Multiagent System Simulations of Sealed-Bid Auctions with Two-Dimensional Value Signals

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1 Deparmen Dscusson Paper DDP77 ISSN Deparmen of Economcs Mulagen Sysem Smulaons of Sealed-Bd Aucons wh Two-Dmensonal Value Sgnals Alan Mehlenbacher Deparmen of Economcs, Unversy of Vcora Vcora, B.C., Canada V8W 2Y2 Absrac Ths sudy uses a mulagen sysem o nvesgae how sealed-bd aucon resuls vary across wodmensonal value sgnals from pure prvae o pure common value. I fnd ha several aucon oucomes are sgnfcanly nonlnear across he wo-dmensonal value sgnals. As he common value percen ncreases, prof, revenue, and effcency all decrease monooncally, bu hey decrease n dfferen ways. Fnally, I fnd ha forcng revelaon by he aucon wnner of he rue common value may have benefcal revenue effecs when he common-value percen s hgh and here s a hgh degree of uncerany abou he common value. JEL classfcaon: C5, C72, D83. Keywords: agen-based compuaonal economcs; mul-dmensonal value sgnals; sealed-bd aucons; mpulse balance learnng Auhor Conac: Alan Mehlenbacher, Dep. of Economcs, Unversy of Vcora, P.O. Box 7, STN CSC, Vcora, B.C., Canada, V8W 2Y2; e-mal: amehlen@uvc.ca; Voce: (25) ; FAX: (25)

2 2 Inroducon Ths sudy endows compuaonal agens wh a learnng model and uses hese agens n compuaonal expermens o make hree conrbuons o knowledge abou mulagen smulaons of sealed-bd aucons. Several emprcal sudes have shown ha mpulse balance learnng explans how human bdders n aucon expermens adjus her bd prce sraeges (Selen and Bucha, 998; Selen e ala, 25; Ockenfels and Selen, 25; Neugebauer and Selen, 26; Garvn and Kagel, 994; Kagel and Levn, 999). Ths makes a promsng mehod o nvesgae as he learnng model n a mulagen sysem. The frs conrbuon s o adap Selen s mpulse balance learnng mehod for use by agens n a mulagen sysem. In real-world aucons (such as hose for mber sales, ol leases, specrum, and servces) he em value ofen has boh a prvae value and a common value componen (Goeree and Offerman, 22). Thus, he second conrbuon s o deermne how prof, revenue, and effcency change as he common value componen ncreases. There are no lab expermens o ndcae wheher hs change s lnear or non-lnear. The mulagen smulaons show ha as he common value percen ncreases, prof, revenue, and effcency all decrease monooncally (and ofen nonlnearly), bu hey decrease a dfferen raes. Prof curves end o decrease faser a hgher common values, revenue curves end o decrease more rapdly a low common value percens, and effcency curves end o say hgh and hen decrease rapdly for hgh percens of common value. The hrd conrbuon s o deermne wheher may be worhwhle for a seller (such as a federal or sae governmen) o enforce ruhful revelaon of he rue common value by aucon wnners. In lab expermens, Kagel and Levn (999) show ha

3 3 revealng nformaon abou he rue common value n frs-prce aucons ncreased or decreased revenue dependng upon he number of bdders and he degree of uncerany abou he common value. The mulagen smulaons show ha forcng revelaon of he rue common value may have benefcal revenue effecs when here s a hgher degree of uncerany abou he common value. In Secon 2, I descrbe he aucon model. Secon 3 provdes deals of he learnng model and s properes of convergence and sensvy. Secon 4 compares he resuls of learnng model wh resuls from lab expermens n oher sudes. Secon 5 demonsraes he nonlnear varaon of revenue and effcency wh he common value percen. Secon 6 shows he resuls of requrng he aucon wnners o reveal he acual common value o he aucon losers. Secon 7 presens conclusons. 2 Aucon Model The mulagen sysem plaform s descrbed n Mehlenbacher (27). In hs secon, I descrbe how he sysem mplemens values and he value sgnals for bdders (2.), he levels of nformaon feedback (2.2), and he number of perods and bdders (2.3). 2. Values and Value Sgnals Before parcpang n a sealed-bd aucon n perod, each bdder deermnes s esmae for he value v of he em, and hs esmae s called a value sgnal, denoed v ˆ. Mos aucon research has nvolved a sngle value sgnal v ˆ ha s eher pure prvae ( v P, ) or pure common ( v ˆ C, ), and hese pure sgnals are called one-dmensonal The noaon s summarzed n Table.

4 4 value sgnals. The bdders value sgnals are pure prvae value when hey base her esmaes on her own value for he em, whou consderng how oher bdders mgh value he em. The value sgnals are pure common value when bdders base her esmaes on an esmaed fuure acual value ha s common o all bdders, for example a resale prce. In he case of pure prvae values, each bdder wll have a dfferen value sgnal and he esmaed value for a bdder s he acual value of he em o ha bdder. In he case of pure common values, he acual common value s unknown o he bdders before and durng he aucon, and s dscovered n he markes afer he aucon only by he wnnng bdder. In mos real-world suaons, a value sgnal s a mxure of prvae and common value componens. A few researchers (Dasgupa and Maskn, 2; Jehel and Moldovanu, 2; Goeree and Offerman, 22) have suded hese mxed value sgnals and desgnaed hem mul-dmensonal (or more precsely, wo-dmensonal ) value sgnals. For example mber sale aucons and ol leases have a common value componen conssng of he volume and marke prce of he resource and a prvae value componen conssng of frm-specfc coss, capaces, and sklls (Ahey and Hale, 22; Hendrcks e ala, 23; Hale e ala, 23). Smlarly, servce procuremen aucons have a common value componen ha s he scope of work and a prvae value componen conssng of producvy, wage coss, and overhead coss. Whn he conex of a unque mxure of prvae and common values, he seller esablshes he aucon rules, he mos fundamenal of whch are he paymen rule and he nformaon o be released o he bdders afer he aucon. In hs case, he value sgnal v ˆ s a funcon of boh ypes of value so ha vˆ ˆ ( ˆ = v vp,, vc, ). Followng Goeree and Offerman (22),

5 5 I use lnear combnaons of prvae values and common value sgnals o produce mxed value sgnals ha range from pure prvae value o pure common value. An agen s value sgnal s vˆ ( ) ˆ θc vp, θcvc, = +, where [,] θ s he fracon of common value. The C acual value, known by he wnner, s herefore v = ( θc ) vp, + θcvc. Two levels of wodmensonal value sgnals ( θ C =.4 and.25) have been nvesgaed n expermens by Goeree and Offerman (22), bu my sudy s he frs o look a he full specrum of wodmensonal sgnals and he varaon n prof and revenue as well as effcency. Values are dsrbued o he agen bdders n a dfferen way han he dsrbuon o human subjecs n lab expermens (Kagel and Levn, 22). In hs sudy, each bdder agen s prvae and common value sgnals, as well as he acual common value, are fxed hroughou he aucons. Ths s an arfcal suaon, bu has he purpose of denfyng he adapvely bes bddng sraegy for each possble value sgnal. The alernave, whch s used n lab expermens, s o provde each bdder wh a random value sgnal for each aucon. Ths resuls n each bdder learnng an average bddng sraegy n response o he full range of value sgnals. However, snce bddng sraeges may be dfferen for dfferen value sgnals, especally n frs-prce aucons, hs average s no very nformave. The expermener specfes he suppor [ S P, S P] of a dsrbuon of he prvae value sgnals v P, and a suppor [ SC, S C] for he common value v C. In mos expermenal sudes and he smulaons n hs paper [ SC, S ] = [ S P, S ]. There are wo mehods of provdng he bddng agens wh value sgnals from hese suppors: random and deermnsc. In he frs mehod, a bdder s prvae value v P, s drawn from a C P

6 6 dsrbuon (usually he unform dsrbuon) on he suppor. Each bdder's common value sgnal v ˆ C, s drawn from a dsrbuon on he suppor cenred on he common value [ vc ε, vc + ε], where he common value s he cenre of he suppor [ SC, S C]. There s uncerany among he bdders abou wha hs common value s, and a larger ε represens more uncerany. Ths mehod s sasfacory for nvesgang a sngle pon n he wo-dmensonal value specrum (.e. 5% common value, pure common value, ec.) However, for smulaons performed across he full wo-dmensonal specrum from pure common o pure prvae value, random draws lead o dfferen value sgnal profles a each common value percen. Ths nroduces some unnecessary nose no he resuls, bu n fac does no change he overall resuls. However, s preferable o have he same profle across he smulaons so ha he resuls are perfecly comparable. Therefore, he second mehod s a smple algorhm ha ses he prvae and common value sgnals. Each agen s provded wh a unque wo-dmensonal value sgnal so ha he collecon of sgnals spans he suppors. The frs mehod s used for he fxed pon smulaons and he second s used for he smulaons ha span he wo-dmensonal value specrum. When usng he frs mehod, I use he Unform dsrbuon of value sgnals over hs suppor, snce hs s commonly used n he expermens n Kagel and Levn (22) and ohers. I expermened wh dfferen dsrbuons (normal, bea(2,2), bea(4,2), and bea(2,4) 2 ) and he resuls are as expeced: he bd prce sraeges for he symmerc dsrbuons (unform, normal, and bea(2,2)) were vrually dencal and he bd prce 2 These dsrbuons are, respecvely, more n he mddle wh als, more n he mddle whou als, more on he hgh end, and more on he low end.

7 7 sraeges for he asymmerc dsrbuons (bea(4,2) and bea(2,4)) shf rgh and lef respecvely. 2.2 Informaon Levels The seller mus decde how much nformaon should be released o he bdders afer he aucon, wh alernaves rangng from each bdder s own nformaon o nformaon abou all bds. Dufwenberg and Gneezy (22) compare he resuls from lab expermens for a wo-person barganng game wh hree ncremenal levels of nformaon abou aucon resuls: no nformaon abou ohers, he wnnng bd prce (sem-full), and all bds (full). Neugebauer and Selen (26) repor he resuls from lab expermens for frs-prce sealed-bd aucon wh hree nformaon levels provded n beween aucons: no nformaon abou ohers, he wnnng bd prce, and he runner-up bd prce. Smlarly, n hs sudy I use hree levels of nformaon (own, wnner, and wnner and runner-up) and desgnae hem I, I2, and I3 respecvely. 3 Bdders do no know oher bdders value sgnals, nor do hey know he acual common value when hey do no wn. The common value v C s unknown ex ane for all bdders, and only he wnnng bdders know v C ex pos. The acual value known o he wnner n a wo-dmensonal value envronmen s v = ( θc ) vp, + θcvc. In hs sudy, I consss enrely of own nformaon: own value sgnal v ˆ, own bd prce b, own rankng r, acual common value upon wnnng, and own paymen p. I2 consss of he 3 The noaon s defned n Table and he nformaon feedback s summarzed n Table 2.

8 8 own I nformaon plus nformaon abou he wnnng bd prce () b 4 and he paymen p. I3 consss of he nformaon from levels I and I2 plus nformaon abou he runner-up bd prce (2) b. The acual value s revealed only o he wnner, and s revealed before he nex eraon so ha he agen can use he nformaon. However, all bdders know he suppor [ SC, S C] so ha he I and I2 agens have an esmae for he gap beween bds (see Secon 3.2 and 3.3). Snce hs mehod s consruced so he agens seek for her opmal bddng sraegy for he value sgnals hey have been gven, he wnnng agen does no carry forward s knowledge of he acual common value. One way o nerpre hs s ha does no know ha he common value wll say he same. I have expermened wh movng he acual common value randomly from perod o perod whn he ε neghbourhood of he cener of he suppor, bu hs has mnmal effec on he resuls. 2.3 Number of Bdders and Perods Four and seven bdders per aucon were chosen o be compable wh lab expermens of Kagel e ala (987) and Levn e ala (996). Tweny-fve smulaneous seller agens are used when here are four bdders per aucon (for a oal of agens) and sxeen when here are seven bdders per aucon (for a oal of 2). These numbers are chosen o provde a good mx of bdder agens and o keep he oals approxmaely equal. Each aucon has he same number of bdders and he bdder agens move randomly from seller o seller on a fve-by-fve or four-by-four orus. Ths mehod 4 Superscrp numbers n parenheses denoe order sascs. In a sealed-bd aucon, ( ) n he aucon and b n s he lowes bd n an aucon wh n bdders. () b s he hghes bd

9 9 maches he bdder agens randomly so ha each agen has he opporuny o opmze s bddng sraeges by bddng agans a wde range of values held by he oher agens. Each agen parcpaes n one aucon per perod. I use 5 perods n order o accommodae learnng, bu on average he agens converge o a seady sae bddng sraegy whn abou 5 aucons (see Fgures 6 and 7). 3 Learnng Model The frs conrbuon of he sudy s o deermne f Selen s mpulse balance learnng mehod s suable for mulagen smulaons. In hs secon, I descrbe he mpulse balance learnng mehod and hen show ha resuls n an unaccepable amoun of negave prof and sensvy o nal values. A few smple modfcaons solve boh problems and produce a learnng mehod ha converges well, s nsensve o he learnng rae, and produces resuls for value-mulpler, prof, revenue, and effcency ha agree closely wh resuls from lab expermens. Ths demonsraes ha a mulagen sysem wh hs learnng mehod can be used as a credble alernave o lab expermens, especally where bddng experence s desrable. There s consderable scope for choosng he learnng model for he agens, ncludng renforcemen learnng, experence-weghed aracon, mpulse balance, and machne learnng mehods. These mehods are revewed and evaluaed n Mehlenbacher (27). Modfed mpulse balance learnng provdes he bes foundaon for learnng n aucons snce s a realsc represenaon of experenced human bdders, ulzes all nformaon feedback, handles connuous bds, and s exendable. The mpulse balance

10 mehod uses foregone prof 5 upon losng as an upward mpulse on a connuous bddng sraegy and money on he able 6 and acual loss upon wnnng as downward mpulses. Several emprcal sudes have shown ha mpulse balance learnng fs he daa for bd adjusmens by lab expermenal subjecs (Selen and Bucha, 998; Selen e ala, 25; Ockenfels and Selen, 25; Negebauer and Selen, 26; Garvn and Kagel, 994; Kagel and Levn, 999). Secon 3. descrbes Selen s mpulse balance learnng mehod. The nex wo secons descrbe he adjusmen rules for he downward mpulses for wnners (Secon 3.2) and he upward mpulses for losers (Secon 3.3). Secon 3.4 presens resuls from usng mpulse balance learnng and an mproved learnng mehod: mpulse learnng wh loss averson (ILA). Secon 3.5 presens convergence and sensvy analyses for he ILA mehod, and Secon 3.6 compares smulaon resuls o resuls from lab expermens. The common value sgnal suppors n hs secon follow Kagel e ala (989). I use fve bdders, [ SC, S C] = [, 3], and ε = Impulse Balance Learnng Ockenfels and Selen (25) apply mpulse balance learnng o frs-prce aucons wh prvae values and Selen e ala (25) apply mpulse balance learnng o frs-prce aucons wh common values. Bds are adjused usng downward a, or upward a +, 5 A losng bdder regres s low bd o he exen ha s value sgnal v ˆ s above he wnner s paymen. π = v p. Ths amoun s called foregone prof and s denoed ˆ, F 6 The wnnng bdder n a frs-prce aucon sacrfces prof unnecessarly o he exen ha s bd exceeds he runner-up bd. Ths s called leavng money on he able and s denoed m = b b. (2)

11 adjusmens or mpulses ha he agen calculaes usng prof π, foregone prof π F,, and money on he able m. For profable wnners, a s money on he able m, and for unprofable wnners s he loss π. For losers, a +, s he foregone prof. A hghvalue agen wns more frequenly han loses so ha ypcally E a > E a + for he hgh-value agen, and a low-value agen loses more frequenly han wns so ha ypcally E a + > E a for a low-value agen. Thus, because he hgher-value agen receves more downward mpulses han upward mpulses, should pu more wegh on an upward mpulse o compensae for s nfrequency. Smlarly, a lower-value agen should pu more wegh on a downward mpulse. Ths s he movaon for he balance aspec of he mpulse balance mehod. Each agen deermnes s balance wegh λ as he rao of s expeced value of he upward mpulse o he expeced value of he downward mpulse: E a + λ =. To deermne s adjused bd, he agen weghs he E a mpulses by a learnng rae φ and he downward mpulse wegh λ. The bd for perod + s hen a revson of he prevous bd b + = b + φ( a+, λ a, ). Ths ype of adjusmen mehod does no requre assumng ha he bddng sraegy s a lnear funcon of he bdder s value sgnal. However, he bd a any me can be expressed as a rao of he bd b o he value esmae, γ =, so ha we can dscuss he value mulpler γ ha can be vˆ compared wh heorecal and expermenal resuls.

12 2 3.2 Downward Impulses for Wnners A wnnng agen s assgned a rank of, r =, and s ordered bd prce denoed b. Smlarly, he runner-up has r = 2 wh ordered bd prce () (2) b, and so on. In calculang s adjusmens, he wnner consders nformaon ( mpulses ) abou s prof π and, when he paymen rule s frs prce, s money on he able m = b b. () (2) Rule W: For all nformaon levels, r =, and π <. : a, = π. Demonsraon: If he agen wns bu has a loss of π = v p, lowers s bd n proporon o he loss n an effor o mprove s expeced prof. Adjusng for acual loss was found o be a sgnfcan facor n bd adjusmen by Garvn and Kagel (994) and Selen e ala (25). Rule W2(I3): For I3, r =, frs-prce paymen, π >. : a, = m. Demonsraon: An agen wh I3 can use nformaon abou he oher bdders, specfcally he runner-up, o make a more nformed adjusmen when wns. When wnnng s profable n a frs-prce aucon, he agen uses he value of he runner-up bd o deermne how much overbd. Ths overbddng resuls when he paymen rule uses he frs-prce snce he wnnng bdder s deal suaon s o have bd jus slghly above he runner-up bdder. Any amoun ha he wnnng bdder bds over he runner-up bdder s called money on he able and s denoed m = b b. For a frs-prce paymen (2) rule, m s used o adjus he bd down. Money on he able has been shown o be a sgnfcan facor n bd adjusmen by Selen and Bucha (998), Selen e ala (25), Ockenfels and Selen (25), and Negebauer and Selen (26).

13 3 Rule W3(I,I2): For I and I2, r =, frs-prce paymen, π >. : a ˆ, = m. Demonsraon: A profable agen wh I and I2 nformaon mus use an approxmaon for money on he able m ˆ o deermne he adjusmen for lowerng s bd o mprove s prof. The alernave of makng no adjusmen s no conssen wh he mpulse balance mehod, snce here would be no downward mpulse. Snce he agen has nformaon abou he number of bdders n and he suppor [ SC, S C], can use hs o creae an esmae for money on he able. The gap beween bds wll decrease n proporon o n, and snce he values are drawn unformly, an upper bound on an esmae for money on he able s SC S n π so a smple esmae for money on he able s C. However, money on he able wll be small wh large S S n C C ˆ = π. m 3.3 Upward Impulses for Losers Rule L(I2,I3): For I2 and I3, r >, when π F,, a+, = π F, Demonsraon: If an agen loses, usually regres s low bd o he exen ha s value sgnal v ˆ s above he wnner s paymen. Ths s he concep of foregone prof used by Camerer e ala (22), Selen and Bucha (998), Selen e ala (25), Ockenfels and Selen (25), and Negebauer and Selen (26) wh π ˆ, = v p. An F agen wh I2 or I3 nformaon knows he paymen and so can calculae s foregone prof. When foregone prof s posve he agen ncreases s bd n proporon o π, snce hs wll mprove s probably of wnnng profably. If a bdder has a low value sgnal, he foregone prof wll end o be negave, and he bdder wll no ncrease s bd. F

14 4 Rule L2(I): For I, r >, when ˆ π F,, a ˆ +, = π F, Demonsraon: Wh one excepon, I agens do no know he paymen and mus esmae foregone prof ˆ π ˆ ˆ, = v p. The excepon s he runner-up bdder, F (2) r = 2, n a second-prce aucon n whch b = b = p so bdder s foregone prof s ˆ ˆ F, π F, v b π = =. For he oher losng bdders, he foregone prof esmae 7 s a fracon of vˆ b, decreasng wh he number of bdders and ncreasng wh he rank. In a second-prce aucon wh 2 vˆ b r > ˆ π F, = ( r ). 8 n vˆ b = and n frs-prce aucon wh n r >, ˆ π F, ( r 2) 3.4 Negave Prof and Sensvy o Inal Values In hs secon, I analyze resuls of smulaons and make changes o he mpulse balance learnng mehod. The resul s a learnng mehod ha uses mpulses, excludes he balance prncple, and ncludes loss averson, so a reasonable name for he mehod s mpulse learnng wh loss averson (ILA). Resul : Usng mpulse balance learnng n compuaonal expermens resuls n a hgh degree of negave prof,.e. loss, and sensvy o nal values. To acheve profably and nsensvy o nal values, I make hree changes o he mpulse- 7 I am usng he erm foregone prof for conssency wh he oher rules, bu s mpossble for an I agen o esmae he paymen and hence he foregone prof. Insead, he agen uses s value gap o calculae he upward adjusmen. 8 Ths rule s less of a foregone prof and more of a value gap adjusmen. Wh larger n, he gap beween agens wll be smaller, so he basc adjusmen sep s nversely proporonal o n. Agens wh larger wll need o adjus more han agens ha are closer o he wnner. r

15 5 balance mehod. Frs, he balance par of he mehod s removed. Second, he loss adjusmen n Rule W s weghed usng a loss averson facor L = E π π < ha s he expeced value of he magnude of he losses. Thrd, when a wnnng agen has an expeced loss ( L > ) and lowers s bd prce o reduce s probably of wnnng, s couner-producve for he agen o ncrease s bd prce when successfully reaches he losng sae. Thus, a losng agen uses foregone prof o rased s bd prce only when L = and he adjusmen can be wren usng he ndcaor funcon,.e., ( L = ) a ˆ +, π ( L = ) F, =. In summary, he ILA mehod s o adjus bds usng b = b + φ( a a ), where he adjusmen rules are: + +,, Rule W: For all nformaon levels, r =, and. π < : a, ( L ) π Rule W2: For I3, r =, frs-prce paymen, π >. : a, = m. = +. Rule W3: For I and I2, r =, frs-prce paymen, π >. : a ˆ, = m. Rule L: For I2 and I3, r >, when π F,, a, = π, + ( L = ) Rule L2: For I, r >, when ˆ π F,, a, = π, ˆ + ( L = ) F Dscusson: The resuls for mpulse balance learnng n Fgure show ha a large proporon of he bdders (especally hose wh hgh value sgnals) experence losses. Ths s a much hgher level of losses han shown n resuls from lab expermens and a level of sensvy o sarng values ha s undesrable n a compuaonal model. For example, bankrupces occur n abou 6% of he aucons wh experenced bdders (Kagel and Rchard, 2). These bankrupces occurred n wo suaons: 8% of bdders wen bankrup wh a $ cash balance wh a suppor of [5, 38], ε = 8, and 7 bdders; 4% F

16 6 of bdders wen bankrup wh a $2 cash balance wh suppor of [25, 225], ε = 8, and 4 bdders. Experenced bdders n real-world aucons would lkely be skllful enough o avod losses and bankrupces alogeher, so he goal of he learnng model should be a mnmal level losses or bankrupces, a leas below he 6% n Kagel and Rchard s expermens. The fac ha he hgh-value bdders are experencng losses ndcaes ha here s a problem wh he learnng model for hgh-value bdders. The balance facor λ, whch vares wh he bdder value, could be expeced o deal wh hs problem bu s no producng sasfacory resuls. Fgure 2 shows ha he values of λ do vary wh he bdder value, and end o be lower for hgh-value bdders han for low-value bdders as expeced from he dscusson n Secon 3. When λ s removed from he model, he resuls mprove slghly as shown n Fgure 3a. I may sll be reasonable o expec he agen o pu more wegh on a downward mpulse han an upward mpulse, even hough he balance facor may no be he approach ha should be used. An agen may oban mproved profs f weghs he downward mpulse from negave prof more han he correspondng ncrease from posve foregone prof. Tversky and Kahneman (992, Table 6) esmae loss averson facors n he range [.97, 2.44], bu makes sense n hs case of bdders wh dfferen value sgnals o have endogenous loss averson. Fgure 3b shows he resuls for an endogenous loss averson where he loss averson facor s L = E π π <. Now 35% of he bdders experence losses bu he overall average prof s up o -.5. I also makes no sense for an opmzng bdder o rase s bd afer losng, when has been experencng losses when s wnnng. Thus, I nroduce a

17 7 prof swch ha he agen uses for s upward mpulses. Ths fnal modfcaon ( ) L = now rases all agens o non-negave prof as shown n Fgure 3c, and an overall average prof level of.5. The overall model mplemenaon s a nonlnear sysem wh he poenal of convergng or no, or convergng o a local opmum nsead of a global opmum. As such, s preferable for he mehod o be nsensve o nal values (Judd, 998) and oher parameer values. Fgure shows ha he resuls from mpulse balance learnng vary sgnfcanly wh he nal values.95 ±.2,.85 ±.2, and.75 ±.2. However, Fgure 4 shows ha he ILA mehod s que nsensve o he nal values. In Fgure, he prof curve for an nal value.95 ±.2 s close o zero for low-value bdders and decreases rapdly o -.7 for hgh-value bdders. As he nal value s decreased o.85 ±.2 and hen o.75 ±.2, he values for md-value and hgh-value bdders ncrease consderably so ha he curve becomes much flaer. In Fgure 4, he paern of prof s much more smlar across he nal values. The low-value and hghvalue bdders end o have prof close o zero, wh abou weny md-value bdders wh profs as hgh as.25 for all hree nal values. One of he man mehodologcal dfferences beween he expermens wh humans n he varous sudes ced n hs paper and hese compuaonal expermens s ha here each bdder s prvae and common value sgnals are consan hroughou he aucons (as explaned n Secon 2.2). The alernave s o provde each agen wh a random value sgnal for each aucon. Ths resuls n each agen learnng an average bd sraegy ha s he adapve bes response o he full range of value sgnals. Perhaps he mpulse balance mehod s more suable o learnng an average bd sraegy. As shown

18 8 n Fgure 5, hs s no he case. For varyng common value sgnals, mpulse-balance learnng resuls n sgnfcan number of agens wh hgh levels of negave prof. However, he ILA mehod resuls n mos agens achevng posve profs, bu wh some achevng small negave profs. 3.5 Convergence and Sensvy o Learnng Rae Resul 2: The ILA mehod resuls n value-mulplers ha converge n less han perods, and hs convergence s ndependen of he nal values and smooher han he convergence of he mpulse-balance mehod. Dscusson: Fgure 6 shows value-mulpler convergence for he mpulse balance and ILA mehods. The mpulse-balance value mulplers converge o que dfferen values (.94,.89, and.83) for he hree nal values, whereas he ILA value mulplers converge o more smlar values of (.92,.9, and.89). In addon, he paern of he convergence s much smooher for he ILA mehod. For he hree nal values, convergence requres abou, 6, and 9 perods. These convergence resuls are mporan, snce s mpossble o nerpre aucon resuls for prof, revenue, and effcency when here s no convergence. For example, whou convergence he resuls n perod 5 are dfferen from he resuls n perod 6, whereas f he resuls from perod 5 o nfny are he same, we can conclude ha hese are he resuls of he aucon. Also, he fac ha convergence occurs n less han perods makes reasonable o nfer ha he bd sraeges of human agens could converge n a realsc number of realworld aucons. Resul 3: The ILA mehod s nsensve o he learnng rae φ.

19 9 Dscusson: Why should we beleve ha a bdder s downward mpulse s all of s loss and no a fracon of he loss, or ha a bdder s upward mpulse s all of s foregone prof and no a fracon of? Usng dfferen values for he learnng rae φ answers hs queson. Fgure 7 shows he value mulpler for frs-prce aucons usng a sample of learnng raes n he nerval [.,. ]. Frs, he resulng value mulpler s very close across all of he learnng raes (.92). Second, he paern of varaon s also very smlar hroughou he range. The dfference s ha he smaller learnng raes end o produce smooher convergence, wh he sandard devaon rangng from abou.7 for φ =. o.24 for φ =.. These resuls are que nsensve o nal value, and I use φ =.5 and nal values of.85 ±.2 n he smulaons as arbrary choces. 4 Comparng ILA Resuls wh Lab Expermens Usng smulaons wh he ILA learnng mehod, I presen resuls and compare hem wh lab expermen resuls n Kagel e ala (989), Kagel e ala (995), Kagel and Rchard (2), and Goeree and Offerman (22). In he lab expermens, he nformaon feedback s equal o or greaer han he I3 nformaon level. Thus, n he fgures relaed o hs secon, we are neresed n only he I3 bdders, represened by he dashdo curves. Also, he mos frequenly-used suppor for he common value sgnals s [25, 225] so ha s wha I use n he compuaonal expermens. For comparably wh mos of he lab expermen resuls, I vary he uncerany usng ε = 8, 2, 8, and 27 and use four and seven bdders for boh frs-prce and second-prce aucons. The resuls ha are llusraed n he fgures are conssen across repeaed smulaons. In he followng sub-secons I compare resuls for value mulpler, prof, and effcency, and hese resuls are summarzed n Table The value mulpler and prof are

20 2 sraghforward and have been dscussed n Secons 3. o 3.3, bu effcency requres some explanaon. Snce effcency refers o he aucon beng won by he bdder wh he hghes value ex pos, all aucons are equally effcen when he value s % common. Thus, n he wo-dmensonal value envronmen, effcency can be consdered only when here s some componen of prvae value,.e., when he common value componen s less han %. When he common value s %, Kagel e ala (989) and Kagel e ala (995) measure effcency by he percen of aucons won by he bdder wh he hghes value sgnal (ex ane). Ths s no really effcency, bu s sll neresng o look a hs hghes-value-sgnal wnnng percen. When he common value percen s less han %, prvae value effcency as used by Goeree and Offerman (22) compares he wnner s prvae value wh he maxmum prvae value among he bdders,.e.,, mn{ vp, } { vp, } mn{ vp, } wnner vp =. = when he wnner has he hghes prvae value, and = max when he wnner s he bdder wh he lowes prvae value. As he common value componen ncreases, he common value sgnal may undermne he prvae value effcency. Consder wo bdders, one wh a hgh prvae value (say correspondng o low coss of producon) and one wh a low prvae value. If he low prvae value bdder has a hgher esmae of he common value han he hgh prvae value bdder, may subm a hgher bd and wn he aucon. Ths leads o prvae value neffcency. 4. Frs-Prce Aucons Value Mulpler: From nuon and heory, we expec ha bdders wll shade her bds n frs-prce aucons,.e., he value mulpler s expeced o be less han.. For pure prvae values, expermenal evdence from Kagel and Levn (993) shows an

21 2 average value mulpler of.92 for frs-prce aucons, averaged over expermens wh fve and bdders wh I3 nformaon. The smulaon resuls for seven I3 bdders n Fgure 8 show value mulplers of abou.9. These pure prvae-value resuls are conssen wh he daa from lab expermens. Kagel and Rchard (2) show value mulplers for frs-prce common values wh ε = 8 of abou.92 for four I3 bdders and.95 for seven I3 bdders n he mddle regon of he suppor. Fgure 8 shows common value mulplers for ε = 8 of abou.92 for four bdders and of abou.94 for seven bdders. Thus, boh he prvae-value and common-value resuls agree very closely wh he expermenal resuls. Prof: Daa n Kagel e ala (989) for frs-prce pure common-value aucons averagng abou seven bdders show ha prof ends o ncrease wh more uncerany n he common value sgnal (hgherε ). Fgure 9 shows resuls for four and seven bdders across he full specrum of wo-dmensonal value sgnals from pure prvae value (% common value) o pure common value (%). For pure common value, prof ncreases sgnfcanly wh uncerany for four bdders (from abou o ), bu ncreases less wh uncerany for seven bdders (from abou o abou 3). Goeree and Offerman (22) show prof ncreasng slghly wh more uncerany n aucons wh wodmensonal value sgnals ha are abou 4% common value (wh sx bdders) and 25% common value (wh hree bdders). A common value percens of 4% and 25%, Fgure 9 shows ha he prof remans he same as he uncerany ncreases. Thus, he agen resuls for he varaon of prof wh uncerany are only parally n agreemen wh he lab expermen resuls.

22 22 Kagel and Rchard (2) fnd average prof s hgher wh four bdders han wh seven bdders n frs-prce common-value aucons. Comparng he lef column (four bdders) of Fgure 9 wh he rgh column (seven bdders) shows ha prof ends o be hgher for four bdders a all levels of uncerany and all nformaon levels across he full range of common-value percen. Thus, he agen resuls for he varaon of prof wh number of bdders are n agreemen wh he resuls from he lab expermens. Fgure shows he prof resuls for 2, 25, 5,, 2, and 3 bdders for frs-prce aucons wh hgh ( ε = 27 ) uncerany. For I2 and I3 bdders, prof furher decreases wh he ncreasng number of bdders, resulng n near-zero profs when here are over 2 bdders. The prof of I agens connues o decrease sgnfcanly below zero as he number of bdders ncreases. Ths hghlghs he mporance o bdder prof of beng nformed abou he paymen. Effcency: The effcency resuls for frs-prce aucons wh less han pure common value are shown n Fgure, and he resuls for hghes-value-sgnal wnnng percen for pure common-value aucons are shown n Table 4. From heory, we expec ha effcency wll decrease when here s a common value componen (Dasgupa and Maskn, 2; Jehel and Moldovanu, 2). Expermen resuls from Kagel e ala (989) for abou seven bdders show ha he hghes-value-sgnal wnnng percen ends o decrease wh more uncerany n frsprce aucons. Table 4 shows ha for I3 agens he hghes-value-sgnal wnnng percen n frs-prce aucons decreases wh ncreased uncerany for boh four bdders (from 94% o 28%) and seven bdders (from 95% o 22%), whch s conssen wh he resuls from he lab expermens.

23 23 Goeree and Offerman (22) show ha effcency s lower wh more uncerany for common value percens of 4% (wh sx bdders) and 25% (wh hree bdders). Fgure shows ha for I3 bdders, effcency ends o say he same or decrease slghly wh ncreased uncerany a boh 4% and 25% common value for boh four and seven bdders, whch s conssen wh he resuls from he lab expermens. 4.2 Second-Prce Aucons Value Mulpler: From nuon and heory for second-prce aucons, we expec ha bdders wll bd her values when he value s prvae and shade her bds when he value s common. For prvae-value second-prce aucons, expermenal evdence from Kagel and Levn (993) shows an average value mulpler of.2 for second-prce aucons, averaged over expermens wh fve and en bdders (assumng an average of abou seven bdders). The resuls n Fgure 2 for seven bdders show value mulplers of abou.99 for prvae-value second-prce aucons. For common-value second-prce aucons, regressons n Kagel e ala (995) show value mulplers of abou.97,.96,.94, and.92 for ε =8, 2, 8, and 27 respecvely. The resuls n Fgure 2 for seven bdders show a smlar magnude and paern of value mulplers, namely abou.98,.97,.96, and.95. Prof: For second-prce common-value aucons, Kagel e ala (995) fnd ha prof ncreases wh ε for four bdders, bu decreases wh ε for seven bdders. The resuls shown n Fgure 3 are conssen wh her resuls for four bdders (prof ncreases wh ε ), bu no for seven bdders (no change n prof wh ε ). Fgure 3 also shows ha prof decreases slghly wh an ncrease n he number of bdders, for all nformaon levels and across he full range of common value percen. Ths s he same

24 24 compuaonal resul ha was obaned for prvae values. I s also conssen wh he expermenal resuls from Kagel e ala (995) who found ha profs were hgher for four bdders han for seven. Effcency: The effcency for second-prce aucons wh less han pure common value s shown n Fgure 4, and he hghes-value-sgnal wnnng percen for pure common-value aucons s shown n Table 4. Kagel e ala (995) found ha he hghesvalue-sgnal wnnng percen was lower n second-prce aucons han n frs-prce aucons, when he level of uncerany s ε = 27. However, Table 4 shows ha he hghes-value-sgnal wnnng percen s hgher n second-prce aucons. Second-prce aucons are more effcen han frs-prce aucons for he agens because hey are bddng closer o her values. Snce hs s wha s expeced from heory, he agens are bddng more lke opmzng agens han lke he nexperenced agens n he expermens. The resuls of Kagel e ala (995) also show ha hghes-value-sgnal wnnng percen n second-prce aucons s slghly lower for seven bdders han for four bdders. Table 4 shows ha he agens produce smlar resuls for levels of uncerany above ε = 8. 5 Varaon of Prof, Revenue, and Effcency wh Common Value Percen The second conrbuon of hs sudy s o deermne how he aucon resuls change as he common value componen ncreases, and specfcally wheher he change s lnear. In he fgures used n he prevous secon, s obvous ha he resuls across he wo-dmensonal value sgnal are usually no lnear. As he common value percen ncreases, prof, revenue, and effcency all decrease monooncally, bu hey decrease n dfferen ways. The seller endeavors o choose he paymen rule and nformaon level

25 25 ha maxmzes s revenue, maxmzes effcency, or maxmzes boh. Therefore, he seller s neresed n wheher he dfferen paymen rules and nformaon levels produce dfferen levels of revenue and effcency, or wheher hey are equvalen. In hs secon, I dscuss he resuls for all hree nformaon levels. 5. Prof Resul 4: Prof curves decrease nonlnearly for frs-prce aucons and lnearly for second-prce aucons. In frs-prce aucons he nonlneary usually nvolves decreasng faser a hgher common value percens. Dscusson: See Fgure 9 for frs-prce aucons and Fgure 3 for second-prce aucons. The man dfference beween learnng n frs-prce aucons and learnng n second-prce aucons s ha he agens use money on he able n frs-prce aucons bu no n second-prce aucons. When he value sgnal s domnaed by prvae value (.e., a low common value percen), he bd reducon from money on he able keeps he prof hgh. As he common value componen ncreases, he conrbuons from money on he able o prof become domnaed by he effecs of he common value sgnal. There s also some neresng varaon wh he level of nformaon feedback. The curves end o be he same for I, I2, and I3 nformaon levels a lower levels of uncerany, bu as uncerany abou he common value ncreases prof s hgher for he more nformed I3 bdders. 5.2 Revenue See Fgures 5 and 6 for revenue resuls for frs-prce and second-prce aucons, respecvely. Revenue ends o decrease wh ncreasng common value.

26 26 Resul 5: Revenue curves decrease nonlnearly for frs-prce aucons and lnearly for second-prce aucons. In frs-prce aucons he nonlneary usually nvolves decreasng faser a lower common value percens. Resul 6: In mos cases, he seller receves less revenue when provdes bdders wh more nformaon feedback. Dscusson: The fgures show ha n mos cases he seller receves less revenue when he bdders have I3 nformaon. However, for frs-prce aucons (Fgure 5) wh lower levels of uncerany ( ε 8 ), I3 agens provde hgher revenue a hgh common value percens (3% o 9%) han he I and I2 agens, alhough hs effec dmnshes wh more bdders. Once agan, he major dfference beween I3 agens and he I and I2 agens s ha he former can calculae money on he able whle he laer can only esmae. The esmae becomes less relable as he uncerany ncreases so ha I3 agens are beer able o keep her bd sraeges profable, akng more of he surplus and yeldng lower revenue for he seller. 5.3 Effcency See Fgure for frs-prce aucons and Fgure 4 for second-prce aucons. As he common value componen ncreases, he common value sgnal dsrups he prvae value effcency. Second-prce aucons end o be more effcen han frs-prce aucons because he agens bd closer o her values. Resul 7: Prvae-value effcency curves end o say hgh a low percen common value and hen decrease rapdly for hgher common value percens.

27 27 Dscusson: The nonlneary s especally pronounced n second-prce aucons (Fgure 4) where effcency remans close o. unl relavely hgh levels of common value percen, and hen decreases rapdly o effcency as low as.8. 6 Revelaon of Common Value o Losers The hrd conrbuon of hs sudy s o deermne wheher may be worhwhle for a seller (such as a federal or sae governmen) o enforce ruhful revelaon of he rue common value by aucon wnners. In he expermens suded so far, and n nearly all real-world aucons, losng bdders do no know he acual common value. The wnner dscovers he rue common values afer he aucon. For example, n mber sale aucons he wnners learn he rue quany and value of mber; n hghway procuremen aucons, he wnner dscovers he rue scope of he projec; n ol lease lcences, he wnner dscovers he rue quany of ol; and so on. These values are carefully guarded company secres (Baldwn e ala, 997) and are no nenonally revealed o oher bdders. However, some expermenal work has suded he effecs on bddng of revealng some nformaon abou he common value o all bdders (Kagel and Levn (999) for frsprce aucons and Kagel e ala (995) for second-prce aucons). Ths rases he queson of wheher a buyer or seller, say he governmen operang procuremen or assesale aucons, should requre he aucon wnners o reveal he common value ha hey dscover afer wnnng. I know of some aemps o do hs n Canadan federal governmen procuremen aucons n whch he governmen asks bdders o reveal her coss. Of course he coss provded are no ruhful! If were worhwhle, he governmen could raonally decde o nves n mplemenng regulaons and enforcemen of ruhful revelaon of he wnner s value. Gven hs nformaon, losng

28 28 agens would use n her calculaon of he amoun o ncrease her bd (n Rules L and L2). To oban a compuaonal answer o hs queson, I perform expermens wh value revelaon and observe he revenue. Fgure 7 shows for I3 bdders he dfferences beween he revenue wh revealed common value and he revenue whou revelaon. The resuls ha are llusraed n he fgures are conssen across repeaed smulaons. Resul 8: For frs-prce aucons, when he common value percen s hgh (>6%) and here s a hgh degree of uncerany n he common value sgnal ( ε > 2, revealng nformaon abou he common value ncreases revenue for I3 nformaon. Dscusson: Expermens by Kagel and Levn (999) for frs-prce commonvalue aucons wh ε = 27 show ha revealng nformaon abou he rue common value ncreased revenue (+2.75) for four bdders bu decreased revenue (-.88) for seven bdders. For he same condons, he op row of Fgure 7 shows ha revenue ncreases (+7) for four bdders and ncreases less (+3) for seven bdders. 9 These resuls are conssen wh he expermenal daa n ha revenue ncreases more for four bdders han for seven, bu s nconssen n he drecon of change for he seven bdders. Fgure 7 also shows ha he revenue effecs are smaller as uncerany decreases. For % common value, he benefs become neglgble when ε = 8. For values wh less common value percen, common value revelaon somemes has a negave mpac on revenue. Thus, for he aucon desgner he percen common value and he degree of uncerany abou he common value all affec he mpac of value revelaon on revenue. 9 Snce oal revenue s approxmaely 25, he revenue ncrease of 7 s approxmaely 5%.

29 29 Resul 9: For second-prce aucons, revealng nformaon abou he common value sgnfcanly ncreases revenue for I3 bdders, especally when here s a hgh degree of uncerany n he common value sgnal. Dscusson: Expermens by Kagel e ala (995) show ha revealng nformaon abou he common value n second-prce aucons ncreased revenue (+.3) for four-fve bdders bu decreased revenue (-2.5) for sx-seven bdders. The boom row of Fgure 7 shows ha revenue ncreases (+4) for four bdders and ncreases less (+2) for seven bdders. Agan, he resuls are n paral agreemen wh he expermenal resuls. 7 Concluson I fnd ha Selen s mpulse balance mehod can be adaped for use n mulagen smulaons of aucons wh values ha have some common value componen (Resul n Secon 3.4). The resulng ILA (mpulse learnng wh loss averson) mehod converges whn perods and s nsensve o he learnng rae (Resuls 2 and 3 n Secon 3.5). I use he ILA mehod n mulagen smulaons for frs-prce and second-prce paymen rules, hree dfferen nformaon levels, and wo-dmensonal value sgnals ha vary from pure prvae value o pure common value. The resuls are compared o daa from lab expermens n oher sudes (summarzed n Table 3), and he agen resuls for he value mulpler, prof, and effcency are usually conssen wh resuls from lab expermens. These conssences suppor he real-world valdy n hs conex of usng mulagen smulaons wh learnng agens. For values n beween pure prvae and pure common value, curves for prof, revenue, and effcency are nonlnear especally when he paymen rule s frs prce. The

30 3 prof curves end o decrease nonlnearly for frs-prce aucons and lnearly for secondprce aucons (Resul 4 n Secon 5.). The nonlnear revenue curves end o decrease more rapdly a low common value percens (Resul 5 n Secon 5.2). The very nonlnear effcency curves end o say hgh and hen decrease rapdly for common value percens (Resul 7 n Secon 5.3). In addon, revenue n mos cases decrease wh ncreasng nformaon feedback o he bdders (Resul 6 n Secon 5.2). Smulaons also show ha forcng revelaon of he rue common value may have benefcal revenue effecs when he common-value percen s hgh and here s a hgh degree of uncerany abou he common value (Resuls 8 and 9 n Secon 6). Usng mulagen smulaons has provded some nsghs no sngle-un sealedbd aucon performance for dfferen levels of nformaon feedback across dfferen levels of common value. The nex paper wll expand he approach o analyze Englsh aucons. 8 Acknowledgemens I am ndebed o Davd Scoones, Lnda Wellng, Don Ferguson, and Tony Marley for her valuable suggesons. I am also graeful o he Socal Scences and Humanes Research Councl of Canada for scholarshp suppor durng he course of hs research.

31 3 9 References Ahey, S., Hale, P., 22. Idenfcaon of Sandard Aucon Models. Economerca. 7: Byde, A., 22. Applyng Evoluonary Game Theory o Aucon Mechansm Desgn. HPL-22-32, Hewle-Packard Company. Camerer, C., Ho, T. H., Chong, J. K., 22. Sophscaed Experence-Weghed Aracon Learnng and Sraegc Teachng n Repeaed Games. Journal of Economc Theory. 4: Dasgupa, P., Maskn, E., 2.. Effcen Aucons. Quarerly Journal of Economcs. 5: Decker, K.S., Lesser, V. R., 995. Desgnng a Famly of Coordnaon Algorhms. U. Mass Compuer Scence Techncal Repor Dufwenberg, M., Gneezy, U., 22. Informaon dsclosure n aucons: an expermen. Journal of Economc Behavor and Organzaon. 48: Garvn, S., Kagel, J.H., 994. Learnng n Common-value aucons: Some Inal Observaons. Journal of Economc Behavor and Organzaon. 25: Goeree, J.K., Offerman, T., 22. Effcency n Aucons wh Prvae and Common Values: An Expermenal Sudy. Amercan Economc Revew. 92: Hale, P. A., Hong, H., Shum, M., 23. Nonparamerc Tess for Common Values n Frs-Prce Sealed-Bd aucons. NBER Workng Paper No. 5. Halu, A., Schlzz, S., 24. Are Aucons More Effcen Than Fxed Prce Schemes When Bdders Learn? Ausralan Journal of Managemen. 29:

32 32 Hendrcks, K., Pnkse, J., Porer, R.H., 23. Emprcal Implcaons of Equlbrum Bddng n Frs-Prce, Symmerc, Common-value aucons. Revew of Economc Sudes. 7: Jehel, P., Moldovanu, B., 2. Effcen Desgn wh Inerdependen Valuaons. Economerca. 69: Judd, K.L., 998, Numercal Mehods n Economcs. The MIT Press. Kagel, J.H., Levn, D., 993. Independen Prvae Value Aucons: Bdder Behavor n Frs-, Second- and Thrd-Prce Aucons wh Varyng Numbers of Bdders. Economc Journal. 8: Kagel, J.H., Levn, D., 999. Common-value aucons wh Insder Informaon. Economerca. 67: Kagel, J.H., Levn, D., 22. Common-value aucons and he Wnner s Curse. Prnceon Unversy Press. Kagel, J.H., Levn, D., Harsad, R.M, 995. Comparave Sac Effecs of Number of Bdders and Publc Informaon on Behavor n Second-Prce Common-value aucons. Inernaonal Journal of Game Theory. 24: Kagel, J.H., Levn, D., Baalo, R.C, Meyer, D.J., 989. Frs-Prce Common-value aucons: Bdder Behavor and he Wnner's Curse. Economc Inqury. 27: Kagel, J.H., Rchard, J.F., 2. Super-Experenced Bdders n Frs-Prce Common- Value Aucons: Rules of Thumb, Nash Equlbrum Bddng, he Wnner s Curse. Revew of Economc Sascs. 83: Km, Y. S., 27. Maxmzng sellers welfare n onlne aucon by smulang bdders proxy bddng agens. Exper Sysems wh Applcaons. 32:

33 33 Mehlenbacher, A., 27. Mulagen sysem plaform for aucon expermens. Unversy of Vcora, Economcs Deparmen Dscusson Paper, DDP76. Neugebauer, T., Selen, R., 26. Indvdual Behavor of frs-prce aucons: The mporance of nformaon feedback n compuerzed expermenal markes. Games and Economc Behavor. 54: Ockenfels, A., Selen, R., 25. Impulse Balance Equlbrum and Feedback n Frs Prce Aucons. Games and Economc Behavor. 5: Selen, R., 998. Feaures of Expermenally Observed Bounded Raonaly. European Economc Revew. 42: Selen, R., Abbnk, K., Cox, R., 25. Learnng Drecon Theory and he Wnner s Curse. Expermenal Economcs. 8: 5-2. Selen, R., Bucha, J., 998. Expermenal Sealed Bd Frs Prce Aucons wh Drecly Observed Bd Funcons. In: Budescu, D. V., Erve, I., Zwck, R. (Eds), Games and Human Behavor: Essays. Lawrence Erlbaum Assocaes, pp Tesfason, L., 23. Agen-based compuaonal economcs: modelng economes as complex adapve sysems. Informaon Scences. 49: Tesfason, L., Judd, K.L., 26, Handbook of Compuaonal Economcs : Agen-Based Compuaonal Economcs. Norh Holland. Tversky, A., Kahneman, D.,992. Advances n Prospec Theory: Cumulave Represenaon of Uncerany. Journal of Rsk and Uncerany. 5:

34 34 Tables Symbol Table. Sealed-Bd Model Noaon Summary b Bd prce of bdder n aucon. n () ( ) Descrpon b,..., b Ordered bd prces n a sealed-bd aucon where ε Radus of he suppor for he common value sgnal.. γ Value mulpler: b γ vˆ =. () b s he hghes bd. λ The balance wegh n he mpulse balance learnng mehod. () (2) m Money lef on he able by a profable wnner for frs-prce paymen: m = b b. p Paymen by wnner n aucon. φ Learnng rae of bdder a perod. p Paymen made by bdder, gven ha wns. π Prof of bdder n aucon. π Foregone prof of bdder n aucon. F, r Rankng of bdder n aucon. The wnner s r =, he runner-up r = 2, ec. S, S, S, S Upper and lower bounds of suppors for he prvae value sgnal and common value. P P θ C C C Common value componen of he value sgnal. v ˆ Value sgnal of bdder n aucon. v Acual value, revealed only o wnner.

35 35 Table 2. Informaon Levels (ncremenal) Level Descrpon Feedback Number of bdders n Value sgnal suppor [ S, S ] C C I Value Sgnal: Own Bd Prce: Own v ˆ b Rankng: Own r Paymen: Own p r = Value: Own v r = I2 Bd Prce: Wnner () b Paymen p I3 Bd Prce: Runner-up (2) b

36 36 Table 3. Summary of Comparson wh Lab Expermens (Secon 3.6) Prvae Value 4%, 25% Common Value % Common Value Paymen Resul Lab Agen Lab Agen Lab Agen Frs Prce Second Prce Value Mulpler Prof Effcency Value Mulpler Prof Effcency.92 ().2 () Kagel and Levn (993) 2 Kagel and Rchard (2) 3 Kagel e ala (989) 4 Goeree and Offerman (22) 5 Kagel e ala (995).9.92,.95 (2) Increases wh (4) Decreases wh (4) Consan wh Decreases slghly wh Increases wh and n (2,3) Decreases wh (3).99.97,.96,.94,.92 (5) Increases wh for n=4 Decreases wh for n=7 Decreases wh n (5) Decreases wh n (5).92,.94 Increases wh and n Decreases wh.98,.97,.96,.95 Increases wh for n=4 No change wh for n=7 Decreases wh n Decreases wh n