Jesse A. Schwartz and Quan Wen

Transcription

1 A SUBSIDIZED VICKREY AUCTION FOR COST SHARING by Jesse A. Schwartz and Quan Wen Workng Paper No. 7-W5 Aprl 27 DEPARTMENT OF ECONOMICS VANDERBILT UNIVERSITY NASHVILLE, TN

2 A Subsdzed Vckrey Aucton for Cost Sharng Jesse A. Schwartz Kennesaw State Unversty Quan Wen Vanderblt Unversty Aprl 27 Abstract We ntroduce a subsdzed Vckrey aucton for cost sharng problems. Although the average, margnal, and seral cost sharng mechansms are budget-balanced, they are not allocatvely effcent and they do not nduce players to truthfully reveal ther values as a domnant strategy. The conventonal Vckrey aucton, on the other hand, s allocatvely effcent and does nduce truthful bddng as a domnant strategy, but also generates an overpayment. Ths paper modfes the conventonal Vckrey aucton so that some of the overpayment s used to subsdze addtonal producton wthout upsettng the players ncentves to bd truthfully. Although ths subsdzed Vckrey aucton s not allocatvely effcent, t always Pareto domnates the conventonal Vckrey aucton and sometmes domnates other exstng cost sharng mechansms. JEL Classfcaton Numbers: C72 (Noncooperatve Games), D44 (Auctons), H42 (Publcly Provded Prvate Goods) Keywords: Cost sharng, domnant strategy mplementaton, Vckrey aucton, subsdzed Vckrey aucton We would lke to thank Laura Razzoln and semnar partcpants at the Tnbergen Insttute and the 26 Internatonal Conference on Game Theory at Stony Brook for ther comments and suggestons. Department of Economcs, Fnance, and Quanttatve Analyss, Kennesaw State Unversty, 1 Chastan Road, Box 43, Kennesaw, GA 3144, U.S.A. Emal: jschwar7@kennesaw.edu Department of Economcs, Vanderblt Unversty, VU Staton B #351819, 231 Vanderblt Place, Nashvlle, TN , U.S.A. Emal: uan.wen@vanderblt.edu

3 1 Introducton A group of players jontly produce a prvate good and decde how much everyone consumes and how much everyone pays. Examples nclude farmers sharng an rrgaton system, offce workers splttng a secretary s tme, and dvsons of a corporaton usng the same advertsng department or tranng faclty. In many respects, a cost-sharng problem s smlar to a typcal aucton stuaton, where uantty s allocated among players who have dfferent valuatons. Unlke an aucton, however, n a cost-sharng problem there s no opposng seller to allocate the goods and collect the payments: the players themselves desgn or choose a cost-sharng mechansm to accomplsh these tasks. In ths paper, we study cost-sharng problems wth ndependent prvate values. In ths settng, many exstng cost-sharng mechansms ncludng the average, margnal, and seral mechansms are not allocatvely effcent because the goods are not always allocated to the players who value them the most. Furthermore, many of these exstng cost-sharng mechansms do not nduce players to truthfully reveal ther values as domnant strateges. The ueston s then why not adopt the Vckrey (1961) aucton to deal wth cost-sharng problems snce t nduces truthful bddng as a domnant strategy and the resultng eulbrum s allocatvely effcent. In an aucton of goods by a seller, the Vckrey aucton generates a proft for the seller because the buyers pay more than the producton cost. In a cost-sharng problem, the Vckrey aucton generates the same proft. But wth no opposng seller to collectthsoverpayment,theplayershavethethornyssueofwhattodowththsoverpayment. Redstrbuton of the overpayment among the players could upset ther ncentves to bd truthfully and hence upset the allocatve effcency. And as we show wth an example, f ths overpayment s smply destroyed, then the Vckrey aucton may gve the players lower payoffs than they would get wth other exstng cost-sharng mechansms. There are mechansms such as those studed by Arrow (1979), d Aspremont and Gerard- Varet (1979a, 1979b) that forgo domnant strategesnordertoacheve allocatveeffcency 1

4 and budget-balancedness; see Krshna (22) for more on these mechansms. However, f the players deem these mechansms too dffcult to mplement, players may nstead pursue ways to reduce the overpayment generated by the Vckrey aucton whle mantanng truthful bddng as a domnant strategy. Referrng to modfcatons to hs aucton, Vckrey (1961, page 13) stated However, t seems that all modfcatons that do dmnsh the [mbalance] of the scheme... rentroduce a drect ncentve for msrepresentaton of the margnal value curves. One am of our paper s to solate those components of the Vckrey aucton that ensure players do have the ncentve to truthfully report ther margnal values as domnant strateges. We fnd that allocatve effcency though certanly a desrable property s not among these components. We ntroduce perfect prce dscrmnatng (PPD) mechansms that operate as follows. Each player reports hs margnal value curve or euvalently hs pece of nformaton whch pns down hs margnal value curve. The mechansm then generates a player specfc supply curve for each player. A player receves the uantty where hs reported margnal value curve ntersects the supply curve generated for hm, and he pays the area under the supply curve up to the uantty he wns. Ths payment rule s the analog of a perfectly prce dscrmnatng monopsonst buyng an nput. As we show n Proposton 1 f the supply curve generated for a player s ndependent of hs reported margnal value curve, then t wll beadomnantstrategyfortheplayertoreporthstruemargnalvaluecurve. Weshowthat the conventonal Vckrey (1961) aucton can be reformulated as a PPD mechansm where the supply curve a player faces s the resdual supply curve left over after the other players demands are satsfed. Snce the resdual supply curve faced by a player s ndependent of hs own reported margnal value curve, the player has a domnant strategy to report hs true margnal value curve, a well-known property of the Vckrey aucton. We then ntroduce a new PPD mechansm for cost-sharng, a mechansm we call the subsdzed Vckrey aucton. In the subsdzed Vckrey aucton, a player s supply curve s constructed ndependently of hs reported margnal value curve n such a way that some of 2

5 the overpayment he generates by dsplacng other players bds s used to subsdze addtonal producton. We show that when players have constant margnal values, the subsdzed Vckrey aucton always balances the budget and nduces truthful bddng as a domnant strategy. Unlke the conventonal Vckrey aucton, ths subsdzed Vckrey aucton s not allocatvely effcent snce the margnal cost wll exceed everyone s margnal value. Although each ndvdual player can be over-subsdzed n the sense that the true margnal cost could be hgher than hs margnal value, the mechansm tself s feasble, always collectng enough payments to cover producton costs. Even though the subsdzed Vckrey aucton s generally allocatvely neffcent, we show that t Pareto domnates the conventonal Vckrey aucton. In some stuatons, t also Pareto domnates the average, margnal, and seral cost-sharng mechansms. Mouln and Shenker (1999) and Fredman and Mouln (1999) characterze the average, margnal, Aumann-Shapley, seral, and other cost-sharng mechansms by some desrable propertes n a cooperatve framework. One such property, for example, s the demandmonotoncty property whch says that the more uantty a player consumes the more he must pay, holdng fxed the uantty consumed by the other players. Although many of the propertes they consder certanly have strategc mplcatons for the players, the focus of these papers s not on strategc behavor or eulbrum n noncooperatve games. For a more recent paper n ths cooperatve ven, see Koster (26) whch has a more complete set of references to ths lterature. Mouln and Shenker (1992) consder some of these cost sharng mechansms n a noncooperatve game, and show that the seral cost sharng mechansm s domnance solvable. Ther paper dffers from ours n that ther game has complete nformaton, and the strategy space lmts each player to choose a fxed uantty regardless of prce. Our paper, on the other hand, consders players wth prvate nformaton and also admts players a rcher strategy space, where players submt what amount to demand curves, beng able to condton the uantty they want on the prce. Mouln and Shenker (21) consder a Clarke-Groves mechansm, and lke what we do n ths paper, wrestle wth the 3

6 tenson between effcency, budget-balance, and domnant strateges. But n ther paper, consumpton s bnary: ether a player consumes or does not. Furthermore, the cost functon n Mouln and Shenker (21) dffers substantally from ours ther cost depends on the actual subset of players who consume so that a drect comparson of ther results to ours s not possble. Chen (23), Chen and Razzoln (25), and Razzoln, Reksulak, Dorsey (24) run experments on the average or seral cost sharng mechansms n envronments smlar to the example we use n our paper where players have constant margnal values and costs are uadratc. However, the games they consdered have ether complete nformaton or lmted nformaton n whch the lab subjects know ther own values, but the subjects are not told the dstrbutons from whch ther opponents draw ther values. We are not aware of research on cost sharng n whch players have prvate nformaton and play a noncooperatve game. Outsde of the lterature on cost sharng, McAfee (1992) has some smlar themes as our paper. Inhspaper,McAfeentroducesanauctonwhchmantansthedomnantstrateges ofthevckreyaucton,andlkeourpapertdealswththebudgetmbalanceproblemby sacrfcng some effcency. Nevertheless, the model and applcaton of hs aucton s much dfferent from ours: hs s a double aucton wth both strategc buyers and sellers, and each of hs players demand or supply only a sngle-unt. The next secton specfes the cost-sharng game. Secton 3 contans a smple example to llustrate the average, margnal, and seral cost mechansms and the conventonal Vckrey aucton. In Secton 4, we ntroduce PPD mechansms for cost-sharng problems and represent the conventonal Vckrey aucton as a PPD mechansm. In Secton 5, we reformulate a PPD mechansm as a subsdzed Vckrey aucton and nvestgate ts propertes. Secton 6 contans our conclusons. 2 The Cost-Sharng Game Asetofn 2 players, denoted as N = {1,...,n}, jontly produce a prvate good and decde how much each player consumes and pays. The cost functon C( ) :R + R + s 4

7 assumed to be strctly convex and twce contnuously dfferentable, and C() = (so that only varable costs are consdered, as typcal n the lterature). Let c( ) =C ( ) denote the margnal cost functon. An outcome (, t) =( 1,..., n,t 1,...,t n ) R+ n R n specfes player s uantty and player s payment t for all N. Outcome (, t) s feasble f t t n C ( n ),andbudget-balanced f t t n = C ( n ). We consder an ncomplete nformaton game wth ndependent prvate values. Player s type θ s a random varable wth support Θ. Types are prvate nformaton but the contnuous dstrbutons of types are common knowledge. Player has a uaslnear utlty that depends only on hs type θ and hs part of the outcome (, t): u (,t θ )=V ( θ ) t. For all θ Θ,player s value functon V ( θ ):R + R + s assumed to be concave, strctly ncreasng, and twce contnuously dfferentable, wth V ( θ )=. Let v ( θ )=V ( θ ) denote player s margnal value functon. A smplfyng assumpton made throughout the paper s that v ( θ ) c() for all N and all θ Θ. Ths assumpton s harmless, servngonlytoelmnatesomeextraneouscaseswhentheoptmalproductonszero. A feasble outcome (, t) s Pareto effcent f there does not exst another feasble outcome (, t ) such that u (,t θ ) u (,t θ ) for all N, andu (,t θ ) >u (,t θ ) for at least one player N. Outcome (, t) s allocatvely effcent f maxmzes the aggregate surplus: arg max à X X V ( θ ) C N N!. Observe that Pareto effcency depends on both and t, but allocatve effcency depends only on. A cost-sharng mechansm specfes a set of permssble bds B for each player N, a uantty rule ( ) : N B R+,andapaymentrulet( ) n : N B R n. For each bd profle b =(b 1,...,b n ) N B, the mechansm selects the outcome ((b), t(b)). A 5

8 cost-sharng mechansm s feasble or budget-balanced f ((b), t(b)) s feasble or budgetbalanced, respectvely, for all b N B. Let b denote the bd profle of player s opponents. A bd b s weakly domnant for player of type θ f for all b j6= B j, b arg max u ( (b, b ),t (b, b ) θ ). b B A cost-sharng mechansm nduces a non-cooperatve cost-sharng game of ncomplete nformaton, where players smultaneously choose ther bds and then obtan the payoffs that result from the outcome the mechansm selects. For all N,player s strategy s a functon that maps hs types Θ nto bds B. Player s strategy s weakly domnant f t maps to a weakly domnant bd for all θ Θ. 1 In ths paper, we sometmes consder a class of cost-sharng games where each B s the set of all possble margnal value functons: B = {v ( θ ):θ Θ }. Insuchacost-sharnggame,player s sad to bd truthfully f for each θ Θ,hsbdsv ( θ ). For the conventonal Vckrey aucton and the subsdzed Vckrey aucton that we ntroduce n ths paper, each player wll have a weakly domnant strategy to bd truthfully. Conseuently, bddng truthfully forms a Bayesan Nash eulbrum. Such an eulbrum s attractve because t does not depend on the dstrbutons of players types and reures only elementary eulbrum calculatons. 3 A Motvatng Example In ths secton, we present an example to motvate our subsdzed Vckrey aucton for cost sharng. We frst examne three promnent cost-sharng mechansms and then show how the Vckrey aucton works n ths context. In the example, there are n =2players wth ndependent prvate values. For {1, 2}, player draws hs type θ from the unform dstrbuton on [, 1], andθ s player s constant margnal value: u (,t θ )=θ t. 1 For expostonal smplcty, we do not dstngush between always optmal and weakly domnant strateges as does Mlgrom (24), who further reures that weakly domnant strateges be unue. 6

9 ThecostfunctonsC() = 2 /2 wth lnear margnal costs c() =. We frst analyze three cost-sharng games nduced respectvely by the average, margnal, and seral costsharng mechansms. In the case of a homogenous good, the Aumann-Shapley cost-sharng mechansm s dentcal to the average cost-sharng mechansm; see Mouln and Shenker (1992). See the same artcle and the references theren for more detals about average, seral, and margnal cost-sharng mechansms. In each of these three mechansms, player bds a uantty b and obtans the uantty he bd;.e, B = R + and (b) =b for {1, 2}. The average, seral, and margnal cost-sharng mechansms dffer n ther payment rules: t A (b) = (b) C (Q(b)) Q(b) = (b) Q(b) 2, t S (b) = = 1 2 C(2 (b)) f b b j 1 C(2 2 j(b)) + C(Q(b)) C(2 j (b)) otherwse (b) 2 f b b j 1 2 Q(b)2 j (b) 2 otherwse, t M (b) = (b)c(q(b)) 1 [Q(b) c(q(b)) C(Q(b))] 2 = (b)q(b) 1 4 Q(b)2, where Q(b) = 1 (b)+ 2 (b) =b 1 +b 2 s the aggregate uantty awarded. It s straghtforward to verfy that each of these mechansms s budget-balanced. In the correspondng cost-sharng games, player s strategy maps hs type θ toabd b (θ ). For each cost-sharng mechansm, Table 1 gves the symmetrc Bayesan Nash eulbrum that we have calculated and each player s ex ante payoff: Z 1 Z 1 θ b(θ ) t (b(θ ),b(θ j )) dθ j dθ, 7

10 where b( ) s the symmetrc Bayesan Nash eulbrum and t (, ) s the payment rule assocated wth the mechansm. Mechansm Eulbrum Bddng Functon b(θ ) ex ante Payoff Average max θ, ª Seral ln 2 ln(2 θ ) Margnal max θ 3 3, ª Table 1: Eulbrum bds and payoffs. In the cost-sharng games nduced by these three mechansms, the eulbra do not nvolve weakly domnant strateges, and eulbrum outcomes are almost never allocatvely effcent. To see that t s not weakly domnant for player to bd as gven n Table 1, supposethatplayerj bds no matter what hs type s. Then player could mprove hs payoff by bddng the uantty where hs margnal value euals margnal cost, notng player wll pay all of the costs. Allocatve effcency n ths example reures that the player wth the lower margnal value always wns zero uantty and the player wth the hgher value wn the uantty where the margnal cost euals hs margnal value, whch n turn reures that b (θ )=θ and b j (θ j )=whenever θ >θ j. However, these bds occur wth zero probablty n the eulbra. The dea of the Vckrey (1961) aucton, whch we call the conventonal Vckrey aucton n order to dstngush t from our subsdzed Vckrey aucton, s that players bd ther margnal value curves, and then by constructon, uantty s awarded to guarantee allocatve effcency presumng the players bd truthfully. Due to hs ngenous payment rule, players do ndeed have an ncentve to bd truthfully, as Vckrey showed. Detals specfc to our example follow. Each player bds b [, 1] so that hs reported margnal value curve s v () =b for all. To guarantee allocatve effcency (and to settle the pesky matter of tes should both players report the same value), the uantty rule n the conventonal Vckrey aucton s: c 1 (b ) f b >b j V 1 (b) = 2 c 1 (b ) f b = b j otherwse. 8

11 In other words, the conventonal Vckrey aucton assgns all producton to the player wth the hgher bd up to where the margnal cost s eual to hs bd. The payment rule for player n the conventonal Vckrey aucton nvolves an ngenous dea. Frst, fnd the effcent assgnment of uantty n the absence of player (or more precsely n ths example, f player reported hs value at ). Then, when uantty s awarded effcently ncludng player, the conventonal Vckrey aucton has player pay player j s margnal value for any uantty that j s dsplaced from and pay any costs of addtonal uantty produced. In partcular, for ths example, R V (b) max{b t V j,c(z)}dz f b b j (b) = otherwse. Fgure 1 below llustrates the wnnng player s uantty and payment ( V (b),t V (b)) n the conventonal Vckrey aucton. p 6 c( ) b b j t V V - Fgure 1: Wnnng player s outcome n the conventonal Vckrey aucton We defne a () =max{b j,c()} as the ask curve that player faces. Observe that the ask curve player faces n no way depends on hs own bd b, but rather on the bd of hs opponent and the exogenously gven margnal cost curve. As Fgure 1 shows, the wnnng player wns the uantty V where hs reported margnal value curve ntersects the ask curve, and pays the area under the ask curve for the uantty he wns. For the th margnal unt, player wll have to pay a (). Wth ths nterpretaton, player s just lke a monopsonst who can 9

12 perfectly prce dscrmnate when buyng hs nput n a market wth (nverse) supply curve a ( ). The best player candostopurchasetheuanttywherehsmargnalvaluecurve ntersects a ( ), and ths s guaranteed when player bds truthfully. In other words, bddng truthfully s weakly domnant for player. To put t another way, t s weakly domnant for player to bd truthfully becausebydongso,hewnsalloftheuanttythatcostshm less than hs value and avods wnnng any uantty that costs more than hs value. Ths ntuton s formalzed n the proof of Proposton 1, gven n the next secton. Conseuently, there s a symmetrc Bayesan Nash eulbrum where each player bds b (θ )=θ. The resultng ex ante payoff to each player s: Z 1 Z " θ Z # θ θ 2 θ 2 j d dθ j dθ = 1 12 =.83333, θ j whch s lower than that from any of the three cost-sharng mechansms consdered n Table 1. Observe that the wnner s payment exceeds the producton cost, and ths overpayment s not accounted for n the players utltes. In ths truthful bddng eulbrum, the expected overpayment s Z 1 Z θ Z θ2 jdθ j + θ 2 θ2 dθ j dθ = 1 12 =.83333, whch s euvalent to per player, or 5% of each player s ex ante payoff. In costsharng stuatons, players jontly produce the good: there s no opposng seller producng the good and pocketng ths overpayment. One possble opton for the players s to sell the rghts for any overpayment to an outsde party, whch nduces a budget-balanced mechansm. However, ths opton may not be practcal n realty, whch may be the reason that the Vckrey aucton has not been consdered much for cost-sharng problems. The ueston that motvates ths paper s whether the overpayment generated n the conventonal Vckrey aucton can be used to mprove players welfare wthout destroyng ther ncentves to bd truthfully. Snce drect refunds do not work, n secton 5, we wll modfy the Vckrey aucton n such a way that the overpayment subsdzes addtonal producton, 1

13 but wthout upsettng the players ncentves to bd truthfully. As we wll show, ths subsdzed Vckrey aucton performs ute well n ths example: a player s ex ante payoff n the eulbrum s.14167, substantally hgher than that from any of the promnent costsharng games consdered n Table 1. Moreover, outsde of ths example, we wll show that the subsdzed Vckrey aucton always outperforms the conventonal Vckrey aucton. Before ntroducng ths subsdzed Vckrey aucton, n the next secton we dstll from the conventonal Vckrey aucton the key elements that ensure truthful bddng s a weakly domnant strategy. 4 Perfect Prce Dscrmnaton Mechansms In ths secton, we frst descrbe a class of mechansms, called perfect prce dscrmnaton (PPD) mechansms, n whch truthful bddng s a weakly domnant strategy. We then show that the conventonal Vckrey aucton s a PPD mechansm. In a PPD mechansm, players bd or report ther margnal value functons;.e, B = {v ( θ ):θ Θ } s the set of allowable bds to player N. From any bddng profle b =(b 1 ( ),...,b n ( )), a non-decreasng ask curve a ( ) :R + R + s constructed for each player N. Player s part of the outcome s then determned as: (b) arg max t (b) = Z (b) Z [b (z) a (z)] dz, (1) a (z)dz. (2) At ths pont, we stay agnostc about how these ask curves are constructed. Dfferent ask curves nduce dfferent PPD mechansms. Assume that (b) from (1) s well-defned. Then by (1), player wns the uantty at whch hs bd curve and ask curve ntersect, and by (2), player pays the area under hs ask curve up to the uantty he wns. We may nterpret the ask curve as the nverse supply curve faced by a monopsonst who can perfectly prce dscrmnate. Accordngly, we refer to such a mechansms as perfect prce dscrmnaton (PPD) mechansms. Fgure 2 llustrates player s outcome (,t ) n a PPD mechansm and 11

14 Proposton 1 gves a suffcent condton that ensures truthful bddng s a weakly domnant strategy. p 6 a ( ) b ( ) t - Fgure 2: Player s outcome n a PPD mechansm Proposton 1 In a PPD mechansm, f a ( ) s ndependent of b ( ), then bddng truthfully s a weakly domnant strategy for every player N. Proof. Gven that a ( ) s ndependent of b ( ), (2) mples that player s bd can only affect hs payment nsofar as t affects the uantty he wns. Ths means that player s payoff s bounded from above by max V ( θ ) Z a (z)dz =max Z [v (z θ ) a (z)] dz, whch s acheved f player bds truthfully due to (1). Snce ths s true for any a ( ) that s ndependent of b ( ), t s weakly domnant for player to bd truthfully. In the remander of ths secton, we reformulate the conventonal Vckrey (1961) aucton as a PPD mechansm. To smplfy the exposton, we separate the case when players have constant margnal values from the case when players have dmnshng margnal values. When every player has a constant margnal value, as n our example of secton 3, every player submts a bd b R + (hs reported margnal value). Gven a bd profle b =(b, b ), 12

15 the uantty and payment rules are: c 1 (b ) f b > max j6= b j V 1 (b) = M c 1 (b ) f b =max j6= b j (3) otherwse, R V (b) max {c(z), max t V j6= b j } dz f b max j6= b j (b) = (4) f b < max j6= b j. where M s the number of players ted wth the hghest bd. The te breakng rule can be arbtrary snce tes occur wth zero probablty n eulbrum. It follows mmedately that the conventonal Vckrey aucton s a PPD mechansm wth the followng ask functons: ½ ¾ a V () =max c(), max b j. (5) j6= Snce player s ask functon s ndependent of hs own bd b, Proposton 1 apples, and truthful bddng s weakly domnant for every player. Therefore, there s an eulbrum where every player bds hs true margnal value. The conventonal Vckrey aucton s feasble but not budget-balanced, snce (4) mples that the wnnng player pays more than the producton cost. On the other hand, the Vckrey aucton s allocatvely effcent snce all producton s awarded to the player wth the hghest margnal value at exactly the uantty where the margnal cost euals hs margnal value. When every player has a decreasng margnal value curve, formulatng the conventonal Vckrey aucton as a PPD mechansm s slghtly more complcated. In ths case, every player N s allowed to submt a dfferentable and strctly decreasng bd functon: b ( ) :R + R +. Ths bd s nterpreted as a reported margnal value curve. Let d ( ) be the demand curve assocated wth b ( ): b 1 (p) f p b () d (p) = otherwse. Gventhebdprofle b =(b 1 ( ),...,b n ( )), denote the correspondng demand profle by d =(d 1 ( ),...,d n ( )). The aggregate demand s then the horzontal sum of ndvdual demand curves: d(p) =d 1 (p)+...+ d n (p). 13

16 Vckrey (1961) sets up a double aucton wth strategc buyers and sellers, and generates the market supply curve by aggregatng the sellers reports of ther margnal costs. Accordngly, for a cost-sharng problem, we take the market supply curve to be the exogenous (nverse) margnal cost curve c 1 (p) f p c() s(p) = otherwse. Assume that there s a unue market eulbrum (p, ) where s(p )=d(p )=. Lkewse, assume for all N that there s a unue eulbrum (p, ) n the market wthout player, such that s(p ) =d (p ) = where d ( ) s the aggregate demand by player s opponents. The outcome n the conventonal Vckrey aucton s V (b) = s(p ) d (p ) d (p ), (6) t V (b) = Z s(p ) d (p ) max {b (z),c(z)} dz, (7) where b ( ) gves the heght of d ( ). The left panel of Fgure 3 llustrates player s outcome n the conventonal Vckrey aucton, as defned by (6) and (7). p 6 s( p a d( @ t d ( ) - {z } V V Fgure 3: Player s outcome n the conventonal Vckrey aucton p 6 Q QQQ s( ) d ( ) p Q QQQ p Q Q d ( ) t V - The left panel of Fgure 3 closely resembles the scheme descrbed by Vckrey (1961, pages 9-14), except that the supply curve n our paper s gven by the margnal cost curve 14

17 of producton. We next reformulate the conventonal Vckrey aucton as a PPD mechansm. For any bd profle, the resdual supply curve to player s gven by s(p) d (p) f p p r V (p) = otherwse. Our next proposton shows that the conventonal Vckrey aucton s a PPD mechansm wth the nverse of r ( ) as the ask functon to player n the PPD mechansm, establshng that the rght panel of Fgure 3 s euvalent to the left panel. 2 Proposton 2 For the case of v ( θ ), the conventonal Vckrey aucton s a PPD mechansm where for all N, B = {v ( θ ):θ Θ },anda ( ) s the nverse of r V (p). Proof. When b () p ths means that p = p and so V (b) =and t V (b) =by (6) and (7). When b () >p, due to the monotoncty and contnuty of b ( ) and r ( ), the unue soluton to (1) where a ( ) s the nverse of r V ( ) s the unue ntersecton of b ( ) and a ( ). But because d(p )=s(p ) d (p )+d (p )=s(p ) d (p )=s(p ) d (p )=r V (p ), player s demand and ask curves ntersect at d (p ). The payment n the conventonal Vckrey aucton can be expressed as: Z V max {b (),c()} d = p V Z p [s (p) d (p)] dp = p Z V a (z)dz, as llustrated n Fgure 3. Ths concludes the proof (llustrated n Fgure 3). Snce player s ask curve s ndependent of player s bd, Proposton 1 mples that n the conventonal Vckrey aucton, bddng truthfully s a weakly domnant strategy, a wellknown feature of the conventonal Vckrey aucton. To conclude ths secton, we collect the well-known propertes of the conventonal Vckrey aucton n the followng proposton. 2 Indeed, the rght panel of Fgure 3 s just how Ausubel (24) nterprets the uantty and payment rule n the Vckrey aucton: see hs Fgure 1. 15

18 Proposton 3 The conventonal Vckrey aucton s feasble, but not budget balanced n general. It s weakly domnant to bd truthfully, and the resultng eulbrum outcome s allocatvely effcent. 5 A Subsdzed Vckrey Aucton We now ntroduce the subsdzed Vckrey aucton for cost-sharng problems. For expostonal convenence, we wll consder separately the cases of constant and decreasng margnal values. To keep the notaton from sprawlng, we wll use the same notaton for these two cases. 5.1 Constant Margnal Values Consder the case that for all N, v ( θ )=θ,whereθ s player s constant margnal value. As n the conventonal Vckrey aucton, player reports hs margnal value: B = {v ( θ ):θ Θ } = R +. Gven a bd profle b, letb j =max k6= b k. Lke n the conventonal Vckrey aucton, for player to wn any uantty he wll have to bd at least b j and by so dong wll pay at least the unt prce of b j for any uantty he receves. But unlke the conventonal Vckrey aucton, the unt prce wll reman b j beyond = c 1 (b j ) up untl all of the overpayment that player wll generate by dsplacng bd b j s exhausted. Specfcally, defne the uantty x mplctly by the followng euaton: Z c 1 (b j ) Z x [b j c(z)] dz + [b j c(z)] dz =, (8) {z } c 1 (b j ) {z } Ω Ψ where Ω and Ψ represent player s overpayment and underpayment, respectvely, should player bd hgher than b j. We can then defne the subsdzed ask functon faced by player as: a SV () = b j c() f x otherwse, as llustrated n Fgure 4 (where the dagonal represents the margnal cost c( )). 16

19 p 6 a SV ( ) c b j Ω Ψ r x - Fgure 4: Player s subsdzed ask curve (Ω = Ψ ) The subsdzed Vckrey aucton s then the PPD mechansm where for all N, player bds a margnal value and faces the subsdzed ask functon a SV ( ). Player s outcome s gven by (1) and (2) wth hs subsdzed ask functon a SV ( ). In the case of a te for the hghest bd, those submttng the hghest bd wll splt x eually. Fgure 5 llustrates two possbltes n determnng player s outcome, and the followng proposton gves the propertes of the subsdzed Vckrey aucton. p b 6 a SV ( ) c p 6 a SV ( ) c b j t SV r SV - Fgure 5: Player s outcome n the subsdzed Vckrey aucton b b j t SV r SV - Proposton 4 In the case of constant margnal values, the subsdzed Vckrey aucton: () s budget-balanced, () gves every player a weakly domnant strategy to bd truthfully, and () n the truthful bddng eulbrum, Pareto domnates the conventonal Vckrey aucton. 17

20 Proof. () Consder any bddng profle b =(b 1,...,b n ), and denote by the player wth the hghest bd. If b = b j max k6= b k, then the total uantty produced s eual to x and the wnnng players pay b j per unt, so that by euaton (8) the total payments collected are x b j = Z x c(z)dz = C(x ), thereby balancng the budget. If b >b j,then ether SV (b) =x t SV (b) =x b j = R x c(z) dz, or SV (b) >x t SV (b) =x b j + R SV x c(z) dz = R SV c(z) dz, agan applyng euaton (8). Ether way, player pays exactly the cost for the uantty he wns, so that the subsdzed Vckrey aucton s budget-balanced. () By constructon, the subsdzed Vckrey aucton s a PPD mechansm where every player s ask functon s ndependent of ths player s bd, so that Proposton 1 apples. () Just as n the Vckrey aucton, players who do not have the hghest value wn and pay. The player wth the hghest value, however, faces an ask curve n the subsdzed Vckrey aucton that s below the ask curve he faces n the conventonal Vckrey aucton, ether wnnng the same amount but payng less (as n the left panel of Fgure 5) or wnnng more uantty at a cost below hs value (as n the rght panel of Fgure 5). These events occur wth probablty one, thereby establshng the desred result. Unlke the conventonal Vckrey aucton, the subsdzed Vckrey aucton s not allocatvely effcent. In the eulbrum where every player bds truthfully, when the wnner s value θ < c(x ), as n the rght panel of Fgure 5, player neffcently wns x, nstead of the effcent uantty c 1 (θ ). Although the subsdzed Vckrey aucton may award too much to the wnnng player(s), n the eulbrum, uantty s never awarded to a player f he does not have the hghest value. We now return to the example consdered n secton 3. Wth the subsdzed Vckrey aucton descrbed above, a player s ex ante payoff n the truthful bddng eulbrum s Z " 1 Z Ã! θ1 /2 θ 1 (2θ 2 ) (2θ 2) 2 Z θ1 µ # dθ 2 + θ 1 θ 1 θ2 1 dθ 2 dθ 1 = =.14167, θ 1 /2 18

21 whch s hgher than a player s ex ante payoff from the conventonal Vckrey aucton, and all three cost-sharng mechansms we consdered n Secton Decreasng Margnal Values Consderthecasewherev ( θ ) s strctly decreasng for all N and all θ Θ. As n the conventonal Vckrey aucton, every player submts a downward slopng margnal value curve: B = {v ( θ ):θ Θ }. We next show how to construct the ask curves that wll be used n the subsdzed Vckrey aucton. A subsdzed ask curve s constructed so that some of the overpayments nherent n the conventonal Vckrey aucton are used to subsdze addtonal producton. As shown n the left panel of Fgure 3, player s payment exceeds the margnal cost of producton over the nterval of [d (p ),d (p )]. In the conventonal Vckrey aucton, for player to obtan any uantty, he must force the market prce above p, whch not only dsplaces other players from uantty they would otherwse wn, but also expands the producton. In the conventonal Vckrey aucton, player must pay the socal opportunty cost of the uantty he obtans. Snce the other players value the uantty that player dsplacesmorethanthe producton costs, ths creates excess revenue over producton costs. Specfcally, we defne player s overpayment functon as Ω (p) =for p<p and Ω (p) = Z d (p ) d (p) [b (z) c (z)] dz, for p p. (9) ThefollownglemmasummarzessomepropertesofΩ ( ). Lemma 5 For all N, Ω ( ) s contnuous and dfferentable almost everywhere, and s strctly ncreasng over p,b (). Proof. The contnuty and dfferentablty of b ( ), d ( ), andc ( ) mply that Ω ( ) s contnuous everywhere and dfferentable at all p/ p ª {bj () : j 6= }, notngthatknks occur when aggregatng the demands of the other players j 6=. For p [p,b ()), (9) 19

22 mples that wherever t exsts, Ω (p) = d (p)[b (d (p)) c (d (p))] = d (p)[p c(d (p))], whch s strctly postve due to d (p) < and p>c(d (p)) for p>p. We next show how to use the overpayment Ω (p) to offset some of the producton costs. Specfcally, for each N, a subsdzed supply functon s (p) s(p) s created as follows (whch s well-defned almost everywhere): s(p) f p b () s (p) = soluton to s (p) [c(s (p)) p] =Ω (p) f p [p,b ()) s(p) f p<p. There are three dfferent segments n player s subsdzed supply functon. For p<p and p b (), s (p) concdes wth the actual supply curve (or margnal cost of producton). But for p [p,b ()), our next lemma shows s (p) les below and to the rght of s(p) far enough so that Ω (p) s exhausted (subsdzng producton). Lemma 6 For all N, s ( ) s strctly ncreasng and for p [p,b ()), Z s (p) (1) c(z) s 1 (z) dz = Ω (p). (11) Proof. Because s( ) s the nverse of c( ), wehave s (p) = s(p) c(s (p)) = p, s (p) > s(p) c(s (p)) >p. Because s (p) s(p) for p [p,b ()), Lemma 5 and euaton (1) mply that s (p) > and s (p) > s(p) for all p where Ω (p) exsts. For p [p,b ()), ntegratng s (p) [c(s (p)) p] =Ω (p) from p up to p yelds (11), dong a change of varable p = s 1 () on the left hand sde of (11) (Alternatvely, dfferentate both sdes of (11) to get (1)). The left hand sde of (11) represents the underpayment by player, whch we denote by Ψ (p). Fgure 6 below llustrates player s subsdzed supply functon s (p) nthecaseof decreasng margnal values. 2

23 p 6 d c( ) p Ψ s ( @@ - Fgure 6: Player s subsdzed supply functon (Ω = Ψ ) Usng the subsdzed supply functon s (p), wedefne player s resdual supply functon as r SV (p) =max{s (p) d (p), }. Then player s subsdzed ask functon s the nverse of the resdual supply player s facng: a SV () = p for = SV 1 r () for >. To summarze, the subsdzed Vckrey aucton s the PPD mechansm where every player submts a downward slopng demand curve b ( ) and player s outcome s determned by b ( ) and a SV ( ) accordng to (1) and (2). Now defne the market clearng prce p that d(p such )=s (p ), whch wll dffer across players because the subsdzed supply functons dffer across players. Our next lemma provdes another way to derve player s outcome n the subsdzed Vckrey aucton, whch we state wthout proof snce t s straghtforward and resembles that for Proposton 2. Lemma 7 The subsdzed Vckrey aucton nduces the followng outcome rule: for all bd profle b, SV (b) =s (p ) d (p ) d (p ) (12) 21

24 and Z d(p t SV ) (b) = d(p ) d (p ) c (z) dz. (13) Observe from euaton (13) that n some sense each player pays the producton costs for theuanttyhewns. Theproofofournextpropostonusesthsfeaturetoshowthatthe subsdzed Vckrey aucton s ndeed feasble. Proposton 8 In the case of decreasng margnal values, the subsdzed Vckrey aucton: () s feasble, () s weakly domnant for every player to bd truthfully, and () n the truthful bddng eulbrum, Pareto domnates the conventonal Vckrey aucton. Proof. Wthout loss of generalty, assume p 1... p n. In the subsdzed Vckrey aucton, the total payment s nx =1 t SV (b) = = Xn 2 =1 n 2 X. =1 Z d(p ) d(p ) d (p ) Z d(p ) d(p ) d (p ) Z d(p n ) Q+ SV 1 d(p Z d(p n ) c () d + c () d + n ) Q c () d + + d(p n ) Q Z Q c () d c () d = C(Q). Z d(p n 1 ) d(p n 1 ) SV n 1 Z d(p n ) n SV d(p n ) n SV c () d + SV n 1 Z d(p n ) n SV d(p n ) SV n c () d + SV n 1 Z d(p n ) d(p n ) n SV Z d(p n ) c () d + d(p n ) n SV Z d(p n ) c () d (14) d(p n ) SV n c () d c () d where Q = P n =1 SV s the total amount awarded n the aucton. The frst eualty s due to (13). The next neualty s the only subtle part of the proof. Observe that for each player, by constructon, d(p ) SV d(p ) d(p ),wherep sthemarketclearngprcen the conventonal Vckrey aucton. From ths t follows that d(p n ) n SV d(p ) d(p n 1), showng that the frst neualty above results by calculatng the second to last ntegral of 22

25 the strctly ncreasng margnal costs over a uantty nterval of the same wdth SV n 1, but for an nterval closer to the orgn (zero uantty). Ths process s repeated for each ntegral, and then the ntegrals are summed up to get the eualty above. The last neualty results by agan evaluatng an ntegral of the same wdth, but closer to the orgn, notng that d(p n ) P d (p n ) P d (p ) Q. Snce the total payment s at least the total cost, the conventonal Vckrey aucton s feasble. () By constructon, the subsdzed Vckrey aucton s a PPD mechansm where every player s ask functon s ndependent of ths player s bd, so that Proposton 1 apples. () No player s worse off n the subsdzed Vckrey aucton because the ask curve he faces n the subsdzed Vckrey aucton les below the ask curve he faces n the conventonal Vckrey aucton. Indeed, any player that wns postve uantty n the Vckrey aucton wll be made strctly better off n the subsdzed Vckrey aucton because of the lower ask curve he faces. Unlke the case of constant margnal values, n the case of dmnshng margnal values the subsdzed Vckrey aucton s generally not budget-balanced. The neualtes n euaton (14) are generally strct. In the event that one player, say 1, pushes the prce p 1 up so hgh that p 1 b j () for players j>1, smlar to what the left panel of Fgure 5 shows for the cases of constant margnal values, the outcome wll be budget balanced and allocatvely effcent, and yet stll Pareto domnate the outcome of the conventonal Vckrey aucton snce the wnnng player wll pay only the producton costs and no more. 6 Concluson In many games of ncomplete nformaton, t s generally not possble to smultaneously acheve truthful bddng as a domnant strategy, budget balance, and allocatve effcency. Thus, choosng a mechansm to solve an allocaton problem wll nvolve tradeoffs. The Vckrey aucton and Vckrey-Clarke-Groves mechansms acheve allocatve effcency and domnant strateges, but are often plagued by budget mbalance. In ths paper, we have shown that the budget surplus need not rule out altogether the Vckrey aucton from use 23

26 as a cost sharng mechansm. Wth some modfcatons, the Vckrey aucton can be made to redstrbute back the surplus to players, not n drect monetary rebates, but rather n terms of addtonal subsdzed producton. Although ths subsdzed Vckrey aucton sacrfces some allocatve effcency, t stll easly Pareto domnates the conventonal Vckrey aucton. Wehaveshownnanexamplenanoncooperatvegamewhereplayershaveunformly dstrbuted values, that our subsdzed Vckrey aucton also outperforms other cost sharng mechansms, lke the average and seral mechansms. We do not beleve that ths rankng between our subsdzed Vckrey aucton and the average and seral cost sharng mechansms holds unversally n noncooperatve games wth ncomplete nformaton because by constructon the subsdzed Vckrey aucton wll grossly overproduce when players values are close together wth hgh probablty. Lookng at the performance of these cost sharng mechansms n noncooperatve games of ncomplete nformaton s a logcal next step. In the perfect prce dscrmnatng mechansms ntroduced n ths paper, we solated the key feature n the conventonal Vckrey aucton that ensures truthful beatng as a domnant strategy. Effcency plays no role n whatsoever. Thus, effcency can be sacrfced to allevate budget mbalance whle mantanng truthful bddng. In future research, we wll explore whether ths dea can help resurrect the Vckrey aucton n other stuatons lke publc goods, bddng rngs (McAfee and McMllan, 1992), and partnershp dssolutons (Cramton, Gbbons, and Klemperer, 1987) where budget mbalance prevents ts use. Although we have lmted dscusson to envronments wth ndependent prvate values, t would not be dffcult to adapt the subsdzed Vckrey aucton to the afflated values of Mlgrom and Weber (1982). Indeed n ths settng, our subsdzed Vckrey aucton could be mplemented wth an ascendng aucton, smlar to the Ausubel (24) aucton. Our subsdzed Vckrey aucton suffers the same drawbacks as conventonal Vckrey auctons: players may want to merge, or a player may want to submt multple bds under multple dentfes. Such behavor can compromse effcency as descrbed by Mlgrom (24) and Ausubel and Mlgrom (22). 24

27 References [1] Arrow, K. (1979): The Property Rght Doctrne and Demand Revelaton under Incomplete Informaton, n M. Boskn (ed.), Economcs and Human Welfare, NewYork: Academc Press. [2] Ausubel, L.M. and P. Cramton (22): Demand Reducton and Ineffcency n Mult- Unt Auctons, Unversty of Maryland Workng Paper 96-7 (revsed July 22). [3] Ausubel, L.M. and P.R. Mlgrom (22): Ascendng Auctons wth Package Bddng, Fronters of Theoretcal Economcs, 1,Artcle1. [4] Ausubel, L. M. (24): An Effcent Ascendng-Bd Aucton for Multple Objects, Amercan Economc Revew, 94, [5] Chen, Y. (23): An Expermental Study of Seral and Average Cost Prcng Mechansms, Journal of Publc Economcs, 87, [6] Chen, Y. and Razzoln (25): Congeston and Cost Allocaton for Dstrbuted Networks: An Expermental Study, Unversty of Mchgan, manuscrpt. [7] Cramton, P., R. Gbbons, and P. Klemperer (1987), Dssolvng a Partnershp Effcently, Econometrca, 55, [8] d Aspremont, C. and L. A. Gerard-Varet (1979a): Incentves and Incomplete Informaton, Journal of Publc Economcs, 11, [9] d Aspremont, C. and L. A. Gerard-Varet (1979b): On Bayesan Incentve Compatble Mechansms, n J.-J. Laffont (ed.), Aggregaton and Revelaton of Preferences, Amsterdam: North-Holland. [1] Fredman, E. and H. Mouln (1999): Three Methods to Share Jont Costs or Surplus, Journal of Economc Theory, 87, [11] Koster, M. (26): Consstent Cost Sharng and Ratonng, Unversty of Amsterdam, manuscrpt. 25

28 [12] Krshna, V.(22): Aucton Theory, New York: Academc Press. [13] McAfee, P. (1992): A Domnant Strategy Double Aucton, Journal of Economc Theory, 56, [14] McAfee, P.R. and J. McMllan (1992): Bddng Rngs, Amercan Economc Revew, 82, [15] Mlgrom, P. (24): Puttng Aucton Theory to Work, Cambrdge, UK: Cambrdge Unversty Press. [16] Mlgrom, P.R. and R.J. Weber (1982): A Theory of Auctons and Compettve Bddng, Econometrca, 5, [17] Mouln, H. and S. Shenker (1992): Seral Cost Sharng, Econometrca, 6, [18] Mouln, H. and S. Shenker (1994): Average Cost Prcng Versus Seral Cost Sharng: An Axomatc Comparson, Journal of Economc Theory, 64, [19] Mouln, H. and S. Shenker (21): Strategyproof Sharng of Submodular Costs: Budget BalanceVersusEffcency, Economc Theory, 18, [2] Razzoln, L., M. Reksulak, and R. Dorsey (24): An Expermental Evaluaton of the Seral Cost Sharng Rule, Vrgna Commonwealth Unversty, manuscrpt. [21] Vckrey, W. (1961): "Counterspeculaton, Auctons, and Compettve Sealed Tenders," Journal of Fnance, 16,