How to Maximize the Profit for Bidder and Seller in a Sealed-Bid Second-Price Auction

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1 Unon College Unon Dgtal Works Honors Theses Student Work How to Maxmze the Proft for Bdder and Seller n a Sealed-Bd Second-Prce Aucton We Yu Unon College - Schenectady, Y Follow ths and addtonal works at: Part of the Economc Hstory Commons Recommended Ctaton Yu, We, "How to Maxmze the Proft for Bdder and Seller n a Sealed-Bd Second-Prce Aucton" (203). Honors Theses Ths Open Access s rought to you for free and open access y the Student Work at Unon Dgtal Works. It has een accepted for ncluson n Honors Theses y an authorzed admnstrator of Unon Dgtal Works. For more nformaton, please contact dgtalworks@unon.edu.

2 How to Maxmze the Proft for Bdder and Seller n a Sealed-Bd Second-Prce Aucton By We Yu * * * * * * * * * Sumtted n partal fulfllment of the requrements for Honors n the Department of Economcs UIO COLLEGE June, 203

3 ABSTRACT YU, WEI How to Maxmze the Proft for Bdder and Seller n a Sealed-Bd Second-Prce Aucton. Department of Economcs, June 203. ADVISOR: Ren, Yufe Wth a hstory of more than 2500 years, auctons have long een used to negotate the exchange of goods and commodtes. In an aucton, dders compete wth rvals y sumttng ds dependng on ther personal evaluatons of the goods. The good s allocated to the dder who offers the hghest d. There are many dfferent types of auctons, ut four maor ones are prmarly concerned y economsts and researchers--the Englsh aucton, the Dutch aucton, the sealed-d frst-prce aucton and the sealed-d second-prce aucton. My thess manly focuses on the characterstcs of the sealed-d second-prce aucton, wth oth contnuous and dscrete ddng. My thess dscusses the dder's strateges that can maxmum the expected payoff and the seller's strategy that can affect the expected revenue. In contnuous ddng, truthful ddng s the domnant strategy. In terms of the dscrete ddng, my thess apples the model from Yu (999) to specfcally dscuss the sealed-d second-prce aucton and fnds out equlrum strategy usng the expected payoff functon from uyers. My thess dscusses the trade-off etween the wnnng proalty and expected payoff for uyers and gves out suggeston on ddng ased on ndvdual's rsk preference. Also, my thess dscusses the expected revenue functon for sellers and the factors affectng the expected revenue. 2

4 TABLE OF COTETS Chapter : Introducton A. A ref revew of aucton...6. The hstory of aucton 2. The mportance of aucton 3. How dd aucton evolve B. The contnuous and dscrete ddngs...8. The contnuous ddng. o restrctons on the d levels. 2. The dscrete ddng. There are restrctons on d levels. C. Purpose and organzaton of ths paper...9. Show that truthful ddng s the domnant strategy n contnuous ddng 2. Expected payoff functon for dders 3. Expected revenue functon for uyers. Chapter 2: Revew of Exstng Lterature A. Games of ncomplete nformaton...0. Foundaton and theory of ncomplete nformaton game. John Harsany s (967; 968a, ) s one of the great papers n modern nformaton economcs. 2. The asymmetrc nformaton. George Akerlof s (970) dscussed nformaton asymmetry and ts nfluence n the market. B. Four types of auctons...2. Introducton of the Englsh aucton, the Dutch aucton, the sealed-d frst-prce aucton and the sealed-d second-prce aucton. 2. Comparsons among four types of auctons n terms of the expected revenue and market effcency C. Dscrete ddng...5. Dscrete ddng space. Vckrey (96) consdered the stuaton where there s only a sngle oect to d. 2. Pure ash equlrum under dscrete ddng. Chwe (989) focused on the frst-prce aucton usng an ndependent prvate-values model wth contnuous ddngs. He showed a ash equlrum ddng strategy under the unform dstruton functon. 3. Dscrete ddng on four types of aucton. Yu (999) examned each of the four common aucton forms (sealed d frst prce, sealed d second prce, Englsh aucton, and Dutch aucton). 4. Modern dscrete ddng. Dscrete auctons are wdely appled n modern onlne-aucton. Easley and Tenoro (2004) dscuss the ump ddng strateges n 3

5 Internet auctons such as ebay. Isaac, Salmon and Zllante (2007) dscuss a theory of ump ddng n ascendng. Chapter 3: The Sealed-Bd Second-Prce Aucton wth Contnuous Bddng A. Statement: truthful ddng s the domnant strategy...20 Chapter 4: Model for Sealed-Bd Second-Prce Aucton wth Dscrete Bddng A. The asc set up of the model The asc model for sealed-d second-prce aucton wth contnuous ddng a) The model ) otaton and descrptons 2. The model for sealed-d second-prce aucton wth dscrete ddng. a) Chwe's (989) assumpton ) Yu's (999) model B. The strategc ehavor Consder the certan ehavors when a dder sumts hs or her d C. Equlrum ponts n ddng levels The exstence of s (, ), the "md-pont" n whch the expected payoff for ddng and for are dentcal D. The Equlrum Strategy n Sealed-Bd Second-Prce Aucton The expected payoff functon 2. The trade-off etween the wnnng proalty and expected payoff E. Seller's revenue...4. Expected revenue 2. Influental factors Chapter 5. Concluson and Applcaton...44 Blography...45 Appendx A

6 Chapter Introducton A. A Bref Revew of Aucton The word "aucton" s derved from the Latn augeō whch means "I ncrease" or "I augment." The record of aucton can e retrospect to early 500 B.C. where the women were arranged for ther marrage. Durng the Romans perod, auctons were used to lqudate the assets of detors whose property had een confscated. owadays, the man functon of the aucton s allocatng and exchangng goods and servces. Most of the common auctons consst of one seller sellng one or more goods and numers of nterested dders. Bdders compete wth ther rvals y sumttng personal ds dependng on ther personal valuatons for the goods. The good s allocated to the dder who offers the hghest d. However, the actual prce pad y the wnner does not always equal the hghest d, ut depends on the aucton type. In regular market, sellers usually set prces of the good or servce to e worthy of ts property. However, sellers n aucton seek the uyer whose personal valuaton of the good or the servce to e the hghest. Therefore, though auctons do sell normal goods, they put more attenton on art works, antques, ewelres and other precous goods n whch ther values vary a lot among dders. The aucton can get a larger proft n the process of acceptng ncreasngly hgher ddngs. owadays, as of the aucton market and regular market has een changed due Krshna, 2002: p2 5

7 to the development of the nternet, whch has led to a sgnfcant rse for the range of avalale uyers and categores of goods. The aucton ndustry had a rapd growth recently. In 2008, the atonal Auctoneers Assocaton reported that the gross revenue of the aucton ndustry n the U.S. was approxmately $268.4 llon. owadays, suctons have een appled n a wde range n socety. The world's most famous wne aucton, known as Hospce de Beaune, greatly promoted the reputaton and the sale of famous wnes. For example, France's former frst lady, Carla Brun, acted as the guest auctoneer n 202 and a Ukranan usnessman offered Brun and Sarkozy a ddng up to $350,000.The U.S. government also has treasury aucton department for short and medum term government onds. Even for onlne games, such as World of Warcraft, the game pulsher Blzzard Entertanment, Inc. apples aucton system to alance wth monetary system wthn the game. Moreover, the E-commerce apples onlne aucton system such as the nternet aucton ste ebay. Theoretcally, researchers treat the aucton as an ncomplete nformaton game. The exchange of good and the determnaton of prce can e relatvely easy to deal wth due to the asence of market nterventon. amely, an aucton can e treated as a sealed-sngle market n whch the prce s not affected y the outsde market. Also, auctons can e treated as an ncomplete nformaton game snce a dder's strategc ehavor s greatly affected y the nformaton of hs compettors' strateges. Hence, the game theory of ncomplete nformaton plays an mportant role n dealng wth auctons. Applyng game theory n aucton helps dders fnd out the optmal strategy for ddng so that they can maxmze ther profts. 6

8 B. Dscrete Bd Auctons Most of the exstng lterature aout auctons focuses on the stuaton n whch there are no restrctons on the ds. Generally speakng, a dder s allowed to d any artrary amount ased on hs or her valuaton for the good. In other words, a dder's choce s determned y hs oserved nformaton and t s from a contnuum of acceptale choces. However, n a real world aucton, restrctons ndeed exst. For example, the dscrete nature of currency makes a restrcton on the ddng choces. Another example would e the Englsh aucton n whch the auctoneer sometmes sets a mnmum prce to reserve the value of the good, or restrcts on the amount n whch the next d must e hgher from the current d. Consderng the onlne aucton ste ebay, a d ncrement wll e mposed to the wnner and ths s also a form of restrcton. Mathematcally, those ddng restrctons make the ddng space to e dscrete. Compared to contnuous ddng, even though dders' valuatons to the good are dentcally and ndependently dstruted, they expected revenues may change. Furthermore, wth dscrete ddng space, the sealed-d second-prce aucton no longer has a domnant strategy, whch s the truthful ddng strategy under contnuous ddng. Ths paper wll dscuss the characterstcs of dscrete ddng as well as the dfference for strategc equlrum etween contnuous and dscrete ddng. 7

9 C. Purpose and Organzaton My thess manly nvestgates the sealed-d second-prce aucton, wth dscrete ddngs. Also, my thess dscusses the equlrum strategy where the proft for the dder wll e maxmzed. Secondly, my thess apples Chwe's (989) assumptons and Yu's (999) model to seek an equlrum strategy for uyers y conductng the expected payoff functon. Fnally, my thess dscusses the expected revenue for sellers and some of the nfluental factors that could affect the expected revenue. Chapter 2 of my thess s the lterature revew, whch focuses on the games of ncomplete nformaton, auctons types and the dscrete ddngs. In Chapter 3, I dscuss the lterature of the sealed-d second-prce aucton n contnuous ddng space. In Chapter 4, I dscuss the model for the sealed-d second-prce aucton n dscrete ddng space. In ths case, my thess dscusses the expected payoff functon for dders and expected revenue functon for sellers. In Chapter 5, I make the concluson for my thess and applcatons to the real world. Fnally, the appendx contans the proofs of my thess 8

10 Chapter 2 Revew of Exstng Lterature Ths chapter revews the exstng lterature aout aucton theory, whch ncludes the game theory of ncomplete nformaton, aucton types, the comparsons etween dfferent types of auctons and auctons wth dscrete ddng. The ncomplete nformaton game s one of the foundatons of the aucton theory. Also, the advantages and dsadvantages among varous type of auctons draw many attenton from the researchers. Lastly, ths chapter dscusses the theoretcal works that focus on the dscrete ddng. A. Games of Incomplete Informaton In an aucton, nformaton s not fully symmetrc among dders snce one can only know hs own strategc ehavor ut not others'. We can treat an aucton as an ncomplete nformaton game snce the nformaton s asymmetrc n the aucton. Hence, t s helpful to apply game theory of ncomplete nformaton to seek the equlrum strategy n an aucton. Harsany s (967; 968) studes are consdered to e one of the great papers leadng the modern nformaton economcs. Though t manly focuses on game theory, ts economc thought plays an mportant role. Informaton n realty s usually unevenly dstruted and hence a ash equlrum cannot e easly acheved. Harsany (968) consders the games wth ncomplete nformaton where players lack some mportant parameters such as payoff functons, rval's 9

11 strategc ehavors and so on. The author sets up a new theory analyzng games wth ncomplete nformaton and shows that, under certan assumpton, such game can e equvalent to a certan game wth complete nformaton, called the "Bayes-equvalent" of the orgnal game, or refly a "Bayesan game." Ths study provdes us a new way of achevng equlrum n the games wth ncomplete nformaton. Harsany (973) also contrutes a theorem solvng the mxed strategy ash equlrum. Gven that each player's personal nformaton s not transparent to hs rvals and only know to hmself, the mxed strategy equlrum can e explaned as the lmt of pure strategy equlrum for a dstured game of ncomplete nformaton. As approaches to the lmt, the player's strateges converge to the predcted ash equlrum, whch s equvalent to the equlrum n the complete nformaton game. Harsany (968) provdes us a fundamental support to fnd out the equlrum n an aucton game, whch s consdered as an ncomplete nformaton game. Harsany (973) theory helps us to fnd out the equlrum strategy n an aucton game n whch the dders' strateges are mxed. Akerlof (970) dscusses the nformaton asymmetry etween the seller and the uyer, and ts nfluence to the market. The author uses the market for used cars as an example to llustrate the prolem of qualty uncertanty. The author ponts out that the asymmetrc nformaton provdes the ncentve for the seller to pass off low-qualty goods as hgher-qualty ones and the uyer prefers to consder the average qualty of the goods. Ths phenomenon s sometmes categorzed as "the ad drvng out the good" n the market. Therefore, such asymmetrc nformaton gves great nfluence on the market effcency. Consder the aucton market n whch 0

12 the Englsh aucton has transparent nformaton ut sealed-d auctons have asymmetrc nformaton, Akerlof's thought nfers a great dfference among aucton types. The transparency of nformaton certanly affects the effcency of the aucton market. In my study, I assume that each dder only knows hs or her own strategc ehavor so that there s no second-thnker n the game. The result wll e entrely dfferent f all dders know ther rvals' strategc ehavors. B. Aucton Types and Comparsons The nsttutonal rule s the man factor to categorze auctons and t also has great nfluence on the ddng ncentves, whch has een mentoned y Vckrey (96), of the dders. There are many dfferent types of auctons, ut four maor ones are prmarly concerned y economsts and researchers. The frst type of aucton s called the Englsh Aucton, whch s also known as the open ascendng prce aucton. Bdders compete wth rvals openly wth each susequent d to e hgher than the prevous one. The good s allocated to the hghest dder and the prce pad equals to the hghest d. Sometmes the seller wll set a mnmum prce as a reserve prce, and sometmes the seller wll set a mnmum amount requrng that the next d much exceed such amount to the current d. The most sgnfcant feature of Englsh aucton s that the current hghest d s always open to any potental dders. Ths type of aucton s argualy the most common form of aucton n use today.

13 The second type of aucton s called the Dutch Aucton, whch s also known as the open descendng prce aucton. The auctoneer egns wth a celng prce and gradually lower t untl some dders are wllng to accept. The good s allocated to the dder who frst accepts the prce and the pad prce s the last announced one. Ths type of aucton s convenent when t s mportant to aucton goods quckly, snce a sale never requres more than one d. The thrd type of aucton s called the Sealed-Bd Frst-Prce Aucton, whch s also known as the frst-prce sealed-d aucton (FPSB). Bdders sumt ther personal sealed ds smultaneously wthout know others ddng nformaton. The good s allocated to the hghest dder and the prce pad equals the amount he or she sumtted. Dfferent from the Englsh aucton, dders can only sumt one d. Also, the nformaton s ntransparent where dders cannot change ther ds accordng to ther rvals' ds. Ths knd of aucton s commonly used for government contracts and mnng leases. The fourth type of aucton s called the Sealed-Bd Second-Prce Aucton, whch s also known as the Vckrey aucton. Ths aucton s nvented y Vckrey (96) and named after hm. Compared to the sealed-d frst-prce aucton, ths s dentcal except the pad prce y the wnner equals to the second hghest d nstead of the hghest one. One example for ths aucton type would e the nternet aucton ste ebay n whch the aucton system s almost dentcal to the sealed-d second-prce aucton ut an extra ddng ncrement. Besdes these four maor types, there are many secondary aucton types such as all-pay aucton, payout aucton and so on. 2

14 There are ndeed advantages and dsadvantages for each of the four maor types of auctons. The expected revenue, for example, s one of the most mportant factors that concerned y the sellers. Whch knd of aucton could offer the hghest expected revenue so that t s the most favorale one to the sellers? It s surprsng that, n a contnuous aucton, all four types of aucton have the dentcal expected revenue for the seller. Under the assumpton of rsk neutralty, ndependence of prvate valuatons and symmetry among dders, Rley and Samuelson (98) and Myerson (98) show that all four types of aucton wll have the same expected revenue for the seller. Though ths result s restrcted n the equlrum where the ncentves to partcpate n the aucton do not change, the four types of aucton would e equvalent to the sellers n terms of expected revenue. Snce the aucton types are dentcal, n terms of the expected revenue, to the sellers, theoretcal works put more attenton on the expected payoff of the dders as well as ther strategc ehavors. In realty, however, those mentoned assumptons--rsk neutralty, ndependence of prvate valuatons and symmetry among dders-- may not e fulflled. For example, n the case of natural resource aucton, the assumpton of ndependence of valuatons fals. Therefore, a more general verson of assumpton requres the value of the good to dders to e dentcal and the dstruton functon for each dder's valuaton s unased. Mlgrom and Weer (982) fnd out that the expected revenue s no longer dentcal among four types: the Englsh aucton provdes a hgher expected revenue than sealed-d aucton. Moreover, when the assumpton changes agan where the rsk neutralty ecomes rsk averson, expected revenue ecomes hgher n the 3

15 sealed-d auctons. Ths result explans the fact that sealed-d auctons are commonly used n offshore ol leases auctons ecause people n the aucton are tend to e rsk averse. Compared to the Englsh aucton and the Dutch aucton, researchers put more attentons on sealed-d auctons nowadays. The Vckrey aucton, whch s also known as the sealed-d second-prce aucton, has een frst descred n Vckrey (96). Vckrey consders an aucton where there s only one ndvsle good s eng sold and the pad prce y the wnner equals to the second hghest d. He fnds that truthful ddng s the domnant strategy for each dder regardless of dders' rsk atttudes. Snce dders n a Vckrey aucton are lkely to d on ther true valuatons on the good, t s qute welcome y the sellers. For example, ebay's proxy ddng system s almost dentcal to the Vckrey aucton except there s a ddng ncrement for each wnner. Also, Google's and Yahoo!'s onlne advertsement system apply the Vckrey aucton. Despte the strengths of Vckrey aucton, t stll has some shortages. Suppose, for example, dders know the valuatons of ther rvals, they could lower ther d whle preservng to wn the good. Moreover, f the ddng level of dders s restrcted to a dscrete ddng space, truthful ddng may not e a domnant strategy anymore. C. Dscrete Bddng. Most of the exstng theoretcal works on auctons focus exclusvely on the stuaton assumng there s no restrctons on the d levels. Ths means that a dder can d any artrary amount on the good ased on hs personal valuaton and oserved nformaton aout hs rvals. 4

16 However, restrctons on ds often exst n the real world. For example, the dscrete nature of currency makes the acceptale d levels restrcted. The mnmum amount of currency one can hold s, let us say, one cent. It s mplausle for a dder to d lower than such amount. Also, for example, n an Englsh aucton, the seller sometmes sets a mnmum prce to preserve the value of the good, whch s also a restrcton on ds. Another example would e ebay, where the wnner pays an extra ddng ncrement vares etween fve cents and one hundred dollars esde the ddng prce. Therefore, restrcted ddng deserves more attenton ecause t reflects the realty. Vckrey (962) llustrates the stuaton n whch each dder only has two dstnct d levels. Under ths assumpton, the expected revenue for sellers are no longer dentcal among four types of auctons. Lke Mlgrom and Weer's (982) result, the Englsh aucton has a hgher expected revenue than that of the second-prce aucton. Though ths knd of dscrete ddng space s smple and unrealstc, t stll gves us some hnt aout the dfference y changng the contnuous ddng space nto a dscrete one. In realty, the stuaton n whch there are only two d levels ndeed exsts. The Chnese government offers the contract for uldng a hghway and the prce s fxed as default. Bdder can only choose to accept or declne the offer. However, such case s relatvely rare n realty. Hence, my thess apples the model n whch there are multple of d levels. Chwe (989) focuses on the sealed-d frst-prce aucton wth dscrete ddng where there are n dders, each wth an ndependent prvate valuaton functon and each dder's valuaton s 5

17 contnuously dstruted. The authors rngs out the thought aout "evenly spaced" dscrete d levels from a set M = {!,!,,! } such that! =!!!. Based on the fact that overddng s the domnant strategy n the sealed-d frst-prce ddng, Chwe (989) puts another d!!!! =!!!!!! = to make the maxmum d equals to the hghest possle dder valuaton. He shows that there exsts a unque symmetrc ash equlrum ddng strategy and converges to the equlrum of the contnuous ddng aucton when M approaches nfnty. (.e. the d ncrement goes to 0) The authors argues that such aucton has less revenue than the contnuous d aucton ut converges to the revenue n contnuous d aucton. In realty, such evenly spaced ddng space s wdely exstng. In Englsh aucton and Dutch aucton, the sellers usually ncrease or decrease the d level y a certan amount, whch makes the ddng space evenly dstruted. In my thess, the model apples the assumpton that the ddng space s evenly dstruted n order to mmc the real world stuaton. Rothkopf and Harstad (994) apply dscrete ddng to an Englsh aucton. Qute nterestngly, wth dscrete d, an Englsh aucton s no longer strategcally equvalent to a sealed-d second-prce aucton where these two are equvalent wth contnuous ddng. The authors assume there are n dders, each wth an ndependent prvate valuaton and m+ dscrete ddng levels. Under the stuaton where the dder valuatons are unformly dstruted, expected revenue of the seller can acheved at the cost of decreasng the expected dfference etween the hghest dder valuaton and the valuaton of the wnner (whch s also called the expected economc neffcency). Also, the equlrum s dynamc ased on the numer of 6

18 dders and dstruton of dders' valuaton functons. Those factors nfluencng the equlrum provdes a clue n my thess aout the determnant for the expected revenue for sellers. For example, the numer of dders ndeed affects the expected revenue for sellers. Yu (999) apples Chwe's (989) assumpton aout "evenly spaced" dscrete values to examne each of the four prmary aucton types. Yu (999) assumes that the valuatons of the n dders are ndependent from a common dstruton functon F(v) such that F(0) = 0 and F() =. In her paper, each type of the auctons has a symmetrc pure strategy equlrum. However, under such assumpton, the truthful ddng s no longer the domnant strategy compared to overddng and underddng. Ths means some dders wll d aove ther true valuaton and some wll d elow. Such phenomenon wll lead to a market neffcency. Also, Yu fnds out that as the ddng ncrement goes to 0 (M goes to nfnty), equlrum converges to the equlrum n contnuous d aucton. Yu also assumes that the mnmum d level equals to 0 and the maxmum d level equals to. My thess apples Yu's assumpton ecause such assumptons smplfy the model n terms of calculatng the expected payoff for dders. owadays, onlne auctons often apply the dscrete ddng space. Hence, researchers are tryng to provde practcal gudance as to how an auctoneer should determne the numer and value of these dscrete d levels. Davd el at (2005) am to provde the optmal aucton desgn for an Englsh aucton wth dscrete d levels. The author descre the dscrete ddng space and to the end, derve an expresson for the expected revenue of the seller. The expresson s a functon contans the actual dscrete d levels mplemented, the numer of dders partcpatng, 7

19 and the dstruton from whch the dders draw ther prvate ndependent valuatons. Specfcally, comparng wth prevous theoretcal work, the authors apply a unform dstruton to test ther solutons. To conclude, the optmal d levels result n mprovements n the revenue, duraton and allocatve effcency of the aucton. It s also worthy to menton another ranch of dscrete ddng n the lterature. There s a phenomenon called ump ddng n ascendng Englsh auctons where dders sometmes d hgher than what s necessary to e the current hghest dder. For example, Isaac et.al.(2007), Easley and Tenoro (2004) examne an aucton form of open out-cry n Englsh auctons. They also apply dscrete ddng spaces that have specal ddng restrctons. Such "ump ddng" dscusses a case n whch dders have ncentve to d aove hs or her true value for the good. My study also mentons ths phenomenon, whch s correlated wth the trade of etween wnnng proalty and expected payoff. 8

20 Chapter 3 The Sealed-Bd Second-Prce Aucton wth Contnuous Bddng In ths chapter, we dscuss the sealed-d second-prce aucton n whch ts ddng space s contnuous and there s only one sngle, ndvsle good s eng sold. The sealed-d second-prce aucton s also known as the Vckrey aucton that was ntally descred y Professor Wllam Vckrey n 96. The sealed-d second-prce aucton provdes dders an ncentve to d on ther true value. In other word, truthful ddng s the domnant strategy for every dders. To clam that truthful ddng s the domnant strategy, we provde the followng statement. Statement: The domnant strategy n a sealed-d second-prce aucton wth contnuous ddngs for a sngle, ndvsle good s for each dder to d on hs or her true value of the good. Proof for the statement: 2 Let v e dder ' s value for the good. Let e dder ' s d for the good. The payoff π for dder s v max f > max = 0 otherwse π 2 Smlar proof can e found at Rley and Samuelson (98) and Myerson (98) 9

21 ow we are gong to dscuss that truthful ddng has a hgher expected payoff than ether overddng and underddng. Case : Overddng Assume that dder ds > v, whch means dder ' s d exceed hs value of the good. If v > max, then the dder wll wn the good snce > v > max wth overddng. Under the crcumstance of a truthful d where = v, the dder stll wns the good snce = v > max. The payoff of dder s ndependent of so that these two strateges have equal payoffs n ths case. If < max, then the dder wll lose the good, oth n overddng and truthful ddng. They payoffs of oth strateges wll equally e 0. If v < max <, then the dder wll wn the good under overddng ut wth a negatve payoff snce π = v max < 0. Under truthful ddng, the dder wll lose the good snce v = < max and hence the payoff s 0. To conclude, the strategy of overddng s domnated y the strategy of truthful ddng n terms of the payoff functon. Case 2: Underddng Assume that dder ds < v, whch means hs d s less than hs value of the good. 20

22 If v < max, then the dder wll lose the good snce < v < max. Under truthful ddng, the dder wll also lose the good snce = v < max. These two strateges offer the same payoff to e 0 n ths case. If max <, then the dder wll wn the good ether n underddng or truthful ddng snce hs d s the hghest. Agan, the payoff of dder s ndependent of so that these two strateges have equal payoffs n ths case. If < max < v, then the dder wll lose the good n underddng and hs payoff wll e 0. However, under truthful ddng, the dder wll wn the good snce max < v =. Also they payoff π = v max > 0 s postve. To conclude, the strategy of underddng s domnated y the strategy of truthful ddng. In oth cases, truthful ddng domnates the other possle strategy. Therefore, n the seal-d second-prce aucton wth contnuous ddng, truthful ddng s an optmal strategy. To conclude n ths case, the sealed-d second-prce aucton s effcent snce t provdes ncentve for dders to d on ther true value of the good. In terms of the dder, ddng on ther true value grant them a non-negatve payoff. Ths s ecause, y ddng on true value of the good, the dder wll ether wn the good gettng a non-negatve payoff, or lose the good gettng a zero payoff. In terms of the seller, the truthful ddng strategy offers the sellers hghest 2

23 expected revenue. These results have een showed n Rley and Samuelson (98) and Myerson (98). 22

24 Chapter 4 Model for Sealed-Bd Second-Prce Aucton wth Dscrete Bd In ths chapter, we wll ntroduce the model descrng the sealed-d second-prce aucton wth dscrete ddng space. In part A, we wll ntroduce the asc set up of the model, ncludng the assumptons, the asc structure and notatons for the model. In part B, we wll dscuss the strategc ehavor for a dder, gven hs valuaton of the good. In part C, we gve out a fundamental concept aout the equlrum pont n ddng levels, whch s mportant n the followng part. In part D, we apply the expected payoff functon for dders to dscuss the equlrum strateges. In part E, we dscuss the expected revenue for sellers. A. The Basc Set up of the Model for Sealed-Bd Second-Prce Aucton wth Dscrete Bddng There are researchers who have ult models for sealed-d second-prce aucton have een studed y several researchers, such as Vckrey (96), Chwe (989) and Yu (999). In my thess, we are nspred y Chwe's (989) assumptons and manpulate Yu's (999) model for the sealed-d second-prce aucton. The actual set up for the model s descred as follows. 23

25 Ths model dscusses an aucton n whch the seller auctons an oect to dders. Each dder's value of the oect s dstruted ndependently wth v [ v, v ] = [0,], wth L H cumulatve contnuous dstruton functon Fv () such that F (0) = 0 and F () =. Each dder only knows hs own value of the oect and ds from a set contanng M+ dscrete d levels B= {, 2,..., M + } where =. Assume that 3 = 0 and M +. Ths means M the d possltes are multples of the ncrement. Follow the requrement of the seal-d M second-prce aucton, the hghest dder receves the good payng a prce equal to the second hghest d. If more than one dders d the same hghest, then randomly and farly select one of them to receve the good. B. The Strategc Behavor for Bdders Here we are gong to dscuss the strategc ehavor for an artrary dder gven hs or her personal value for the good. The strategc ehavor n aucton markets descres the preference of ddng for a certan dder. Suppose there are fnte numer of dscrete d levels. There exsts at least an acceptale d level. Ths followng Proposton s nspred from Mathews' (2008) Proposton. Proposton : 3 Ths follows Chwe's (989) and Yu's (999) assumptons. 24

26 Consder an artrary acceptale d level. For dder whose value v, ddng weakly domnates ddng aove ; for dder whose value v, ddng weakly domnates ddng elow. Consder a dder whose value v (, ). Proposton mples that all ds other than and are weakly domnated. From Proposton, we have () A dder wth v wll have hgher payoff from ddng than from d levels elow. (2) A dder wth v wll have hgher payoff from ddng than from d levels aove. Proposton has restrcted the plausle d levels to and, gven that the value for a dder v (, ). Ths helps us to narrow down the range of d levels that are could e dscussed. C. Equlrum Pont n Bddng Levels In ths part, we wll dscuss that there exsts an equlrum pont, or so-called the "md-pont", etween any two consecutve d levels and that t s dentcal to d on or n terms of the expected payoff for dders. 25

27 Frst consder dder wth v (, ) from ddng ether or. Let a two-tuple Eπ ( v, ) denote the expected payoff for dder, where v s dder 's value of the good and s the d he offers. The expected payoff for dder from ddng s Eπ ( v, ) = ( v ) p + ( v ) p (3.) t t t= k= 2 k where k stands for the numer of dders who d on and p s the condtonal proalty that the second hghest d equals to. The descrpton for the expected payoff functon s as follows. The two-tuple Eπ ( v, ) represents the expected payoff for a dder whose value for the good s v and ds on. Recall that n the sealed-d second-prce aucton, the actual pad prce equals to the second hghest d nstead of the hghest one. Then the frst term of the equaton (), ( v t) pt, t= descres the expected payoff n whch the second hghest d s less than. The second term of the equaton (), k = 2 ( v ) p, ndcates the stuaton n whch there are more than one k dders d on the hghest d. In ths case, the wnner s chosen randomly and farly. For the sake of smplfyng the queston, we frst look = 2. So the expected payoff functon would e Eπ ( v, ) = ( v ) p + ( v ) p (3.2) t t t= 2 The dfference etween the expected payoff from ddng and s 26

28 Eπ( v, ) Eπ( v, ) = ( v ) p + ( v ) p (3.3) 2 2 Consder the stuaton v = v = 0. We have L Eπ (0, ) = p p t t t= 2 and 2 Eπ (0, ) = p p t t t= 2 We have that Eπ(0, ) < Eπ(0, ) (3.4) Ths means that for dder whose personal value for the good s 0, he would lke to d as low as possle. Then we consder the partal dervatve of the expected payoff functon n terms of v and E Eπ ( v, ) = pt + v t= 2 2 π ( v, ) = pt + p = pt p v t= 2 t= 2 p Snce p > 0for all, we have Eπ( v, ) Eπ( v, ) < v v (3.5) Equaton (3.5) tells us that the expected revenue wth a hgher ddng s more senstve to the change of personal value for the good. 27

29 Consder the stuaton that dder's value for the good falls exactly on v =., Susttute v = n (3), we get So we have Eπ(, ) Eπ(, ) = ( ) p + ( ) p > Eπ(, ) > Eπ(, ) (3.6) Equaton (3.6) llustrates the same property as Proposton -(), whch says that a dder wth v wll have hgher payoff from ddng - than from d levels elow -. Smlarly, consder v =. Susttute v = n (3.3), we get Eπ(, ) Eπ(, ) = ( ) p + ( ) p < So we have Eπ(, ) < Eπ(, ) (3.7) Equaton (3.7) llustrates the same property as Proposton -(2), whch says that a dder wth v wll have hgher payoff from ddng - than from d levels aove -. ow, consder equaton (3.6) Eπ(, ) > Eπ(, ) and equaton (3.7) Eπ(, ) < Eπ(, ). Snce the expected payoff functon s lnear, then the Intermedate-Value Theorem suggests that there exst an unque s (, ) such that Eπ(s, ) = Eπ(s, ). To generalze ths, let = k. We have that there exsts an unque sk ( k, k) for all B. Ths s also true for sk (v H, k) = (, k). 28

30 Here we call such s to e the equlrum pont etween and snce t the expected payoff for ddng on and are the same, gven that dder's value for the good s s. We can treat s to e the "md-pont" for the expected payoff etween and. In Proposton, we have shown that dder wth v (, ) wll only d or snce other d levels are weakly domnated. Snce v >, then dder wll get a postve payoff y ddng. Consder, whch s greater than v. In a sealed-d second-prce aucton, a dder wth v could stll d ecause the actual prce he pays s equal to the second hghest prce nstead of. By ddng, the dder ncreases the correspondng wnnng proalty y sufferng a rsk of gettng negatve payoff. For any dder wth v (, ), we know that s (, ) gves a equlrum pont at whch the expected payoff from ddng and are equal. If v [, s ), then ddng wll offer a hgher payoff than ddng. If v ( s, ], then ddng wll offer a hgher payoff than ddng. Recall that we assume the set of d levels contans M+ dscrete ponts, B= {, 2,..., M + }. Then for any (, ), there exsts a s (, ) such that Eπ( s, ) < Eπ( v, ). Let S = { s0, s, s2,..., s r } such that 0 = v = s < s < s <... < s = v = wth r M +. L 0 2 r H Therefore, a dder wth v [ s, s) wll d snce v < sand a dder wth v = s wll d r. Those explanatons provde us a clear ncentve to construct the expected r payoff functon for dders n the followng part. 29

31 D. The Equlrum Strategy n Sealed-Bd Second-Prce Aucton In ths part, we wll construct a functon descrng the optmal strategy for a dder n terms of hs or her expected payoff. Here we frst dscuss the strategy functon for a dder. A dder's strategy s a functon from [0,] to B= {, 2,..., M + } returnng the dders optmal, gven the dder's personal value v of the oect. Denote ths functon as v ( ) :[0,] {, 2,..., M + }. A dder's strategy v () s an equlrum strategy f t satsfes Eπ(v, ) Eπ (v, ), B= {, 2,..., M+ } (4.) and Eπ (v, ) 0 (4.2) Bascally these two nequaltes gve out the restrctons of eng an equlrum. amely, (4.) nfers that an equlrum strategy offers the hghest expected payoff among all strateges, and (4.2) nfers that an equlrum strategy should not let the dder lose proft. Recall that n a sealed-d second-prce aucton, the good s allocated to the dder who offers the hghest d and the actual payment s equal to the second hghest d. Let P denote the payment that a dder ddng needs to pay. Denote "the second hghest d" as SHB and denote "the hghest d" as HB. We have the expected payoff functon for ddng : 30

32 Eπ (, v ) = ( v P)Pro( s the HBddng ) The aove functon lterally tells us that the expected payoff s the dfference etween one's personal value to the good and the actual pad prce, tmes the correspondng proalty. A game n normal form 4 s symmetrc f all agents have the same strategy set, and the payoff s depends on the gven strategy, not on the agents. Hence, an equlrum strategy n a symmetrc game s called a symmetrc equlrum strategy. The followng four condtons allow us to construct the form of the symmetrc equlrum strategy v ()(proofs for condton, c, d can e found n Appendx) 5. a) (0) = 0 It s easy to show that, under truthful ddng, a dder's optmal d wll e 0 gven that hs value of the good s 0. ) B = { v [0,] ( v) = } s convex n v. c) v ( ) = 0, v [0,] s not an equlrum strategy d) v () s monotoncally ncreasng n v. The symmetrc equlrum strategy (v) we consdered wll e of the followng form: f v [ s, s ), r v () = r f v= sr (4.3) where S = { s0, s, s2,..., s r } s the strategy space for the dder wth 0 = v = s < s < s <... < s = v = and r s an nteger such that r M +. Equaton (4.3) L 0 2 r H can e explaned nto two separate cases. For dder whose personal value s not as hgh as, 4 Adapted from Defnton 7.D.2 n Mas-Colell et al. (995) 5 Condtons, c, d are the Lemma, 2, 3, respectvely, n Yu's (999) page 20. 3

33 then there exsts a closest "md-pont" to the dder's personal value. In ths case, the dder should choose the d level havng the same suscrpt wth that "md-pont". For dder whose personal value s, the dder should ust d. We can show that the strategy form (4.3) s a symmetrc (pure strategy) ash equlrum. The exstence of such pure strategy equlrum has een dscussed n many theoretcal works. From the Purfcaton Theorem n Mlgrom and Weer (985), there exsts a symmetrc equlrum strategy n any fnte symmetrcal strategc-form game. Yu (999) has dscussed that all the four maor types of aucton are symmetrcal games and she proved the exstence of a symmetrc pure strategy equlrum. 6 Recall that the ddng strategy v () s an equlrum strategy f equaton (4.) and (4.2) are satsfed at the same tme. ote that n the sealed-d second-prce aucton, the expected payoff for any dder would e greater than 0. Therefore, equaton (4.2) wll e satsfed..e. Eπ ( v, ) 0, v [0,] Suppose a dder d r nstead of r +, then t must e ether Eπ(v, r) Eπ (v, r+ ) or + s not avalale..e. = and r = M +. ow we ntroduce two lemmas 7 that are r r mportant n later proofs. Lemma 8 : 6 Yu (999) proved the exstence n Proposton. 7 Lemma &2 are Lemma 4&5 n Yu (999) 8 Proof for Lemma s n Yu (999) page

34 Gven that Eπ(v, ) Eπ (v, ), B= {, 2,..., M+ } for each =, 2,..., r, a dder wth value s s ndfferent from ddng and Eπ( s, ) = Eπ ( s, + ) +..e. Lemma s ascally follows the defnton of S = { s0, s, s2,..., s r }. It tells us that f the dder's personal value falls exactly on some "md-pont," then the expected payoff would e the same from ddng aove or elow that "md-pont." Lemma 2 9 : Eπ( sr, r) Eπ ( sr, r+ ) (4.4) for r M +, and, for =, 2,..., r are the only ndng constrants. Eπ( s, ) = Eπ ( s, + ) (4.5) Wth these two lemmas, we can then determne that the ddng strategy v () s an equlrum strategy, stated n the followng Proposton 2. Proposton 2 0 : The ddng strategy v () of the form (4.3) s an equlrum strategy f Eπ( sr, r) Eπ ( sr, r+ ) for r M + and Eπ( s, ) = Eπ ( s, + ) for =, 2,..., r. 9 Proof of Lemma 2 s n Yu (999) page Proof of Lemma 2 s n Yu (999) page

35 Recall that n the sealed-d second-prce aucton, each dder sumts a sealed d to the seller y hs own value of the good. The good wll e allocated to the dder who ds the hghest, and the prce he pays equals to the second hghest d. We have the expected payoff functon n the followng form: Eπ (v, ) = (v P)Pro( s the HB ddng ) Gven that the dder wns the good y ddng, we need to calculate the value of P, whch stands for the actual prce the dder needs to pay and s determned y the second hghest d among dders. We can calculate the proalty of each to e the second hghest d and ts correspondng payoff. Then we can get the expected payoff functon y summng the calculated payoffs. Suppose that n the case of te, the wnner wll e chosen randomly and farly. So we expand the expected payoff functon as follows: Eπ ( v, ) = ( v )Pro( SHB = = 0 HB= ) + ( v )Pro( SHB = HB= ) ( v )Pro( SHB = HB= ) + ( v ) Pro(t 2 dders d HB= ) t= 2 t (4.6) Also, we have the expected payoff functon for ddng + : 34

36 E ( v, ) ( v )Pro( SHB 0 HB= ) π + = = = + + ( v )Pro( SHB = HB= ) ( v )Pro( SHB = HB= ) + + ( v )Pro( SHB = HB= ) + + ( v ) Pro(t 2 dders d HB= ) t= 2 t (4.7) Recall from the proposton that v () s an equlrum strategy f t satsfes condton (4.4): Eπ( sr, r) Eπ ( sr, r+ ) for = r, and (4.5): Eπ( s, ) = Eπ ( s, + ) for =, 2,..., r are the only ndng constrants. Plug (4.6) and (4.7) nto condton (4.5), we have that: ( s )pro(shb= = 0 HB = ) = ( s )pro(shb= = 0 HB = + ) for k =, 2,..., r. Hence, (4.7) - (4.6) wll get: ( s ) Pro( t 2 dders d HB= ) t= 2 t + ( s ) Pro(SHB = HB = ) + ( s ) Pro( t 2 dders d HB= ) = 0 t= 2 t (4.8) The reason why we comne equaton (4.6) and (4.7) s that we can cancel many smlar terms ased on the fact that the expected payoff y ddng on and + are same, gven that the personal value for the good falls exactly on s. ow we are gong to dscuss the condtonal proalty presented n the equaton aove. For Pro( t 2 dders d + HB= + ). Gven that the hghest d s + and there are t 2 dders d +. Then esdes the wnner, there are t dders ddng + among 35

37 dders. Snce s < + < s, then the proalty for those dders wth value + + would e ( ( ) ) ( ) t Fs+ Fs. For the rest ( ) ( ) t = t dders, ther d levels are less than s. So the proalty s ( ) Fs ( ) t. To conclude, we have that Pro( t 2 dders d + HB= + ) = F( s+ ) F( s) F( s) t t ( ) ( ) For Pro(SHB = HB = + ). Gven that the hghest d s +, then for the rest t dders, ther proalty s F ( s ). Gven the second hghest d s, then for the rest dders, ther proalty s F s ( ). Therefore, we have Pro(SHB = HB = ) = F ( s) F ( s ) + For Pro( t 2 dders d HB= ). Gven that the hghest d s and there are t 2 dders d. Then esdes the wnner, there are t dders ddng among dders. Snce s < < s, then the proalty for those dders wth value would e ( ( ) ( )) t Fs Fs. For the rest ( ) ( ) s. So the proalty s ( ) t = t dders, ther d levels are less than ( ) Fs t. To conclude, we have that Pro( t 2 dders d HB= ) = F( s) F( s ) F( s ) t Susttute these proaltes nto equaton (8) we get t ( ) ( ) t ( s + ) F( s) F( s ) F( s ) t= 2 t t ( ) + ( s ) F ( s ) F ( s ) t ( ) ( ) ( s ) F( s) F( s ) F( s ) t= 2 t t = 0 t ( ) ( ) 36 t t (4.9)

38 Equaton (4.9) contans the condtonal proalty so that we can derve an equaton ased on the proalty densty functon. The followng steps are the process of solvng (4.9). otce that = t t and a = ( a + ) t = ( a ) t= 0 t t We have that = = = t= 2 t= 2 t ( a ) t t t t ( a ) t t a t= 0 ( ) ( ) ( a ) ( a ) t t 0 a ( ) ( ) a a a a = a Hence, equaton (4.8) can e rewrtten as t t t ( ) 37

39 ( s + ) F( s) F( s ) F( s ) t= 2 t t ( ) + ( s ) F ( s ) F ( s ) t ( ) ( ) ( s ) F( s) F( s ) F( s ) t= 2 t t = F ( s ) F ( s ) t ( ) ( ) + ( s + ) F ( s) ( F( s+ ) F( s) ) F ( s) F ( s ) + ( s ) ( F( s) F( s ) ) F ( s) F ( s ) ( s + ) F ( s ) ( F( s) F( s ) ) = F ( s ) F ( s ) = + ( s + ) F ( s) ( F( s+ ) F( s) ) + ( s ) F ( s ) 0 F ( s ) F ( s ) ( ( ) F( s )) F s t t (4.0) From Yu (999), we have the followng Lemma 3 and the proof s n Appendx. Lemma 3 F ( s) F ( s ) F ( s+ ) F ( s) < F ( s ) < F s F s F s F s From Lemma 3, we smplfy equaton (4.9) ( ( ) ( )) ( ( ) ( )) + F ( s ) F ( s ) + ( s + ) F ( s) ( F( s+ ) F( s) ) F ( s) F ( s ) + ( s ) F ( s) ( F( s) F( s ) ) = ( s ) k + ( s ) k + 2 Yu (999) page 26 Lemma 6 38

40 where k, k 2 > 0 represent the proalty of wnnng. Recall from the defnton of s, we have that < s < +. Therefore, from equaton (4.0) and Lemma 3, we have the followng two condtons ) As dder's value approaches, k wll ncrease to fulfll equaton (4.9) 2) As dder's value approaches +, k 2 wll decrease to fulfll equaton (4.9) These two condtons mply the trade-off etween the payoff and the proalty of wnnng. As dder's value approaches, the dder sumts a lower d to ensure a postve payoff ut the correspondng proalty of wnnng decreases. On the other hand, as dder's value approaches +, the dder sumts a hgher d n whch may lead to a negatve payoff ut the correspondng proalty of wnnng ncreases. Recall that n a sealed-d second-prce aucton, they actual pad prce equals to the second hghest d. Hence, ddng hgher than one's personal to the good may stll lead to a postve payoff. We can also conclude that, unlke the contnuous ddng n whch the truthful ddng s the domnant strategy, there s no domnant strategy n dscrete ddng. Ths s ecause for (, ) v +, t s possle for a dder to choose ether or + to get a postve payoff, shown n equaton (4.0). Ths means oth underddng and overddng are not weakly domnated. In real world, one can choose a strategy of overddng or underddng ased on hs or her estmaton of proalty. For a person who overestmates the proalty of wnnng wth value v, he or she mght prefer to d over s n order to get a hgher payoff. On the other 39

41 hand, a person who overestmates the proalty of wnnng wth value v mght prefer to d elow s to avod potental rsk y gvng up some amount of payoff. E. The Expected Revenue for Sellers In the prevous secton, we have dscussed the expected payoff for dders. The expected payoff s the most mportant thng that dder cares and the payoff functon s related to the proalty densty functon, the condtonal proalty of wnnng and the numer of dders. In ths secton, we wll dscuss the most mportant thng for sellers, whch s the expected revenue. In a sealed-d second-prce aucton, the revenue of the seller equals to the second hghest d. Therefore, the expected revenue for seller s determned y the second hghest d level and ts correspondng proalty. Let M e the proalty mass functon for each dder to d. ote that the proalty mass functon M has no drect relatonshp wth the proalty densty functon Fv () n the last secton. Let EΠ denote the expected revenue of the seller, then we have EΠ= Pro(SHB= ) + Pro(SHB= ) Pro(SHB= ) (5.) 0 0 M+ M+ Same as the prevous notatons, SHB stands for the second hghest d. ow we are gong to dscuss the condtonal proalty. Consder Pro(SHB= ), there are two cases such that SHB=. 40

42 Case : there s only one dder ds +, one or more dders d. Ths means that the hghest d s unque and there exsts at least one second hghest d. Case 2: two or more dders d and no dder ds +. Ths means that there s a te n the hghest d, n other word, the hghest d equals to the second hghest d. Case descres the stuaton n whch there are k dders wth valuaton n range of [, ) +, one dder wth valuaton n range of M [ +, + ] and other k dders wth valuaton n range of [ 0, ). For those k dders, the proalty s denoted as ( M M ) ( ) ( ) k + ; for k dders, the proalty s denoted as M( ) k + ; for that one dder, the proalty s denoted as M ( + ). Hence, the proalty of case turns out to e Pro(case) = ( + ) ( ) ( + ) ( + ) k= k k k ( M M ) M ( M ) Case 2 descres the stuaton n whch there k dders wth valuaton n range of [, + ) and other k dders wth valuaton n range of [ 0, ). For those k dders, the proalty s denoted as ( M M ) M ( ) k. Hence, the proalty n case 2 s ( ) ( ) k + ; for k dders, the proalty s denoted as k Pro(case2)= ( M ( + ) M ( ) ) M ( + ) k= 2 k Therefore, we can update equaton (5.) y pluggng these proaltes as k 4

43 EΠ = M + = 0 0 ( Pro(case)+Pro(case2) ) M+ k k = 0 ( M( + ) M( ) ) M( + ) ( M( + ) ) = 0 k= k + M+ k k 0 ( M( + ) M( ) ) M( + ) = 0 k= 2 k (5.2) Equaton (5.2) s the updated verson of equaton (5.) havng the proalty mass functon. In order to ncrease the second hghest d, whch s equvalent to seller's revenue, the seller could make the ncrement of ddng levels, M, to e smaller so that the second hghest d may ncrease relatve to the orgnal one. As M approaches 0, the equlrum of dscrete ddng wll converge to contnuous ddng n whch the truthful ddng domnates. Therefore, on average, seller would lke to provde an ncentve for dder to d on ther true value for the good nstead of underddng. Hence, y decreasng the ddng ncremental wll certanly help seller to rse the expected revenue snce dders are more lkely to d on ther true value for the good. Another mportant factor that could affect the expected revenue for sellers s the numer of dders. In ths case, EΠ>0 Hence, t s etter for seller to call for more dders to attend the aucton. 42

44 Chapter 5 Concluson and Applcaton In my thess, I studed the sealed-d second-prce aucton wth oth contnuous ddng and dscrete ddng. In the case where the ddng space s contnuous, truthful ddng s the domnant strategy for dders. Bdders prefer to d on ther true value for the good to maxmze ther expected payoff and at the same tme, sellers could maxmze ther expected revenue snce all dders wll not underd. In the case where the ddng space s dscrete, we dscussed the expected payoff for dders and the expected revenue for sellers. Truthful ddng s no longer the domnant strategy and dders wll consder the trade-off etween expected payoff and wnnng proalty. On seller's sde, the ncremental sze and the numer of dders affect the expected revenue. Based on these results, we can make changes to those factors to ncrease the expected revenue for sellers. We have shown that there exsts a trade-off etween the expected payoff and the correspondng proalty of wnnng. Bddng lower grants the dder a lower chance of wnnng ut ensure a non-negatve expected payoff. Bddng hgher enhances the wnnng proalty for dder along wth a rsk of havng negatve payoff. The dders, knowng ther proalty densty functon, could calculate the equlrum pont s and compare t to ther own value for the good, v. By knowng the relatonshp etween s and v, a dder could 43