Sequential equilibria of asymmetric ascending auctions: the case of log-normal distributions 3

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1 Sequental equlbra of asymmetrc ascendng auctons: the case of log-normal dstrbutons 3 Robert Wlson Busness School, Stanford Unversty, Stanford, CA , USA Receved: ; revsed verson. Summary: The sequental equlbrum of an ascendng-prce aucton of a sngle tem s derved explctly for the case of log-normal dstrbutons and a multplcatve valuaton model comprsng both common and prvate factors, and allowng asymmetres. If the pror dstrbuton on the common factors s dffuse, or of the form obtaned by Bayesan updatng from a dffuse pror dstrbuton, then the equlbrum strateges are log-lnear wth coeffcents obtaned by solvng a set of lnear equatons. A smlar constructon apples to normal dstrbutons and addtve terms n the valuaton model. An example llustrates the predctons derved from the model. JEL Classfcaton Number: C7. Introducton Theresasubstantalbodyoftheoryaboutdynamcauctonsbutthereseemstobe no fully developed class of examples that can be used for teachng and expermental purposes, and as a test case for the study of theoretcal and polcy problems. To fll the gap, ths note derves the bddng strateges for a sequental equlbrum of an ascendngprce aucton of a sngle tem. In the specal case that the probablty dstrbutons are log-normal these strateges are log-lnear. The analogous constructon of lnear strateges 3 Research support receved from Natonal Scence Foundaton grant SBR9509.

2 for the case of normal dstrbutons s omtted here. APL programs for both cases are avalable from the author. In one respect the class of examples developed here s consderably rcher than those addressed n the man theoretcal treatments, such as Mlgrom and Weber (98), n that the bdders can be asymmetrc both n ther valuatons and n the precson of ther estmates. On the other hand, the model of bdders valuatons s more restrctve n that each bdder s valuaton s assumed to be the product (or the sum, f normal dstrbutons are used) of a bdder-specfc commonly-known factor, a bdder-specfc prvately-known factor, and a common unknown factor. The prvately-known value factor and an estmate of the common unknown value factor are drawn ndependently from condtonal dstrbutons whose means are a common dstrbutonal factor and the common value factor, respectvely. A partcular advantage of the model s that t makes explct the Bayesan updatng process by whch bdders learn from the prces at whch other bdders drop out n order to obtan posteror dstrbutons for the two common factors based on the current hstory of the aucton. Formulaton In ths secton we lst the assumptons that defne the model. x. If bdder wns the tem then ts realzed value has the form v = A P V; before subtractng the cost of ts wnnng bd. The three factors n the realzed value v are nterpreted as follows: A s a publc bdder-specfc factor that represents that part of ts valuaton that s common knowledge. P s a prvate bdder-specfc factor that s known only to bdder. V s a common value factor that s not observed drectly by any bdder. x. Before the aucton, bdder s ntal mean estmate of the tem s value v s x = A P E : In ths representaton, E s nterpreted as an estmate of the common value factor V. Here and elsewhere we wrte such an estmate as E V exp(s ) or ln(e ) ln(v )+s ;

3 where the log-error s wrtten n terms of the standard devaton s and the standardzed random varable. The log-error s assumed to be ndependent of all other random varables n the model, and has a normal dstrbuton wth mean 0 and varance. Ths s equvalent to sayng that the condtonal dstrbuton of each estmate E gven the common factor V has a log-normal dstrbuton around V.NotethatE s bdder s ntal estmate before the aucton; later n the aucton, observatons of the bddng hstory enable mproved estmates. Bdder observes only ts estmate E,not V or any j. x3. From the perspectve of any other bdder, bdder s prvate valuaton factor has the form P M exp(t ),nwhch M s the mean of the dstrbuton of prvate-value factors and the log-error reflects the departure of P from ths mean. Agan, the random varable has an ndependent normal dstrbuton wth mean 0 and varance, and no bdder observes ether M or. From the perspectve of bdder, ts prvate value factor P s also an estmate of the common dstrbutonal factor M. Bdder observes only P,not M or any j. x4. The bdders publc bdder-specfc factors A and the standard devatons s and t of ther log-errors are common knowledge, as are ther pror dstrbutons of the common factors. x5. The bdders share a common pror dstrbuton for the par (M;V ) of unknown common dstrbutonal and value factors. However, ths dstrbuton s also assumed to be dffuse (.e., wth nfnte varances and zero correlaton), or equvalently, ts nformatonal content s swamped by the bdders observatons P and E. 3 Assumpton x5 s essental to ensurng that the bdders equlbrum strateges are log-lnear n ther estmates. It enables us to bypass the pror dstrbutons by workng entrely n terms of the bdders condtonal dstrbutons gven ther observatons and the sequence of observed drop outs from the aucton n the lmt form obtaned by shrnkng the precson of the pror dstrbuton. We use ths notatonal conventon to make clear the many assumptons of ndependence and condtonal ndependence that pervade the model. Otherwse, one easly gets lost tracng the roles of the ndependence assumptons. 3 A non-dffuse pror dstrbuton can be ncluded by usng the methods developed below for handlng observatons of drops by other bdders. That s, the pror s represented by the nformaton gleaned from observng other potental bdders droppng out at varous prce levels. In unpublshed work I have computed explctly the nonlnear strateges for smlar sealed-tender auctons to verfy that lnearty s approached smoothly as the precson of the pror dstrbuton approaches zero. 3

4 Prelmnares From bdder s perspectve, ts only uncertanty about the value of the tem concerns the common value factor V. In partcular, ts realzed value v s related to ts estmate x by the equaton v = x exp(0s ) ; n whch s an unknown error term. Consequently, a chef am of s bddng strategy s to take account of observatons durng the aucton to mprove ts estmate of ths error term, and thereby to mprove ts estmate of the tem s value. Suppose that, at some stage of the aucton, bdder s condtonal dstrbuton gven the aucton s hstory H s such that has a normal dstrbuton wth mean and varance. For nstance, ntally = 0 and = by assumpton. Then the condtonal expectaton of the tem s value gven ths hstory s E[v jh]=x exp(0s + s ) ; when the pror s dffuse. As the prce rses durng the aucton, t s always optmal for bdder to contnue bddng up to the condtonal expectaton of v based on all the nformaton nferred from s ntal estmate x and from the progress of the aucton so far. In practce, ths can nvolve a varety of nformaton; here, however, we develop only the mplcatons of each bdder s observatons of other bdders who dropped out of the aucton, and the prces at whch they dropped out. 4 Our am, therefore, s to determne and as functons of the hstory of the aucton as summarzed by the sequence of bdders dropout prces. Determnng these allows calculaton of the maxmum prce at whch bdder s wllng to contnue so long as no other current bdder drops out; that s, to calculate the prce B (x ) at whch drops out f no other bdder drops out. As we shall see, at each stage of the aucton s generally a lnear functon of ln(x ). Ths mples that after each hstory H the bd lmt for bdder has the powerfuncton form B (x )=E[v jh]= x : Our am s to determne the parameters and at each stage, or equvalently and, at each stage. 4 We are excludng from consderaton here such other observatons as jump bds, and n multple smultaneous auctons (such as the FCC spectrum auctons) prce levels n parallel auctons, bdders swtchng among tems, etc. 4

5 Dervaton of the Bddng Lmts The observatons avalable at a gven stage consst of (a) the set of observatons (`; p`) ndcatng each bdder ` L whohasdroppedoutandtheprce p` at whch t dropped out; and (b) the set J of those other bdders j J besdes who reman n the aucton. Mlgrom and Weber (98) establsh that so long as these observatons reman unchanged, bdder s bddng lmt as the prce rses s ts condtonal expectaton of v gven ts ntal estmate x, the observatons, and the supposton that B (x )=B j (x j ) for all j J where these other actve bdders also condton on the same observatons and a smlar supposton that all other bdders have reached ther bddng lmts. To derve the mplcatons of ths condton that determne the optmal bddng lmt, we break the analyss nto several parts. Recall that the model specfes that P = M exp(t ), and therefore P E = MV exp(r ) ; wherewedefnetheuntnormalrandomvarable = s + t r and r = s + t : Because and are ntally drawn ndependently from a standard normal dstrbuton wth mean 0 and varance, the random varable s also ntally normal wth mean 0 and varance, and ndependent of the correspondng varates ` and j for other bdders. These defntons mply further that the condtonal dstrbuton of gven s normal wth mean and varance E[ j ]= s =r and V[ j ]=t =r : The next two steps determne the condtonal dstrbuton of gven the observatons and suppostons mentoned above. Note that from the drop prce p` of a bdder ` n some prevous stage we can nfer ts ntal estmate x` by nvertng the strategy t used at that stage. That s, we suppose that p` s the bddng lmt at whch the optmal strategy ndcates that bdder ` should drop out: p` = `x ` ; ` 5

6 usng the parameters ` and ` pertanng to that prevous stage. For the analyss of the current stage, therefore, we assume that ths nference has been made and that we are gven as data the nferred ntal estmate x` for each bdder ` L who prevously dropped out. s determned from the n- Gven these nferences, the posteror dstrbuton of ference that x`=a` = MV exp(r`` ) =[x =A ] exp(0r ) exp(r`` ) ; and therefore x =A ln = 0 [r`=r ]` : r x`=a` That s, the expresson on the left has a condtonal dstrbuton that s normal wth mean and standard devaton r`=r. Ths mples that the posteror dstrbuton of based on all these observatons s normal wth mean " X # " # x =A r X ln = +r r x`=a` r ` =r ` ; `L and precson (recprocal of the varance) +r X `L =r ` : The next step s to obtan the further posteror mpled by the supposton that all those bdders j J are at ther bddng lmts. Ths supposton can be nterpreted as specfyng that n the current stage, `L A MV exp(r ) 3 = j Aj MV exp(r j j ) 3 j for each other bdder j J.Thats, 0 r j ln A j A j j x 0j! A = 0 [r j =r ] j : The resultng normal posteror dstrbuton of, condtonal only on ths supposton, can be constructed usng formulas smlar to the prevous ones. Combnng these results, we obtan the posteror dstrbuton of condtonal on both the pror drops and the supposton that all other actve bdders are at ther bddng 6

7 lmts. In partcular, the posteror dstrbuton of s normal wth mean 8 < m (x )= X r h : ln x =A X A 0j! x x`=a` r ` 0 ln j A j A j `L jj and precson h = r h where X h = =r j : jj Lastly, we can combne these results wth the prevous characterzaton of the condtonal dstrbuton of gven to obtan (x )=m (x )s =r and = s =r r h + t : To dentfy the parameter exponent of x r j j 9 = ; we notce that the precedng results mply that the n the resultng formula for the bddng lmt s =0 Alternatvely, ths can be wrtten as g = s 4 X`L r h 4 0 [(s =r ) =h] r ` 0 X jj X jj r j j 5 : j g j =r 5 j =[ 0 (s =r ) ] ; where g ==. Ths equaton, one for each actve bdder J,defnesasetof lnear equatons that determne the recprocals g of the exponents that appear n the formulas for the bddng lmts. Havng solved for the exponents, the last step s to solve for the bd factor of each actve bdder. Ths can be done by adoptng the expedent of assumng that x =, nwhchcaseweobtan,fromtheprevousformulafor = B () = E[v j x =],the equaton =exp(0s () + s ) =exp(0s [m ()s =r ]+ s s =r h + t ) ; from whch t follows that, f b ln( ),then b = [s =r ] h X`L ln x`=a` =A r ` + X r jj [(b 0 b j )= j +ln(a =A j )] r j 7 + [s =r + ht ] :

8 Thus, these lnear equatons, one for each bdder J, provde a means of computng the parameter b, and thereby the coeffcent of the bd lmt. As a last step, we note that t s usual to use as the estmate ˆx = x exp( s ) so that E[ ˆx j v ]=v ;thats, ˆx s an unbased estmator of v. Therefore, f the bddng lmt s expressed as ˆB (ˆx )= ˆ ˆx Example then ˆ = exp(0 s ) : The accompanyng table provdes an llustratve example. There are sx bdders ntally, labeled A through F, and part () shows ther parameters A, s,and t. Part () shows n the second row a typcal sequence of drop prces, ncreasng from $0 to $7 for bdders F to B n that order, rght to left. The frst row shows the ntal estmates x nferred from these drop prces, and the thrd shows the bddng lmt for bdder A pror to the assocated drop. The drop prces n the second and thrd rows are each shown only for the hstory mmedately pror to the drop of the bdder ndcated for that column. For nstance, bdder A s ntal estmate s $40.00 but t s wllng to contnue bddng up the lmt $65.8 f all fve of the other bdders do so too; and after F drops out at $0.00, A s wllng to bd up to $53.3 f all four of the other remanng bdders do so too but n fact before that E drops out at the prce $4.00, leadng A to nfer E s estmate was $8.46 and so A reduces ts drop prce to $48.. The aucton ends when bdder B drops out at the prce $7.00, whch s optmal gven ts ntal estmate $35.7 and the observed dropout prces of bdders C through F. At ths pont A s payment s $7.00 for an tem whose expected value, condtonal on the bddng hstory, s $ Part (3) shows the bd factor and the exponent that characterze the bd-lmt functon B (x ) for each actve bdder at each stage. Reference Mlgrom, Paul, and Robert Weber (98), A Theory of Auctons and Compettve Bddng, Econometrca, 50:

9 A Numercal Example Bdder: A B C D E F () Parameters A : s : t : () Aucton Hstory from A s Perspectve Estmate x : s Drop Prce B (x ) : A s Drop Prce B A (x A ) : (3) Parameters of the Bddng Strateges Stage Parameters 0 : : : : : : : : : : : 0.58 :

10 Runnng Head: Asymmetrc Ascendng Auctons Journal of Economc Lterature Classfcaton Number: C7. Author s Malng Address: Professor Robert Wlson Stanford Busness School Stanford, CA USA 0