Efficiency Loss in a Network Resource Allocation Game: The Case of Elastic Supply

Transcription

1 Efficiency Loss in a Netwok Resouce Allocation Game: The Case of Elastic Supply axiv:cs/ v1 [cs.gt] 14 Jun 2005 Ramesh Johai (johai@stanfod.edu) Shie Manno (shie@mit.edu) John N. Tsitsiklis (jnt@mit.edu) Febuay 1, 2008 Abstact We conside a esouce allocation poblem whee individual uses wish to send data acoss a netwok to maximize thei utility, and a cost is incued at each link that depends on the total ate sent though the link. It is known that as long as uses do not anticipate the effect of thei actions on pices, a simple popotional picing mechanism can maximize the sum of uses utilities minus the cost (called aggegate suplus). Continuing pevious effots to quantify the effects of selfish behavio in netwok picing mechanisms, we conside the possibility that uses anticipate the effect of thei actions on link pices. Unde the assumption that the links maginal cost functions ae convex, we establish existence of a Nash equilibium. We show that the aggegate suplus at a Nash equilibium is no wose than a facto of times the optimal aggegate suplus; thus, the efficiency loss when uses ae selfish is no moe than appoximately 34%. 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. This has led to a suge of inteest in congestion picing, whee the netwok is teated as a maket, and pices ae set to mediate demand and supply of netwok esouces; see, e.g., [5, 9]. We investigate a specific pice mechanism consideed by Kelly et al. in [16] (motivated by the poposal made in [14]). Fo simplicity let us fist conside the special case of a single link; in this case the mechanism woks as follows. Each use submits a bid, o total willingness-to-pay, to the link manage. This epesents the total amount the use expects to pay. The link manage then chooses a total ate and pice such that the poduct of pice and ate is equal to the sum of the bids, and the pice is equal to maginal cost; note, in paticula, that the supply of the link is elastic, i.e., it is not fixed in advance. Finally, each use eceives a faction of the allocated ate in popotion to thei bid. It is shown in [16] that if uses do not anticipate the effect of thei bid on the pice, such 1

2 a scheme maximizes the sum of uses utilities minus the cost of the total allocated ate, known as the aggegate suplus (see [20], Chapte 10). The picing mechanism of [16] takes as input the bids of the uses, and poduces as output the pice of the link, and the esulting ate allocation to the uses. Kelly et al. [16] continue on to discuss distibuted algoithms fo implementation of this maket-cleaing pocess: given the bids of the uses, the authos pesent two algoithms which convege to the maket-cleaing pice and ate allocation. Indeed, much of the inteest in this maket mechanism stems fom its desiable popeties as a decentalized system, including both stability and scalability. Fo details, we efe the eade to [11, 13, 27, 28]. One impotant intepetation of the pice given to uses in the algoithms of [16] is that it can povide ealy notification of congestion. Building on the Explicit Congestion Notification (ECN) poposal [22], this intepetation suggests that the netwok might chage uses poactively, in hopes of avoiding congestion at links late. Fom an implementation standpoint, such a shift implies that athe than a had capacity constaint (i.e., a link is oveloaded when the ate though it exceeds the capacity of the link), the link has an elastic capacity (i.e., the link gadually begins to signal a buildup of congestion befoe the link s tue capacity is actually met). Many poposals have been made fo active queue management (AQM) to achieve good pefomance with Explicit Congestion Notification; see, e.g., [2, 15, 18, 19]. This issue is of seconday impotance to ou discussion, as we do not concen ouselves with the specific intepetation of the cost function at the link. (An insightful discussion of the elationship between active queue management and the cost function of the link may be found in [10].) In this pape, we investigate the obustness of the maket mechanism of [16] when uses attempt to manipulate the maket. Fomally, we conside a model whee uses anticipate the effects of thei actions on the link pices. This makes the model a game, and we ask two fundamental questions: fist, does a Nash equilibium exist fo this game? And second, how inefficient is such an equilibium elative to the maximal aggegate suplus? We show that Nash equilibia exist, and that the efficiency loss is no moe than a facto of the maximal aggegate suplus (appoximately 34%) when uses ae pice anticipating. Such an investigation foms pat of a boade body of wok on quantifying efficiency loss in envionments whee paticipants ae selfish. Results have been obtained fo outing [7, 17, 21], taffic netwoks [6, 25] and netwok design poblems [1, 8]. Ou wok is most closely elated to that of [12], whee the same maket mechanism as in this pape was consideed fo the case whee the supply of a link is fixed, o inelastic; this was the mechanism fist pesented in [14]. Johai and Tsitsiklis show the efficiency loss when uses ae pice anticipating is no wose than 25% [12]. The outline of the emainde of the pape is as follows. We stat by consideing a single link in isolation. In Section 1, we descibe the maket mechanism fo a single link, and ecapitulate the esults of Kelly et al. [16]. In Section 2, we descibe a game whee uses ae pice anticipating, and establish the existence of a Nash equilibium. We also establish necessay and sufficient conditions fo a stategy vecto to be a Nash equilibium. These conditions ae used in Section 3 to pove the main esult of the pape fo a single link: that when uses ae pice anticipating, the efficiency loss that is, the loss in aggegate suplus elative to the maximum is no moe than 34%. In Section 4, we compae the settings of inelastic and elastic supply. In paticula, we conside 2

3 a limit of cost functions which appoach a had capacity constaint. We show that if these cost functions ae monomials and we let the exponent tend to infinity, then the efficiency loss appoaches 25%, which is consistent with the esult of [12]. In Section 5, we extend the esults to geneal netwoks. This extension is done using the same appoach as [12]. We conside a game whee uses submit individual bids to each link in the netwok, and establish existence of a Nash equilibium. Using techniques simila to the esults poven in a netwok context in [12], we show that the efficiency loss is no moe than 34% when uses ae pice anticipating, matching the esult of Section 3. Some conclusions ae offeed in Section 6. 1 Backgound Suppose R uses shae a single communication link. Let d 0 denote the ate allocated to use. We assume that use eceives a utility equal to U (d ) if the allocated ate is d. In addition, we let f = d denote the total ate allocated at the link, and let C(f) denote the cost incued at the link when the total allocated ate is f 0. We will assume that both U and C ae measued in the same monetay units. A natual intepetation is that U (d ) is the monetay value to use of a ate allocation d, and C(f) is a monetay cost fo congestion at the link when the total allocated ate is f. We make the following assumptions egading U and C. Assumption 1 Fo each, ove the domain d 0 the utility function U (d ) is concave, stictly inceasing, and continuously diffeentiable, and the ight diectional deivative at 0, denoted U (0), is finite. Assumption 2 Thee exists a continuous, convex, stictly inceasing function p(f) ove f 0 with p(0) = 0, such that fo f 0: C(f) = f Thus C(f) is stictly convex and stictly inceasing. 0 p(z)dz. Concavity in Assumption 1 coesponds to elastic taffic, as defined by Shenke [26]; such taffic includes file tansfes such as FTP connections and pee-to-pee connections. Note that Assumption 2 does not equie the pice function p to be diffeentiable. Indeed, assuming smoothness of p would simplify some of the technical aguments in the pape. Howeve, we late equie the use of nondiffeentiable pice functions in ou poof of Theoem 8. Given complete knowledge and centalized contol of the system, a natual poblem fo the netwok manage to ty to solve is the following [14]: 3

4 SYSTEM: ( ) maximize U (d ) C d (1) subject to d 0, = 1,...,R. (2) Since the objective function (1) is continuous, and U inceases at most linealy while C inceases supelinealy, an optimal solution d S = (d S 1,...,d S R ) exists fo (1)-(2); since the feasible egion is convex and C is stictly convex, if the functions U ae stictly concave, then the optimal solution is unique. We efe to the objective function (1) as the aggegate suplus; this is the net monetay benefit to the economy consisting of the uses and the single link [20]. Fo convenience, we define a function suplus(d) which gives the aggegate suplus at an allocation d: suplus(d) ( ) U (d ) C d. (3) Due to the decentalized natue of the system, the esouce manage may not have an exact specification of the utility functions [14]. As a esult, we conside the following picing scheme fo ate allocation. Each use makes a payment (also called a bid) of w to the esouce manage. Given the vecto w = (w 1,..., w ), the esouce manage chooses a ate allocation d(w) = (d 1 (w),..., d R (w)). We assume the manage teats all uses alike in othe wods, the netwok manage does not pice diffeentiate. Thus the netwok manage sets a single pice µ(w); we assume that µ(w) = 0 if w = 0 fo all, and µ(w) > 0 othewise. All uses ae then chaged the same pice µ(w), leading to: 0, if w = 0; d (w) = w µ(w), if w > 0. Associated with this choice of pice is an aggegate ate function f(w), defined by: f(w) = 0, if w = 0; d (w) = w µ(w), if. (4) w > 0. We will assume that w is measued in the same monetay units as both U and C. In this case, given a pice µ > 0, use acts to maximize the following payoff function ove w 0: ( ) w P (w ; µ) = U w. (5) µ 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 4

5 units, the payoff is quasilinea in money, a typical assumption in modeling maket mechanisms [20]. Notice that as fomulated above, the payoff function P assumes that use acts as a pice take; that is, use does not anticipate the effect of his choice of w on the pice µ, and hence on his esulting ate allocation d (w). Infomally, we expect that in such a situation the aggegate suplus will be maximized if the netwok manage sets a pice equal to maginal cost, i.e., if the pice function satisfies: µ(w) = p(f(w)). (6) We show in the following poposition that a joint solution to (4) and (6) can be found; we then use this poposition to show that when uses optimize (5) and the pice is set to satisfy (6), aggegate suplus is maximized. Poposition 1 Suppose Assumption 2 holds. Given any vecto of bids w 0, thee exists a unique pai (µ(w), f(w)) 0 satisfying (4) and (6), and in this case f(w) is the unique solution f to: w = fp(f). (7) Futhemoe, f( ) has the following popeties: (1) f(0) = 0; (2) f(w) is continuous fo w 0; (3) f(w) is a stictly inceasing and stictly concave function of w ; and (4) f(w) as w. Poof. Fix a vecto w 0. Fist suppose thee exists a solution to (4) and (6). Then fom (4), we have: w = f(w)µ(w). Afte substituting (6), the peceding elation becomes (7). Convesely, if f(w) solves (7), then defining µ(w) accoding to (6) makes (7) equivalent to (4). Thus, it suffices to check that thee exists a unique solution f to (7). By Assumption 2, p is stictly inceasing, and since p is convex, p(f) as f ; thus defining g(f) = fp(f), we know that g(0) = 0; g is stictly inceasing, stictly convex, and continuous; and g(f) as f. Thus g is invetible, and cosses the level w at a unique value f(w) = g 1 ( w ). Fom this desciption and the popeties of g it is not had to veify that f has the fou popeties stated in the poposition. Obseve that we can view (7) as a maket-cleaing pocess. Given the total evenue w fom the uses, the link manage chooses an aggegate ate f(w) so that the evenue is exactly equal to the aggegate chage f(w)p(f(w)). Due to Assumption 2, this maket-cleaing aggegate ate is uniquely detemined. Kelly et al. pesent two algoithms in [16] which amount to dynamic pocesses of maket-cleaing; as a esult, a key motivation fo the mechanism we study in this pape is that it epesents the equilibium behavio of the algoithms in [16]. Kelly et al. show in [16] that when uses ae non-anticipating, and the netwok sets the pice µ(w) accoding to (4) and (6), the esulting allocation solves SYSTEM. This is fomalized in the following theoem, adapted fom [16]. 5

6 Theoem 2 (Kelly et al., [16]) Suppose Assumptions 1 and 2 hold. Fo any w 0, let (µ(w), f(w)) be the unique solution to (4) and (6). Then thee exists a vecto w such that µ(w) > 0, and: P (w ; µ(w)) = max w 0 P (w ; µ(w)), = 1,...,R. (8) Fo any such vecto w, the vecto d(w) = w/µ(w) solves SYSTEM. If the functions U ae stictly concave, such a vecto w is unique. Poof. Let d S be any solution to SYSTEM; as discussed above, at least one such solution exists. Let f S = ds, and define ws ds p(fs ) fo each. Obseve that with this definition, we have ws = ds p(f S ) = f S p(f S ); thus f S satisfies (7), and we have f(w S ) = f S, d(w S ) = d S. Given Assumptions 1 and 2, obseve that any solution to SYSTEM is identified by the following necessay and sufficient optimality conditions: ) U (ds ) = p ( s U (0) p ( s d S s d S s ), if d S, if d S > 0; (9) = 0. (10) Now, since p(0) = 0 but U (0) > 0 fo all, we must have fs = ds > 0; thus µ(w) = p(f S ) > 0. But then d S = w /p(f S ) fo each, so the peceding optimality conditions become: ( ) U w = p(f S ), if w p(f S > 0; ) U (0) p(f S ), if w = 0. These conditions ensue that (8) holds. Convesely, suppose we ae given a vecto w such that µ(w) > 0, and (8) holds. Then we simply evese the agument above: since (8) holds, we conclude that the optimality conditions (9)-(10) hold with d(w) = w/µ(w) = w/p(f(w)), so that d(w) is an optimal solution to SYSTEM. Finally, if the functions U ae each stictly concave, then the solution d S to SYSTEM is unique, so the pice p(f S ) is uniquely detemined as well. As a esult, fo each the poduct d S p(f S ) is unique, so the vecto w identified in the theoem must be unique as well. Theoem 2 shows that with an appopiate choice of pice function (as detemined by (4) and (6)), and unde the assumption that the uses behave as pice takes, thee exists a bid vecto w whee all uses have optimally chosen thei bids w, with espect to the given pice µ(w); and at this equilibium, the aggegate suplus is maximized. Howeve, when the pice taking assumption is violated, the model changes into a game and the guaantee of Theoem 2 is no longe valid. We investigate this game in the following section. 6

7 2 The Single Link Game We now conside an altenative model whee the uses of a single link ae pice anticipating, athe than pice taking, and play a game to acquie a shae of the link. Thoughout the emainde of this section and the next, we will assume that the link manage sets the pice µ(w) accoding to the unique choice pescibed by Poposition 1, as follows. Assumption 3 Fo anyw 0, the aggegate ate f(w) is the solution to (7): w = f(w)p(f(w)). Futhemoe, fo each, d (w) is given by: d (w) = 0, if w = 0; w p(f(w)), if w > 0. Note that we have f(w) > 0 and p(f(w)) > 0 if w > 0, and hence d is always well defined. We adopt 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 0 to maximize: (11) Q (w ;w ) U (d (w)) w, (12) ove nonnegative w. The payoff function Q is simila to the payoff function P, except that the use now anticipates that the netwok will set the pice accoding to Assumption 3, as captued by the allocated ate d (w). 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. (13) In the next section, we show that a Nash equilibium always exists, and give necessay and sufficient conditions fo a vecto w to be a Nash equilibium. In Section 2.2, we outline a class of pice functions fo which the Nash equilibium is unique. 2.1 Existence of Nash Equilibium In this section we establish that a Nash equilibium exists fo the game defined by (Q 1,..., Q R ). We stat by establishing cetain popeties of d (w) in the following poposition. Poposition 3 Suppose that Assumptions 1-3 hold. Then: (1) d (w) is a continuous function of w; and (2) fo any w 0, d (w) is stictly inceasing and concave in w 0, and d (w) as w. Poof. We fist show (1): that d (w) is a continuous function of w. Recall fom Poposition 1 that f(w) is a continuous function of w, and f(0) = 0. Now at any vecto w such that s w s > 0, we have p(f(w)) > 0, so d (w) = w /p(f(w)); thus continuity of d at w follows by continuity of f and p. Suppose instead that w = 0, and conside a sequence w n such that w n 0 as 7

8 n. Then d (w n ) = f(w n ) 0 as n, fom pats (1) and (2) of Poposition 1; since d (w n ) 0 fo all n, we must have d (w n ) 0 = d (0) as n, as equied. We now show (2): that d (w) is concave and stictly inceasing in w 0, with d (w) as w. Fom Assumption 3, we can ewite the definition of d (w) as: d (w) = 0, if w = 0; w s w f(w), if w > 0. s Fom this expession and Poposition 1, it follows that d (w) is stictly inceasing in w. To show d (w) as w, we only need that f(w) as w, a fact that was shown in Poposition 1. It emains to be shown that fo fixed w, d is a concave function of w 0. Since we have aleady shown that d is continuous, we may assume without loss of geneality that w > 0. We fist assume that p is twice diffeentiable. In this case, it follows fom (7) that f is twice diffeentiable in w. Since w > 0, we can diffeentiate (14) twice to find: 2 d (w) w 2 = 2 s w s ( s w s) f(w) + 2 s w s 3 ( s w s) f(w) + w 2 w s w s 2 f(w). w 2 Fom Poposition 1, f is a stictly concave function of s w s; thus the last tem in the sum above is nonpositive. To show that d is concave in w, theefoe, it suffices to show that the sum of the fist two tems is negative, i.e.: f(w) s w f(w). s w By diffeentiating both sides of (7), we find that: f(w) w = On the othe hand, fom (7), we have: 1 p(f(w)) + f(w)p (f(w)). f(w) s w s = 1 p(f(w)). Substituting these elations, and noting that f(w)p (f(w)) 0 since p is stictly inceasing, we have: f(w) 1 s w = s p(f(w)) 1 p(f(w)) + f(w)p (f(w)) = f(w), w as equied. Thus d (w) is concave in w, as long as p is twice diffeentiable. Now suppose that p is any pice function satisfying Assumption 2, but not necessaily twice diffeentiable. In this case, we may choose a sequence of twice diffeentiable pice functions p n satisfying Assumption 2, such that p n p pointwise as n (i.e., p n (f) p(f) as n, (14) 8

9 fo all f 0). 1 Let d n be the allocation function fo use when the pice function is p n ; then by the agument in the peceding paagaph, d n (w) is concave in w, fo each n. In ode to show that d (w) is concave in w, theefoe, it suffices to show that d n d pointwise as n. Fom (14), this will be tue as long as f n f pointwise as n, whee f n is the solution to (7) when the pice function is p n. Fix a bid vecto w; we now poceed to show that f n (w) f(w) as n. Fo each n, define g n (f) = fp n (f), and let g(f) = fp(f). By continuity, g n (f) g(f) as n, fo all f 0. Futhemoe, fom (7), w = g(f(w)). Fix ε > 0, and choose δ > 0 so that: { δ < min w g(f(w) ε), g(f(w) + ε) } w. (Note that such a choice is possible because g is stictly inceasing.) Now fo sufficiently lage n, we have: g n (f(w) ε) g(f(w) ε) < δ, and g(f(w) + ε) g n (f(w) + ε) < δ. Fom the definition of δ, this yields: g n (f(w) ε) < w < g n (f(w) + ε). Since g n is stictly inceasing, and f n (w) satisfies g n (f n (w)) = w, we conclude that f n (w) f(w) < ε fo sufficiently lage n, as equied. The pevious poposition establishes concavity and continuity of d ; this guaantees existence of a Nash equilibium, as the following poposition shows. Poposition 4 Suppose that Assumptions 1-3 hold. Then thee exists a Nash equilibium w fo the game defined by (Q 1,...,Q R ). Poof. We begin by obseving that we may estict the stategy space of each use to a compact set, without loss of geneality. To see this, fix a use, and a vecto w of bids fo all othe uses. Given a bid w fo use, we note that d (w) d (w ;0 ), whee 0 denotes the bid vecto whee all othe uses bid zeo. This inequality follows since w = d (w)p(f(w)); and if s w s deceases, then p(f(w)) deceases as well (fom Poposition 1), so d (w) must incease. We thus have Q (w ;w ) U (d (w ;0 )) w. By concavity of U, fo w > 0 we have: ( Q (w ;w ) U (0) + U (0)d U (w ;0 ) w = U (0) + w (0) ) p(f(w ;0 )) 1. (15) Now obseve fom (7) that: w = f(w ;0 )p(f(w ;0 )). 1 Define p(f) = 0 fo f 0, and conside a sequence of twice diffeentiable functions φ n, such that φ n has suppot on [ 1/n, 0], and φn (z) dz = 1. Then it is staightfowad to veify the sequence p n defined by p n (f) = p(z)φn (z f) dz has the equied popeties. 9

10 Since p is convex and stictly inceasing, we have lim f p(f) = ; thus we conclude that p(f(w ;0 )) as w. Consequently, using (15), thee exists B > 0 such that if w B, then Q (w ;w ) < U (0). Since Q (0;w ) = U (0), use would neve choose to bid w B at a Nash equilibium. Thus, we may estict the stategy space of use to the compact inteval S = [0, B ] without loss of geneality. The game defined by (Q 1,...,Q R ) togethe with the stategy spaces (S 1,...,S R ) is now a concave R-peson game: applying Poposition 3, each payoff function Q is continuous in the composite stategy vecto w, and concave in w (since U is concave and stictly inceasing, and d (w) is concave in w ); and the stategy space of each use is a compact, convex, nonempty subset of R. Applying Rosen s existence theoem [24], we conclude that a Nash equilibium w exists fo this game. In the emainde of this section, we establish necessay and sufficient conditions fo a vecto w to be a Nash equilibium. Because the pice function p may not be diffeentiable, we will use subgadients to descibe necessay local conditions fo a vecto w to be a Nash equilibium. Since the payoff of use is concave, these necessay conditions will in fact be sufficient fo w to be a Nash equilibium. We begin with some concepts fom convex analysis [3, 23]. An extended eal-valued function is a function g : R [, ]; such a function is called pope if g(x) > fo all x, and g(x) < fo at least one x. We say that a scala γ is a subgadient of an extended eal-valued function g at x if fo all x R, we have g(x) g(x) + γ(x x). The subdiffeential of g at x, denoted g(x), is the set of all subgadients of g at x. Finally, given an extended eal-valued function g, we denote the ight diectional deivative of g at x by + g(x)/ x and left diectional deivative of g at x by g(x)/ x (if they exist). If g is convex, then g(x) = [ g(x)/ x, + g(x)/ x], povided the diectional deivatives exist. Fo the emainde of the pape, we view any pice function p as an extended eal-valued convex function, by defining p(f) = fo f < 0. Ou fist step is a lemma identifying the diectional deivatives of d as a function of w ; fo notational convenience, we intoduce the following definitions of ε + (f) and ε (f), fo f > 0: ε + (f) f p(f) + p(f), ε (f) f f p(f) p(f). (16) f Note that unde Assumption 2, we have 0 < ε (f) ε + (f) fo f > 0. Lemma 5 Suppose Assumptions 1-3 hold. Then fo all w with s w s > 0, d (w) is diectionally diffeentiable with espect to w. These diectional deivatives ae given by: ( + d (w) 1 = 1 d ) (w) w p(f(w)) f(w) ε + (f(w)) ; (17) 1 + ε + (f(w)) ( d (w) 1 = 1 d ) (w) w p(f(w)) f(w) ε (f(w)). (18) 1 + ε (f(w)) Futhemoe, + d (w)/ w > 0, and if w > 0 then d (w)/ w > 0. 10

11 Poof. Existence of the diectional deivatives is obtained because d (w) is a concave function of w (fom Poposition 3). Fix a vecto w of bids, such that w > 0. Since f is an inceasing concave function of w, and the convex function p is diectionally diffeentiable at f(w) ([23], Theoem 23.1), we can apply the chain ule to compute the ight diectional deivative of (7) with espect to w : 1 = + f(w) w p(f(w)) + f(w) + p(f(w)) f Thus, as long as w > 0, + f(w)/ w exists, and is given by: + f(w) w. ( ) + 1 f(w) = p(f(w)) + f(w) + p(f(w)), w f We conclude fom (11) that the ight diectional deivative of d (w) with espect to w is given by: + d (w) w = 1 p(f(w)) w ( ) 2 + p(f(w)) + f(w). p(f(w)) f w Simplifying, this educes to (17). Note that since d (w) f(w) and ε + (f(w))/[1+ε + (f(w))] < 1, we have + d (w)/ w > 0. A simila analysis follows fo the left diectional deivative. Fo notational convenience, we make the following definitions fo f > 0: β + (f) ε+ (f) 1 + ε + (f), β (f) ε (f) 1 + ε (f). (19) Unde Assumption 2, we have 0 < β (f) β + (f) < 1 fo f > 0. The next poposition is the cental esult of this section: it povides simple local conditions that ae necessay and sufficient fo a vecto w to be a Nash equilibium. Poposition 6 Suppose that Assumptions 1-3 hold. Then w is a Nash equilibium of the game defined by (Q 1,..., Q R ), if and only if w > 0, and with d = d(w), f = f(w), the following two conditions hold fo all : ( ) U (d ) 1 β + (f) d p(f); (20) f ( ) U (d ) 1 β (f) d p(f), if d > 0. (21) f Convesely, if d 0 and f > 0 satisfy (20)-(21), and d = f, then the vecto w = p(f)d is a Nash equilibium with d = d(w) and f = f(w). Poof. We fist show that if w is a Nash equilibium, then we must have w > 0. Suppose not; then w = 0 fo all. Fix a use ; fo w > 0, we have d (w ;w )/w = f(w ;w )/w = 11

12 1/p(f(w ;w )), which appoaches infinity as w 0. Thus + d (w)/ w =, and thus we have: + Q (w ;w ) w = U (0) + d (w) w 1 =. In paticula, an infinitesimal incease of w stictly inceases the payoff of use, so w = 0 cannot be a Nash equilibium. Thus, if w is a Nash equilibium, then w > 0. Now let w be a Nash equilibium. We established in Lemma 5 that d is diectionally diffeentiable in w fo each, as long as s w s > 0. Thus, fom (13), if w is a Nash equilibium, then the following two conditions must hold: + Q (w ;w ) w Q (w ;w ) w = U (d (w)) + d (w) w 1 0; = U (d (w)) d (w) w 1 0, if w > 0. We may substitute using Lemma 5 to find that if w is a Nash equilibium, then: ( U (d (w)) 1 β + (f(w)) d(w) ) p(f(w)); f(w) ( U (d (w)) 1 β (f(w)) d(w) ) p(f(w)), if w > 0. f(w) Since the condition w > 0 is identical to the condition d (w) > 0, this establishes the conditions in the poposition. Convesely, if w > 0 and the peceding two conditions hold, then we may evese the agument: since the payoff function of use is a concave function of w fo each (fom Poposition 3), (20)-(21) ae sufficient fo w to be a Nash equilibium. Finally, suppose that d and f > 0 satisfy (20)-(21), with d = f. Then let w = d p(f). We then have w > 0 (since f > 0); and w = fp(f), so that f = f(w). Finally, since f > 0, we have d = w /p(f) = w /p(f(w)), so that d = d (w). Thus w is a Nash equilibium, as equied. Note that the peceding poposition identifies a Nash equilibium entiely in tems of the allocation made; and convesely, if we find a pai (d, f) which satisfies (20)-(21) with f > 0 and d = f, then thee exists a Nash equilibium which yields that allocation. In paticula, the set of allocations d which can aise at Nash equilibia coincides with those vectos d such that f = d > 0, and (20)-(21) ae satisfied. 2.2 Nondeceasing Elasticity Pice Functions: Uniqueness of Nash Equilibium In this section, we demonstate that fo a cetain class of diffeentiable pice functions, thee exists a unique Nash equilibium of the game defined by (Q 1,...,Q R ). We conside pice functions p which satisfy the following additional assumption. 12

13 Assumption 4 The pice function p is diffeentiable, and exhibits nondeceasing elasticity: fo 0 < f 1 f 2, thee holds: f 1 p (f 1 ) f 2p (f 2 ) p(f 1 ) p(f 2 ). To gain some intuition fo the concept of nondeceasing elasticity, conside a pice function p satisfying Assumption 2. The quantity fp (f)/p(f) is known as the elasticity of a pice function p [20]. Note that the elasticity of p(f) is the deivative of ln(p(f)) with espect to ln f. Fom this viewpoint, we see that nondeceasing elasticity is equivalent to the equiement that ln(p(f)) is a convex function in ln f. (Note that this is not equivalent to the equiement that p is a convex function of f.) Nondeceasing elasticity can also be intepeted by consideing the pice function as the invese of the supply function s(µ) = p 1 (µ); the supply function gives the amount of ate the povide is willing to supply at a given pice µ [20]. In this case, nondeceasing elasticity of the pice function is equivalent to noninceasing elasticity of the supply function. Nondeceasing elasticity captues a wide ange of pice functions; we give two common examples below. Example 1 (The M/M/1 Queue) Conside the cost function C(f) = af/(s f), whee a > 0 and s > 0 ae constants; then the cost is popotional to the steady-state queue size in an M/M/1 queue with sevice ate s and aival ate f. (Note that we must view p as an extended eal-valued function, with p(f) = fo f > s; this does not affect any of the analysis of this pape.) It is staightfowad to check that, as long as 0 < f < s, we have: fp (f) p(f) = 2f s f, which is a stictly inceasing function of f. Thus p satisfies Assumption 4. Example 2 (M/M/1 Oveflow Pobability) Conside the function p(f) = a(f/s) B, whee a > 0, s > 0, and B 1 is an intege. Then the pice is set popotional to the pobability that an M/M/1 queue exceeds a buffe level B, when the sevice ate is s and the aival ate is f. In this case we have fp (f)/p(f) = B, so that p satisfies Assumption 4. We now pove the key popety of diffeentiable nondeceasing elasticity pice functions in the cuent development: fo such functions, thee exists a unique Nash equilibium of the game defined by (Q 1,...,Q R ). Poposition 7 Suppose Assumptions 1-3 hold. If in addition p is diffeentiable and exhibits nondeceasing elasticity (Assumption 4 holds), then thee exists a unique Nash equilibium fo the game defined by (Q 1,...,Q R ). Poof. We use the expessions (20)-(21) to show that the Nash equilibium is unique unde Assumption 4. Obseve that in this case, fom (19), we may define β(f) = β + (f) = β (f) fo 13

14 f > 0, and conclude that w is a Nash equilibium if and only if s w s > 0 and the following optimality conditions hold: ( U (d (w)) 1 β(f(w)) d(w) ) = p(f(w)), if w > 0; (22) f(w) U (0) p(f(w)), if w = 0. (23) Suppose we have two Nash equilibia w 1, w 2, with 0 < s w1 s < s w2 s ; then p(f(w1 )) < p(f(w 2 )), and f(w 1 ) < f(w 2 ). Note that U (d ) is noninceasing as d inceases; and β(f) is nondeceasing as f inceases (fom Assumption 4), and theefoe β(f(w 1 )) β(f(w 2 )). Futhemoe, if w 2 > 0, then fom (22) we have U (0) > p(f(w2 )); thus U (0) > p(f(w1 )), so w 1 > 0 as well (fom (23)). Now note that the ight hand side of (22) is stictly lage at w 1 than at w 2 ; thus the left hand side must be stictly lage at w 1 than at w 2 as well. This is only possible if d (w 1 )/f(w 1 ) > d (w 2 )/f(w 2 ) fo each use, since we have shown in the peceding paagaph that f(w 1 ) < f(w 2 ); U (d ) is noninceasing as d inceases; and β(f(w 1 )) β(f(w 2 )). Since f(w) = d (w), we have: 1 = d (w2 ) f(w 2 ) < d (w1 ) f(w 1 ) = 1, :w 2>0 :w 2>0 which is a contadiction. Thus at the two Nash equilibia, we must have s w1 s = s w2 s, so we can let f 0 = f(w 1 ) = f(w 2 ), p 0 = p(f 0 ), and β 0 = β(f 0 ). Then all Nash equilibia w satisfy: ( ) U (d d (w) (w)) 1 β 0 = p 0, f 0 if w > 0; (24) U (0) p 0, if w = 0. (25) But now we obseve that the left hand side of (24) is stictly deceasing in d (w), so given p 0, thee exists at most one solution d (w) to (24). Since w = d (w)p 0, this implies the Nash equilibium w must be unique. We obseve that uniqueness of the Nash equilibium implies an additional desiable popety in the case of symmetic uses. If two uses shae the same utility function, and the pice function p is diffeentiable, we conclude fom Poposition 7 that at the unique Nash equilibium, these uses submit exactly the same bid (and hence eceive exactly the same ate allocation). 3 Efficiency Loss: The Single Link Case We let d S denote an optimal solution to SYSTEM, defined in (1)-(2), and let w denote any Nash equilibium of the game defined by (Q 1,..., Q R ). We now investigate the efficiency loss of this system; that is, how much aggegate suplus is lost because the uses attempt to game the system? To answe this question, we must compae the aggegate suplus U (d (w)) C( d (w)) 14

15 obtained when the uses fully evaluate the effect of thei actions on the pice, and the aggegate suplus U (d S ) C( ds ) obtained by choosing an allocation which maximizes aggegate suplus. The following theoem is the main esult of this pape: it states that the efficiency loss is no moe than appoximately 34%, and that this bound is essentially tight. Theoem 8 Suppose that Assumptions 1-3 hold. Suppose also that U (0) 0 fo all. Let d S be any solution to SYSTEM, and let w be any Nash equilibium of the game defined by (Q 1,..., Q R ). Then we have the following bound: ( suplus(d(w)) 4 ) 2 5 suplus(d S ), (26) whee suplus( ) is defined in (3). In othe wods, thee is no moe than appoximately a 34% efficiency loss when uses ae pice anticipating. Futhemoe, this bound is tight: fo evey δ > 0, thee exists a choice of R, a choice of (linea) utility functions U, = 1,..., R, and a (piecewise linea) pice function p such that a Nash equilibium w and a solution d S to SYSTEM exist with: ( suplus(d(w)) 4 ) δ suplus(d S ). (27) Poof. The poof of (26) consists of a sequence of steps: 1. We show that the wost case atio occus when the utility function of each use is linea. 2. We estict attention to games whee the total allocated Nash equilibium ate is f = We compute the wost case choice of linea utility functions, fo a fixed pice function p( ) and total Nash equilibium ate f = We pove that it suffices to conside a special class of piecewise linea pice functions. 5. Combining Steps 1-3, we compute the wost case efficiency loss by minimizing the atio of Nash equilibium aggegate suplus to maximal aggegate suplus, ove the wost case choice of games with linea utility functions (fom Step 2) and ou esticted class of piecewise linea pice functions (fom Step 3). Step 1: Show that we may assume without loss of geneality that U is linea fo each use ; i.e., without loss of geneality we may assume U (d ) = α d, whee α 1 = 1 and 0 < α 1 fo > 1. The poof of this claim is simila to the poof of Lemma 4 in [12]. Let d S denote any solution to SYSTEM, and let w denote a Nash equilibium, fo an abitay collection of utility functions (U 1,...,U R ) satisfying the assumptions of the theoem. We let d = d(w) denote the allocation vecto at the Nash equilibium. Fo each use, we define a new utility function U (d ) = α d, whee α = U (d ); we know that α > 0 by Assumption 1. Then obseve that if we eplace the utility functions (U 1,...,U R ) with the linea utility functions (U 1,...,U R ), the vecto w emains a Nash equilibium; this follows fom the necessay and sufficient conditions of Poposition 6. 15

16 We fist show that α d C(f) > 0. To see this, note fom (21) that α > p(f) fo all such that d > 0. Thus α d > d p(f) fo such a use, so α d > fp(f) C(f), by convexity (Assumption 2). Next, we note that U (d S ) C( ds ) > 0. This follows since U is stictly inceasing and nonnegative, while C (0) = p(0) = 0; thus if d is sufficiently small fo all, we will have U (d ) C( d ) > 0, which implies U (d S ) C( ds ) > 0 (since d S is a solution to SYSTEM). Using concavity, we have fo each : U (d S ) U (d ) + α (d S d ). Expanding the definition of suplus( ), we have: suplus(d) suplus(d S ) = U (d ) C( d ) U (d S ) C( ds ) ( ) U (d ) α d + α d C( d ) ( ) U (d ) α d + α d S C( ds ) ( ) U (d ) α d + α d C( d ) ( ) U (d ) α d + maxd 0 ( α d C( d ) ). (Note that all denominatos ae positive, since we have shown that U (d S ) C( ds ) > 0.) Since we assumed U (0) 0, we have U (d ) U (d )d 0 by concavity; and since 0 < α d C(f) max d 0 ( α d C( d )), we have the inequality: U (d ) C( d ) U (d S ) C( ds ) α d C( d ) max d 0 ( α d C( d ) ). Now obseve that the ight hand side of the pevious expession is the atio of the Nash equilibium aggegate suplus to the maximal aggegate suplus, when the utility functions ae (U 1,...,U R ); since this atio is no lage than the same atio fo the oiginal utility functions (U 1,...,U R ), we can estict attention to games whee the utility function of each use is linea. Finally, by eplacing α by α /(max s α s ), and the cost function C( ) by C( )/(max s α s ), we may assume without loss of geneality that max α = 1. Thus, by elabeling the uses if necessay, we assume fo the emainde of the poof that U (d ) = α d fo all, whee α 1 = 1 and 0 < α 1 fo > 1. Befoe continuing, we obseve that unde these conditions, we have the following elation: ( ( )) ( ) max α d C d = max f C(f). d 0 f 0 16

17 To see this, note that at any fixed value of f = d, the left hand side is maximized by allocating the entie ate f to use 1. Thus, the atio of Nash equilibium aggegate suplus to maximal aggegate suplus becomes: α d C( d ) ( ). (28) max f 0 f C(f) Note that the denominato is positive, since C (0) = p(0) = 0; and futhe, the optimal solution in the denominato occus at the unique value of f > 0 such that p(f) = 1. Step 2: Show that we may estict attention to games whee the total allocated ate at the Nash equilibium is f = 1. Fix a cost function C satisfying Assumption 2. Let w be a Nash equilibium, and let d = d(w) be the esulting allocation. Let f = d be the total allocated ate at the Nash equilibium; note that f > 0 by Poposition 6. We now define a new pice function ˆp accoding to ˆp( ˆf) = p(f ˆf), and a new cost function Ĉ( ˆf) = ˆf ˆp(z) dz; note that Ĉ( ˆf) = C(f ˆf)/f. 0 Then it is staightfowad to check that ˆp satisfies Assumption 2. We will use hats to denote the coesponding functions when the pice function is ˆp: ˆβ + ( ˆf), ˆβ ( ˆf), ˆd (w), ˆf(w), etc. Define ŵ = w /f. Then we claim that ŵ is a Nash equilibium when the pice function is ˆp. Fist obseve that: ŵ = w = p(f) = ˆp(1). f Thus ˆf(ŵ) = 1. Futhemoe: ˆd (ŵ) = ŵ ŵ = ˆp( ˆf(ŵ)) ˆp(1) = w fp(f) = d f. Finally, note that: +ˆp(1) ˆf = f + p(f), f fom which we conclude that ˆβ + (1) = β + (f), and similaly ˆβ (1) = β (f). Recall that w is a Nash equilibium fo the pice function p; thus, if we combine the peceding conclusions and apply Poposition 6, we have that ŵ is a Nash equilibium when the pice function is ˆp, with total allocated ate ˆf = 1 and allocation ˆd = d/f. To complete the poof of this step, we note the following chain of inequalities: α d C( d ) max ˆf 0 ( ˆf C( ˆf) ) = = α ˆd ( Ĉ(1) ) (29) max ˆf 0 ˆf/f C( ˆf)/f α ˆd Ĉ(1) max g 0 ( g Ĉ(g) ), (30) whee we make the substitution g = ˆf/f. But now note that the ight hand side is the atio of Nash equilibium aggegate suplus to maximal aggegate suplus fo a game whee the total allocated 17

18 ate at the Nash equilibium is equal to 1. Consequently, in computing the wost case efficiency loss, we may estict ou attention to games whee the Nash equilibium allocated ate is equal to 1. Step 3: Fo a fixed pice function p, detemine the instance of linea utility functions that minimizes Nash equilibium aggegate suplus, fo a fixed Nash equilibium allocated ate f = d = 1. Note that fixing the pice function p fixes the optimal aggegate suplus; thus minimizing the aggegate suplus at Nash equilibium also yields the wost case efficiency loss. We will optimize ove the set of all games whee uses have linea utility functions (satisfying the conditions of Step 1), and whee the total Nash equilibium ate is f = 1. We use the necessay and sufficient conditions of Poposition 6. Note that by fixing the pice function p and the total ate f > 0, the Nash equilibium pice is fixed, p(1), and β + (1) and β (1) ae fixed as well (fom the definition (19)); fo notational convenience, we abbeviate p = p(1), C = C(1), β + = β + (1), and β = β (1) fo the duation of this step. Since α 1 = 1, fo a fixed value of R the game with linea utility functions that minimizes aggegate suplus is given by solving the following optimization poblem (with unknowns d 1,...,d R, α 2,...,α R ): minimize d 1 + R α d C (31) =2 subject to α ( 1 β + d ) p, = 1,...,R; (32) ( ) α 1 β d p, if d > 0, = 1,...,R; (33) R d = 1; (34) =1 0 < α 1, = 2,...,R; (35) d 0, = 1,...,R. (36) (Note that we have applied Poposition 6: if we solve the peceding poblem and find an allocation d and coefficients α, then thee exists a Nash equilibium w with d = d(w).) The objective function is the aggegate suplus given a Nash equilibium allocation d. The conditions (32)-(33) ae equivalent to the Nash equilibium conditions established in Poposition 6. The constaint (34) ensues that the total allocation made is equal to 1, and the constaint (35) follows fom Step 1. The constaint (36) ensues the ate allocated to each use is nonnegative. We solve this poblem though a sequence of eductions. We fist show we may assume without loss of geneality that the constaint (33) holds with equality fo all uses = 2,..., R. The esulting poblem is symmetic in the uses = 2,...,R; we next show that a feasible solution exists if and only if 1 β + p < 1 and R is sufficiently lage, and we conclude using a convexity agument that d = (f d 1 )/(R 1) at an optimal solution. Finally, we show the wost case occus in the limit whee R, and calculate the esulting Nash equilibium aggegate suplus. We fist show that it suffices to optimize ove all (α,d) such that (33) holds with equality fo = 2,..., R. Note that if (α,d) is a feasible solution to (31)-(36), then fom (33)-(36), and the fact that 0 < β < 1, we conclude that p < 1. Now if d > 0 fo some = 2,...,R, but the coesponding constaint in (33) does not hold with equality, we can educe α until the constaint 18

19 in (33) does hold with equality; by this pocess we obtain a smalle value fo the objective function (31). On the othe hand, if d = 0 fo some = 2,...,R, we can set α = p; since p < 1, this peseves feasibility, but does not impact the tem α d in the objective function (31). We can theefoe estict attention to feasible solutions fo which: α = p 1 β d, = 2,..., R. (37) Having done so, obseve that the constaint (35) that α 1 may be witten as: d 1 p, = 2,...,R. β Finally, the constaint (35) that α > 0 becomes edundant, as it is guaanteed by the fact that d 1 (fom (34)), β < 1 (by definition), and (37). We now use the peceding obsevations to simplify the optimization poblem (31)-(36) as follows: minimize d 1 + p R =2 d 1 β d C (38) subject to 1 β + d 1 p 1 β d 1 ; (39) R d = 1; (40) =1 d 1 p, = 2,...,R; (41) β d 0, = 1,...,R. (42) The objective function (38) equals (31) upon substitution fo α fo = 2,...,R, fom (37). We know that d 1 > 0 when p(f) < 1 (fom (32)-(33)); thus the constaint (39) is equivalent to the constaints (32)-(33) fo use 1 with d 1 > 0. The constaint (32) fo > 1 is edundant and eliminated, since (33) holds with equality fo > 1. The constaint (40) is equivalent to the allocation constaint (34); and the constaint (41) ensues α 1, as equied in (35). We fist note that fo a feasible solution to (38)-(42) to exist, we must have 1 β + p < 1. We have aleady shown that we must have p < 1 if a feasible solution exists. Futhemoe, fom (39) we obseve that the smallest feasible value of d 1 is d 1 = (1 p)/β +. We equie d 1 1 fom (40) and (42), so we must have (1 p)/β + 1, which yields the estiction that 1 β + p. Thus, thee only exist Nash equilibia with total ate 1 and pice p if: 1 β + p < 1. (43) We will assume fo the emainde of this step that (43) is satisfied. We note that if d = (d 1,...,d R ) is a feasible solution to (38)-(42) with R uses, then letting d R+1 = 0, the vecto (d 1,...,d R+1 ) is a feasible solution to (38)-(42) with R + 1 uses, and with 19

20 the same objective function value (38) as d. Thus, the minimal objective function value cannot incease as R inceases, so the wost case efficiency loss occus in the limit whee R. We now solve (38)-(42) fo a fixed feasible value of d 1. Fom the constaints (40)-(41), we obseve that a feasible solution to (38)-(42) exists if and only if the following condition holds in addition to (43): d 1 + (R 1) 1 p 1. (44) β In this case, the following symmetic solution is feasible: d = 1 d 1, = 2,...,R. (45) R 1 Futhemoe, since the objective function is stictly convex and symmetic in the vaiables d 2,...,d R, and the feasible egion is convex, the symmetic solution (45) must be optimal. If we substitute the optimal solution (45) into the objective function (38) and take the limit as R, then the constaint (44) is vacuously satisfied, and the objective function becomes d 1 +p(1 d 1 ) C. Since we have shown that p < 1, the wost case occus at the smallest feasible value of d 1 ; fom (39), this value is: d 1 = 1 p β. (46) + The esulting wost case Nash equilibium aggegate suplus is: p + (1 p)2 β + C. To complete the poof of the theoem, we will conside the atio of this Nash equilibium aggegate suplus to the maximal aggegate suplus; we denote this atio by F(p) as a function of the pice function p( ): F(p) = p(1) + (1 p(1))2 /β + (1) C(1). (47) max f 0 (f C(f)) Note that hencefoth, the scala p used thoughout Step 3 will be denoted p(1), and we etun to denoting the pice function by p. Thus F(p) as defined in (47) is a function of the entie pice function p( ). Fo completeness, we summaize in the following lemma an intemediate tightness esult which will be necessay to pove the tightness of the bound in the theoem. Lemma 9 Suppose that Assumptions 2 and 3 ae satisfied. Then thee exists R > 0 and a choice of linea utility functions U (d ) = α d, whee α 1 = max s α s = 1, with total Nash equilibium ate 1, if and only if (43) is satisfied, i.e.: 1 β + (1) p(1) < 1. (48) In this case, given δ > 0, thee exists R > 0 and a collection of R uses whee use has utility function U (d ) = α d, such that d is a Nash equilibium allocation with d = 1, and: α d C(1) max d 0 ( α d C( d F(p) + δ. (49) )) 20

21 Poof of Lemma. The poof follows fom Step 3. We have shown that if thee exists a Nash equilibium with total ate 1, then (48) must be satisfied. Convesely, if (48) is satisfied, we poceed as follows: define d 1 accoding to (46); choose R lage enough that (44) is satisfied; define d accoding to (45); and then define α accoding to (37) with p = p(1). Then it follows that (d, α) is a feasible solution to (31)-(36), which (by Poposition 6) guaantees thee exists a Nash equilibium whose total allocated ate equals 1. The bound in (49) then follows by the poof of Step 3. The emainde of the poof amounts to minimizing the wost case atio of Nash equilibium aggegate suplus to maximal aggegate suplus, ove all valid choices of p. A valid choice of p is any pice function p such that at least one choice of linea utility functions satisfying the conditions of Step 1 leads to a Nash equilibium with total allocated ate 1. By Lemma 9, all such functions p ae chaacteized by the constaint (48). We will minimize F(p), given by (47), ove all choices of p satisfying (48). Step 4: Show that in minimizing F(p) ove p satisfying (48), we may estict attention to functions p satisfying the following conditions: { af, 0 f 1; p(f) = (50) a + b(f 1), f 1; 0 < a b; (51) 1 a + b 1 < 1 a. (52) Obseve that p as defined in (50)-(52) is a convex, stictly inceasing, piecewise linea function with two pats: an initial segment which inceases at slope a > 0, and a second segment which inceases at slope b a. In paticula, such a function satisfies Assumption 2. Futhemoe, we have + p(1)/ f = b, so that ε + (1) = b/a. This implies β + (1) = b/(a + b); thus, multiplying though (52) by a yields (48). To veify the claim of Step 4, we conside any function p such that (48) holds. We define a new pice function p as follows: fp(1), 0 f 1; p(f) = p(1) + + p(1) (f 1), f 1. f (See Figue 1 fo an illustation.) Let a = p(1), and let b = + p(1)/ f. Then a > 0; and since p(0) = 0, we have + p(1)/ f p(1) by convexity of p, so that b a. Futhemoe, since p(1) < 1 fom (48), we have 1/a > 1. Finally, we have: 1 a + b = 1 p(1) (1 β+ (1)) 1, (53) 21

22 p(f) p(f) p(1) 1 f Figue 1: Poof of Theoem 8, Step 4: Given a pice function p (solid line) and Nash equilibium ate 1, a new pice function p (dashed line) is defined accoding to (53). whee the equality follows fom the definition of β + (1) and the inequality follows fom (48). Thus p satisfies (50)-(52). Obseve also that p(1) = p(1), and + p(1)/ f = + p(1)/ f, and thus β + (1) = β + (1). We now show that F(p) F(p). As an intemediate step, we define a new pice function ˆp( ) as follows: { p(f), 0 f 1; ˆp(f) = p(f), f 1. Of couse, ˆp(1) = p(1) and +ˆp(1)/ f = + p(1)/ f = + p(1)/ f, so that (48) is satisfied fo ˆp. Let Ĉ(f) = f ˆp(z) dz denote the cost function associated with ˆp( ). Obseve that (by convexity 0 of p), we have fo all f that ˆp(f) p(f), so that Ĉ(f) C(f). Thus: max f 0 (f Ĉ(f)) max(f C(f)). Futhemoe, Ĉ(1) = C(1) so that F(ˆp) F(p). Next, we let C(f) = f p(z) dz denote the cost function associated with p( ). By convexity 0 of p, we know p(1) p(1) fo 0 f 1; thus C(f) C(f) in that egion. We let f 0 22

23 C(1) C(1) 0. Then we have the following elationship: F(ˆp) = p(1) + (1 p(1))2 /β + (1) C(1) max f 0 ( f Ĉ(f) ) (54) p(1) + (1 p(1))2 /β + (1) (C(1) + ) max f 0 ( f (Ĉ(f) + ) ) (55) = F(p). (56) The last equality follows by obseving that since ˆp(1) = p(1) < 1, the solution to max f 0 (f Ĉ(f)) occus at ˆfS > 1 whee ˆp( ˆf S ) = 1; and at all points f 1, we have the elationship Ĉ(f) + = C(f). Combining the peceding esults, we have F(p) F(p), as equied. Step 5: The minimum value of F(p) ove all p satisfying (50)-(52) is We fist show that given p satisfying (50)-(52), F(p) is given by: 1 2 F(p) = a + ( ) 1 + a b (1 a) 2 ab + 2(a + b)(1 a)2 1 1a + =. (57) 1 (1 a) 2 2b ab + (1 a) b The numeato esults by simplifying the numeato of (47), when p takes the fom descibed by (50)-(52). To aive at the denominato, we note that the solution to max f 0 (f C(f)) occus at f S satisfying p(f S ) = 1. Since a < 1, we must have f S > 1 and: Simplifying, we find: a + b(f S 1) = 1. f S = a. (58) b The expession f S C(f S ), upon simplification, becomes the denominato of (57), as equied. Fix a and b such that 0 < a b, and 1/(a + b) 1 < 1/a, and define p as in (50). We note hee that the constaints 0 < a b and 1/(a + b) 1 < 1/a may be equivalently ewitten as 0 < a < 1, and max{a, 1 a} b. Define H(a, b) F(p); fom (57), note that fo fixed a, H(a, b) is a atio of two affine functions of b, and thus the minimal value of H(a, b) is achieved eithe when b = max{a, 1 a} o as b. Define H 1 (a) = H(a, b) max{a,1 a}, and H 2 (a) = lim b H(a, b). Then: H(a, b) b=1 a = 2 a, if 0 < a 1/2; 3 2a H 1 (a) = (59) H(a, b) b=a = a 2 + 4a(1 a) 2, if 1/2 a < 1; H 2 (a) = lim b H(a, b) = a + 2(1 a)2. (60) 2 a 23