Single-Item Auctions. CS 234r: Markets for Networks and Crowds Lecture 4 Auctions, Mechanisms, and Welfare Maximization

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1 CS 234r: Markets for Networks and Crowds Lecture 4 Auctons, Mechansms, and Welfare Maxmzaton Sngle-Item Auctons Suppose we have one or more tems to sell and a pool of potental buyers. How should we decde who gets whch tems? What protocol should we follow? Ths dffers from our prevous study of two-sded markets, snce we are assumng the role of the seller. We ll start wth a smple settng, where we have just one tem to sell. Many dfferent procedures (e.g., auctons) can be used. For example: Englsh Aucton. Ask f anyone s wllng to purchase at an ntal, low prce. If so, call out ncreasngly hgher prces untl only one bdder remans. That bdder wns, and pays the last amount called out. Dutch Aucton. Start wth a hgh prce, and slowly lower the offered prce untl someone says they are wllng to buy. That bdder wns at the fnal prce. Slent Aucton. Each bdder wrtes down a prvate bd and passes t to the auctoneer. The hghest bd wns and pays ther bd. Ths lecture: a formal theory for reasonng about dfferent aucton formats, usng the language of Mechansm Desgn. Def: A mechansm s a protocol between a central authorty (the auctoneer) and one or more agents (bdders). The protocol nvolves one or more rounds of communcaton between the agents and the central authorty. At the concluson of the protocol, the mechansm outputs an allocaton and payments owed by the agents. The agents are assumed to have valuaton functons that assgn value to the possble allocatons. For example, an Englsh aucton nvolves many rounds of communcaton where the auctoneer rases the current prce, and agents specfy whether they are wllng to bd at the current prce. The aucton concludes wth a choce of who wns, and what payment the wnner owes. A slent aucton has just a sngle round of communcaton: the agents send bds to the auctoneer, who then determnes the wnner and payments. The slent aucton s an example of a drect mechansm. Def: In a drect (or drect revelaton) mechansm, each agent declares a valuaton, and the mechansm maps those reports to a choce of outcome and payments. Some notaton: we ll tend to wrte b = (b 1,..., b n ) for the reports of the agents, to dfferentate them from the real valuatons v. We wrte b to mean the reports of everyone except agent. We also tend to wrte x (b) and p (b) for the allocaton assgned to agent, and the payment of agent, respectvely, when agents declare valuatons b. The utlty of agent when the agents report b s the value of the allocaton receved mnus the 1

2 payment: u (b) = v (x (b)) p (b). Def: A drect mechansm s domnant strategy ncentve compatble (DSIC) f for each agent, and for every possble report of the other agents b, agent weakly maxmzes utlty by reportng b = v. That s, for all possble reports b, u (v, b ) u (b, b ). We sometmes say truthful to mean the same thng as DSIC. Note: f a mechansm s DSIC, then t s optmal for each agent to report ther valuaton truthfully, regardless of what the other agents are dong. In other words, agents do not need to know anythng about the market or the preferences of others n order to bd optmally. The followng mportant result n aucton theory mples that we can focus on DSIC drect mechansms. Ths s useful because t means we can restrct our attenton to smple mechansms defned by allocaton and payment functons, rather than arbtrarly complex mult-round protocols. Theorem 1 (Revelaton Prncple) Any outcome that can be mplemented at equlbrum by some mechansm can be mplemented by a DSIC drect mechansm. For example, the Englsh aucton for a sngle tem s outcome-equvalent to the followng Second-Prce Aucton: each agent declares ther value to the auctoneer, the agent wth the hghest declared value wns, and they pay the second-hghest declared value. Ths aucton s also known as the Vckrey Aucton. Theorem 2 The Vckrey aucton s DSIC and mplements the socally effcent outcome. Ths s a specal case of a more general theorem we ll prove later for VCG auctons. But t s a good exercse to try to prove t drectly. Note: Why s the second-hghest bd the rght prce? Note that ths s the mnmum amount that the wnner could have bd and stll won. We call ths the threshold (or crtcal) payment, and t shows up frequently n the desgn of truthful auctons. Note: The Vckrey aucton mplements the socally effcent outcome even f the auctoneer knows nothng about the bdders values. Note: The Vckrey aucton sn t the only truthful sngle-tem aucton. For example, the aucton that gnores the reported valuatons and gves away the tem to a random agent, for free, s also DSIC (but not socally effcent). Also, there are some allocaton rules that cannot be made DSIC; for example, an aucton that gves the tem to the agent wth the smallest bd s not truthful, regardless of what payment rule s used (exercse: why?). We ll revst ths queston of what allocaton rules can be made truthful, when we study revenue-maxmzng auctons n a few weeks. General Auctons The Vckrey aucton s DSIC and mplements the socally effcent outcome, for a sngletem aucton. Can we hope for a smlarly nce result n more general settngs? Here s a very general allocaton problem setup: a space Ω of possble outcomes n agents, where agent has a valuaton functon v : Ω R 0 that assgns a value to each outcome 2

3 For example, n a sngle tem aucton Ω s the set of possble wnners of the tem. In a combnatoral aucton (recall lecture 2), Ω s the set of all parttons of m tems among the bdders. Note that ths formulaton s general enough that an agent s value mght depend not only what they wn, but also what others wn. We can also model publc goods n ths framework: for example, Ω could be a set of shared outcomes, lke a set of possble locatons for a brdge, and v encodes agent s value for each possble brdge locaton. In each of these examples, a mechansm wll choose an outcome ω Ω, and also choose payments to collect from each of the agents. Surprsngly, we can extend the Vckrey aucton to cover ths extremely general setup, and obtan an aucton that s DSIC and mplements the socally effcent outcome. Ths s known as the Vckrey-Clarke-Groves (VCG) mechansm. Def: The VCG Mechansm s a drect mechansm defned as follows. There are n bdders and a space Ω of potental outcomes. Agents declare valuaton functons (b 1,..., b n ) The mechansm returns an outcome ω argmax ω Ω { b (ω)} Agent pays ( ) ( ) p = b j (ω) b j (ω ) max ω Ω j j Note: The outcome chosen, ω, maxmzes the socal welfare, gven the declared valuatons. Note: The payment of agent s the externalty mposed by agent on the other agents. In other words, t s how much happer the other agents would be f agent were not present, and we nstead selected an outcome that maxmzed the total welfare enjoyed by everyone other than. Example: Suppose Ω s a set of 3 possble brdge locatons, {A, B, C}. There are n = 3 agents. The agents report the followng valuatons: b 1 (A) = 10, b 1 (B) = 4, b 1 (C) = 2 b 2 (A) = 3, b 2 (B) = 5, b 2 (C) = 6 b 3 (A) = 1, b 3 (B) = 6, b 3 (C) = 1 Then the VCG mechansm would choose locaton B, snce that s the choce that maxmzes total declared value. The payments are as follows: If agent 1 were not present, the mechansm would stll choose B, snce (5+6) s bgger than ether (3+1) or (6+1). So p 1 = (5 + 6) (5 + 6) = 0. If agent 2 were not present, the mechansm would choose A, snce 10+1 > 4+6. So p 2 = (10 + 1) (4 + 6) = 1. If agent 3 were not present, the mechansm would agan choose A. So p 3 = (10 + 3) (4 + 5) = 4. Theorem 3 The VCG mechansm s DSIC and mplements the socally effcent outcome. Proof: The fact that the VCG mechansm maxmzes socal welfare follows mmedately from truthfulness, snce t s defned to maxmze declared socal welfare. So t suffces to prove DSIC. 3

4 Fx and b. Suppose ω s the outcome when agent bds b. Then agent s utlty when bddng b s v (ω ) p = v (ω )+ j b j (ω ) max ω Ω b j (ω) j The last term n ths expresson does not depend on b. So for agent, maxmzng utlty s equvalent to maxmzng v (ω ) + j b j (ω ). (1) But note that f declares b = v, then the VCG mechansm chooses ω precsely to maxmze expresson (1), over all possble outcomes n Ω. It s therefore utlty-maxmzng for agent to declare b = v. Crtques of VCG The VCG mechansm s very general, but has numerous drawbacks especally n practce. Some common crtques: 1. Hgh communcaton cost: agents need to express values for every possble outcome, up front. 2. Computatonal cost of fndng the welfare-optmal outcome. For some problems, such as combnatoral auctons, ths s NP-hard. 3. Payments are unntutve and dffcult to explan. It s not obvously strategyproof. 4. Lack of prvacy: auctoneer learns a lot about the agents preferences. 5. Vulnerable to colluson between bdders. 6. Not renegotaton proof a losng agent mght be wllng to offer the auctoneer more money than they collected from the wnners. Issues 1 and 2, the complexty of VCG, are central problems n computatonal economcs. A hgh-level queston s: when can a smple and approxmately-optmal allocaton algorthm be made nto a DSIC mechansm? We wll be explorng ths n problem set 2. Example: For an example of ssue 5, colluson, suppose we have two cars for sale, one red and one blue, and two bdders. The frst bdder bds $4 on the red car and $5 on the blue car, and $9 on the par. (I.e., hs valuaton s addtve.) The second bdder bds $5 on the red car and $4 on the blue car, and s also addtve across the cars. Then the VCG mechansm gves the blue car to the frst bdder and the red car to the second bdder, and charges each of them $4. But f the agents colluded wth each other, they could each change ther bds to $0 on the car they lke least, and then they would get the same outcome but each pay $0. Example: For an example of ssue 6, renegotaton, suppose we have two cars for sale, one red and one blue, and three bdders. The frst bdder bds $5 on the red car and the second bdder bds $5 on the blue car. The thrd bdder declares that they are wllng to pay $6 to get both cars; but they would not pay anythng for ether car by tself. Then the VCG mechansm would gve the red car to the frst bdder and the blue car to the second bdder, and each of them pays $1. Whle ths s DSIC, the thrd bdder mght be unhappy wth ths outcome; after all, the seller refused to sell to the thrd bdder and receved $2 n total, but the thrd bdder would be wllng 4

5 to pay more than $2 to get both cars! Inter- Practcal Example: net Advertsng Advertsng space on search engnes tends to be sold by aucton. Ths has the advantage that advertsers can bd on keywords that target the users they want to reach, and the advertsng platform (e.g., Google, Bng, etc.) does not have to predct how valuable each possble search query s n advance. But what aucton rule should be used? A toy model: n bdders (advertsers), k advertsng slots. advertsers get assgned to slots, and a user mght clck on one or more of the ads shown. Advertsers want to be clcked. slot has a clck rate α, wth α 1 α 2... α k. bdder has value v per clck, so the value of beng assgned to slot s v α. we can assume k n wthout loss of generalty, by addng slots wth clck rate 0. We ll consder drect auctons, where the agents declare values b 1,..., b n. For convenence we ll order the bdders so that b 1 b 2... b n. Each of the auctons we consder wll return the allocaton that maxmzes declared socal welfare. Ths allocaton assgns each bdder to slot (recallng that slots are ordered from hghest clck rate to lowest, and bdders are ordered from hghest bd to lowest). What payments should be used? We ll consder three optons: Pay-your-bd: bdder pays b per clck, for a total expected payment of b α. VCG: each bdder pays ther externalty. (Exercse: calculate the VCG payment of bdder.) Generalzed Second-prce Aucton (GSP): bdder pays b +1 per clck, for a total expected payment of b +1 α. The GSP aucton s commonly used n practce, as an alternatve to the frst-prce payment rule. (Some platforms use VCG nstead.) Whle GSP looks smlar to the Vckrey aucton, t s not truthful. Example: Suppose there are 3 slots and 3 bdders. The clck rates are α 1 = 1.0, α 2 = 0.9, α 3 = 0.8. The bdders have value-perclck of v 1 = 20, v 2 = 19, and v 3 = 5. If we run a GSP aucton and the bdders bd truthfully, then bdder 1 gets the frst slot for a value of , and pays , for a total utlty of 1. Suppose bdder 1 nstead changed her bd to 18. Then she would wn the second slot for a value of , but would only pay 5 0.9, for a total utlty of So bdder 1 would be much better off by bddng 18 nstead of truthfully bddng 20. Does ths mean that GSP s a bad aucton format? Not necessary. To argue about what happens n a GSP aucton, we need to reason about what bdders do n a non-truthful aucton. One approach to ths s to study equlbra of the GSP aucton. Def: Gven valuatons v, a set of bds b forms a (pure) Nash equlbrum f for all and all potental bds b, u (b, b ) u (b, b ). 5

6 Note that, unlke the defnton of DSIC, ths defnton fxes a partcular profle of bds b, rather than quantfyng over all possble b. Def: The Prce of Anarchy (PoA) of an aucton s max x max max v (x ) v b a NE v (x (b)) where the max over x s over all possble allocatons, and the max over b s over all Nash Equlbra for valuaton profle v. In other words, the prce of anarchy s the worst-case rato between the optmal socal welfare achevable for a valuaton profle, and the welfare actually acheved at a worst-case equlbrum. It s mportant to note that we take a worst case over equlbra, snce there may be multple Nash equlbra and the PoA always consders the worst one. It turns out that GSP always has a welfareoptmal Nash equlbrum, but may have nonoptmal Nash equlbra as well. It also turns out that, under some mld assumptons on the agents bddng behavor, the prce of anarchy of GSP s qute small; the current best-known bound s We ll prove somethng smpler: that the aucton wth the pay-your-bd payment rule has PoA at most 2. Theorem 4 The advertsng aucton descrbed above wth pay-your-bd payment rule has PoA 2. Proof: Fx v and let b be a Nash Equlbrum. Order the agents so that v 1... v n, so that n the optmal allocaton agent wns slot. Consder what would happen f bdder changed ther bd from b to b = v /2. Case 1: bdder wns slot or better. That s, bdder wns slot j. Then we have u (b, b ) = (v v /2) α j v α /2. Snce b s a Nash equlbrum, t must be that u (b, b ) u (b, b ). So we therefore have that u (b) v α /2. (2) Case 2: bdder does not wn slot or better. That means that there are at least other bdders who are bddng v /2 or more n b. In partcular, the bdder who wns slot under the bd profle b must be bddng at least v /2, so s payng at least v α /2. Wrte R j (b) for the expected payment of the bdder who wns slot j when the bdders declare b. Then we ve argued that R (b) v α /2. Each bdder falls nto ether case 1 or case 2. Wrte W for the set of agents who fall nto case 1. Note that v (x (b)) ( = v (x (b)) = u (b) + R (b) ) ( ) p (b) + p (b) whch s to say that the total socal welfare can be splt nto the utlty obtaned by the buyers, and the revenue obtaned by the seller. In the second equalty, we also used the fact that the total payment of all buyers s the same as the total payment obtaned from each slot. But now, usng our cases from above, we have v (x (b)) = u (b) + R (b) W W = u (b) + W R (b) v α /2 + W v α /2 v α /2 6

7 whch s exactly half of the optmal socal welfare. So the welfare obtaned at equlbrum s at least half of the optmal socal welfare, and hence the PoA s at most 2. Crtques of the model and PoA result Real advertsng auctons are more complcated than ths, wth advertser qualty scores, ads of dfferent types / szes, etc. Ths analyss gnores budget constrants of advertsers PoA analyss assumes advertsers are playng at equlbrum. Is ths a reasonable assumpton? The model consders only a sngle aucton nstance. In realty, advertsers are partcpatng n many many auctons per day, often aganst the same compettors over and over. Ths can nfluence agent behavor n ways we aren t capturng here. E.g., bddng strateges that try to use up a compettor s budget, or otherwse nfluence compettor behavor n subsequent rounds. 7

Applications of Myerson s Lemma

Applications of Myerson s Lemma Applcatons of Myerson s Lemma Professor Greenwald 28-2-7 We apply Myerson s lemma to solve the sngle-good aucton, and the generalzaton n whch there are k dentcal copes of the good. Our objectve s welfare