A Game Theoretic View of Efficiency Loss in Resource Allocation

Transcription

1 A Game Theoetic View of Efficiency Loss in Resouce Allocation Ramesh Johai 1 and John N. Tsitsiklis 2 1 Stanfod Univesity, Stanfod, CA amesh.johai@stanfod.edu 2 MIT, Cambidge, MA jnt@mit.edu Dedicated to Pavin Vaaiya on the occasion of his 65th bithday Summay. Motivated by esouce allocation poblems in communication netwoks as well as powe systems, we conside the design of maket mechanisms fo such settings which ae obust to gaming behavio by maket paticipants. Recent esults in this wok ae eviewed, including: (1) efficiency loss guaantees fo a data ate allocation mechanism fist poposed by Kelly, both when link capacities ae fixed and when they ae elastic; (2) chaacteization of mechanisms that minimize the efficiency loss, within a cetain class of simple mechanisms; (3) extensions to geneal netwoks; and (4) mechanism design fo supply function bidding in electic powe systems. 1 Intoduction This pape addesses a poblem at the nexus of engineeing, compute science, and economics: in lage scale, decentalized systems, how can we efficiently allocate scace esouces among competing inteests? On one hand, constaints ae imposed on the system designe by the inheent achitectue of any lage scale system. These constaints ae countebalanced by the need to design mechanisms that efficiently allocate esouces, even when the system is being used by paticipants who have only thei own inteests at stake. We conside two main classes of esouce allocation poblems. Fist, we conside a setting whee a esouce in scace supply must be allocated among multiple competing consumes. Second, we discuss a setting whee multiple poduces compete to satisfy a fixed demand. The fome model is motivated by applications to communication netwoks, while the latte is motivated by electic powe maket design. What goals might we have fo makets in such settings? We would of couse like the equilibia of mechanisms designed fo such settings to be desiable; a common equiement is that equilibia should be Paeto efficient.

2 2 Ramesh Johai and John N. Tsitsiklis In othe contexts, we want the equilibia to satisfy a pedetemined notion of fainess; o we may wish the esulting vecto of monetay tansfes to satisfy cetain popeties, such as pofit maximization fo the maket opeato. Beyond such constaints on the popeties at equilibium, howeve, we ae also concened with the complexity of such mechanisms. In paticula, we may desie mechanisms which have elatively low infomation ovehead: the stategy spaces of the playes should be simple, and the feedback fom the maket to the playes should be simple as well. Often, such complexity issues aise in a discussion of the dynamic behavio of maket mechanisms, in tying to detemine whethe equilibia ae actually achieved ove time by playes. In this pape, we will focus on efficiency of mechanisms which maintain low complexity, appopiately defined. We focus on efficiency pimaily as a fist test of feasibility. Taditionally, economics has focused on selection of efficient mechanisms because mechanisms with inefficient equilibia ae less likely to be useful in pactice. Indeed, the classical theoy of mechanism design is lagely devoted to detemining when fully efficient equilibia can be guaanteed (see, e.g., Chapte 23 of [23] fo an oveview). The landmak contibution of mechanism design is the Vickey-Clake- Goves class of mechanisms, which guaantee efficient allocations at dominant stategy equilibia [4, 11, 32]; unfotunately, implementing VCG mechanisms is geneally a vey complex poposition with many possible pitfalls [2, 27]. The task is futhe complicated by the fact that the VCG class of mechanisms ae essentially the only class which guaantee fully efficient outcomes as dominant stategy equilibia [8]. Thus, to make pogess, eithe the notion of equilibium must be weakened, o some efficiency must be lost. Pevious esults in the economics liteatue have consideed weakening the notion of equilibium; fo example, Maskin has shown that if we only conside Nash equilibia, efficiency can be guaanteed if cetain conditions ae satisfied by playes chaacteistics [24]. Howeve, no guidance is available as to how to design such mechanisms with low complexity. We conside an altenate appoach, by weakening of the equiement of efficiency. The basic technique we conside is one of esticting the stategy spaces of the playes (eithe buyes o selles). With the pope choice of estiction, we can achieve two goals simultaneously. Fist, by ensuing that stategy spaces ae elatively simple, we can estict attention to mechanisms with low complexity. Second, if stategies of playes ae esticted, we can educe thei oppotunities to game the system; this will lead to povable bounds on efficiency loss at Nash equilibia. In the emainde of the pape, we povide an oveview of the pogess made in ou ealie wok [17, 16, 15]. In Section 2, we conside a setting of multiple consumes and inelastic supply, motivated by ate allocation in communication netwoks. Fo a single link of fixed capacity, we investigate a esouce allocation mechanism poposed by Kelly [18]. Netwok uses choose bids, which denote the total amount they ae willing to pay. A pice is then chosen to clea the maket; fo the case of a single link, this allocation mech-

3 A Game Theoetic View of Efficiency Loss in Resouce Allocation 3 anism allocates factions of the esouce to the uses in popotion to thei bids. Kelly has peviously shown that if uses ae pice taking that is, if they do not anticipate the effects of thei actions on the maket-cleaing pice the esulting competitive equilibium allocation is fully efficient. Ou key esult in this section is that when uses ae pice anticipating, aggegate utility falls by no moe than 25% elative to the maximum possible. In Section 3, we conside the same basic mechanism as in Section 2, but now conside a setting whee supply is elastic; this is the model consideed by Kelly et al. [19]. In this case the link is chaacteized by a cost depending on the total allocated ate, athe than a fixed capacity. Again, Kelly et al. have peviously shown that if uses ae pice taking, this mechanism maximizes aggegate suplus (i.e., aggegate utility minus cost). Fo this setting we establish that when uses ae pice anticipating, aggegate suplus falls by no moe than appoximately 34% elative to the maximum possible. Sections 2 and 3 establish efficiency loss esults fo a specific maket mechanism. In Section 4, we chaacteize the mechanism studied in Section 2 as the best choice of mechanism unde easonable assumptions. Fomally, we show that in a class of maket-cleaing mechanisms satisfying cetain simple mathematical assumptions and fo which thee exist fully efficient competitive equilibia, the mechanism of Section 2 uniquely minimizes efficiency loss when maket paticipants ae pice anticipating. These esults justify the attention devoted to undestanding the paticula maket mechanism studied in Sections 2 and 3; futhemoe, they clealy delineate conditions which must be violated if we hope to achieve highe efficiency guaantees than those povided by the esults of Sections 2 and 3. In Section 5, we summaize two futhe diections of eseach. Fist, in Section 5.1, we discuss the genealization of the models of Sections 2 and 3 to netwoks with abitay topology. We conside games whee uses submit individual bids to each link in the netwok. Such games ae then poven to have the same efficiency loss guaantees as the single link games consideed in Sections 2 and 3. Next, in Section 5.2, motivated by powe systems, we discuss a setting whee multiple poduces bid to satisfy an inelastic demand D. We conside a maket mechanism whee poduces submit supply functions esticted to lie in a cetain one-paamete family, and a maket-cleaing pice is chosen to ensue that aggegate supply is equal to the inelastic demand. We establish that when poduces ae pice anticipating, aggegate poduction cost ises by no moe than a facto 1 + 1/(N 2) elative to the minimum possible poduction cost, whee N > 2 is the numbe of fims competing. Finally, we conclude with some open issues in Section 6.

4 4 Ramesh Johai and John N. Tsitsiklis 2 Multiple Consumes, Inelastic Supply 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 make the following assumptions on the utility function. Assumption 1. Fo each, ove the domain d 0 the utility function U (d ) is concave, stictly inceasing, and continuous; and ove the domain d > 0, U (d ) is continuously diffeentiable. Futhemoe, the ight diectional deivative at 0, denoted U (0), is finite. Given complete knowledge and centalized contol of the system, it would be natual fo the link manage to ty to solve is the following optimization poblem [18]: maximize subject to U (d ) (1) d C; (2) d 0, = 1,..., R. (3) Note that the objective function of this poblem is the aggegate utility. Since the objective function is continuous and the feasible egion is compact, an optimal solution d = (d 1,..., d R ) exists. If the functions U ae stictly concave, then the optimal solution is unique, since the feasible egion is convex. 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 submits a payment (also called a bid) 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 wods, the link 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, 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 possible pice µ > 0, given by the pevious equation.

5 A Game Theoetic View of Efficiency Loss in Resouce Allocation 5 We can intepet this mechanism as a maket-cleaing pocess by which a pice is set so that demand equals supply. To see this intepetation, note that when a use submits a total payment w, it is as if the use has submitted a demand function D(p, w ) = w /p fo p > 0. The demand function descibes the ate that the use demands at any given pice p > 0. The link manage then chooses a pice µ so that D(µ, w ) = C, i.e., so that the aggegate demand equals the supply C. Fo the specific fom of demand functions we conside hee, this leads to the expession fo µ given in (4). Use then eceives a ate allocation given by D(µ, w ), and makes a payment µd(µ, w ) = w. This intepetation of the mechanism we conside hee will be futhe exploed in Section 4, whee we conside othe maket-cleaing mechanisms with the uses submitting demand functions fom a family paametized by a single scala. 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 povide a esult, due to Kelly [18], on the existence of a competitive equilibium. Futhemoe, this competitive equilibium leads to an allocation which is an optimal solution to (1)-(3). In Section 2.2, we change the model and assume uses ae pice anticipating, and povide a esult (due to Hajek and Gopalakishnan [12]) on the existence and uniqueness of a Nash equilibium. In Section 2.3, we then conside the loss of efficiency at this Nash equilibium, elative to the optimal solution to (1)-(3). 2.1 Pice Taking Uses and Competitive Equilibium In this section, we conside a competitive equilibium between the uses and the link manage [23], following the development of Kelly [18]. 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 [23]. We say that 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)

6 6 Ramesh Johai and John N. Tsitsiklis Kelly shows in [18] that when uses ae pice takes, thee exists a competitive equilibium, and the esulting allocation is an optimal solution to (1)-(3). This is fomalized in the following theoem, adapted fom [18]. Theoem 1 (Kelly, [18]). Suppose that Assumption 1 holds. 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 an optimal solution to (1)-(3). If the functions U ae stictly concave, then w is uniquely detemined as well. Theoem 1 shows that unde the assumption that the uses of the link 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 /C; and at this equilibium, aggegate utility is maximized. Howeve, when the pice taking assumption is violated, the model changes into a game and the guaantee of Theoem 1 is no longe valid. 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 (8) U (0), if w = 0. ove all 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. (9) 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 (1)-(3), but with modified utility functions. This is fomalized in the following theoem, adapted fom [12].

7 A Game Theoetic View of Efficiency Loss in Resouce Allocation 7 Theoem 2 (Hajek and Gopalakishnan, [12]). Suppose that R > 1, and that Assumption 1 holds. 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 = w s w C, = 1,..., R, (10) s is the unique optimal solution to the following optimization poblem: maximize subject to Û (d ) (11) d C; (12) d 0, = 1,..., R, (13) whee Û (d ) = ( 1 d ) U (d ) + C ( ) ( ) d 1 d U (z) dz. (14) C d 0 Theoem 2 shows that the unique Nash equilibium of the game is chaacteized as the solution to the above optimization poblem. Othe games have also pofited fom such elationships notably taffic outing games, in which Nash-Wadop equilibia can be found as solutions to a elated global optimization poblem. Roughgaden and Tados use this fact to thei advantage in computing efficiency loss fo such games [28]; Coea, Schulz, and Stie Moses also use this elationship to conside outing games in capacitated netwoks [5]. Finally, we note that fo the game pesented hee, seveal authos have deived esults simila to Theoem 2 [7, 21, 22], though not as geneal. 2.3 Efficiency Loss We let d S denote an optimal solution to (1)-(3), and let d G denote the unique optimal solution to (11)-(13). We now investigate the efficiency loss of this system; that is, the utility loss caused by the pice anticipating behavio of the uses. Moe pecisely, we will compae the utility U (d G ) obtained when the uses fully evaluate the effect of thei actions on the pice, and the maximum possible aggegate utility U (d S ). (We know, of couse, that U (d G ) U (d S ), by definition of d S.) Accoding to the following theoem, the wost case efficiency loss is exactly 25%; the poof may be found in [17]. Theoem 3. Suppose that R > 1, and that Assumption 1 holds. Suppose also that U (0) 0 fo all. If d S is any optimal solution to (1)-(3), and d G is the unique optimal solution to (11)-(13), then:

8 8 Ramesh Johai and John N. Tsitsiklis U (d G ) 3 U (d S ). 4 Futhemoe, this bound is tight: fo evey ε > 0, thee exists a choice of R, and a choice of (linea) utility functions U, = 1,..., R, such that: ( ) ( ) 3 U (d G ) 4 + ε U (d S ). We povide some comments on the method fo poving a esult such as Theoem 3. The fist step is to show that the wost case efficiency loss occus when the utility functions belong to a cetain finite-dimensional family; in the cuent context, it is the family of linea utility functions. Identifying the wost case utility functions amounts to minimizing an efficiency measue ove all possible choices of the coefficients of the linea utility functions. It tuns out that this minimization can be cast as a sequence of finite-dimensional nonlinea optimization poblems (each poblem in the sequence coesponding to a diffeent numbe R of uses), which can be studied analytically. In the context of Theoem 3, the wost efficiency loss coesponds to 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. As R, at the Nash equilibium of the game, use 1 eceives a ate 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 [28]. They found that the atio of wost case Nash equilibium cost to optimal cost was 4/3. Howeve, it is questionable 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 efficiency loss due to selfish uses in taffic outing may be abitaily high if link latency functions ae nonlinea. 3 Multiple Consumes, Elastic Supply In this section, we allow the supply of the scace esouce to be elastic, athe than fixed as in the pevious section. Rathe than being chaacteized by a capacity, we will chaacteize the esouce though a cost function that gives the cost incued by the esouce as a function of the flow though it. We continue to assume that R uses shae a single communication link, and that use eceives a utility U (d ) if the allocated ate is d. 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

9 A Game Theoetic View of Efficiency Loss in Resouce Allocation 9 allocation d, and C(f) is a monetay cost fo congestion at the link when the total allocated ate is f. We continue to assume the utility functions U satisfy Assumption 1. In addition, we make the following assumption on the cost function C. 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 0 p(z)dz. Thus C(f) is stictly convex and stictly inceasing. Given complete knowledge and centalized contol of the system, it would be natual fo the link manage to ty to solve the following optimization poblem[18]: maximize ( ) U (d ) C d (15) subject to d 0, = 1,..., R. (16) We efe to the objective function (15) as the aggegate suplus. This is the net monetay benefit to the economy consisting of the uses and the single link. Since the objective function is continuous, and U inceases at most linealy, while C inceases supelinealy, an optimal solution d S = (d S 1,..., d S R ) exists; 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 conside the following picing scheme fo ate allocation, a natual analogue of the mechanism pesented in Section 2. Each use submits a payment (o bid) of w to the esouce manage. Given the composite 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. Notice that, with this fomulation, the ate allocated to use is simila to the ate allocated to use in the model of Section 2. The key diffeence in this setting is that the aggegate ate is not constained to an inelastic supply; athe, associated with the choice of pice µ(w) is an aggegate ate function f(w), defined by:

10 10 Ramesh Johai and John N. Tsitsiklis f(w) = 0, if w = 0; d (w) = w µ(w), if. (17) w > 0. Let us assume fo now that given a pice µ > 0, use wishes to maximize the following payoff function ove w 0: ( ) w P (w ; µ) = U w. (18) 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. 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; that is, if the pice function satisfies: µ(w) = p(f(w)). (19) The well-posedness of such a picing mechanism is the subject of the following poposition. Poposition 1. Suppose Assumption 2 holds. Given any vecto of bids w 0, thee exists a unique pai (µ(w), f(w)) 0 satisfying (17) and (19), and in this case f(w) is the unique solution f to: w = fp(f). (20) 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. Obseve that we can view (20) 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 [19] which amount to dynamic pocesses of maket-cleaing; as a esult, a key motivation fo the mechanism we study in this section is that it epesents the equilibium behavio of the algoithms in [19]. Fo the emainde of this section, we will assume that µ(w) is set accoding to the choice pescibed in Poposition 1, as follows.

11 A Game Theoetic View of Efficiency Loss in Resouce Allocation 11 Assumption 3. Fo all w 0, the aggegate ate f(w) is the unique solution to (20): w = f(w)p(f(w)). Futhemoe, fo each, d (w) is given by: 0, if w = 0; d (w) = w p(f(w)), if w > 0. (21) Note that we have f(w) > 0 and p(f(w)) > 0 if w > 0, and hence d is always well defined. In the emainde of this section, we conside two diffeent models fo how uses might inteact with this pice mechanism. In Section 3.1, we conside a model whee uses do not anticipate the effect of thei bids on the pice, in which case thee exists a competitive equilibium. Futhemoe, this competitive equilibium leads to an allocation which is an optimal solution to (15)-(16). In Section 3.2, we change the model and assume uses ae pice anticipating, in which case thee exists a Nash equilibium. Finally, Section 3.3 consides the loss of efficiency at Nash equilibia, elative to the optimal solution to (15)-(16). 3.1 Pice Taking Uses and Competitive Equilibium Kelly et al. show in [19] that when uses ae pice takes, and the netwok sets the pice µ(w) accoding to (17) and (19), the esulting allocation is an optimal solution to (15)-(16). This is fomalized in the following theoem, adapted fom [19]. Theoem 4 (Kelly et al., [19]). Suppose Assumptions 1, 2, and 3 hold. 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. (22) Fo any such vecto w, the vecto d(w) = w/µ(w) is an optimal solution to (15)-(16). If the functions U ae stictly concave, such a vecto w is unique as well. Theoem 4 shows that with an appopiate choice of pice function (as detemined by (17) and (19)), and unde the assumption that the uses of the link 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 4 is no longe valid.

12 12 Ramesh Johai and John N. Tsitsiklis 3.2 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 taking, and play a game to acquie a shae of the link. Thoughout the emainde of this section as well as in Section 3.3, we will assume that the link manage sets the pice µ(w) accoding to the unique choice pescibed by Poposition 1, as follows. 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 ). Then given w, each use chooses w 0 to maximize: Q (w ; w ) = U (d (w)) w, (23) 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. (24) The poof of the following poposition can be found in [16]. Poposition 2. Suppose that Assumptions 1, 2, and 3 hold. Then thee exists a Nash equilibium w fo the game defined by (Q 1,..., Q R ). 3.3 Efficiency Loss We let d S denote an optimal solution to (15)-(16), and let w denote any Nash equilibium of the game defined by (Q 1,..., Q R ). We now investigate the associated efficiency loss. In paticula, we compae the aggegate suplus U (d (w)) C( d (w)) 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. Accoding to the following theoem, the efficiency loss is no moe than appoximately 34%, and this bound is essentially tight; the poof can be found in [16]. Theoem 5. Suppose that Assumptions 1, 2, and 3 hold. Suppose also that U (0) 0 fo all. If d S is any optimal solution to (15)-(16), and w is any Nash equilibium of the game defined by (Q 1,..., Q R ), then: ( ) U (d (w)) C d (w) ( ) ( ( U (d S ) C (25) In othe wods, thee is no moe than appoximately a 34% efficiency loss when uses ae pice anticipating. d S )).

13 A Game Theoetic View of Efficiency Loss in Resouce Allocation 13 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 exists with: ( ) U (d (w)) C d (w) ( δ ) ( ( U (d S ) C d S )) (26) Let us emak hee that, accoding to the poof of Theoem 5, the wost possible efficiency loss is achieved along a sequence of games whee: 1. The pice function p has the following fom, with b : { (2 2)f, if 0 f 1; p(f) = b(f 1), if f 1; 2. The numbe of uses becomes lage (R ); and 3. Use 1 has linea utility with U 1 (d 1 ) = d 1, and all othe uses have linea utility with U (d ) = α d, whee α p(1) = 2 2. (Note that fomally, we must take cae that the limits of R and b ae taken in the coect ode; in paticula, in the poof we fist have R, and then b.) In this limit, we find that at the Nash equilibium the aggegate allocated ate is 1, and the Nash equilibium aggegate suplus conveges to A Chaacteization Theoem In this section we evisit the esouce allocation poblem of Section 2, and addess the following question: can we identify a mechanism that minimizes the efficiency loss, in the pesence of pice anticipating uses, within a class of mechanisms with cetain desiable popeties? Fomally, we conside a collection of uses bidding to eceive a shae of a finite, infinitely divisible esouce of capacity C. Each use has a utility function U : R + R + (whee R + = [0, )) that satisfies Assumption 1. Moe specifically, U belongs to the set U utility functions defined by U = { U : R + R + U is continuous, stictly inceasing, concave on [0, ), and continuously diffeentiable on [0, ), with U (0) < }. We let R denote the numbe of uses, and let U = (U 1,..., U R ) denote the vecto of utility functions, whee U is the utility function of use. We call a pai (R, U), whee R > 1 and U U R, a utility system; ou goal will be to design a esouce allocation mechanism with attactive efficiency guaantees fo all utility systems.

14 14 Ramesh Johai and John N. Tsitsiklis We assume once moe that utility is measued in monetay units; thus, if use eceives a ate allocation d, but must pay w, his net net payoff is: U (d ) w. Given a utility system U U R, the social objective is to maximize aggegate utility, as defined in the poblem (1)-(3); we epeat that poblem hee, and efe to it as the poblem SYSTEM(C, R, U), to emphasize that the poblem is specified by C, R, and the utility system (U). maximize subject to R U (d ) (27) =1 R d C; (28) =1 d 0. (29) We will say that d solves SYSTEM(C, R, U) if d is an optimal solution to (27)-(29), given the utility system (R, U). In geneal, the utility system (R, U) is unknown to the mechanism designe, so a mechanism must be designed to elicit infomation fom the uses. We will focus on mechanisms in which each use i submits a demand function, within a one-paamete family of admissible demand functions. In paticula, each use has a one-dimensional stategic vaiable, denoted by θ i. Definition 1. Given C > 0, a smooth maket-cleaing mechanism fo C is a diffeentiable function D : (0, ) [0, ) R + such that fo all R, and fo all nonzeo θ (R + ) R, thee exists a unique solution p > 0 to the following equation: R D(p, θ ) = C. =1 We let p D (θ) denote this solution, and efe to it as the maket-cleaing pice. Note that the maket-cleaing pice is undefined if θ = 0. As we will see below, when we fomulate a game between consumes fo a given mechanism D, we will assume that the payoff to all playes is if the composite stategy vecto is θ = 0. Note that this is slightly diffeent fom the definition in Section 2, whee the payoff to a playe who submits θ = 0 is set to zeo. We will discuss this distinction futhe late; we simply note fo the moment that it does not affect the esults of this section. Ou definition of smooth maket-cleaing mechanisms genealizes the mechanism discussed in Section 2. We ecall that in that development, each use submits a demand function of the fom D(p, θ) = θ/p, and the link manage chooses a pice p D (θ) to ensue that R =1 D(p, θ ) = C. Thus, fo this mechanism, we have p D (θ) = R =1 θ /C if θ 0. Anothe elated example

15 A Game Theoetic View of Efficiency Loss in Resouce Allocation 15 is povided by D(p, θ) = θ/ p; in this case it is staightfowad to veify that p D (θ) = ( R =1 θ /C) 2, fo θ 0. We will futhe estict attention to a paticula class of mechanisms denoted D, which we define as follows. Definition 2. The class D consists of all functions D(p, θ) such that the following conditions ae satisfied: 1. Fo all C > 0, D is a smooth maket-cleaing mechanism fo C (cf. Definition 1). 2. Fo all C > 0, and fo all U U, the payoff of a pice anticipating use is concave; that is, fo all R, and fo all θ (R + ) R, the function: U (D(p D (θ), θ ) p D (θ)d(p D (θ), θ ) is concave in θ > 0 if θ = 0, and concave in θ 0 if θ The demand functions ae nonnegative; i.e., fo all p > 0 and θ 0, D(p, θ) 0. The fist condition equies a mechanism in D to be a smooth maket-cleaing mechanism fo any C > 0; in paticula, the maket-cleaing pice p D (θ) must be uniquely defined fo any C > 0. (Note that in the notation we suppess the dependence of the maket-cleaing pice p D (θ) on the capacity C.) The second condition allows us to chaacteize Nash equilibia in tems of only fist ode conditions; indeed, some such assumption needs to be in place in ode to guaantee existence of pue stategy Nash equilibia [26]. Finally, the thid condition is a nomalization condition, which ensues that a use is neve equied to supply some quantity of the esouce (which would be the case if we allowed D(p, θ) < 0). In ode to state the main esult of this section, we must define competitive equilibium and Nash equilibium. Given a utility system (R, U), a capacity C > 0, and a smooth maket-cleaing mechanism D D, we say that a nonzeo vecto θ (R + ) R is a competitive equilibium if µ = p D (θ) satisfies: [ θ ag max U (D(µ, θ )) µd(µ, θ ) ],. (30) θ 0 Similaly, we say that a nonzeo vecto θ (R + ) R is a Nash equilibium if: whee θ ag max Q (θ ; θ ),, (31) θ 0 Q (θ ; θ ) = { U (D(p D (θ), θ )) p D (θ)d(p D (θ), θ ), if θ 0;, if θ = 0. (32) Notice that the payoff is if the composite stategy vecto is θ = 0, since in this case no maket-cleaing pice exists.

16 16 Ramesh Johai and John N. Tsitsiklis Ou inteest is in the wost-case atio of aggegate utility at any Nash equilibium to the optimal value of SYSTEM(C, R, U) (temed the the pice of anachy by Papadimitiou [25]). Fomally, fo D D and a capacity C > 0 we define a constant ρ(c, D) as follows: { R =1 ρ(c, D) = inf U (D(p D (θ), θ )) R =1 U (d ) R > 1, U U R, } d solves SYSTEM(C, R, U) and θ is a Nash equilibium Note that since all U U ae stictly inceasing and nonnegative, and C > 0, the aggegate utility R =1 U (d ) is stictly positive fo any utility system (R, U) and any optimal solution d to SYSTEM(C, R, U). Howeve, Nash equilibia may not exist fo some utility systems (R, U); in this case we set ρ(c, D) =. The following theoem shows that among smooth maket-cleaing mechanisms fo which thee always exists a fully efficient competitive equilibium, the mechanism poposed in Section 2 minimizes efficiency loss when uses ae pice anticipating. The poof can be found in Chapte 5 of [15]. Theoem 6. Let D D be a smooth maket-cleaing mechanism such that fo all capacities C > 0 and utility systems (R, U), thee exists a competitive equilibium θ such that (D(p D (θ), θ ), = 1,..., R) solves SYSTEM(C, R, U). Then fo any capacity C and utility system (R, U), thee exists a unique Nash equilibium. Futhemoe, ρ(c, D) 3/4 fo all C > 0 and all D D, and this bound is met with equality if and only if D(p, θ) = θ/p fo some > 0. Theoem 6 suggests that the best efficiency guaantee we can hope to achieve is 75%, if we ae esticted to maket-cleaing mechanisms with scala stategy spaces. A key estiction in the mechanisms we conside is that a single pice is chosen to clea the maket. If the maket designe is ganted the latitude to pice disciminate (i.e., to chage a diffeent pice to each use), bette efficiency guaantees ae possible. The most famous mechanisms which ensue such a guaantee ae the Vickey-Clake-Goves class of mechanisms, fo which fully efficient dominant stategy equilibia exist [4, 11, 32]. Moe ecently, in a netwoking context, Sanghavi and Hajek [30] have shown that if uses choose thei payments (as in the Kelly mechanism), but the link manage is allowed to choose the allocation to uses as an abitay function of the payments, it is possible to ensue no wose than a 13% efficiency loss. Futhemoe, Yang and Hajek [34] have shown that if a mechanism allocates esouces in popotion to the uses stategies (i.e., use eceives a faction θ /( R s=1 θ s) of the esouce), then by using diffeentiated picing, it is possible to guaantee abitaily small efficiency loss at the Nash equilibium. The mechanisms poposed by both Sanghavi and Hajek [30] as well as Yang

17 A Game Theoetic View of Efficiency Loss in Resouce Allocation 17 and Hajek [34] equie pice discimination, since the atio of payment to allocation is not necessaily identical fo all uses (as must be the case in the maket-cleaing mechanisms studied hee). 5 Futhe Diections In addition to the esults outlined above, seveal additional theads ae included in this body of eseach. In this section, we descibe two extensions: (1) esouce allocation in geneal netwoks; and (2) a setting of multiple poduces competing to satisfy an inelastic demand. 5.1 Geneal Netwoks The models pesented in Sections 2 and 3 only conside esouce allocation fo a single link. We now conside extensions to the netwok case, following [16] and [17]. We conside netwoks consisting of a set of links; each use has a set of paths available though the netwok to send taffic, and each path uses a subset of the links. In a setting of inelastic supply, each link j is chaacteized by a fixed capacity C j. In a setting of elastic supply, each link j is chaacteized by a cost function C j ( ). We continue to assume that each use eceives a utility U (d ) fom a total ate allocation d ; howeve, note that in a netwok context d is the total ate deliveed to use acoss all paths available to use though the netwok. We extend the single link maket mechanisms to multiple links by teating each link as a sepaate maket. Thus we conside a game whee each use equests sevice fom multiple links by submitting an individual bid to each link. Links then allocate ates using the same scheme as in the single link model, and each use sends the maximum ate possible, given the vecto of ates allocated fom links in the netwok. Although this definition of the game is natual, we demonstate that Nash equilibia may not exist in the setting of inelastic supply, 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 at least 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. In the setting of elastic supply, Nash equilibia ae always guaanteed to exist, without having to extend the stategy space. Finally, we show that in this netwok setting, if link capacities ae inelastic then the total utility achieved at any Nash equilibium of the game is no less than 3/4 of the maximum possible aggegate utility; and if link supplies ae elastic then the aggegate suplus achieved at any Nash equilibium of the game is no less than a facto of the maximal aggegate suplus. These

18 18 Ramesh Johai and John N. Tsitsiklis esults extend the efficiency loss esults fom the single link setting to geneal netwoks. The mechanisms we have studied equie each use to submit a sepaate bid fo each link that the use may use. An altenative mechanism had been poposed ealie by Kelly [18] wheeby a use submits a single total payment, and the netwok detemines both the ate allocations, as well as the divisions of the uses total payments among the links; in the single link case, this scheme educes to that studied in Section 2. But Hajek and Yang [13] have shown that Kelly s mechanism can esult in Nash equilibia in which the aggegate utility is an abitaily small faction of the optimal aggegate utility. It emains an open poblem whethe thee exists a netwok esouce allocation mechanism in which each use submits a single numbe, epesenting total payment, and which has some nontivial efficiency guaantees. 5.2 Multiple Poduces, Inelastic Demand The models pesented thus fa conside consumes competing fo esouces in scace supply. Motivated by cuent poblems in maket design fo electic powe systems, we conside a model whee multiple poduces compete to satisfy an inelastic demand. Demand fo electicity, paticulaly in the shot un, is chaacteized by low elasticity with espect to pice, i.e., changes in pice do not lead to significant changes in the level of demand; see, e.g., [31], Section A basic model fo electicity maket opeation involves supply function bidding [20]: each geneato submits a supply function expessing thei willingness to poduce electicity as a function of the maket cleaing pice. A single pice is then chosen to ensue that supply matches the inelastic demand. Most pevious wok on supply function bidding has focused almost entiely on using the supply function equilibium (SFE) famewok of Klempee and Meye [20] fo its pedictive powe. In such models, geneatos can submit nealy abitay supply functions; the Nash equilibia of the esulting game ae used to give insight into expected behavio in cuent makets. In othe wods, by solving the SFE model fo an appopiate set of assumptions, most pevious wok hopes to lend insight into the opeation of powe makets which equie geneatos to submit complete supply schedules as bids [1, 6, 9, 10, 29, 33]. But because thee may be a multiplicity of equilibia, an explicit undestanding of efficiency losses in these games has not been developed. Papes such as the wok of Rudkevich et al. [29] do suggest, howeve, that in the pesence of inelastic demand, pice anticipating behavio can lead to significant deviations fom pefectly efficient allocations. Fo this eason we take a diffeent appoach (see Chapte 4 of [15]). We conside estictions on the supply functions which can be chosen by the geneatos, and aim to design these estictions so that nealy efficient allocations ae achieved even if fims ae pice anticipating. Fomally, we assume that each

19 A Game Theoetic View of Efficiency Loss in Resouce Allocation 19 fim n has a convex cost function C n, as a function of the electicity geneated. An efficient poduction vecto minimizes the aggegate cost n C n(s n ), subject to the constaint that the total poduced electicity n s n must equal the demand D. We then conside the following maket. Each fim submits a supply function of the fom S(p, w) = D w/p, whee D is the fixed (exogenous) demand and w is a nonnegative scala chosen by the fim. The maket then chooses a pice so that aggegate supply is equal to demand. If we assume that fims ae pice taking, it is possible to show that thee exists a competitive equilibium; futhemoe, at this competitive equilibium the esulting allocation minimizes aggegate poduction cost. If we assume instead that fims ae pice anticipating, we can establish existence of a Nash equilibium and uniqueness of the esulting poduction vecto, as long as moe than two fims compete. Next, we conside the aggegate poduction cost at a Nash equilibium elative to the minimal possible aggegate poduction cost. As long as moe than two fims ae competing, we show that the atio of Nash equilibium poduction cost to the minimal poduction cost is no wose than 1 + 1/(N 2), whee N is the numbe of fims in the maket. Futhemoe,, we demonstate that this efficiency loss esult caies ove even to a setting whee demand is inelastic but stochastically detemined, by showing that in such an instance it is as if fims play a game with deteministic demand but diffeent cost functions. Finally, a chaacteization theoem, simila to the one in Section 4, is also available, indicating that the mechanism unde study has the best possible efficiency guaantees, within a class of mechanisms in which the geneatos ae esticted to submitting a supply function chosen fom within a esticted, one-paamete family. These esults, which have been established in [15], suggest that maket powe can be contolled, and efficient allocations guaanteed, by esticting the supply functions available to geneatos in electicity makets. Resticted families of supply functions have also been consideed elsewhee in the liteatue, e.g., in [3]. Howeve, these models ae typically used as appoximations to unconstained supply function bidding, and thus the esulting efficiency loss has not been studied. Still, this wok leaves many open questions; in paticula, the dynamics of powe systems, togethe with thei complex netwok stuctue, has not been captued in the models developed (in contast to the telecommunications models peviously discussed). Futhemoe, the wok descibed hee depends on convexity assumptions on the cost functions of the poduces, and such assumptions may geneally not hold in electicity makets [14]. Finally, away fom a Nash equilibium, e.g., if some geneatos do not act ationally, the emaining geneatos may have to poduce electicity at highly undesiable o even impossible levels. Addessing these types of questions is the subject of cuent eseach.

20 20 Ramesh Johai and John N. Tsitsiklis 6 Open Issues We have discussed the efficiency popeties of Nash equilibia associated with cetain esouce allocation mechanisms. Fo the case whee thee is a single available esouce (espectively, a single demand to be satisfied), the mechanisms involve the submission of a demand (espectively, a supply) function, which can be specified in tems of a single paamete, followed by maketcleaing. In each case, we have povided a tight bound on the wost case efficiency loss. It emains to undestand the wost case efficiency loss when mechanisms belonging to boade classes ae consideed. Fo example, in the context of Section 2 what efficiency guaantees ae possible if uses can choose a demand function fom within a two-paamete family of demand cuves? Anothe eseach diection elates to the study of natual adjustment dynamics in the context of vaious mechanisms. Indeed, a desiable mechanism should not only have efficiency guaantees fo the esulting Nash equilibia. It should also allow fo simple adjustment algoithms wheeby the diffeent playes can convege, in a stable manne, to such a Nash equilibium. 7 Acknowledgment This eseach was suppoted by the National Science Foundation unde a Gaduate Reseach Fellowship and gant ECS , by the Defense Advanced Reseach Pojects Agency unde the Next Geneation Intenet Initiative, and by the Amy Reseach Office unde gant DAAD The mateial of Section 3 was epoted in a pape jointly authoed with Shie Manno; the authos would also like to acknowledge helpful convesations with Fank Kelly. Refeences 1. E. J. Andeson and A. B. Philpott. Using supply functions fo offeing geneation into an electicity maket. Opeations Reseach, 50(3): , L. M. Ausubel and P. Milgom. The lovely but lonely Vickey auction In pepaation. 3. R. Baldick, R. Gant, and E. P. Kahn. Theoy and application of linea supply function equilibium in electicity makets. Jounal of Regulatoy Economics, To appea. 4. E. H. Clake. Multipat picing of public goods. Public Choice, 11:19 33, J. R. Coea, A. S. Schulz, and N. Stie Moses. Selfish outing in capacitated netwoks. Mathematics of Opeations Reseach, To appea. 6. C. J. Day, B. F. Hobbs, and J.-S. Pang. Oligopolistic competition in powe netwoks: a conjectued supply function appoach. IEEE Tansactions on Powe Systems, 17(3): , 2002.

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