Posted-Price, Sealed-Bid Auctios Professors Greewald ad Oyakawa 207-02-08 We itroduce the posted-price, sealed-bid auctio. This auctio format itroduces the idea of approximatios. We describe how well this auctio does agaist a secod price auctio, ad give a lower boud o the expected welfare geerated. The Posted-Price, Sealed-Bid Auctio I our auctio desig space, we have see that there is a large differece i our methodology of aalyzig strategies whe we go from a pricig scheme that charges the secod highest bid, to oe i which bidders pay their bid. We ow itroduce a differet pricig ad allocatio scheme: suppose the auctioeer aouces the price of a good, ad ay bidder that bids at least the posted price has uiform probability of wiig. The wier is charged the posted price P, ad all other bidders are charged othig. Give a posted price P, we ca ask: How should bidders bid? How good is this outcome? Good i the secod questio implies that we are measurig somethig. I this case, it will be the total expected welfare, which is the sum of everyoe s expected utility. Recall that the utility of each bidder is u i b i, b i ) = v i x i b i, b i ) p i b i, b i ), i N, ) ad that the utility of the auctioeer is u 0 b i, b i ) = p i b i, b i ). 2) The total expected welfare is the sum of everyoe s expected utility: ] E v F u 0 b i, b i ) + u i b i, b i ). Computatioal Complexity = E v F p i b i, b i ) + v i x i b i, b i ) p i b i, b i ) 3) ] = E v F v i x i b i, b i ). 4) With bidders, determiig the wier takes O) time, as this is the complexity of a arg max fuctio. Give the wier, we ca ]
posted-price, sealed-bid auctios 2 determie paymets i O) time. Therefore, this auctio ca be ru i polyomial time. 2 Domiat Strategies The aalysis required to determie what strategy each bidder should use is similar to that of the secod-price, sealed-bid auctio. Sice utilities are quasi-liear, we ca reaso what the utility of a wiig bidder will be based o what she bids. The case aalysis is described graphically i Figure ad Figure 2, ad summarized i Figure 3. v i P Figure : The utility of bidder i if the posted price is smaller tha v, as a fuctio of what she bids. Utility 0 P v i Bid, b i Figure 2: The utility of bidder i if the posted price is larger tha v, as a fuctio of what she bids. Utility 0 v i P v i P Bid, b i Thus, we ca see that the posted-price, sealed-bid auctio format is DSIC: regardless of what ay other bidder does, submittig oes
posted-price, sealed-bid auctios 3 v i Figure 3: The utility of bidder i if she wis, as a fuctio of the posted price. Utility 0 0 v i = P Posted Price, P true valuatio as a bid is a domiat strategy. Bidders oly have a strictly positive probability of beig allocated a good whe they exceed the posted price the auctioeer has set, ad the radomized ature of wier selectio meas that each bidder ca do othig to try ad block other bidders from wiig beyod placig a bid at least as large as the posted price. Successfully prevetig aother bidder from wiig would mea havig placed a bid b i p, ad it is of bidder i s iterest to oly do so if v i p. Now, observe that because paymets for the wier are predetermied by the auctioeer, a bid is i fact describig a biary sigal. Placig a bid b i P is tellig the auctioeer I am willig to purchase the good at that price. Placig a bid b i < P is tellig the auctioeer I am ot willig to purchase the good at that price. Thus, a strategy which places the followig bids is also a domiat strategy: P, ), if v i P b i 5), P), otherwise. This meas that there are multiple domiat strategies i this auctio format. The followig two strategies would do just as well as biddig truthfully i the posted price auctio: P, if v i P b i = 6) 0, otherwise, if v i P b i =, otherwise. 7)
posted-price, sealed-bid auctios 4 Sice there are multiple domiat strategies, it is ot the case that the auctioeer may always kow exactly what valuatios the bidders have. This may be beeficial i settigs where bidders do ot wat to divulge their true prefereces to a auctioeer: iformatio is ofte a very useful trade secret etities try to guard. 3 Expected Welfare Ulike the first ad secod price auctios formats, which ca always produce a wier amogst the participats, the posted price mechaism has o such guaratee. For example, if the posted price is larger tha ay possible bidder type, the o oe will ever wi. Thus, based o distributioal kowledge of bidder types, the auctioeer must reaso about what a appropriate posted price is i order to maximize the expected welfare. Suppose the bidders are symmetric, ad have iid draw valuatios from some distributio F. Oe reasoable way to set the posted price is to set it so that, i expectatio, there will be a wier. That is, we ca set P so that the probability of a bidder with type v i P is : If Pr v P) =, 8) the Pr v P) =, 9) so P = F ). 0) Now, with such a posted price, we ca try to fid what the expected welfare is. To do this, we will upper-boud what the expected optimal welfare, OPT, is, ad lower-boud the expected welfare of this auctio format, APX. You ca thik of OPT as the expected welfare a secod price auctio would geerate. Notice that i the secod-price auctio, it is the case that each bidder has probability of wiig. I expectatio, there will be a wier with valuatio v i P, as Prv i P) =. The welfare geerated from a expected profile is at least P, but o larger tha whatever the highest type may be. 3. Posted Price with Goods The upper boud o OPT will be costructed by aalyzig the total expected welfare of the posted price auctio with goods. That is, i a settig where everyoe ca be served. The reaso why we have this upper boud is because while a posted-price auctio may ot always sell a good, there are also istaces where we ca award multiple
posted-price, sealed-bid auctios 5 bidders a good, which will make up for those cases where there is o sale. To gai some ituitio for why this is, we will go over a few examples, where there are = 2 bidders, ad each draw valuatio from discrete distributio F, where there are two possible types. The low type is draw with probability q = 2, ad the high type is draw with probability q = 2. Example 3.. Suppose there are two types, T = {0, }, ad each are draw with equal probability. There are four possible bidder profiles, each of which occur with equal probability: 0, 0), Prv) = 0.25 0, ), Prv) = 0.25 v = ), 0), Prv) = 0.25, ), Prv) = 0.25. The expected welfare a secod-price auctio will geerate is 3 4. Now cosider a posted-price auctio with P 0, ). The expected welfare this auctio will geerate is 3 4. Fially, cosider a posted-price auctio with P 0, ) where there are copies of the good. The expected welfare this auctio will geerate is. The first example tells us that the total expected welfare a postedprice auctio with goods is strictly larger tha the expected welfare a secod-price auctio ca geerate. We ow show that they ca be made to be arbitrarily close. Example 3.2. Suppose there are two types, T = { 2ɛ, }, ad each are draw with equal probability. There are four possible bidder profiles, each of which occur with equal probability: 2ɛ, 2ɛ), Prv) = 0.25 2ɛ, ), Prv) = 0.25 v = 2), 2ɛ), Prv) = 0.25, ), Prv) = 0.25. Let the posted price be P = ɛ. The expected welfare a secod-price auctio will geerate is 4 2ɛ + + + ) = 4 2ɛ). 4 The expected welfare a posted-price auctio will geerate is 4 0 + + + ) = 4 3).
posted-price, sealed-bid auctios 6 The expected welfare a posted-price auctio with copies of a good is 4 0 + + + 2) = 4 4). Oce we set ɛ = 0, the there is oly oe type. A posted-price auctio will geerate the same expected welfare as a secod-price auctio, ad a posted-price auctio with copies of a good ca do strictly better tha the secod-price auctio format. Now, you might reaso the followig: if the probability of the low types appearig is sigificatly larger tha the probability of the high types appearig, the a posted-price auctio with copies of a good should do worse whe P = ɛ. This is certaily true. However, by shiftig probability mass towards the low types, we must reaso about what the appropriate posted price P should be. Recall that we had set the posted price so that i expectatio, there is oe bidder that will wi i.e., the iverse CDF at half, F 2 ). By shiftig probability mass towards the low types, we must adjust the posted price so that, i expectatio, we have at least wier. Example 3.3. Suppose there are two types, T = { ɛ, }, ad with probability q q, a bidder has type ɛ. There are four possible bidder profiles, ad the profile where both bidders have low types is sigificatly more probable: ɛ, ɛ), Prv) = q 2 ɛ, ), Prv) = q q) v = 3), ɛ), Prv) = q)q, ), Prv) = q) 2. Let the posted price be P 0, ɛ]. The expected welfare a secod-price auctio will geerate is q 2] ɛ) + 2 q q)] ) + q) 2] ) = q 2 ɛ. The expected welfare a posted-price auctio will geerate is q 2] ) ɛ) + 0) + 2 q q)] + q) 2] ) = q 2 ɛ q 2 qɛ +. 2 The expected welfare a posted-price auctio with copies of a good is q 2] ɛ)2 + 2 q q)] 2 ɛ)) + q) 2] )2 = 2 2qɛ. Each term i the sum for the copy case is at least as large as each term i the secod-price auctio case, so we coclude that the total expected welfare of a posted-price auctio with copies of a good does at least as well as that of a secod-price auctio.
posted-price, sealed-bid auctios 7 3.2 Bouds Based o our ituitio about a upper boud o OPT, we ow derive upper ad lower bouds, as well as a approximatio ratio for the total expected welfare the posted price auctio ca geerate. Lemma 3.4. The optimal welfare is upper-bouded by OPT E v v P]. 4) Proof. We will upper-boud OPT by cosiderig a posted-price auctio where there are copies of the good beig sold. This meas it is possible for every bidder to be allocated, so the welfare geerated i this istace is at least as large as the expected welfare the optimal welfare a secod price auctio ca geerate. Sice bidders are ot competig agaist each other, we ca cosider each bidder s impact o welfare idepedet of what ay other bidder s impact is. Just like i the secod-price auctio, we expect someoe to wi, ad she will have valuatio v i P. However, ulike i the secod price auctio, there ca be more tha oe wier, makig the welfare upper boud times the highest type ay bidder ca have. The expected welfare such a auctio ca geerate is OPT E v i v i P] Pr v i P). 5) i= Sice Pr v i P) = ad all bidders are symmetric, OPT E v v P]. 6) Lemma 3.5. The lower boud o the expected welfare the posted-price auctio format achieves is at least APX ) E v v P] 7) e Proof. The probability that a aget has type v i P is, so the probability that a aget has type v i P is. This meas that the probability of a good ot begi sold is i= Pr v i P) = ad that the probability of a good beig sold is ), 8) ). 9)
posted-price, sealed-bid auctios 8 The expected welfare geerated by this auctio the must be at least APX ) ) E v v P]. 20) ) The term ca be bouded to form a slightly simpler ). I the limit, as teds lookig expressio. Let f ) = toward ifiity, lim f ) = e. 2) Also otice that f ) is a icreasig fuctio whe, as show i Figure 4. Thus, we ca coclude that ) e, 22) ad APX e ) ) E v v P] 23) ) E v v P]. 24) If you are ot satisfied with proof by picture, you may fid Sectio A of iterest, which derives this more formally. f ) ad /e 0.8 0.6 0.4 f ) /e Figure 4: A compariso betwee ) f ) = ad /e. Notice that f ) is a icreasig fuctio whe, ad approaches e from below. 0.2 0 0 0 20 30 40 50 Now that we have a upper boud, ad lower boud, o the expected welfare that the posted price mechaism geerates, we ca derive a approximatio ratio. Theorem 3.6. The approximatio ratio of the welfare geerated by the posted price mechaism whe all valuatios are iid draws from a cotiuous distributio is. APX OPT ) e 25)
posted-price, sealed-bid auctios 9 Proof. Combie the results we have derived for the upper ad lower bouds o welfare geerated to get: E v v P] OPT APX ) E v v P]. 26) e Now divide by OPT: E v v P] OPT OPT OPT APX OPT ) E v v P] e OPT. 27) Sice E v v P] OPT, we ca costruct a weaker lower boud o the right had side to get APX OPT ) E v v P] 28) e OPT ) E v v P] 29) e E v v p] = ) 30) e 0.63. 3) The result is rather surprisig. What this meas is that by simply lettig buyers purchase goods usig a askig price, we ca obtai at least.63 of the optimal welfare. Thus, the posted price auctio format e e is a.58 approximatio to the optimal welfare maximizig auctio. Oe way to iterpret this is that competitio, a key igrediet dowplayed i this auctio format, accouts for just.37 of the optimal welfare. A Let The Expoetial Fuctio f x, ) = + x ). 32) Let s see what happes i the limit, as approaches, Ax) = lim + x ). 33) We begi by takig logs, log Ax) = lim log + x ) ) log + x = lim 34). 35) Apply L Hopital s rule 2 to get 2 f x) lim x c gx) = lim x c f x) g x)
posted-price, sealed-bid auctios 0 lim log + x ) )] log + x = lim ] 36) x = 2 lim + x 2 37) x = lim + x 38) = x. 39) Therefore, log Ax) = x, 40) ad so e x = Ax) 4) = lim + x ), 42) e = lim ). 43) Now, if f, ) is icreasig for, the log f, ) must be icreasig, as log is a icreasig fuctio: log f, ) = log ). 44) ) log is a icreasig fuctio, so we coclude that f, ) is also a icreasig fuctio. 3 3 The trasformatio with logarithms meas that the domai is restricted to > if we wat to restrict ourselves to real umbers. However, i) the limit, as approaches, log =.