Lecture 8. v i p i if i = ī, p i otherwise.

Transcription

1 CS-621 Theory Gems October 11, 2012 Lecture 8 Lecturer: Aleksander Mądry Scrbes: Alna Dudeanu, Andre Gurgu 1 Mechansm Desgn So far, we were focusng on statc analyss of games. That s, we consdered scenaros n whch the game and utltes of all the players are fxed and known and our task s only to predct possble outcomes of that game when some (or all) the players are actng ratonally. Today, we turn the tables: there s no predefned game, only players that have some utltes. However, the key pont s that these utltes are prvate. That s, we have no access to them (we only know a unverse they are comng from) so, n partcular, when players clam to have some utlty functon there s no way for us to know f they are tellng the truth. Our goal now s to desgn a game that compels players that are actng ratonally (wth respect to ther prvate utltes) to choose an outcome that maxmzes the socal welfare,.e., a one that maxmzes the sum of (prvate) utltes of all the players. (Note that an outcome that maxmzes the socal welfare mght not necessarly be optmal from the pont of vew of any partcular player. So, the dffculty here s to ensure that the socal-welfare outcome s stll the preferable one for all the players and, furthermore, to do t n a way that does not even requre us to know what ther actual utltes are.) 2 Vckrey Aucton Let s start wth a motvatng example. Consder the followng setup: we have one tem to aucton and n bdders, each of them has a prvate valuaton v of the tem. Our goal s to desgn a way of auctonng the tem that ensures that the tem goes to the person that values t most ths can be seen as correspondng to maxmzaton of socal welfare. Note that n the aucton settng that s probably most famlar to us, an auctoneer cares for somethng else: to maxmze hs/her own revenue. Ths s not the case here, and there actually are real-world stuatons where a socal-welfare objectve makes sense. For example, when the government s auctonng rado frequences, ts man goal (nstead of just makng proft) s to ensure that whoever gets these frequences wll be wllng and capable of utlzng them to greatest extent. Another real-world scenaro s auctonng blocks of unused IP addresses (that due to runnng out of IP space become a sought-after commodty). The not-for-proft body that oversees ths process s obvously nterested n gvng them to an organzaton that needs t (and thus values t) most and gettng a revenue s actually undesrable. Before we proceed, let us cast the above scenaro nto a more formal framework. The way we wll vew an aucton s as a process n whch frst, each bdder submts a bd b. Next, there s a publc (.e., known to every bdder before placng hs/her bds) outcome functon f(b 1,..., b n ) = (ī, p 1,..., p n ) that based on these bds determnes the wnner ī of the aucton (to whom the tem s gven), as well as, payments p that for each bdder that he/she has to pay. (Note that, n prncple, we allow here stuatons n whch bdder has to pay some amount even f he/she dd not wn an tem.) Now the resultng utlty u of bdder s defned as { v p f = ī, u (ī, p 1,..., p n ) = p otherwse. So, our task here s to choose the outcome functon f n such a way that bdders that are ratonal wth respect to ther utlty functons (and f) are compelled to favor the outcome that maxmzes the socal welfare,.e., gves out the tem to the bdder that has maxmum valuaton v of t. Also, to make sure that any ratonal bdder wll be nterested n partcpatng n the aucton, we mpose an addtonal condton that every bdder can always make hs expected utlty non-negatve. In our case, ths bols 1

2 down to requrng that only the wnner can be charged wth non-zero payment,.e., p s zero whenever ī. At ths pont, the queston s: what s the rght choce of the wnner and what payment should he/she be charged wth? Gven that we are not nterested n gettng a revenue and want to just gve the tem to the bdder that has the hghest valuaton for t, a temptng approach would be to always gve the tem to the hghest bdder and not charge anyone. The hope here would be that the bds wll reflect the actual valuatons and thus ths ndeed wll lead to the desred outcome. Unfortunately, t s easy to see that such approach wll fal mserably n ths settng. As none of the bdders s requred to be truthful about ther prvate valuatons, the ratonal strategy s for every bdder to just le by bddng + rregardless of the actual valuaton. Clearly, that s not the rght soluton. One natural attempt to fxng the above over-bddng problem s to make the bdders accountable for ther bds. That s, one could consder so-called frst-prce aucton n whch stll the hghest bdder gets the tem (wth tes broken arbtrarly), but the payment of the wnner has to be equal to hs/her stated bd. Ths prevents over-bddng, as wnnng an aucton wth nflated bd results n negatve utlty, but leads to an opposte problem - underbddng. Namely, n ths case nobody has an ncentve to bd hs/her true valuaton of the tem, as dong so guarantees zero utlty (no matter f the tem s won or not). So, the resultng dynamcs would be that each bdder tres to underbd n hope that the resultng bd wll stll be hgh enough to wn the aucton (provded the orgnal valuaton s suffcently large), whle leavng some postve margn of utlty n case of the wn. As no bdder has any pror nformaton about the valuatons of the other bdders, ths dynamcs s completely unpredctable and mpossble to analyze wthn our framework (.e., wthout any assumptons on the prors of the bdders). Even more mportantly, as an auctoneer we would never be sure f the resultng outcome of the aucton s ndeed optmal from socal welfare pont of vew, or just some bdder was more lucky wth hs/her choce of the bd. Ths motvates lookng for a better soluton. The crucal nsght here comes from consderng what would happen n frst-prce aucton f the valuatons of the bdders were actually publc. It s not hard to see that n ths case, f k s the bdder wth the k-th largest valuaton then the best underbd for each bdder k s the valuaton v k+1 of the next bdder n ths orderng. In fact, one can show (we essentally prove t n Lemma 1) that such bds consttute a domnant strategy n ths settng. The above observaton motvates usng so-called second-prce (or Vckrey) aucton, n whch the tem s stll awarded to the hghest bdder, but the payment of ths bdder s equal to the second hghest bd. As t turns out, once we make ths modfcaton, the game-theoretc propertes of the resultng aucton mprove dramatcally. In partcular, as we prove below, ths aucton s ncentve compatble (IC),.e., bddng the true valuaton s a domnant strategy for all the bdders. Lemma 1 Second-prce aucton s ncentve compatble (IC). Proof Let us focus on the perspectve of a bdder n ths aucton and, for the sake of the argument, assume he/she submts a bd b that s dfferent than hs/her true valuaton v. Let us frst consder the case of b > v. If someone outbds the bdder then he/she could have as well bd b = v, snce he/she would not wn anyway. On the other hand, f he/she does wn wth ths bd, then the only stuaton n whch just bddng v would not lead to wnnng too would be f some other bdder, say j, had hs/her bd below b, but stll above v. However, n ths case, bdder wll be forced to pay more than hs/her valuaton, and the resultng utlty wll be negatve. So, we see that bddng v s always not worse (and sometme actually better) than bddng b > v. Now, to consder the complementary case of b < v. If wns wth ths bd, then so would he/she wth bddng v and obvously the payment would be the same. On the other hand, f he/she loses, then let us consder the wnnng bd b j of some bdder j. If b j v then bddng v would not make any dfference n resultng utlty (t stll would be zero). However, f b j s bgger than b, but smaller than v then actually bddng v would lead to wnnng and thus gettng postve utlty. 2

3 So, we see that ndeed bddng exactly v always leads to the best outcome, rrespectvely of other bds, and sometme devatng from ths bd can actually lower the resultng utlty. Ths means that bddng one s own true valuaton s a (strctly) domnant strategy, as desred. Note that one consequence of second-prce aucton beng ncentve compatble s that we, as an auctoneer, are certan that as long as all the bdders are ratonal they are bound to bd truthfully. As a result, we can be sure that the tem s ndeed allocated n socally optmal way,.e., t always s gven to the bdder that values t most. So, by settng up ths aucton we managed to acheve qute remarkable feat. We managed to leverage the bdders own ratonalty to make them dsclose to us ther prvate valuatons and choose a socally optmal outcome even though from perspectve of everyone but the wnner, ths outcome s very suboptmal. 3 Mechansm Desgn Wthout Money After fndng the soluton for the aucton problem above, t s natural to wonder for what other type of problems a smlar soluton can be obtaned. Also, as we already mentoned, havng to use money payments s undesrable n some scenaros, therefore we would lke to nvestgate a possblty of dong wthout them. To ths end, let us frst defne more precsely our goal here. Once agan, we wll be nterested n desgnng a game (mechansm). Let A be the set of ts possble outcomes. There wll be n players wth prvate utltes u 1,..., u n, where each of these utltes comes from the same unverse and utlty u provdes correspondng player s valuaton of every possble outcome from A. Now, the dynamcs of the game s that each player submts hs alleged utlty functon u (that mght or mght not be true) and then there s a publc (.e., known to everyone beforehand) functon f that maps all u s nto an outcome a of the game,.e., f(u 1,..., u n) = a. Clearly, functon f that we wll call the socal choce functon s the core of the game descrpton and our goal s to choose t n a way that makes the resultng game have a domnant strategy ū such that f(ū ) = arg max u (a),.e., we want that f all the players are ratonal then they are compelled to follow ths strategy ū and ths strategy wll lead to choosng an outcome a that maxmzes the socal welfare u (a). (Note that the socal welfare s measured n terms of the prvate utltes of the players, not the submtted ones.) To relate the above framework to our aucton example from above, note that there one can thnk that the set A corresponds to all possble choces of the wnner, and the utlty of each player s equal to hs/her valuaton v of the tem f he/she wns the tem and s zero otherwse. Then, submttng a bd can be vewed as declaraton of havng utlty functon wth correspondng valuaton, and maxmzng the socal welfare functon corresponds exactly to gvng the tem to a player that values ts most. (Note, however, that we do not have payments here.) Before proceedng further, we note that although above we just requre that there exsts some domnant strategy ū that leads to maxmzaton of the socal welfare, we can actually wthout loss of generalty requre that ths strategy conssts of each player beng truthful about hs/her utlty functon. Lemma 2 (Revelaton prncple) If there exsts a socal choce functon f that facltates a socalwelfare-maxmzng domnant strategy ū, then there also exsts a socal choce functon f that s ncentve compatble,.e., n whch ths domnant strategy s just truthful submsson of everyone s utlty functons. Proof Let us fx some game wth socal choce functon f whose socal-welfare-maxmzng domnant strategy ū s not truthful,.e., ū ū. Observe that whenever player s ratonal, there s a fxed reasonng that leads hm/her to play u gven hs/her prvate utlty functon s u,.e., we can exactly 3

4 model the way n whch the players choose to le. We can then consder a new game n whch the socal choce functon f frst apples ths reasonng to the submtted utlty functons and then apples the functon f to the output of that reasonng. It s not hard to see that n ths new game beng truthful consttutes a domnant strategy, as desred. In the lght of the above, from now on, we can always constrant ourselves to lookng for mechansms that are ncentve compatble and ths does not reduce the generalty of our nvestgaton. Now, as we already argued, beng able to come up wth an ncentve compatble socal choce functon for a large class of useful problems would be very powerful. Unfortunately, as the followng theorem states, ths goal s too dealstc and essentally n any nterestng settng, such a functon has to be necessarly not too useful. Theorem 3 (Gbbard-Satterthwate) If the socal choce functon f s IC, s onto A and A 3, then f s a dctatorshp,.e. f(u 1,..., u n) = f(u ), for some fxed ī. ī We dd not prove ths theorem n the class. The proof can be found, e.g, n [1]. Roughly speakng, the proof of t s based on applcaton of Arrow s Impossblty Theorem, whch (roughly) states that any reasonable votng system wth at least three alternatves has to be a dctatorshp. Gbbard and Satterthwate showed that exstence of a non-dctatorshp socal functon as n the statement of ther theorem, would also mply exstence of non-dctatorshp reasonable votng system. Thus, the desred mpossblty statement follows. 4 Mechansm Desgn wth Money The Gbbard-Satterthwate Theorem effectvely klls our dreams of desgnng mechansm wthout money. So, we now turn our attenton to the settng when payments are allowed. In ths settng, we agan have n players and a set of possble outcomes A. Also, each player has a prvate preference v (comng from some fxed unverse) that ranks/evaluates from the perspectve of that player all the possble outcomes n A. Smlarly to the prevous case, the dynamcs of the game s that each player wll declare frst hs/her alleged preference v (that agan mght or mght not be true) and there s a publc functon f (whch we wll now smply call a mechansm) that gven the vector (v 1,..., v n) of declared preferences produces an outcome a, as well as, a vector (p 1,..., p n ) of payments for all the players. Now, gven the outcome and the payments, the utlty u of player s equal to hs/her preference v (a) of the obtaned outcome mnus the payment p that he/she needs to pay,.e., u (a) = v (a) p. Our task, agan, s to come up wth the mechansm f that s ncentve compatble and whose correspondng truthful strategy results n maxmzng the socal welfare,.e., declarng v = v by all the players s a domnant strategy and n ths case our mechansm should produce an outcome a such that a = arg max v (a). Note that the socal welfare s based only on the (prvate) preferences v of the players and not on ther utltes u (that nclude the effect of payments). Ths dstncton s crucal, as n ths way we are able to ncentvze the players to the desred behavor by nfluencng ther utlty va money, whle not affectng our objectve functon (socal welfare). 4.1 The Vckrey-Clarke-Grove (VCG) Mechansm After the unpleasant falure of mechansm desgn wthout money, one mght also become skeptcal about the mechansm desgn wth money. Maybe the second-prce aucton s just one of very few tasks for whch mechansm desgn s possble? 4

5 Fortunately, t turns out that, once money payments are allowed, there s a very elegant, versatle and essentally automatc way of obtanng good mechansms: the Vckrey-Clarke-Grove (VCG) Mechansm. To descrbe the VCG mechansm, let us fx some set of outcomes A, a set of n players, and ther prvate preferences v 1,..., v n. We should frst note that as we want our mechansm to be ncentve compatble, the choce of outcome a for a gven vector of declared preferences (v 1,..., v n) s already predefned. Namely, t has to be that a = arg max v (a), (1) wth arbtrary te breakng. Otherwse, our mechansm would not be maxmzng socal welfare when players are truthful (whch we want to be the case). So, the only (but crucal!) desgn choce that we need to make s how to set the vector of payments (p 1,..., p n ). In the VCG mechansm, the payment p of player, gven the declared preferences (v 1,..., v n), s p (v 1,..., v n) := v j(a ) + h (v ), (2) where a s the outcome gven by (1) and h (v ) s certan quantty called Clarke s potental that depends on the declared preferences v j of all the players but (we wll make t precse later). To gan some ntuton regardng ths choce of payments, note that the utlty of player wth respect to ths payment becomes u (a, v 1,..., v n) = v (a ) + v j(a ) h (v ), (3) where, agan, a s gven by (1). Now, the crucal thng to notce s that, once we gnore the term h (v ) (that does not depend on the choce of v nor a and thus s beyond control of player ), submttng v = v by player makes the outcome a chosen va (1) become exactly the outcome that maxmzes ths player s utlty (3) (once all the other v j are fxed)! So, player has no ncentve to submt any other choce of v than v - f he/she s truthful he/she s guaranteed to get maxmum utlty that s possble n ths stuaton anyway. We see now that the key property of the choce of payments (2) s that t made our goal (gettng an outcome that maxmzes socal welfare) and goal of every player (maxmzng hs/her utlty) perfectly algned. In the lght of ths, we can conclude wth the followng lemma. Lemma 4 For any choce of Clarke s potentals h 1,... h n, the VCG payment rule (2) results n an ncentve compatble mechansm that maxmzes the socal welfare. 4.2 Clarke s Pvot Rule Although the VCG mechansm, as presented above, meets all the requrements of our model, t stll has one shortcomng t mght make all the payments negatve,.e., make all the players receve money for ther partcpaton. Needless to say, havng a mechansm that loses money s not deal, so let us try to fx that. Ideally, we would lke our mechansm to have two addtonal propertes: (Indvdual Ratonalty) Ratonal players should always get a non-negatve utlty. In our settng, ths means that submttng v = v should never result n a payment that s greater than the preference of the obtaned outcome. (Ths ensures that ratonal players have ncentve to partcpate n the mechansm.); (No Postve Transfers) No players s ever pad any money,.e., all p are always non-negatve; 5

6 To acheve these propertes, we wll use the crank that was not utlzed so far n the VCG mechansm: Clarke s potentals h 1,..., h n. Specfcally, we wll set these potentals accordng to so-called Clarke s Pvot Rule: h (v ) := max v j(a), (4) for each player. Note that ths defnton ndeed depends only on the submtted preferences of all the other players except. Furthermore, we can prove the followng lemma. Lemma 5 The VCG mechansm wth Clarke s pvot rule (4) s ndvdually ratonal and, as long as, all v s are non-negatve, there s no postve transfers. Observe that we can guarantee no postve transfers only f all preferences are non-negatve (.e., all the players vew the outcomes of the mechansm as potentally proftable to them). In fact, one can prove that ths restrcton s unavodable. Proof Indvdual ratonalty follows snce max a v j(a) v j(a ) 0, for any outcome a. To see that the no postve transfer property holds too when v s are non-negatve, note that u (a, v 1,..., v n) = v (a ) + v (a ) max v j(a) 0, snce a s chosen so as to maxmze v (a ), v = v when the player s truthful, and - due to non-negatvty of v s - the maxmum socal welfare can only decreases when there s one less player n the game. Fnally, note that after applcaton of Clarke s pvot rule, we can express the payment p of player correspondng to an outcome a as p (v 1,..., v n) := v j(a ) + max v j(a). Ths quantty has a very ntutve nterpretaton. Namely, note that when all the players are truthful, v j (a ) becomes equal to the socal welfare that all the other players get out of the game, whle max v j (a) s the socal welfare that these players would get f player was not partcpatng. So, the payment of player s equal to the total loss n the socal welfare of the other players that resulted from hs/her partcpaton. 4.3 Examples Let us now take a look at two examples of applcaton of the VCG mechansm The Vckrey Aucton We frst show how the Vckrey/second-prce aucton that we ntroduced at the begnnng of the lecture, can be obtaned drectly from the VCG mechansm. To ths end, let us choose the set A of outcomes to be A = {1,..., n}, wth an outcome a = ī beng just the dentfer of the wnner of the aucton. The preferences of the users are functons of the form { w, f = ī v (ī) = 0, otherwse, 6

7 where w s the prvate valuaton of the tem by player. The outcome and payment of the resultng VCG mechansm are gven by ( ) f(v 1,..., v n) = ī = arg max v, p 1,..., p n. and p (ī, v 1,..., v n) = v j(ī) + max j v j( j), for each player. Clearly, the wnner s always the player that declares hghest valuaton/bd. Now, to understand the payments, note that when s not the wnner, the j that maxmzes the second sum s exactly ī, so p = 0, as desred. (In other words, partcpaton of player n the aucton dd not nfluence the outcome and thus he/she does not owe anythng.) Next, let us consder the case when s the wnner. The frst sum wll be equal to 0, because none of the other players wns. In the second sum, j wll be the player wth the second hghest bd (as he/she would wn f player would not partcpate), so hs valuaton v j s exactly what player owes. Thus ndeed we recovered the second-prce aucton Publc Project Now, consder a stuaton n whch government wants to decde whether to buld a publc project that could beneft n dfferent partes (.e., each party has a beneft w from havng the project bult). As the project s qute costly et us say ts cost s C the government wants to go ahead wth t only f the total beneft to all the partes s at least that large,.e., only f C w. How can t be done, when the benefts w are prvate? (In partcular, the partes mght try to le about ther benefts just to encourage the government to go ahead wth the project.) To cast ths problem nto the VCG framework, let us set A to be A := {Buld, Not buld}. For every player, let us defne hs/her preference to be { w f the project s bult v (a) = 0 otherwse. To make sure that we buld only f the total beneft s bgger than the cost of the project, we ntroduce an addtonal dummy player, who has a negatve beneft C f somethng s bult and 0 otherwse. It s not hard to see that the resultng mechansm wll provde a soluton for our task. However, nterestngly, one can check that the only tme a player owes somethng s when that player makes the dfference between the project beng bult and not beng bult. Ths, n turn, means that unless w s exactly C, the total sum of payments from all the partes wll not cover the cost C of buldng the project. (Agan, one can show that ths s n some sense unavodable.) References [1] Noam Nsan, Tm Roughgarden Eva Tardos, and Vjay V. Vazran, Algorthmc Game Theory, Cambrdge Unversty Press,