DISCRIMINATION VIA SYMMETRIC AUCTIONS: ONLINE APPENDIX

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1 DISCRIMINATION VIA SYMMETRIC AUCTIONS: ONLINE APPENDIX RAHUL DEB AND MALLESH M. PAI. IMPLEMENTATION USING RANDOMIZATION In ths secton, we demonstrate that f the mplementaton crteron s weakened, the prncpal can acheve the outcomes correspondng to certan non-herarchcal mechansms usng randomzaton. The prncpal can randomze by choosng amongst a set of mechansms va a lottery. After choosng one such mechansm from the set, the prncpal can announce t to the buyer. For nstance, the prncpal could toss a con and choose between a frst- and second-prce aucton. Havng chosen, the buyer s nformed of the aucton format and the game proceeds. Such randomzaton s approprate for a prncpal concerned about expected outcomes as n??). Randomzaton can be useful n achevng the outcomes of both unmplementable herarchcal allocaton mechansms and nonherarchcal mechansms. A smple two-buyer example of a nonherarchcal mechansm s one where rrespectve of the values, buyer gets the good 5% of the tme and buyer gets t 75%. Clearly, ths s not a herarchcal allocaton snce our defnton of the latter requres the equal breakng of tes. Another example s a mechansm n whch the seller randomly allocates the good 50% of the tme and runs a second prce aucton the remanng 50%. A mechansm a d, p d ) s defned to be a randomzaton over a set of mechansms M, f there s a measure ζ defned on M such that a dv ) = a v )dζa, p)) and p dv ) = p v )dζa, p)). M The lemma below shows that all IC and IR drect mechansms can be obtaned as a randomzaton over herarchcal mechansms. Ths lemma follows from results n? and?. Lemma. Every IC and IR drect mechansm s a randomzaton over the set of herarchcal mechansms. Proof. Defne the set of non-decreasng nterm allocaton rules acheved by some ndex rule as H M the set of all feasble, non-decreasng nterm allocaton rules by F M and the set of all feasble nterm allocaton rules by F. By feasble, we mean that ths nterm allocaton rule can result from some feasble ex-post allocaton rule. The proof follows from two observatons. Observaton. F M s a compact subset of L n n the weak/ weak topology σl n, Ln ). Proof. Lemma 8 of? shows that the set of feasble nterm allocaton rules F s a compact convex subset of L n n ths topology. M DEPARTMENT OF ECONOMICS, UNIVERSITY OF TORONTO DEPARTMENT OF ECONOMICS, UNIVERSITY OF PENNSYLVANIA E-mal addresses: rahul.deb@utoronto.ca, mallesh@econ.upenn.edu. Date: May 7, 06.

2 RAHUL DEB AND MALLESH M. PAI By observaton, F M s convex. We now argue that F M s also compact n ths topology. By the Eberlen-Smulan theorem Theorem 6.34,?), sequental compactness and compactness concde n ths topology. It s therefore enough to show that f for some sequence {a n } n= F M, a n a, then a F M. Snce each a n s monotone, t s a functon of bounded varaton and therefore by Helly s selecton theorem, there exsts a subsequence whch converges pontwse. Therefore a s also non-decreasng, and a F M, concludng our argument. Therefore, we have that the closure of the convex hull of H M s a subset of F M or convh M ) F M. Observaton. For any ndex functon I : V R n, there exsts a herarchcal allocaton rule a h H M whch solves ) max a F M a j v)i j v) f j v) dv. I-OPT-M) V j Proof. If I s non-decreasng,.e. I j v) s non-decreasng n v for each j N, then the soluton to I-OPT-M) s n H M. Ths follows easly from the defnton of herarchcal allocaton rule. Snce at every profle of values, the good s allotted to the buyer wth the hgher ndex, the rule maxmzes the ndex revenue profle-by-profle. Therefore t solves the maxmzaton problem I-OPT-M). So let us consder the soluton to I-OPT-M) for other ndex functons. We can re-wrte the problem as ) max a j v)i j v) f j v) a F V j a s non-decreasng. In ths case, we can relax the non-decreasng constrant nto the objectve functon. By the ronng procedure of?, there exsts an roned non-decreasng ndex rule Î such that the soluton to the above problem s the same as ) max a j v)î j v) f j v) a F V j Note that the correspondng herarchcal rule for ndex rule Î les n H M. dv, dv. To conclude the proof, suppose by way of contradcton that convh M ) F M. Then there exsts a F M such that a convh M ). By Corollary 7.47 of? there exsts an I L n such that a, I > max a, I, a convh M ) where a, I s the standard nner product ) V j a j v)i j v) f j v) dv.

3 DISCRIMINATION VIA SYMMETRIC AUCTIONS: ONLINE APPENDIX 3 By Observaton, the herarchcal allocaton rule correspondng to I solves I-OPT-M), mplyng that a, I > max a F M a, I. Snce a F M, ths s a contradcton. It follows that convh M ) = F M. The Lemma follows. Clearly, the outcome from any mechansm that s a randomzaton over mplementable herarchcal mechansms can be acheved n such an ex-ante sense. The auctoneer can just randomly choose usng measure ζ) from the symmetrc auctons that correspond to the mplementable herarchcal mechansms. Note that, strctly speakng, ths s not nterm mplementaton as we defned t. However, for practcal applcatons, t serves the same purpose, as randomzaton s done before the chosen symmetrc aucton s announced to the buyers. The next corollary summarzes ths dscusson, and n t, we use the termnology outcomes are achevable to clarfy the dstncton from nterm mplementaton. Corollary. The outcomes from an IC and IR drect mechansm are achevable f t s a randomzaton over mplementable herarchcal mechansms. Fnally, we dscuss the two examples of the unmplementable mechansms and examne whether ther outcomes can be acheved va randomzaton. EXAMPLE Contnued): Recall that, n ths example, the seller assgns the good at random wth equal probablty), buyer s never asked to pay anythng, whle buyer s always asked to pay 0.5. The outcome from the mechansm can be acheved by randomzng wth equal probablty over two mplementable herarchcal mechansms. In the frst herarchcal mechansm, buyer s always awarded the good rrespectve of value and s not asked to pay anythng. In the second herarchcal mechansm, buyer s always gven the good rrespectve of value and s asked to pay 0.5. EXAMPLE 3 Contnued): Recall that, n ths example, buyer wns the good f and only f her value exceeds that of buyer by. The outcome of ths mechansm cannot be acheved usng randomzaton. Consder the ndex functon I v) = I v + ) = v. By observaton, the allocaton rule a h correspondng to these ndex functons s the unque almost everywhere) maxmzer of V j {,} a d j v j, v j )I j v j ) f j v j ) dv, amongst all IC drect allocatons a d. Therefore, for any herarchcal allocaton rule ã h = a h that dffers from a h at a postve measure subset of values, t must be that V j {,} ã h j v j, v j )I j v j ) f j v j ) dv < V j {,} a h j v j, v j )I j v j ) f j v j ) dv.

4 4 RAHUL DEB AND MALLESH M. PAI Moreover, any allocaton rule that s equal to a h almost everywhere s not mplementable. Therefore, a h s not a randomzaton over mplementable herarchcal allocatons, so ts outcome s not achevable.. ADDITIONAL DESIDERATA In ths appendx, we buld on the results n Secton IV and show that restrctons n addton to symmetry can be placed on the admssble formats n order to constran the seller. We separately consder three such desderata contnuty, monotoncty of the payment rule n the bds and the requrement that the mplementaton have an ex-post IR equlbrum. Throughout ths appendx, we consder the case of two bdders n = ) prmarly for the sake of brevty and tractablty. A key takeaway from ths secton s that the optmal aucton s no longer always mplementable under these addtonal requrements. We provde a short summary of the results before the formal presentaton that follows. We show that a herarchcal mechansm has a contnuous mplementaton when, loosely speakng, the nterm allocaton s contnuous. A key consequence of ths s that the optmal aucton may not have a contnuous mplementaton. Ths s because, when the dstrbutons are rregular that s, φ ) s non-monotone), the optmal allocaton rule after ronng ) may be dscontnuous. We consder two types of monotoncty of the payment rule separately: the payments are ncreasng n your opponent s bd as n a second prce aucton) and n your own bd as n a frst prce aucton). In each of these cases, we show that the optmal aucton may not mplementable wth ths addtonal restrcton. In fact, ths s true even for the smple Example wth unform value dstrbutons. Fnally, for the case of ex-post IR mplementaton, we begn by observng that f an equlbrum bd s made by two dfferent values of the two bdders, the payment correspondng to ths bd can never be hgher than the lower of the two values. Ths provdes a smple necessary condton the complete characterzaton s far more complex) whch can be used to demonstrate that, once agan, the optmal aucton correspondng to the unform value dstrbuton case of Example does not have an ex-post IR mplementaton. We should note n advance that all the proofs that follow are wrtten dscussng the possblty or mpossblty of pure strategy mplementatons satsfyng the addtonal desderata. One may wonder about mxed strategy mplementatons. Observaton above showed that n any mxed strategy mplementaton, an at most probablty 0 set of values for any buyer can be mxng over bds that acheve dfferent probabltes of wnnng. It follows that to nduce the same nterm allocaton rule, any mxed strategy mplementaton must nduce the same dstrbuton over bds as some pure strategy ndex rule mplementng the allocaton rule. Therefore our results extend to mplementaton n mxed strateges as well. We omt the detals n the nterests of brevty... Contnuty The basc constructon we used n the example of Secton II conssted of dscontnuous payment rules where a buyer s payment dscontnuously changed dependng on whether ther opponent bd above or below the cutoff bd ˆb. Of course, condtonal on wnnng, the payments n frst- and Barrng Propostons 3 and 4, we can extend the results n ths secton to more than two bdders.

5 DISCRIMINATION VIA SYMMETRIC AUCTIONS: ONLINE APPENDIX 5 second-prce auctons are contnuous n the profle of bds snce losers do not pay, payments n these auctons are not contnuous uncondtonally). We now examne the effect that the addtonal requrement of contnuty has on the set of mplementable mechansms. A herarchcal mechansm has a symmetrc, contnuous mplementaton a s, p s ) f p s b, b j ) s contnuous n both b, b j. We show that the exstence of a contnuous mplementaton s equvalent to there beng no nontrval atoms n the herarchcal mechansm. A non-trval atom s one n whch there s a postve measure of values of buyer whch have the same ndex b, and ths ndex also les n the support of the bd space of buyer j. Formally, a nontrval atom exsts f, for a buyer, there are two dstnct values v, v V such that I v ) = I v ) = b and b B j. Such atoms can occur n natural applcatons such as the optmal aucton when the buyers value dstrbutons do not satsfy the monotone hazard rate condton. The absence of nontrval atoms s a necessary condton for a contnuous mplementaton. To see ths, note that, n any symmetrc mplementaton, at such an atom, t must be the case that σ v ) = σ v ) = σ jv j ) = b for all v j I j b). In other words, ths says that, n any mplementaton, t must be that all types at the nontrval atom make the same bd. However, f the payment p s s contnuous, buyer j has an ncentve to bd slghtly hgher than b. Bddng slghtly hgher would lead to a contnuous ncrease n payment but a dscontnuous ncrease n the probablty of wnnng, so σ j v j ) s not a best response for v j. The result below shows that ths s the only addtonal condton requred for the exstence of a contnuous mplementaton. Proposton. Suppose that n =. An mplementable herarchcal mechansm I, p h ) has a contnuous mplementaton f and only f t has no non-trval atoms. The ntuton for ths result can be easly seen by revstng the example of the regular optmal aucton of Secton II. When the vrtual values are ncreasng, the resultng optmal aucton does not have non-trval atoms. The payment rule we constructed was dscontnuous n the opponent s bd b j but can easly be smoothed around the pont of the dscontnuty whle ensurng that the nterm payments reman the same. The smplest way to do ths s lnearly, whch s llustrated below n Fgure. Addtonally, when the herarchcal mechansm has no non-trval atoms, t s also possble to acheve contnuty n b. The formal proof s constructve. Proof of Proposton. We argue suffcency frst. Suppose the herarchcal mechansm I, p h ) has no non-trval atoms. To begn, note that there exsts a herarchcal mechansm I, p h ) whch mplements exactly the same ex-post allocaton rule and nterm payment rule, such that I s strctly ncreasng for each =,. To see ths, consder any atom of buyer over bd b. By assumpton, I j b) = φ. Defne I v ) = I v ) + ɛ for =, and v s.t. I v ) > b and some ɛ > 0. Further, contnuously stretch the I v) for v I b) over [b, b + ɛ ] for some ɛ < ɛ. Proceed nductvely for each atom n I. Note that by constructon the ex-post allocaton rule mplemented by I s the same as that by I. Therefore, we now may now suppose I s strctly ncreasng wlog. Observe that we do not requre the ndex rules to be contnuous n values, and therefore there may be jump dscontnutes.

6 6 RAHUL DEB AND MALLESH M. PAI p u b ) p u b ) p s b, b j ) p s b, b j ) p l b ) p l b ) ˆb ˆb ˆb + ε b j b j FIGURE. Contnuous Symmetrc Implementaton To ensure contnuty n the bd space, we therefore need to be careful about the outcomes at these dscontnutes. Formally, note that snce I and I are strctly ncreasng, G and G do not have any atoms. Further let b = I v ), b = I v ). Defne B = [b, b ], the smallest nterval that contans the set of equlbrum bds B of buyer. Observe that for any b I V ), we requre the nterm expected payment to be exactly p hi. We wll now extend the desred nterm payments for any b B \B prevously when contnuty was not a concern, we set t to be some large payment to deter a devaton. Defne t as: ) p b) = sup β B,β<b p hi β)) + vb) G b) sup G b) β B :β<b where vb) = nf{i β) : β b, β B }. Note that by constructon, p b) s contnuous n b the strct ncreasng-ness of I and I guarantees that G and G are contnuous n b. Further note that by constructon for b B, p b) = ph I. We now use ths contnuous p ) to construct an ex-post payment rule that s contnuous n both own bd and the opponent s bd. Consder the constructon of Theorem, wth the provso that the nterm expected payment for buyer bddng b B \B s p b ). Frst consder the case that b B, b B j. In ths case the constant payment rule p s b, b j ) = p b ) for any b j B j s contnuous n both arguments. Next consder the followng payment rule for bds b B B. Let ˆb be a bd such that G ˆb) = G ˆb). For a gven ε > 0, we defne e ε) as e ε) = E [b ˆb ˆb b ˆb + ε].,

7 DISCRIMINATION VIA SYMMETRIC AUCTIONS: ONLINE APPENDIX 7 Consder the payment rule defned as: p u b ) f b j ˆb + ε, p s b, b j ) = p l b ) + b j ˆb ε p u b ) p l b ) ) f ˆb b j < ˆb + ε, p l b ) f b j < ˆb. Note that ths payment rule s contnuous n b j. The expected payment for a bd b by bdder s then p u b )[ G j ˆb + ε)] + e jε) ε whch then yelds the followng system of equatons [ G ˆb + ε) + e ε) ε G ˆb) e ε) ε G ˆb + ε) + e ε) ε G ˆb) e ε) ε ) p u b ) p l b ) + G j ˆb)p l b ), ] [ p u b ) p l b ) ] = [ p b ) p b ) Note that snce G ˆb) G ˆb) = G ˆb) G ˆb) and e ε) s contnuous, there exsts a small enough ε > 0 such that G ˆb + ε) + e ε) ε G ˆb + ε) + e ε) ε = G ˆb) e ε) ε G ˆb) e ε) ε Fnally, note by constructon p b ) s contnuous n b. Ths n turn mples that p u and p l are contnuous n b whch completes the proof of suffcency. Next to argue necessty. Consder any herarchcal mechansm I, p h ) such that I has a nontrval atom. Suppose that buyer has a postve mass on bd b, and some buyer of value v V also bds b. Bddng b + ɛ for ɛ > 0, ɛ small wll result n a jump n buyer s probablty of wnnng the good. However, by contnuty of payments requres that buyer s payment must ncrease contnuously. Ths results n a contradcton.. ]... Monotoncty Incentve compatblty mples that a buyer s nterm payments n any symmetrc aucton must be nondecreasng n hs value. However, the payment rule need not be monotone n an ex-post sense. For nstance, n the payment ) we constructed for the example n Secton II, we make nether the restrcton that p u b ) p l b ) nor that p u and p l are ncreasng n b. In other words, we so far have not restrcted ex-post payments from our symmetrc mplementatons to be monotone n ether n a buyer s bd or ther opponent s bd. Of course, condtonal on wnnng, the payment rules for both frst- and second- prce auctons are monotone n such an ex-post sense. We frst examne the effect of mposng monotoncty n the opponent s bd, a property of second-prce auctons. We say a herarchcal mechansm has a symmetrc, monotone n opponent s bd mplementaton a s, p s ) f p s b, b j ) s nondecreasng n b j. The next result provdes necessary and suffcent condtons for such an mplementaton to exst. Proposton. Suppose that n =. An mplementable herarchcal allocaton mechansm I, p h ) has a monotone n opponent s bd mplementaton f and only f whenever G frst order stochastcally domnates G j, t s the case that p I p j I j for all b B B.

8 8 RAHUL DEB AND MALLESH M. PAI Intuton for the suffcency of the above condtons can be understood by examnng payments p u and p l n the example of Secton II. For ths partcular constructon, monotoncty n the opponent s bd requres that p u b ) p l b ) for all b. From 3), ths happens f and only f ) p I b ) p I b ) 0. G ˆb) G ˆb) The suffcency of our condton s mmedate and ts necessty s easly establshed n the proof n the appendx. Observe that f nether dstrbuton frst order stochastcally domnates the other, there must exst pvot bds ˆb and ˆb such that G ˆb) > G ˆb) and G ˆb ) < G ˆb ). For bds at whch p I b )) > p I b )), we can construct the payments p u and p l n 3) by pvotng around ˆb and smlarly we can pvot around ˆb when p I b )) < p I b )). Ths payment rule would satsfy the above nequalty and would hence be monotone n the opponent s bd. Note that the condton s not satsfed genercally. Intutvely, ths s because t s possble to perturb a herarchcal mechansm that volates t to get another mechansm that contnues to volate ths condton. Below, we show that Example volates fals ths condton. Addtonally, we modfy Example 3 slghtly to show that there are cases where one dstrbuton frst order stochastcally domnates the other, but the condton s satsfed. EXAMPLE Contnued): In ths example, G U[0, 4] frst order stochastcally domnates G U[0, ]. However, for bd b = the payments gven by equatons 6, 7) are volatng the condton of the proposton. p φ )) = 9 8 > 5 6 = p φ )), EXAMPLE 3 Contnued): Recall that, n ths example, buyer s value dstrbuton F U[0, ] and buyer s value dstrbuton F U[, ]. Now, unlke prevously, suppose that the seller subsdzes the bd of buyer by one and a half dollars nstead of one dollar). In ths case, G U[.5,.5] strctly frst order stochastcally domnates G U[, ], so ths mechansm s mplementable. Addtonally, t s easy to show that p I p I for all bds b. Therefore, there s a monotone n opponent s bd mplementaton for ths mechansm. Proof of Proposton 3. Consder the payment rule we constructed whch pvots around a pont ˆb: [ ] [ ] G ˆb) G ˆb) p u b ) G ˆb) G ˆb) p l = ph I b ). b ) p h I b ) Invertng, we get [ ] [ ] p u b ) G p l = ˆb) G ˆb) ph I b ) b ) G ˆb) G ˆb) G ˆb)) G ˆb) p h I b ) G ˆb)p h I = b ) G ˆb)p h I b ) G ˆb) G ˆb) G ˆb))p h I b ) G ˆb))p h I b ).

9 DISCRIMINATION VIA SYMMETRIC AUCTIONS: ONLINE APPENDIX 9 For monotoncty n the opponents bd, we requre that p u b ) p l b ) 0 or, equvalently, that ) p h I b ) p h I b ) 0. ) G ˆb) G ˆb) We can now consder two cases. Case ): Nether dstrbuton frst order stochastcally domnates the other. Ths mples that there exst ˆb and ˆb such that G ˆb) > G ˆb) and G ˆb ) < G ˆb ). Ths mmedately mples that there exsts a symmetrc mplementaton. We can now use these to construct a monotone payment rule. For all b where p h I b ) p h I b ) > 0, we pvot the payment around ˆb. Smlarly, for all b where p h I b ) p h I b ) < 0, we pvot the payment around ˆb. Ths wll ensure that ) s satsfed and hence that payments are monotone. Case ): One of the dstrbutons frst order stochastcally domnates the other, wlog G frst order stochastcally domnates G. We frst show that n ths case there exsts a symmetrc monotone mplementaton. If G = G, then we are done. If not then we can take any ˆb such that G ˆb) < G ˆb) and construct the usual payment rule. Clearly, ths condton mples that ) wll be satsfed. We now show the converse. Assume wthout loss of generalty that G strctly frst order stochastcally domnates G. Now suppose, n contradcton, that there s a b B B such that p hi > p hi and that there s a symmetrc and monotone mplementaton p s. Frst order stochastc domnance would then mply that p h I = p s b, b )dg b ) B whch sn t possble. Ths completes the proof. B p s b, b )dg b ) = p h I, We now examne the effect of requrng monotoncty n a buyer s own bd. We say a herarchcal mechansm has a symmetrc, monotone n own bd mplementaton a s, p s ) f p s b, b j ) s nondecreasng n b. Ths requrement restrcts the relatve rates at whch the nterm payments of both buyers can ncrease n ther bds for any mplementable mechansm. Put dfferently, f one buyer s payment ncreases very rapdly, then ths monotoncty requrement wll place a lower bound on the rate at whch the other buyer s payment must ncrease. For smplcty, the characterzaton restrcts attenton to herarchcal mechansms wth strctly ncreasng and dfferentable ndex functons ths ensures that the mpled dstrbuton over bds for any buyer has a densty. Moreover, the characterzaton nvolves slghtly dfferent necessary and suffcent condtons, as we have been unable to derve a sngle characterzng condton. The suffcent condton nvolves the slopes dph I b )) of the payments on the common part of the supports of the bd spaces B B. Snce B B s a closed nterval, these dervatves refer to the left rght) dervatve at the upper lower) bound of the support. Proposton 3. Suppose that n =. Consder an mplementable herarchcal mechansm I, p h ) wth dfferentable and strctly ncreasng ndex functons. I, p h ) has a monotone n own bd mplementaton f,

10 0 RAHUL DEB AND MALLESH M. PAI for all dstnct, j {, } and for all b B B for whch dph I > 0, we have { } g b) nf g j b) : b B B, g b) + g j b) > 0 dp h j I j dp hi sup { g b) g j b) : b B B, g b) + g j b) > 0 }, ) wth a strct lower upper) nequalty unless the correspondng nfmum supremum) s reached on a set of bds wth postve G j G ) mass. Conversely, I, p h ) has a monotone n own bd mplementaton only f, for all dstnct b, b B B, we have nf ph j I j p hi sup { g b) g j b) : b B B, g b) + g j b) > 0 b )) p h j I j b )) p hi { g b) g j b) : b B B, g b) + g j b) > 0 } }, 3) wth a strct lower upper) nequalty unless the correspondng nfmum supremum) s reached on a set of bds wth postve G j G ) mass. The astute reader mght observe that, on the surface, t seems that the necessary condton s stronger than the suffcent condton n the above proposton dvde the numerator and denomnator of the central term of 3) by b b and take the lmt b b to get the same central term of )). However, consder a case where nether the nfnmum nor supremum s acheved on a set of postve mass. Further, suppose both nequaltes n 3) are satsfed strctly for every par b, b. It may stll be the case that for some b, one of the nequaltes n ) may be satsfed only as an equalty. In ths case, the suffcent condton wll be volated, whle the necessary condton s satsfed. Ths dscusson also demonstrates that the gap between these condtons s loosely speakng) qute small. Note that the above condtons are always satsfed whenever nether bd space B or B s a subset of the other. In that case, for dstnct, j {, }, we get { } { } g b) nf g j b) : b B B, g = b) 0 and sup g b) + g j b) > 0 g j b) : b B B, =, g b) + g j b) > 0 because there are bds b B, b j B j that do not le n the ntersecton b, b j / B B j. Note that, once agan, the necessary condton s not genercally satsfed the ntuton s dentcal to the monotone n opponent s bd case) and below, we show that t s, n partcular, not satsfed by Example.

11 DISCRIMINATION VIA SYMMETRIC AUCTIONS: ONLINE APPENDIX EXAMPLE Contnued): Snce G U[0, 4] and G U[0, ], we have { } { g b) nf g b) : b B B, g b) = 0 and sup g b) + g b) > 0 g b) : b B B, g b) + g b) > 0 For bds b = and b = 0, the necessary condton s volated because p φ )) p φ 0)) 9/8 p φ )) = p φ 0)) 5/6 >. } =. Proof of Proposton 4. Frst note that: dp hi = dph v ) dv v =I b) = v da v ) dv I v ) = v dg j I v )) dv di b) v =I b) I v ) = v g j I v )) v =I b) = I b)g j b). v =I b) Snce I j s strctly ncreasng and dfferentable by assumpton, g j b) exsts and therefore, so does dp hi. Suffcency. To see that the condtons are suffcent, recall that any symmetrc mplementaton p s b, b ) must be such that for any b B B p s b, b )g b ) = p hi, B p s b, b )g b ) = pi h. B Note that p hi s non-decreasng n b by assumpton for =,. Therefore for an mplementaton that s monotone n own bd, t s suffcent to fnd p s such that: where dps b,b ) 0. If both dph I loss suppose dp I b) and dph I dp s b, b ) g b ) = dph I, B 4a) dp s b, b ) B, 4b) equal 0, settng dps b,b ) = 0 solves 4). So suppose not. Wthout = 0, the other case follows symmetrcally.

12 RAHUL DEB AND MALLESH M. PAI Pck a set B wth G B) > 0 such that g b ) g b ) dph G B) > 0 such that dp hi dp hi Fnally consder a p s s.t.: Substtutng nto 4): g b ) g b ) for all b B. dp s b, b ) I dp h I x b B = x b B 0 otherwse xg B) + xg B) = dph I, xg B) + xg B) = dph I. for all b B. Smlarly pck a set B wth By constructon, therefore x and x must be postve. Formally, notce that f ths system does not have a non-negatve soluton, then the Farkas alternatve of: y G B) G B) + y 0, y G B) G B) + y 0, dp I y dp I + y < 0, must have a soluton. Clearly ths s mpossble by assumpton snce Therefore, by constructon, we have shown that dps b,b ) 0 for all b. dp I dp I [ ] G B) G B), G B). G B) Necessty. Once agan, recall that any symmetrc mplementaton p s b, b) must be such that for any b B B p s b, b)g b) = p hi b )), B p s b, b)g b) = pi h b )). B Suppose the condton 3) s volated for some b > b. Any symmetrc mplementaton must satsfy: p s b, b) p s b, b))g b) = p hi b ) p hi b )), 5a) B p s b, b) p s b, b))g b) = pi h b ) pi h b )). 5b) B Analogous to the Farkas Lemma argument above, when 3) s volated, there cannot exst a soluton to 5) such that p s b, b) p s b, b) 0, for all b B B.

13 DISCRIMINATION VIA SYMMETRIC AUCTIONS: ONLINE APPENDIX 3.3. Ex-Post IR Whle the symmetrc mplementatons we construct for Theorem are by defnton IR, they are IR n an nterm sense. As we have argued above, the equlbrum however need not be IR n an ex-post sense: certan bd profles may result n losng bdders havng to make payments or wnners havng to pay more than ther valuaton. Ths s unappealng and may result n certan bdders choosng not to partcpate. Perhaps more mportantly, ths may result n non-payment by budget-constraned bdders. Ths s because a bdder s valuaton may reflect her ablty to pay for the good. Addtonally, certan bdders who plan to pay by obtanng a loan may be unable to obtan credt upon losng the aucton. Formally, we say that a herarchcal mechansm I, p h ) has a symmetrc, ex-post IR mplementaton a s, p s ) wth assocated equlbrum strateges σ, f for all v V and N, we have p s σ v ), σ v )) v a s σ v ), σ v )). Ths states that at any bd profle that occurs n equlbrum, wnnng buyers are never charged more than ther value and losers do not have to make payments, although they may receve subsdes whch mples that losers may not be nactve). Notce that, when there are tes, the above nequalty mples that buyers only have to pay n the event that they wn. The ex-post IR requrement places a bound on the payments that the symmetrc aucton can requre buyers to make at both wnnng and losng bds. As n the case wth nactve losers mplementaton, the optmal aucton may not have an ex-post IR mplementaton. In fact, ths can be demonstrated by once agan revstng Example. EXAMPLE Contnued): Recall that buyer wth value v = 3 has a vrtual value of φ 3) =, always wns the good, and pays p 3) = 5. For there to be a symmetrc ex-post IR mplementaton, there must exst a symmetrc payment p s such that p s, b )dg b ) = 5 whch n turn mples that there must exst at least one b [0, ] such that 0 p s, b) 5. However, note that a buyer wth value v = also has the vrtual value φ ) =. Snce there s a b [0, ] such that p s, b) 5, there wll be a bd profle n the support of the equlbrum bds at whch buyer s payng more than her value. Ths volates the ex-post IR requrement. We derve necessary and suffcent condtons for a herarchcal mechansm to admt a symmetrc ex-post IR mplementaton. Due to the complexty of the characterzaton, we need to make two addtonal assumptons. As n Proposton 3, we frst restrct attenton to herarchcal mechansms I, p h ) n whch the ndex functons I are dfferentable and strctly ncreasng. Second, we further restrct attenton to the case where the lower bounds of the supports of the bd space do not concde, or I v ) = I v ). The characterzaton for ths case s easer to state. In the appendx, we present the characterzaton for allocaton rules n whch I v ) = I v ). If we were to take the procurement nterpretaton of our model, the ex-post IR requrement would ensure that frms can cover ther costs and complete the project.

14 4 RAHUL DEB AND MALLESH M. PAI Wthout loss of generalty, we assume that bdder s bd space has the lower support, or I v ) = b < b = I v ). Addtonally, we defne I v ) = b for {, } and vb) mn{i b), I b)} for b B B, as the lower of the values of the two buyers correspondng to a bd b that les n both bd spaces. Recall that, snce we have restrcted attenton to strctly ncreasng ndex functons, ths nverse s well defned. We can now state a smple frst necessary condton that a herarchcal mechansm I, p h ) must satsfy to have an ex-post IR mplementaton. Condton C: The dstrbuton of values F and F nduce dstrbutons G and G such that b B B : vb)g b) p h I b). C) Ths s an ntutve necessary condton. vb) s the maxmum amount that can be charged to a wnnng buyer who bds b B B and whose opponent bds b B B, b b. Snce the aucton s symmetrc, such a profle of bds wll not reveal the dentty of the wnnng bdder, so the ex-post IR requrement restrcts the payment to be lower ) than both possble values of the wnnng bdder. Hence, bdder s nterm payment p h I b) cannot be hgher than vb)g b) for any bd b B B. Note that the necessty of ths condton does not hnge on the lower bounds of the supports of the bd spaces beng dfferent, and C wll contnue to reman necessary when b = b. We revst Example yet agan and show that t volates ths condton. EXAMPLE Contnued): Once agan, consder buyer wth value v = 3, at whch the nterm payment s p 3) = 5. At the bd φ 3) =, Condton C s volated because and hence, v) = mn{φ v)g ) = < p ), φ )} = mn{3, } =, φ ) ) = 5. It remans to derve a smlar condton for the nterm payment of buyer, whch accounts for the fact that the lower bounds of the supports of the bd dstrbutons dffer b < b ). Suppose that one buyer bds b B B whle the other bd s n [b, b ). Then, t s clear that the buyer bddng b s buyer, so payments n ths range of bds can be chosen to be up to her value I b) whch may be hgher than vb). By contrast, when buyer bds b, she can never be charged more than vb) even f her value I b) s strctly greater. Ths argument yelds an analogous necessary condton for buyer. Condton C : The dstrbuton of values F and F nduce dstrbutons G and G such that b B B : vb) G b) G b )) + I b)g b ) p h I b). C ) However, condtons C and C together need not be suffcent. Ths s because ensurng the approprate nterm payment for buyer places a bound on the amount that can be extracted from

15 DISCRIMINATION VIA SYMMETRIC AUCTIONS: ONLINE APPENDIX 5 buyer from bds that ) le n the common support B B. Suppose that, at a bd b, the nterm ) payment p h I b) of buyer s substantally lower than that of buyer, whch s p h I b). Ths may prevent the seller from extractng the entre expected payment vb)[g b) G b )] from buyer when buyer s bds le n the range [b, b]. Hence, we need to derve the maxmum payment ηb) vb)[g b) G b )] that can be extracted symmetrcally from buyer when ) she bds b B B, ) postve payments are only taken when b s the wnnng bd.e., the other ) buyer s bds are n the range [b, b]) and ) buyer s expected payment from bd b s p h I b). In words, we need to defne payments for bds b B B n a way that maxmzes the amount ) extracted from buyer whle ensurng that buyer s expected payment remans p h I b). If ths amount extracted s greater than the requred payment p h I b) for buyer, subsdes can always be provded when buyer s bds le n the range [b, b ) because, n equlbrum, such bds can only come from buyer. We now need some addtonal notaton. Frst, we defne the followng functon for b B, whch depends on the ratos of the denstes: f g b) = g b) = 0, Lb) = g b) otherwse. g b) That s, L ) s the lkelhood rato of a buyer bddng b beng buyer versus buyer. Further, we defne l mn b B { Lb) }. Ths s the lowest value of the lkelhood rato for bds n B. Snce ndex functons are assumed to be dfferentable and strctly ncreasng, denstes g and g are well defned and contnuous on B and B respectvely. As a result, l s well defned and postve when b b and 0 when b > b. Addtonally, we defne the sets γl) { } b B Lb) l as the set of bds less than b where the lkelhood rato s at most l, and { } = γl) b B Lb) = l smlarly as the set of bds less than b where the lkelhood rato s exactly l. These sets wll be useful to descrbe payment rules that derve ηb). To obtan ηb), we concentrate the maxmum payment vb) on bds that are more lkely to le n the bd space of buyer relatve to that of buyer, and buyer s nterm payment s then guaranteed by provdng a subsdy at bds that are least lkely.

16 6 RAHUL DEB AND MALLESH M. PAI When Condton C holds, that s, when vb)g b) p h I b), the followng two cases are mutually exclusve and exhaustve for any b B B. 3 G γl)) = > 0 OR vb)g b) = p h I b). Case ) G γl)) = = 0 AND vb)g b) > p h I b). Case ) ηb) needs to be derved separately for each of these two cases, and hence, we analyze them separately below. Case. Let ˆB be a subset of γl) = such that vb)g [b, b]\ ˆB ) p h I b). If vb)g b) = p h I b), then ˆB must be a G -null set else consder any set ˆB that satsfes the above nequalty and has a strctly postve measure. We now defne a payment rule, vb) for b [b, b]\ ˆB, ˆpb, b ) = s for b ˆB, 0 for b B and b / [b, b] ˆB ) C,P). where s s chosen to solve Note that s here s a subsdy. We set vb)g [b, b]\ ˆB ) + sg ˆB ) = p h ηb) = b I b). b ˆpb, b )dg b ). 6) Observe that ηb) does not depend on the choce of ˆB. In addton, observe that, when b > b, then ˆB b, b ] and ηb) = vb). Case. Snce G = γl)) = 0, t must be that b b. Here, we defne the payment rule ˆp l for l > l as follows: where s s chosen to solve vb) for b [b, b]\γl), ˆp l b, b ) = s for b γl), 0 for b B and b / [b, b] ˆB ). vb)g [b, b]\γl)) + sg γl)) = p h I b). Note that, for l close to l, s s negatve, so the payment rule ˆp l s ex-post IR. Defne: η l b) = b C,P) b ˆp l b, b )dg b ), 7) 3 Recall that we use G to represent both a measure and a CDF.

17 DISCRIMINATION VIA SYMMETRIC AUCTIONS: ONLINE APPENDIX 7 and let We can now defne the second condton. ηb) = lm l l [η l b)]. Defnton. Condton C). The dstrbuton of values F and F nduce dstrbutons G and G such that b B B : ηb) + I b)g b ) p h I b), C) wth the nequalty holdng strctly for any b such that G = γl)) = 0 and vb)g b) > p h I b). The followng proposton states that the two condtons C and C are necessary and suffcent for a symmetrc ex-post IR mplementaton. Proposton 4. Suppose that n =. Consder an mplementable herarchcal allocaton mechansm I, p h ) wth dfferentable and strctly ncreasng ndex functons such that the lower bounds of the supports of the bd dstrbutons dffer; that s, b < b. Then, Condtons C and C are necessary and suffcent for there to exst a symmetrc, ex-post IR mplementaton of I, p h ). We end ths secton by observng that Proposton 4 can be adapted to accommodate entry fees. In many practcal stuatons, auctons are often conducted n two steps: buyers frst pay to partcpate, followng whch the aucton s conducted. Such entry fees can relax ex-post IR constrants of the aucton tself, as buyers are makng a part of the payment before partcpatng. In partcular, f the seller could charge a hgh enough entry fee, he would not need the buyers to make payments n the aucton and could offer rebates nstead. Havng sunk the entry cost, ex-post IR would then be obtaned automatcally. Condtons C and C can be approprately weakened to accommodate a gven entry fee; the constructon n ths secton can smply be altered so that wnnng bdder never pays more than her value plus the fee and the loser never has to pay more than the fee. Proof of Proposton 5. We wll frst prove the theorem as stated for dfferentable and strctly ncreasng ndex rules that s, no atoms n G ). Later, we wll extend to the more general case. Proof. We frst demonstrate suffcency, and then argue necessty. Suffcency. For smplcty, we wll only defne payments for equlbrum bds; off-equlbrum bds can be dscouraged n the same way as n the proof of Theorem. We consder the two cases of Condton C separately. For Case ) we consder the followng payment rule: { [ ] p s b, b p ) = G b ) hi ηb) for b [b, b ) ˆpb, b ) for b [b, b ) where ˆp s gven by C,P), and ηb) s gven by 6).

18 8 RAHUL DEB AND MALLESH M. PAI Smlarly, Case ) we consder the followng payment rule: { [ ] p s b, b p ) = G b ) hi η l b) for b [b, b ) ˆp l b, b ) for b [b, b ) where n ths case ˆp l s gven by C,P) for a gven l > l, and η l b) s as defned n 7). From Condton C and contnuty there s a l close enough to l for whch p s b, b ) < vb) for b [b, b ). By constructon, for each buyer, hs expected payment wll equal hs nterm payment n the herarchcal mechansm. Condton C) guarantees that n the range b [b, b ), the mplementaton stll satsfes ex-post IR, p s b, b ) I b). Indeed t mght be the case that p s b, b ) < 0. Necessty. We frst verfy these condtons are necessary for mplementaton n a symmetrc aucton where bddng the actual ndex rule s each buyer s equlbrum strategy. We then show that the same condtons also rule out other mplementatons as well. Let us verfy that C s necessary. So suppose not,.e. suppose: vb)g b) < p hi for some b B B. Note that f a buyer bds b, and the other bdder bds b B B, the maxmum she can be asked to pay wthout volatng ex-post IR s vb). But now, for bdder, t follows that the maxmum expected payment that she can be asked to make s vb)g b). If her requred payment, p hi, exceeds ths, then there cannot be a symmetrc, ex-post IR mplementaton. Wth buyer, there s a lttle more wrggle room. When buyer bds b B B, she could be a wnner n some asymmetrc profles;.e. when buyer bds n the range [b, b ). At these bd profles, a potentally hgher payment up to I can be extracted from buyer. Condton C) then guarantees that the requred nterm payment, p hi can be extracted. Note that n the constructon of ether ˆp or ˆp l, ether the maxmum permssble amount vb) s beng pad by the wnnng buyer, or a rebate of s s beng returned to the buyer who bds b. The rebates are beng pad when the other buyer s bd b has the lowest possble value of Lb ). Ths means that the rebates are worth the lowest possble n expectaton to a wnnng buyer, because they occur where L ) s mnmzed. We begn by consderng the followng maxmzaton problem for b B B and any gven s 0: b m s b) = max ϱb )dg b ), Max-P) ϱ ) b s.t. b b ϱb )dg b ) = p h I, λ) s ϱb ) vb), b [b, b], δb ), κb )) s ϱb ) 0, b b, max{b, b }]. δb ), κb )) To understand ths optmzaton program n words, fx a bd b. Thnk of ϱ ) as the payment made by the buyer n ths case as a functon of the other buyer s bd. The program asks what the maxmum expected payment that can be extracted from buyer s subject to constrants we descrbe next. The frst constrant requres that the expected payment of buyer under ϱ ) s hs correct nterm payment. The latter two constrants requre that ϱ ) s pontwse bounded below

19 DISCRIMINATION VIA SYMMETRIC AUCTIONS: ONLINE APPENDIX 9 by s and bounded above by the maxmum possble ex-post IR payment vb) when wnnng and 0 when losng. The terms n the parentheses to the rght of the constrants denote the correspondng dual co-state) varables. We clam that lm s m s b) = ηb). When vb)g b) = p h I, then ϱb ) = vb) for all b [b, b] s the only feasble functon, so ths case s trval. Hence, we focus on the case vb)g b) > p h I. The Hamltonan n ths case s: wth complementary slackness condtons: g b ) λg b ) + δb ) κb ) = 0, = g b ) g b ) λ + g b ) δb ) κb )) = 0 δb )s ϱb )) = 0, for b [b, b], κb )vb) ϱb )) = 0, for b b, max{b, b }], κb )ϱb ) = 0, and δb ), κb ) 0. By observaton, the soluton to ths for any s s bang bang,.e. s f Lb ) λ, ϱb ) = vb) f Lb ) > λ, b [b, b], 0 otherwse. wth λ selected such that the correspondng prmal equaton bnds for ϱ ) selected thus. The corner case that needs care s when G = γl)) > 0. In ths case, there s a postve measure of b [b, b ] such that Lb ) = l. Here, the soluton s bang bang, but possbly dependng on s), there s ˆB = γl) such that s f b ˆB γl), = ϱb ) = vb) f b [b, b]\ ˆB, 0 otherwse. It follows by constructon, therefore, that lm s m s b) = ηb). Therefore, subject to the payment rule extractng the approprate nterm payment p hi when buyer bds b, ηb) s the maxmum expected payment that can be extracted from buyer when she bds b and buyer makes a bd hgher than b. It follows therefore that f nequalty C) s volated, there cannot be an mplementaton satsfyng both symmetry and ex-post ndvdual ratonalty. Next, consder any other mechansm I, p h ) wth a dfferentable ndex rule, that mplements the same mechansm. Then, t must be that I v ) = ΓI v )).

20 0 RAHUL DEB AND MALLESH M. PAI for some dfferentable and strctly ncreasng functon Γ : R R. Note that the resultng dstrbuton on bds, whch we shall denote by G, s Note that ths mples that G Γ = G b). g ΓΓ b) = g b). Our prevous arguments already mply that Condtons C and C, wrtten n terms of G s are necessary for an mplementaton. By the equatons above, we see that for b B B Also, for any b B, vb)g Γ p h I = vb)g b) p h I. Lb) = g b) g b) = g Γ g Γ. Therefore our condtons n terms of the orgnal G s are necessary for any pure strategy mplementaton. Weakly ncreasng ndex rules. So far we have only consdered strctly ncreasng ndex rules. If the ndex rules are not strctly ncreasng, the correspondng bd dstrbutons wll have atoms. Denote by B the atoms n G. For b B, the sze of the atom s G {b }) recall that ths s a measure and not a densty. Further, I ) may be correspondence v ) may not be well defned. Redefne vb) as vb) = nf{v I b) I b)}. Note that when I b) and I b) are sngletons, ths s the same as the old defnton of vb). Now Condton C wll be as before wth ths extended defnton of v ). Next, note that Condton C depends on g /g, whch agan may not be well defned. We redefne L ) as follows g b) b g B b) and b B B, G Lb) = {b}) b G B {b}) B, 0 b B \B. We can now redefne ηb) wth ths defnton Lb). It should be clear that Condtons C and C thus extended are necessary and suffcent..4. Symmetrc Ex-Post IR Implementaton wth Common Lower Bound of Bd Space Support We now use the prevous ntuton to derve axoms for the case where b = b. Ths adds a lttle more complexty to our analyss. To see why, recall that our prevous mplementaton heavly used the fact that b < b. In partcular, profles of the sort b, b ) where b B B and b < b were used as a sort of resdual clamant. The payment of the wnnng buyer n profles

21 DISCRIMINATION VIA SYMMETRIC AUCTIONS: ONLINE APPENDIX could be set as hgh v to make up for any shortfall n buyer s expected payment vs-a-vs her nterm payment. Conversely, she can be gven a rebate to make up for any surplus. Snce G b ) = 0, Condton C rewrtten n ths case reflects the fact that there s no such regon to make up for any shortfall: Defnton. Condton C ). Condton C requres that for all b n B B, wth b = b wth the nequalty holdng strctly for any b such that: ηb) p h I 8) G = γl)) = 0 and vb)g b) > p h I. Intutvely, Condton C requres that the maxmum expected payment ηb) that can be extracted from buyer when she bds b, among all payment rules that extract exactly p hi from buyer n expectaton, s more than p hi. In the prevous secton ths was enough, because any excess ηb) p hi can be rebated to buyer when the other buyer bds n the range [b, b ). Now, ths s no longer enough. We need an addtonal condton to account for the fact that there s no lower regon to rebate any surplus to. We now wrte down the exact analog condton,.e. that the mnmum expected payment ζb) that can be extracted from buyer when she bds b, among all payment rules that extract exactly p hi from buyer n expectaton, s at most p hi. If both condtons hold, there clearly exsts a payment rule whch wll acheve the requred mplementaton, snce the set of all payment rules that extract exactly p hi from buyer n expectaton s convex. We consder two cases dependng on the orderng of the upper bound of the possble bds, b and b. If b > b, we can rebate money to buyer smlarly as before n ths case when the other bdder bds n the range b, b ]. In ths case defne ζb) = 0 for all b B B. Now let us consder the other case,.e. that b b n ths case B B. We need some addtonal notaton. Frst, we defne l = max b [b,b ]\{b:g b)=g b)=0}) Lb ). As before l s well defned. As before, there are two sub-cases. The frst sub-case s when G = γl)) > 0. Let ˆB = γl > 0 be a potentally empty) subset such that: vb)g [b, b]\ ˆB ) p h I. We now defne a payment rule vb) for b [b, b]\ ˆB, ˆp b, b ) = s for b ˆB, 0 o.w. C3,P)

22 RAHUL DEB AND MALLESH M. PAI where s s chosen to solve Notce that s here s a subsdy. We set: vb)g [b, b]\ ˆB)) + sg ˆB) = p h I. The second sub-case s when ζb) = b b ˆp b, b )dg b ). G = γl)) = 0, we defne the payment rule for l < l vb) for b [b, b], Lb ) l, ˆp l b, b ) = s for Lb ) > l, 0 otherwse. C3,P) where s s chosen to solve Here we set vb)g {b : b [b, b], Lb ) l}) + sg {b : Lb ) > l}) = p h I. [ b ] ζb) = lm ˆp l b, b )dg b ). l l b Defnton.3 Condton C3). Condton C3 requres that wth the nequalty holdng strctly when: We can now state the proposton ζb) p h I G = γl)) = 0 and vb)g b) > p h I. Proposton 5. Suppose there are buyers. Consder a herarchcal allocaton mechansm I, p h ) wth dfferentable and strctly ncreasng ndex functons such that the lower bounds of the supports of the bd dstrbutons are the same, that s, b = b. Then Condtons C, C and C3 are necessary and suffcent for there to exst a symmetrc, ex-post IR mplementaton of I, p h ). The proof follows from also consderng the analogous mnmzaton problem to Max-P) and s omtted.