Notes o Epected Reveue from Auctios Professor Bergstrom These otes spell out some of the mathematical details about first ad secod price sealed bid auctios that were discussed i Thursday s lecture You ca fid a eve more detailed discussio i Steve Matthews Techical Primer, which is liked to the class website Although there is quite a bit of detail, you will see that the oly mathematics used is simple probability theory, a little algebra ad some easy calculus of a sigle variable di eretiatig ad itegratig polyomial fuctios The aalysis for 2 bidders is fairly simple ad I urge you to try to follow the argumet here for this case The bidder case requires some more calculatio, but uses the same road map to its results Eve if you do t follow the etire argumet for the bidder case, it is valuable to compare these results with the two bidder case ad to uderstad what happes as you add more bidders Private Value Auctios Suppose that a supplier has a sigle object for sale ad that there are possible buyers The value of the object to perso i is some umber v i that is kow to i but ot kow to ayoe else If Perso i gets the object for price p, hisprofitwillbev i pletussupposethateverybodyeceptperso i thiks that i s value for the object is a radom variable draw from the same probability distributio If radom variable is cotiuous, we let F () deote the probability that this radom variable is less tha or equal to ad f() =F () bethecorrespodigdesityfuctio We will ofte work with the eample of a uiform distributio o the iterval [, ] Sometimes it is coveiet to thik of a discrete versio of this radom variable i which the radom variable takes o oly iteger values ad where the probability the that = for ay iteger betwee adisf() = / The the probability that apple must be F () = / Here we will work with the cotiuous uiform distributio o [, ] I this case, for ay real umber betwee ad, we have F () =/ ad f() =F () =/
How to bid i a secod-price sealed bid auctio I a secod-price sealed bid auctio, the object is sold to the high bidder The high bidder does ot pay his ow bid, but istead pays the amout bid by the secod-highest bidder If your bid is ot the high bid, you do t get the object ad you do t have to pay aythig I this auctio, your ow bid does ot a ect the amout that you would pay if you got the object It oly determies whether you get the object A remarkable feature of the secod-price auctio is that o matter what you believe about other people s strategies, the best thig for you to do is to bid your true value Why is this so? Whatever happes i the auctio, there will be some bid z that is the highest bid made by the other guys If your ow value is v, thewhatever the value of z is, your profit will be v z if your bid is greater tha z (because you get the object ad pay z for it) ad it will be if you bid less tha z (because i this case you do t get the object ad do t pay for it) What di erece would it make if you tried to overbid by biddig b>v?ifz<v, it would t matter at all, because you would wi the object ad pay z just as you would if you bid your true value v But what if b>z>v? I this case, you would ot wi the object if you bid your true value v, butyou will wi the object with your bid of b Butifb>z>v,theyourprofitis v z< You will be payig more for the object tha it is worth to you ad will be worse o tha if you had bid the truth ad ot wo the object This meas that you ca ever make yourself better o ad you may make yourself worse o by overbiddig tha by biddig your true values What di erece would it make if you tried uderbiddig, by biddig less tha your true valuatio? If your bid is b<vad if z<b,itwillmakeodi erecethat you uderbid, sice you will still wi the object ad you will still pay z for it But if it happes that b<z<v,thewithabidofb, youwillotwithe object ad will get profit If you had bid your true value v, youwouldhave wo the object ad had a profit of v z> So you caot help yourself ad you may hurt yourself by biddig less tha your true value It follows that o matter what the other bidders are doig, the best thig for you to do is to bid your true value Biddig aythig else caot help you ad may hurt you A strategy for which this is the case is called a weakly domiat strategy Thus biddig your true value i a secod-price sealed bid auctio is a weakly domiat strategy 2
How to bid i a first-price sealed bid auctio I a first-price sealed bid auctio, the object is sold to the high bidder at the high bidder s bid I this case, it is easy to see that there will ot be a weakly domiat strategy (A strategy is weakly domiat if oe could ot do better eve if oe kew what the other players are doig I the case of a first-price auctio, the best thig to do depeds o what the other bidders are doig For eample if the value to you is $ ad the highest bid by ayoe else is $4, you are best o biddig just a tiy bit more tha $4, but if the highest bid by ayoe else is $5, you are best o biddig just a bit more tha $5 So if there is o weakly domiat strategy, what do you do whe you do t kow what the other bidders are doig? A reasoable approach is to maimize the epected value of your profit If you bid b ad your value is v, yourepectedprofitwill(v b)p (b) wherep (b) istheprobabilitythatyou get the object if you bid b If higher bids are more likely to wi the object, the P (b) willbeaicreasigfuctioofbwhile v b will be a decreasig fuctio of b You are goig to have to make a tradeo Let s see how this would work if there are oly two bidders ad each of them believes that the other s value is a radom variable draw from the uiform distributio o the iterval [, ] You realize that it would be foolish to bid your true value, sice i this case your profit is whether you wi the object or ot To get started, assume that the other guy will bid some fractio c of his true value The if you bid b, the probability that you will wi the object will be the probability that c < b where is the other bidder s value This is the same as the probability that <b/cwhere is the other guy s value Sice have assumed that the other guy s value is a radom variable from the uiform distributio o [, ], it follows that the other guy s value is less tha b/c is F (b/c) = b c SowehaveP (b) = b c ad epected profits of b (v b) c = vb b 2 c You wat to choose b to maimize this epected profit calculus coditio for this maimizatio is d vb b 2 = (v 2b) = db c c The first order (You should check the secod order coditio to see that this is really a maimum) The solutio of this equatio is v =2b, orequivaletly,b = v 2 3
So if you believe that the other guy is goig to bid a fractio c of his value, the best thig for you to do is to bid /2 of your value If he believes the same thig about you, the best thig for him is to bid /2 of his value So i the case of two bidders with values uiformly distributed o a iterval, there is a equilibrium i which each bids half of his value We ote that the object would the always be sold to the bidder with the higher buyer value ad it would be sold at a price equal to half of that perso s value Let s work out the case for bidders Suppose that you bid b What is the probability that your bid is the highest oe? It is the probability that all oftheotherguysbidlessthab Suppose that each of them bids some fractio c of his true value The the probability that ay oe of them bids less tha b is F (b/c) = b ad the probability that all of them bid less c tha b is! F (b/c) b = = b c (c) The if you bid b, yourepectedprofitis b (v b) (c) = vb (c) b To maimize your epected profit, you would choose b so that d vb db (c) This is equivalet to b = (c) ( )b 2 v b = ( )b 2 v b =, which is equivalet to b = v Thus we see, that as the umber of bidders gets larger, each bidder bids closer to his actual value If there are 3 bidders, the i equilibrium, each bids 2/3 of his value If there are 4 bidders, each bids 3/4 of his value If there are bidders, each bids 9/ of his value ad so o Epected Reveue If he does t kow the values of the bidders, a seller will ot kow i advace whether oe type of auctio will give him more reveue tha aother His 4
retur from either type of auctio is a radom variable But give his beliefs about the probability distributio of bidders values, he ca calculate his epected reveue from ay type of auctio Epected Reveue from First-Price Sealed Bid Auctio We will work this out first for the case of 2 bidders where the values of bidders are idepedet draws from a uiform distributio o the iterval [, ] From our earlier discussio, we have leared that i equilibrium, each bidder will bid half of his value Thus, the price at which the object will be sold is /2 of the higher of the two bidders values To fid the probability distributio of the seller s reveue, we first wat to fid the probability distributio of the higher of the two bidder s values Let us fid the probability g(, 2) that the highest of 2 bids is Now ca be the highest bid i two possible ways Oe way is that bidder has value ad bidder 2 has value less tha or equal to The probability of this evet is f()f () Sice f() = ad F () =,theprobabilityofthisoutcomeis The other way i which ca be the highest bid is that bidder 2 has value ad bidder has value less tha or equal to Thisalsohappeswithprobability Thereforetheprobabilitythatis the highest value from two bidders is 2 g(, 2) = We ca calculate the epected reveue of the seller as follows Recall that with two bidders, each bidder bids half of his value So the epected reveue of the seller must be /2 of the epected value of the radom variable which is the highest value bidder s value Sice g(, 2) is the probability that the highest value is, the epected value of the highest value is Z g(, 2)d = Z 2 d = 2 Z 2 d 2 Now Z 2 d = 3 3 = 3 3 Therefore the epected value of the highest bidder value is Z g(, 2) = 2 3! = 2 2 3 3 5
Recall that i equilibrium, bidders bid oly half their values, so the epected reveue of the seller is /2 of the epected value of the higher bidder value Hece the epected reveue of the seller is just 3 What if there are bidders? What is the probability g(, ) that the high bid is if there are bidders? The could be the highest buyer value i di eret ways, oe way for each perso who could be high bidder Each of these ways has probability ad so it must be that g(, ) = The epected value of the high bid will be the itegral Z Now, sice g(, )d = Z Z Z d = d d = + + = + +, we have Z g(, )d = + Recall that that whe there are bidders, each bids the fractio of his value Thus where there are bidders, the epected value of the high bidder s bid is times the epected value of the highest value Thus the seller s epected reveue must be Z g(, )d = + = + This is cosistet with our previous result for 2 bidders, where epected reveue was (/3) Now we see that if there are 3 bidders, epected reveue will be (/2), if there are 4 bidders, epected reveue will be (3/5) ad if there are bidders, epected reveue will be (9/) 6
Epected Reveue from Secod-price sealed bid auctio Recall that i a sealed bid secod-price auctio, the best strategy for ay player is to bid her true value Thus the epected reveue of the seller will be the epected value of the secod highest value Agai, we will work this out first for the case of 2 bidders where the values of bidders are idepedet draws from a uiform distributio o the iterval [, ] Let h(, 2) be the probability that the secod highest bid is This ca happe i two di eret ways Bidder ca have a value of, while the value of the other bidder is greater tha or equal to The probability that a bidder has a value greater tha or equal to is F () wheref () isthe probability that a bidder has value less tha or equal to Therefore the probability that bidder has value ad bidder 2 has value at least is f()( F ()) = The other way i which the secod highest bid ca be is if bidder 2 has value ad bidder has value at least This also happes with probability f()( F ()) ad so h(, 2) = 2f()( F ()) = The epected value of h(, 2) is Z Now h(, 2)d = Z Z 2 2! d = 2 2 2 d = 2 Z 3 3 = 6 2 Substitutig ito our previous epressio, we see that Z h(, 2)d = 3 2! d Notice that this is eactly the same as the epected reveue of the seller from a first-price auctio i which both bidders use equilibrium strategies Sayig this i aother way, whe there are two bidders, the epected value of the secod highest buyer value is equal to half of the epected value of the highest buyer value 7
What if there are bidders? Calculatig the epected reveue is a little more complicated ow If perso is the secod highest bidder, the oe of the otherbiddersmustbidatleastasmuchashedoesadallofthe 2otherbiddershavetobidomorethahedoes Theprobabilityof this happeig is f()( )( F ())F () 2 Sicetherearedi eret people who could be the secod highest bidder, the probability that the secod highest bid is must be h(, ) =( )f()( F ())F () 2 Now the epected value of h(, ) willbe Z Z h(, )d = ( ) Simplifyig this epressio, we have Z h(, )d = ( )! Z d 2 d Now Z d =! = + ( +) Substitutig ito the previous epressio, we fid that the epected value of the secod highest bidder s value is Z h(, )d = + Sice i equilibrium for the secod-bidder auctio, everyoe bids his value, this is also the epected reveue of the seller Note that this is eactly the same as the epected reveue of a seller i a first-price sealed bid auctio The Reveue Equivalece Theorem Most people are very surprised to lear that i the eamples we looked at, if bidders are i equilibrium, the reveue from a first-price auctio is eactly the same as that from the secod-price auctio Is this just a fluke? Would the result disappear if the distributios of values were ot uiform o some 8
iterval? It turs out that this result is a special case of a much more geeral theorem about private value auctios The equilibrium epected reveue from first price ad secod price auctios would be the same so log as the distributios of values are cotiuous ad idepedet betwee idividuals Eve more remarkably, reveue from may other kids of auctios would be the same as that from the first ad secod price auctios, so log as distributios of values are idepedet ad the auctios have the followig features ) The good always goes to the perso with highest buyer value 2) The perso with the lowest buyer value is guarateed to make zero profits i the auctio This result is kow as the reveue equivalece theorem 9