Efficiency Loss in a Network Resource Allocation Game

Transcription

1 Efficiency Loss in a Netwok Resouce Allocation Game Ramesh Johai johai@mit.edu) John N. Tsitsiklis jnt@mit.edu) June 11, 2004 Abstact We exploe the popeties of a congestion game whee uses of a congested esouce anticipate the effect of thei actions on the pice of the esouce. When uses ae shaing a single esouce, we establish that the aggegate utility eceived by the uses is at least 3/4 of the maximum possible aggegate utility. We also conside extensions to a netwok context, whee uses submit individual payments fo each link in the netwok which they may wish to use. In this netwok model, we again show that the selfish behavio of the uses leads to an aggegate utility which is no wose than 3/4 the maximum possible aggegate utility. We also show that the same analysis extends to a wide class of esouce allocation systems whee end uses simultaneously equie multiple scace esouces. These esults fom pat of a gowing liteatue on the pice of anachy, i.e., the extent to which selfish behavio affects system efficiency. 1 Intoduction The cuent Intenet is used by a widely heteogeneous population of uses; not only ae diffeent types of taffic shaing the same netwok, but diffeent end uses place diffeent values on thei peceived netwok pefomance. As a esult, chaacteizing good use of the netwok is difficult: how should esouces be shaed between a file tansfe and a pee-to-pee connection? Patly in esponse to this heteogeneity, a vaiety of models fo congestion picing in the futue Intenet have emeged. These models popose a taditional economic solution to the poblem of heteogeneous demand: they teat the collection of netwok esouces as a maket, and pice thei use accodingly. The last decade has witnessed a damatic ise in eseach suggesting the use of maket mechanisms to manage congestion in netwoks; see, e.g., [1] fo an ealy oveview of some of the issues involved, and [2, 3] fo moe ecent discussion. The poposals have vaied widely in appoach and simplicity, including applications of auction theoy [4] as well as fixed ate picing mechanisms [5]. In this pape, we will conside a famewok with a single netwok manage, who wishes to allocate netwok capacity efficiently among a collection of uses, each endowed with a utility function depending on thei allocated ate. In [6], a maket is poposed whee each use submits a bid, o willingness-to-pay pe unit time, to the netwok; the netwok accepts these submitted bids and detemines the pice of each netwok link. A use is then allocated ate in popotion to his 1

2 bid, and invesely popotional to the pice of links he wishes to use. Unde cetain assumptions, it is shown in [6] that such a scheme maximizes aggegate utility. In the special case whee the netwok consists of only a single link, a given use is allocated a faction of the link equal to his bid divided by the sum of all uses bids. This popotional allocation mechanism has been consideed in a vaiety of othe contexts as well. Hajek and Gopalakishnan have consideed such a mechanism in the context of Intenet autonomous system competition [7]. They suggest that smalle Intenet povides might bid fo esouces fom lage Intenet povides upsteam using the popotional allocation mechanism. In the economics liteatue, such a mechanism is efeed to as a affle ; it has been analyzed in the context of chaitable fundaising [8]. In the compute science community, this mechanism is known as the popotional shae mechanism, whee it has been investigated fo time-shaing of esouces [9]. In this pape, we wish to undestand the extent to which the analysis poposed in [6] accuately models the inteactions of netwok uses. Specifically, a fundamental assumption in the model of [6] is that each use acts as a pice take; that is, uses do not anticipate the effect of thei actions on the pices of the links. In contast, we elax this assumption, and ask whethe pice anticipating behavio significantly wosens the pefomance of the netwok. If we assume that uses can anticipate the effects of thei actions, then the model becomes a game; we will show that the Nash equilibia of this game lead to allocations at which total utility is no wose than 3/4 the aggegate system utility. The fact that Nash equilibia of a game may not achieve full efficiency has been well known in the economics liteatue [10]. Recent eseach effots have focused on quantifying this loss fo specific game envionments; the esulting degee of efficiency loss is known as the pice of anachy [11]. Most of the esults on pice of anachy have focused on outing [12], taffic netwoks [13, 14], and netwok design [15, 16], as well as a special class of submodula games including facility location games [17]. Stated in the language of this liteatue, the cental esult of ou pape is that the pice of anachy of the netwok picing mechanism studied is an efficiency loss of no moe than 25%. The investigation of the pice of anachy povides a foundation fo design of engineeing systems with obustness against selfish behavio; in paticula, ou esults suggest that selfish behavio of individual netwok uses need not degade netwok pefomance abitaily, povided the netwok picing mechanism is caefully chosen. The emainde of the pape is oganized as follows. In Section 2 we give backgound on the model fomulation. We ecapitulate the key esults of [6], and pecisely define the notion of pice taking. We pove the main theoem of [6] fo a single link: if uses ae pice taking, then aggegate utility is maximized. We then conside a game whee uses ae pice anticipating. We give a poof of a esult due to Hajek and Gopalakishnan establishing existence and uniqueness of a Nash equilibium, by showing that at a Nash equilibium, it is as if aggegate utility is maximized but with modified utility functions [7]. In Section 3, we conside the loss of efficiency at the Nash equilibium of the single link game. Theoem 3 is key esult of this pape: when uses ae pice anticipating, the pice of anachy is a 25% efficiency loss. In Section 4, we extend the ealie analysis to netwoks. We conside a game whee each use equests sevice fom multiple links by submitting a bid to each link. Uses have multiple outes available to them fo sending taffic, so that this is a model including altenative outing. Links then allocate ates using the same scheme as in the single link model, and each use sends the maximum flow possible, given the vecto of ates allocated fom links in the netwok. Although 2

3 this definition of the game is natual, we demonstate that Nash equilibia may not exist, due to a discontinuity in the payoff functions of individual playes. This poblem also aises in the single link setting, but is ielevant thee as long as moe than two playes shae the link.) To addess the discontinuity, we extend the stategy space by allowing each use to equest a nonzeo ate without submitting a positive bid to a link, if the total payment made by othe uses at that link is zeo; this extension is sufficient to guaantee existence of a Nash equilibium. Futhemoe, if a Nash equilibium exists in the oiginal game, it coesponds natually to a Nash equilibium of the extended game. Finally, we show that in this netwok setting, the total utility achieved at any Nash equilibium of the game is no less than 3/4 of the maximum possible aggegate utility. This extends the pice of anachy esult fom the single link case to the netwok setting. In Section 5, we conside a moe geneal esouce allocation game. We suppose that uses bid fo multiple esouces, as in Section 4; but we no longe define utility as a function of the maximum flow that a use can send. Rathe, we allow the use s utility to be any concave function of the vecto of esouces allocated. Such a game can also be intepeted moe geneally; fo example, each esouce may be a aw mateial, and each end use may be a manufactuing facility that takes these aw mateials as input. We show that such a game can be analyzed using the same methods as Section 4, and in paticula pove once again that the efficiency loss is no wose than 25% elative to the system optimal opeating point. We conclude in Section 6. 2 Backgound Suppose R uses shae a communication link of capacity C > 0. Let d denote the ate allocated to use. We assume that use eceives a utility equal to U d ) if the allocated ate is d ; we assume that utility is measued in monetay units. We also assume the utility function U d ) is concave, stictly inceasing, and continuously diffeentiable, with domain d 0; concavity coesponds to the assumption of elastic taffic, as defined by Shenke [18]. Given complete knowledge and centalized contol of the system, a natual poblem fo the netwok manage to ty to solve is the following optimization poblem [6]: SYSTEM: maximize subject to U d ) 1) d C; 2) d 0, = 1,..., R. 3) Since the objective function is continuous and the feasible egion is compact, an optimal solution d = d 1,..., d R ) exists; since the feasible egion is convex, if the functions U ae stictly concave, then the optimal solution is unique. In geneal, the utility functions ae not available to the link manage. As a esult, we conside the following picing scheme fo ate allocation. Each use gives a payment also called a bid) of w to the link manage; we assume w 0. Given the vecto w = w 1,..., w ), the link manage chooses a ate allocation d = d 1,..., d ). We assume the manage teats all uses alike in othe 3

4 wods, the netwok manage does not pice disciminate. Each use is chaged the same pice µ > 0, leading to d = w /µ. We futhe assume the manage always seeks to allocate the entie link capacity C; in this case, following the analysis of [6], we expect the pice µ to satisfy: w µ = C. The peceding equality can only be satisfied if w > 0, in which case we have: µ = w C. 4) In othe wods, if the manage chooses to allocate the entie available ate at the link, and does not pice disciminate between uses, then fo evey nonzeo w thee is a unique pice µ > 0 which must be chosen by the netwok, given by the pevious equation. In the emainde of the section, we conside two diffeent models fo how uses might inteact with this pice mechanism. In Section 2.1, we conside a model whee uses do not anticipate the effect of thei bids on the pice, and establish existence of a competitive equilibium a esult due to Kelly [6]). Futhemoe, this competitive equilibium leads to an allocation which solves SYSTEM. In Section 2.2, we change the model and assume uses ae pice anticipating, and establish existence and uniqueness of a Nash equilibium a esult due to Hajek and Gopalakishnan [7]). Section 3 then consides the loss of efficiency at this Nash equilibium, elative to the optimal solution to SYSTEM. 2.1 Pice Taking Uses and Competitive Equilibium In this section, we conside a competitive equilibium between the uses and the link manage [19], following the development of Kelly [6]. A cental assumption in the definition of competitive equilibium is that each use does not anticipate the effect of thei payment w on the pice µ, i.e., each use acts as a pice take. In this case, given a pice µ > 0, use acts to maximize the following payoff function ove w 0: P w ; µ) = U w µ ) w. 5) The fist tem epesents the utility to use of eceiving a ate allocation equal to w /µ; the second tem is the payment w made to the manage. Obseve that since utility is measued in monetay units, the payoff is quasilinea in money, a typical assumption in modeling maket mechanisms [19]. We now say a pai w, µ) with w 0 and µ > 0 is a competitive equilibium if uses maximize thei payoff as defined in 5), and the netwok cleas the maket by setting the pice µ accoding to 4): P w ; µ) P w ; µ) fo w 0, = 1,..., R; 6) µ = w C. 7) 4

5 Kelly shows in [6] that when uses ae pice takes, thee exists a competitive equilibium, and the esulting allocation solves SYSTEM. This is fomalized in the following theoem, adapted fom [6]; we also pesent a poof fo completeness. Theoem 1 Kelly, [6]) Assume that fo each, the utility function U is concave, stictly inceasing, and continuously diffeentiable. Then thee exists a competitive equilibium, i.e., a vecto w = w 1,..., w R ) 0 and a scala µ > 0 satisfying 6)-7). In this case, the scala µ is uniquely detemined, and the vecto d = w/µ is a solution to SYSTEM. If the functions U ae stictly concave, then w is uniquely detemined as well. Poof. The key idea in the poof is to use Lagangian techniques to establish that optimality conditions fo 6)-7) ae identical to the optimality conditions fo the poblem SYSTEM, unde the identification d = w/µ. Step 1: Given µ > 0, w satisfies 6) if and only if: ) U w µ = µ, if w > 0; 8) U 0) µ, if w = 0. 9) Indeed, since U is concave, P is concave as well; and thus 8)-9) ae necessay and sufficient optimality conditions fo 6). Step 2: Thee exists a vecto d 0 and a unique scala µ > 0 such that: U d ) = µ, if d > 0; 10) U 0) µ, if d = 0; 11) d = C. 12) The vecto d is then a solution to SYSTEM. If the functions U ae stictly concave, then d is unique as well. As discussed above, at least one optimal solution to SYSTEM exists since the feasible egion is compact and the objective function is continuous. We fom the Lagangian fo the poblem SYSTEM: Ld, µ) = ) U d ) µ d C Hee the second tem is a penalty fo the link capacity constaint. The Slate constaint qualification [20], Section 5.3) holds fo the poblem SYSTEM at the point d = 0, since then 0 = d < C; this guaantees the existence of a Lagange multiplie µ. In othe wods, since the objective function is concave and the feasible egion is convex, a feasible vecto d is optimal if and only if thee exists µ 0 such that the conditions 10)-12) hold. Since thee exists at least one optimal solution d to SYSTEM, thee exists at least one pai d, µ) satisfying 10)-12). Since C > 0, at least one d is positive, so µ > 0 since U is stictly inceasing). We now claim that µ is uniquely detemined. Suppose not; then thee exist d, µ), d, µ) that satisfy 10)-12), whee without loss of geneality) µ < µ. Fo any such that d > 0, we will have 5

6 U d ) µ < µ = U d ), which implies that d > d > 0. Summing ove all, we obtain d > d, which contadicts the feasibility condition d = C = d. Thus µ is unique. Step 3: If the pai d, µ) satisfies 10)-12), and we let w = µd, then the pai w, µ) satisfies 6)-7). By Step 2, µ > 0; thus, unde the identification w = µd, 12) becomes equivalent to 7). Futhemoe, 10)-11) become equivalent to 8)-9); by Step 1, this guaantees that 6) holds. Step 4: If w and µ > 0 satisfy 6)-7), and we let d = w/µ, then the pai d, µ) satisfies 10)-12). We simply evese the agument of Step 3. Unde the identification d = w/µ, 8)-9) become equivalent to 10)-11); and 7) becomes equivalent to 12). Step 5: Completing the poof. By Steps 2 and 3, thee exists a vecto w and a scala µ > 0 satisfying 6)-7); by Step 4, µ is uniquely detemined, and the vecto d = w/µ is a solution to SYSTEM. Finally, if the utility functions U ae stictly concave, then by Steps 2 and 4, w is uniquely detemined as well since the tansfomation fom w, µ) to d, µ) is one-to-one). 2.2 Pice Anticipating Uses and Nash Equilibium We now conside an altenative model whee the uses of a single link ae pice anticipating, athe than pice takes. The key diffeence is that while the payoff function P takes the pice µ as a fixed paamete in 5), pice anticipating uses will ealize that µ is set accoding to 4), and adjust thei payoff accodingly; this makes the model a game between the R playes. We use the notation w to denote the vecto of all bids by uses othe than ; i.e., w = w 1, w 2,..., w 1, w +1,..., w R ). Given w, each use chooses w to maximize: ) w U Q w ; w ) = s w C w, if w > 0; s 13) U 0), if w = 0. ove nonnegative w. The second condition is equied so that the ate allocation to use is zeo when w = 0, even if all othe uses choose w so that s w s = 0. The payoff function Q is simila to the payoff function P, except that the use anticipates that the netwok will set the pice µ accoding to 4). A Nash equilibium of the game defined by Q 1,..., Q R ) is a vecto w 0 such that fo all : Q w ; w ) Q w ; w ), fo all w 0. 14) Note that the payoff function in 13) may be discontinuous at w = 0, if s w s = 0. This discontinuity may peclude existence of a Nash equilibium, as the following example shows. Example 1 Suppose thee is a single use with stictly inceasing utility function U. In this case, the use is not playing a game against anyone else, so any positive payment esults in the entie link being allocated to the single use. The payoff to the use is thus: { UC) w, if w > 0; Qw) = U0), if w = 0. 6

7 Since U has been assumed to be stictly inceasing, we know UC) > U0). Thus, at a bid of w = 0, a pofitable deviation fo the use is any bid w such that 0 < w < UC) U0). On the othe hand, at any bid w > 0, a pofitable deviation fo the use is any bid w such that 0 < w < w. Thus no optimal choice of bid exists fo the use, which implies that no Nash equilibium exists. We will find the pevious discontinuity plays a lage ole in the netwok context, whee an extended stategy space is equied to ensue existence of a Nash equilibium. In the single link setting, Hajek and Gopalakishnan have shown that thee exists a unique Nash equilibium when multiple uses shae the link, by showing that at a Nash equilibium it is as if the uses ae solving anothe optimization poblem of the same fom as the poblem SYSTEM, but with modified utility functions. This is fomalized in the following theoem, adapted fom [7]; we also pesent a poof fo completeness. Theoem 2 Hajek and Gopalakishnan, [7]) Assume that R > 1, and that fo each, the utility function U is concave, stictly inceasing, and continuously diffeentiable. Then thee exists a unique Nash equilibium w 0 of the game defined by Q 1,..., Q R ), and it satisfies w > 0. In this case, the vecto d defined by: d = is the unique solution to the following optimization poblem: w s w C, = 1,..., R, 15) s GAME: maximize subject to Û d ) 16) d C; 17) d 0, = 1,..., R, 18) whee Û d ) = 1 d ) U d ) + C ) d 1 d ) U z) dz. 19) C d 0 Poof. The poof poceeds in a numbe of steps. We fist show that at a Nash equilibium, at least two components of w must be positive. This suffices to show that the payoff function Q is stictly concave and continuously diffeentiable fo each use. We then establish necessay and sufficient conditions fo w to be a Nash equilibium; these conditions look simila to the optimality conditions 8)-9) in the poof of Theoem 1, but fo modified utility functions defined accoding to 19). Mioing the poof of Theoem 1, we then show the coespondence between these conditions and the optimality conditions fo the poblem GAME. This coespondence establishes existence and uniqueness of a Nash equilibium. Step 1: If w is a Nash equilibium, then at least two coodinates of w ae positive. Fix a use, and suppose w s = 0 fo evey s. If w > 0, use can maintain the same ate allocation and educe his payment by educing w slightly; and since U is stictly inceasing, if w = 0, then 7

8 use can pofitably deviate by infinitesimally inceasing his bid w and captuing the entie link capacity C. Thus at a Nash equilibium, w s > 0 fo some s. Since this holds fo evey use, at least two coodinates of w must be positive. Step 2: If the vecto w 0 has at least two positive components, then the function Q w ; w ) is stictly concave and continuously diffeentiable in w, fo w 0. This follows fom 13), because when s w s > 0, the expession w /w + s w s) is a stictly inceasing function of w fo w 0); in addition, U ) is a stictly inceasing concave, and diffeentiable function by assumption. Step 3: The vecto w is a Nash equilibium if and only if at least two components of w ae positive, and fo each, the following conditions hold: ) U w s w C 1 w ) s s s w = w s s C, if w > 0; 20) U 0) s w s C, if w = 0. 21) Let w be a Nash equilibium. By Steps 1 and 2, w has at least two positive components and Q w ; w ) is stictly concave and continuously diffeentiable in w 0. Thus w must be the unique maximize of Q w ; w ) ove w 0, and satisfy the following fist ode optimality conditions: Q w w ; w ) { = 0, if w > 0; 0, if w = 0. Afte multiplying though by s w s/c, these ae pecisely the conditions 20)-21). Convesely, suppose that w has at least two stictly positive components, and satisfies 20)- 21). Then we may simply evese the agument: by Step 2, Q w ; w ) is stictly concave and continuously diffeentiable in w 0, and in this case the conditions 20)-21) imply that w maximizes Q w ; w ) ove w 0. Thus w is a Nash equilibium. If we let µ = w /C, note that the conditions 20)-21) have the same fom as the optimality conditions 8)-9), but fo a diffeent utility function given by Û. We now use this elationship to complete the poof in a manne simila to the poof of Theoem 1. Step 4: The function Û defined in 19) is stictly concave and stictly inceasing ove 0 d C. The poof follows by diffeentiating, which yields Û d ) = U d )1 d /C). Since U is concave and stictly inceasing, we know that U d ) > 0, and that U is noninceasing; we conclude that Û d ) is nonnegative and stictly deceasing in d ove the egion 0 d C, as equied. 8

9 Step 5: Thee exists a unique vecto d and scala ρ such that: U d ) 1 d ) = ρ, if d > 0; 22) C U 0) ρ, if d = 0; 23) d = C. 24) The vecto d is then the unique solution to GAME. By Step 4, since Û is continuous and stictly concave ove the convex, compact feasible egion fo each, we know that GAME has a unique solution. This solution d is uniquely identified by the stationaity conditions 22)-23), togethe with the constaint that d C. Since Û is stictly inceasing fo each, the constaint 24) must hold as well. That ρ is unique then follows because at least one d must be stictly positive at the unique solution to GAME. Step 6: If d, ρ) satisfy 22)-24), then the vecto w = ρd is a Nash equilibium. We fist check that at least two components of d ae positive, and that ρ > 0. We know fom 24) that at least one component of d is stictly positive. Suppose now that d > 0, and d s = 0 fo s. Then we must have d = C. But then by 22), we have ρ = 0; on the othe hand, since U s is stictly inceasing and concave, we have U s0) > 0 fo all s, so 23) cannot hold fo s. Thus at least two components of d ae positive. In this case, it is seen fom 22) that ρ > 0 as well. By Step 3, to check that w = ρd is a Nash equilibium, we must only check the stationaity conditions 20)-21). We simply note that unde the identification w = ρd, using 24) we have that: ρ = w C ; and d = w s w C. s Substitution of these expessions into 22)-23) leads immediately to 20)-21). Thus w is a Nash equilibium. Step 7: If w is a Nash equilibium, then the vecto d defined by 15) and scala ρ defined by ρ = w )/C ae the unique solution to 22)-24). We simply evese the agument of Step 6. By Step 3, w satisfies 20)-21). Unde the identifications of 15) and ρ = w /C, we find that d and ρ satisfy 22)-24). By Step 5, such a pai d, ρ) is unique. Step 8: Thee exists a unique Nash equilibium w, and the vecto d defined by 15) is the unique solution of GAME. This conclusion is now staightfowad. Existence follows by Steps 5 and 6, and uniqueness follows by Step 7 since the tansfomation fom w to d, ρ) is one-to-one). Finally, that d solves GAME follows by Steps 5 and 7. Theoem 2 shows that the unique Nash equilibium of the single link game is chaacteized by the optimization poblem GAME. Othe games have also pofited fom such elationships notably taffic outing games, in which Nash equilibia can be found as solutions to a global optimization poblem. Roughgaden and Tados use this fact to thei advantage in computing the pice of anachy fo such games [13]; Schulz and Stie-Moses also use this elationship to conside outing games in capacitated netwoks [14]. 9

10 Theoem 2 is also closely elated to potential games [21], whee best esponses of playes ae chaacteized by changes in a global potential function. In such games, the global minima of the potential function coespond to Nash equilibia, as we obseved fo the poblem GAME. Howeve, we note that despite this coespondence the objective function of the poblem GAME is not a potential function. Finally, we note that fo the congestion game pesented hee, seveal authos have deived esults simila to Theoem 2. Gibbens and Kelly [22] consideed the special case whee all the functions U ae linea, and demonstated existence and uniqueness of the Nash equilibium in this setting. The fist esult fo geneal utility functions was given by La and Ananthaam [23], who showed that if the uses stategies ae esticted to a compact set [W min, W max ], whee 0 < W min < W max <, then thee exists a unique Nash equilibium. Maheswaan and Basa conside a model whee a fixed value of ɛ > 0 is added to the pice of the link [24]; the allocation to use is thus d = w / s w s + ɛ), which avoids the possible discontinuity of Q when w = 0. The authos demonstate existence and uniqueness of the Nash equilibium in this setting. It is possible to use the model of [24] to show existence but not uniqueness) of the Nash equilibium of the congestion game defined by Q 1,..., Q R ), by taking a limit as ɛ 0; indeed, such a limit foms the basis of ou poof of existence of Nash equilibia in the netwok context see Theoem 6). 3 Pice of Anachy of the Single Link Game We let d S denote an optimal solution to SYSTEM, and let d G denote the unique optimal solution to GAME. We now investigate the pice of anachy of this system [11]; that is, how much utility is lost because the uses attempt to game the system? To answe this question, we must compae the utility U d G ) obtained when the uses fully evaluate the effect of thei actions on the pice, and the utility U d S ) obtained by choosing the point which maximizes aggegate utility. We know, of couse, that U d G ) U d S ), by definition of d S.) An easy lowe bound on Ûd G ) may be constucted by using the modified utility functions Û defined in 19). Notice that Ûd ) may be viewed as the expectation of U with espect to a pobability distibution which places a mass of 1 d /C on the ate d the fist tem of 19)), and unifomly distibutes the emaining mass of d /C on the inteval [0, d ] the second tem of 19)). Fom this intepetation and the fact that U is stictly inceasing, it follows that Ûd ) U d ) if 0 d C. Futhemoe, if we assume that U 0) 0, then using concavity of U, it is staightfowad to establish that Ûd ) U d )/2 fo all d such that 0 d C. Recalling that d G solves GAME, and assuming that U 0) 0 fo all, we can bound U d G ) as follows: 1 2 U d S ) Û d S ) Û d G ) U d G ). The peceding agument shows that the pice of anachy is no moe than a 50% efficiency loss when uses ae pice anticipating. Howeve, this bound is not tight. As we show in the following theoem, the efficiency loss is exactly 25% in the wost case. Theoem 3 Assume that fo each, the utility function U is concave, stictly inceasing, and continuously diffeentiable. Suppose also that U 0) 0 fo all. If d S is any solution to SYSTEM, 10

11 and d G is the unique solution to GAME, then: U d G ) 3 U d S ). 4 Futhemoe, this bound is tight: fo evey ɛ > 0, thee exists a choice of R, and a choice of linea) utility functions U, = 1,..., R, such that: ) ) 3 U d G ) 4 + ɛ U d S ). In othe wods, fo this system the pice of anachy is a 25% efficiency loss. Poof. We fist show that because of the assumption that U is concave and stictly inceasing fo each, the wost case occus with linea utility functions. This is summaized in the following lemma. Lemma 4 Suppose that U 0) 0 fo all. Let d = d 1,..., d ) satisfy d C, and let d S be any solution to SYSTEM. Then the following inequality holds: U d ) U d S ) U d )d max U d ) ) C. 25) Poof of Lemma. Using concavity, we have U d S ) U d ) + U d )d S d ). Thus: U d ) U d S ) U d ) U d )d ) + U d )d U d ) U d )d ) + U. d )d S Futhemoe, since ds = C, we have the following tivial inequality: ) U d )d S max U d ) C. This yields: U d ) U d S ) U d ) U d )d ) + U d )d U d ) U d )d ) + max U d ) ) C. Now notice that because we have assumed U 0) 0, we again have by concavity that U d )d U d ). Thus the expession U d ) U d )d ) is nonnegative, so we conclude that: U d ) U d S ) U d )d max U d ) ) C, since the ight hand side of the expession above is less than o equal to 1. Let d G be the unique Nash equilibium of the game with utility functions U 1,..., U R. We define a new collection of linea utility functions by: U d ) = U d G )d. 11

12 Notice that the stationaity conditions 22)-24) only involve the fist deivatives of the utility functions U, = 1,..., R, at d G ; thus, the unique Nash equilibium of the game with utility functions U 1,..., U R is given by d G as well. Fomally, d G satisfies the stationaity conditions 22)-24) fo the family of utility functions U 1,..., U R. Futhemoe, the system optimal aggegate utility fo this family of utility functions is given by max U d G ) ) C. Applying Lemma 4 with d = d G, we thus see that the wost case pice of anachy occus in the case of linea utility functions. We now poceed to calculate this pice of anachy. Assume fo the emainde of the poof, theefoe, that U is linea, with U d ) = α d, whee α > 0. Let d G be the Nash equilibium of the game with these utility functions. Fom the discussion in the peceding paagaph, the atio of aggegate utility at the Nash equilibium to aggegate utility at the social optimum is given by: α d G max α ) C. By scaling and elabeling if necessay, we assume without loss of geneality that max α = α 1 = 1, and C = 1. To identify the wost case situation, we would like to find α 2,..., α R such that d G 1 + R =2 α d G, the total utility associated with the Nash equilibium, is as small as possible; this esults in the following optimization poblem with unknowns d G 1,..., d G R, α 2,..., α R ): minimize d G 1 + R α d G 26) =2 subject to α 1 d G ) = 1 d G 1, if d G > 0; 27) α 1 d G 1, if d G = 0; 28) d G = 1; 29) 0 α 1, = 2,..., R; 30) d G 0, = 1,..., R. 31) This optimization poblem chooses linea utility functions with slopes less than o equal to 1 fo playes 2,..., R. The constaints in the poblem equie that given linea utility functions U d ) = α d fo = 1,..., R, the allocation d G must in fact be the unique Nash equilibium allocation of the esulting game. As a esult, the optimal objective function value is pecisely the lowest possible aggegate utility achieved, among all such games. In addition, since C = 1, and the lagest α is α 1 = 1, the system optimal aggegate utility is exactly 1; thus, the optimal objective function value of this poblem also diectly gives the pice of anachy. Suppose now α, d) is an optimal solution to 26)-31) in which n < R uses, say uses = R n + 1,..., R, have d G = 0. Then the fist R n coodinates of α and d must be an optimal solution to the poblem 26)-31), with R n uses. Theefoe, in finding the wost case game, it suffices to assume that d G > 0 fo all = 2,..., R, and then conside the optimal objective function value fo R = 2, 3,.... This allows us to conside only the constaint: α 1 d G ) = 1 d G 1. 32) 12

13 This constaint then implies that α = 1 d G 1 )/1 d G ). We will solve the esulting educed optimization poblem by decomposing it into two stages. Fist, we fix a choice of d G 1 and optimize ove d G, = 2,..., R; then, we choose the optimal value of d G 1. Given these obsevations, we fix d G 1, and conside the following, simple optimization poblem: minimize d G 1 + subject to R =2 d G 1 d G 1 ) 1 d G R d G = 1 d G 1 ; =2 0 d G d G 1, = 2,..., R. Notice that we have substituted fo α in the objective function. The constaint α 1 becomes equivalent to d G d G 1 unde the identification 32). This simple poblem is only well defined if d G 1 1/R; othewise the feasible egion is empty in othe wods, thee exist no Nash equilibia with d G 1 < 1/R. If we assume that d G 1 1/R, then the feasible egion is convex, compact, and nonempty, and the objective function is stictly convex in each of the vaiables d G, = 2,..., R. This is sufficient to ensue that thee exists a unique optimal solution as a function of d G 1 ; futhe, by symmety, this optimal solution must be: d G = 1 dg 1 R 1, fo = 2,..., R. We now optimize ove d G 1. Afte substituting, we have the following optimization poblem: minimize d G d G 1 ) dg 1 R 1 1 subject to R dg 1 1. The objective function fo the peceding optimization poblem is deceasing in R fo evey value of γ; in the limit whee R, the wost case pice of anachy is given by the solution to: minimize d G d G 1 ) 2 subject to 0 d G 1 1. It is simple to establish that the solution to this poblem occus at d G 1 = 1/2, which yields a wost case aggegate utility of 3/4, as equied. This bound is tight in the limit whee the numbe of uses inceases to infinity; using this fact, we obtain the tightness claimed in the theoem. The peceding theoem shows that in the wost case, aggegate utility falls by no moe than 25% when uses ae able to anticipate the effects of thei actions on the pice of the link. Futhemoe, this bound is essentially tight. In fact, it follows fom the poof that the wost case consists of a link of capacity 1, whee use 1 has utility U 1 d 1 ) = d 1, and all othe uses have utility U d ) d /2 when R is lage). As R goes to infinity, at the Nash equilibium of this game use 1 eceives a ate 13 ) 1

14 d G 1 = 1/2, while the emaining uses unifomly split the ate 1 d G 1 = 1/2 among themselves, yielding an aggegate utility of 3/4. We note that a simila bound was obseved by Roughgaden and Tados fo taffic outing games with affine link latency functions [13]. They found that the atio of wost case Nash equilibium cost to optimal cost was 4/3. Howeve, it emains an open question whethe a elationship can be dawn between the two games; in paticula, we note that while Theoem 3 holds even if the utility functions ae nonlinea, Roughgaden and Tados have shown that the pice of anachy in taffic outing may be abitaily high if link latency functions ae nonlinea. 4 Geneal Netwoks In this section we will conside an extension of the single link model to geneal netwoks. We conside a netwok consisting of J links, numbeed 1,..., J. Link j has a capacity given by C j > 0; we let C = C 1,..., C J ) denote the vecto of capacities. As befoe, a set of uses numbeed 1,..., R shaes this netwok of links. We assume thee exists a set of paths though the netwok, numbeed 1,..., P. By an abuse of notation, we will use J, R, and P to also denote the sets of links, uses, and paths, espectively. Each path p P uses a subset of the set of links J; if link j is used by path p, we will denote this by witing j p. Each use R has a collection of paths available though the netwok; if path p seves use, we will denote this by witing p. We will assume without loss of geneality that paths ae uniquely identified with uses, so that fo each path p thee exists a unique use such that p. Thee is no loss of geneality because if two uses shae the same path, that is captued in ou model by ceating two paths which use exactly the same subset of links.) Fo notational convenience, we note that the links equied by individual paths ae captued by the path-link incidence matix A, defined by: { 1, if j p; A jp = 0, if j p. Futhemoe, we can captue the elationship between paths and uses by the path-use incidence matix H, defined by: { 1, if p ; H p = 0, if p. Note that by ou assumption on paths, fo each path p we have H p = 1 fo exactly one use. Let y p 0 denote the ate allocated to path p, and let d = p y p 0 denote the ate allocated to use ; using the matix H, we may wite the elation between d = d, R) and y = y p, p P ) as Hy = d. Any feasible ate allocation y must satisfy the following constaint: y p C j, j J. p:j p Using the matix A, we may wite this constaint as Ay C. We continue to assume that use eceives a utility U d ) fom an amount of ate d, whee the utility function U is concave, nondeceasing, and continuous, with domain d 0. Obseve 14

15 that we no longe equie that U be stictly inceasing o diffeentiable, as in the pevious development.) The natual genealization of the poblem SYSTEM to a netwok context is given by the following optimization poblem: SYSTEM: maximize U d ) 33) subject to Ay C; 34) Hy = d; 35) y p 0, p P. 36) Since the objective function is continuous and the feasible egion is compact, an optimal solution y exists; since the feasible egion is also convex, if the functions U ae stictly concave, then the optimal vecto d = Hy is uniquely defined though y need not be unique). As in the pevious section, we will use the solution to SYSTEM as a benchmak fo the outcome of the netwok congestion game. We now define the esouce allocation mechanism fo this netwok setting. The natual extension of the single link model is defined as follows. Each use submits a bid w j fo each link j; this defines a stategy fo use given by w = w j, j J), and a composite stategy vecto given by w = w 1,..., w R ). We then assume that each link takes these bids as input, and uses the picing scheme developed in the pevious section. Fomally, each link sets a pice µ j w), given by: µ j w) = w j. 37) C j As befoe, we assume the ate allocated to a use is popotional to his payment. We denote by x j w) the ate allocated to use by link j; we thus have: x j w) = { wj µ j w), if w j > 0; 0, othewise. 38) We define the vecto x w) by: x w) = x j w), j J). Now given any vecto x = x j, j J), we define d x ) to be the optimal value of the following optimization poblem: maximize y p 39) subject to p p :j p y p x j, j J; 40) y p 0, p. 41) Given the stategy vecto w, we then define the ate allocated to use as d x w)). To gain some intuition fo this allocation mechanism, notice that when the vecto of bids is w, use is allocated 15

16 PSfag eplacements Uses 1 2 C 1 C 2 R Figue 1: Example 2) Link 1 has capacity C 1, and link 2 has capacity C 2, whee C 1 < C 2. Each one of R uses equies sevice fom both links. a ate x j w) at each link j. Since the utility to use is nondeceasing in the total amount of ate allocated, use s utility is maximized if he solves the peceding optimization poblem, which is a max-flow poblem constained by the ate x j available at each link j. In othe wods, use is allocated the maximum possible ate d x w)), given that each link j has ganted him ate x j w). Define the notation w = w 1,..., w 1, w +1,..., w R ). Based on the definition of d x w)) above, the payoff to use is given by: Q w ; w ) = U d x w)) ) j w j. 42) A Nash equilibium of the game defined by Q 1,..., Q R ) is a vecto w 0 such that fo all : Q w ; w ) Q w ; w ), fo all w 0. 43) Although this picing scheme is vey natual, the fact that the payoff Q may be discontinuous can pevent existence of a Nash equilibium, as we fist obseved in Example 1. Although we wee able to pove a Nash equilibium exists with R > 1 uses fo the single link case, the following example shows that Nash equilibia may not exist in the netwok context even if R > 1. Example 2 Conside an example consisting of two links, labeled j = 1, and j = 2. The fist link has capacity C 1, and the second link has capacity C 2 > C 1, as depicted in Figue 1. The system consists of R uses, whee we assume that each use has a stictly inceasing, concave, continuous utility function U. Fo this example, we will assume each use is identified with a single path consisting of both links 1 and 2. This simplifies the analysis, since the solution to the poblem 39)-41) is then given by: d x w)) = min{x 1 w), x 2 w)}. We will show that no Nash equilibium exists fo this netwok. Suppose, to the contay, that w is a Nash equilibium. We fist show that w j > 0, fo j = 1, 2. If not, then all uses ae allocated zeo ate. Fist suppose that w j = 0 fo both j = 1, 2. Then any use can pofitably deviate by infinitesimally inceasing w 1 and w 2, say by > 0; this deviation will give use 16

17 ate min{c 1, C 2 } = C 1, and incease the total payment by 2. Fo small enough, this will stictly impove the payoff of playe ; thus no Nash equilibium exists whee w j = 0 fo both j = 1, 2. A simila agument follows if w 1 = 0, but w 2 > 0: in this case, fo any use such that w 2 > 0, a pofitable deviation exists whee w 2 is educed to zeo; this leaves use s ate allocation unchanged at zeo, while educing his total payment to the netwok. Symmetically, the same agument may be used when w 1 > 0, and w 2 = 0. As a esult, we conclude that if w is a Nash equilibium, we must have w j > 0 fo both j = 1, 2. Now note that tivially) we have the elations: w 1 s w 1s C 1 = C 1 ; and w 2 s w C 2 = C 2. 2s Since C 1 < C 2, thee must exist at least one use fo whom w 1 C 1 )/ s w 1s) < w 2 C 2 )/ s w 2s). Recall that use is allocated a total ate equal to: { } w1 w min 2 s w C 1, 1s s w C 2. 2s As a esult, use can pofitably deviate by educing w 2, since this educes his payment, without alteing his ate allocation. Thus no such w can be a Nash equilibium. As will be seen in the following development, the issue in the pevious example is that link 2 is not a bottleneck in the netwok since C 1 < C 2, link 2 will neve be fully utilized). As a esult, as long as the total payment s w 2s to link 2 is stictly positive, thee will always be some use who is ovepaying i.e., this use could pofitably deviate by educing w 2. Thus the only equilibium outcome is one whee the total payment to link 2 becomes zeo; but in this case, because of the discontinuity in the payoff function defined in 42) o, moe pecisely, the discontinuity in 38)), all uses ae allocated zeo ate. We will see in the following section that a esolution to this poblem can be found if uses ae allowed to equest and be allocated a nonzeo ate fom links fo which the total payment is zeo. We show that Nash equilibia ae always guaanteed to exist fo this extended game; futhemoe, we show that any Nash equilibium fo the game defined by Q 1,..., Q R ) coesponds in a natual way to a Nash equilibium of the extended game. Finally, in Subsection 4.2, we show that the aggegate utility at any Nash equilibium of the extended game is no less than 3/4 times the SYSTEM optimal aggegate utility, matching the esult achieved fo the single link game. 4.1 An Extended Game In this section, we conside an extended game, whee uses not only submit bids, but also ate equests. We conside an allocation mechanism unde which the ate equests ae only taken into account by a link when the total payment to that link is zeo. This behavio ensues that when a link is not congested as in Example 2), o is not in sufficient demand as in Example 1), uses may still be allocated a nonzeo ate on that link. In paticula, this modification addesses the degeneacies which aise due to the discontinuity of Q in the oiginal definition of the netwok game. We will show that Nash equilibia always exist fo this extended game. We note that extended stategy spaces have also poven fuitful fo othe games with payoff discontinuities; see, e.g., [25].) 17

18 Fomally, we suppose that the stategy of use includes a ate equest φ j 0 at each link j; that is, the stategy of use is a vecto σ = φ, w ), whee φ = φ j, j J), and w = w j, j J), as befoe. We will wite σ = σ 1,..., σ R ) to denote the composite stategy vecto of all playes; and we will wite σ = σ 1,..., σ 1, σ +1,..., σ R ) to denote all components of σ othe than σ. We now suppose that each link j povides a ate x j σ) to use, which is detemined as follows: 1. If s w js > 0, then: 2. If s w js = 0, but s φ js C j, then: 3. If s w js = 0 and s φ js > C j, then: x j σ) = w j s w C j. 44) js x j σ) = φ j. 45) x j σ) = 0. 46) In the fist instance, when link j eceives a positive payment fom the uses, ate is allocated in popotion to the bids. The second two cases apply only when the total payment to link j is zeo; in this event, if the total equested ate is less than the capacity C j, then the equests ae ganted. Howeve, if the total equested ate exceeds capacity, no ate is allocated. We note hee that the pecise definition in case 3 above is not essential; any mechanism which splits the capacity C j accoding to a peset deteministic ule will lead to the same esults below. Fo example, if equests exceed capacity, a link may choose to allocate the same ate to all uses who shae the link; o the link may choose to allocate all the entie capacity to some pedetemined pefeed use. As befoe, we define: x σ) = x j σ), j J). The ate of use is then d x σ)) whee d is defined as the optimal value to the optimization poblem 39)-41)). The payoff T to use is given by: T σ ; σ ) = U d x σ)) ) j w j. 47) Note that this is an abuse of notation in the definition of x and x j, since we peviously had defined them as functions of w. Howeve, the definition in use will be clea fom the agument of the function.) A Nash equilibium of the game defined by T 1,..., T R ) is a vecto σ 0 such that fo all : T σ ; σ ) T σ ; σ ), fo all σ 0. 48) We stat with a theoem which states that the game defined in this subsection is indeed an extension of the oiginal netwok game, defined by Q 1,..., Q R ). 18

19 Theoem 5 Assume that fo each, the utility function U is concave, nondeceasing, and continuous. Suppose that w is a stategy vecto fo the game defined by Q 1,..., Q R ). Fo each use, define: { w j φ j = s w C j, if w j > 0; js 0, othewise. Fo each use, let σ = φ, w ). Then use eceives the same payoff in eithe game: T σ ; σ ) = Q w ; w ). Futhemoe, if w is a Nash equilibium of the game defined by Q 1,..., Q R ), then σ is a Nash equilibium of the game defined by T 1,..., T R ). Poof. We will efe to the game defined by Q 1,..., Q R ) as the oiginal game, and the game defined by T 1,..., T R ) as the extended game. We fist note that given the definition of φ j above, we have the identity x j σ) = x j w) fo each link j; that is, the allocation fom link j to use in the extended game is identical to the allocation made by link j in the oiginal game. Futhemoe, the total payment made by use emains unchanged in the extended game. Thus the payoff to use is the same in both games, unde the mapping fom w to σ defined in the statement of the theoem. Now suppose that w is a Nash equilibium of the oiginal game, and define σ as in the statement of the theoem. Fo each link j and each use, define W j = s w js. Suppose thee exists a stategy vecto σ = φ, w ) such that: U d x σ, σ )) ) j w j > U d x σ)) ) j w j. Fix ɛ > 0. Fo each j, we define ŵ j = w j if W j > 0, and ŵ j = ɛ if W j = 0. Then: x j ŵ, w ) x j σ, σ ). The peceding inequality follows because fom each link j with W j = 0, use is allocated the entie capacity C j in etun fo the payment of ɛ > 0. Fom this we may conclude that: d x ŵ, w )) d x σ, σ )). Now as ɛ 0, we have j ŵj j w j. Thus fo sufficiently small ɛ > 0, we will have: U d x ŵ, w )) ) j ŵ j U d x σ, σ )) ) j ŵ j > U d x σ)) ) j = U d x w)) ) j w j w j. Thus the vecto ŵ = ŵ j, j ) is a pofitable deviation fo use in the oiginal game, a contadiction. Theefoe no pofitable deviation exists fo use in the extended game. We conclude 19

20 σ is a Nash equilibium fo the extended game, as equied. The peceding theoem shows that any Nash equilibium of the oiginal game coesponds natually to a Nash equilibium of the extended game. To constuct a patial convese to this esult, suppose that we ae given a Nash equilibium σ = φ, w) of the extended game, but that w j > 0 fo all links j. We fist note that fo each link j, at least two distinct uses submit positive bids. If not, then thee is some link j whee a single use submits a positive bid but this use can leave his ate allocation unchanged and educe his payment by loweing the bid submitted to link j. Thus we conclude that fo each link j and each use, the payment by all othe uses s w js is positive. This ensues the ate equests φ do not have any effect on the ate allocation made to use, so that the payoffs ae detemined entiely by the bid vectos w, fo R. This is sufficient to conclude that w must actually be a Nash equilibium fo the oiginal game. To summaize, we have shown that wheneve all link pices ae positive at a Nash equilibium in the extended game, then in fact we have a Nash equilibium fo the oiginal game as well. We now tun ou attention to showing that a Nash equilibium always exists fo the extended game. Theoem 6 Assume that fo each, the utility function U is concave, nondeceasing, and continuous. Then a Nash equilibium exists fo the game defined by T 1,..., T R ). Poof. Ou technique is to conside a petubed vesion of the oiginal game, whee a vitual use submits a bid of ɛ > 0 to each link j in the netwok. Fomally, this means that at a bid vecto w, use is allocated a ate x ɛ jw) at link j, given by: x ɛ jw) = w j ɛ + s w C j. js We define the vecto x ɛ w) = x ɛ jw), j J), and the ate attained by use is then d x ɛ w)), whee d is the optimal value to the optimization poblem 39)-41). The modified allocation defined by x ɛ was also consideed by Maheswaan and Basa in the context of a single link [24]; we will use this allocation mechanism to pove existence fo ou game by taking a limit as ɛ 0. Ou appoach will be to fist apply standad fixed point techniques to establish existence of a Nash equilibium w ɛ fo this petubed game, with an associated allocation to each use given by x ɛ w ɛ ). We will then show that w ɛ and x ɛ w ɛ ) fo each ) lie in compact sets, espectively. If we then choose w and φ = φ, R) as limit points when ɛ 0, we will find that w, φ) is a Nash equilibium of the extended game. Step 1: A Nash equilibium w ɛ exists in the petubed game. We fist obseve that since ɛ > 0, x ɛ jw) is a continuous, stictly concave, and stictly inceasing function of w j 0 in paticula, thee is no longe any discontinuity in the ate allocation at w j = 0). Futhemoe, since d is defined as the maximal objective value of a linea pogam, d x ) is concave and continuous as a function of x [26], Section 5.2); and if x j x j fo all j, then clealy d x ) d x ), i.e., d is nondeceasing this follows immediately fom the poblem 39)-41)). We will now combine these facts to show that use s payoff in this petubed game is concave as a function of w, and continuous as a function of the composite stategy w. The payoff in the 20

21 petubed game, denoted Q ɛ, is given by: Q ɛ w ; w ) = U d x ɛ w)) ) j w j. Continuity of Q ɛ as a function of w follows immediately fom continuity of x ɛ j, d, and U. To show that Q ɛ is concave as a function of w, it suffices to show that U d x ɛ w, w ))) is a concave function of w. Since fo each j the function x ɛ j is concave in w j, and does not depend on w k fo k j, we conclude that each component of x ɛ w, w ) is a concave function of w. If we fix the bids of the othe playes as w, then since d is nondeceasing and concave in its agument, we have fo any two bid vectos w, w, and δ such that 0 δ 1: d x ɛ δw + 1 δ)w, w )) d δx ɛ w, w ) + 1 δ)x ɛ w, w )) δd x ɛ w, w )) + 1 δ)d x ɛ w, w )). We now apply the fact that U is nondeceasing and concave to conclude that: U d x ɛ δw + 1 δ)w, w )) ) U δd x ɛ w, w )) + 1 δ)d x ɛ w, w )) ) δu d x ɛ w, w )) ) + 1 δ)u d x ɛ w, w )) ). Thus use s payoff function Q ɛ w ; w ) is concave in w. Finally, we obseve that in seaching fo a Nash equilibium of the petubed game defined by Q ɛ 1,..., Q ɛ R ), we can estict the stategy space of each use to a compact, convex subset of RJ. To see this, fix a use, and choose B > U j C j) U 0). When use sets w = 0, his payoff is U 0). On the othe hand, the maximum ate use can be allocated fom the netwok is bounded above by j C j; and thus, if use chooses any stategy w such that j w j > B, then egadless of the stategies w of all othe playes, we have: U d x ɛ w, w )) j w j U j C j ) B < U 0). Thus, if we define the compact set S = {w : j w j B }, we obseve that use would neve choose a stategy vecto that lies outside S ; this allows us to estict the stategy space of use to the set S. The game defined by Q ɛ 1,..., Q ɛ R ) togethe with the stategy spaces S 1,..., S R ) is then a concave R-peson game: each payoff function is continuous in the composite stategy vecto w; Q ɛ is concave in w ; and the stategy space of each use is a compact, convex, nonempty subset of R J. Applying Rosen s existence theoem [27] poven using Kakutani s fixed point theoem), we conclude that a Nash equilibium w ɛ exists fo this game. Step 2: Thee exists a limit point σ = φ, w) of the Nash equilibia of the petubed games. Fo each use, define φ ɛ j = x ɛ jw ɛ ). Let φ ɛ = φ ɛ j, j J), and φ ɛ = φ ɛ, R). We now note that fo all ɛ > 0, the pai φ ɛ, w ɛ ) lies in a compact subset of Euclidean space. To see this, note that w ɛ lies in the compact set S 1 S R, and that 0 φ ɛ j C j fo all j and. Thus, thee exists a sequence ɛ k 0 such that the sequence φ ɛ k, w ɛ k ) conveges to some σ = φ, w), whee w S 1 S R and 0 φ j C j. 21