Non-bossy Single Object Auctions

Transcription

1 Non-bossy Sngle Object Auctons Debass Mshra and Abdul Quadr January 3, 2014 Abstract We study determnstc sngle object auctons n the prvate values envronment. We show that an allocaton rule s mplementable (n domnant strateges) and non-bossy f and only f t s a strongly ratonalzable allocaton rule. Wth a mld contnuty condton, we show that an allocaton rule s mplementable and non-bossy f and only f t s a smple utlty maxmzer (wth approprate te-breakng). All our characterzatons extend the semnal result of Roberts (1979) from the unrestrcted doman to the restrcted doman of sngle object auctons. JEL Classfcaton Codes: D44 Keywords: sngle object aucton; mplementaton n domnant strateges; ratonalzablty; non-bossness. An earler verson of ths paper was crculated under the ttle Determnstc Sngle Object Auctons wth Prvate Values. We are grateful to an anonymous referee for hs thoughtful comments. We also thank Sushl Bkhchandan, Drk Bergemann, Shurojt Chatterj, Rahul Deb, Johannes Horner, Matthew Jackson, Vjay Krshna, Takash Kunmuto, Rchard McLean, Herve Mouln, Mallesh Pa, Davd Parkes, Tm Roughgarden, Souvk Roy, Arunava Sen, Dres Vermulen, and numerous semnar audence for useful comments and suggestons. Indan Statstcal Insttute; Correspondng author. Emal: dmshra@sd.ac.n. Indan Statstcal Insttute. 1

2 1 Introducton We study sngle object auctons n the prvate values model. We restrct attenton to determnstc sngle object auctons,.e., auctons where the probablty of allocatng the object to any agent s ether zero or one. An allocaton rule for sngle object aucton s mplementable f we can fnd payments such that truth-tellng s a domnant strategy for every agent. A central result n mechansm desgn s that the effcent allocaton rule n the sngle object aucton prvate values model s mplementable usng the Vckrey aucton (Vckrey, 1961; Clarke, 1971; Groves, 1973). On the other hand, a revenue maxmzng aucton n the ndependent prvate values model maxmzes the vrtual valuatons of the agents (Myerson, 1981). Englsh aucton wth a reserve prce s popular n practce (seen on EBay and other Internet stes) and n theory, for nstance, n desgnng approxmately optmal auctons (Hartlne and Roughgarden, 2009; Dhangwatnota et al., 2010). Such an aucton mplements a constraned effcent allocaton rule wth a reserve prce - t does not allocate the object f the valuaton of each bdder s less than the reserve prce, but when t allocates the object t does so to the hghest bdder. Whle the set of mplementable allocaton rules s qute rch, we encounter only these partcular smple class of mplementable allocaton rules n theory and practce. Hence, t s mportant to understand how these allocaton rules dstngush themselves from the remanng mplementable allocaton rules. A prmary motvaton of ths paper s to carry out a systematc analyss of ths queston axomatcally. Common features of all these auctons are that the allocaton rules are determnstc, domnant strategy mplementable, and nvolve maxmzaton of some form. If tes n these maxmzatons are broken carefully, then the allocaton rules mentoned above satsfy another appealng property - non-bossness. Non-bossness s the followng requrement. Suppose agent s not wnnng the object at a partcular valuaton profle (v, v ) and we go to another valuaton profle (v, v ), where the valuaton of only agent changes, such that agent stll does not wn the object. Then, the agent who was wnnng the object at the valuaton profle (v, v ) contnues to wn the object at (v, v ). In other words, f an agent cannot change hs own outcome, then t cannot change the outcome of any other agent. 1 We provde a complete characterzaton of mplementable and non-bossy allocaton rules. For ths characterzaton, we ntroduce a novel noton of ratonalzablty n the sngle object allocaton model, and use t to defne a class of allocaton rules that we call the strongly rato- 1 The use of non-bossness axom n socal choce theory wth prvate good allocatons, specally matchng problems, s extensve - t was frst used by Satterthwate and Sonnenschen (1981), and subsequently n matchng problems (Svensson, 1999; Papa, 2000; Ehlers, 2002; Hatfeld, 2009) and cost sharng problem (Mutuswam, 2005). 2

3 nalzable allocaton rules. Our characterzaton says that an allocaton rule s mplementable and non-bossy f and only f t s a strongly ratonalzable allocaton rule. Under a mld contnuty condton, we sharpen our characterzaton. We defne the noton of a smple utlty functon, whch s any non-decreasng functon that maps the set of possble valuatons of an agent to the set of real numbers. A smple utlty maxmzer s an allocaton rule that chooses a smple utlty functon for every agent. Then, at every valuaton profle (a) t does not allocate the object f every agent has negatve smple utlty and (b) f at least one agent has postve smple utlty, then t allocates the object to an agent wth the hghest smple utlty. We show that f an allocaton rule satsfes a mld contnuty condton, then t s mplementable and non-bossy f and only f t s a smple utlty maxmzer allocaton rule (supplemented wth an approprate te-breakng rule). All the commonly used allocaton rules n sngle object auctons (e.g., effcent allocaton rule, effcent allocaton rule wth a reserve prce, the optmal aucton allocaton rule n Myerson (1981)) are smple utlty maxmzer allocaton rules. Hence, our results provde an axomatc foundaton for a rch class of commonly used allocaton rules. Although we characterze mplementable and non-bossy allocaton rules, usng revenue equvalence (Myerson, 1981), we can pn down the payments that wll mplement these allocaton rules. Thus, we get a complete characterzaton of mechansms that use non-bossy allocaton rules. Our characterzatons have a common feature - mplementablty and non-bossness s equvalent to some form of maxmzaton by the seller at every valuaton profle. These results relate to two fundamental results n mechansm desgn and aucton theory. A benchmark result n prvate value mechansm desgn n quas-lnear envronments s the Roberts affne maxmzer theorem (Roberts, 1979). It consders general multdmensonal type spaces wth fnte set of alternatves. A type of an agent n such models s a vector n R A, where A s the set of alternatves. Roberts (1979) showed that f there are at least three alternatves and the type space s unrestrcted (.e., R A ), then every onto mplementable allocaton rule s an affne maxmzer. It can be shown that every affne maxmzer s mplementable. 2 An affne maxmzer can be thought to be a lnear smple utlty functon. The sngle object aucton model has a restrcted type space. As a result, Roberts result does not apply. Our characterzatons can be thought as extenson of Roberts affne maxmzer result to the sngle object aucton model. Further, n a semnal result, Border (1991) showed that the nterm allocaton probablty obtaned by every Bayesan and randomzed allocaton rule can be obtaned by takng con- 2 Carbajal et al. (2012) show that f there are at least three alternatves and the type space of every agent s unrestrcted, then an onto allocaton rule s mplementable f and only f t s a lexcographc affne maxmzer. Lexcographc affne maxmzers contan a partcular class of affne maxmzers where tes are broken carefully. 3

4 vex combnaton of certan domnant strategy mplementable allocaton rules that he called herarchcal allocaton rules - see also Manell and Vncent (2010); Deb and Pa (2013). As we dscuss later, a herarchcal allocaton rule can be wrtten as a convex combnaton of smple utlty maxmzer allocaton rules that we dentfy (whch are determnstc, domnant strategy mplementable, and non-bossy allocaton rules). Hence, the set of domnant strategy mplementable and non-bossy determnstc allocaton rules occupy a pvotal role n the set of all randomzed and Bayesan mplementable allocaton rules. Fnally, we extend our dea of smple utlty maxmzer allocaton rule to defne an even larger class of allocaton rules that we call generalzed utlty maxmzer allocaton rules. We show that mplementablty s equvalent to these allocaton rules. Whle ths result s also n the sprt of Roberts affne maxmzer theorem, the proof s a smple consequence of Myerson s monotoncty characterzaton of mplementable allocaton rule, whch we dscuss below. Generalzed utlty maxmzers are more complex allocaton rules than smple utlty maxmzers. Ths shows how a natural condton lke non-bossness helps us to separate complex aucton rules from smple and commonly used aucton rules. 1.1 Relatonshp wth Lterature Myerson (1981) shows that mplementablty s equvalent to a monotoncty property of the allocaton rules. 3 The monotoncty property s equvalent to requrng that for every agent and for every valuaton profle of other agents, there s a cutoff valuaton of agent below whch he does not get the object and above whch he gets the object. 4 The relatonshp between our results and the monotoncty characterzaton can be best llustrated by reference to parallel results n the strategc votng lterature. Muller and Satterthwate (1977) show that Maskn monotoncty, the counterpart of monotoncty n the strategc votng models, s necessary for domnant strategy mplementaton, and f the doman s unrestrcted then t s also suffcent. However, the semnal results of Gbbard (1973) and Satterthwate (1975) show that dctatorshp s the only domnant strategy mplementable votng rule satsfyng unanmty. In the quas-lnear prvate values models, Roberts theorem can be thought of as the counterpart of the Gbbard-Satterthwate theorem (Gbbard, 1973; Satterthwate, 1975). After the result of Gbbard (1973) and Satterthwate (1975), a vast lterature n socal choce the- 3 See also extensons of ths characterzaton to the multdmensonal prvate values models n Bkhchandan et al. (2006); Saks and Yu (2005); Ashlag et al. (2010); Cuff et al. (2012); Mshra and Roy (2012). 4 The results n Myerson (1981) are more general. In partcular, he consders mplementaton n Bayes- Nash equlbrum and allows for randomzaton. But the expected revenue maxmzng allocaton rule he dentfes s a determnstc and domnant strategy mplementable allocaton rule. 4

5 ory has pursued the characterzaton of mplementable allocaton rules n restrcted votng domans, e.g., the medan votng rule and ts generalzatons characterze mplementable allocaton rules n sngle-peaked domans (Mouln, 1980; Barbera et al., 1993). Indeed, these characterzatons of mplementable allocaton rules are all n the sprt of Roberts theorem - they descrbe the precse parameters that are requred to desgn an mplementable allocaton rule. In ths sprt, our results gve explct characterzaton of mplementable allocaton rules for the sngle object aucton model. There have been extensons of Roberts theorem to certan envronments. For nstance, Mshra and Sen (2012) show that Roberts theorem holds n certan bounded but full dmensonal type spaces under an addtonal condton of neutralty. Ther neutralty condton s vacuous n the sngle object aucton model. Moreover, the type space n the sngle object aucton model s not full dmensonal. Carbajal et al. (2012) extend Roberts theorem to certan restrcted type spaces whch satsfy some techncal condtons. Though t covers many nterestng models, ncludng those wth nfnte set of alternatves, the sngle object aucton model does not satsfy ther techncal condtons. Marchant and Mshra (2012) extend Roberts theorem to the case of two alternatves. Snce the number of alternatves n the sngle object aucton model s more than two, ther results do not hold n our model. Jehel et al. (2008) show that a verson of the Roberts theorem holds even n the nterdependent values model (they requre mplementaton n ex-post equlbrum). They also requre the complete doman assumpton lke Roberts (1979), and remark that ther result does not hold n restrcted one-dmensonal settngs lke the sngle object aucton. Two related work n computer scence lterature deserve specal menton. Lav et al. (2003) focus on a partcular restrcted doman, whch they call order-based domans (ths ncludes some aucton domans). Under varous addtonal restrctons on the allocaton rule (whch ncludes an ndependence condton), they show that every mplementable allocaton rule must be an almost affne maxmzer - roughly, almost affne maxmzers are affne maxmzers for large enough values of types of agents. Next, Archer and Tardos (2002) consder the sngle object aucton model and show that f the object s always allocated then the only mplementable allocaton rules satsfyng nonbossness and three more addtonal condtons are mn functon allocaton rules. 5 functon allocaton rules are smple utlty maxmzer allocaton rules, but wth some addtonal lmtng and contnuty propertes. Though our characterzaton of smple utlty maxmzer s related to ther result, t has several mportant dfferences. Frst, ther result 5 Archer and Tardos (2002) consder a more general envronment than ours n whch a planner needs to select a path n a graph, where each edge represents an agent. Informally, ther three addtonal condtons are varous range and te-breakng condtons, and called edge autonomy, path autonomy, and senstvty. The non-bossy condton s called ndependence by them. Mn 5

6 requres that we always sell the good. Ths rules out any allocaton rule wth a reserve prce, such as Myerson s revenue maxmzng allocaton rule. Further, our proof shows that allowng the object to be not sold adds several non-trval complcatons n dervng our results. Second, they seem to requre dfferent types of range and te-breakng condtons than our contnuty requrement. On the other hand, our characterzaton of smple utlty maxmzer makes t explct the way tes need to be broken. Fnally, they have no analogue of our other characterzatons. There have been many smplfcatons of the orgnal proof of Roberts (Jehel et al., 2008; Lav, 2007; Dobznsk and Nsan, 2009; Vohra, 2011; Mshra and Sen, 2012). But none of these proofs show how Roberts theorem can be extended to a restrcted doman lke the sngle object aucton model. Unlke most of the lterature, our goal s not to characterze affne maxmzers - ndeed, all our characterzatons capture a larger class of mplementable allocaton rules than affne maxmzers. An alternate approach s to characterze the set of domnant strategy mechansms drectly by mposng condtons on mechansms rather than just on allocaton rules. A contrbuton along ths lne s Ashlag and Serzawa (2011). They show that any mechansm whch always allocates the object, satsfes ndvdual ratonalty, non-negatvty of payments, anonymty n net utlty, and domnant strategy ncentve compatblty must be the Vckrey aucton. Ths result s further strengthened by Mukherjee (2012), who shows that any strategy-proof and anonymous (n net utlty) mechansm whch always allocates the object must use the effcent allocaton rule. Further, Saka (2012) characterzes the Vckrey aucton wth a reserve prce usng varous axoms on the mechansm (ths ncludes an axom on the allocaton rule whch requres a weak verson of effcency). By placng mnmal axoms on allocaton rules, we are able to characterze a broader class of mechansms (usng revenue equvalence) than these papers. 2 The Sngle Object Aucton Model A seller s sellng an ndvsble object to n potental agents (buyers). The set of agents s denoted by N := {1,..., n}. The prvate value of agent for the object s denoted by v R ++. The set of all possble prvate values of agent s V R ++ - note that we do not allow zero valuatons. We wll use the usual notatons v and V to denote a profle of valuatons wthout agent and the set of all profles of valuatons wthout agent respectvely. Let V := V 1 V 2... V n. The set of alternatves s denoted by A := {e 0, e 1,...,e n }, where each e s a vector n R n. In partcular, e 0 s the zero vector n R n and e s the unt vector n R n wth -th component 1 and all other components zero. The j-th component of the vector e wll be denoted by 6

7 e j. The alternatve e 0 s the alternatve where the seller keeps the object and for every N, e s the alternatve where agent gets the object. Notce that our model focuses on determnstc alternatves. Every agent N gets zero value from any alternatve where he does not get the object. An allocaton rule s a mappng f : V A. For every v V and for every N, the notaton f (v) {0, 1} wll denote f agent gets the object (f (v) = 1) or not (f (v) = 0) at valuaton profle v n allocaton rule f. Payments are allowed and agents have quas-lnear utlty functons over payments. A payment rule of agent N s a mappng p : V R. Defnton 1 An allocaton rule f s mplementable (n domnant strateges) f there exst payment rules (p 1,...,p n ) such that for every agent N and for every v V v f (v, v ) p (v, v ) v f (v, v ) p (v, v ) v, v V. In ths case, we say (p 1,...,p n ) mplement f and the mechansm (f, p 1,...,p n ) s ncentve compatble. Notce that we focus on determnstc domnant strategy mplementaton. Myerson (1981) showed that the followng noton of monotoncty s equvalent to mplementablty - see also Laffont and Maskn (1980) for a smlar characterzaton. Defnton 2 An allocaton rule f s monotone f for every N, for every v V, and for every v, v V wth v < v and f (v, v ) = 1, we have f (v, v ) = 1. Myerson (1981) shows that an allocaton rule s mplementable f and only f t s monotone - ths result does not requre any restrcton on the space of valuatons (see Vohra (2011), for nstance). Throughout the paper, our results wll be drven by the monotoncty condton. 3 Implementaton, Non-Bossness, and Ratonalzablty We now provde the man results of ths paper. We wll defne the noton of a non-bossy allocaton rule. Then, we wll provde a complete characterzaton of non-bossy and mplementable allocaton rules. Fnally, we wll add a mld contnuty-lke condton to sharpen ths characterzaton even further. The backbone of ths result s a noton of ratonalzablty n our model, and ths reveals an elegant structure of mplementable and non-bossy allocaton rules. We ntroduce ths dea of ratonalzablty n the sngle object auctons next. 7

8 3.1 Ratonalzablty To defne ratonalzablty n our context, we vew the mechansm desgner as a decson maker who s makng choces usng hs allocaton rule. Notce that at every profle of valuatons, by choosng an alternatve, the mechansm desgner assgns values to each agent - zero to all agents who do not get the object but postve value to the agent who gets the object. Denote by 1 v the vector of valuatons n R n +, where all the components except agent has zero and the component correspondng to agent has v. Further, denote by 1 0 the n-dmensonal zero vector. For convenence, we wll wrte 1 0 as 1 v0 at any valuaton profle. Usng ths notaton, at a valuaton profle (v 1,...,v n ), a mechansm desgner s choce of an alternatve n A can lead to the selecton of one of the followng (n + 1) vectors n R n + to be chosen - 1 v0,1 v1,...,1 vn. We wll refer to these vectors as utlty vectors. Any allocaton rule f can alternatvely thought of choosng utlty vectors at every valuaton profle. The doman of valuatons V of agent gves rse to a set of feasble utlty vectors where only agent gets postve value. In partcular defne for every N, D := {1 v : v V }. Further, let D 0 := {1 v0 } and V 0 = {0}. Denote by D := D 0 D 1 D 2... D n the set of all utlty vectors consstent wth the doman of profle of valuatons V. To defne the noton of a ratonal allocaton rule, we wll use orderngs (reflexve, complete, and transtve bnary relaton) on the set of utlty vectors D. For any orderng on D, let be the asymmetrc component of and be the symmetrc component of. A strct lnear orderng s an ant-symmetrc orderng wth no symmetrc component. An orderng on D s monotone f for every N, for every v, v V wth v > v, we have 1 v 1 v. Our noton of ratonal allocaton requres that at every profle of valuatons t must choose a maxmal element among the utlty vectors at that valuaton profle, where the maxmal element s defned usng a monotone orderng on D. An example wth three agents wll clarfy some of the concepts. Example 1 Let N = {1, 2, 3}. So, the set of alternatves s A = {e 0, e 1, e 2, e 3 }. Let V 1 = V 2 = V 3 = {1, 2, 3}. In that case, the utlty vectors are vectors n R 3 +. In partcular, D 0 contans the orgn, D 1 = {(1, 0, 0), (2, 0, 0), (3, 0, 0)}, D 2 = {(0, 1, 0), (0, 2, 0), (0, 3, 0)}, and D 3 = {(0, 0, 1), (0, 0, 2), (0, 0, 3)}. Fgures 1(a) and 1(b) show D 0, D 1, D 2, D 3 wth two valuaton profles (shown n dark crcles n each fgure). A valuaton profle corresponds to four ponts n D (D 0 D 1 D 2 D 3 ). The valuaton profle (v 1, v 2, v 3 ) correspondng to Fgure 1(a) s (2, 3, 1) (the correspondng utlty vectors are shown n dark blue dots n the fgure) and that correspondng to Fgure 1(b) s (2, 1, 1). Now, consder the followng orderng defned on D: (0, 0, 3) (0, 3, 0) (0, 2, 0) (3, 0, 0) (0, 0, 0) (0, 0, 2) (0, 1, 0) (2, 0, 0) (1, 0, 0) (0, 0, 1). Note that s 8

9 monotone. Consder an allocaton rule f, whch chooses the -maxmal utlty vector at every valuaton profle. For nstance, consder the utlty vectors correspondng to valuaton profle (2, 3, 1) (shown n Fgure 1(a)). The -maxmal utlty vector at ths valuaton profle s (0, 3, 0) and hence, f allocates the object to agent 2. Smlarly, consder the utlty vectors correspondng to valuaton profle (2, 1, 1) (shown n Fgure 1(b)). The -maxmal utlty vector at ths valuaton profle s (0, 0, 0) and hence, f does not allocate the object to any agent. We call such allocaton rules ratonalzable allocaton rules. (0,0,3) (0,0,3) (0,0,2) D_3 (0,0,2) D_3 (0,0,1) (0,0,1) D_0 D_0 (1,0,0) (2,0,0) (3,0,0) (1,0,0) (2,0,0) (3,0,0) (0,1,0) D_1 (0,1,0) D_1 (0,2,0) D_2 (0,2,0) D_2 (0,3,0) (0,3,0) (a) (b) Fgure 1: Illustraton of ratonalzable allocaton rule We now formally defne a ratonalzable allocaton rule. For every allocaton rule f, let G f : V D be a socal welfare functon nduced by f,.e., for all v V, G f (v) = 1 vj f f(v) = e j for any j {0, 1,..., n}. Defnton 3 An allocaton rule f s ratonalzable f there exsts a monotone orderng on D such that for all v V, G f (v) 1 vj for all j {0, 1,..., n}. In ths case, we say ratonalzes f. An allocaton rule f s strongly ratonalzable f there exsts a monotone strct lnear orderng on D such that for all v V, 1 v 1 vj for all j {0, 1,..., n} \ {}, where G f (v) = 1 v. In ths case, we say strongly ratonalzes f. We wll nvestgate the relatonshp between (strongly) ratonalzable allocaton rules and mplementable allocaton rules. The followng lemma establshes that a ratonal allocaton rule s mplementable. Lemma 1 Every ratonalzable allocaton rule s mplementable. Proof : Consder a ratonalzable allocaton rule f and let be the correspondng orderng on D. Fx an agent and valuaton profle v. Consder two valuatons of agent : v and 9

10 v wth v < v wth f(v, v ) = e. By defnton of, 1 v 1 vj for all j (N {0}) \ {}. Snce s monotone, 1 v 1 v. By transtvty, 1 v 1 vj for all j (N {0}) \ {}. Then, by the defnton of, f(v, v ) = e. Hence, f s monotone, whch further mples that t s mplementable (Myerson, 1981). The converse of Lemma 1 s not true. The followng example establshes that. Example 2 Suppose there are two agents: N = {1, 2}. Suppose V 1 = V 2 = R ++. Consder an allocaton rule f defned as follows. At any valuaton profle (v 1, v 2 ), f max(v 1 2v 2, v 2 v 1 ) < 0, then f(v 1, v 2 ) = e 0. Else, f v 1 2v 2 < v 2 v 1, then f(v 1, v 2 ) = e 2 and f v 1 2v 2 v 2 v 1, then f(v 1, v 2 ) = e 1. It s easy to verfy that f s monotone, and hence, mplementable. We argue that f s not a ratonalzable allocaton rule. Assume for contradcton that f s a ratonalzable allocaton rule and s the correspondng monotone orderng. Consder the profle of valuaton (v 1, v 2 ), where v 1 = 1 and v 2 = 2. For ǫ > 0 but arbtrarly close to zero, f(v 1, v 2 ǫ) = e 2. Hence, 1 v2 ǫ 1 v0. By monotoncty, 1 v2 1 v0. Now, consder the profle of valuatons (v 1, v 2), where v 1 = 2 + ǫ and v 2 = 2. Note that f(v 1, v 2) = e 0. Hence, 1 v0 1 v2. Ths s a contradcton. A feature of ths example s that at valuaton profle (v 1, v 2 ), the allocaton rule was choosng e 2. But when valuaton of agent 1 changed to v 1, t chose e0 at valuaton profle (v 1, v 2). Hence, agent 1 could change the outcome wthout changng hs own outcome. As we show next, such allocaton rules are ncompatble wth ratonalzablty. 3.2 Non-bossy Sngle Object Auctons In ths secton, we wll show that the set of mplementable and non-bossy allocaton rules are characterzed by strongly ratonalzable allocaton rules. Defnton 4 An allocaton rule f s non-bossy f for every N, for every v V and for every v, v V wth f (v, v ) = f (v, v ), we have f(v, v ) = f(v, v ). Non-bossness requres that f an agent does not change hs own allocaton (.e., whether he s gettng the object or not) by changng hs valuaton, then he should not be able to change the allocaton of anyone. It was frst proposed by Satterthwate and Sonnenschen (1981). As dscussed n the ntroducton, t s a plausble condton to mpose n prvate good allocaton problems and has been extensvely used n the strategc socal choce theory lterature. We gve an example of a bossy and a non-bossy allocaton rule n Fgure 2(a) and Fgure 2(b) respectvely. These fgures ndcate a scenaro wth two agents. The possble outcomes 10

11 of the allocaton rules at dfferent valuaton profles are depcted n the Fgures. In Fgure 2(a), the allocaton rule s bossy snce f we start from a regon where alternatve e 2 s chosen and agent 1 ncreases hs value, then we can come to a regon where alternatve e 0 s chosen (.e., agent 1 can change the outcome wthout changng hs own outcome). However, such a problem s absent for the allocaton rule n Fgure 2(b). e_2 e_2 valuaton of agent 2 e_0 valuaton of agent 2 e_1 e_0 e_1 valuaton of agent 1 valuaton of agent 1 (a) (b) Fgure 2: Bossy and non-bossy allocaton rules Lemma 2 A strongly ratonalzable allocaton rule s non-bossy. Proof : Let f be a strongly ratonalzable allocaton rule wth beng the correspondng orderng on D. Fx an agent and v V. Consder v, v V such that f(v, v ) = e j e. By defnton, 1 vj 1 vk for all k (N {0}) \ {j}. Suppose f(v, v ) = e l e. By defnton, 1 vl 1 vk for all k (N {0}) \ {l}. Assume for contradcton e l e j. Then, we get that 1 vj 1 vl and 1 vl 1 vj, whch s a contradcton. Ths leads to the formal connecton between mplementablty and ratonalzablty. Theorem 1 An allocaton rule s mplementable and non-bossy f and only f t s strongly ratonalzable. The proof of Theorem 1 s n the appendx. Theorem 1 reveals a surprsng connecton between ratonalzablty and sngle object aucton desgn. Such a connecton of ratonalzablty and mechansm desgn was frst establshed n Mshra and Sen (2012). They consder general quas-lnear envronments wth prvate values. They show that f the type space s a multdmensonal open nterval, then every mplementable and neutral allocaton rule s ratonalzable. Note that ratonalzablty s weaker than strong ratonalzablty n the sense that t does not requre the underlyng orderng to be a strct lnear orderng. Our results depart from those n Mshra and Sen (2012) n many ways. Frst, as dscussed earler, ther 11

12 doman condton s not satsfed n our model, and neutralty s vacuous n the sngle object aucton models. Second, we show that mplementablty and non-bossness s equvalent to strong ratonalzablty. Mshra and Sen (2012) do not provde any such equvalence. Indeed, the non-bossness that we use, s a condton that s specfc to prvate good allocaton problems, and cannot be used n general mechansm desgn problems. Notce that Theorem 1 does not requre any restrcton on V. If the strct lnear orderng we constructed n the proof of Theorem 1 can be represented usng a utlty functon, then the characterzaton wll be even more drect. If for every agent N, V s fnte, then t s possble. But, as the next example llustrates, ths s not always possble. Example 3 Suppose N = {1, 2} and V 1 = V 2 = R ++. Consder the allocaton rule f such that for all valuaton profles (v 1, v 2 ), f(v 1, v 2 ) = e 1 f v 1 1, f(v 1, v 2 ) = e 2 f v 1 < 1 and v 2 1, and f(v 1, v 2 ) = e 0 otherwse. It can be verfed that f s mplementable (monotone) and non-bossy. By Theorem 1, f s strongly ratonalzable. Now, consder the strct lnear order defned n the proof of Theorem 1 that strongly ratonalzes f - denote t by f. If v 1 = v 2 = 1, we have f(v 1, v 2 ) = e 1. Hence, 1 v1 f 1 v2. Now, consder the followng defnton. Defnton 5 An orderng on the set D s separable f there exsts a countable set Z D such that for every x, y D wth x y, there exsts z Z such that x z y. It s well known that an orderng on D has a utlty representaton f and only f t s separable - the result goes back to at least Debreu (1954) (see also Fshburn (1970) for detals). We show that f s not separable. Consder v 1 = v 2 = 1. By defnton of f, 1 v1 f 1 v2 f 1 v0. Note that snce f s monotone, any utlty vector between 1 v1 and 1 v2 (accordng to f ) wll be of the form 1 v2 +ǫ or 1 v1 ǫ for some ǫ > 0. But, f(v 1, v 2 + ǫ) = e 2 mples that 1 v2 +ǫ f 1 v1 for all ǫ > 0. Also, f(v 1 ǫ, v 2 ) = e 2 mples that 1 v2 f 1 v1 ǫ for all ǫ > 0. Hence, there cannot exst z D such that 1 v1 f z f 1 v Smple Utlty Maxmzaton We saw that the strct lnear orderng that strongly ratonalzes an allocaton rule may not have a utlty representaton. The am of ths secton s to explore mnmal condtons that allow us to defne a new orderng for any mplementable and non-bossy allocaton rule whch has a utlty representaton. Ths allows us to sharpen our characterzaton, and relate t to a semnal result of Border (1991). Our extra condton s a contnuty condton. 12

13 Defnton 6 An allocaton rule f satsfes Condton C f for every, j N ( j) and for every v j, for every ǫ > 0, there exsts a δ ǫ,v j > 0 such that for every v, v j wth f(v, v j, v j ) = e, we have f(v + ǫ, v j + δ ǫ,v j, v j ) = e. Condton C requres some verson of contnuty of the allocaton rule. It says that f some agent s wnnng the object at a valuaton profle, for every ncrease n value of agent, there exsts some ncrease n value of agent j such that agent contnues to wn the object. Later, we provde an example to show that Condton C and non-bossness do not mply mplementablty. If f s monotone (mplementable) and non-bossy, then Condton C mples that for every, j N ( j) and for every v j, for every ǫ > 0, there exsts a δ ǫ,v j > 0 such that for every v, v j wth f(v, v j, v j ) = e, we have f(v + ǫ, v j + δ, v j ) = e for all 0 < δ < δ ǫ,v j. To see ths, choose some δ (0, δ ǫ,v j ) and assume for contradcton, f(v +ǫ, v j +δ, v j ) = e k for some k. If k = j, then by monotoncty, f(v + ǫ, v j + δ ǫ,v j, v j ) = e j, whch s a contradcton to Condton C. If k {, j}, then by non-bossness, f(v +ǫ, v j +δ ǫ,v j, v j ) {e j, e k }, agan a contradcton to Condton C. Snce we wll use Condton C along wth mplementablty and non-bossness, we can freely make use of ths mplcaton. We wll now ntroduce a new class of allocaton rules. Defnton 7 An allocaton rule f s a smple utlty maxmzer (SUM) f there exsts a non-decreasng functon U : V R for every N {0}, where U 0 (0) = 0, such that for every valuaton profle v V, f(v) = e j mples that j arg max N {0} U (v ). Notce that an SUM allocaton rule s smpler to state and, hence, more sutable for practcal use than a strongly ratonalzable allocaton rule. The am of ths secton s to show that the SUM allocaton rules are not much dfferent from the strongly ratonalzable allocaton rules. It can be easly seen that not every SUM allocaton rule s non-bossy. For nstance, consder the effcent allocaton rule that allocates the good to an agent wth the hghest value. Suppose there are three agents wth valuatons 10, 10, 8 respectvely and suppose that the effcent allocaton rule allocates the object to agent 1. Consder the valuaton profle (10, 10, 9) and suppose that the effcent allocaton rule now allocates the object to agent 2. Ths volates non-bossness. As we wll show that such volatons can happen n case of tes (as was the case here wth tes between agents 1 and 2), and when tes are broken carefully, an SUM allocaton rule becomes non-bossy. Smlarly, not every SUM allocaton rule s mplementable. For nstance, consder an example wth two agents {1, 2} wth V 1 = V 2 = R ++. Let U 1 (v 1 ) = 1 and U 1 (v 2 ) = v 2. Now, suppose we pck agent 1 as the wnner of the object at valuaton profle (1, 1) but pck agent 13

14 2 as the wnner of the object at valuaton profle (2, 1). Note that ths s consstent wth smple utlty maxmzaton but volates monotoncty, and hence, not mplementable. Now, consder the followng modfcaton of the SUM allocaton rule. Defnton 8 An allocaton rule f s a smple utlty maxmzer (SUM) wth orderbased te-breakng f there exsts a non-decreasng functon U : V R for every N {0}, where U 0 (0) = 0, and a monotone strct lnear orderng on D such that for every valuaton profle v V, f(v) = e j mples that j arg max N {0} U (v ) and 1 vj 1 vk for all k j and k arg max N {0} U (v ),.e., 1 vj s the unque smple utlty maxmzer accordng to. The te-breakng rule that we specfed s very general. It covers some ntutve tebreakng rules such as havng an orderng over N {0} and breakng the te n smple utlty maxmzaton usng ths orderng. Lemma 3 An SUM allocaton rule wth order-based te-breakng s mplementable. Proof : Suppose f s an SUM allocaton rule wth order-based te-breakng. Let the correspondng smple utlty functons be U 0, U 1,...,U n and be the orderng used to break tes. At any valuaton profle v, let W(v) = {j N {0} : U j (v j ) U k (v k ) k N {0}}. Fx an agent and the valuaton profle of other agents at v. Consder v, v such that v < v and f(v, v ) = e. Then, by SUM maxmzaton, W(v, v ). Further, by order-based te-breakng 1 v 1 vj for all j W(v, v ). Snce U s non-decreasng, U (v ) U j(v j ) for all j (N {0}) \ {}. Hence, W(v, v ). Agan, by order-based te-breakng, 1 v 1 v 1 vj for all j W(v, v ). Ths mples that f(v, v ) = e. So, f s monotone, and hence, mplementable. An SUM allocaton rule wth order-based te-breakng s also non-bossy. Lemma 4 An SUM allocaton rule wth order-based te-breakng s non-bossy. Proof : Let f be an SUM allocaton rule wth order-based te-breakng and v be a valuaton profle such that f(v) e j for some j N. Suppose f(v j, v j) e j. Then, by defnton, the unque smple utlty maxmzer of f remans the same n (v j, v j ) and (v j, v j ). So, f(v j, v j ) = f(v j, v j), and hence, f s non-bossy. We are now ready to state the man result of ths secton. 14

15 Theorem 2 Suppose V = (0, β ), where β R ++ { }, for all N and f s an allocaton rule satsfyng Condton C. Then, the followng statements are equvalent. 1. f s an mplementable and non-bossy allocaton rule. 2. f s a smple utlty maxmzer allocaton rule wth order-based te-breakng. The proof of Theorem 2 s gven n the Appendx. The non-trval part of the proof s to establsh that under Condton C, mplementablty and non-bossness mply smple utlty maxmzaton. Ths part of the proof s long and tedous, but reveals beautful structure of mplementable and non-bossy allocaton rules. Once ths s establshed, we use Theorem 1 to conclude how the tes must be broken. As we dscussed earler, the strct lnear orderng nduced by an mplementable and non-bossy allocaton rule on the set of utlty vectors D may not have a utlty representaton. Hence, we cannot nvoke Theorem 1 drectly to show Theorem 2. The proof of Theorem 2 constructs another orderng (whch s not a lnear order) and shows that ths has a utlty representaton under Condton C. We provde some remarks on Theorem 2 below. Some smple utlty maxmzers. An effcent allocaton rule s also an SUM allocaton rule, where U (v ) = v for all N and for all v V. Smlarly, we can defne for every N and for every v V, U (v ) = λ v + κ for some λ 0 and κ R, and ths SUM wll correspond to the affne maxmzer allocaton rules of Roberts (1979). The smple utlty functon n Myerson (1981) takes the form U (v ) = v 1 F (v ) f (v ), where F and f are respectvely the cumulatve densty functon and densty functon of the dstrbuton of valuaton of agent. Payments. It s well known that revenue equvalence (Myerson, 1981) mples that for any mplementable allocaton rule, the payments are determned unquely up to an addtve constant. Suppose V s an nterval for all N. For any mplementable allocaton rule f, defne the cutoff for agent and valuaton profle v as κ f (v ) = nf{α V : f(α, v ) = e }, where κ f (v ) = 0 f f(α, v ) e for all α V. It s well known that for every N and for every (v, v ) V, p f (v, v ) = κ f (v ) f f(v, v ) = e and p f (v, v ) = 0 f f(v, v ) e s a payment rule whch mplements f. Further, by revenue equvalence, any payment rule p whch mplements f must satsfy for every N and for every (v, v ), p (v, v ) = p f (v, v ) + h (v ), where h : V R s any functon. Thus, by characterzng mplementable allocaton rules, we characterze the class of domnant strategy ncentve compatble mechansms. 15

16 Other versons of non-bossness. Another verson of non-bossness, whch seems appealng s the utlty non-bossness. Utlty non-bossness s a condton on mechansms rather than on allocaton rules only. In partcular, an ncentve compatble mechansm (f, p) satsfes utlty non-bossness f for every N, for every v, and for every v, v V, such that v f (v, v ) p (v, v ) = v f (v, v ) p (v, v ), we have v j f j (v, v ) p j (v, v ) = v j f j (v, v ) p j (v, v ) for all j N. In words, f an agent changes hs valuaton such that hs net utlty does not change, then the net utlty of every agent must reman unchanged. We do not mpose such verson of utlty non-bossness because ths s a condton on mechansms, and we are nterested n condtons on allocaton rules. Further, utlty nonbossness s not satsfed by many canoncal mechansms. For nstance, the second-prce Vckrey aucton s not utlty non-bossy. To see ths, consder an example wth two agents wth valuatons 10 and 7 respectvely. Note that the allocaton rule n a second-prce Vckrey aucton s an effcent allocaton rule. The net utltes of agents 1 and 2 n the second-prce Vckrey aucton are 3 and 0 respectvely. Now, consder the valuaton profle (10, 8). At ths valuaton profle, agent 2 contnues to get zero net utlty n the second prce Vckrey aucton, but the net utlty of agent 1 s reduced to 2. Ths shows that the second-prce Vckrey aucton s not utlty non-bossy. On the other hand, the effcent allocaton rule wth order-based te-breakng s a non-bossy allocaton rule. Condton C. We gve an example of an allocaton rule whch s non-bossy and satsfes Condton C but not mplementable. The example llustrates that Condton C and nonbossness do not make mplementablty a redundant condton. In other words, these two condtons together are not stronger than monotoncty. Example 4 Let N = {1, 2}. Suppose V 1 = V 2 = R ++. Let U 1 (v 1 ) = v 1 and U 2 (v 2 ) = v 2. The allocaton rule f s defned as follows. It chooses e 0 (not allocatng the object) f U 1 (v 1 ) and U 2 (v 2 ) are less than 1. Else, t allocates the object to the agent wth the hghest U (v ), breakng tes n favor of agent 1. Formally, f max(u 1 (v 1 ), U 2 (v 2 )) 1, then f(v 1, v 2 ) = e 0. Else, f U 1 (v 1 ) max(u 2 (v 2 ), 1), then f(v 1, v 2 ) = e 1 and f U 2 (v 2 ) > U 1 (v 1 ) and U 2 (v 2 ) 1, then f(v 1, v 2 ) = e 2. Clearly, ths allocaton rule s not monotone, and hence, not mplementable. However, t s non-bossy and satsfes Condton C. 3.4 Randomzaton and Bayesan Implementaton va 16

17 Border s Herarchcal Allocaton Rules We relate our results to Border s herarchcal allocaton rules (Border, 1991). 6 Border consdered allocaton rules whch are not necessarly determnstc and Bayesan mplementable. To descrbe hs results, we consder randomzed allocaton rules n ths secton. A randomzed allocaton rule s a map f : V A, where A denotes the convex hull of the (n + 1) vectors {e 0, e 1,...,e n } n R n. Hence, f (v) wll now denote the probablty of agent gettng the object at valuaton profle v. Border (1991) consders ndependent prvate values settng. Each bdder has a probablty dstrbuton G usng whch t draws ts value from V. Denote by G (v ) j G j (v j ). The nterm allocaton probablty of an allocaton rule f for agent s a f (v ) = f (v, v )dg (v ). V Border also consders Bayesan mplementaton. An allocaton rule f s Bayesan mplementable f there exsts a payment rules (p 1,...,p n ) such that for every N, for every v, v V v a f (v ) p (v, v )dg (v ) v a f (v ) p (v, v )dg (v ). V V Defnton 9 An allocaton rule f h s a herarchcal allocaton rule f there exsts nondecreasng functons I : V R for all N such that at every valuaton profle v V { 1 f h {j N:I (v) = (v )=I j (v j f I )} (v ) 0 and I (v ) I j (v j ) for all j N 0 otherwse In a semnal result, Border showed that for every Bayesan mplementable allocaton rule f, there exst a set of herarchcal allocaton rules whose randomzaton gves the same nterm allocaton probablty as f - see also Manell and Vncent (2010); Merendorff (2011); Deb and Pa (2013). 7 Now, notce that a herarchcal allocaton rule s a randomzaton over smple utlty maxmzers (whch are determnstc allocaton rules). To see ths, we defne (n + 1)! order based te-breakng rules. Take any strct lnear orderng P of the set of alternatves n A. Defne an orderng on the set of utlty vectors D as follows. For any N, f 1 v,1 v D wth v > v, then 1 v 1 v. If e Pe j, then for every 1 v D and every 1 vj D j, 1 v 1 vj. Note that can be defned exactly (n + 1)! ways, one for each P. Let P be the set of all such orderngs of D. Now, gven a herarchcal allocaton rule wth (I 1,..., I n ), we can 6 I am grateful to Mallesh Pa for motvatng the contents of ths secton. 7 Although Border (1991) does not consder ncentve constrants, t s clear how hs results can be modfed n the presence of ncentve constrants. 17

18 construct (n + 1)! smple utlty maxmzers wth U = I for all N and takng as tebreakng rule one of the orderngs n P. Clearly, unform randomzaton over these smple utlty maxmzers produce the herarchcal allocaton rule. Hence, randomzaton over the herarchcal allocaton rules s equvalent to randomzaton over smple utlty maxmzers. Thus, smple utlty maxmzers occupy a central role n the theory of prvate value sngle object auctons. By characterzng smple utlty maxmzers, Theorem 2 ndrectly provdes an axomatc foundaton for Border s herarchcal allocaton rules. In partcular, the nterm allocaton probablty of any mplementable allocaton rule can be obtaned by randomzng over the set of mplementable and non-bossy allocaton rules satsfyng Condton C. 3.5 Extenson of Roberts Theorem Consder a general mechansm desgn set up wth prvate values and quas-lnear utlty. Let A be a fnte set of alternatves. Suppose A 3. The type of agent s denoted as v R A and v (a) denotes the valuaton of agent for alternatve a. Roberts (1979) shows that f type space of every agent s R A, then for every onto and mplementable allocaton rule f, there exsts λ 1,...,λ n 0, not all of them equal to zero, and κ : A R such that at every valuaton profle v, f(v) arg max [ λ v (a) + κ(a)]. a A N Such allocaton rules are called affne maxmzer allocaton rules. Theorems 1 and 2 can be thought of as the analogue of Roberts affne maxmzer theorem n the sngle object aucton model (under non-bossness). It shows how much the set of mplementable allocaton rule expands n a restrcted doman lke the sngle object aucton doman. 4 The Complete Characterzaton Theorems 1 and 2 characterze mplementable allocaton rules under addtonal assumptons. In ths secton, we drop these addtonal assumptons and provde a complete characterzaton of mplementable allocaton rules. These characterzatons are n the sprt of extendng the Roberts affne maxmzer theorem. In partcular, we show that an mplementable allocaton rule s equvalent to a generalzed utlty maxmzer allocaton rule. A generalzed utlty functon (GUF) of agent N s a functon u : V R. Notce that the generalzed utlty of an agent may be negatve also. Further, a smple utlty functon s a GUF. We wll need the followng verson of sngle crossng property. Defnton 10 The GUFs (u 1,...,u n ) satsfy top sngle crossng f for every N, for every v V, and for every v, v V wth v > v and u (v, v ) max(0, max k N\{} u k (v, v )), 18

19 we have u (v, v ) > max(0, max k N\{} u k (v, v )). The top sngle crossng condton s a very general nter-agent crossng condton. Such crossng condtons are extensvely used n the lterature of nterdependent value auctons - see for nstance, Cremer and McLean (1985); Maskn (1992); Dasgupta and Maskn (2000); Perry and Reny (2002). For the fnte type space, Cremer and McLean (1985) use condtons smlar to our top sngle crossng to establsh mplementaton (n ex post equlbrum) of the effcent allocaton rule n the nterdependent values model. The standard defnton of a sngle crossng property, whch mples top sngle crossng, s the followng. Defnton 11 GUFs (u 1,..., u n ) satsfy sngle crossng f for every, j N, for every v V, for every v, v V wth v > v, we have u (v, v ) u (v, v ) > u j (v, v ) u j (v, v ). A GUF u s ncreasng f for every v V and for every v, v V wth v > v we have u (v, v ) > u (v, v ). Lemma 5 If GUFs (u 1,...,u n ) satsfy sngle crossng and u s ncreasng for every N, then they satsfy top sngle crossng. Proof : Consder N and v V. Let v, v V such that v > v and u (v, v ) max(0, max k N\{} u k (v, v )). Snce u s ncreasng, u (v, v ) > u (v, v ) 0. Further, by sngle crossng, u (v, v ) u (v, v ) > u j (v, v ) u j (v, v ) for all j. Usng the fact that u (v, v ) u j (v, v ) for all j, we get that u (v, v ) > u j (v, v ) for all j. Hence, u (v, v ) > max(0, max k N\{} u k (v, v )). We are now ready to ntroduce a new class of mplementable allocaton rules. Defnton 12 An allocaton rule f s a generalzed utlty maxmzer f there exst GUFs (u 1,...,u n ) satsfyng top sngle crossng such that for every v V, f(v) = e mples that arg max N {0} u (v), where u 0 (v) = 0. Generalzed utlty maxmzers are mplementable. The proof s smlar to the proof n Cremer and McLean (1985), who establsh mplementaton (n ex post equlbrum) of effcent allocaton rule n an nterdependent values model. Lemma 6 If f s a generalzed utlty maxmzer, then t s mplementable. 19

20 Proof : Fx a generalzed utlty maxmzer f, and let (u 1,...,u n ) be the correspondng GUFs satsfyng top sngle crossng. Consder agent and v V. Also, consder any v, v V wth v > v and f(v, v ) = e. By defnton, u (v, v ) max(0, max k N\{} u k (v, v )). By top sngle crossng, u (v, v ) > max(0, max k N\{} u k (v, v )). Hence, f(v, v ) = e. So, f s monotone, and hence, mplementable. Ths leads to the man result of ths secton. Theorem 3 Suppose V R ++ s bounded for every N. Then, f s mplementable f and only f t s a generalzed utlty maxmzer. Proof : Lemma 6 showed that every GUF maxmzer s mplementable. Now, for the converse, suppose f s mplementable. Fx an agent N and v V. If f(v, v ) e for all v V, then defne κ f (v ) = sup{v : v V }. Else, defne κ f (v ) = nf{v V : f(v, v ) = e }. Snce V s bounded, κ f (v ) s well defned. Further, snce f s monotone, for every agent N, for every v, and for every v V, f v > κ f (v ), we have f(v, v ) = e and for every v < κ f (v ) we have f(v, v ) e. Defne for every N and for every (v, v ), u (v, v ) := v κ f (v ). By defnton, f f(v) = e, then v κ f (v ) 0 and v j κ f j (v j) 0 for all j. Hence, arg max k N {0} u k (v), where u 0 (v) = 0. To show that (u 1,...,u n ) satsfy top sngle crossng, consder N and v V. Let v, v V such that v > v and u (v, v ) max(0, max k N\{} u k (v, v )). Notce that u (v, v ) > u (v, v ) 0. By defnton of u 1,...,u n, f u (v, v ) > 0, then v > κ f (v ), and hence, f(v, v ) = e. But, ths mples that u k (v, v ) = v k κ f k (v k) 0 for all k. Hence, u (v, v ) > max(0, max k N\{} u k (v, v )). Our characterzaton of mplementablty shows that mplementablty s equvalent to maxmzng generalzed utltes. Generalzed utltes transform the orgnal valuaton of an agent to a new utlty, whch depends on the valuatons of all the agents. In contrast to smple utlty functons, generalzed utlty functons are much harder to construct. Ths llustrates how a natural axom lke non-bossness helps to smplfy the class of mplementable allocaton rules. Generalzed utlty maxmzers are smlar to mplementng the effcent allocaton rule n an nterdependent values model wth the qualfcaton that we allow generalzed utltes to be negatve, whch s precluded n the standard nterdependent value model. It s well known that the effcent allocaton rule s not generally mplementable n the nterdependent values sngle object aucton unless some nter agent crossng condton holds (Cremer and McLean, 1985; Maskn, 1992; Dasgupta and Maskn, 2000; Perry and Reny, 2002; Jehel et al., 2006). Our top sngle crossng condton s smlar to these condtons n the nterdependent values 20

21 lterature. Our result reveals a surprsng and nterestng connecton between these seemngly dfferent models. 5 Dscussons We conclude by dscussng some of the open questons that reman. Randomzaton and Bayesan Implementaton. Although we focus on determnstc domnant strategy mplementaton, randomzaton s a natural extenson of our model. Indeed, the monotoncty characterzaton of Myerson (1981) extends to sngle object auctons wth randomzaton. Extendng characterzatons of determnstc allocaton rules to randomzed allocaton rules present several challenges. A natural way to thnk of randomzaton s that of doman restrcton - the utlty from a lottery alternatve s restrcted to be the expected utlty from the determnstc alternatves n ts support. Thus, the challenges of gong from determnstc to randomzed allocaton rules s smlar to that of gong from a larger doman to a restrcted doman. For nstance, a counterpart of Roberts semnal result wth randomzaton s stll not known n the unrestrcted doman. However, we provded a relatonshp of our smple utlty maxmzer and Border s herarchcal allocaton rules that can be used to obtan nterm allocaton probablty of every Bayesan and randomzed allocaton rule. Hence, our characterzatons can be used n an ndrect way to characterze nterm allocaton probabltes of Bayesan mplementable randomzed allocaton rules. However, the drect characterzaton remans an open queston. Optmzng payments. A popular research theme n aucton theory and mechansm desgn s to optmze over the set of ncentve compatble mechansms. Ths usually nvolves optmzng over payments and assumes some pror dstrbuton over valuatons of agents by the mechansm desgner. The mplcatons of such optmzatons n the sngle object auctons s farly well understood. Clearly, our results do not contrbute to ths lterature. Our characterzatons are more talored towards understandng the nherent structure of determnstc sngle object auctons n prvate values set up. They completely descrbe the set of optons avalable to a mechansm desgner (wthout botherng about the dstrbutonal assumptons) n the sngle object auctons. Our man characterzatons provde axomatc foundatons to varous commonly used auctons. We also beleve that ths opens a door for carryng out smlar exercses n multdmensonal mechansm desgn models, ncludng the mult-object aucton model. The problem of fndng an expected revenue maxmzng mechansm n such models s consdered a df- 21