Social Decision with Minimal Efficiency Loss: an Automated Mechanism Design Approach

Transcription

1 Socal Decson wth Mnmal Effcency Loss: an Automated Mechansm Desgn Approach ABSTRACT Mngyu Guo, Hong Shen School of Computer Scence Unversty of Adelade, Australa {mngyu.guo, We study the problem where a group of agents need to choose from a fnte set of socal outcomes. We assume every agent s valuaton for every outcome s bounded and the bounds are publc nformaton. For our model, no mechansm smultaneously satsfes strategy-proofness, ndvdual ratonalty, non-defct, and effcency. In lght of ths, we am to desgn mechansms that are strategy-proof, ndvdually ratonal, non-defct, and mnmze the worst-case effcency loss. We propose a famly of mechansms called the shfted Groves mechansms, whch are generalzatons of the Groves mechansms. We frst show that f there exst mechansms that are strategy-proof, ndvdually ratonal, and non-defct, then there exst shfted Groves mechansms wth these propertes. Our man result s an Automated Mechansm Desgn (AMD) approach for dentfyng the (unque) optmal shfted Groves mechansm, whch mnmzes the worst-case effcency loss among all shfted Groves mechansms. Fnally, we prove that the optmal shfted Groves mechansm s globally optmal among all determnstc mechansms that are strategy-proof, ndvdually ratonal, and non-defct. Categores and Subect Descrptors I..11 [Dstrbuted Artfcal Intellgence]: Multagent Systems General Terms Algorthms, Economcs, Theory Keywords Groves Mechansms, Automated Mechansm Desgn, Socal Decson 1. INTRODUCTION Socal decson-makng s a fundamental problem n multagent systems. Ths paper studes how to desgn economc mechansms for mplementng good socal outcomes. We study the problem where a group of agents need to choose Appears n: Proceedngs of the 14th Internatonal Conference on Autonomous Agents and Multagent Systems (AAMAS 015), Bordn, Elknd, Wess, Yolum (eds.), May 4 8, 015, Istanbul, Turkey. Copyrght c 015, Internatonal Foundaton for Autonomous Agents and Multagent Systems ( All rghts reserved. Tak Todo, Yuko Sakura, Makoto Yokoo Graduate School of ISEE Kyushu Unversty, Japan from a fnte set of socal outcomes. The agents valuatons for the outcomes are prvate nformaton. We assume every agent s valuaton for every outcome s bounded and the bounds are publc nformaton (e.g., the bounds are prescrbed by the nature of the outcomes). One example settng s the publc proect problem: Example 1. Publc Proect Problem n agents need to decde whether or not to buld a publc proect (e.g., a brdge accessble by everyone). The cost of the proect s 1. There are two outcomes: Outcome 1: not buld Every agent s valuaton for ths outcome equals 0. Outcome : buld, and every agent s responsble for an equal share of the cost, whch equals 1 n Dfferent agents value the proect dfferently. We assume an agent s valuaton for the proect s between 0 (not apprecatng at all) and 1 (as hgh as the cost of the whole proect). That s, takng the cost share nto consderaton, every agent s valuaton for ths outcome s between 1 n 1 and. n n For socal decson problems lke the above, deally, we prefer mechansms that satsfy the followng propertes: Strategy-proofness: Every agent s domnant strategy s to report truthfully, regardless of the other agents reports. It s wthout loss of generalty to requre strategy-proofness due to the revelaton prncple. (Ex post) ndvdual ratonalty: Every agent s utlty (valuaton of the chosen outcome mnus payment) s non-negatve. Indvdual ratonalty s necessary when partcpaton s voluntary. Non-defct property: The agents total payment s nonnegatve. 1 That s, no external subsdy s ever needed. Another related but stronger property s budget balance, whch requres that the agents total payment s exactly 0. We requre non-defct, otherwse the mechansm s not sustanable. Effcency: The mechansm always chooses the outcome that maxmzes the agents total valuaton. (For the publc proect problem, t means the proect s bult f and only f the agents total valuaton for outcome s hgher. 1 An ndvdual agent may stll pay negatve payment.

2 For resource allocaton settngs, the above propertes can be smultaneously acheved, e.g., by the VCG redstrbuton mechansms [1, 5, 13]. However, when t comes to socal decson settngs, sometmes an agent has to face an undesrable outcome that s forced onto hm by the others (e.g., an agent who does not apprecate the publc proect may be forced to chp n to buld the proect). The VCG redstrbuton mechansms, when appled to the publc proect problem, are not ndvdually ratonal (unless we change the defnton of ndvdual ratonalty)[7, 8, 14]. Actually, as we wll dscuss n more detals later on, for socal decson settngs, no mechansms can be smultaneously strategy-proof, ndvdual ratonal, non-defct, and effcent. In lght of the above, we choose to sacrfce effcency. Our am s to desgn mechansms that are strategy-proof, ndvdually ratonal, non-defct, and as close to effcency as possble. Specfcally, we try to mnmze the worst-case effcency loss (the loss of effcency n the worst case, as the result of usng an neffcent mechansm). We adopt a(computatonally Feasble) Automated Mechansm Desgn approach [6, ]. We propose a famly of mechansms called the shfted Groves mechansms, whch are generalzatons of the Groves mechansms. We frst show that f there exst mechansms that are strategy-proof, ndvdually ratonal, and non-defct, then there exst shfted Groves mechansms wth these propertes. We use AMD to dentfy the (unque) optmal shfted Groves mechansm, whch mnmzes the worst-case effcency loss among all shfted Groves mechansms. Fnally, we prove that the optmal shfted Groves mechansm s globally optmal among all determnstc mechansms that are strategy-proof, ndvdually ratonal, and non-defct. Besdes proposng optmal mechansms for a fundamental mechansm desgn settng, our approach also demonstrates the effectveness of usng AMD to delver new economc results.. FORMAL MODEL DESCRIPTION There are n agents who need to choose from k outcomes. Weuseθ = (v 1,v,...,v k )todenoteagent stype, where v represents agent s valuaton for outcome. We make the followng assumptons: v s bounded below and above by known constants L and U. It should be noted that both L and U may be negatve. For every, agent s type space Θ s the whole of [L 1,U 1] [L,U ]... [L k,u k ]. The set of type profles s the whole of Θ 1 Θ... Θ n. There do not exst two outcomes, so that the agents always prefer one outcome than the other for all type profles. That s, for every par of and, we have U L. Our analyss throughout the paper reles on the above assumptons (e.g., we need the above to prove that our AMD approach produces optmal mechansms). However, The Faltngs mechansm [3] also sacrfces effcency. The mechansm works for both resource allocaton settngs and socal decson settngs, but agan, t s not ndvdually ratonal for our problem. t should be noted that only the frst assumpton s needed to apply the AMD approach, whch takes the bounds as nput. For example, t could be that certan type values do not exst (e.g., v s possble values do not form an nterval or form a much narrower nterval than [L,U ]) and/or certan types/type profles do not exst (e.g., due to nterdependence). For these cases, the AMD approach stll lead to a feasble mechansm. Of course, the mechansm s generally not optmal, because we fal to recognze that certan profles are mpossble. As mentoned earler, t s mpossble to smultaneously have strategy-proofness, ndvdually ratonalty, non-defct, and effcency. Below we present the detals. Accordng to Holmström [9], for convex type space (our settng), the Groves mechansms [4] are the only mechansms that are effcent and strategy-proof. The Groves mechansms are defned as follows: Themechansmchoosesanoutcomefromargmax v. That s, the mechansm chooses an outcome that maxmzes the agents total valuaton. Let be the chosen outcome. For every, agent receves v. That s, the amount an agent receves s equal to the other agents total valuaton for. For every, agent then pays an amount that s ndependent of s own type, whch s denoted by h (θ ). 3 Dfferent sets of h functons correspond to dfferent Groves mechansms. Every Groves mechansm s effcent and strategy-proof, but no Groves mechansm s both ndvdually ratonal and non-defct. We need the h (θ ) term to be small enough, so that agent s utlty s non-negatve. We also need the h (θ ) term to be large enough, so that the agents total payment s non-negatve. However, we cannot satsfy both. We use E(θ,θ ) to denote the agents total valuaton for the effcent outcome. Under the Groves mechansms, agent s utlty s smply E(θ,θ ) h (θ ), whch needs to be non-negatve for ensurng ndvdual ratonalty. Hence, we have h (θ ) E(θ,θ ) for all θ and all θ. The lefthand sde does not depend on θ, and the rght-hand sde s mnmzed when θ = (L 1,L,...,L k ) (we use θ to denote ths type). Hence, ndvdual ratonalty s equvalent to for all and all type profles, h (θ ) E(θ,θ ) = max(l + v ) It does not hurt to set h (θ ) to exactly max (L + v ). As mentoned earler, we want ths term to be large enough for ensurng non-defct. If we set t any larger, ndvdual ratonalty s volated. Once we fx the h (θ ) term, there s one Groves mechansm left. If ths mechansm s not non-defct, then no Groves mechansms are both ndvdually ratonal and non-defct. We use the followng example to show ths. Example. Let us consder a publc proect problem (descrbednexample1)wthagents. WehaveL 1 = U 1 = 0, 3 When we run the mechansm, we smply combne the payment terms n step and 3. Throughout ths paper, they are presented separately as we analyze them separately.

3 L = 1, and U = 1. Let θ1 = θ = (0, 1 ). That s, v 1 = 0 and v = 1. For ths type profle, the decson s to buld, both agents receve 1 (here, h(θ ) = 0). The non-defct property s volated here. In concluson, n lght of the above mpossblty result, our am s to desgn socal decson mechansms that are strategy-proof, ndvdually ratonal, non-defct, and mnmze the worst-case effcency loss. 4 Defnton 1. Let M be a mechansm. Let M(θ,θ ) be the agents total valuaton under M for type profle (θ,θ ). We recall that E(θ,θ ) represents the agents total valuaton for the effcent outcome. Mechansm M s worst-case effcency loss s defned as: max θ,θ (E(θ,θ ) M(θ,θ )) 3. SHIFTED GROVES MECHANISMS We revst the publc proect example. As we have seen, f outcome (buld) s chosen, defct may occur. On the other hand, f outcome 1 s chosen, defct never occurs. 5 One dea s then to dscourage outcome by askng the agents to pay addtonal payments when outcome s chosen (essentally changng the agents valuatons for outcome ). For example, we may shft the outcome buld to buld, and every agent pays x n addton to mechansm payments and buldng cost. Ths helps reducng defct for two reasons. Frst, after ntroducng these addtonal payments, less type profles are mapped to outcome, so the maxmal defct when outcome s chosen may potentally decrease. Second, the addtonal payments also help offset the defct. If outcome s chosen, we know that we have nx addtonal payment to offset the defct. The above dea leads to the shfted Groves mechansms: For every agent, every outcome, we ntroduce an addtonalpayment t. Thats, f outcome s chosen, agent needs to pay t n addton to her mechansm payment. The t are constants set by the mechansm desgner. The total addtonal payment for outcome s P = t. Apply the Groves mechansms on the modfed outcomes. The mechansm chooses an outcome from arg max (v t ) = argmax{ P + v } That s, the mechansm chooses an outcome that maxmzes the agents total valuaton, consderng the addtonal payments. Let be the chosen outcome. For every, agent receves (v t ). That s, the amount anagentrecevessequaltotheotheragents total valuaton for the chosen outcome, consderng 4 For mechansms that need to deal wth te-breakng, we assume te-breakng s done randomly, and the worst-case effcency loss consders all possble te-breakng scenaros. 5 If the chosen outcome s 1 (not buld), then both agents receve 0 and h (θ ) = 0 (e.g., h 1(θ 1) = max{0, 1 + v } = 0). the addtonal payments. We also keep n mnd that agent needs to pay the addtonal payment t. Therefore, agent s net ncome at ths pont should be (v t ) t = P + v For every, agent then pays an amount that s ndependent of s own type, whch s denoted by h (θ ). Smlar to the case wthout addtonal payments, we can fnd an upper bound on h (θ ). Agent s utlty equals max { P + v} h(θ ), whch must be non-negatve forensurngndvdualratonalty. Therefore,wehaveh (θ ) max { P + v} for all θ. By settng θ = θ = (L 1,L,...,L k ), we have h (θ ) max{ P +L + v } Agan, t does not hurt to set h (θ ) to be exactly the rghthand sde. (Ths value does not affect strategy-proofness and allocatve effcency. We need t large for achevng the non-defct property, but any larger value volates ndvdual ratonalty.) Once we fx the h (θ ) terms, we observe that the shfted Groves mechansms are completely characterzed by the total addtonal payments for the outcomes (the P ). Both the outcome and the payments do not depend on the ndvdual addtonal payments (the t ). Ths leads to a concser defnton of shfted Groves mechansms. Defnton. A shfted Groves mechansm s characterzed by k constants P 1,P,...,P k. The mechansm chooses an outcome from arg max{ P + v } Let be the chosen outcome. For every, agent pays max{ P +L + v }+P v WeuseM(P 1,P,...,P k )todenotetheshftedgrovesmechansm characterzed by the P. It should be noted that the shfted Groves mechansms are specal cases of Affne Maxmzer Auctons (AMA) [10, 11]. There are many works on characterzng strategy-proof mechansms n socal choce settngs. Roberts [15] shows that when the type space of agents s unrestrcted (and there exst three or more choces), the only strategy-proof mechansms are affne maxmzers (of whch the shfted Groves mechansms are specal cases). In our settng, the type space of agents s bounded so the results n [15] do not apply. There also exst works on characterzng strategyproof mechansms n a restrcted doman. For example, [1] showed that all strategy-proof mechansms are affne maxmzers f the agents valuatons are drawn from ntervals. However, [1] requres that the ntervals be open (we use close ntervals n ths paper), and more mportantly, t also requres that an agent s nterval for all outcomes are dentcal

4 (we do not requre ths, e.g., for the publc proect problem, the agents obvously have dfferent ntervals for not buld and buld). Therefore, [1] also does not apply. For our settng, t remans to see whether all strategy-proof mechansms are affne maxmzers. Nevertheless, n ths paper, we show that the optmal shfted Groves mechansm s optmal among all strategy-proof mechansms, wthout relyng on such characterzaton results. 6 Based on how they are defned, all shfted Groves mechansms are strategy-proof and ndvdual ratonal. Unlke the Groves mechansms, some shfted Groves mechansms can be non-defct (whle beng ndvdually ratonal). Theorem 1. Gven a settng, f there exst mechansms that are strategy-proof, ndvdually ratonal, and non-defct, then there must exst shfted Groves mechansms wth these propertes. To prove the above theorem, we ntroduce the followng lemma. We recall that L s the agents lowest possble total valuaton for outcome. Lemma 1. If L 0, then under all shfted Groves mechansms (for all values of the P ), when outcome s chosen, defct never occurs. Proof. When outcome s chosen, agent s payment equals max{ P +L + v }+P v P +L + v +P v = L Therefore, the agents total payment s at least L, whch s non-negatve. Wthout loss of generalty, we assume L1 L... L k. Now we are ready to prove the theorem. Proof. If L1 < 0, then for some type profles, the agents total valuaton for every outcome s negatve. For these type profles, ndvdual ratonalty and non-defct cannot coexst. If L1 0, then under any shfted Groves mechansm, outcome 1 never ncurs defct. We may smply shft down every outcome except for 1 (for all > 1, set P to be really large), so that only outcome 1 can be possbly chosen (e.g., ensure that for all > 1, P + U < L1). The resultng mechansm smply always pcks outcome 1, whch guarantees the non-defct property accordng to Lemma 1. We already know that all shfted Groves mechansms are strategy-proof (they are part of the AMA famly) and ndvduallyratonal(thewaywesettheh (θ )termguarantees ndvdual ratonalty). In concluson, f L1 < 0, no mechansms can be both ndvdually ratonal and non-defct. If L1 0, there exst shfted Groves mechansms that are strategy-proof, ndvdually ratonal, and non-defct. 6 It should be noted that even f all strategy-proof mechansms are affne maxmzers for our settng, we stll need to prove that only shfts are needed to reach the optmal mechansm, snce affne maxmzers also rely on weghts (multplcaton). Based on the proof of Theorem 1, from ths pont on, we assume L1 0. Otherwse, no feasble mechansms exst. For many settngs, there exsts an outcome whch corresponds to the status quo (e.g., not buld n the publc proect problem). The agents valuatons for the status quo are all equal to 0. If the status quo belongs to the set of outcomes, then accordng to Theorem 1, there exst mechansms that are strategy-proof, ndvdually ratonal, and non-defct. Gven L1 0, we also have that for all, U 0, because our assumpton s that there do not exst two outcomes, so that the agents always prefer one to the other for all type profles. We do not have to study all shfted Groves mechansms. Frst of all, we only consder those that are non-defct (we call these shfted Groves mechansms feasble). We also gnore those that are not onto. Defnton 3. AshftedGrovesmechansmM(P 1,P,...,P k ) s onto ff under t, for every outcome, there exsts at least one type profle for whch s chosen by the mechansm (or may be chosen by the mechansm assumng random te-breakng). Formally, ths means that for every par of outcomes a and b, we have P a + U a P b + Lemma. Let M(P 1,P,...,P k ) be a shfted Groves mechansm that s feasble but not onto. There must exst another feasble shfted Groves mechansm that s onto, and has the same or hgher allocatve effcency for all type profles. L b Proof. IfM(P 1,P,...,P k )sfeasblebutnotonto, then there exst two outcomes a and b, so that P a + U a < P b + We can smply decrease P a so that P a + U a = P b + Ths modfcaton does not change the allocatve effcency of any type profle, except for maybe when v a = U a and v b = L b for all. For ths case, a may be chosen nstead of b. We have that U > L for every par of and. That s, the above modfcaton never decreases the allocatve effcency. The modfcaton also does not change the payments of any agents when outcome a s not chosen. So for outcomes other than a, the modfed mechansm s stll non-defct. Fnally, when a s ndeed chosen, agent s payment equals L b L b max{ P +L + v }+P a v a P b +L b + P a + v b +P a v a U a +P a v a U a The total payment s then at least Ua, whch s at least 0 (at least L1). In summary, the modfed mechansm

5 remans feasble and has the same or hgher allocatve effcency. We can repeat the above process untl the mechansm becomes onto. Theorem. Let M(P 1,P,...,P k ) be a shfted Groves mechansm that s both feasble and onto. M(P 1,P,...,P k ) s worst-case effcency loss s max P mn P. Theorem drectly follows from the followng two lemmas: Lemma 3. Let M(P 1,P,...,P k ) be a shfted Groves mechansm. M(P 1,P,...,P k ) and M(P 1,P,...,P k ) are equvalent ( s an arbtrary constant). The above lemma drectly follows from Defnton. It mples that M(P 1,P,...,P k ) and M(P 1 mn P,P mn P,...,P k mn P ) are equvalent. That s, t s wthout loss of generalty to focus on cases where the P are non-negatve, and at least one of the P s 0. Lemma 4. Let M(P 1,P,...,P k ) be a shfted Groves mechansm that s feasble and onto. The P are non-negatve and mn P = 0. M(P 1,P,...,P k ) s worst-case effcency loss s max P. Proof. If max P = 0, then the mechansm s smply a Groves mechansm, whch s effcent (worst-case effcency loss s 0). We then analyze cases wth max P > 0. Frst of all, t s easy to see that the worst-case effcency loss s at most max P. We only need to prove that t s also at least ths much. Let P a = 0 and P b = max P. a and b are dfferent. We construct the followng type profle: v = L for all and all a,b. Due to the onto property, we have Ua = P a + Ua max( P + L). We certanly have La = Pa+ La max( P + L). Therefore, we can set the v a so that va = max( P + L). Ths ensures that a s at least ted wth all b. Agan, due to the onto property, we have P b + U b max ( P + L) P b + L b. We set the v b so that outcome b s ted wth a. For ths type profle, snce we assume random te-breakng, a may be chosen ahead of b, whch corresponds to an effcency loss of max P. Next we try to fnd the optmal shfted Groves mechansm. As mentoned earler, we only need to consder shfted Groves mechansms that are feasble and onto. Also, we only need to consder cases where the P are non-negatve and at least one of the P s 0. Defnton 4. AshftedGrovesmechansmM(P 1,P,...,P k ) s optmal, f there does not exst another shfted Groves mechansm M(P 1,P,...,P k), so that P P for all. Our am s to mnmze the worst-case effcency loss. Based on Lemma 4. the optmal shfted Groves mechansm should mnmze max P. We wll prove later on that ths s ndeed the case, and the optmal mechansm s unque. Before that, we frst propose a lnear program based AMD approach for dentfyng the optmal mechansm. Gven a shfted Groves mechansm M(P 1,P,...,P k ), we can calculate the maxmal defct for outcome usng the followng lnear program: Constants: L, U, P, Varables: v Dummy Varables: d (d represents agent s payment) Maxmze: d (maxmze defct) Subect to: For all, For all, P + d ( P +L + For all and, v P + v (1) v )+P v () L v U Constrant 1 ensures that we are dealng wth type profles where outcome s chosen. Constrant ensures that d represents agent s payment. That s, d = max{ P +L + v }+P v Ths s true because the lnear program wll mnmze the d n order to maxmze the obectve. The above lnear program can be sgnfcantly smplfed. We may dvde Constrant nto the followng two: For all, d ( P +L + For =, d ( P +L + v )+P v (3) v )+P v = L (4) Constrant 4 has nothng to do wth the v. For Constrant 1 and 3, they are the most relaxed f we set v = L for all and all, and we set v = U for all. After smplfcaton, we have Constants: L, U, P, Dummy Varables: d (d represents agent s payment) Maxmze: d (maxmze defct) Subect to: For all, P + For all and all, U P + L (5) d ( P + L )+P U (6) d L (7) Constrant 5 s already guaranteed by the onto property. The obectve value of the above lnear program s smply max{max + L { P }+P U,L }

6 We use D (P 1,P,...,P k ) to denote the above obectve value. That s, D (P 1,P,...,P k ) represents the maxmal defctundertheshftedgrovesmechansmm(p 1,P,...,P k ), when outcome s chosen, We notce that D (P 1,P,...,P k ) s non-ncreasng n P and non-decreasng n P for. Ths leads to the followng AMD approach. Start from P 1 = P =... = P k = 0. That s, we start from the orgnal Groves mechansm, whch s onto but not necessarly feasble. For every outcome, f D (P 1,P,...,P k ) s postve, thenncreasep untld (P 1,P,...,P k )reaches 0. Ths move mantans the onto property. If after the modfcaton, t s no longer onto, then we have P + U < max { P + L}. For D (P 1,P,...,P k ), we then have max{max { P+ L }+P U,L } < (max + L { P }+P U ) ( U U ) = U 0 That s, D (P 1,P,...,P k ) s negatve, whch contradcts wth the algorthm descrpton. Theorem 3. The AMD process produces a unque optmal shfted Groves mechansm that s feasble, onto, and has mnmal worst-case effcency loss among all shfted Groves mechansms. Proof. If we follow the AMD steps, then P 1 must reman 0, because accordng to Lemma 1, D 1(P 1,P,...,P k ) s never postve. Ths mples that n all calculatons, for > 1, P 1 + L 1 = L 1 L P + L Ths fact further smplfes D (P 1,P,...,P k ) for > 1: max{max + L { P }+P U,L } = max{ L 1 +P U,L } That s, the maxmal defct for outcome depends only on P (the other P values are rrelevant). Therefore, the AMD algorthm does produce a feasble mechansm that s onto. Next, we prove the optmal mechansm s unque. Let M(P 1,P,...,P k) be an optmal mechansm. When we start the AMD process, we have P 1 = P =... = P k = 0. That s, we have P P for all. Durng the AMD, f D (P 1,P,...,P k ) > 0. The AMD process ncreases P to P + so that D (P 1,P,...,P k ) = 0. We recall that D (P 1,P,...,P k )snon-ncreasngnp andnon-decreasng n P for. Therefore, 0 = D (P 1,...,P 1,P +,P +1,...,P k ) We also have D (P 1,...,P 1,P +,P +1,...,P k) D (P 1,...,P 1,P,P +1,...,P k) 0 Therefore, wehavep P +. Thats, aftertheamd,we havep P forall. SnceweassumedM(P 1,P,...,P k) to be optmal, we should have P = P for all after the AMD. Snce M(P 1,P,...,P k) s arbtrarly chosen, the optmal mechansm must be unque. Ifthereexstsanon-optmalmechansmM(P 1,P,...,P k) that mnmze the worst-case effcency loss. Then there must exst an optmal mechansm M(P 1,P,...,P k) satsfyngp P forall. ThenbyLemma4, M(P 1,P,...,P k) should have the same or less worst-case effcency loss. Fnally, based on Lemma 1, for all wth L 0, we have D (P 1,P,...,P k ) 0 for all the P. Ths mples that when conductng the AMD, f L 0, we smply leave P to be GLOBAL OPTIMALITY Theorem 3 only shows that the optmal shfted Groves mechansm s optmal wthn the famly of shfted Groves mechansms. Here, we show that t s also optmal among all determnstc mechansms that are strategy-proof, ndvdually ratonal, and non-defct. Lemma 5. When there are only two outcomes, the optmal shfted Groves mechansm has the lowest worst-case effcency loss among all determnstc mechansms that are strategy-proof, ndvdually ratonal, and non-defct. Proof. We denote outcome 1 by a and outcome by b. We have the usual assumptons that La L b and La 0. Let H be a determnstc mechansm that s strategy-proof, ndvdually ratonal, and non-defct. Let beh sworst-caseeffcencyloss. Weshowthatthereexstsa shfted Groves mechansm wth the same or lower worst-case effcency loss. If U b La, then we can smply construct a trval shfted Groves mechansm that always chooses a (choosng a never results n defct), and ths mechansm s worst-case effcency loss s at most. Therefore, we only need to consder < U b La. We consder the shfted Groves mechansm M(0, + ǫ), where ǫ s a small postve value ( +ǫ < U b La). For smplcty, we smply call ths mechansm M. M s worstcase effcency loss s at most +ǫ. If we can show that M s non-defct, then that means one feasble shfted Groves mechansm has a worst-case effcency loss of at most +ǫ. ǫ can be made arbtrary small, whch means that there must exst a feasble shfted Groves mechansm wth a worst-case effcency loss of at most. Ths proves that the optmal shfted Groves mechansm s globally optmal. Now we prove that M s non-defct. When outcome a s chosen, by Lemma 1, defct wll not occur. We only need to prove that when M chooses b, there wll not be defct. Whenever M chooses b, we know that the agents prefer b to a by at least +ǫ, whch means that H must also choose b. Let (θ,θ ) be a type profle for whch M chooses b. For ths type profle, H must also choose b. We analyze agent s payment under H and M.

7 Case 1: Under M, gven θ, no matter what agent reports, the outcome s always b. If ths s the case, then under H, gven θ, no matter what agent reports, the outcome s always b. (Whenever b s chosen under M, H must also choose b.) H s determnstc. Agent s payment under H must be fxed (does not change wth the report). Otherwse, H s not strategy-proof. Let x be agent s payment. x must be at most L b. Otherwse, H s not ndvdually ratonal. Agent s payment under M s exactly L b accordng to the defnton. That s, for Case 1, agent pays the same or more under M. Case : Under M, gven θ, both a and b may be chosen (dependng on θ ). When b s chosen, agent pays max{l a + (v a v b)+ +ǫ,l b } We dvde Case nto the followng sub-cases. Case : If under H, gven θ, no matter what agent reports, the outcome s always b, then as analyzed earler, agent pays at most L b. That s, for Case, agent pays the same or more under M. Case : Under H, gven θ, for some θ, outcome a s chosen, and for some other θ, outcome b s chosen. We recall that H s determnstc. When a s chosen, agent s payment must be a fxed amount that s ndependent of θ. Otherwse, H s not strategy-proof. We use p H a to denote ths amount. Smlarly, we use p H b to denote the fxed payment pad by whenever b s chosen. Let C H = p H b p H a. Ths C H value s a crtcal value: whenever agent favors outcomebby more than C H, outcomebwll be chosen (otherwse, a s chosen). Ths crtcal value C H must be less than (v a v b)+ +ǫ. Otherwse, H may choose a when the agents favor b by more than, whch contradcts wth the fact that H s worst-case effcency loss s. H s ndvdually ratonal, whch means that we ether have p H a L a or p H b L b. Otherwse, agent may end up not able to afford ether outcome. So we have p H b C H L a or p H b L b. That s, p H b max{l a +C H,L b } max{l a + (v a v b)+ +ǫ,l b } Therefore, for Case, agent pays the same or more under M when b s chosen. Theorem 4. The optmal shfted Groves mechansm has the lowest worst-case effcency loss among all determnstc mechansms that are strategy-proof, ndvdually ratonal, and non-defct. Proof. If there s only one outcome, then the optmal shfted Groves mechansm s the orgnal Groves mechansm, whch obvously mnmzes the worst-case effcency loss (as t s effcent). Lemma 5 already proved the case wth two outcomes. We then consder cases wth more than two outcomes. We frst analyze the worst-case effcency loss of the optmal shfted Groves mechansm M(P 1,P,...,P k). Reusng theanalyssntheproofoftheorem3, wemusthavep 1 = 0. For > 1, P = 0 f Otherwse, P s the mnmal value that makes max{ L 1 +P U,L } = 0 Accordng to Lemma 4, the worst-case effcency loss s smply max P. Let P b = max P. If P b = 0, then the shfted Groves mechansm s effcent, whch obvously mnmzes the worst-case effcency loss. We only consder P b > 0. When P b > 0, P b s the mnmal value that makes max{ L 1 +P b U b,l b } = 0 Let us then consder the followng settngs: Settng 1 s the orgnal settng. That s, for all and, v s n [L,U ]. Settng s a restrcted verson of settng 1. Outcome 1: For all, let v 1 = L 1. Outcome b: For all, let v b s range be the same as before, that s, n [L b,u b ]. Outcome other than 1 and b: For all, let v = C. C s a constant chosen from [L,U ]. We need C = L1. The C exst because L L1 U. Settng 3 s obtaned by droppng all outcomes other than 1 and b. Outcome 1: For all, let v 1 = L 1. Outcome b: For all, let v b s range be [L b,u b ]. Let H be the strategy-proof, ndvdually ratonal, and non-defct mechansm that mnmzes the worst-case effcency loss n settng. Let α be H s worst-case effcency loss. α 1 α : Settng s a restrcted verson of settng 1, so H 1 apples n settng, whch means that the optmal worst-case effcency loss n settng s at most α 1. α α 3: We frst consder settng and H. It s wthout loss of generalty to assume that H only chooses outcome 1 and b, because gven a type profle, f H chooses outcome 1,b, then we may smply change H so that t chooses 1 nstead. Then for all, ask agent to pay an addtonal payment equal to L 1 C. Ths makes the agents as f they are choosng outcome. Ths change does not affect the agents utltes (strategy-proofness and ndvdual ratonalty are not affected). Ths change does not affect the total payment, as (L1 C) = 0 (non-defct s not affected). Ths change does not affect the allocatve effcency, as the agents total valuatons for outcome 1 and are the same. In concluson, we can apply H to settng 3, whch mples that the optmal worst-case effcency loss n settng 3 s at most α. Above, we have proved that α 1 α 3. Accordng to Lemma 5, n settng 3, α 3 s acheved by the optmal shfted Groves mechansm. We run the AMD process on settng 3. We get that α 3 s the mnmal value that makes max{ L 1 +α 3 U b,l b } = 0 max{ L 1 U,L } 0 Now earler we have shown that the optmal shfted Groves mechansm n settng 1 (the orgnal settng) has the same

8 worst-caseeffcencylossasα 3. Gventhatα 1 α 3, wemust have that the optmal shfted Groves mechansm s optmal n the orgnal settng, whch concludes the proof. Corollary 1. There exsts a mechansm that s strategyproof, ndvdually ratonal, non-defct, and effcent, ff for all > 1, we have max{ L 1 U,L } 0 Proof. Accordng to Theorem 4, there exsts a mechansm that s strategy-proof, ndvdually ratonal, non-defct, and effcent, ff the optmal shfted Groves mechansm s effcent. Now the optmal shfted Groves mechansm s effcent ff t s M(0,0,...,0). That s, the optmal shfted Groves mechansm s effcent ff the AMD process carres out no updates at all, whch s exactly when for all > 1, we have max{ L 1 U,L } 0 If for all > 1, L 0, then the above corollary apples. Inthscase,theorgnalGrovesmechansmM(0,0,...,0) (effcent) s the optmal shfted Groves mechansm. 5. NUMERICAL EXPERIMENTS Gven a settng, let the optmal shfted Groves mechansm be M(P 1,P,...,P k). We recall that we must have P 1 = 0. For > 1, P = 0 f Example 3. There are n agents. We need to choose at most wnners (e.g. there are Ad slots for n advertsers). There are 1 + n + n(n 1) outcomes. There s 1 outcome that selects 0 wnners. There are n outcomes that select 1 wnner. Fnally, there are n(n 1) outcomes that select wnners. Agent s valuaton for an outcome conssts of two parts. One part s her valuaton for wnnng, whch s a value from 0 to 1. The other part s her externalty toward the other agent selected by the outcome, f she s not the only wnner. The externalty value s from 1 to 1. For outcome {} (no wnners), the agents valuatons are all 0. For outcome {a} (the only wnner s a), agent a s valuaton s from 0 to 1, and the other agents valuatons are all 0. For outcome {a,b} (the wnners are a and b), agent a s valuaton s from 1 to (same for b), and the other agents valuatons are all 0. As usual, we sort the outcomes accordng to the agents mnmal total valuatons. So outcome 1 should be {}. We have L 1 = U 1 = 0. The next n outcomes should be {a} for a = 1,,...,n. For outcome among these outcomes, we have L = 0, so these outcomes never cause defct and can be gnored. Fnally, the next n(n 1) outcomes should be {a,b} for every par of a and b from 1,,...,n. The worst-case effcency loss s caused by one of the outcomes that selects wnners. Due to symmetry, we only consder the outcome {1, }. We denote ths outcome by. We have that the worst-case effcency loss for ths outcome s P, where P s the mnmal value that makes max{ L 1 +P U,L } = 0 max{ L 1 U,L } 0 Pluggng n the numbers, we have Otherwse, P s the mnmal value that makes max{p U,L } = 0 max{ L 1 +P U,L } = 0 At the end, the maxmal P defnes the optmal worst-case effcency loss. We revst Example 1 (the publc proect problem). There are only two outcomes. The optmal shfted Groves mechansm s M(0,P ). For =, we have = max{ L 1 U,L } max{ n 1 n, 1 n } > 0 We then need to fnd the mnmal P that makes max{p n 1 n, 1 n } = 0 We get that P = (n 1), whch s then the optmal worstcase effcency loss of the publc proect problem. n Resource allocatons wth externaltes can also be modelled as socal decson problems, for whch our AMD approach can be appled. Below we present a varant of the k-wnner selecton problem descrbed n [16]. Smplfy t, we get max{p, 1} (n )max{p 4,0} = 0 Solvng the above equaton, we get that P should be. That s, the optmal worst-case effcency loss s. 6. CONCLUSION For the problem where a group of agents need to choose from a fnte set of outcomes, we proposed the optmal shfted Groves mechansm, whch s strategy-proof, ndvdually ratonal, non-defct, and mnmzes the worst-case effcency loss. One mmedate future research drecton s to study other obectves, such as mnmzng the expected effcency loss gven a pror dstrbuton. We may also consder settngs where we drop certan assumptons (e.g., an agent s valuaton space s not a closed nterval, or the agents valuatons are nterdependent). 7. ACKNOWLEDGMENTS Ths work was partally supported by JSPS KAKENHI Grant Number and , and JST PRESTO program.

9 REFERENCES [1] R. Cavallo. Optmal decson-makng wth mnmal waste: Strategyproof redstrbuton of VCG payments. In Proceedngs of the Internatonal Conference on Autonomous Agents and Mult-Agent Systems (AAMAS), pages , Hakodate, Japan, 006. [] V. Contzer and T. Sandholm. Complexty of mechansm desgn. In Proceedngs of the 18th Annual Conference on Uncertanty n Artfcal Intellgence (UAI), pages , Edmonton, Canada, 00. [3] B. Faltngs. A budget-balanced, ncentve-compatble scheme for socal choce. In Agent-Medated Electronc Commerce (AMEC), LNAI, 3435, pages 30 43, 005. [4] T. Groves. Incentves n teams. Econometrca, 41: , [5] M. Guo and V. Contzer. Worst-case optmal redstrbuton of VCG payments n mult-unt auctons. Games and Economc Behavor, 67(1):69 98, 009. [6] M. Guo and V. Contzer. Computatonally feasble automated mechansm desgn: General approach and case studes. In Proceedngs of the Natonal Conference on Artfcal Intellgence (AAAI), pages , Atlanta, GA, USA, 010. NECTAR track. [7] M. Guo, E. Markaks, K. R. Apt, and V. Contzer. Undomnated groves mechansms. Journal of Artfcal Intellgence Research, 46:19 163, 013. [8] M. Guo, V. Narodtsky, V. Contzer, A. Greenwald, and N. R. Jennngs. Budget-balanced and nearly effcent randomzed mechansms: Publc goods and beyond. In Proceedngs of the Seventh Workshop on Internet and Network Economcs (WINE), Sngapore, 011. [9] B. Holmström. Groves scheme on restrcted domans. Econometrca, 47(5): , [10] A. Lkhodedov and T. Sandholm. Methods for boostng revenue n combnatoral auctons. In Proceedngs of the Natonal Conference on Artfcal Intellgence (AAAI), pages 3 37, San Jose, CA, USA, 004. [11] A. Lkhodedov and T. Sandholm. Approxmatng revenue-maxmzng combnatoral auctons. In Proceedngs of the Natonal Conference on Artfcal Intellgence (AAAI), Pttsburgh, PA, USA, 005. [1] D. Mshra and A. Sen. Roberts theorem wth neutralty: A socal welfare orderng approach. Games and Economc Behavor, 75(1):83 98, 01. [13] H. Mouln. Almost budget-balanced VCG mechansms to assgn multple obects. Journal of Economc Theory, 144(1):96 119, 009. [14] V. Narodtsky, M. Guo, L. Dufton, M. Polukarov, and N. R. Jennngs. Redstrbuton of VCG payments n publc proect problems. In Proceedngs of the Eghth Workshop on Internet and Network Economcs (WINE), Lverpool, 01. [15] K. Roberts. The characterzaton of mplementable socal choce rules. In J.-J. Laffont, edtor, Aggregaton and Revelaton of Preferences. North-Holland Publshng Company, [16] Y. Sakura, T. Okmoto, M. Oka, and M. Yokoo. Strategy-proof mechansms for the k-wnner selecton problem. In Prncples and Practce of Mult-Agent Systems (PRIMA), Dunedn, New Zealand, 013.