Efficiency of a Two-Stage Market for a Fixed-Capacity Divisible Resource

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1 Effcency of a Two-Stage Market for a Fxed-Capacty Dvsble Resource Amar Prakash Azad, and John Musaccho Abstract Two stage markets allow partcpants to trade resources lke power both n a forward market (so consumpton or producton can be planned n advance), and n a spot market (allowng adjustments to be made contemporaneously wth consumpton and producton.) However, two-stage markets ntroduce the possblty for a player to manpulate the market by creatng an arbtrage between the two stages. We nvestgate the effcency of two stage markets compared to that of a sngle stage market wth strategc users. We show that the subgame perfect equlbrum effcency of a two-stage market for a fxed, dvsble resource and buyers wth lnear utlty functons can be no worse than 82.8% compared to 75% for sngle stage markets. We also study the performance of two-stage market n the presence of uncertanty. I. INTRODUCTION Forward markets allow buyers and sellers of a resource delvered over tme to learn the tradng outcome n advance so that they can plan ther consumpton or producton of the resource respectvely. However, random shocks may happen between the tme the forward market s run and when servce s delvered that can affect both the cost of supplyng and utlty of consumng a resource. Therefore t s useful to have a spot market, so that traders can act after the realzaton of these random shocks are known. However, a spot market alone may lead to nadequate tme for plannng resource producton and consumpton. To get the benefts of both forward and spot markets, two stage markets are used n domans such as electrcty and also have potental for allocatng other constraned resources such as network capacty. However, two-stage markets possess a potental weakness the possblty that players can manpulate the market by creatng an arbtrage opportunty. If a buyer can manpulate the prces so that the spot market prce s hgher than the forward market prce, that buyer can buy n the forward market and sell n the spot market and earn a tradng proft. In contrast, f all users were prce takers and bought up to the pont that margnal beneft equals the market prce, the market would acheve an effcent allocaton. Thus the ncentve buyers have to manpulate prces whch can potentally lead to neffcency. In ths work, we focus our nvestgaton around the fundamental queston of how neffcent a two-stage market can be, and whether the neffcency s better or worse than a sngle stage market for a fxed capacty resource. Capacty allocaton usng sngle stage markets n communcaton networks has been wdely studed and there s a very large lterature. Two partcularly related works to ours are [] and [2]. In [], t s shown that allocaton can be neffcent Amar Prakash Azad (amarazad@gmal.com) s wth Supélec, France, and John Musaccho (johnm@soe.ucsc.edu) s wth the UC Santa Cruz, USA. wth prce antcpatng users: users that can antcpate the effects of ther actons (then the model becomes a game). Later on, Johar et. al. [2] showed that the neffcency can be no worse than 3/4, whch s a tght and achevable bound when utlty functons are lnear. Two-stage markets are routnely used n electrcty markets and have been wdely studed. In the most common settng, energy retalers use forward markets to acqure part of ther energy requrements. The spot market allows retalers and generators to adjust to fluctuatons n consumpton and producton. A large number of theoretcal studes are avalable n the lterature, and we refer to some of the work that s most related. Allaz et. al. [3] nvestgate the forward-spot market as a blevel game wth a Nash-Cournot model and show that the exstence of the contract market ncreases the effcency of spot markets. Moreover, Yao and Oren [4] ncorporate these blevel games nto some spatal electrcty markets wth transmsson constrants. More recently, Zhang et. al. [5] studed two-stage markets n a stochastc equlbrum settng. In the context of cogntve rados, some recent studes [6], [7] explore the use of prmary and secondary contract prcng to acheve the optmal allocaton n spectrum markets. However, most of the studes consder supply fluctuatons whle varatons n the demand are just n response to the changng market prce. To the best of our knowledge, we are the frst to derve an effcency bound on a two-stage sum-bd style market for a dvsble, fxed capacty resource. In our model, we suppose that users can resell capacty acqured n the forward market n the spot market. Our analyss approach s to dentfy condtons for sub-game perfect Nash equlbra. For smplcty of analyss we restrct buyers to have lnear utlty functons. Our man result shows that the worst case effcency s bounded by as compared to 75% for a sngle stage market [2]. In secton II, we descrbe the two-stage market model and equlbrum concept. We llustrate some key behavour of a two stage market wth an example n secton refs:exa. In secton IV, we derve the bound on effcency. We then descrbe the uncertanty model and nvestgate the arbtrage related aspects n secton VI va example. Fnally, we summarze and conclude n secton VII. II. SYSTEM MODEL We consder a two-stage market n whch n users trade for the capacty c of a dvsble resource such as the capacty of communcaton lnk. Wthout loss of generalty, we assume c. Users are assumed to be strategc. As n [2], [], each player selects a bd and then each player pays ther bd

2 2 and s gven an allocaton proportonal to ther bd. Unlke [2], there are two rounds of bddng. In the forward round, whch we also call the frst stage market, each player of n players chooses bd w and receves an ntal allocaton of x w. Snce each player pays the full amount of ther wj n j frst stage bd w to the network manager, ths s equvalent to chargng each user a per-unt prce of µ c n j w j n j w j. In the second round, the entre lnk capacty s agan on the market and each user selects a bd v. The second stage prce s then set to be ρ c n j v j n j v j, and each user gets a fnal allocaton y v /ρ. The net payment each user makes n the second round s v ρx ρ(y x ). The reasonng s that s ntal allocaton x s treated as an asset n the second stage, wth a value determned by the second stage prce. Thus each user s net payment s the second stage prce ρ tmes the net change n ther allocaton. Snce the capacty remans fxed between rounds, the sum of all these net payments s 0. Consequently, the money n the second round s exchanged between the players, wth no net money receved by the network manager. We assume players have perfect nformaton,.e., each player knows hs own utlty functon and that of hs opponents, and each player can see all the actons that were taken n the frst stage before he selects a second stage acton. Each user s utlty depends on hs fnal allocaton y, and thus hs utlty s U (y ) where U ( ) :R + R + s a strctly concave ncreasng twce dfferentable functon. Hs net payoff for the game, ncludng frst stage payment, and second stage net payment s A. Equlbrum concept J (w, v) U (y ) v + ρx w. Let w and v be the acton profle n the frst and second stages respectvely. A pure strategy for player s a par (w,v ( ) : w v ) that specfes a frst stage bd and a functon mappng the frst stage acton profle to a second stage bd. Later, we wll suppose that observable random events or shocks can happen between stages. When we add that feature, a player wll be able to make hs second stage bd a functon of these observatons as well. ) Sub-game perfect equlbrum: A sub-game perfect equlbrum (SPE) strategy profle specfes a Nash equlbrum strategy for each substage of the orgnal multstage game. The standard way to construct an SPE s backwards nducton [8, chap 3]. Defnton : (w, v ( )) s a subgame perfect equlbrum for the two stage market, f for each v (w ) arg max J (w, {v,v }), and v R + w arg max J ({w,w }, v ({w,w })). w R + B. Allocaton effcency The socal optmal allocaton s defned by: SO: Maxmze U (y ), subject to y ; y 0,. Because the objectve functon s contnuous and the feasble reason s compact, an optmal soluton exsts. Snce the feasble reason s convex and functons U (.) are strctly concave, the optmal soluton s unque. We defne the effcency rato of an SPE of the game to socal optmum as E Non-cooperatve welfare Socal optmal welfare U (y ) U (y SO ). () where {y } s assumed to be an SPE allocaton vector and {y SO } the socally optmal soluton. Later n the paper, we wll compare ths rato of the two stage game to the same rato for a sngle stage market. We denote the sngle stage Nash equlbrum allocaton as y SS. The worst case effcency rato s wdely known as Prce of Anarchy (PoA). III. MOTIVATING EXAMPLE: ONE USER VS. MANY SYMMETRIC USERS Before we begn the analyss, we attempt to unvel some mportant behavour of a two stage market wth the help of a smple example. We consder two types of strategc users wth lnear utlty: a sngle user havng utlty U (x) a x, and N symmetrc users wth utlty U (x) a x. (Notaton N s chosen to be dstnct from n, the total number of players.) We choose ths example because t s known to have the worst-case effcency for a sngle stage market [2]. For comparson, we numercally analyze the two-stage market and the analogous sngle stage market sde by sde. Numercal analyss suggests that there exsts a unque SPE for ths example. In Fg., we depct the equlbrum behavour for the a and lettng a n vary n the range [0, ]. When a n 0, user has a much larger utlty functon and the other users can be thought of as small. As a n approaches t becomes more and more lke havng (N + ) symmetrc users. We pck N to be 00. We make the followng mportant observatons from Fg. :. Effcency: The two-stage market appears more effcent compared to the sngle-stage market n Fg. (a). Further, observe that the user wth larger slope (user ) acqures more resources (n (c)) n the 2 stage market than n the one-stage market and hence the system s more effcent. (Note the most effcent allocaton would be to gve the player wth the hghest slope the entre resource.). Non-equal stage prces: The spot market prce s just slghtly hgher than the forward prce n Fg. (b). Therefore, snce users are strategc, users mght want to buy at a lower prce n forward market and sell n spot market to make proft. However, users know that f they try to explot ths arbtrage opportunty by buyng more n the forward market, the forward market prce would grow and the arbtrage opportunty would lessen. Ths tradeoff keeps the two prces from becomng exactly equal.. Prce manpulaton by the bg player: In a sngle stage market, the player wth the largest slope ( bg player) bds so that the prce s less than the slope of ther utlty functon so that they get a postve payoff. Ths behavor keeps the prce low enough so that users

3 3 Effcency Allocaton Two Stage Sngle Stage a n (a) Effcency Market Prce 0.5 TS 2nd Stage TS st Stage SS a n (b) Market prce y TS y n TS x TS x n TS y SS y n SS a n (c) Rate allocaton Fg.. Comparson: Sngle stage market vs Two stage market; SO: Socal optmal, SS: Sngle stage market, TS: Two stage market. wth lesser slopes buy as well, even though t s not socally optmal for them to have any allocaton. A smlar effect appears to happen n the two-stage market, wth a notable dfference. In the two-stage market the small users buy some of the resource n the frst stage, but n the second stage they sell some of t back to the bg player. Ths trade amongst the players results n the bg player gettng a larger allocaton than he would n sngle stage market. v. Arbtrage: Clearly, the possblty of arbtrage n such a smple example, opens up an avenue of arbtrage related nvestgaton. Can one construct an example for whch the two-stage market performs worse than the one-stage market? We nvestgate ths key queston n ths work. In the followng sectons, we dscuss the exstence of equlbra and ther effcency. For the smplcty of analyss, we restrct our analyss to the lnear utlty functons of the form U (y ) a y wth the assumpton a > 0 for all. The treatment of nonlnear, concave utlty functons s a topc of future work. IV. EQUILIBRIUM ANALYSIS A. Exstence of equlbrum n the spot market game We frst show the spot market game has a unque equlbrum, whch enables us to explctly express the user s payoff n terms of user s forward stage actons. Defnton 2 (Spot equlbrum): Let G s (x) denote the subgame that follows a frst stage that results n an ntal allocaton vector x. In ths sub-game each user chooses v to maxmze J (w, {v, v })U (v /ρ) v + ρx w where recall ρ N j v j. Note that the spot market game s strategcally equvalent to a game n whch each user chooses hs bd v to maxmze r (x, {v, v }) U (v /ρ) v + ρx over nonnegatve v. Thus a Nash equlbrum of the spot market sub-game s a vector v 0 such that for all r (x, {v, v }) r (x, {ṽ, v }) for all ṽ 0. Notce that the spot market game s smlar to the sngle stage allocaton game n [2]. The key dfference s that users have addtonal value because of carred over resource from the frst stage. The followng lemma notes that gven a vector x there exst a unque equlbrum n the spot market sub-game. Lemma 4.: Assume n >, and for each, the utlty functon U (.) s concave, ncreasng and contnuously dfferentable. Then there exst a unque Nash equlbrum v 0 of the spot sub-game G s (x) whch satsfes s v s > 0. The proof s just a mnor adaptaton of the result n [2], so we omt t to save space. Moreover, by evaluatng the frst order condtons of each player, the spot equlbrum can be shown to satsfy the followng: U (y ) y ρx f y > 0, (2) U (0) ρx f y 0, and y where the quantty U (y )a snce we are only consderng lnear utlty functons at ths pont. B. On SPE Exstence n the Forward-Spot Game Explotng the fact that a frst stage allocaton vector x leads to a unque sub-game equlbrum n the second stage, we may defne the equlbrum of the overall two stage game n the followng way. Defnton 3 (Forward-Spot SPE): Let G be the forwardspot game. A Forward-Spot SPE s a strategy profle (w, v ( )), v ( ) :x R + such that v (x) s a spot-market equlbrum for each x {χ R n + : n χ } and for each, w maxmzes J ({w,w }, v (w / w )) over nonnegatve real numbers. Snce the equlbrum n the spot subgame s unque, t s convenent to use the smplfed notaton J(w) to be the payoff vector when the acton profle w s played n the frst stage and the unque v (w / w ) s played n the second stage. Whle J(w) s contnuous (except at w 0) ts dervatves wth respect to w are not contnuous. Ths s because the dervatves depend on whch subset of players fnsh the fnal stage wth a postve allocaton. Ths makes t dffcult to prove that an SPE always exsts. To make progress, we look for a weaker result. We consder a modfed game n whch we essentally know ahead of tme whch players wll fnsh wth a postve allocaton and whch wll not. Ths s acheved by mposng a set of constrants on the players frst stage bds that cause a partcular set of players to fnsh the fnal stage wth a postve allocaton. From (2), we know that a y ρx for any player that fnshes wth postve allocaton. By addng these denttes, we get the relaton ρ n p H j P P n p (n p ) w (3) a j x j n p x p j P a j w j

4 4 where P be the set of players wth postve fnal allocaton, and n p P, H P s the weghted hyperbolc mean of the numbers {a j } j P wth weghts {x j } j P, H P s the unweghted hyperbolc mean of the numbers {a j },j P,j, and x p j P x j. (See Lemma A. n the appendx for detals.) Snce a ρx < for players that fnsh wth a postve allocaton, and a ρx for those that fnsh wth a zero allocaton, we can wrte a set of nequalty constrants on w that enforce whch players end up fnshng wth a postve allocaton. That s what we do n the followng defnton. Defnton 4: The coupled constrant forward-spot (CCFS) game G c (P) s defned for any set P 2 {...n} such that P 2. G c (P) s a coupled constrant game wth dentcal payoff functons and player-move structure as the orgnal forward-spot game G wth the followng added constrants on the frst stage bds: for P: [H P ] w (a j P,j for P c : 0 (a j P,j [H P j] )w j + a [H P j] )w j + a j P c w j, (4) j P c w j, (5) w w j for all, j {...n} such that a >a j, (6) [H j] P w j. (7) j P The set of vectors w satsfyng the above constrants we denote as S(P) R n. Note that (6) requres that players wth steeper utlty functons bd more n the frst stage and turns out to be needed to show our equlbrum exstence result for the CCFS game. Condton (7) can be shown to be equvalent to enforcng that ρ µ. One queston that mmedately arses s whether one can choose the set of postve players P so that S(P) s nonempty. It turns out one always can as stated n the lemma that follows. Lemma 4.2: There exsts a dscrete set P wth P 2 such that S(P)\0 s nonempty. The proof s by constructon and found n the appendx. The followng result establshes the exstence of an equlbrum n the CCFS game. Theorem 4. (Equlbrum exstence n the CCFS game): Consder the CCFS game G c (P). If feasble regon S(P)\0 s nonempty, then G c (P) has a pure subgame perfect (forward-spot) equlbrum. V. EFFICIENCY OF A TWO STAGE MARKET We are now ready to state our man result on equlbrum effcency. The result apples to any SPE of the orgnal unconstraned game (provded such an equlbrum exsts) as well as to SPE of the CCFS game, provded the equlbrum meets the condton n the theorem statement. Theorem 5.: [Prce of Anarchy] The prce of anarchy s (2 2 2) 82.8% n () any SPE of the game G or () any SPE of the the CCFS game G c for whch players n P choose frst stage bds satsfyng djs dw s (w) 0. The last condton n the theorem statement says roughly that players wth postve allocaton would not beneft by lowerng ther bds slghtly f they were allowed to by the constrants. The proof of the theorem follows. Proof: Wthout loss of generalty we assume player has the largest slope a. We analyze the effcency rato of an arbtrary Subgame Perfect Nash Equlbrum (SPE) wth equlbrum frst stage bd vector w. Let P be the set of players wth postve fnal allocaton n SPE, and n p P. From Lemma A.3 n the appendx, the rght dervatve of J wth respect to w exsts and must be nonpostve n any SPE of the unconstraned game. If we are dealng wth the CCFS game there s no upper bound n force for player n equlbrum, and hence agan the rght dervatve of J wth respect to w exsts and must be nonpostve. Moreover, there must be a well defned set P+ that s the set of players wth postve fnal allocaton when the frst stage bd vector s (w,w + ) for a suffcently small and postve. Let n p+ P+. Usng the FOCs from Lemma A.3 we have µ ρ 2ρ[HP ] (x y )+y 2 n p+ n p+ x p H P+ [H P+ ] (x y )+y (8) where H P+ s the weghted hyperbolc mean of the numbers {a j } j P+ wth weghts {x j } j P+, H P+ s the unweghted hyperbolc mean of the numbers {a j },j P+,j, and x p j P+ x j. Note that the weghted arthmetc mean of the numbers {H P+ } wth weghts { x x p } s equal to n p+ x p (n p+ )x p H P+, and H P+ s the largest of the numbers n {H P+ }. Thus H P+ H P+ n p+ x p x p(n +p ). For more detals on the algebra behnd these denttes see Lemma A. of the appendx. Ths leads to the relaton x µ 2 ρ + 2 y. (9) The sum of the players utlty functons satsfes ρx a y a ( y ) a P P P P a ρ(n p x p )a + (a ρ) ρx z. P P, where x z x p s the total frst stage allocaton to players that fnsh the second stage wth zero allocaton. The effcency rato satsfes E a y a a + P, (a ρ) ρx z a x y + y (y x )+ y (y x ) + (0 x z ) B S x y where B { : y >x, P}and S { : y x, P} Let y s be the largest of y s wth S. Replacng each y n the summaton over S wth y s makes the expresson smaller (larger magntude negatve) snce each term n the.

5 5 summaton s negatve and y s y for each S. Smlarly replacng each y n the summaton over B wth y s makes the expresson smaller snce each term n the summaton s postve and y s <y for each B. Ths gves E x y + y s(y x ) x y. Note we collected n the (0 x z ) term when wrtng the above by notng that y s. The above s decreasng n x. Now consder player s. Ths player has a postve allocaton and thus s n P. The dervatve djs dw s (w) 0 by the assumptons of the theorem for the CCFS case whle ths follows from the frst order optmalty condton for the unconstraned game snce x s > 0 by Lemma A.7 n the appendx. Rearrangng the expresson for djs dw s from Lemma A.3 yelds y s µ ρ 2ρ[ H P ] (y s x s ) µ ρ where the last nequalty follows snce y s x s. Usng ths result and (9) we have x 2 y s + 2 y. Substtutng ths upper bound we get E y s + y 2 y s y s + y. The dervatve of the above wth respect to y s can be shown to be negatve. Snce y s, we have E +y2 +y. Ths expresson s mnmzed when y 2 and at that pont t has a value of VI. DEMAND UNCERTAINTY A key feature of a two-stage market s ts ablty to respond to a demand shock. In ths secton, we compare a two-stage market to a forward only market n the presence of random events or shocks that effect demand. The most nterestng types of shocks are the shocks that effect dfferent users dfferently, snce these shocks change the optmal allocaton of the resource amongst the users. We consder a smple nose model: the so called dfferental mode nose the effect of nose s dfferent on each user s demand. In the followng we brefly defne the equlbrum model for each market type wth nose. We then consder a set of parameter to perform evaluaton. We suppose that not all the users are exposed to shock the so called dfferental shock. Let the random nose or shock be denoted by the random vector β. The utlty of user takes the form U β (.) β (ω)u (.). The random shock β :Ω Ξ s a contnuous random vector defned on the probablty space (Ω, F, P) wth known dstrbuton. To smplfy the notaton, we wrte β(ω) as β. We suppose that players know the dstrbuton of β when they choose ther forward market bd, and they know ts realzaton when choosng ther spot market bd. Hence ths nformaton structure leads to the second stage acton profle v(, ) beng a functon of both the frst stage acton profle w and the realzaton of β. Effcency Fg Two Stage Sngle Stage a 2 (a) No Shock E[ Effcency ] Two Stage Sngle Stage a 2 (b) Wth Shock Comparson: Impact of shock n a Sngle and Two-stage market. A natural equlbrum extenson for the model wth random shock s the subgame perfect Bayesan equlbrum (PBE) defned as follows. Defnton 5: (w, v (, )) s a PBE for the two stage market, f for each and each β n the support of the dstrbuton of β, v (w,β) arg max J β (w, {v,v }) v R + and also for each, w arg max E β [J β ({w,w }, v ({w,w }, β)) w R + where J β (w, v) β U (y ) v + ρx w. We llustrate the performance by consderng a smple example wth two users, competng for unt capacty. Let the shock β be unformly dstrbuted n the range [0, 2] and β 2, hence only user faces a random shock. For the sake of exposton, we fx a and compute the effcency at equlbrum for dfferent choces a 2 from 0 to. A complete performance evaluaton would also compare a two-stage market to a spot-only market. A spot-only market would be more responsve, but t would also have the dsadvantage of not allowng the buyers to know anythng n advance about ther fnal allocaton. Buldng a model to evaluate ths tradeoff s a topc of future work. VII. REMARKS AND CONCLUSION In ths work, we have studed the worst case effcency when two stage market s deployed for the allocaton of a dvsble, constraned resource. We showed that the effcency can be no worse than for players wth lnear utlty functon. Ths ndcates that the two-stage market has mproved effcency as compared to the analogous sngle stage market, whch s known to have a worst case effcency of 3/4 (whch s achevable wth lnear utlty functons). Wth the help of a smple example, we have seen the potental mproved responsveness of a two-stage market vs a forwardonly market. In contnung work, we ntend to develop a more general analyss of the two-stage market s responsveness to shocks, and a framework to compare two-stage markets to spot-only markets. VIII. ACKNOWLEDGMENTS Ths work was supported by NSF Grant CNS The authors acknowledge helpful dscussons wth Professors Jean Walrand and Patrck Loseau.

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Prelmnary Lemmas APPENDIX Lemma A.: In any equlbrum of the spot game wth players n P havng postve allocaton, equaton (3) and the followng holds: A {[H P ] } P, { x } P x p where A {[H P ] } P, { x x p } P n p x p x p (n p ) [HP ] (0) s the weghted arthmetc mean of the numbers {[H P ] } P wth weghts { x x p } P, and x p j P x j. Proof: The second stage FOCs requre that ρx j a j y j f y j > 0 and ρx j a j otherwse. The former relaton s equvalent to a j x j ρ y j. Summng the equalty condtons across j Pgves the relaton n the frst part of (3). The weghted hyperbolc average of a set of numbers s found by summng the recprocals tmes the weghts, and dvdng that total from the sum of the weghts. Thus H P j P x j n p x p j P a j x j P ρ n p x p j P a j x j n p. Now turnng to prove (0), we wrte the term on the rght sde of (0) as [H ] P x a j x x p (n p )x p (n p )x p P P j a x n p x p (n p )x p [H P ]. Lemma A.2: The functon J (w) s contnuous n w and almost everywhere dfferentable. Proof: We omt the full proof here due to space lmtatons. It s easy to verfy that x s contnuous n w > 0. After ths note that the spot market subgame has a unque equlbrum gven an ntal allocaton vector x. The soluton satsfes a system of FOCs, and n turn the prce ρ that satsfes ths can be shown to satsfy a fxed pont equaton n mn(, ρa x )n. The left sde s monotone ncreasng, Lpschtz contnuous n ρ, and ncreases from 0 to n as ρ ncreases. Therefore there s a unque soluton, and that soluton s Lpschtz contnuous n x, and thus almost everywhere dfferentable. From ths, t s also easy to verfy that y s contnuous n x and thus n w. From these facts the contnuty of the functon J can be establshed. Lemma A.3: If player has a postve fnal allocaton n an SPE of the forward-spot game, then the dervatve of hs payoff wth respect to w s dj ρ 2ρ[H P µ ] (x y )+y f the dervatve exsts. At ponts that the functon s not dfferentable, the rght dervatve and left dervatve exst and are found by takng the above expresson and substtutng H P+ or H P respectvely for H P where P+ lm 0{j : y j (w,w + )} and P lm 0 {j : y j (w,w + )}. Proof: Dfferentatng expresson (3) for ρ wth respect to w yelds dρ ρ µ [ ρ[hp ] ]. Note that ths expresson s vald only where the dervatve exsts, whch henceforth we refer to as a regular pont. The dervatve jumps at ponts the set P changes. From the above form, we see that n any nterval of w that results n the same set P, ρ monotoncally approaches [H P ]. Snce x w dx w, dfferentatng yelds P, y a ρx. Dfferentatng yelds ρ x µ. For dy µ [HP ] y on regular ponts and consequently y s monotone non-decreasng wth respect to y. Smlar analyss shows dy that j a ρ j µ [y ja j [H P ] ]. Consequently, f y j s decreasng and approachng 0 as w s ncreased, t must be that a j <H P. After y j becomes 0, the set P looses player j and H P ncreases snce a below average player has left the set of whch the hyperbolc average s taken. Smlarly f y j ncreases from zero as w s ncreased, t must be that a j >H P. When y j becomes postve, the set P gans player j and H P ncreases relatve to ts value before player j became postve snce an above average player has joned the set of whch the hyperbolc average s taken. Thus H P ncreases monotoncally n w. From ths and the above expresson for dy j, f y j decreases to zero as w ncreases, t cannot later ncrease from zero. Consequently, H P changes only a fnte number of tmes wth respect to w, and thus t s always possble to fnd a left and rght dervatve of J wth respect to w. The objectve functon s J a y + ρx ρy w a y + ρ(y x ) w. Dfferentatng gves dj ρ a µ [HP ] y + ρ µ [ ρ[hp ] ](y x )+ ρ( ρ µ [HP ] y + x µ ). at regular ponts, whch smplfes to the expresson n the lemma statement. Snce the left and rght dervatves exst at non-regular ponts, the above calculatons can be done wth

7 7 the left and rght dervatves respectvely to get the result n the lemma statement. B. Analyss of CCFS game Before startng the proof, we frst prove a needed lemmas. The followng s a proof of Lemma 4.2 stated n the man text. Proof of Lemma 4.2: Set w 0for any player not n P. Frst consder the case n whch the there are 2 players whose slopes are larger than those of any others and w.l.o.g. suppose these have ndces,2 and a a 2. Set P {, 2}. Now we show there exsts a feasble w. Set w 0for any, 2. Snce for, 2, a [H P ] a a 0, thus (4) holds for any nonnegatve w and w 2. For j > 2, [H P ] 0 so (5) holds. By constructon (6) holds and one can always choose w and w 2 small enough for (7) to hold. Next, suppose there are m>2players wth the hghest slope and w.l.o.g. suppose they are ndexed,...,m. Let P a j {,...,m}. For m, a [H P ] a a 0, thus (4) holds for any nonnegatve w. For j>2, a j [H P ] 0 so (5) holds. By constructon (6) holds and one can always choose w,...,w m small enough for (7) to hold. Fnally, suppose there s player (ndexed ) wth the largest slope and m > players wth the second hghest slope (ndexed 2,...,m+). Let P {,...,m+}. Note a [H j P ] 0, thus (4) holds for any nonnegatve w. Now suppose that w w for 2,...,m +. Note that H P s the same for all such players, so call that value H m/(a +(m )a 2 ). (4) requres H w (a 2 a 2 )w +(m )(a 2 H )w, whch just smplfes to w 0. By constructon (6) holds and one can always choose w,w small enough for (7) to hold. (5) can also be shown to hold. We need the followng to later prove quasconcavty. Lemma A.4: Suppose that frst stage bd vector w S(P) and ρ s determned by the unque subgame equlbrum v followng bds w n the frst stage. If Pand ρ>[h P ] then a >ρ. Proof: Summng the second stage FOCs yelds ρ (n )[ P a j x j + P c ã k x k] where ã k ρx k. Let H l be the hyperbolc mean of the set ({a j } j P {ã k } j P c)\a l (or... \ã l nstead f l P c ). By rearrangng sums we observe that ρ A({ H j }, {x j}) where the notaton on the rght sde of the equalty denotes the weghted arthmetc mean of the numbers H,..., H n wth respectve weghts x,...,x n. Then we see that ρ A({ H j }, {x j}) A n H a j n n n H n A({a j }, {x j }), {x j } where wth some abuse of notaton the set { } above has n elements and uses the ã k s for those ndces n P c. The notaton H s the hyperbolc mean of {a j } j P {ã k } j P c. Ths reduces to ρ H + a A({a j }, {x j }). n H <H P snce the former adds numbers to the average that are all not bgger than any number ncluded n the average H P. Thus ρ H >. From the above equaton, t s necessary that a <A({a j }, {x j }) A({a j }, {x j })ρ where the 2nd nequalty s because the arthmetc average on the rght weghts bgger a j values more than arthmetc average on the left as a consequence of (6). The followng result s needed n the proof of Theorem 4.. Lemma A.5: The functon J(w, w ) s quasconcave wth respect to w on S(P) {w : w j 2 max a } for any >0. Proof: Frst we consder P. By the lemma assumptons the dervatve dj (w ; w ) exsts and s equal to ρ 2ρ[H P µ ] (x y )+y () 2 ρ2 µ 2 [HP ] ( ρa )w + ρ2 µ 2 a w. Multplyng both sdes by µ2 ρ 2 µ 2 ρ 2 dj gves (w ; w ) 2[H dw ] P ( ρa )w + a w. Defnng ψ(w ; w ) µ2 dj ρ 2 (w ; w ) and dfferentatng wth respect to w gves d ψ(w ; w ) 2a w [H P dρ dw ] 2 µ ρ [HP ], 2[H ] P y ρ[h P ] + µ ρ The last expresson comes from substtutng an expresson for dρ and the 2nd stage FOCs. The quantty ψ(w ; w ) must always have the same sgn as dj (w ; w ). Snce ψ(w ; w ) s dfferentable and has contnuous dervatves on S(P) {w : w j 2 max a }, any zero crossngs must occur after: () ψ(w ; w ) s postve and decreasng or () ψ(w ; w ) s negatve and ncreasng. We now show that () can not happen. Suppose the opposte and d ψ(w ; w ) > 0 then from the equaton above y ρ[h P ] + µ ρ < 0 so. ρ[h ] P > +y µ ρ > 0 (2) leadng to [H P ] >. Thus by Lemma A.4, ρa < and hence y >x by the 2nd stage FOC, and hence (x y ) > 0. Thus ρ[h P ] has a postve coeffcent n equaton (), so substtutng a lower bound for ρ[h P ] gves a lower bound for dj (w ; w ). Usng bound (2) we have dj (w ; w ) > 2 ρ µ ( + y µ ρ )(x y )+ ρ µ y ( ρ µ ) + (y x ) + 2(y x ) The frst term of three n the above expresson s nonnegatve (as a consequence of (7) and the other are postve. Thus dj (w ; w ) s postve and hence ψ(w ; w ) s postve. Ths rules out possblty (). Thus all zero crossngs of

8 8 ψ(w ; w ) occur when t s postve and decreasng. Thus ψ may: () cross zero once at one pont, () ht zero, stay there, and then go below, () always be above zero, (v) always be below zero. Case () corresponds to a strctly quasconcave functon, Case () corresponds to a non-strct quas concave functon. Cases () and (v) mply that there s a unque local maxmum at an edge of the allowed regon at 2 max a or 0 respectvely. The proof of the P c case s smlar but less complex so t s omtted to save space. We are now ready to prove Theorem 4.. Proof of Theorem 4.: Consder the feasble acton set S(P). The set s convex, nonempty and contans more than just the vector 0. We augment the constrants to create the set D S(P) {w : w j 2 max a }. Set D s compact, convex, and nonempty for suffcently small. By Lemma A.5, J(w, w ) s quasconcave w.r.t w and contnuous w.r.t. w on D. By Berge s theorem, the best response set w BR (w ) arg max w D J(w, w ) s a nonempty, closed, convex-valued, upper-hemcontnuous correspondence [9]. By Kakutan s fxed pont theorem there s a fxed pont w eq, of the set-valued functon w BR (w) {z : z w BR (w )}. Ths fxed pont s a Nash equlbrum n the coupled constrant game wth acton space D. It remans to show that ths same pont s a Nash equlbrum of the game wth acton space S(P). We clam that for small enough, w eq, >by the argument that follows, whch was adapted from an argument n [0]. Suppose ths were not true. By (3), ρ 2 mn a. For players wth postve fnal allocaton, consder expresson () for dj. Take the weghted average of these expressons across P wth weghts H P. Ths procedure yelds ρ µ [2ρ( x p )+ H y P ] P HP P a ay a2 (n p ) 2n p ā 2ān p P where a and ā are the mnmum and maxmum of the a values respectvely. Also note that t can be shown that n p 2 because 2 players must always bd postvely n an equlbrum of the spot game. Thus, at least one player must have a value of dj as large as the last expresson. Snce for small enough, the expresson s postve, we conclude that when all the players frst stage bds add to, there wll always be one player who could mprove ther payoff by ncreasng ther bd. If such a player can do so wthout volatng the feasble regon S(P), then the pont s not an equlbrum. If he s unable to rase hs bd because a larger player s bddng the same amount (constrant (6)), one can show that ths larger player would also want to ncrease hs bd. We now argue that w eq, s an equlbrum of the game wth constrants S(P) {w : w j 2 max a }. Suppose ths were not true. Then at least one player could mprove ther payoff by reducng ther bd to w so that j weq, j + w < and ( w, w eq, ) S(P). Constrants (5) and (6) could not be tght for player n w eq, or otherwse the player would not be able to lower hs bd. Also the constrant w s not tght by the argument n the precedng paragraph. dj Thus, for ths player t must be that (w eq, ) 0. Havng J ( w, w eq, ) >J (w eq,, w eq, ) would thus volate the quasconcavty property that we have already establshed. Fnally, we argue that w eq, s an equlbrum of the game wth constrants S(P). Suppose ths were not true. Then at least one player could mprove ther payoff so that the new frst stage prce µ 2 max a. Snce ρ µ ths player pays an amount per unt capacty larger than the steepest utlty, so he must fnsh wth a nonpostve payoff, whch cannot be an mprovement over hs payoff when the bd vector s w eq, snce one can show that each player gets a payoff of at least 0 n that stuaton. C. Lemmas Used n the Proof of Theorem 5. Lemma A.6: In any SPE of the forward-spot game (unconstraned verson), µ ρ. Proof: Note that there exst at least one user j wth y j x j, because f not the the resource constrant x y s volated. If ths user has a postve fnal allocaton, the FOC requres that the left dervatve of the payoff wth respect w be greater than or equal to zero. From Lemma A.3, we have 0 d J ρ 2ρ[H P µ ] (x y )+y, µ ρ 2ρ[HP ] (x y )+y. The last nequalty follows because y and the other term n the summaton s non-postve. The proof n the case that y 0s smlar and omtted for space. Lemma A.7: In the unconstraned forward-spot game, f user fnshes wth postve spot allocaton y > 0, then he wll have postve forward allocaton x > 0 at forward-spot equlbrum. Proof: Suppose user fnshes wth y > 0, and hs forward poston s x 0, whch mples w 0. Then, after substtutng ρ a x y from the 2nd stage FOC and wth some rearrangements we have d + J (w, w ) 2[H P+ ] ρ2 µ [ y ρ ]+y. (3) µ From the 2nd stage FOCs for player, a ρy >ρ. Note, ρ[h P+ ] k a k k a k x k k P+ a k a (n p )ρ k a k (n p )ρ ρ a (n p ) ρ a (n p ) > n p n p 2 n p where n the above n p P+. From the last nequalty we have ρ[h P ] 2 for n p 3. Substtutng ths nto (3) yelds us d+j 0 at w 0. Hence, w 0 s not an equlbrum for player. Now consder the case of n p 2. Agan a ρy >ρ. Snce ρ s equal to a weghted hyperbolc average of the two a s tmes a constant less than, t must be that a j a for j. Also note that [H P+ ] a j. Hence [H P+ ] >ρ. Substtute ths wth the fact that ρ/µ from Lemma A.6 nto (3), we agan note d+j 0. Hence, w 0s not an equlbrum for n p 2also. Hence proved.