [13], [28], [29], and again, encouragingresults exist.Though the study of tolling in congestion games is exten-sive, there are few results regarding subsidies as incentives a r X i v : . [ c s . G T ] F e b in this context, especially in the presence of uncertain userheterogeneity. In [30], the authors investigate budget-balancedtolls in which the sum of all monetary transactions is zero, butthe authors only consider homogeneous users. The authors of[31] give the first formal analysis of subsidies in congestiongames and provide an algorithm that computes optimal rebateswhen users are homogeneous and the network structure isknown. The authors of [32] consider more general incentives,but in an evolutionary setting. From a system designer’sperspective, subsidies may be a feasible method of influencinguser behavior; the performance guarantees of subsidies is thusof interest as well as how this performance compares to taxincentives.Though there is a clear disparity in the breadth of resultsin the literature on tolls and subsidies, we bridge this gap byproving fundamental relationships between the performanceand robustness of subsidies and tolls. Namely, subsidies of-fer better performance guarantees than tolls under budgetaryconstraints but are inherently less robust to user heterogeneity.The manuscript is outlined as follows: Section III: Performance of Incentives.
In Theorem 1, itis shown in the nominal setting, where users behavesimilarly and predictably, that subsidies give better per-formance guarantees under similar budgetary constraints.
Section IV: Incentives with Heterogeneity.
In Theorem 2 itis shown that tolls can effectively mitigate the negativeeffects of player heterogeneity while in Theorem 3 it isshown subsidies cannot.
Section V: Robustness of Incentives.
It is shown that tollsare more robust to uncertainty in the user populationthan subsidies. In the presence of a budgetary constraint,Theorem 4 shows that uncertainty degrades subsidy per-formance more rapidly than it degrades toll performance.
Section VI: Trade-off in Performance and Robustness.
Given the contrast in the nominal performance of subsi-dies and the robustness of tolls to user heterogeneity, thisfundamental relationship is analyzed between the two inparallel-affine congestion games by finding the level ofuncertainty at which the robustness of tolls gives superiorperformance guarantees than subsidies.In addition to finding general performance and robustnessrelationships between subsidies and tolls, we additionally findexplicit price of anarchy bounds for optimal tolls and subsidiesin several classes of congestion games to show that the differ-ences in performance can be significant. We introduce toolsto construct optimal incentives and corresponding performanceguarantees. II. P
RELIMINARIES
A. System Model
Consider a directed graph ( V, E ) with vertex set V , edgeset E ⊆ ( V × V ) , and k origin-destination pairs ( o i , d i ) .Denote by P i the set of all simple paths connecting origin o i to destination d i . Further, let P = ∪ ki =1 P i denote the setof all paths in the graph. A flow on the graph is a vector f ∈ R |P|≥ that expresses the mass of traffic utilizing each path.The mass of traffic on an edge e ∈ E is thus f e = (cid:80) P : e ∈ P f P , e e e e e v v v v e (cid:96) ( f ) = 4 f e (cid:96) ( f ) = 1 / e (cid:96) ( f ) = 1 / e (cid:96) ( f ) = 2 f e (cid:96) ( f ) = 1 / Figure 1:
An example network routing problem G with two origin-destination pairs: ( o , d ) = ( v , v ) with r = 1 / , and ( o , d ) =( v , v ) with r = 1 / . and we say f = { f e } e ∈ E . A flow f is feasible if it satisfies (cid:80) P ∈P i f P = r i for each source-destination pair, where r i isthe mass of traffic traveling from origin o i to destination d i .Each edge e ∈ E in the network is endowed with a non-negative, non-decreasing latency function (cid:96) e : R ≥ → R ≥ that maps the mass of traffic on an edge to the delay userson that edge observe. The system cost of a flow f is the totallatency , L ( f ) = (cid:88) e ∈ E f e · (cid:96) e ( f e ) . (1)A routing problem is specified by the tuple G =( V, E, { (cid:96) e } e ∈ E , { r i , ( o i , d i ) } ki =1 ) as illustrated in Fig. 1, andwe let F ( G ) denote the set of all feasible flows. We definethe optimal flow f opt as one that minimizes the total latency,i.e., f opt ∈ arg min f ∈F ( G ) L ( f ) . (2)We denote a family of routing problems by G . A family ofrouting problems is any set of routing problems, often spec-ified by a specific network topology (e.g., parallel networks)and/or edge latency function types (e.g., polynomial latencyfunctions) but can also be a singleton. B. Incentives
In this paper, we consider the problem of selfish routing,where each user in the system chooses a path as to minimizetheir own observed delay. Let N i be the set of users travelingfrom origin o i to destination d i . Each non-atomic user x ∈ N i is thus free to choose between paths P ∈ P i . Let each N i bea closed interval with Lebesgue measure µ ( N i ) = r i that isdisjoint from each other set of users, i.e., N i ∩ N j = ∅ ∀ i, j ∈{ , . . . , k } , i (cid:54) = j . The full set of agents is thus N = ∪ ki =1 N i whose mass is µ ( N ) = (cid:80) ki =1 r i .It is well known that selfish routing can lead to sub-optimal system performance [25]. It is therefore up to a systemdesigner to select a set of incentive functions τ e : R ≥ → R ∀ e ∈ E to influence the behavior of the users in the systemto more closely align with the system optimal flow. Theseincentives can be regarded as monetary transfers with the usersdependent on the paths they choose.A user x ∈ N i traveling on a path P x ∈ P i observes cost J x ( P x , f ) = (cid:88) e ∈ P x (cid:96) e ( f e ) + τ e ( f e ) . (3) A flow f is a Nash flow if J x ( P x , f ) ∈ arg min P ∈P i (cid:40)(cid:88) e ∈ P (cid:96) e ( f e ) + τ e ( f e ) (cid:41) ∀ x ∈ N i , i ∈ { , . . . , k } . (4)A game is therefore characterized by a routing problem G and a set of incentive functions { τ e } e ∈ E , denoted by the tuple ( G, { τ e } e ∈ E ) . It is shown in [33] that a Nash flow exists ina congestion game of this form if the latency and incentivefunctions are Lebesgue-integrable. C. Incentive Mechanisms & Performance Metrics
To determine the manner in which incentive functions areapplied to edges, we investigate incentive mechanisms . Toformalize this notion, let L ( G ) := { ( (cid:96) e , e, G ) } e ∈ E ( G ) be the set of links or edges in the routing problem G . Anelement in L ( G ) is a tuple of the latency function (cid:96) e , edgeindex e , and routing problem G for each link in the edge set E ( G ) . Further, for a family of problems, we denote L ( G ) = ∪ G ∈G L ( G ) as the set of links that occur in the family of games G .For each edge e in the routing problem G with la-tency function (cid:96) e , an incentive mechanism T assigns anincentive T ( (cid:96) e ; e, G ) , i.e. τ e ( f e ) = T ( (cid:96) e ; e, G )[ f e ] , where T ( (cid:96) e ; e, G )[ f e ] is the incentive evaluated at f e . This mappingis denoted by T : L ( G ) → T where T is some set of allowableincentive functions. For brevity, an incentive mechanism willbe written simply as T ( (cid:96) e ) , but it is assumed that, unlessotherwise stated, the incentive designer has knowledge ofthe exact edge and full network structure when assigning anincentive T ( (cid:96) e ) ; these are termed network-aware incentivemechanisms [14], [17], and are the focus of Theorem 1 andTheorem 4.In the case where the incentive mechanism must be designedfor a family of routing problems and without knowledge ofthe full network structure, we add the implied constraint thattwo edges with the same latency function are indistinguishableand must have the same assigned incentive; we highlight suchcases by terming the mechanism network-agnostic . The useof network agnostic incentive mechanisms has been studiedin [16], [29] and are useful for their robustness in settingswith frequent changes to the system structure (i.e., commerce,supply-chain-management, and even traffic when consideringaccidents and emergencies), where partial changes to thenetwork structure or edge latencies need not require globalredesign of the incentive mechanism. One such incentive thatfits this framework is the classic Pigouvian or marginal costtax, T mc ( (cid:96) )[ f ] = f · ddf (cid:96) ( f ) , (5)which is known to incentivize users to route optimally inmany classes of congestion games [24]. This is only truehowever, when there is no bound on the incentive and usersare homogeneous [29]. We use the price of anarchy to evaluate the performance ofa taxation mechanism, defined as the worst case ratio betweentotal latency in a Nash flow and an optimal flow, exemplifiedin Fig. 1. Let L nf ( G, T ) be the highest total latency in a Nashflow of the game ( G, T ( L ( G ))) . Additionally, let L opt ( G ) bethe total latency under the optimal flow f opt . The inefficiencycan be characterized by PoA(
G, T ) = L nf ( G, T ) L opt ( G ) . (6)We extend this definition to a family of instances PoA( G , T ) = sup G ∈G L nf ( G, T ) L opt ( G ) , (7)where T is used in each routing problem. The price ofanarchy is now the worst case inefficiency over all such routingproblems while using incentive mechanism T . The objectiveof such incentive mechanisms is to minimize this worst caseinefficiency, thus the optimal incentive mechanism is definedas, T opt ∈ arg inf T : L ( G ) →T PoA( G , T ) , (8)such that it minimizes the price of anarchy for a class ofrouting problems G . D. Tolls & Subsidies
We differentiate between two forms of incentives, tolls τ + e : R ≥ → R ≥ and subsidies τ − e : R ≥ → R ≤ . With tolls, theplayer’s observed cost is strictly increased, i.e., the systemdesigner levies taxes for the users to pay depending on theirchoice of edges. With subsidies, the players cost is strictlyreduced, i.e., the system designer offers some payments tousers for their choice of action. The main focus of this work isto assess which is more effective in influencing user behavior,tolls or subsidies.A tolling mechanism is one which only assigns tollingfunctions, defined as T + : L ( G ) → T + where T + is the setof all non-negative, integrable functions on R + . An optimaltolling mechanism is one that minimizes the price of anarchyratio, i.e., T opt+ ∈ arg inf T : L ( G ) →T + PoA( G , T ) . (9)An optimal subsidy mechanism is defined analogously withnon-positive subsidy functions. In the following sections, wecompare the price of anarchy ratio associated with the optimaltoll and optimal subsidy.The following example, illustrated in Fig. 1, highlights thenotation and the difference between tolls and subsidies. Example 1.
Consider the network G in Fig. 1 with two origindestination pairs: ( o , d ) = ( v , v ) with r = 1 / , and ( o , d ) = ( v , v ) with r = 1 / . The optimal flow in G ,that minimizes (1), is f opt ≈ { . , . , . , . , . } with a total latency of L ( f opt ) ≈ . . With no tolling,the Nash flow is f nf = { / , , , / , } with total latency L ( f nf ) = 1 producing a price of anarchy of PoA( G, ∅ ) ≈ . . Under a scaled marginal-cost toll, the cost incurred bya user for utilizing edge e is (cid:96) e ( f e )+ f e · ddf e (cid:96) e ( f e ) and the Nash flow becomes the same as f opt , leading to a price of anarchyof PoA(
G, T mc ) = 1 . Similarly, under a subsidy mechanism T − ( (cid:96) e ) = f e · ddf e (cid:96) e ( f e ) − (cid:96) e , the Nash flow is again theoptimal, and PoA(
G, T − ) = 1 .This example highlights that subsidies and tolls are botheffective at reducing the inefficiencies associated with selfishrouting. In this work, we study how this performance changesunder budgetary constraints and user price-heterogeneity. E. Summary of Our Contributions
We start by addressing the nominal homogeneous setting, inwhich all users react to incentives identically. In Theorem 1, inany congestion game, we show that under a similar budgetaryconstraint, the optimal subsidy offers better performance thanthe optimal toll; the magnitude of this difference is exemplifiedin Proposition 1.1 by deriving explicit price of anarchy boundsfor optimal tolls and subsidies in affine congestion games.Next, we look at the efficacy of each incentive in mitigatingthe effect of user heterogeneity as the budgetary constraintis lifted. In Theorem 2, we show that tolls can effectivelyeliminate the effect of user heterogeneity when the boundon incentives is lifted. However, in Theorem 3, it is shownthat even in congestion games with convex, non-decreasing,continuously-differentiable latency functions, it is impossiblefor subsidies to mitigate the effect of user heterogeneity, evenwith the ability to give arbitrarily large payments.When budgetary constraints do exist and users are hetero-geneous in their response to incentives, we show in Theo-rem 4 that, for tolls and subsidies bounded to give similarperformance in the homogeneous setting, the performance ofsubsides is worse than tolls when users become heterogeneous,i.e., the performance of subsidies degrades more significantlyfrom player heterogeneity than tolls. This is exemplified inProposition 4.1, giving price of anarchy bounds for robustincentives in affine congestion games.Finally, because subsides offer better performance undersimilar budgetary constraints in the homogeneous setting andtolls offer better robustness in the face of user heterogeneity,we investigate what level of user heterogeneity allows tollsto outperform similarly bounded subsides. In Theorem 5, arelationship between the incentive bound and level of het-erogeneity is derived in a class of parrallel-affine congestiongames that leads to similar performance guarantees of theoptimal toll and subsidy.III. B
OUNDED I NCENTIVES
We first look at the case where users are homogeneousin their response to incentives. This setting has been thefocus of study for many incentive related works [14], [16],[24], [26], [27]. For these reasons, we start by comparingthe effectiveness of subsidies and tolls in this setting whenadditional budgetary constraints are added. Subsidies and tollsboth serve as mechanisms for influencing user behavior andcan be implemented by similar methods. The act of applyingconstraints on either is of little difference to the system de-signer, however the reasoning for these constraints may differ. For instance, budgetary constraints on subsidies can serve tolimit the monetary obligation of the system operator, whilebounding tolls can prevent scenarios where users may avoidusing the network entirely. Though any specific budgetaryconstraint on either incentive is heavily influenced by theproblem setting, here we seek to understand more generallyhow limits on the magnitude of incentives comparatively affectsubsidies and tolls.To explore this, we introduce bounded tolls and subsidies.A bounded toll satisfies τ + e ( f e ) ∈ [0 , β · (cid:96) e ( f e )] for f e ≥ andeach e ∈ E , where β is a bounding factor. A bounded tollingmechanism is denoted by T + ( (cid:96) e ; β ) . Similarly, a boundedsubsidy satisfies τ − e ( f e ) ∈ [ − β · (cid:96) e ( f e ) , for f e ≥ andeach e ∈ E , and a bounded subsidy mechanism is denotedby T − ( (cid:96) e ; β ) . This form of bounded incentive functions re-sembles the bounded path deviations studied in [9]. Thoughmany forms of bounding constraint can be considered, thisform is chosen as it can be applied to network-aware and-network agnostic incentive mechanisms, captures the idea thatlarger delays can be incentivized more significantly, and avoidstrivialities caused by arbitrarily large delays. Additionally,these constraints can be represented as the total incentive in arouting problem being within a multiplicative factor β of thetotal latency, i.e., (cid:80) e ∈ E f e | τ e ( f e ) | ≤ β L ( f ) .For some bounding factor β , let T + β denote the set of taxa-tion mechanisms appropriately bounded by β . More formally, T + β = { T | T : L ( G ) → T + ( β ) } , where T + ( β ) = { τ + e ∈ T + | τ + e ( f e ) ∈ [0 , β · (cid:96) e ( f e )] ∀ f e ≥ } is the set of all tolling functions bounded by β . To compare theefficacy of bounded tolls and subsidies, we define an optimalbounded tolling mechanism as T opt+ ( β ) ∈ arg inf T + ∈ T + β PoA( G , T + ) . (10)The optimal bounded subsidy mechanism T opt − ( β ) is definedanalogously. For brevity, bounded mechanisms are often writ-ten T ( β ) when being discussed without reference to their useon a specific edge and T ( (cid:96) e ; β ) when they are referenced toa specific edge latency function.Though we consider any incentive bound β ≥ , we offerthe following definition to differentiate from cases where thebound is very large or trivially zero. Definition 1.
A toll (subsidy) is tightly bounded if τ ( f ) = β(cid:96) ( f ) , (if τ ( f ) = − β(cid:96) ( f ) ) for some f ≥ . When an optimal incentive is tightly bounded, the budgetaryconstraint is active.
A. General Relation of Performance
We first consider the relationship between bounded subsi-dies and tolls in general for congestion games (i.e., arbitrarylatency functions and network topologies). Theorem 1 statesthat bounded subsidies outperform similarly bounded tollswith respect to the price of anarchy, and strictly outperformwhen the budgetary constraint is active. (cid:96) ( f ) = f p (cid:96) ( f ) = 1 o d Figure 2:
Two link parallel congestion game. One edge possessesa polynomial latency function, the other a constant latency function.This routing problem realizes the worst case price of anarchy forpolynomial congestion games [34].
Theorem 1.
For a congestion games G , under a boundingfactor β ≥ the optimal subsidy mechanism T opt − ( β ) has nogreater price of anarchy than the optimal tolling mechanism T opt+ ( β ) , i.e., PoA (cid:0)
G, T opt+ ( β ) (cid:1) ≥ PoA (cid:0)
G, T opt − ( β ) (cid:1) ≥ . (11) Additionally, if every optimal subsidy is tightly bounded , thenthe first inequality in (11) is strict. The proof of Theorem 1 appears at the end of this subsec-tion; we first discuss the implications of this result. Theorem 1implies that when limiting the size of monetary transactionswith homogeneous users, subsidies are more effective thantolls at influencing user behavior. This result holds for anycongestion game. Though (11) need not be strict in general,there does exist a gap between the performance of tollsand subsidies in many non-trivial settings. To illustrate this,we offer the following example to highlight that boundedsubsidies may strictly outperform bounded tolls and outlinethe proof structure.
Example 2.
Polynomial Congestion Network.
Consider acongestion game, depicted in Fig. 2, possessing two nodesforming a source destination pair with unit mass of traffic andtwo parallel edges between them, one with latency function (cid:96) ( f ) = f p , where p is a positive integer, and the other (cid:96) ( f ) = 1 . This example has been shown to demonstratethe worst case inefficiency among polynomial congestiongames [34]. Step 1: Identify an Optimal Incentive.
When users arehomogeneous in their sensitivity to incentives, an optimal tollfor this class of games is the marginal cost toll in (5), proven toincentivize optimal routing [24]. Notice that the marginal-costtoll will manifest in this network as τ mc1 ( f ) = pf p , τ mc2 ( f ) = 0 , (12)and indeed incentivize the Nash flow to be the system optimalof f = 1 / p √ p + 1 . Step 2: Find Incentives with similar performance.
It can beshown that any incentive mechanism in the set { T ( (cid:96) ) = λT mc ( (cid:96) ) + ( λ − (cid:96) | λ > } , (13) T opt − ( (cid:96) e ; β ) satisfies Definition 1 with bounding factor β for each (cid:96) e ∈ L ( G ) . has the same performance as the marginal cost taxationmechanism. This observation can be proven from the laterLemma 1. Step 3: Identify Bounded Subsidies and Tolls.
For a bound-ing factor β ≥ p the marginal cost taxation mechanism gives aprice of anarchy of one; however, for β ∈ [0 , p ) , there exists notaxation mechanism in the set defined in (13) which possessesall optimal mechanisms. The similar subsidy mechanism T − ( (cid:96) ) = (1 / ( p + 1) − (cid:96) + (1 / ( p + 1)) T mc ( (cid:96) ) , (14)which manifests in the network as τ − ( f ) = 0 , τ − ( f ) = − pp +1 , (15)is in the set of optimal incentive mechanisms and is validunder bounding factors β ≥ pp +1 . Thus, for bounding factors β ∈ [ pp +1 , p ) , there exists a subsidy mechanism that gives priceof anarchy one, but there does not exist a tolling mechanismthat does the same. For other bounding factors, the sameprinciples can be followed. In Section III-B, the magnitudeof the difference of performance between subsidies and tollsis further explored in the context of affine congestion games.Having concluded Example 2, in Lemma 1 we show a trans-formation on incentive mechanisms that does not affect theprice of anarchy under homogeneous user sensitivities. Thistransformation gives us the important relationship betweenincentive mechanisms that their performance is not unique andsimilar performance can be garnered with different magnitudesof transactions. Lemma 1.
Let T : L ( G ) → T be an incentive mechanismover the family of congestion games G . If another influencingmechanism is defined as T λ ( (cid:96) e ) = λT ( (cid:96) e ) + ( λ − (cid:96) e for any λ > , then PoA( G , T ) = PoA( G , T λ ) . (16)The proof of Lemma 1 appears in the appendix. Proof of Theorem 1:
First, observe that if β = 0 theonly permissible incentive function for tolls and subsidies is τ + e ( f e ) = τ − e ( f e ) = 0 , i.e., there is no incentive. Therefore,the left and right hand side of (11) equate to the unincentivizedcase and (11) holds with equality.Let j e ( f e ) = (cid:96) e ( f e ) + τ e ( f e ) denote the cost a playerobserves for utilizing an edge e when a mass of f e users areutilizing it. The observed cost of a player x ∈ N can be rewrit-ten as J x ( P x , f ) = (cid:80) e ∈ P x j e ( f e ) . In the case where β > ,a bounded tolling function on an edge must exist between τ + e ( f e ) ∈ [0 , β · (cid:96) e ( f e )] , and the edges observed cost satisfies j + e ( f e ) ∈ [ (cid:96) e ( f e ) , (1+ β ) · (cid:96) e ( f e )] . Similarly, a subsidy functionon an edge must exist between τ − e ( f e ) ∈ [ − β · (cid:96) ( f e ) , , and theedges observed cost satisfies j − e ( f e ) ∈ [(1 − β ) · (cid:96) e ( f e ) , (cid:96) e ( f e )] .Let T + ( (cid:96) e ; β ) be a bounded tolling mechanism with edgecosts of j + e ( f e ) . Now, define T λ ( (cid:96) e ) = λT + ( (cid:96) e ; β )+( λ − (cid:96) e ;from Lemma 1, T + and T λ have the same price of anarchyfor any λ > . Let ˆ j e be the edge cost under influencingmechanism T λ , from the construction of T λ ˆ j e = (cid:96) e + T λ ( (cid:96) e ) = (cid:96) e + λT + ( (cid:96) e ; β )+( λ − (cid:96) e = λj + e . (17) (a) Price of anarchy withbounded incentives (b) Price of anarchy with playerheterogeneity Figure 3:
Price of Anarchy bounds for comparable tolls and sub-sidies in affine congestion games. (Left) Price of Anarchy underoptimal toll and subsidy respectively bounded by a factor β fromProposition 1.1. (Right) Price of Anarchy of a nominally equivalenttoll and subsidy with heterogeneity of user sensitivity introducedfrom Proposition 4.1; S U /S L expresses the amount of possibleheterogeneity in the population. We now look at the cases where β ∈ (0 , and β ≥ respectively. When β ∈ (0 , , let λ = (1 − β ) . Now, ˆ j e ( f e ) = (1 − β ) j + e ( f e ) ∈ [(1 − β ) (cid:96) e ( f e ) , (1 − β ) (cid:96) e ( f e )] ⊂ [(1 − β ) (cid:96) e ( f e ) , (cid:96) e ( f e )] , thus the edge costs are sufficiently bounded such that T λ is apermissible subsidy mechanism bounded by β with the sameprice of anarchy as T + . If β ≥ let λ = 1 / (1 + β ) and get ˆ j e ( f e ) = 1(1 + β ) j + e ( f e ) ∈ (cid:20) β ) (cid:96) e ( f e ) , (cid:96) e ( f e ) (cid:21) ⊂ [(1 − β ) (cid:96) e ( f e ) , (cid:96) e ( f e )] , and again T λ is a permissible subsidy mechanism bounded by β . By letting T + = T opt+ we obtain (11).We have proven that, for β > , if PoA( G , T opt − ( β )) =PoA( G , T opt+ ( β )) , then there exists a T opt − ( β ) that doesnot achieve the bound. The contrapositive of this is thatif every optimal subsidy achieves the bound, the price ofanarchy guarantees are not equal. In this case, the optimalsubsidies are each tightly bounded and PoA( G , T opt − ( β )) < PoA( G , T opt+ ( β )) , proving the final part of Theorem 1. B. Bounded Incentives in Affine Congestion Games
In Proposition 1.1, we explicitly give the price of anarchybounds of optimal bounded tolls and subsidies in affine con-gestion games with homogeneous users, again demonstratingthe strictly superior performance of subsidies as well asillustrating the magnitude of this difference in performance.Observe that the optimal subsidy outperforms the optimal tollfor each incentive bound, matching the results from Theo-rem 1.As a means of illustrating Theorem 1, we look at the wellstudied class of affine congestion games, denoted by G aff := { G | (cid:96) e ( f e ) = a e f e + b e , a e , b e ≥ , ∀ e ∈ E ( G ) } . We include this result to highlight the appreciable gap inperformance between subsidies and tolls in this setting.
Proposition 1.1.
The optimal bounded network-agnostictolling mechanism in G aff is T opt+ ( af + b ; β ) = (cid:40) βax β ∈ [0 , ,ax β ≥ , (18) with a price of anarchy bound of PoA( G aff , T opt+ ( β )) = (cid:40) β − β β ∈ [0 , , β ≥ . (19) Additionally, the optimal bounded network-agnostic subsidymechanism in G aff is T opt − ( af + b ; β ) = (cid:40) − βb β ∈ [0 , / , − b/ β ≥ / , (20) with a price of anarchy bound of PoA( G aff , T opt − ( β )) = (cid:40) β − ˆ β β ∈ [0 , / , β ≥ / , (21) where ˆ β = 1 / (1 − β ) − . Accordingly, for any β ∈ (0 , , PoA( G aff , T opt+ ( β )) > PoA( G aff , T opt − ( β )) . (22)The proof of Proposition 1.1 appears in the appendix.Fig. 3a illustrates the price of anarchy for tolls and subsidiesrespectively over various incentive bounds. Though this resultis only for a specific class of games, it helps to quantify thebroader notion of Theorem 1: when users are homogeneousin their response to incentives, a subsidy can consistently giveprice of anarchy closer to one and often by a significantmargin. In the following sections, we further inspect thisrelationship when user heterogeneity is introduced.IV. I
NCENTIVES WITH H ETEROGENEITY
Section III showed that, when users are homogeneous intheir response to incentives, subsidies offer better performanceguarantees than tolls under budgetary constraints. We now seekto understand how each type of incentive performs when usersdiffer in their price sensitivity.Specifically, each user x ∈ N is associated with a sensitivity s x > to incentives. We call s : N → R > a sensitivitydistribution . We highlight the case where s x = c ∀ x ∈ N forsome known constant c as a homogeneous distribution of usersensitivities , in which each user behaves similarly; any otherdistribution is referred to as a population of heterogeneous users.A user x ∈ N i traveling on a path P x ∈ P i observes cost J x ( P x , f ) = (cid:88) e ∈ P x (cid:96) e ( f e ) + s x τ e ( f e ) . (23)A flow f is a Nash flow if J x ( P x , f ) ∈ arg min P ∈P i (cid:40)(cid:88) e ∈ P (cid:96) e ( f e ) + s x τ e ( f e ) (cid:41) ∀ x ∈ N i , i ∈ { , . . . , k } . (24)A game is now denoted by the tuple ( G, s, { τ e } e ∈ E ) .To quantify the robustness of an incentive mechanism, wealso consider that the system designer may be unaware of Without loss of generality, we use s x = 1 for a homogeneous population,as was the case in Section III users’ response to incentives. We denote a set of sensitivitydistributions by S = { s : N → [ S L , S U ] } , where S L > is alower bound on users’ sensitivity to incentives and S U ≥ S L is an upper bound; we include these bounds to quantify therange of users responses, signifying the amount of possibleuser heterogeneity.We extend the prior definition of the price of anarchy to in-clude the heterogeneity of users. Let L nf ( G, s, T ) be the high-est total latency in a Nash flow of the game ( G, s, T ( L ( G ))) .Now we define, PoA( G , S , T ) = sup G ∈G sup s ∈ S L nf ( G, s, T ) L opt ( G ) , (25)where the price of anarchy ratio is now the worst caseinefficiency over all routing problem, sensitivity distributionpairs using the incentive mechanism T .To illustrate this notation, we revisit Example 1, also de-picted in Fig. 1, but now with user heterogeneity. Example 3.
In the routing problem G , depicted in Fig. 1,consider the user sensitivity distribution s = { s x = 2 ∀ x ∈ N , s x = 1 / ∀ x ∈ N } . As a reminder, the optimal flowin G is f opt ≈ { . , . , . , . , . } with a totallatency of L ( f opt ) ≈ . , and with no tolling, the Nashflow is f nf = { / , , , / , } with total latency L ( f nf ) = 1 producing a price of anarchy of PoA(
G, s, ∅ ) ≈ . . Witha marginal cost toll T mc as defined in (5), the Nash flowbecomes f nf ≈ { . , . , . , . , . } produc-ing a price of anarchy of PoA(
G, s, T mc ) ≈ . . Witha subsidy mechanism T − ( (cid:96) e ) = f e · ddf e (cid:96) e ( f e ) − (cid:96) e asdefined in (14) with p = 2 , the Nash flow becomes f nf ≈{ , . , . , . , . } producing a price of anarchy of PoA(
G, s, T sub ) ≈ . .This example shows that user heterogeneity can have anotable impact on the effectiveness of incentives and can effecttheir relative performance. In the remainder of this paper, weconsider the setting where users are heterogeneous in theirprice sensitivity when discussing the relative performance ofsubsidies and tolls. We start by looking at tolls and subsidiesindependently and investigate their performance in the limit ofallowable incentives, i.e., as the budgetary constraint is lifted,how does each type of incentive fare?In Theorem 2 we look at the performance of tolls first andfind that, when the budgetary constraint is lifted, tolls caneliminate the negative effect of user heterogeneity. Theorem 2.
For a class of congestion games G , let T ∗ ∈ arg inf T PoA( G , T ) be an optimal incentive mechanism forhomogeneous populations, then lim β →∞ inf T + ∈ T + β PoA( G , S , T + ) = PoA( G , T ∗ ) . (26) Furthermore, if G is any class of non-atomic congestion gamesthat has convex, non-decreasing, and continuously differen-tiable latency functions, then lim β →∞ inf T + ∈ T + β PoA( G , S , T + ) = 1 . (27) The proof of Theorem 2 appears in the appendix . The proofof Theorem 2 follows closely from Lemma 2 and the notionof responsiveness to heterogeneity presented in the followingsection. The result follows from the idea that larger incentivesare less impacted by user heterogeneity.After observing positive results for the use of tolls with userheterogeneity, we next seek to understand the effectivenessof subsidies in the same situation. In Theorem 3, we showthat, even in a restricted class of congestion games, subsidiescannot effectively mitigate the effect of player heterogeneityin the same way tolls do. Theorem 3.
Let G be any class of non-atomic congestiongames that has convex, non-decreasing, and continuouslydifferentiable latency functions, the set of latency functionsis closed under nonnegative scalar multiplication, and hasat least one network where the untolled price of anarchy isgreater than one. There exists no network-agnostic subsidymechanism T that gives price of anarchy of 1, i.e., lim β →∞ inf T − ∈ T − β PoA( G , S , T − ) > . (28)The proof of Theorem 3 appears in the appendix.Though the class of routing problems has a more strictdefinition than in Theorem 2, the result is still very general andholds for most cases other than singleton networks and thosewhere the price of anarchy is always 1. From Theorem 2 andTheorem 3 we conclude that without the presence of budgetaryconstraints, tolls can mitigate the effect of player heterogeneitywhile subsidies cannot. However, this relationship was shownonly as the budgetary constraint was lifted; in the next section,we further investigate the effect of user heterogeneity on sub-sidies and tolls while budgetary constraints on the incentivesremain. V. R
OBUSTNESS OF I NCENTIVES
In Section IV, user heterogeneity was discussed in the senseof whether incentives could or could not fully mitigate theeffect of non-uniform user behavior. In many cases the verylarge incentives needed to completely eliminate the negativeeffects of user heterogeneity are not possible, particularly inthe presence of budgetary constraints. It is thus of interestwhat the performance guarantees are when the effects ofuser heterogeneity cannot be entirely overcome and how thiscompares when using subsidies or tolls.To compare the robustness of bounded tolls and subsidies,we define an optimal bounded tolling mechanism as T opt+ ( β, S ) ∈ arg inf T + ∈ T + β PoA( G , S , T + ) . (29) This result is reminiscent of [24] stating that there exist tolls that influenceoptimal selfish routing in some settings. In this paper, we extend the resultfrom [24] to cases where users are heterogeneous and classes of games wherea price of anarchy of one may not be achievable. We note that Theorem 2is more general than of [29, Theorem 1], as this result is given for generalincentives and is not reliant on marginal cost taxes nor is it limited to thefamily of congestion games in which they are optimal. Further, the results of[17] cover the case in which the system designer is fully aware of the users’price sensitivities (or value of time in their case) and applies fixed tolls. Incontrast, in this paper the toll designer is unaware of the users’ exact pricesensitivities but is still able to provide a flow-varying tolling scheme that givesa price of anarchy of one as the bounding constraint is lifted.
The optimal bounded subsidy mechanism T opt − ( β, S ) is de-fined analogously. For notational convenience, we will omitthe dependence on S in the homogeneous setting.Often, increased user heterogeneity causes performance ofan incentive mechanism to diminish. We give the followingdefinition for classes of congestion games with this property. Definition 2.
A class of congestion games is responsive toplayer heterogeneity if PoA( G , S , T ∗ ) is strictly increasingwith S U /S L > for an optimal bounded incentive mechanism T ∗ ∈ arg inf T PoA( G , S , T ) . These classes of games are those that have a degradation inperformance from increased player heterogeneity, even whilethe optimal incentive mechanism is in use; many classes ofwell studied congestion games possess this property [29].
A. General Relation of Robustness
In Theorem 4, we give a robustness result that shows theperformance of subsidies degrades more quickly than tolls asplayer heterogeneity is introduced.
Theorem 4.
For a class of congestion games G , define twoincentive bounds β + and β − such that PoA (cid:0) G , T opt − ( β − )) = PoA( G , T opt+ ( β + ) (cid:1) , (30) then at the introduction of player heterogeneity, PoA (cid:0) G , S , T opt − ( β − , S )) ≥ PoA( G , S , T opt+ ( β + , S ) (cid:1) ≥ . (31) Additionally, each inequality in (31) is strict if G is responsiveto player heterogeneity and S L < S U . Intuitively, this result stems from the fact that subsidies aremore finely tuned to give performance guarantees, as guar-anteed in Theorem 1. Essentially, applying a small, negativeincentive to an edge’s increasing latency function will have amore significant impact on the shape of the users’ cost functionthan a larger, positive toll. This fact causes the same amountof player heterogeneity to have a larger effect on Nash flowscaused by subsidies than with an equivalent toll. Thus, whenincreased player heterogeneity escalates the inefficiency, thisrelationship is strict. Though the relationship isn’t strict forgeneral classes of congestion games, it is for many well studiedcases, including the aforementioned polynomial congestiongames.We show in Lemma 2 a relation between nominallyequivalent incentives in the heterogeneous population setting;specifically, we show that the heterogeneous price of anarchydecreases as incentives increase costs to the users.
Lemma 2.
For a class of congestion games G , let T bean incentive mechanism. If T λ ( (cid:96) ) = ( λ − (cid:96) + λT , then PoA( G , S , T λ ) is non-increasing with λ and strictly decreasingif G is responsive to user heterogeneity and S L < S U . The proof of Lemma 2 appears in the appendix.
Proof of Theorem 4:
First, we give the following definition forincentives that have the same performance in the homogeneoussetting.
Definition 3.
For any incentive mechanism T and λ > , eachincentive mechanism satisfying T λ ( (cid:96) e ) = ( λ − (cid:96) e + λT ( (cid:96) e ) is termed nominally equivalent. From Lemma 1, nominallyequivalent incentives satisfy PoA( G , T ) = PoA( G , T λ ) . (32)The theorem follows closely from Lemma 1 and Lemma 2.First, suppose T opt+ ( β + ) is an optimal tolling mechanismbounded by β + . From Lemma 1 there exists a nominallyequivalent subsidy T − λ . If T − λ (cid:54)∈ T − β − , then there must exist a T + λ ∈ T + β + that is nominally equivalent to T opt − ( β − ) fromthe monotonicity and invertability of the transformation inLemma 1. From (30), this implies there exists a nominallyequivalent T opt+ ( β + ) and T opt − ( β − ) .Now, let T opt − ( β − , S ) be the optimal subsidy with playerheterogeneity bounded by β − . From the fact before, weknow there exists a toll T + that is nominally equivalent to T opt − ( β − , S ) and bounded by β + . From Lemma 2, we obtainthat PoA( G , S , T + ) ≤ PoA( G , S , T opt − ( β − , S )) , (33)and by the definition of T opt+ ( β + , S ) , we get PoA( G , S , T opt+ ( β + , S )) ≤ PoA( G , S , T + ) . (34)Combining (33) and (34) gives (31). If the class of gamesis responsive to player heterogeneity, then PoA( G , S , T λ ) isstrictly decreasing with λ and the relationship is strict. B. Robustness of Incentives in Affine Congestion Games
Theorem 4 states that the performance of subsidies degradesmore quickly than tolls when users differ in their response toincentives. Further, if a subsidy and a toll perform the samein the homogeneous setting, the subsidy performs worse thanthe toll with any level of user heterogeneity. To illustrate thisfact, we again look at the class of affine congestion games. Inthis section, we specifically look at G pa , defined as the classof parallel-network affine-latency congestion games in whicheach edge has positive traffic in the untolled Nash flow. Weassign taxes using the optimal scaled marginal cost toll withplayer heterogeneity , T smc ( af + b ) := ( √ S L S U ) − af . Thistolling mechanism was first introduced in [35], and was shownto minimize the price of anarchy in parallel affine congestiongames with sensitivity distributions in S bounded by S L and S U . In Proposition 4.1, we give price of anarchy bounds onthe optimal scaled marginal cost toll as well as a nominallyequivalent subsidy T nes . Proposition 4.1.
Let G pa be the set of fully-utilized paral-lel affine congestion games with sensitivity distributions in S . The optimal scaled marginal cost tolling mechanism is T smc ( af + b ) = af √ S L S U with price of anarchy PoA( G pa , S , T smc ) = 43 (cid:18) − √ q (1 + √ q ) (cid:19) . (35) where q := S L /S U . Additionally, a nominally equivalentsubsidy is T nes ( af + b ) = − √ S L S U b , with price of anarchy PoA( G pa , S , T nes ) = 43 (cid:18) − √ ˆ q (1 + √ ˆ q ) (cid:19) , (36) where ˆ q = λq − q + λq < q, and λ = √ S L S U / (1 + √ S L S U ) . The proof of Proposition 4.1 appears in the appendix.Observe that, because ˆ q < q in (35) and (36) the nominallyequivalent subsidy has greater price of anarchy when playerheterogeneity is introduced. This can be seen in Fig. 3b.Intuitively, the same amount of player heterogeneity has alarger effect on the subsidy than the toll.VI. B OUNDED & R
OBUST I NCENTIVES
In the previous sections, it was shown that when usersare homogeneous in their response to incentives, subsidiesoffer better performance guarantees than tolls under similarbudgetary constraints; however, as users become heteroge-neous in their response to incentives, the performance ofsubsidies degrades more quickly than that of tolls. The logicalnext question we address is, how much heterogeneity causesbounded tolls to outperform bounded subsidies? In general,this question is difficult to answer. We therefore look atthe case of affine congestion games on parallel networkswhile using network-agnostic affine incentive functions. InTheorem 5, we find the incentive bound β ∗ that causes theprice of anarchy of the optimal bounded toll and subsidy withuser heterogeneity to be equal. Without loss of generality(because we assume S L and S U are known to the systemdesigner), we normalize to S L S U = 1 . Theorem 5.
Let T opt+ ( β, S ) and T opt − ( β, S ) be an optimal,affine toll and subsidy mechanism for G pa with incentive bound β and player sensitivities between S L and S U . An incentivebound of β ∗ = 1 /S U = S L gives PoA( G pa , S , T opt − ( β ∗ , S )) = PoA( G pa , S , T opt+ ( β ∗ , S )) . (37)As illustrated in Fig. 4, for lower levels of user heterogeneity(i.e., β ∗ < /S U ), the optimal subsidy offers price of anarchycloser to one than the optimal toll. When there is a largeramount of user heterogeneity (i.e., β ∗ > /S U ) the optimal tollhas a lower price of anarchy bound than the optimal subsidy.The proof of Theorem 5 appears at the end of this sectionand is supported by the following two propositions. Proposi-tion 5.1 (originally introduced in [29]) gives the optimal affinetolling mechanism and the accompanying price of anarchyguarantee. Proposition 5.1. (Brown & Marden [29])
Let T + ( k , k ) denote an affine taxation mechanism that assigns tollingfunctions τ + e ( f e ) = k a e f e + k b e . For any β > , the optimalcoefficients k ∗ and k ∗ satisfying ( k ∗ , k ∗ ) ∈ arg min ≤ k ,k ≤ β PoA (cid:0) G pa , S , T + ( k , k ) (cid:1) , (38) are given by k ∗ = β, (39) k ∗ = max (cid:26) , β S L S U − S L + S U + 2 βS L S U (cid:27) . (40) Figure 4: Price of anarchy under optimal bounded tolls and subsidieswith heterogeneous users in parallel-affine congestion games with β = 0 . . When the amount of user heterogeneity is low (i.e. S U /S L close to one), subsidies offer better performance guarantees thantolls as stated in Theorem 1; however, as the level of heterogeneityincreases, the performance of subsidies degrade more quickly thantolls, stated in Theorem 4. When the incentive bound is β = 1 /S U theperformance of subsidies and tolls is equal, as stated in Theorem 5. Furthermore, for any G ∈ G pa , PoA( G, S , T + ( k ∗ , k ∗ )) isupper bounded by the following expression: (cid:18) − βS L (1 + βS L ) (cid:19) if β < √ S L S U (41) (cid:32) − (1 + βS L )( S L S U + βS L )(1 + 2 βS L + S L S U ) (cid:33) if β ≥ √ S L S U . (42)The proof of Proposition 5.1 appears in the appendix. Theprice of anarchy bound is shown in Fig. 4. Similarly, inProposition 5.2 the optimal affine subsidy is given along withits price of anarchy guarantee. Proposition 5.2.
Let T − ( k , k ) denote an affine subsidymechanism that assigns subsidy functions τ − e ( f e ) = k a e f e + k b e . For any β > , the optimal coefficients k ∗ and k ∗ satisfying ( k ∗ , k ∗ ) ∈ arg min − β ≤ k ,k ≤ PoA (cid:0) G pa , S , T − ( k , k ) (cid:1) , (43) are given by k ∗ = 0 , (44) k ∗ = − min (cid:26) β, S L + S U (cid:27) . (45) Furthermore, for any G ∈ G pa , PoA( G, S , T − ( k ∗ , k ∗ )) isupper bounded by the following expression:
43 (1 − βS L (1 − βS L )) if β < S L + S U (46) (cid:18) − S L /S U (1 + S L /S U ) (cid:19) if β ≥ S L + S U . (47)The proof of Proposition 5.2 appears in the appendix. Theprice of anarchy bound is shown in Fig. 4. The price of anarchybounds equate at β = 1 /S U , as substantiated by Theorem 5,and for β < /S U the subsidy price of anarchy bound is lower,while for β > /S U the toll price of anarchy bound is lowerand converging to one. Proof of Theorem 5:
Proposition 5.1 and Proposition 5.2 givethe price of anarchy bounds for the optimal affine incentives.By inspection, when β ∈ [ S L + S U , √ S L S U ] , the optimal tolland subsidy price of anarchy bounds fall in the domain of(41) and (46) respectively. Additionally, when β = 1 /S U , wecan see that the optimal toll is T + ( S U , and the optimalsubsidy is T − (0 , − S L + S U ) ; furthermore, these incentives havethe same price of anarchy bound, i.e., PoA (cid:18) G pa , S , T + (cid:18) S U , (cid:19)(cid:19) = PoA (cid:18) G pa , S , T − (cid:18) , − S L + S U (cid:19)(cid:19) . (48)It is easy to see from (41),(42),(46), and (47) that for β > /S U , PoA (cid:0) G pa , S , T opt+ (cid:1) < PoA (cid:0) G pa , S , T opt − (cid:1) , and for β < /S U , PoA (cid:0) G pa , S , T opt+ (cid:1) > PoA (cid:0) G pa , S , T opt − (cid:1) . Therefore, β = 1 /S U is the unique incentive bound thatgives equal price of anarchy for subsidies and tolls withheterogeneous users in the class of parallel, affine congestiongames. VII. C ONCLUSION
In this work, the effectiveness of subsidies and tolls incongestion games were compared in the presence of budgetaryconstraints on incentives and user heterogeneity. The results ofthis manuscript show that, in a nominal setting, smaller subsi-dies offer better performance guarantees than tolls; however, inthe face of unknown user heterogeneity, tolls are more robustthan subsidies. These results hold for general classes of non-atomic congestion games, and future work will show the mainconclusions hold for atomic congestion games as well. Futurework may look at more general notions of user sensitivitiesas well as other realistic emergent behavior for the society ofusers. R
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We prove Lemma 1 using the definition of the Nash flow,and by showing this transformation does not affect userpreferences.
Proof of Lemma 1:
Let f (cid:48) be a Nash flow for a game G ∈ G under influencing mechanism T . User x ∈ N i observes cost J x ( P x , f (cid:48) ) = (cid:88) e ∈ P x (cid:96) e ( f (cid:48) e ) + τ e ( f (cid:48) e ) , (49)and by the definition of Nash flow, will have preferencessatisfying J x ( P x , f (cid:48) ) ≤ J x ( P (cid:48) , f (cid:48) ) , ∀ P (cid:48) ∈ P i . (50)In the same flow f (cid:48) , but now under an influencing mechanism ˆ T ( (cid:96) e ) = λT ( (cid:96) e ) + ( λ − (cid:96) e where λ > , user x observescost ˆ J x ( P x , f (cid:48) ) = (cid:88) e ∈ P x (cid:96) e ( f (cid:48) e ) + λτ e ( f (cid:48) e ) + ( λ − (cid:96) e ( f (cid:48) e ) , (51) = (cid:88) e ∈ P x λ ( τ e ( f (cid:48) e ) + (cid:96) e ( f (cid:48) e )) (52) = λJ x ( P x , f (cid:48) ) . (53)Observe that through the same process, it can be shown that ˆ J x ( P, f (cid:48) ) = λJ x ( P, f (cid:48) ) for every P ∈ P i . From (50), (1 /λ ) ˆ J x ( P x , f (cid:48) ) ≤ (1 /λ ) ˆ J x ( P (cid:48) , f (cid:48) ) , ∀ P (cid:48) ∈ P i (54) ˆ J x ( P x , f (cid:48) ) ≤ ˆ J x ( P (cid:48) , f (cid:48) ) , ∀ P (cid:48) ∈ P i . (55)(55) holds for all x ∈ N , satisfying that f (cid:48) is a Nashequilibrium in G under ˆ T . It is therefore the case that anyequilibrium in any game G ∈ G under T is also an equilibriumunder ˆ T , thus L nf ( G, T ) = L nf ( G, ˆ T ) , (56)and, because this holds for every game G ∈ G , it certainlyholds for the supremum over the set which is the same as(16) by definition. Proof of Proposition 1.1:
We first look at the optimal boundedtoll and its associated price of anarchy bound. Trivially, when β > the optimal toll is the marginal cost toll that gives priceof anarchy of one. For a bounding factor β ∈ [0 , , a feasiblebounded toll must satisfy τ + e ( f e ) ∈ [0 , β · (cid:96) e ] = [0 , βa e f e + βb e ] . (57)Because the tolls are network-agnostic, and must satisfy anadditivity property discussed in [29] as well as in the proof ofTheorem 3, we can therefore reduce the search for an optimalbounded toll to τ + e ( f e ) = k a e f e + k b where k , k ∈ [0 , β ] .We first show that the optimal toll will have k = 0 . Let T + be a tolling mechanism that assigns bounded tollswith some k , k ∈ [0 , β ] . A player x ∈ N i utilizing path P x in a flow f observes cost J x ( P x , f ) = (cid:88) e ∈ P x (1 + k ) a e f e + (1 + k ) b e . (58)Now, consider an incentive mechanism ˆ T where edges areassigned tolls τ e ( f e ) = ( k k − a e f e . Under this newincentive, the same player as before now observes cost ˆ J x ( P x , f ) = (cid:88) e ∈ P x k k a e f e + b e . (59)Because the player’s cost in (58) and (59) are proportional,the players preserve the same preferences and the Nash flowsremains unaltered. Because ( k k − ≤ k ≤ β the newincentive is bounded by β . Note that any toll that improvesthe price of anarchy satisfies k > k ; this can be seen byconsidering the worst-case example depicted in Fig. 2 with p = 1 . Because < k < β , we need only consider tolls of theform τ e ( f e ) = ka e f e when in search of the optimal boundedtoll. When k < the price of anarchy is at least / and isindeed not optimal .For a tolling mechanism T + ( af + b ) = kaf with k ∈ [0 , β ) ⊆ [0 , , a player’s cost takes the form J x ( P x , f ) = (cid:88) e ∈ P x (1 + k ) a e + b e . (60)When player cost functions take this form, the game is similarto that of an altruistic game (introduced in [36]) and has priceof anarchy of PoA( G aff , T + ) = 43 + 2 k − k . (61)The price of anarchy is decreasing with k ∈ [0 , and thusthe optimal toll occurs when k is maximized at k = β .For the optimal subsidy, we now note that incentives must bebounded by τ e ( f e ) ∈ [ − β(cid:96) e ( f e ) , . From Lemma 1, we canmap any such subsidy to an equivalent toll, now constrained tothe region ˆ τ e ( f e ) ∈ [0 , ˆ β(cid:96) e ( f e )] where ˆ β = ( − β − . It wasshown prior that the optimal tolling mechanism in this regionis ˆ T ( af + b ) = ˆ βaf . Finally, we can again use Lemma 1 tomap back to the optimal bounded subsidy, T opt − ( af + b ) = ( λ − af + b ) + λ ˆ T ( af + b ) , (62)with λ = 1 − β . The result is an optimal subsidy of the form T opt − ( af + b ) = − βb for β ∈ [0 , / . The price of anarchybound comes from considering the equivalent toll. Proof of Lemma 2:
First, we assume without loss of generality,that S L = 1 . To see this, we make an equivalent problemwhere this is true and show the same price of anarchy boundholds. Let T be any incentive mechanism and S be a familyof sensitivity distributions with lower bound S L and upperbound S U . In any game G ∈ G , a player x ∈ N i observes Consider the classic Pigou network, as in Fig. 2 with p = 1 . It is wellknown this network gives the worst case price of anarchy of / with Nashflow of f = 1 . Consider using a taxation mechanism T ( af + b ) = kaf forsome k < and observe that the Nash flow is unchanged, thus not reducingthe price of anarchy for the class of affine congestion games. costs as expressed in (3). Observe that if we normalize everysensitivity distribution s ∈ S by multiplying by /S L andcorrespondingly scale the incentive by S L the player cost re-mains unchanged. It is therefore the case that any equilibriumis preserved and unchanged, enforcing that PoA( G , S , T ) = PoA ( G , S /S L , S L · T ) . (63)Accordingly, we will consider that S L = 1 throughout.Let f be a flow in G ∈ G induced by sensitivity distribution s ∈ S , and let T be an incentive mechanism that assigns tolls τ + e . From Lemma 1 a nominally equivalent incentive mech-anism can be found by using the transformation ˆ T ( (cid:96) e ; λ ) =( λ − (cid:96) e + λT ( (cid:96) e ) , where choosing λ sufficiently close tozero causes ˆ T to be a subsidy mechanism. We will show thatfor any λ ∈ (0 , , the incentive mechanism ˆ T performs worsethan T at the introduction of player heterogeneity.Let ˆ s be a new sensitivity distribution such that ˆ s x = g ( s x , λ ) = s x λ + s x − s x λ , (64)for all x ∈ N . Now, consider an agent’s cost in flow f withsensitivity ˆ s under incentive mechanism ˆ T . An agent x ∈ N i utilizing path P x in f experiences cost, ˆ J x ( P x , f ) = (cid:88) e ∈ P x (cid:96) e ( f e ) + ˆ s x ˆ T ( (cid:96) e ( f e ); λ )= (cid:88) e ∈ P x (cid:96) e ( f e ) + ˆ s x [( λ − (cid:96) e + λτ + e ( f e )]= λλ + s x − s x λ (cid:88) e ∈ P x ( (cid:96) e ( f e ) + s x τ e ( f e )) , which is proportional to J x ( P x , f ) . By observing proportionalcosts, players preserve the same preferences over paths, pre-serving the same Nash flows.Finally, we show that ˆ s is a feasible sensitivity distributionin S . From the original bounds S L and S U , any generateddistribution ˆ s exists between g ( S L , λ ) and g ( S U , λ ) . Frombefore, S L = 1 , thus from (64), g ( S L = 1 , λ ) = 1 = S L , forany λ ∈ (0 , . Now, observe that any generated distributionsatisfies g ( S U , λ ) = S U λ + S U − S U λ ≤ S U , (65)for any λ ∈ (0 , . Thus any generated distribution ˆ s is suffi-ciently bounded by S L and S U and is a feasible distribution in S . By choosing f to be a Nash flow, we can see that any Nashflow that can be induced by some s ∈ S while using T cansimilarly be induced by ˆ s ∈ S while using ˆ T . It is therefore thecase that the price of anarchy with user heterogeneity is non-decreasing as λ decreases, showing the monotonicity. Further,if S L < S U , then S L ≤ g ( S L , λ ) ≤ g ( S U , λ ) < S U , and if G is responsive to user heterogeneity, the price of anarchy isstrictly increasing with λ . Proof of Theorem 2:
Lemma 2 states that though two incentivemechanisms have the same price of anarchy when users arehomogeneous (from Lemma 1), they need not perform thesame when users are heterogeneous. Further, by increasing λ , one can lower the heterogeneous price of anarchy withoutaltering the performance in the homogeneous setting. The proof of Theorem 2 is a simple extension of Lemma 2.Increasing λ reduces the effect of player heterogeneity on theprice of anarchy, and by letting λ → ∞ we can construct anincentive that recovers (26).In Theorem 1 of [29], the authors propose a realization ofthis result when using marginal cost taxes. In the class ofcongestion games where marginal cost taxes are optimal in thehomogeneous setting, they show that the taxation mechanism T u ( (cid:96) e ; k )[ f e ] = k (cid:18) (cid:96) e ( f e ) + f e · ddf e (cid:96) e ( f e ) (cid:19) has a price of anarchy of 1 as k approaches infinity, i.e., lim k →∞ PoA( G , S , T u ( k )) = 1 . (66)This same result can be recovered using Theorem 2. Themarginal cost taxation mechanism defined in (5) has the sameperformance as T λ ( (cid:96) e )[ f e ] = ( λ − (cid:96) e ( f e ) + λT mc ( (cid:96) e )[ f e ] . By taking the limit as λ approaches infinity, this incentivebecomes T λ ( (cid:96) e )[ f e ] = λ (cid:18) (cid:96) e ( f e ) + f e · ddf e (cid:96) e ( f e ) (cid:19) = T u ( (cid:96) e ; λ )[ f e ] . Not only does this give us the same toll, but by Lemma 2, weknow that lim λ →∞ PoA( G , S , T λ ) = PoA( G , T mc ) = 1 , (67)giving the final statement in Theorem 2. Proof of Theorem 3:
First, consider a game G ∈ G that has aunique equilibrium and optimal flow respectively, to obtain aheterogeneous price of anarchy of one, the equilibrium mustbe the same for any sensitivity distribution s ∈ S . If a taxationmechanism is agnostic of the users’ sensitivities, the only waythis can be accomplished is by letting the magnitude of thesubsidies become large compared to the latency function; fora player x ∈ N this causes J x ( P x , f ) ≈ (cid:80) e ∈ P x s x T ( (cid:96) e )[ f e ] .With this subsidy, the users price sensitivity does not affecttheir preference over paths.Each of the following three conditions is necessary for asufficiently large subsidy to incentivize optimal routing (wejustify each but note the proof that any one is necessary istrivial).1) Additivity.
A network-agnostic incentive mechanism mustsatisfy T ( α(cid:96) + β(cid:96) ) = αT ( (cid:96) )+ βT ( (cid:96) ) . A proof of thisappears in [16]; intuitively, a single latency function canbe represented as multiple in series and the total incentivemust be the same in both cases to guarantee the same totalcost.2) Incentives are Unbounded. | T ( (cid:96) )[ f ] | > M ∀ M ∈ [0 , ∞ ) ∀ (cid:96) ∈ L ( G ) , f > . Any bounded incentivemay allow different sensitivity distributions to inducedifferent equilibrium flows. When the optimal flow isunique, a bounded incentive is incapable of enforcingeach equilibrium flow be optimal. Related by Marginal Cost.
For any two edges (cid:96) i , (cid:96) j with respective flow f i , f j , if (cid:96) mc i ( f i ) ≤ (cid:96) mc j ( f j ) then T ( (cid:96) i )[ f i ] ≤ T ( (cid:96) j )[ f j ] , where (cid:96) mc i ( f i ) = (cid:96) i ( f i ) + f i ddf i (cid:96) i ( f i ) is the marginal cost on edge i . Recall that T isdefined as a cost and therefore negative for subsidies, thusthis condition states that users must receive less subsidyon edges with higher marginal cost. It is shown in [25]that when users observe the marginal cost, the equilibriumflow is optimal.Note that condition 2 implies player costs are negative: (cid:96) ( f )+ T ( (cid:96) )[ f ] < ∀ (cid:96) ∈ L ( G ) , f > . Similarly, condition 3 impliesthat incentives are non-decreasing: if f > f then T ( (cid:96) )[ f ] ≥ T ( (cid:96) )[ f ] ∀ (cid:96) ∈ L ( G ) .We now show that no network-agnostic subsidy mechanismcan satisfy each of these three conditions. Assume T is anoptimal subsidy mechanism. By the symmetry of condition 3,we see that if (cid:96) mc i ( f i ) = (cid:96) mc j ( f j ) , then T ( (cid:96) i )[ f i ] = T ( (cid:96) j )[ f j ] .Consider a unit mass of traffic traversing a two link parallelnetwork with edges possessing latency functions (cid:96) and (cid:96) thatare strictly increasing. Let f be the solution to (cid:96) mc1 ( f ) = (cid:96) mc2 (1 − f ) , and by condition 3, T ( (cid:96) )[ f ] = T ( (cid:96) )[1 − f ] .Now, consider a similar network, but (cid:96) is replaced by ascaled latency function (cid:96) . Now, define f (cid:48) as the solutionto (cid:96) mc1 ( f (cid:48) ) = (cid:96) mc2 (1 − f (cid:48) ) ; from (cid:96) , (cid:96) strictly increasing, f (cid:48) < f . Implied by condition 3, T ( (cid:96) )[ f (cid:48) ] < T ( (cid:96) )[ f ] and T ( (cid:96) )[1 − f ] < T ( (cid:96) )[1 − f (cid:48) ] . From conditions 1 and 2, T ( (cid:96) )[1 − f (cid:48) ] < T ( (cid:96) )[1 − f (cid:48) ] = T ( (cid:96) )[1 − f (cid:48) ] . Put togetherthis gives, T ( (cid:96) )[ f (cid:48) ] < T ( (cid:96) )[ f ] = T ( (cid:96) )[1 − f ] < T ( (cid:96) )[1 − f (cid:48) ] < T ( 12 (cid:96) )[1 − f (cid:48) ] , implying T ( (cid:96) )[ f (cid:48) ] (cid:54) = T ( (cid:96) )[1 − f (cid:48) ] , contradicting condition3. Proof of Proposition 4.1:
The first part of the propositioncomes from [35]. We thus find the nominally equivalentsubsidy mechanism and find the associated price of anarchybound.For notational convenience, let k = 1 / √ S L S U ; the robustmarginal cost toll is thus T smc ( af + b ) = kaf . From Lemma 1,we can derive a nominally equivalent subsidy by T nes ( af + b ) = ( λ − af + b ) + λ ( kaf ) , for any λ > . By letting λ = 1 / (1 + k ) , we get the nominally equivalent subsidy to be T nes ( af + b ) = − kb/ (1 + k ) = − √ S L S U b .To determine the price of anarchy of T nes with playerheterogeneity, we use the result of Theorem 4 to determine theequivalent level of heterogeneity on the nominally equivalenttoll, T smc . Let s ∈ S be a feasible sensitivity distribution,bounded by S L and S U . As it is defined above, we seek to findthe preimage of [ S L , S U ] under the function g ( S, / (1 + k )) .Without loss of generality, we normalize [ S L , S U ] , to [ q, andlook for its preimage. Because g is continuous on S ∈ [0 , ,we look at the endpoints of the region. We first note that g (1 , λ ) = 1 for any λ > . Next, we determine ˆ q such that g (ˆ q, λ ) = q as ˆ q = λq − q + λq , and by setting λ = 1 / (1+ k ) = √ S L S U / (1+ √ S L S U ) recoverthe equivalent amount of heterogeneity, ˆ q , on T smc as theoriginal subsidy T nes with heterogeneity q . By replacing q with ˆ q in (35) we obtain the price of anarchy for T nes withheterogeneity. Proof of Proposition 5.2:
The proof follows similar steps tothat of Proposition 5.2, which appears in [29]. Let G ∈ G pa bea game instance and user be distributed with sensitivity s ∈ S .Because each network in G is parallel, each path constitutes asingle edge. Under an affine subsidy mechanism T − ( k , k ) ,player x ∈ N utilizing edge e observes cost J x ( e, f ) = (1 − k s x ) a e f e + (1 − k s x ) b e , where k , k > . Note that scaling users cost functions doesnot alter their preference over their paths, thus without loss ofgenerality we can write player costs as J x ( e, f ) = (1 − k s x )(1 − k s x ) a e f e + b e . (68)We define a new incentive mechanism T (cid:48) ( af + b ) = k (cid:48) af .Now, let s (cid:48) be a new sensitivity distribution such that playersobserve the same cost under T (cid:48) as they did in (68) withsensitivity distribution s , i.e., (1 − k s x )(1 − k s x ) = (1 + k (cid:48) s (cid:48) x ) . (69)The new distribution can be realized by the transformation s (cid:48) x = s x ( k − k ) k (cid:48) (1 − k s x ) . (70)The taxation mechanism T (cid:48) constitutes a scaled marginal costtoll, for which, the following result exists: Theorem 6. [Brown & Marden [35]]:
For any network G ∈ G with flow on all edges in an un-tolled Nash flow, and any s ∈ S ,any scaled marginal cost taxation mechanism reduces the totallatency of any Nash flow when compared to the total latencyof any Nash flow associated with the un-tolled case, i.e., forany k > L nf ( G, s, T A ( k, < L nf ( G, s, ∅ ) . (71) Furthermore, the unique optimal scaled marginal-cost tollingmechanism uses the scale factor k ∗ = 1 √ S L S U = arg min k ≥ { PoA( G , S , T A ( k, } . (72) Finally, the price of anarchy resulting from the optimal scaledmarginal-cost taxation mechanism is
PoA( G , S , T A ( k ∗ , − (cid:112) S L /S U (cid:16) (cid:112) S L /S U (cid:17) ≤ . (73)Because of this, we set k (cid:48) = √ S (cid:48) L S (cid:48) U to be the optimalscaled marginal cost taxation mechanism over the new family of sensitivity distributions S (cid:48) , generated by transforming eachsensitivity distribution in S as in (70).Now, in the original subsidy mechanism T ( k , k ) , let k = k (cid:48) = 1 (cid:112) S (cid:48) L S (cid:48) U . (74)Combining (70) and (74) gives an expression for the accom-panying choice of k to satisfy (69), k = k S L S U − k S L S U − S L − S U . (75)Observe that (73) is decreasing with S L /S U < . For thesimilar taxation mechanism T (cid:48) , S (cid:48) L S (cid:48) U = S L (1 − k S U ) S U (1 − k S L ) , (76)for S (cid:48) found by (70). Notice (76) is decreasing with k
Philip N. Brown is an Assistant Professor in theDepartment of Computer Science at the Universityof Colorado, Colorado Springs. Philip received theBachelor of Science in Electrical Engineering in2007 from Georgia Tech, after which he spent sev-eral years designing control systems and processtechnology for the biodiesel industry. He receivedthe Master of Science in Electrical Engineering in2015 from the University of Colorado at Boulderunder the supervision of Jason R. Marden, wherehe was a recipient of the University of ColoradoChancellor’s Fellowship. He received the PhD in Electrical and ComputerEngineering from the University of California, Santa Barbara under thesupervision of Jason R. Marden. He was finalist for the Best Student PaperAward at the 2016 and 2017 IEEE Conferences on Decision and Control,and received the 2018 CCDC Best PhD Thesis Award from UCSB. Philip isinterested in the interactions between engineered and social systems.