Weak Monotonicity and Bayes-Nash Incentive Compatibility

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1 Weak Monotoncty and Bayes-Nash Incente Compatblty Rudolf Müller, Andrés Perea, Sascha Wolf Abstract An allocaton rule s called Bayes-Nash ncente compatble, f there exsts a payment rule, such that truthful reports of agents types form a Bayes-Nash equlbrum n the drect reelaton mechansm consstng of the allocaton rule and the payment rule. Ths paper prodes a characterzaton of Bayes-Nash ncente compatble allocaton rules n socal choce settngs where agents hae mult-dmensonal types, quas-lnear utlty functons and nterdependent aluatons. The characterzaton s dered by constructng complete drected graphs on agents type spaces wth cost of manpulaton as lengths of edges. Weak monotoncty of the allocaton rule corresponds to the condton that all 2-cycles n these graphs hae non-negate length. For the case that type spaces are conex and the aluaton for each outcome s a lnear functon n the agent s type, we show that weak monotoncty of the allocaton rule together wth an ntegrablty condton s a necessary and suffcent condton for Bayes-Nash ncente compatblty. 1 Introducton Ths paper s concerned wth the characterzaton of Bayes-Nash ncente compatble allocaton rules n socal choce settngs where agents hae ndependently dstrbuted, mult-dmensonal types and quas-lnear utlty functons, that s, utlty s the aluaton of an allocaton mnus a payment. We allow for nterdependent aluatons across agents. The central task addressed n ths paper s the followng: gen such type dstrbutons and aluatons, characterze precsely those allocaton rules for whch there exsts a payment rule such that truthful reportng of agent s types forms a Bayes-Nash equlbrum n the drect reelaton mechansm consstng of the allocaton rule combned wth the payment rule. In addton, we am for a framework that lets us construct a payment rule, f any, whch makes a partcular allocaton rule Bayes-Nash ncente compatble. For example, gen an allocaton rule whch decdes n a combnatoral aucton for each set of bds for each agent whch set of tems he wns, we want to be able to decde whether there exsts a prcng scheme for wnnng bds that makes truthful bddng a Bayes-Nash equlbrum. If the answer s yes, we would lke to hae means to construct such a prcng scheme. 1.1 Related Work An allocaton rule s domnant strategy ncente compatble, f there exsts a payment rule such that for any report of the other agents an agent maxmzes

2 hs own utlty by reportng truthfully hs type. Roberts 1979) mplctly uses a monotoncty condton on the allocaton rule n order to dere hs characterzaton of domnant strategy ncente compatble mechansms n terms of affne maxmzers for unrestrcted preference domans. For a selecton of restrcted preference domans, Bkhchandan et al. 2003) and La et al. 2003) characterze domnant strategy ncente compatblty drectly n terms of a monotoncty condton on the allocaton rule. Gu et al. 2004) extend these results to larger classes of preference domans by makng a lnk to network theory. The most general results are by Saks and Yu 2005), who show that preous results extend to any conex mult-dmensonal type space. The enronment consdered by Saks and Yu 2005) features quas-lnear utltes and mult-dmensonal types. The allocaton rule maps agents type reports nto a fnte set of m possble outcomes. An agent s type s a ector n R m reflectng hs aluaton of the dfferent possble outcomes, that s, the agent s aluaton of some outcome a s gen by the a th element of hs type ector. Agents type spaces are assumed to be conex. Saks and Yu 2005) show that domnant strategy ncente compatble allocaton rules n ths settng can be characterzed n terms of weak monotoncty, a term ntroduced by La et al. 2003). In order to dere ths result they construct complete drected graphs n the followng way: Take some agent and fx a profle of type reports for the others. Now, a drected graph s constructed by assocatng a node wth each outcome and puttng a drected edge between each ordered par of nodes. Take two outcomes a and b. Consder the dfference of the aluaton of a and the aluaton of b wth respect to eery type for whch truthfully reportng ths type yelds outcome a. The length of the network edge from a to b s defned as the nfmum of all these dfferences. In ths fashon a graph s constructed for eery agent and eery possble report profle of the other agents. Weak monotoncty states that for any two dfferent outcomes a and b, the sum of the two edge lengths from a to b and from b to a s non-negate. Earler, Rochet 1987) characterzed domnant strategy mplementaton n cases where the set of outcomes s not necessarly fnte; an assumpton that s crucal to the work of Saks and Yu 2005). He consders a settng where agents hae mult-dmensonal, conex type spaces and aluaton functons whch are lnear w.r.t. ther own true types. Makng some addtonal dfferentablty assumptons, Rochet 1987) shows that n ths case domnant strategy ncente compatblty can be characterzed n terms of a monotoncty condton on the allocaton rule plus an ntegrablty condton. Monotoncty has also been used to characterze Bayes-Nash ncente compatble allocaton rules. Jehel et al. 1999) and Jehel and Moldoanu 2001) deelop characterzatons for socal choce settngs where agents hae multdmensonal, conex type spaces and aluaton functons whch are lnear w.r.t. ther true types. Ther characterzatons of Bayes-Nash ncente compatblty nclude a monotoncty condton on the allocaton rule as well as an ntegrablty condton comparable to the one presented by Rochet 1987).

3 1.2 Our Contrbuton Smlar to the network approach of Gu et al. 2004) and Saks and Yu 2005) we construct graphs. If an allocaton rule s Bayes-Nash ncente compatble, then there exsts a payment rule such that an agent s expected utlty for truthfully reportng hs type t s at least as hgh as hs expected utlty for msreportng some type s. Smlarly, an agent s expected utlty for truthfully reportng type s s at least as hgh as hs expected utlty for msreportng type t. From combnng these two condtons we get a weak monotoncty condton on the allocaton rule. Ths condton s the expected utlty equalent of the monotoncty condton mentoned n the context of domnant strategy ncente compatble allocaton rules. Weak monotoncty s a necessary condton for Bayes-Nash ncente compatblty. It expresses that the expected gan n aluaton for truthfully reportng t nstead of msreportng s should be at least as bg as the expected gan n aluaton for msreportng t nstead of truthfully reportng s. Recognzng that the constrants nherent n the defnton of Bayes-Nash ncente compatblty hae a natural network nterpretaton we buld complete drected graphs for agents type spaces. To do so we assocate a node wth each type and put a drected edge between each ordered par of nodes. The length of the edge gong from the node assocated wth type s to the node assocated wth type t s defned as the cost of manpulaton, that s, the expected dfference n an agent s aluaton for truthfully reportng t nstead of msreportng s. Note that unlke the network approach of Gu et al. 2004) and Saks and Yu 2005) see descrpton aboe) we construct only one graph for each agent snce we work n terms of expectatons and do not consder each possble type profle of the other agents separately. Furthermore, each of these graphs contans an nfnte number of nodes as we assocate a node wth each possble type of the agent. One could also construct outcome based graphs as done by Gu et al., 2004; Saks and Yu, 2005) by assocatng a node wth each possble probablty dstrbuton oer outcomes. Howeer, these graphs also contan an nfnte number of nodes wheneer the dfferent possble type reports of an agent nduce an nfnte number of probablty dstrbutons oer outcomes. The outlne of the paper s as follows: In Secton 2 we state some basc assumptons and defntons. Throughout the paper we assume that agents hae quas-lnear utlty functons and ndependently dstrbuted, prately known, mult-dmensonal types. Furthermore, we allow for nterdependent aluatons. We do not put any restrctons on the number of possble outcomes. In Secton 3 we show that an allocaton rule s Bayes-Nash ncente compatble f and only f the graphs descrbed aboe contan no fnte, negate length cycles. Rochet 1987) shows that domnant strategy ncente compatblty can be characterzed n terms of the absence of fnte, negate length cycles n smlar graphs. Our result s the Bayes-Nash equalent for hs fndng. In Secton 4 agents type spaces are assumed to be conex and ther aluaton functons are assumed to be lnear w.r.t. to ther own true types. Een under

4 these restrctons, weak monotoncty alone s not suffcent for Bayes-Nash ncente compatblty. Howeer, we show that weak monotoncty together wth an ntegrablty condton s both necessary and suffcent for Bayes-Nash ncente compatblty. The settng of a sngle-tem aucton wth externaltes consdered n Jehel et al. 1999) and the socal choce settng consdered n Jehel and Moldoanu 2001) are specal cases of the framework presented n ths secton. Compared to ther settngs, our mult-dmensonal framework allows for a broader class of possble nterdependences between agents aluatons. The man contrbuton of ths paper s thus to dere for the settng descrbed aboe a complete characterzaton of Bayes-Nash ncente compatblty n terms of weak monotoncty and an addtonal ntegrablty condton. Thereby we achee a characterzaton that depends purely on the aluatons and the allocaton rule. The characterzaton resembles the one dered by Rochet 1987) for domnant strategy ncente compatblty. Howeer, our result does not follow from Rochet 1987) mmedately, as we coer nterdependent aluatons. 2 The Model and Basc Defntons There s a set of agents N = {1,..., n}. Each agent has a type t T wth T R k. T denotes the set of all type profles t = t 1,..., t n), and T denotes the set of all type profles t = t 1,..., t 1, t +1,..., t n). A payment rule s a functon P : T R n, so gen a report profle r of the others, reportng a type r results n a payment P r, r ) for agent. Denotng the set of outcomes by Γ, an allocaton rule s a functon f : T Γ. We allow for nterdependent aluatons across agents, that s, agents aluatons do not only depend on ther own types but on the types of all agents. As an example one can thnk of an aucton for a pantng see Klemperer, 1999) where agents types reflect how much they lke the pantng. An agent s aluaton for ownng the pantng depends on the types of the others as they affect the possble resale alue of the pantng and the owner s prestge. Take agent hang true type t and reportng r whle the others hae true types t and report r. The alue that agent assgns to the resultng allocaton s denoted by f r, r ) t, t ). Utltes are quas-lnear, that s, an agent s utlty s hs aluaton of an allocaton mnus hs payment. Agents types are ndependently dstrbuted. Let π denote the probablty densty on T. The jont densty π on T s then gen by π t ) = π j t j). j N j

5 Assume that agent belees all other agents to report truthfully. If agent has true type t, then hs expected utlty for makng a report r s gen by U r t ) = f r, t ) t, t ) P r, t )) π t ) dt T = E f r, t ) t, t ) P r, t )]. 1) We assume E f r, t ) t, t )] to be fnte r, t T. An allocaton rule f s Bayes-Nash ncente compatble f there exsts a payment rule P such that N and r, r T : Symmetrcally, we hae also E f r, t ) r, t ) P r, t )] E f r, t ) r, t ) P r, t )]. 2) E f r, t ) r, t ) P r, t )] E f r, t ) r, t ) P r, t )]. 3) By addng 2) and 3) we get the followng monotoncty condton: 1 Defnton 1 Weak Monotoncty) An allocaton rule f monotoncty f N and r, r T : E f r, t ) r, t ) f r, t ) r, t )] E f r, t ) r, t ) f r, t ) r, t )]. satsfes weak Ths condton s the expected utlty equalent to the weak monotoncty W- MON) condton of La et al. 2003), the non-decreasng n margnal utlty condton NDMU) of Bkhchandan et al. 2003) and the 2-cycle nequalty of Gu et al. 2004). The ratonale for namng the aboe condton weak monotoncty becomes edent once we consder aluaton functons that are lnear wth respect to agents types n Secton 4. Obously, weak monotoncty s a necessary condton for Bayes-Nash ncente compatblty. In Secton 4 we present a settng where weak monotoncty together wth an ntegrablty condton s also a suffcent condton. 3 A Network Interpretaton We begn ths secton by brefly reewng a well-known result from the feld of network flow theory. 2 Let X = {x 1,..., x k } be a fnte set of arables. Consder the followng system of constrants: x x j w j, j {1,..., k}, 4) 1 Expected payments cancel snce we work under the assumpton of ndependently dstrbuted types. 2 A comprehense ntroducton to network flows can be found n Ahuja et al. 1993).

6 where w j s some constant specfc to the ordered par, j). The system can be assocated wth a network by constructng a drected, weghted graph whose nodes correspond to the arables. A drected edge s put between each ordered par of nodes. The length of the edge from the node correspondng to x to the node correspondng to x j s gen by w j. It s a well-known result see e.g. Shostak, 1981) that the system of lnear nequaltes n 4) s feasble, that s, there exsts an assgnment of real alues to the arables such that the constrants n 4) are satsfed, f and only f there s no negate length cycle n the assocated network. Furthermore, f the system s feasble then one feasble soluton s to assgn to each x the length of a shortest path from the node assocated wth x to some arbtrary source node. 3 In order to see that the constrants n 2) hae a natural network nterpretaton t s useful to rewrte 2) as follows: E P r, t ) P r, t )] E f r, t ) r, t ) f r, t ) r, t )]. 5) Consderng a specfc allocaton rule, the rght-hand sde of 5) s a constant. Thus, we hae a system of dfference constrants as descrbed n 4) except that we are now dealng wth a potentally nfnte number of arables). Gen ths obseraton, we assocate the system of nequaltes 5) wth a network n the same way as s descrbed aboe. For each agent we buld a complete drected graph Tf. A node s assocated wth each type and a drected edge s put between each ordered par of nodes. For agent the length of an edge drected from r to r s denoted l r, r ) and s defned as the cost of manpulaton: l r, r ) = E f r, t ) r, t ) f r, t ) r, t )]. 6) Gen our preous assumptons, the edge length s fnte. For techncal reasons we allow for loops. Howeer, note that an edge drected from r to r has length l r, r ) = 0. Usng ths defnton of the edge lengths, the weak monotoncty condton can be wrtten as l r, r ) + l r, r ) 0 N, r, r T. So weak monotoncty corresponds to the absence of negate length 2-cycles n the graphs descrbed aboe. Rochet 1987) obsered that domnant strategy ncente compatblty can be characterzed n terms of the absence of fnte, negate length cycles n smlar graphs. Usng the same proof technque, we can dere such a characterzaton for Bayes-Nash ncente compatblty as well. 3 In order to be consstent wth the exstng lterature we defned the system of constrants as n 4). Howeer, n network theory the constrants are commonly defned as x j x w j. In ths case, f the system s feasble then one feasble soluton s to assgn to each x the length of a shortest path from some arbtrary source node to the node assocated wth x.

7 Theorem 1 An allocaton rule f s Bayes-Nash ncente compatble f and only f there s no fnte, negate length cycle n Tf, N. Proof Adapted from Rochet, 1987.) Take some agent and let C = r1,..., rm, rm+1 = r1) denote a fnte cycle n Tf. Let us assume that f s Bayes-Nash ncente compatble. Ths mples, usng 5) and the edge length defnton 6), that for eery j {1,..., m}, E P r j, t ) P r j+1, t )] l r j, r j+1). Addng up these nequaltes yelds 0 l rj, rj+1), so C has non-negate length. Conersely, let us assume that there exsts no fnte, negate length cycle n Tf, N. For each agent we pck an arbtrary source node r 0 T and defne r T p r ) = nf l rj, rj+1), where the nfmum s taken oer all fnte paths A = ) r1 = r,..., rm+1 = r0 n Tf, that s, all fnte paths that start at r and end at r0. Absence of fnte, negate length cycles mples that p r0) = 0. Furthermore, r T we hae p r0 ) p r ) + l r0, r ) whch mples that p r ) s fnte. For eery par r, r T we also hae p r ) p r ) + l r, r ). Thus, by settng 4 P r, t ) = p r ), t T, and usng 6) we get E P r, t ) P r, t )] E f r, t ) r, t ) f r, t ) r, t )]. Hence, the constrants n 5) are satsfed and f s Bayes-Nash ncente compatble. Let us conclude ths secton wth a condton for the costs of manpulaton that s used n the deraton of the characterzaton theorem presented n the followng secton. 4 Note that t s suffcent f P s set such that E ˆP `r, t = p `r + c. Ths allows for a arety of payment rules yeldng the same expected payments up to an addte constant.

8 Defnton 2 Decomposton Monotoncty) The costs of manpulaton are decomposton monotone f r, r T and r T s.t. r = 1 α)r +α r, α 0, 1) we hae l r, r ) l r, r ) + l r, r ). So lookng at a par of nodes, f decomposton monotoncty holds then the drect edge between those nodes s at least as long as any path connectng the same two nodes a nodes lyng on the lne segment between them. 4 Weak Monotoncty and Path Independence In ths secton we restrct the rather general settng presented n Secton 2. We assume that T s conex for each agent. Furthermore, we now assume that an agent s aluaton functon s lnear n hs own true type. So f agent has true type t and reports r whle the others hae true types t and report r, hs aluaton for the resultng allocaton s f r, r ) t, t ) = α f r, r ) t ) + f r, r ) t ) t. 7) Note that α : Γ T R and : Γ T R k,.e. α assgns to eery γ, t ) Γ T a alue n R, whereas assgns to eery γ, t ) Γ T a ector n R k. Smlarly, assumng he belees all other agents to report truthfully, agent s expected aluaton for reportng r whle hang true type t s E f r, t ) t, t )] = E α f r, t ) t )] + E f r, t ) t )] t. 8) Usng 8), the weak monotoncty condton becomes: N, r, r T E f r, t ) t ) f r, t ) t )] r r ) 0. 9) In ths restrcted settng weak monotoncty mples that the costs of manpulaton are decomposton monotone: Lemma 1 Suppose that eery agent has a aluaton functon whch s lnear n hs true type: If f satsfes weak monotoncty then the costs of manpulaton are decomposton monotone. Proof Take some agent and let r, r T. Let r T such that r = 1 α)r + α r for some α 0, 1). Weak monotoncty mples that E f r, t ) t ) f r, t ) t )] r r ) 0. Note that r r s proportonal to r r, specfcally r r = α 1 α r r ). Snce α 0, 1), the aboe nequalty mples that E f r, t ) t ) f r, t ) t )] r r ) 0.

9 Addng E f r, t ) t ) f r, t ) t )] r on both sdes of the latter nequalty and rearrangng terms yelds E f r, t ) t ) f r, t ) t )] r +E f r, t ) t ) f r, t ) t )] r E f r, t ) t ) f r, t ) t )] r +E f r, t ) t ) f r, t ) t )] r. Notce that the frst and the last term on the left-hand sde of the nequalty cancel. Hence, usng 6), the aboe can be wrtten as l r, r ) l r, r ) + l r, r ), so the costs of manpulaton are decomposton monotone. It can be shown Müller et al. 2005) that f agents type spaces are onedmensonal then weak monotoncty s a suffcent condton for Bayes-Nash ncente compatblty. Unfortunately, f type spaces are mult-dmensonal then weak monotoncty alone s not suffcent anymore as s llustrated n Müller et al. 2005). Howeer, n the followng we are gong to show that weak monotoncty together wth an ntegrablty condton s suffcent. Defnton 3 Path Independence) Let ψ: T R k be a ector feld. ψ s called path ndependent f for any two r, r T the path ntegral of ψ from r to r r r,s s ndependent of the path of ntegraton S. Note that E f r, t ) t )] s a ector feld T R k. Theorem 2 Suppose that eery agent has a conex type space and a aluaton functon whch s lnear n hs true type. Then the followng statements are equalent: 1) f s Bayes-Nash ncente compatble. 2) f satsfes weak monotoncty and for eery agent, E f r, t ) t )] s path ndependent. 5 Proof 1) 2): Let us assume that f s Bayes-Nash ncente compatble. As mentoned n Secton 2, the necessty of weak monotoncty follows trally. Furthermore, from Theorem 1 t follows that for eery agent the graph Tf has no 5 That weak monotoncty of f and path ndependence of E ˆ `f `r, t t do not mply one another s llustrated n Müller et al ψ

10 fnte, negate length cycles. Let C = r1,..., rm, rm+1 = r1) denote a fnte cycle n Tf. Absence of fnte, negate length cycles mples that l rj, rj+1) 0 whch can be rewrtten usng 6) and 8) as E f rj, t ) t ) f rj+1, t ) t )] rj 0. Ths mples that E f rj+1, t ) t )] rj+1 rj) 0. Thus, E f r, t ) t )] s cyclcally monotone. 6 From Rockafellar 1970), Theorem 24.8, t follows that there exsts a conex functon ϕ: T R such that E f r, t ) t )] s a selecton from ts subdfferental mappng, that s, E f r, t ) t )] ϕ r ), r T. Ths mples see Krshna and Maenner, 2001, Theorem 1) that for any smooth path S n T jonng r and r the followng holds: r r,s E f r, t ) t )] = ϕ r ) ϕ r ), so E f r, t ) t )] s path ndependent. 2) 1): Let us assume that f satsfes weak monotoncty and that for eery agent, E f r, t ) t )] s path ndependent. Take any edge from Tf and denote ts startng node r and ts endng node r. Let L denote the lne segment between r and r,.e. L = { r T r = 1 α)r + α r, α 0, 1] }. Now we pck any r L and substtute the orgnal edge wth the path A = r, r, r ) whch has length l r, r ) + l r, r ). By Lemma 1 we hae l r, r ) l r, r ) + l r, r ), 10) that s, the orgnal edge s at least as long as the path A. By repeated substtuton we can generate a new path à = r1 = r,..., rm, rm+1 = r ) where rj L, j {1,..., m + 1}. Then 10) mples that the orgnal edge s at least as long as Ã, that s, l r, r ) l rj, rj+1). 6 The noton of cyclcal monotoncty was ntroduced by Rockafellar 1966).

11 Note that = l rj, rj+1 ) E f rj, t ) rj, t ) f rj+1, t ) rj, t )] = E f r 1, t ) r 1, t ) f r m+1, t ) r m, t )] E f rj+1, t ) rj+1, t ) f rj+1, t ) rj, t )] m 1 + = E f r1, t ) r1, t ) f rm+1, t ) rm+1, t )] + E f rj+1, t ) rj+1, t ) f rj+1, t ) rj, t )] = E f r, t ) r, t ) f r, t ) r, t )] + E f rj+1, t ) t )] rj+1 rj). The frst equalty follows from the defnton of the edge length gen n 6). The second equalty follows from rearrangng the terms of the summaton. The thrd equalty s dered by addng and subtractng E f rm+1, t ) rm+1, t )]. To dere the last equalty we use 8) and that r1 = r, rm+1 = r. By repeated substtuton we can generate paths wth more and more edges. In the lmt the dstance between neghborng nodes goes to zero and E f rj+1, t ) t )] rj+1 rj ) r E f r, t ) t )]. r,l Thus, the length of à goes to E f r, t ) r, t ) f r, t ) r, t )] + r r,l E f r, t ) t )], 11) as m. Now, let C = r1,..., rm, rm+1 = r1) denote a fnte cycle n T f. Furthermore, let L j denote the lne segment between rj and r j+1. The result n 11) and the path ndependence of E f r, t ) t )] mply for the

12 length of C that l rj, rj+1 ) E f rj, t ) rj, t ) f rj+1, t ) rj+1, t )] + = 0, r j+1 r j,lj E f r, t ) t )] that s, C has non-negate length. In order to see the equalty relaton, note the followng: the terms of the frst summaton cancel each other out. Furthermore, the second summaton descrbes an ntegral oer a closed path n T whch, due to path ndependence, equals zero. If f s Bayes-Nash ncente compatble, the correspondng payments can be constructed by usng shortest path lengths as descrbed n the proof of Theorem 1). For each N, let us pck some a as the source node n T f. Thus, f agent reports t, he has to make a payment P t ) = nf l rj, rj+1), 12) where the nfmum s taken oer all fnte paths from t to a. Take any fnte path A = r 1 = t,..., r m+1 = a ) n T f. Let L j denote the lne segment between r j and r j+1, whereas L t denotes the lne segment between the source and t. Followng the repeated substtuton approach presented n the second part of the proof of Theorem 2, we can construct paths that are shorter or as long) by lettng them st the same nodes as A and also addtonal nodes along the lne segments n between. In the lmt, as the number of nodes goes to nfnty, the dstance between neghborng nodes goes to zero and the length of the paths goes to E f rj, t ) rj, t ) f rj+1, t ) rj+1, t )] r j+1 + E f r, t ) t )] ). 13) rj,lj

13 Usng path ndependence n 13) we hae that 7 r j+1 r j,lj E f r, t ) t )] = Applyng the aboe to 12) yelds a t,l t E f r, t ) t )]. P t ) = E f t, t ) t, t ) f a, t ) a, t )] t a,l t E f r, t ) t )], 14) mplyng that the expected utlty see 1) for defnton) for truthfully reportng t s 8 U t t ) = U a a ) t + E f r, t ) t )]. 15) a,l t 5 Acknowledgements The authors are grateful to the partcpants of the Second World Congress of the Game Theory 2004) and the Frst Span Italy Netherlands Meetng on Game Theory 2005) for helpful dscussons. We especally thank Phlp J. Reny, Rakesh V. Vohra and an anonymous assocate edtor of Games and Economc Behaor for ther useful comments. References 1] R. K. Ahuja, T. L. Magnant, and J. B. Orln. Network Flows: Theory, Algorthms and Applcatons. Prentce-Hall, New Jersey, ] S. Bkhchandan, S. Chatterj, and A. Sen. Incente compatblty n multunt auctons. Lene s Bblography, , UCLA Department of Economcs, ] H. Gu, R. Müller, and R. V. Vohra. Domnant strategy mechansms wth multdmensonal types. METEOR Research Memorandum, 04/046, ] P. Jehel and B. Moldoanu. Effcent desgn wth nterdependent aluatons. Econometrca, 695): , ] P. Jehel, B. Moldoanu, and E. Stacchett. Multdmensonal mechansm desgn for auctons wth externaltes. Journal of Economc Theory, 852): , The lne segment L t for the path of ntegraton s pcked for conenence. Due to path ndependence, t can be replaced wth any other path connectng the source and t. 8 In order to dere 15) one can use that by constructon P `a = 0 and thus add ths term to the rght-hand sde of 14).

14 6] P. Klemperer. Aucton theory: A Gude to the lterature. Journal of Economc Sureys, 133): , ] V. Krshna and E. Maenner. Conex potentals wth an applcaton to mechansms desgn. Econometrca, 694): , ] R. La, A. Mu alem, and N. Nsan. Towards a characterzaton of truthful combnatoral auctons. In Proc. 44th Annual IEEE Symposum on Foundatons of Computer Scence FOCS-2003). IEEE Computer Socety, ] R. Müller, A. Perea, and S. Wolf. Weak monotoncty and Bayes-Nash ncente compatblty. METEOR Research Memorandum, 05/040, ] K. Roberts. The characterzaton of mplementable choce rules. In J. J. Laffont Ed.), Aggregaton and Reelaton of Preferences. North-Holland, Amsterdam, ] J.-C. Rochet. A necessary and suffcent condton for ratonalzablty n a quas-lnear context. Journal of Mathematcal Economcs, 162): , ] R. T. Rockafellar. Characterzaton of the subdfferentals of conex functons. Pacfc Journal of Mathematcs, 173): , ] R. T. Rockafellar. Conex Analyss. Prnceton Unersty Press, Prnceton, ] M. Saks and L. Yu. Weak monotoncty suffces for truthfulness on conex domans. In Proc. 6th ACM Conference on Electronc Commerce EC-2005). ACM Press, ] P. Shostak. Decdng lnear nequaltes by computng loop resdues. Journal of the ACM, 284): , Rudolf Müller Department of Quanttate Economcs, Unersty Maastrcht P.O. Box 616, 6200 MD Maastrcht, The Netherlands Emal: r.muller@ke.unmaas.nl Andrés Perea Department of Quanttate Economcs, Unersty Maastrcht P.O. Box 616, 6200 MD Maastrcht, The Netherlands Emal: a.perea@ke.unmaas.nl Sascha Wolf Department of Quanttate Economcs, Unersty Maastrcht P.O. Box 616, 6200 MD Maastrcht, The Netherlands Emal: s.wolf@ke.unmaas.nl