Efficiency Loss and Uniform-Price Mechanism

Transcription

1 Proceedngs of the 47th IEEE Conference on Decson and Control Cancun, Mexco, Dec. 9-11, 28 Effcency Loss and Unform-Prce Mechansm Mchał Karpowcz*,** Krzysztof Malnowsk*,** Abstract We nvestgate the propertes of the class of resource allocaton mechansms wth unform prcng scheme. An mportant consequence of the assumpton that agents maxmze ther profts s that both the resource prce and the quanttes assgned to agents can be vewed as the mplct functons of the messages communcated to the mechansm. As a result agents ndvdual decsons become nterdependent, whch makes each agent capable of antcpatng the effects of ndvdual actons on the prce of the resource. Focusng our attenton on the Nash equlbrum soluton concept we dscuss the effcency of equlbrum allocatons of the game defned by the allocaton mechansm both from the perspectve of the global system goals and the ndvdual objectves of agents. Our frst contrbuton s the rankng of three varatons of the unform-prce mechansm. We demonstrate the sgnfcant role of strategc varables used by agents and analyze mechansm desgner s best response to agents expected prce-antcpatng behavor. Snce the resource allocaton model beng subject to our consderatons can very well serve as a descrpton of the unform-prce aucton for dvsble resources, the results of ths paper can be vewed as an nqury nto propertes of ths aucton format. As a second contrbuton we show how sgnals exchanged between agents and the mechansm can be successfully used to reach an equlbrum pont n an teratve bddng process. I. INTRODUCTION Ths work bulds upon Kelly s [1], [2] and Johar s [3], [4] analyss of the unform-prce resource allocaton mechansm desgned for dynamc bandwdth allocaton n communcaton networks. In general, however, the results presented below hold for the settngs where a dvsble resource must be dstrbuted between a small number of consumers maxmzng ther profts. For such an envronment we nvestgate the consequences of agents strategc bddng behavor, whch s argued to result from agents percepton of the marketclearng resource allocaton rules computng a sngle prce to balance the aggregate demand and supply n the system. The prce and allocatons are determned by the messages reported to the mechansm by agents. The messages reveal agents demand at the gven prce of resource and are requred to belong to the set of postve real numbers. Snce the resource allocaton model beng subject to our consderatons can very well serve as a descrpton of the unform-prce aucton for dvsble resources, the results of ths paper can be vewed as an extenson of the results presented by Ausubel and Cramton [5], Engelbrecht-Wggans and Kahn [6]. As we are about to show, a defnton of the resource allocaton rules wth unform prcng has a truly remarkable *Insttute of Control and Computaton Engneerng, Warsaw Unversty of Technology, Nowowejska 15/19, -665 Warsaw, Poland, **NASK Research and Academc Computer Network, Wawozowa 18, Warsaw, Poland We thank Professor W. Ogryczak for helpful comments. mplcaton. Namely, n lght of the assumpton that agents maxmze ther profts, t makes both the prce and the quanttes the mplct functons of the messages communcated to the mechansm. As a consequence, agents ndvdual decsons become nterdependent, whch makes each agent capable of antcpatng the effects of hs ndvdual actons on the prce of the resource. If the number of agents s small enough, so that each ndvdual can have a consderable nfluence on the total cost of allocaton, then the decson problem faced by each agent becomes a noncooperatve game of ncomplete nformaton. In the paragraphs whch follow we nvestgate the propertes of solutons to the game. Focusng our attenton on the pure strategy Nash equlbrum soluton concept we dscuss the effcency of equlbrum allocatons both from the perspectve of the global system goals and the ndvdual objectves of agents. In partcular, we present a rankng of dfferent varatons of the unformprce mechansm wth elastc supply and emphasze the role of strategc varables used by agents. We also show when the sgnals exchanged between agents and the mechansm can be successfully used to reach an equlbrum pont n an teratve bddng process. In Secton II we defne the resource allocaton settng beng subject to our nvestgatons. Next we dscuss the Nash equlbrum condtons gven by Johar [3], [4] for the class of unform-prce scalar strategy mechansms. These condtons are then used to establsh the result of Proposton 2. It characterzes the equlbrum allocatons n terms of the relatonshp between margnal utlty gans and prce elastcty. The relatonshp s next shown n Proposton 3 to mply the rankng of allocatons attanable n equlbra of the games nduced by the unform prce mechansm wth payments and allocaton levels as strategc varables. We show that from the vewpont of a prce-takng resource manager t s reasonable to requre from the prce-antcpatng agents to submt n ther bds the total payments they are wllng to make for allocatons, rather than the demanded allocaton levels. On the other hand, from the vewpont of each agent, the best way to mprove profts s to bd strategcally n the game where sgnals reported to the resource manager represent the amount of resource demanded at a gven prce. Fnally, n Secton VI we gve a bddng procedure whch s proved n Proposton 4 to converge to Nash equlbrum of the game defned by resource allocaton rules of Secton II. II. MODEL Consder a case of L ratonal preference maxmzng agents competng for a sngle dvsble resource. Let x R /8/$ IEEE 4466

2 denote the amount of the resource allocated to the agent and suppose that: Assumpton II-A: For each L, over the doman x the utlty functon U x s concave, strctly ncreasng, and contnuous, and over the doman x >, U x s contnuously dfferentable. Furthermore, the rght drectonal dervatve at, denoted U +, s fnte. The total cost of assgnng the amount y = x of the resource to the agents s gven by Cy, whch s assumed to be ncurred by a sngle, strategcally neutral resource manager,.e. a manager actng as prce-taker maxmzng hs profts: Πµ,y = µy Cy, 1 where µ s a fxed unt prce of the resource establshed wthn the market-clearng process accordng to the rules of a resource allocaton mechansm selected by the manager. Snce manager s profts are maxmzed when margnal producton cost s equal to the prce, t s natural to assume that: Assumpton II-B: There exsts a contnuous, convex and strctly ncreasng functon py such that p and such that: Cy = y psds. 2 Thus total cost functon Cy s strctly convex and strctly ncreasng over y. For further development t s also necessary to defne the prce elastcty: εy y py 3 py y measurng the margnal change n logpy resultng from the margnal change n logy. Furthermore, we defne: βy εy 1 + εy. 4 If py s not dfferentable at y, then the correspondng rght and left drectonal dervatves of p defne ε + y y/py + py/ y and ε y y/py py/ y respectvely. Defntons of β + and β follow mmedately. Notce that, by the assumpton of contnuty and convexty of p, drectonal dervatves exst [7], [8]. There are three key assumptons that we follow. Frst, the aggregate utlty U x of the consumers less the aggregate cost Cy of allocaton gves a value nterpreted as a total welfare measure related to the allocaton y. Second, the mechansm sets a sngle market-clearng prce for a resource unt that s assumed to be equal to the margnal producton cost and that ensures that demand equals supply,.e. that total amount of resource avalable to agents at the prce s allocated to them. Thrd, agents antcpate the effects of ther actons on the market-clearng prce. III. PAYMENT BIDDING Consder a resource allocaton game n whch, havng w = w 1,...,w 1, w +1,...,w L fxed, each player selects a sgnal w to maxmze hs payoff functon: Q x w,w, w = U x w,w w. 5 The sgnal w can be nterpreted as the total payment that the agent s wllng to make for the amount x of the resource. Agent does not know w, but knows that the prce pyw and the amount of the resource he receves x w depend on t. The followng assumpton s crucal from the vewpont of the developed framework. It states that the allocatons assgned to agents consttute a soluton to the market-clearng equaton,.e. the equaton accordng to whch supply of the resource equals aggregate demand at the market-clearng prce. Assumpton III-A: For all w = w 1,...,w L, the aggregate allocaton yw = x w s the unque soluton to the market-clearng equaton: w = ywpyw. 6 Furthermore, for each L: w w > ; x w = pyw w =, and there exsts k L such that U k + > p. The followng result s due to Johar [4]: Proposton 1 Johar: If assumptons II-A-III-A hold, then w s a Nash equlbrum of the game defned by Q x w, w f and only f w >, and wth y yw and x x w the followng two condtons hold: U x 1 β + y x py; 8 y U x 1 β y x py. 9 y Conversely, f x =,...,x L and y > satsfy 8-9 then w = pyx s a Nash equlbrum wth x x w and y yw = x w. The result shows that the solutons to the payoff maxmzaton problem: 7 w arg maxq x s,w, s, 1 s R + calculated ndvdually by each agent L subject to the constrants 6 and 7, consttute a Nash equlbrum of the game nduced by the market-clearng equaton gven by 6. Indeed, snce x = w /pyw 1,...,w,...,w L whenever w >, t s straghtforward to conclude that each agent can approxmate hs nfluence on the total supply of the resource. By assumpton II-B functon p s contnuous, convex and strctly ncreasng, whch mples that gy = ypy s strctly ncreasng, strctly convex, contnuous and thus nvertble. As 4467

3 a consequence yw = g 1 w s strctly ncreasng and strctly concave functon of w, and therefore drectonally dfferentable. Margnal changes n supply, caused by agent s unlateral devaton from the consumpton level determned by hs wllngness to pay w, are therefore gven by: + 1 yw = pyw + yw + pyw ; w w 1 yw = pyw + yw pyw, w w whch follows from drectonal dfferentaton of 6 wth respect to w. It s mportant to observe that no assumpton on dfferentablty of py has been made, so rght and left drectonal dervatves are not necessarly equal; see [4] for the general treatment. Snce the total producton level yw determnes the prce of the resource, an agent capable of calculatng the above dervatves of yw s also able to antcpate the margnal prce changes and, as a consequence, to mprove hs ndvdual profts. At ths pont t should be notced that the prce of the resource and the total producton level are observable varables; ther values, as well as the form of the market-clearng equaton 6, are a common knowledge. It s therefore reasonable to assume that each agent wll expect other agents to perform the smlar prce-antcpatng reasonng. As a result, decson problem faced by each agent ndeed has a nature of noncooperatve game. From the frst order optmalty condtons for payoff maxmzaton problem 1 t mmedately follows that gven a fxed vector w = w 1,...,w 1, w +1,...,w L agent s best response to other agents decsons must satsfy the next two condtons: + Q x w,w, w w Q x w,w, w w = U x w + x w w 1 ; = U x w x w w 1. Proposton 1 shows that under assumptons II-A-III-A the above best response condtons, admttng optmal soluton to be stuated at a nondfferentable pont of py, consttute a Nash equlbrum w of the resource allocaton game; f w satsfes the condtons of Proposton 1 then no agent can mprove hs payoff by unlaterally devatng from w. Notce that wth w k =, by assumpton III-A, f w k + then x k w/w k 1 p < U k +, whch mples that at least one agent k can mprove hs payoff by devatng from w k =. Dfferentatng 7 wth respect to w and substtutng drectonal dervatves of yw yelds 8-9. EFFICIENCY LOSS The drect mplcaton of Proposton 1 s that the allocatons obtaned at Nash equlbrum pont defned by 8-9 are not Pareto-optmal. To see ths, consder the resource allocaton problem faced by the resource manager: [ ] U x Cy, s.t. y = x. 11 max x R L + By assumpton II-A, functon U s concave, strctly ncreasng and dfferentable over x > for all L. By assumpton II-B, functon C s convex and dfferentable. As a result, the necessary and suffcent condtons for optmalty of x =,...,x L are gven, for every L, by: { [ ] x U x py = ; U 2 py. If agent s assgned a postve amount x of the resource, then the correspondng margnal ncrease n utlty U x he receves must be equal to the market-clearng prce pȳ >, where ȳ = x. If, on the other hand, no amount of the resource s allocated to agent, then the market-clearng prce must be greater than or equal to the margnal utlty of agent at zero. Pareto-optmalty of the allocatons satsfyng 12 s mpled by the followng result: Lemma 1: Optmal soluton to: { m } max f x: x R n 13 =1 s an an effcent soluton to: max {f 1 x,...,f m x: x R n } 14 where f : R n R for = 1,...,m. Proof: See e.g. [9], [1]. Thus, by Lemma 1 the optmal soluton to 11 s also an effcent soluton to the multobjectve problem: max U 1,...,U L x L, p j Lx j : x R L +. Furthermore, by concavty of 11 and the fundamental theorems of welfare economcs, the correspondng par pȳ, x consttutes a compettve equlbrum as well. Our frst proposton demonstrates that these solutons are substantally dfferent from those that arse at Nash equlbra of the game wth payoffs defned by 5. The proposton s techncally very useful and wll be used to derve our subsequent nsghts. Proposton 2: Suppose that py s dfferentable and exhbts nondecreasng elastcty εy for y. If Assumptons II-A-III-A hold and ŵ s Nash equlbrum of the game defned by Q x w, w, then: pyŵ < U x ŵ < pyŵ1 + εyŵ 15 for all L such that x ŵ >. 4468

4 Proof: By the dfferentablty of p and Proposton 1, f ŵ s a Nash equlbrum, then: 8 >< w»u x ŵ 1 βyŵ xŵ «pyŵ = yŵ >: U + pyŵ. 16 Fx ŵ and consder ŵ >. Snce U s strctly ncreasng, from 7 and the fact that βyŵ > for yŵ > observe that β y = ε y/1 + εy 2 t follows, that: U x ŵ pyŵ = U x ŵβyŵ x ŵ yŵ >. Now, suppose U x ŵ > pyŵ1 + εyŵ. Then, for every L such that ŵ > : pyŵ1 + εyŵ 1 βyŵ x ŵ < pyŵ. yŵ Ths, however, mples that: εyŵ 1 x ŵ <. yŵ whch s a contradcton, snce εyŵ > and yŵ = j L x jŵ x ŵ n equlbrum. As a result, for all L such that x ŵ > : pyŵ < U x ŵ < pyŵ1 + εyŵ. Condton 15, the key result of ths paper, shows that the margnal gans from postve allocatons assgned to agents n Nash equlbrum of the game nduced by the unform-prce mechansm exceed the market-clearng prce of resource. As a result they are not Pareto-optmal and do not maxmze the effcency measure 11 appled by the resource manager nterested n optmal utlzaton of system resources. Notceably, however, from the vewpont of each agent prce-antcpatng bddng provdes an opportunty to obtan an addtonal margnal revenue equal to the dfference between the margnal utlty from the consumpton level acheved n equlbrum and the correspondng prce equal to the margnal cost of allocaton ncurred by the system; t mproves agents profts. IV. RESOURCE BIDDING Consder now the classcal Cournot competton model [11], [12] and the related resource allocaton game wth payoffs defned for all L by: Q x = U x x p j Lx j. 17 Instead of reportng to the mechansm a value of the payment w each agent s now requred to report the allocaton level x demanded at the prce p j L x j. The pontwse characterzaton of ndvdual demand for the resource s therefore descrbed here drectly by the value of x rather than ndrectly by the value of agent s wllngness to pay w. The market-clearng equaton n ths settng s defned as follows: x = Spy, 18 where y = x and Sµ = p 1 µ s a supply correspondence settng the total producton level y to the value maxmzng the proft Πµ,y. We defne a Cournot-Nash equlbrum x =,..., x L as a vector of the solutons to: x arg max s R + Q s,x. 19 To see that the vector x exsts, notce that Q s concave and contnuous n x X = [, R ]. The strategy space X, defned by the rules of resource allocaton, s compact and convex for each L; by assumptons II-A and II-B there exsts a postve number R > such that U R R pr + j x j, so t s ratonal for each agent L to bd x [, R ]. Snce agent s best response x to x s a soluton to 19, a mappng H: X X wth X = [, R 1 ] [, R L ] and H x arg max s X Q,...,s,...,x L } s upper semcontnuous. Exstence of Cournot equlbrum results, as a consequence, from the Kakutan fxed pont theorem, as proved by Rosen n [13]. EFFICIENCY LOSS We wll now examne the relatonshp between the allocatons arsng at the Cournot-Nash equlbrum and at the Nash equlbrum defned by condtons 8-9. Proposton 3: Consder payoffs Q x defned by 17 and let x be a vector of the solutons to 19. Suppose also that ŵ s a Nash equlbrum of the game defned by payoffs Q x w, w. If the assumptons of Proposton 2 hold, then for all L: x ŵ x. 2 Proof: Consder frst the followng system of equatons: x w α =, where α [, for all L. By assumpton III-A, the system has a unque trval soluton w = ff α = for all L. Suppose then that α k > and that α = for all k. We clam that the system stll has a unque soluton, but now w. Indeed, snce x k w s contnuous, strctly ncreasng and concave n w k, and snce x k w as w k see Proposton 3.3 [4], there exsts a soluton w, such that w = for k and w k >. Next, take an arbtrary set L 1 L such that α > for all L 1 and consder a mappng H: W W, where W [, ω1 1 + ǫ]... [, ω 1 L + ǫ] wth ω w x w,w α for both α and w fxed, and ǫ. Let H w = ω 1,.e. for an arbtrarly chosen L 1, gven α and w, let H return the value of w such that ω w =. Observe that H w [, ω 1 + ǫ] for all L. Snce W s nonempty, compact and convex, and the mappng Hw = H 1 w,...,h L w s contnuous, by the Brouwer s fxed pont theorem there exsts w = H w, such 4469

5 that x w = α for all L wth α = x beng soluton to 19. Next, from the frst order optmalty condtons for 19 t follows that: x p ỹ = U x pỹ whenever x > and ỹ = x. Substtutng rght hand sde of the above equaton to the partal dervatve of Q x w, w yelds: Q x w, w = U x w pỹ = U x pỹ 1 βỹ x ỹ 1 U x pỹ pỹ1 + εỹ 1. 1 Suppose now that < x ŵ < x for some L 1 L and x k ŵ x k for k L \ L 1. Ths mmedately mples that pyŵ < pỹ, w > ŵ and U x ŵ U x for L 1. Snce p s assumed to be dfferentable and exhbts nondecreasng elastcty, from the fact that Q s strctly concave n w see Proposton 3.7 [4] and Q x ŵ,ŵ / w = for ŵ >, t follows that: Q x w, w = U x w pỹ From ths we obtan: U x pỹ U x 1 U x pỹ pỹ1 + εỹ < U x pỹ pỹ1 + εỹ, 1 <. whch mples that: U x ŵ U x > pỹ1+εỹ > py1 + εyŵ. By Proposton 2 ths s a contradcton, snce n equlbrum U x ŵ < pyŵ1 + εyŵ whenever x ŵ >. We have, therefore, demonstrated that Q x w, w / w for w such that x = x w. Ths proves that x ŵ x for all L. V. QUANTITATIVE COMPARISON OF MARKET STRATEGIES In ths secton we compare the mechansms studed above wth respect to the generated effcency and allocaton levels. Observe that an mmedate corollary to condton 15 of Proposton 2 s that yŵ ȳ = x, where x s a vector of Pareto-optmal allocatons satsfyng condtons 12. Furthermore, concavty and monotoncty of U, together wth convexty and monotoncty of p mply that pyŵ pȳ. As a consequence, profts of resource manager are reduced n comparson to those of compettve equlbrum. To see ths assume, to the contrary, that: pyŵyŵ Cyŵ > pȳȳ Cȳ. Ths mples that: Cȳ > Cyŵ + C ȳȳ yŵ, whch s a contradcton, snce C s strctly convex. From the vewpont of mechansm desgn t s mportant to notce, though, that condton 6 guarantees that allocaton costs are covered. Indeed, by convexty pyy y pada. Ths 1 w 8 E w E x w 5 1 E x w E w w 5 1 Fg. 1. Optmal response curves for the settng wth two agents: px = 1 y 1, U x = γ logx + 1 left and U x = γ arctanx rght. Optmal response curves are tangent to the so-payoff curves at the ponts beng soluton to 1 and 19. In equlbrum E w of the game wth payments w as strateges each agent s assgned more resources then n equlbrum E x of the game wth allocatons x as strateges. condton need not hold for other classes of mechansms. Ths, therefore, gves the followng result: Corollary 1: Let µ, x be a compettve equlbrum defned by condtons 12 and let ŵ be a Nash equlbrum of the game wth payoffs defned by 5. Then: Πpyŵ, yŵ Πpȳ,ȳ. 21 The results of Proposton 2 and Corollary 1 stem from the fact that the goals of resource manager, expressed by condtons 12, are not compatble wth the myopc goals of agents maxmzng ther payoffs, expressed by condtons 8-9. Equvalently, the rules of a unform-prce mechansm, appled to the settng of strategc agents capable of exertng ther market power, are characterzed by the undesrable property of creatng ncentves to msrepresent preferences. The observaton s straghtforward because for x ŵ > : < 1 βyŵ x ŵ yŵ < 1, whch mples that: U x ŵ 1 βyŵ x ŵ < U yŵ x ŵ. Thus, from the perspectve of the mechansm processng messages submtted by agents, t s as though agent L were characterzed by the margnal gans descrbed by the left hand sde of the equaton above, whch are lower then the real ones defned by U. Ths s precsely the effect of demand reducton dscussed by Ausubel and Cramton n [5]. 447

6 Each agent s optmal strategy s to shade bds n order to reduce the market-clearng prce of the resource and acheve profts that are hgher n comparson to those that would be acheved wth prce-takng behavor n compettve equlbrum. Ths brngs us to the concluson that n the consdered settng of strategcally nterdependent agents and elastc supply the unform-prce mechansm optmzes utlzaton of resources for the manpulatvely declared preference model. The next result demonstrates that the postve allocatons arsng n the Nash equlbrum of the game where agents report ther wllngness to pay are at least as hgh as those arsng n the Cournot-Nash equlbrum of the game where agents declare the demanded amount of resource. Corollary 2: If the assumptons of Proposton 2 hold, then: U x Cỹ, 22 U x ŵ Cyŵ where yŵ = j L x jŵ and ỹ = j L x j. Proof: Frst, notce that from Proposton 2 we have: U x pỹ, U x ŵ pyŵ. Furthermore, from the Proposton 2 and Proposton 3 we obtan: U x pyŵ. As a result, by convexty of p: xŵ yŵ ỹ U ada x U ada = psds = Cyŵ Cỹ, whch completes the proof. Corollary 2 shows that the total welfare of the allocatons attanable n the case of payment bddng s at least as hgh as the welfare of the allocatons attanable n the case of resource bddng. Notce, though, that both sgnals w and x provde the unque characterzaton of the agent s demand for resource. Ths characterzaton s gven by: pywx w = w. However, dfferent strategy spaces make agents maxmze dfferent payoff functons, whch n turn leads to dfferent solutons. Clearly, by Corollary 2, from the vewpont of resource manager t s more reasonable to apply the mechansm where payments w are reported. Our next result shows that t s not what agents would prefer. Corollary 3: Suppose that assumptons of Proposton 2 hold. Defne Q x, µ U x µx and let µ, x denote compettve equlbrum beng soluton to 11. Then for all L: Q x, µ Q x ŵ,ŵ Q x. 23 Proof: Suppose frst, to the contrary, that we have Q x ŵ,ŵ > Q x. As a consequence, from Proposton 2, we obtan for x ŵ > and x > : U x ŵ U x > x ŵpy x pỹ U x x ŵ x x p ỹx ŵ x, where the last nequalty follows from the fact that pỹ = U x x p ỹ. We, therefore, conclude that: U x ŵ + x p ỹx ŵ x > U x + U x x ŵ x, whch s a contradcton. Indeed, by concavty of U, we must have: U x ŵ U x + U x x ŵ x. However, snce x p ỹ+pỹ = U x n equlbrum, t follows that x p ỹ < U x. Smlarly, suppose that Q x, µ > Q x ŵ,ŵ. By Proposton 1 and 2, snce µ > py, for x ŵ > and x > : U x + U x ŵβyŵ x ŵ yŵ x x ŵ > U x ŵ + U x ŵ x x ŵ. By the concavty argument, ths consttutes a contradcton as well. Therefore, Q x, µ Q x ŵ,ŵ and Q x ŵ,ŵ Q x, whch completes the proof Fg. 2. Proof of Corollary 3. Sold lne s a tangent to U x at x = x. Dashed lne represents the functon αx = U x + λx x, where λ < U x. Corollary 3 provdes a rankng of the unform-prcng schemes for three dfferent settngs. From the vewpont of resource manager the best scenaro s the one where agents act as prce-takers and compettve equlbrum s acheved; resultng allocatons are then Pareto-optmal and marketclearng prce s equal to margnal cost of producton, whch maxmzes manager s proft. Unfortunately, the manager cannot guarantee that the prce-takng assumpton wll hold. In fact, Corollary 3 shows that there s a ratonale for agents to vew the resource allocaton problem as a game wth payoffs defned by 5 or 17. Prce-antcpatng bddng, resultng n demand reducton, s a way to mprove profts and therefore s a reasonable strategy. Thus, from the perspectve of the agents the most desrable settng s the one descrbed by the Cournot competton model leadng to the hghest ndvdual payoffs. However, t s the mechansm desgner, not agents, that defnes the model by the rules of resource allocaton. Corollary 2 demonstrates that the potental neffcency of 4471

7 outcomes makes the Cournot settng the most undesrable one from the resource manager vewpont. As a result, agents should not expect from the ratonal resource manager to apply t. In the consdered settng of unform-prcng scheme desgner s best response to the agents expected prce-antcpatng behavor s to apply the soluton where agents report ther wllngness to pay as t reduces the adverse effects of agents msrepresentaton of preferences. VI. BIDDING ALGORITHM To fnalze the paper we wll now nvestgate the stablty of Nash equlbra and the convergence condtons of bddng process n the unform-prce share aucton wth prceantcpatng agents reportng ther wllngness to pay. Aucton rules correspond to the competton model defned by the market-clearng equaton 6, where allocaton rule satsfes 7 and payoffs are defned by 5. We focus our attenton on the famly of smooth prce functons characterzed by nondecreasng elastcty,.e. β y. Lemma 2: If the assumptons of Proposton 2 hold, then: pỹ pȳ ȳ ỹ, 24 where ỹ = x w and ȳ = x w. Proof: Suppose that pỹ pȳ. Then, by assumptons II-B and III-A, ỹ ȳ. Ths mples that pỹỹ pỹȳ and pȳỹ pȳȳ. Subtractng the frst nequalty from the latter yelds 24. The same follows for pỹ pȳ. Fnally, 24 follows from the fact that p s s strctly ncreasng,.e. pȳ pỹ/ȳ ỹ <. Proposton 4: Let Fwt pywtˆxt wt, where ˆx t {[x ] + : U βywtx /ywt = pywt} for every L. If the assumptons of Proposton 2 hold, then every Nash equlbrum of the game wth payoffs defned by 5 s an asymptotcally stable fxed pont of the system: ẇt = Fwt. 25 Proof: We wll show that pywt converges to the equlbrum prce ˆp when agents modfy ther bds accordng to Fwt. Snce for w : yw w = pyw + yw pyw w 1, we conclude that yw/ w pyw 1. Ths mples that: ṗywt = p ywt X p ywt X ywt ẇ t w pywtˆx t w t pywt = p ywt X ˆx t x wt. Now, defne V t pywt ˆp 2 /2 and observe that: V t = pywt ˆpṗywt p ywtpywt ˆpŷ ywt, for ŷ ˆx t and ywt x wt. Snce ŷ satsfes the equlbrum condton ŷˆp = ŵ, from Lemma 2 t follows that V t, wth equalty only for pywt = ˆp defnng a unque statonary pont of Fwt. Thus, V s a Lyapunov functon for system Fwt. Under assumptons of the proposton ths mples that wt converges to ŵ for w = w. Fg. 3. Communcaton scheme of mechansm Proposton 4 proves the convergence of a bddng process n the unform-prce share aucton under the assumptons of Proposton 2. Fgure VI explans the procedure. Resource manager begns by settng an ntal value of the prce p > and the correspondng total producton level y. These two values are then communcated to all bdders elgble to partcpate n the aucton. In response to the receved sgnals each prce-antcpatng bdder calculates a proft-maxmzng consumpton level ˆx and translates the value to the sgnal requred by aucton rules: ˆx k w k+1 [ = Φ w k, p k kˆx where Φa,b a + ρb a wth ρ,1 and: { [x ] + : U x 1 βk x y k ], 26 = p k }. 27 It s assumed that each agent can approxmate the prce elastcty measure β k correspondng to the total resource utlzaton y k to calculate ˆx k. The amount of resource x k assgned to the agent n the teraton k = 1,..,T 1 can be ether communcated to each agent or calculated by each agent ndvdually: x k = w k /p k. Next, when optmzed bds are set, the vector w k+1 1,..., s submtted to the manager to clear the market,.e.: w k+1 L x k+1 where p k+1 = py k+1 and: { y k+1 s: = [wk+1 ] +, 28 p k+1 w k+1 = sps }

8 Havng calculated the prce and the allocatons, resource manager sends the values to the agents and wats for ther response. The bddng procedure stops when predefned stoppng condtons are satsfed. Fnal outcomes of the aucton are communcated to all partcpants. Example 1: Consder a dvsble resource characterzed by the followng prce functon py = C y 1 wth elastcty εy = yc y. Suppose that there are L = 4 agents competng for the resource and let U 1 = γ 1, U 2 = γ 2 log1 +, U 3 x 3 = γ 3 arctanx 3 and U 4 x 4 = γ 4 x4. Fgure 1 demonstrates the bddng process. Observe that allocatons x k decrease as prce p k ncreases wth k T = Fg. 4. Convergence of the unform-prce mechansm The sequences of bds {w k } k=, allocatons {xk } k= and prces {p k } k= are proved to converge to a Nash equlbrum pont f the condtons of Proposton 4 are met. It s mportant to notce that the proof of convergence relates on the property 24, whch can be nterpreted as the law of demand agents aggregate demand y and resource prce move n opposte drectons [12]. Furthermore, the allocatons are entrely determned by a pontwse characterzaton of agents preferences. Messages reported to the mechansm contan scalar values, whch reveal enough nformaton to calculate the resource assgnments and the correspondng payments. Ths s an extremely desrable property of a mechansm, especally n the network envronments of large scale systems wth communcaton constrants and lmted computatonal power. On the other hand, the model assumes synchronous communcaton pattern,.e. allocatons are determned based on sgnals receved from all agents wthn a predefned tme perod. Ths s a restrctve constrant whch should be relaxed n many real-lfe settngs asynchronous communcaton should be admtted. We have also consdered all agents contnuous dynamcs evolvng at the same rate, whch may be a smplfed and potentally harmful assumpton as well; see Shenker [14] for an nterestng comment. Furthermore, the model gnores the problem of agents droppng off when payoff values become negatve ndvdual ratonalty constrant. These and other ssues related to the best response dynamcs are the subject of further nvestgatons. p x 3 x 4 VII. SUMMARY The above dscusson llustrates the consequences of ncompatblty of the overall system goals and the ndvdual objectves of the agents the unform-prce market-clearng mechansm creates ncentves to reveal to the system a reduced value of demand. On one hand, ths mproves agents profts, but on the other t gves rse to allocatons that do not maxmze utlzaton of the resource n the system. Theoretcal results establshed n the paper can therefore serve as a gudelne for lmtng the adverse effects of strategc behavoral patterns wthn the unform-prcng framework t relates to the choce of strateges agents are allowed to play. For other gudelnes, resultng from the exploraton of the relatonshp between condtons 12 and 16, see [15], [16], [17]. Snce both the allocatons and the payments are arguments of the payoff functons, the structure of the mechansm s rules determnes the optmalty condtons a ratonal agent wll try to acheve. As a result, an approprate choce of prcng functons and allocaton rules can make both the resource manager and agents approach the smlar goals. REFERENCES [1] F. P. Kelly, Chargng and rate control for elastc traffc, European Transactons on Telecommuncatons, vol. 8, pp , [2] F. P. Kelly, A. K. Maulloo, and D. K. Tan, Rate control for communcaton networks: shadow prces, proportonal farness, and stablty, Journal of the Operatonal Research Socety, vol. 49, pp , [3] R. Johar and J. N. Tstskls, Effcency loss n a network resource allocaton game, Mathematcs of Operatons Research, vol. 29, pp , 24. [4] R. Johar, Effcency loss n market mechansms for resource allocaton, Ph.D. dssertaton, MIT, 24. [5] L. Ausubel and P. Cramton Demand reducton and neffcency n mult-unt auctons. [Onlne]. Avalable: deas.repec.org/p/pcc/pccumd/98wpdr.html [6] R. Engelbrecht-Wggans and C. Kahn, Mult-unt auctons wth unform prces, Economc Theory, [7] R. Rockafellar, Convex Analyss. Prnceton Unversty Press, 197. [8] I. Ekeland and R. Temam, Convex Analyss and Varatonal Problems. SIAM, [9] W. Ogryczak, Wspomagane decyzj w warunkach ryzyka, 2, manuscrpt. [1] M. M. Kostreva, W. Ogryczak, and A. Werzbck, Equtable aggregatons and multple crtera analyss. European Journal of Operatonal Research, vol. 158, no. 2, pp , 24. [Onlne]. Avalable: [11] D. Fudenberg and J. Trole, Game Theory. The MIT Press, [12] A. Mas-Colell, M. Whnston, and J. Green, Mcroeconomc Theory. Oxford Unversty Press, [13] J. Rosen, Exstence and unqueness of equlbrum ponts for concave n-person games, Econometrca, vol. 33, no. 3, July [14] S. Shenker, Makng greed work n networks: a game-theoretc analyss of swtch servce dscplnes, Networkng, IEEE/ACM Transactons on, vol. 3, no. 6, pp , Dec [15] M. Karpowcz and K. Malnowsk, On effcent resource allocaton n communcaton networks. n SOFSEM 1, ser. Lecture Notes n Computer Scence, J. van Leeuwen, G. F. Italano, W. van der Hoek, C. Menel, H. Sack, and F. Plasl, Eds., vol Sprnger, 27, pp [16], Characterzaton of scalar strategy mechansms for effcent resource allocaton: demand sde case, n Proceedngs of 11th IFAC/IFORS/IMACS/IFIP Symposum on Large Scale Systems, July 27. [17], Characterzaton of scalar strategy mechansms for effcent resource allocaton: supply sde case, n Proceedngs of 46th IEEE Conference on Decson and Control,