Efficiency Loss in Maket Mechanisms fo Resouce Allocation by Ramesh Johai A.B., Mathematics, Havad Univesity (1998) Cetificate of Advanced Study in Mathematics, Univesity of Cambidge (1999) Submitted to the Depatment of Electical Engineeing and Compute Science in Patial Fulfillment of the Requiements fo the Degee of Docto of Philosophy in Electical Engineeing and Compute Science at the Massachusetts Institute of Technology June 2004 c 2004 Massachusetts Institute of Technology. All ights eseved. Signatue of Autho... Depatment of Electical Engineeing and Compute Science May 18, 2004 Cetified by... John N. Tsitsiklis Pofesso of Electical Engineeing and Compute Science Thesis Supeviso Accepted by... Athu C. Smith Chaiman, Committee on Gaduate Students Depatment of Electical Engineeing and Compute Science
Efficiency Loss in Maket Mechanisms fo Resouce Allocation by Ramesh Johai Submitted to the Depatment of Electical Engineeing and Compute Science on May 18, 2004, in patial fulfillment of the equiements fo the degee of Docto of Philosophy in Electical Engineeing and Compute Science Abstact This thesis 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 best inteests at stake. We conside the design of esouce allocation mechanisms in such envionments. The analytic appoach we pusue is chaacteized by fou salient featues. Fist, the monetay value of esouce allocation is measued by the aggegate suplus (aggegate utility less aggegate cost) achieved at a given allocation. An efficient allocation is one which maximizes aggegate suplus. Second, we focus on maket-cleaing mechanisms, which set a single pice to ensue demand equals supply. Thid, all the mechanisms we conside ensue a fully efficient allocation if maket paticipants do not anticipate the effects of thei actions on maket-cleaing pices. Finally, when maket paticipants ae pice anticipating, full efficiency is geneally not achieved, and we quantify the efficiency loss. We make two main contibutions. Fist, fo thee economic envionments, we conside specific maket mechanisms and exactly quantify the efficiency loss in these envionments when maket paticipants ae pice anticipating. The fist two envionments addess settings whee multiple consumes compete to acquie a shae of a esouce in eithe fixed o elastic supply; these models ae motivated by esouce allocation in communication netwoks. The thid envionment addesses competition between
4 multiple poduces to satisfy an inelastic demand; this model is motivated by maket design in powe systems. Ou second contibution is to establish that, unde easonable conditions, the mechanisms we conside minimize efficiency loss when maket paticipants anticipate the effects of thei actions on maket-cleaing pices. 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 mechanisms we conside uniquely minimize efficiency loss when maket paticipants ae pice anticipating. Thesis Supeviso: Title: John N. Tsitsiklis Pofesso of Electical Engineeing
Acknowledgments Fist and foemost, I am deeply indebted to my adviso, Pofesso John Tsitsiklis. His encouagement, suppot, and advice have been immensely valuable, both in pesonal and pofessional tems. I am paticulaly gateful fo his emphasis on simplicity and elegance in eseach, and fo the genuine concen he has shown fo my development as an academic. I am also gateful to Pofesso Fank Kelly fo his enthusiasm and counsel, a constant thoughout my gaduate caee. I fist became familia with netwok picing poblems in convesations with him, and this thesis beas the hallmak of lessons leaned duing my stay in Cambidge unde his tutelage. Special thanks go to my thesis committee membes, Pofesso Abhijit Banejee and Pofesso Robet Gallage. Each devoted significant time and effot to my thesis, and thei suggestions and comments led to substantial impovement in the final poduct. The Laboatoy fo Infomation and Decision Systems (LIDS) povided an ideal wok envionment fo the intedisciplinay wok of this thesis. I am paticulaly gateful to Pofesso Dimiti Betsekas, fo helping me navigate the subtleties of convex analysis; to Pofesso Vincent Chan, fo his steady hand in diecting the lab; and to Pofesso Sanjoy Mitte, fo his intellectual stewadship of my time at MIT. I am in the debt of these faculty and many othes at MIT fo thei suppot ove the last fou yeas. I have also benefited fom time spent with colleagues at MIT. I would especially like to thank Shie Manno, whose dedication to eseach helped keep me disciplined; and Emin Matinian, a fequent countepat fo lunch convesations, both academic and othewise. Finally, I was paticulaly fotunate to have had Constantine Caamanis as an officemate, with whom I ve shaed many inteesting discussions since my days as an undegaduate. Last, I owe a geat debt to Hsin Chau, fo he suppot, encouagement, and sense of humo; he companionship has been invaluable in all facets of life. This eseach was suppoted by a National Science Foundation Gaduate Reseach Fellowship, by the Defense Advanced Reseach Pojects Agency unde the Next Geneation Intenet Initiative, and by the Amy Reseach Office unde gant DAAD10-00-1-0466. 5
Peliminaies Notation We use R to denote the eal numbes, and R + to denote [0, ). Italics will be used to denote scalas, e.g., x. Boldface will be used to denote vectos, e.g., x = (x 1,...,x n ). When x is a scala, the notation (x) + will be used to denote the positive pat of x; i.e., (x) + = x if x 0, and (x) + = 0 if x 0. If x 1,...,x n R m, and x = (x 1,...,x n ), we will use x i to denote the components of x othe than x i ; that is, x i = (x 1,...,x i 1,x i+1,...,x n ). Thoughout the thesis, if f : (R m ) n R is a ealvalued function of n vectos x 1,...,x n R m, we let f(x i ;x i ) denote the function f as a function of x i while keeping the components x i fixed. Convex analytic methods play a key ole in this thesis, and we collect some equied notions hee [14, 103]. An extended eal-valued function is a function f : R n [, ]; such a function is called pope if f(x) > fo all x, and f(x) < fo at least one x. We say that a vecto γ R n is a subgadient of an extended eal-valued function f at x if fo all x R n, we have: f(x) f(x) + γ (x x). The subdiffeential of f at x, denoted f(x), is the set of all subgadients of f at x. We say that f is subdiffeentiable at x if f(x). We will typically be inteested in subgadients of a convex function f, and supegadients of a concave function f. A vecto γ is a supegadient of f if γ is a subgadient of f; thus we denote the supediffeential of f at x by [ f(x)]. Fo extended eal-valued functions f : R [, ], we will equie some additional concepts. We denote the ight diectional deivative of f at x by + f(x)/ x and left diectional deivative of f at x by f(x)/ x (if these exist). If f is convex, then f(x) = [ f(x)/ x, + f(x)/ x], povided the diectional deivatives exist. 7
8 Peequisites The main peequisites fo this thesis ae a backgound in eal analysis at the level of Rudin [110], as well as some facility with convex optimization and elementay convex analysis. Souces fo backgound on convex optimization include the books by Whittle [145], Betsekas [13], and Boyd and Vandenbeghe [17], while backgound on convex analysis may be found in the texts by Betsekas et al. [14] and Rockafella [103]. Micoeconomics (paticulaly maket theoy) and game theoy also play a key ole in this thesis, and some basic knowledge of the two fields is helpful. The text by Vaian povides a concise intoduction to micoeconomic theoy [137], while the textbook by Mas-Colell et al. povides deepe coveage [82]. As to game theoy, in this thesis we will only use elementay concepts fom game theoy, paticulaly Nash equilibium; howeve, some undestanding of the modeling issues is helpful. Fo this pupose, see the books by Fudenbeg and Tiole [43], Myeson [89], and Osbone and Rubinstein [96] (whee the last efeence is a concise intoduction fo the uninitiated eade). Bibliogaphic Note Potions of the content of Chapte 2 appea in the pape by Johai and Tsitsiklis [60]; exceptions ae Sections 2.1.3, 2.3, 2.4.3, and 2.5.1. Sections 3.1, 3.2, 3.3, and 3.4 will appea in the pape by Johai et al. [58].
Contents Abstact 3 Acknowledgments 5 Peliminaies 7 Notation........................................ 7 Peequisites...................................... 8 Bibliogaphic Note.................................. 8 List of Figues 13 1 Intoduction 15 1.1 Consumes, Poduces, and Aggegate Suplus............... 16 1.2 Maket-Cleaing Mechanisms......................... 19 1.3 Pice Taking Behavio and Competitive Equilibium............ 22 1.4 Pice Anticipating Behavio and Nash Equilibium............. 23 1.5 An Example................................... 25 1.5.1 A Maket-Cleaing Mechanism.................... 26 1.5.2 Pice Taking Consumes........................ 27 1.5.3 Pice Anticipating Consumes.................... 28 1.5.4 Efficiency Loss............................. 30 1.6 Contibutions of This Thesis.......................... 31 2 Multiple Consumes, Inelastic Supply 33 2.1 Peliminaies................................... 37 2.1.1 Pice Taking Uses and Competitive Equilibium......... 40 2.1.2 Pice Anticipating Uses and Nash Equilibium.......... 42 2.1.3 Coollaies................................ 47 2.2 Efficiency Loss: The Single Link Case..................... 50 2.3 Pofit Maximizing Link Manages...................... 57 9
10 CONTENTS 2.4 Geneal Netwoks................................ 60 2.4.1 An Extended Game........................... 64 2.4.2 Efficiency Loss............................. 72 2.4.3 A Compaison to Popotionally Fai Picing............ 81 2.5 Extensions.................................... 84 2.5.1 Stochastic Capacity........................... 84 2.5.2 A Geneal Resouce Allocation Game................ 87 2.6 Chapte Summay................................ 89 3 Multiple Consumes, Elastic Supply 91 3.1 Peliminaies................................... 94 3.1.1 Pice Taking Uses and Competitive Equilibium......... 97 3.1.2 Pice Anticipating Uses and Nash Equilibium.......... 99 3.1.3 Nondeceasing Elasticity Pice Functions.............. 106 3.2 Efficiency Loss: The Single Link Case..................... 108 3.3 Inelastic Supply vs. Elastic Supply...................... 119 3.4 Geneal Netwoks................................ 124 3.5 Counot Competition.............................. 135 3.5.1 Models with Bounded Efficiency Loss................ 142 3.5.2 Geneal Netwoks........................... 153 3.6 Counot Competition with Latency...................... 155 3.7 Chapte Summay................................ 161 4 Multiple Poduces, Inelastic Demand 163 4.1 Peliminaies................................... 168 4.1.1 Pice Taking Fims and Competitive Equilibium......... 170 4.1.2 Pice Anticipating Fims and Nash Equilibium.......... 173 4.2 Efficiency Loss.................................. 179 4.3 Negative Supply................................ 181 4.4 Stochastic Demand............................... 183 4.5 Chapte Summay................................ 186 5 Chaacteization Theoems 189 5.1 Multiple Consumes, Inelastic Supply.................... 191 5.1.1 A Fist Chaacteization Theoem.................. 192 5.1.2 A Second Chaacteization Theoem................. 208 5.1.3 A Two Use Mechanism with Abitaily Low Efficiency Loss.. 211 5.2 Multiple Poduces, Inelastic Demand.................... 214 5.3 Chapte Summay................................ 229
CONTENTS 11 6 Conclusion 231 6.1 Two-Sided Makets............................... 231 6.2 Dynamics.................................... 232 6.3 Mechanism Design and Distibuted Systems................ 233 6.4 Futue Diections................................ 236 Refeences 237
List of Figues 1-1 The maket-cleaing pocess.......................... 20 2-1 The maket-cleaing pocess with inelastic supply............. 39 2-2 Poof of Lemma 2.7............................... 53 2-3 Example 2.2................................... 63 3-1 The maket-cleaing pocess with elastic supply.............. 98 3-2 Poof of Theoem 3.8, Step 4.......................... 117 3-3 The function g(b) in Theoem 3.10...................... 121 3-4 The maket-cleaing pocess fo Counot competition........... 138 3-5 Poof of Theoem 3.20............................. 143 4-1 The maket-cleaing pocess with inelastic demand............ 169 5-1 The function g(ε) in Theoem 5.1....................... 195 13
C H A P T E R 1 Intoduction But man has almost constant occasion fo the help of his bethen, and it is in vain fo him to expect it fom thei benevolence only. He will be moe likely to pevail if he can inteest thei self-love in his favou, and show them that it is fo thei own advantage to do fo him what he equies of them.... It is not fom the benevolence of the butche, the bewe, o the bake that we expect ou dinne, but fom thei egad to thei own inteest. Adam Smith, The Wealth of Nations, Book I, Chapte II [125] T his thesis 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 best inteests at stake. Ou inspiation is dawn pimaily fom communication netwoks and powe systems. Communication netwoks, paticulaly the Intenet, have a distibuted stuctue which pohibits the implementation of sophisticated centalized mechanisms to allocate netwok esouces among end uses. On the othe hand, the gowth of the Intenet has led to inceasingly divese taffic shaing the same netwok, making efficient allocation of netwok esouces difficult to ensue. Powe systems ae also chaacteized by lage scale, but by contast, typically un a collection of makets that detemine cleaing pices fo electicity at nodes thoughout egional netwoks. In this setting demand is highly inelastic, lagely because the vaiation in pices seen by most powe consumes is on a slow timescale. In geneal, the chosen maket designs fo lage scale powe netwoks do not potect against the execise of maket powe in the pesence of highly inelastic demand. Ou appoach in this thesis is to design mechanisms which take into account achitectual featues of these systems, while ensuing that efficient esouce allocation is achieved. The analytic appoach we have chosen is chaacteized by fou salient featues, descibed in detail in each of the next fou sections. In Section 1.1, we de- 15
16 CHAPTER 1. INTRODUCTION scibe ou metic of efficiency, the aggegate suplus of consumes and poduces. In Section 1.2, we estict attention to maket-cleaing mechanisms which set a single pice to clea the maket between supplies and consumes of esouces. In Section 1.3, we note that all the mechanisms we conside achieve efficient allocations when maket paticipants act as pice takes. Finally, in Section 1.4, we outline ou measuement of efficiency loss when maket paticipants ae pice anticipating. In Section 1.5, we develop a detailed example that highlights each of the points discussed in Sections 1.1 to 1.4. We biefly outline the contibutions of the thesis in Section 1.6. 1.1 Consumes, Poduces, and Aggegate Suplus In this section we will descibe the geneal esouce allocation setting we will conside, and ou definition of efficiency. All the esouce allocation poblems we descibe consist of two types of maket paticipants: consumes and poduces. Consumes demand an allocation of a esouce; and poduces supply that esouce. We identify each consume with a utility function U (d ), which defines the monetay value to consume of eceiving d units of the scace esouce, whee d 0. Similaly, we identify each poduce n with a cost function C n (s n ), which defines the monetay cost to poduce n of supplying s n units of the scace esouce, whee s n 0. We will fomalize detailed assumptions on these constucts thoughout the thesis; but fo the moment, we simply note that we will always assume both utility and cost ae nondeceasing functions. The implication is that consumes value lage amounts of esouces, while supplies incu highe costs fo poducing lage amounts of esouces. A key assumption we have made is that both utility and cost ae measued in monetay units. This assumption implies that thee ae actually two types of goods in the esouce allocation settings we conside: the fist is the scace esouce unde consideation (data ate, electic powe, etc.), and the second is money. Suppose then that a consume with utility function U eceives a esouce allocation d, but makes a payment w; then the net payoff to this consume is: U(d) w. (1.1) On the othe hand, suppose that a poduce with cost function C poduces a supply s, but eceives evenue w; then the net payoff to this poduce is: w C(s). (1.2) Thus payoffs give the net monetay benefit to consumes and poduces, taking into account both the money paid o eceived, and the esouce allocation eceived o poduced. The sepaable fom of the payoffs we see hee is a diect consequence of the fact that utility and cost ae measued in monetay units. Envionments whee payoffs
SECTION 1.1. CONSUMERS, PRODUCERS, AND AGGREGATE SURPLUS 17 have this sepaable fom ae known as quasilinea envionments [46, 82]. We ae seaching fo mechanisms that achieve efficient allocation of esouces. We adopt as ou notion of efficiency the well known concept of Paeto efficiency: an allocation is Paeto efficient if the benefit to one maket paticipant cannot be stictly inceased without simultaneously stictly deceasing the benefit to anothe playe. Thoughout this thesis, we always use the tem efficient allocation to efe to a Paeto efficient allocation. We now chaacteize the implications of Paeto efficiency on ou esouce allocation model. Fo simplicity, we stat by consideing a model consisting of only two consumes with utility functions U 1 and U 2, and a single esouce of inelastic supply S; that is, the maximum available supply is fixed at S units. We can intepet inelastic supply in tems of a poduce with a discontinuous cost function: if the supply of the esouce available is exactly S units, then it is as if a single poduce supplies the esouce, with cost function C(s) given by: C(s) = { 0, if s S;, if s > S. Thus the poduce incus zeo cost if at most S units must be supplied, and infinite cost othewise. In paticula, in seaching fo a Paeto efficient allocation, the assumption of inelastic supply implies a constaint that the total allocation made to the two consumes cannot exceed the available supply S. The key assumption we make is that the two consumes may feely exchange cuency. We claim that unde this cicumstance, any Paeto efficient allocation d = (d 1, d 2 ) to the two consumes must be an optimal solution to the following optimization poblem: maximize U 1 (d 1 ) + U 2 (d 2 ) (1.3) subject to d 1 + d 2 S; (1.4) d 1, d 2 0. (1.5) The intuition is clea: if a Paeto efficient allocation is not an optimal solution to (1.3)- (1.5), then thee must exist a solution to (1.3)-(1.5) and a vecto of monetay tansfes between the consumes that collectively leave both consumes bette off than the oiginal allocation. We now demonstate this fact fomally. Suppose that (d 1, d 2 ) is Paeto efficient, but that fo anothe feasible solution (d 1, d 2 ) to (1.3)-(1.5), we have: U 1 (d 1 ) + U 2 (d 2 ) < U 1 (d 1) + U 2 (d 2). We can assume without loss of geneality that U 1 (d 1 ) < U 1 (d 1 ). In this case, suppose the consumes shift to d fom d, but that in addition playe 1 pays playe 2 an amount
18 CHAPTER 1. INTRODUCTION w = (U 2 (d 2 ) U 2 (d 2 ))+. Then the payoff to playe 1 is now: U 1 (d 1) w = U 1 (d 1) (U 2 (d 2 ) U 2 (d 2)) + > U 1 (d 1 ), while the payoff to playe 2 becomes: U 2 (d 2) + w = U 2 (d 2) + (U 2 (d 2 ) U 2 (d 2)) + U 2 (d 2 ). Thus playe 1 is stictly bette off than befoe, and playe 2 is no wose off than befoe; so d could not have been Paeto efficient. The peceding simple stoy is in fact quite geneal: as long as we allow abitay monetay tansfes between maket paticipants, any Paeto efficient allocation must maximize aggegate utility less aggegate cost. This quantity is known as the aggegate suplus, o Mashallian aggegate suplus (afte the economist Alfed Mashall, though an ealy pecuso was consideed by Dupuit; see [35, 81, 82, 134]). Aggegate suplus denotes the net monetay benefit to the economy unde a chosen allocation. In this wok we will conside thee instances of the aggegate suplus maximization poblem, which we efe to as the SYSTEM poblem thoughout the thesis. In Chapte 2, we conside a model whee multiple consumes bid fo a single esouce in inelastic supply; in this case the SYSTEM poblem educes to maximization of aggegate utility subject to the supply constaint. In Chapte 3, we conside a model whee multiple consumes bid fo a single esouce in elastic supply, and in this case the SYSTEM poblem is witten diectly as maximization of aggegate utility less aggegate cost. Finally, in Chapte 4, we conside a model whee multiple poduces bid to satisfy an inelastic demand. In this case it is as if thee exists a single consume with utility if less than the fixed demand is poduced, and utility zeo othewise. Thus the SYSTEM poblem educes to minimization of aggegate cost subject to the demand constaint. We note that in an engineeing context, efficient allocation of esouces is often simply intepeted as a equiement that all available esouces be allocated, without any specification of the distibution of the allocation ove playes. In the example illustated above, an efficient allocation would then be any allocation whee the capacity constaint (1.4) holds with equality. Indeed, this is pecisely Paeto efficiency if consumes have utilities that ae stictly inceasing in thei allocation, and if no monetay tansfes ae available to the consumes. Howeve, as we have seen, measuing cost and utility in monetay units educes welfae measuement to a simple and convenient quantity, the aggegate suplus. When we intoduce money into the system, a Paeto efficient allocation must not only fully allocate available esouces, but also maximize aggegate suplus. Fo the example above, a Paeto efficient allocation should not only satisfy the constaint (1.4), but also maximize the objective function (1.3). Howeve, we note that the use of aggegate utility, and aggegate suplus moe geneally, as a welfae metic has taditionally
SECTION 1.2. MARKET-CLEARING MECHANISMS 19 been a point of geat debate in both economics and philosophy. The heat of this debate can be taced to the fact that the vey notion of aggegate utility pesupposes the ability to compae the utilities of diffeent membes of society, even though such a compaison may not be possible. One might assume that incompaability of pefeences is esolved by measuing all utilities and costs in monetay units; howeve, membes of society fom diffeent income classes place diffeent values on cuency itself, and thus one cannot claim that a dolla is a dolla is a dolla. The notion of maximization of aggegate utility as a desiable goal fo society was fist poposed by Bentham, the fathe of utilitaianism [11]. Mashall advanced the application of quasilinea payoffs to maket theoy, and developed the esulting notion of maximization of aggegate suplus as a Paeto efficient allocation ule [81]. But the quasilinea payoff model was citiqued fo its pesumption that compaison of utilities of diffeent maket paticipants is possible; an illuminating discussion of these issues in utility theoy is povided in the pai of suveys by Stigle [128, 129]. The assumption that utility was inheently incompaable eventually led to the famous impossibility theoem of Aow in social choice theoy [3, 117]. Fo the puposes of this thesis, we only advise the eade that ou focus on quasilinea envionments, though well motivated, is by no means canonical. The boade issue in social choice is that maket paticipants may not be motivated by only thei monetay payoff, and in addition, as a community they may not be inteested in achieving a Paeto efficient allocation. Instead, fainess concens may be paamount, such as ensuing that the numbe of consumes o poduces who choose to paticipate in the maketplace is as lage as possible. Such questions suggest inteesting depatues fom the quasilinea payoff models used in this thesis. Fo futhe study, a thoough discussion of pefeences and utility involving elements of both philosophy and economics may be found in the volume of Sen [118]. 1.2 Maket-Cleaing Mechanisms Having defined the maximal aggegate suplus as the benchmak we hope to attain, we now conside the poblem of defining esouce allocation mechanisms which each that goal. Ou focus in this thesis will be on maket-cleaing mechanisms. These mechanisms choose a single pice so that demand equals supply, i.e., to clea the maket. Why conside single pice maket mechanisms, as opposed to pice disciminatoy solutions? Fist, ou hope in this thesis is to advance the theoy of esouce allocation mechanisms which ae feasible in lage scale, distibuted systems. In such systems, paticulaly the Intenet, the fine-scale distinction of uses needed to implement a sophisticated mechanism of pice discimination does not seem viable; fo this eason, we ae led to simple picing schemes. A mechanism which sets a single pice has the advantage of anonymity: to detemine the maket-cleaing pice, the mechanism needs
20 CHAPTER 1. INTRODUCTION pice AD(p) AS(p) p quantity Figue 1-1. The maket-cleaing pocess: Each consume submits a demand function D (p) to the maket mechanism, and each poduce n submits a supply function S n(p). These define the aggegate demand cuve AD(p) = D(p), and the aggegate supply cuve AS(p) = n Sn(p). The pice p is chosen so that supply equals demand, i.e., so that AD(p ) = AS(p ). (Hee and thoughout the thesis, fo maket-cleaing diagams we adopt the standad convention fom economics that quantity appeas on the hoizontal axis and pice on the vetical axis.) no knowledge of individual consumes and poduces only the aggegate supply and aggegate demand ae equied. (We will etun to the discussion of scalability of efficient esouce allocation mechanisms in the context of ou eview of mechanism design in the conclusion to the thesis, Chapte 6.) A second potential motivation fo single pice mechanisms comes fom thei use in pactice, paticulaly in electicity makets; the equity of setting a single pice fo all maket paticipants seems to have appeal fom a social and political standpoint, and has led to widespead use of bidding systems which set a single pice pe node in an electicity gid [131]. We descibe the basic opeation of such a mechanism fo the case of a single esouce. Each consume chooses a demand function D (p), which descibes his demand fo the esouce as a function of the pice p of that esouce. Analogously, each poduce n chooses a supply function S n (p), which descibes the quantity the poduce is willing to supply as a function of the pice of the esouce. The mechanism then chooses a single pice p so that aggegate demand equals aggegate supply: D (p ) = n S n (p ). Each consume eceives a esouce allocation of D (p ), while each poduce n poduces a quantity S n (p ). This pocess is gaphically depicted in Figue 1-1.
SECTION 1.2. MARKET-CLEARING MECHANISMS 21 In this thesis, we will geneally be inteested in eithe competition between consumes, o competition between poduces; we leave open the analysis of models that simultaneously allow competition among both buyes and selles. In Chapte 2, we conside a maket-cleaing mechanism whee each consume submits a demand function, and a pice is chosen so that aggegate demand equals a peset inelastic supply. In Chapte 3, we conside a simila model, but whee instead a pice is chosen so that demand matches supply accoding to a peset elastic supply function. Finally, in Chapte 4, we conside a maket-cleaing mechanism whee each poduce submits a supply function, and a pice is chosen so that aggegate supply equals a peset inelastic demand. Models have peviously been developed to undestand maket behavio when eithe supplies submit supply functions, o consumes submit demand functions. In a seminal pape, Klempee and Meye chaacteized equilibia in makets whee supplies submit supply functions [69]. Competition among buyes who submit demand schedules was consideed by Wilson [146]. Both these models allow maket paticipants to submit nealy abitay supply o demand functions. By contast, we will conside mechanisms whee consumes and poduces ae esticted to choose fom paametized demand o supply functions, whee the paamete is a eal scala. Fo example, in Chaptes 2 and 3, we allow a consume to choose a single scala w 0, and assume the esulting fom of the demand function fo that consume is D(p, w ) = w /p. The intepetation is that w is the total willingness-topay of consume, since egadless of the maket-cleaing pice p, the payment made by consume will be p D(p, w ) = w. (We investigate a simple example of such a mechanism in Section 1.5.) We have two pimay motivations fo consideing maket mechanisms whee paticipants submit paametized demand o supply functions. Fist, in lage scale decentalized systems such as moden communication netwoks, it seems uneasonable to expect the netwok to suppot tansmission of abitay demand functions to widely dispesed esouces unning maket-cleaing pocesses. Instead, scalable maket solutions fo allocation of netwok esouces must ensue the stategy space of uses is simple which we intepet hee as a eduction in the dimension of the space of demand functions possible. (See also Kelly [63] fo a simila agument egading netwok picing.) A second motivation fo the types of maket mechanisms we conside is that educing the stategy space of maket paticipants might educe inefficiency due to the execise of maket powe. This is a point most focefully made in the electicity makets, whee cuently fims may submit abitay supply functions, as poposed in the pape by Klempee and Meye [69]. In geneal, equilibia in supply functions may be highly inefficient when fims manipulate the maket; and thus we ae motivated to conside estictions in the class of supply functions fims ae allowed to submit, in
22 CHAPTER 1. INTRODUCTION hopes of impoving the efficiency of the maket mechanism. (A moe detailed discussion of this issue may be found in the intoduction to Chapte 4.) 1.3 Pice Taking Behavio and Competitive Equilibium A cental featue of all the maket mechanisms we conside is that when maket paticipants act as pice takes, full efficiency can be achieved. Pice takes ae maket paticipants who do not anticipate the effect of thei stategic choices (i.e., thei demand o supply function) on the eventual maket-cleaing pice. An altenative situation is that maket paticipants do in fact anticipate the effects of thei actions on the maketcleaing pice a situation we take up in the following section. Fo simplicity, we will fomalize pice taking hee only fo a special case whee multiple consumes compete fo a single esouce. In this case, if a consume has a utility function U and submits a demand function D (p) to the maket mechanism, the payoff to the consume when the maket-cleaing pice is p is given by: U (D (p )) p D (p ). The fist tem is the monetay value, o utility, to the consume of the allocation D (p ); the second tem is the payment made by the consume when the pice is p pe unit consumed. Of couse, the maket-cleaing pice depends on the demand functions submitted by all consumes. In paticula, as consume vaies his demand function, the maket-cleaing pice p will vay as well. Pice taking behavio assumes this elationship is unknown to the consumes: all consumes take the pice p as fixed, and then choose a demand function which optimizes thei payoff given this fixed pice p. In this thesis, we will establish that fo all the maket mechanisms unde consideation, pice taking behavio leads to a Paeto efficient allocation. The fomal statement of this poposition is that thee exists a pai consisting of a pice and a vecto of stategies fo all maket paticipants such that: (1) the pice is the maket-cleaing pice given the composite stategy vecto of the maket paticipants; and (2) each of the paticipants has chosen thei stategy to maximize thei payoff given the fixed pice. Such a pai is known as a competitive equilibium. We will establish existence of competitive equilibia, and then establish that at competitive equilibia the esulting allocations maximize aggegate suplus; fo the models of Chaptes 2 and 3, these ae esults of Kelly [62] and Kelly et al. [65], espectively. Infomally, these esults state that a single pice can be chosen so that individual optimization by maket paticipants yields a globally efficient outcome. The fact that competitive equilibia yield Paeto efficient allocations is a cental esult in maket theoy, the fist fundamental theoem of welfae economics [82]. (The second fundamental theoem states that unde sufficient assumptions, any Paeto efficient al-
SECTION 1.4. PRICE ANTICIPATING BEHAVIOR AND NASH EQUILIBRIUM 23 location can be achieved as a competitive equilibium.) These fundamental theoems ae the conestone esults of geneal equilibium theoy [5, 31], fist developed by Walas [142]; fo this eason competitive equilibia ae also efeed to as Walasian equilibia. Mashall discussed Paeto efficiency of competitive equilibia in the special case whee utilities ae sepaable [81, 82, 130, 137]; this line of development is often efeed to as patial equilibium analysis, because the assumption that utility is sepaable may be intepeted as a eflection of the fact that consumes spend a small faction of thei income on the goods unde consideation, and that the pices of all othe goods ae held constant. In this case it is easonable to assume that all membes of society will value cuency identically elative to the goods unde consideation (see the discussion in Section 1.1, as well as Chapte 10 of [82]). The fist fundamental theoem povides a fist justification fo the attactiveness of maket mechanisms. Assuming that no maket paticipants anticipate the effects of the thei actions, maket mechanisms povide a simple and decentalized method to ensue efficient allocation of esouces. But the assumption of limited pice anticipation is quite a stong one, paticulaly if only a few maket paticipants compete at any given time. In communication netwoks, one agument in favo of competitive equilibia is that the numbe of end uses is enomous, and each use competes fo only a small faction of oveall netwok esouces. Howeve, if we expect that picing of netwok esouces (such as tansmission capacity) occus only at high levels of aggegation, then only a few sevice povides may be competing with each othe to acquie netwok esouces and in this case the execise of maket powe becomes possible. Similaly, in electicity makets only a few fims typically compete at any given node of a egional electicity gid, calling into question the assumption that pice taking behavio and competitive equilibia will esult. 1.4 Pice Anticipating Behavio and Nash Equilibium Because we cannot guaantee that maket paticipants will be pice takes, we tun ou attention to the possibility that they may anticipate the effect of thei actions on maket-cleaing pices; in economic teminology, the maket paticipants ae said to have maket powe. 1 In this case, each paticipant views the maket-cleaing pice as a function of the composite stategy vecto of all maket paticipants. Thus the competition between maket paticipants who ae pice anticipating is a game: the payoff of a given playe is diectly expessed as a function of his own stategy, as well as the stategies of all othe playes. 1 The tem maket powe can be somewhat misleading, because some maket paticipants may actually achieve a lowe payoff when they ae pice anticipating instead of pice taking. Fo this eason, we typically use the moe pecise phase pice anticipating in most of the fomal development of the thesis.
24 CHAPTER 1. INTRODUCTION We will study these games though thei Nash equilibia. A Nash equilibium [90] is a stategy vecto fom which no playe has a unilateal incentive to deviate; that is, keeping the stategies of othe playes othe than i fixed, the stategy chosen by playe i maximizes his payoff. We make two obsevations egading the choice of Nash equilibium as ou solution concept. Fist, of couse, it is a static concept: that is, it gives no diection as to the dynamic pocess by which equilibium is eached. Second, the Nash equilibium solution concept assumes a geat deal of knowledge on the pat of the playes of the game, and a key point of debate involves whethe o not playes possess sufficient infomation to implement the Nash equilibium. In egads to the second point, a limited ebuttal may be offeed by noting that fo the maket mechanisms consideed in this thesis, a maket paticipant geneally needs infomation only on his own stategy and the pice of the esouce to detemine whethe his chosen stategy is payoff maximizing. Howeve, both the objections aised hee ae of citical impotance in maket modeling and game theoy in geneal, and we will take up futhe discussion on both topics in the conclusion to the thesis, Chapte 6. While competitive equilibia ae guaanteed to achieve full efficiency, Nash equilibia when maket paticipants ae pice anticipating do not geneally ensue full efficiency. Howeve, the aggegate suplus at a Nash equilibium still captues the net monetay benefit to the economy at the allocation that the equilibium achieves. In this thesis, we will focus on the following basic question: how inefficient ae the allocations achieved when maket paticipants ae pice anticipating, elative to Paeto efficient allocations? Fomally, this question is answeed by consideing the atio of aggegate suplus achieved at a Nash equilibium to the maximum possible aggegate suplus. Thus we investigate the pecentage of monetay benefit lost to the economy because maket paticipants ae able to anticipate the effects of thei actions. The fact that Nash equilibia of a game may not achieve full efficiency has been well known in the economics liteatue [33]. The economist Pigou obseved ealy on that thee may be a gap between the optimal pefomance of a system and that achieved by selfish optimization [101]. Pehaps the simplest such example is the well known game of the Pisones Dilemma, whee the Nash equilibium chooses payoffs to the two playes which ae stictly lowe than payoffs each playe would obtain at anothe stategy vecto [96]. This insight has been a cental theme of the theoy of industial oganization as well, paticulaly fo oligopoly models; see Tiole [134] fo a suvey of these issues. In ecognition of the effects of maket powe, heuistic measues ae often used to detemine the efficiency loss in an envionment whee some paticipants may be pice anticipating. Fo example, the U.S. Depatment of Justice uses the Hefindahl index, the sum of the squaes of pecentage maket shaes of all fims in a maket, as a measue of maket concentation [135]. Infomally, a Hefindahl index in excess of 1800 is intepeted as a sign that significant maket powe may be pesent in an industy, and by implication this situation is consideed to yield high efficiency loss. (A
SECTION 1.5. AN EXAMPLE 25 fomal investigation of the elationship between the Hefindahl index and efficiency is attempted by [25], and suveyed by Shapio [119].) A ecent suge of eseach, pimaily diven by the compute science community, has focused on quantifying efficiency loss fo specific game envionments. Most of the esults have focused on netwok outing, stating with the initial wok of Koutsoupias and Papadimitiou [70]. In that pape the authos intoduce the notion of a coodination atio, which Papadimitiou late efes to as the pice of anachy [99]; this is pecisely the atio of a given pefomance metic at the Nash equilibium of a game elative to the optimal value of that pefomance metic. Subsequent woks on outing models include the papes by Mavonicolas and Spiakis [84]; Czumaj and Voecking [24]; Roughgaden and Tados [106, 107, 108, 109]; Coea, Schulz, and Stie Moses [21, 113, 114]; and Peakis [100]. In addition to these woks, othe papes exploe efficiency loss in netwok design poblems [2, 32, 37], as well as a special class of submodula games including facility location games [138]. The key advance made by this eseach is the quantification of the loss of efficiency at Nash equilibia in specific game envionments, and the goal of this thesis is to establish a quantitative undestanding of efficiency loss in maket mechanisms. 1.5 An Example In this section we will wok though an example in detail to illustate the concepts peviously discussed in this chapte. Ou pupose is to elucidate the meaning of the tems and assumptions used though a simple model, as pepaation fo the mathematically igoous teatment offeed in the emainde of the thesis. In woking though this example, we will follow the same ode of pesentation of concepts as Sections 1.1 to 1.4. We conside a model consisting of two consumes competing fo a esouce in inelastic supply, as discussed in Section 1.1; this is a special case of the envionment consideed in much geate detail in Chapte 2. We assume the inelastic supply is fixed at S = 1 unit. Futhemoe, we assume that each consume has a linea utility function: U (d ) = α d, = 1, 2, whee α 1 > α 2 > 0. We ecall that as shown in Section 1.1, a Paeto efficient allocation must solve the following optimization poblem: maximize α 1 d 1 + α 2 d 2 (1.6) subject to d 1 + d 2 1; (1.7) d 1, d 2 0. (1.8) This poblem is identical to (1.3)-(1.5), but whee we have substituted fo the utility functions of the two consumes in the objective function (1.3), and whee we have set the inelastic supply S equal to one unit in (1.4).
26 CHAPTER 1. INTRODUCTION Since we have assumed α 1 > α 2, the unique optimal solution d to (1.6)-(1.8) allocates the entie supply to consume 1, so that d 1 = 1, d 2 = 0; this is theefoe the unique Paeto efficient allocation. This yields aggegate utility α 1 d 1 + α 2d 2 = α 1. Thus, the maximum possible aggegate monetay benefit to the system consisting of two consumes and a single esouce of unit supply is exactly equal to α 1. In Section 1.5.1, we develop a maket-cleaing mechanism fo allocation of this esouce. In Section 1.5.2, we analyze the pefomance of the mechanism when the consumes ae pice takes; we will see that thee exists a unique competitive equilibium, and that the esulting allocation is Paeto efficient. In Section 1.5.3, we conside the possibility that consumes ae pice anticipating; we show thee exists a unique Nash equilibium, and note that it is not Paeto efficient. Finally, in Section 1.5.4, we quantify the efficiency loss when consumes ae pice anticipating by compaing the aggegate utility at the Nash equilibium to the maximal aggegate utility (i.e., α 1 ). 1.5.1 A Maket-Cleaing Mechanism In this section we develop a maket-cleaing mechanism fo allocation of the scace esouce. Suppose that the supplie of the esouce, o esouce manage, wishes to efficiently allocate the unit supply among the two consumes. We will analyze the following simple scheme, an analogue of the mechanism poposed by Kelly [62]: 1. Each consume = 1, 2 submits a total payment, o bid, w that the consume is willing to make. The intepetation of w is that egadless of the maket-cleaing pice, consume will always consume an amount of the esouce which makes his payment exactly equal to w. Fomally, if we denote the demand of consume at a pice p > 0 by D(p, w ), then the equality that must be satisfied is: pd(p, w ) = w, fo all p > 0. In othe wods, we can intepet the bid w as submission of a demand function D(p, w ) = w /p. 2. The esouce manage chooses a pice to clea the maket, i.e., so that the entie unit supply is allocated. Fomally, the manage chooses a maket-cleaing pice p so that: D(p, w 1 ) + D(p, w 2 ) = 1. If we substitute D(p, w) = w/p, we find that the maket-cleaing pice p = p (w) is given by: p (w) = w 1 + w 2. (1.9) (Fo technical simplicity, we ignoe the bounday case whee w 1 + w 2 = 0; such details ae addessed in geate mathematical depth in Chapte 2.)
SECTION 1.5. AN EXAMPLE 27 3. The allocation made to consume is now: D(p (w), w ) = w w 1 + w 2, while the payment made by consume is exactly p (w)d(p (w), w ) = w. In examining the opeation of this mechanism, we have defined it in tems of demand functions. Howeve, because of the special stuctue of the demand functions, the eventual opeation of the mechanism is actually quite simple: each consume pays an amount w, and eceives a faction of the esouce in popotion to his payment. Note that the total evenue to the esouce manage is equal to w 1 + w 2, the sum of the payments fom the two consumes. 1.5.2 Pice Taking Consumes In this section we will assume the consumes do not anticipate the effects of thei actions on the maket-cleaing pice. To undestand this point concetely, conside the mechanism fom the point of view of consume 1. Consume 1 wishes to choose his bid w 1 to maximize his payoff, defined in (1.1). Thee ae two possibilities: eithe consume 1 ealizes that changing w 1 will change the maket-cleaing pice; o consume 1 does not anticipate this effect, and takes the maket-cleaing pice as fixed when choosing an optimal bid w 1. In the latte case we say consume 1 is a pice take. We analyze the pice taking model in this section; we conside pice anticipating consumes in the next section. If consume 1 assumes the maket-cleaing pice stays fixed at p as w 1 vaies, then he solves: [ ] max α 1 w1 w 1 0 p w 1. (1.10) To pase this expession, obseve that when consume 1 submits a bid w 1, if the maketcleaing pice is p he eceives an allocation D(p, w 1 ) = w 1 /p; this yields in tun the utility to consume 1, α 1 w 1 /p. This is the fist tem in the expession (1.10). The second tem is the payment w 1 made by consume 1. Thus, (1.10) expesses the fact that consume 1 chooses w 1 to maximize his payoff (cf. (1.1)) while taking the maket-cleaing pice p as given and invaiant. A symmetic expession holds fo consume 2: [ ] max α 2 w2 w 2 0 p w 2. (1.11) We ae seaching fo a competitive equilibium, as descibed in Section 1.3. In ou setting, a competitive equilibium is a vecto w = (w 1, w 2 ) whee each consume has optimally chosen his bid, while taking the maket-cleaing pice p = p (w) as fixed. Fomally, we want a pai (w 1, w 2 ) such that: (1) the maket-cleaing pice is
28 CHAPTER 1. INTRODUCTION p = p (w) = w 1 + w 2 ; (2) the bid w 1 is an optimal solution to (1.10), given the pice p; and (3) the bid w 2 is an optimal solution to (1.11), given the pice p. Since α 1 > α 2 > 0, it is staightfowad to establish that thee exists a vecto (w 1, w 2 ) satisfying these conditions, given by w 1 = α 1, w 2 = 0. To see this, note that when (w 1, w 2 ) = (α 1, 0), the maket-cleaing pice is p = p (w) = w 1 + w 2 = α 1. Thus the payoff to consume 1 at a bid w 1, given by αw 1 /p w 1, is identically zeo; in paticula, w 1 = α 1 is an optimal choice fo consume 1 given the pice p. On the othe hand, the payoff to consume 2 at a bid w 2 is α 2 w 2 /p w 2. Since α 2 < α 1 = p, we have α 2 /p 1 < 0, so the unique optimal choice fo use 2 is w 2 = 0. Thus the pai w 1 = α 1, w 2 = 0 is a competitive equilibium, with maket-cleaing pice p (w) = α 1. (In fact, it is possible to show that this is the unique competitive equilibium.) Futhemoe, obseve that at this equilibium, consume 1 eceives the entie esouce: D(p (w), w 1 ) = 1, while D(p (w), w 2 ) = 0. In paticula, in light of ou pevious discussion, the allocation at the competitive equilibium is an optimal solution to (1.6)-(1.8). We conclude that the competitive equilibium allocation is Paeto efficient a special case of the fist fundamental theoem of welfae economics (see Section 1.3). 1.5.3 Pice Anticipating Consumes Now suppose that each consume anticipates the effect of a change in his bid on the maket-cleaing pice. Again, fo simplicity, let us conside the mechanism fom the point of view of consume 1. Suppose that consume 2 submits a bid of w 2. When consume 1 submits a bid of w 1, the maket-cleaing pice is p (w) = w 1 + w 2, and the esulting allocation to consume 1 is D(p (w), w 1 ) = w 1 /(w 1 + w 2 ). Now suppose consume 1 anticipates that the maket-cleaing pice will change when w 1 changes; that is, athe than fixing the maket-cleaing pice p and then optimally choosing w 1, as in (1.10), consume 1 now takes into account the functional dependence of p (w) on w 1. Thus, given w 2, consume 1 chooses w 1 to solve: [ ] w 1 max α 1 w 1. (1.12) w 1 0 w 1 + w 2 Note that this payoff is identical to (1.10), except that we have eplaced the allocation w 1 /p with the tem w 1 /(w 1 + w 2 ), eflecting the dependence of the maket-cleaing pice on w 1. Symmetically, given w 1, consume 2 chooses w 2 to solve: [ ] w 2 max α 2 w 2. (1.13) w 2 0 w 1 + w 2 (In the discussion to follow, we ignoe bounday effects whee w 1 = 0 o w 2 = 0; again, such details ae addessed in Chapte 2.)
SECTION 1.5. AN EXAMPLE 29 Note that the payoff of each consume is dependent on the choice made by the othe consume; thus the equations (1.12)-(1.13) define a game. We will seach fo a Nash equilibium of this game, i.e., a vecto (w 1, w 2 ) such that: (1) w 1 is an optimal solution to (1.12) given w 2 ; and (2) symmetically, w 2 is an optimal solution to (1.13) given w 1. Fo technical simplicity, we seach only fo a Nash equilibium such that w 1 > 0, w 2 > 0. Notice that given w 2 > 0, the payoff (1.12) is concave in w 1 ; and given w 1 > 0, the payoff (1.13) is concave in w 2. Thus if we diffeentiate (1.12) with espect to w 1, and (1.13) with espect to w 2, a Nash equilibium is identified by the following two necessay and sufficient optimality conditions: ( ) 1 w 1 α 1 w 1 + w 2 (w 1 + w 2 ) 2 = 1; ( ) 1 w 2 α 2 w 1 + w 2 (w 1 + w 2 ) 2 = 1. It is staightfowad to check that these equations have a unique solution (w1 NE, w2 NE ), given by: w NE 1 = α1 2α 2 wne (α 1 + α 2 ) 2; 2 = α 1 α 2 2 (α 1 + α 2 ) 2. Thus the vecto (w NE 1, w NE 2 ) is a Nash equilibium. We now chaacteize the maket-cleaing pice, allocation, payoffs, and evenue at this Nash equilibium. It is staightfowad to check that the maket-cleaing pice is: p (w NE ) = w NE 1 + w NE 2 = α 1α 2 α 1 + α 2. Note that this is also the evenue to the esouce manage at the Nash equilibium. Recall that the evenue to the esouce manage at the competitive equilibium was w 1 + w 2 = α 1. Since α 1 > α 2, we conclude the evenue to the esouce manage is lowe at the Nash equilibium than at the competitive equilibium. This esult can be shown to hold moe geneally; see Coollay 2.3. The allocation to consume 1 at the Nash equilibium, denoted d NE 1, is: d NE 1 = D(p (w NE ), w1 NE ) = wne 1 p (w NE ) = α 1. (1.14) α 1 + α 2 Symmetically, the allocation d NE 2 to consume 2 is: d NE 2 = D(p (w NE ), w1 NE ) = wne 2 p (w NE ) = α 2. (1.15) α 1 + α 2