Combinatorial Auctions Using Rule-Based Bids

Transcription

1 Cominatorial Auction Uing Rule-Baed Bid Joni L. Jone Michael R. and Mary Kay Hallman Fellow and Aitant Profeor of Computer and Information Sytem Univerity of Michigan Buine School Gary J. Koehler John B. Higdon Eminent Scholar and Profeor of Deciion and Information Science Warrington College of Buine Univerity of Florida Atract The migration of auction to the Internet provide a unique opportunity to harne the power of computing to create new auction form that were previouly impoile. We decrie a new type of cominatorial auction that accept rule-aed id. Allowing id in the form of high-level rule relieve the uyer from the urden of enumerating all poile acceptale undle. The allocation of good require olving a complex cominatorial prolem, a tak that i completely impractical in a conventional auction etting. We decrie implifying winner determination heuritic developed in thi tudy to make large prolem of thi nature manageale. Keyword:(Cominatorial Auction, Heuritic, Contraint Satifaction, Integer Programming) Joni L. Jone Univerity of Michigan Buine School 70 Tappan St. D520 Ann Aror, MI 4809 Tel: (734) Fax: (734) jonij@umich.edu

2 . Introduction and Background Electronic or Internet aed auction have garnered a great deal of interet in recent year. The renewed popularity of auction tem from variou characteritic unique to thi form of commerce. We aed auction enjoy a road audience due to their acceiility to anyone with an Internet connection, therey growing the uyer pool, increaing competition and thu enhancing profit [0, ]. Alo, the expene and logitic of gathering at one location, a major deterrent to conventional auction, ha een replaced with inexpenive weite. Barrier to entering the electronic auction market have een lowered for all participant. Finally, electronic auction provide an opportunity, through harneing the power of computing, to etalih more complex trading rule and handle more complex good. Thi reearch preent a complex auction mechanim that capitalize on thi opportunity and the aociated winner determination algorithm. Traditional ingle item auction take everal form. The mot familiar i the Englih auction, a progreive acending open auction. Other form include the Dutch (open decending), the Firt Price and Vickrey econd price ealed id auction. The firt multipleitem auction wa propoed y Vickrey [3] and allowed everal identical unit to e old to idder who deired ut one item. Two variation of the progreive auction, imultaneou and equential, are mot commonly ued to facilitate multi-unit ale. How the auction i conducted within the multi-unit environment (i.e., equentially or imultaneouly) propagate a variety of alternative deign. For example, the popular Internet aed Yankee Auction [30] and the Grove/Vickrey deign [3], in which a pecified numer of identical item are offered for ale imultaneouly with the item going to the top idder whoe aggregate demand equal the numer of item for ale, have een extenively tudied. The FCC Spectrum licene ale which 2

3 egan in the mid 990 have hown that y executing ingle item auction imultaneouly, uyer could aggregate a deired collection of complementary good [7, 8]. None of the aove deign allow the idder to umit a ingle id for a comination of heterogeneou item. Allowing idder to create a undle of deired good for which a ingle id price i umitted, referred to a a package or undled id, i a logical extenion and ha een propoed and tudied y other [2, 4, 25, 33]. When complementaritie, indiviiilitie, or other complication exit etween the different item in a undle, cominatorial conideration mut e explored. Such cominatorial auction allow the uyer to create and umit id for one or more ( excluive or ) undle and are often uperior to other multi-oject ale mechanim [8]. Winner determination in cominatorial auction involve electing a collection of id that maximize eller revenue without exceeding the availaility of the good eing auctioned. Thi i a NP-complete cominatorial optimization prolem [27]. Several reearcher have tried circumventing thi challenge y either retricting the id tructure to reduce the computational intractaility [27] or y extending ingle item mechanim in ome heuritic fahion [4, 5, 23, 34]. In general, fat heuritic are almot alway required. Price alone may not e ufficient to ditinguih etween undle. Other attriute of a undle may e important and not eaily priced. In many auction etting [6, 33], the very proce of determining acceptale undle and id price i non-trivial. For example, a uyer may want a certain andwidth etween two point at a certain time and quality level. Spellingout the different connection that might atify thi ojective itelf require exploring a cominatorial pace with many idder contraint. We elieve cominatorial auction mechanim that place the urden of determining all acceptale undle on the uyer are 3

4 inherently impractical for many application. To date, complicated tranaction uch a thee have often een handled through negotiated ale ut auction may provide an effective mean of price dicovery, epecially for product hard to price a priori or when information aymmetrie are preent [5, 9, 2]. Thi reearch explore a new auction mechanim that directly addree cominatorial auction without requiring idder to enumerate their deired undle, price, and attriute. Intead, idder provide a high-level pecification or et of rule decriing their ultimate requirement and the auction mechanim directly determine oth the formation of acceptale undle and a revenue-maximizing allocation of reource. Thi malleale id along with the aility of the idder to dictate variou apect of the final undle, eyond her willingne to purchae unit at a particular price, make it poile to ue thi type of mechanim to replace or enhance a negotiated environment. To illutrate our idea, we focu on a market traditionally reliant on negotiated ale. Specifically, we look at the ale of primetime televiion advertiing. Thi paper i organized a follow. We decrie the overall deign of the propoed auction mechanim in Section 2. In Section 3 we provide a rief overview of the TV advertiing prolem which i modeled a an integer program in Section 4 for preciene. The heuritic methodology to facilitate winner determination within the propoed auction mechanim i defined in Section 5. Section 6 and 7 dicu the experiment deigned to tet the efficacy of the mechanim. Limitation of the tudy are identified in Section 8 and we conclude with a ummary and future reearch propoal in Section Auction Mechanim The overall deign of our auction, referred to a the Incompletely Specified Cominatorial Auction (ISCA), i hown in Tale. We propoe a progreive emi-ealed 4

5 auction format that allow the idder to provide high-level requirement that guide the auction mechanim to create feaile undle while maximizing eller revenue. In many market, uyer and eller jealouly protect information on product availaility and price. To accommodate thi we employ a emi-ealed format where uggetion are provided to the idder to help them formulate new id, ut active pricing and overall allocation are not dicloed to other idder. [29]. Bidder are given the flexiility to change their id after each round until a topping criterion ha een reached. INSERT TABLE HERE Three potential topping rule are propoed and teted for ISCA. The Activity Rule reemle the progre of a claic Englih auction allowing the auction to continue until no further id are umitted. The two other topping rule terminate the auction when a predetermined revenue goal ha een reached or a deignated round ha een completed. The predefined condition in the lat two alternative topping rule are not revealed to the uyer to entice the idder to participate in early round. 3. Illutrative Example - Primetime Advertiing We illutrate the potential of the ISCA y applying it in the highly complex multidimenional proce of media uying, pecifically, the ale of televiion advertiing airtime. Airtime i a perihale, commodity product that i currently old through negotiation frequently aed on long-term relationhip and editorial and demographic ynergie, not jut getting the lowet price [32, pp.]. Unit are typically allocated on a firt come firt erve ai rather than dictated y competitive force that could enhance the network aility to achieve an equilirium aed ditriution of good. The complexity of determining an allocation that imultaneouly atifie the market participant demand for different demographic expoure, 5

6 advertiement placement, variou commercial length, etc., while maximizing revenue retrict the eller aility to promote competitive argaining. Similarly, uyer mut determine an advertiing mix that meet their demographic campaign goal while atifying other conideration uch a image, frequency (how often a commercial appear), and reach (how may viewer expoed to the commercial) within a predetermined udget. An indication of the deire and aility to promote competition in thi market i the appearance of a numer of we-aed auction elling exce lat minute advertiing inventory. Thee ite, uch a AdOutlet.com and OneMediaPlace.com, are implitic in nature, elling ingle unit of time that are conidered fire ale pot or unold airtime within cloe proximity to airdate. Thee ite have een criticized for their limited offering and their focu on ditreed inventory [6, 28]. 4. Auction Decription We explore an auction for "up-front" ale. Thi market i a large one-time ale of pot advertiing time for annual TV campaign. That i, a typical week advertiing chedule i determined, the pattern of which will e repeated throughout ome pecified numer of week. We concentrate on primetime advertiing where approximately 300 to 350 uyer compete for commercial airtime in the upcoming eaon. Primetime extend from 8 p.m. to p.m. with how varying in length from 30 minute to 2 hour. An hour how generally contain 5 to 7 commercial reak referred to a pod and roughly 4 to 8 5-econd lot or unit per pod. The ae unit of allocation in thi auction i the 5-econd unit. To decrie accurately the primetime ale auction, we model it a an integer program. A ummary of the notation i preented in Appendix A for eae of reference. Our main deciion variale i x u,,, a inary variale et to if uyer i allotted a 5-econd unit u in pod p for 6

7 how. The model i not meant to e a direct precription of the input into an IP olver ince many known formulation alternative would produce tighter LP relaxation [2], ut i given merely to define accurately and imply the prolem at hand. Additionally, a detailed decription of the pecific prolem tackled in thi reearch erve to illutrate the level of complexity our auction can accommodate. The eller olve prolem P which include the following ojective and contraint ()- (d). Ojective Function B max a y (P) = The ojective function maximize the total revenue from accepted id a, where y = indicate idder ha a winning id. The reervation requirement in the televiion indutry i aed on an un-pulihed lit price for each how, L. The um of the accepted id mut e greater than the um of the internal lit price for the unit old. Thi contraint operate on an aggregate level an individual idder may actually pay le than the um of the lit price of the unit he purchae. Reervation Requirement B = a y B S P U = = p= u= x u,, L () Not all of the advertiing inventory i releaed to "up-front" ale. Network often reerve a portion of their upply for ale in later market. The maximum coverage contraint limit the numer of commercial old in how to C, the numer of unit made availale. Maximum Coverage B P U = p= u= x u,, C =,..., S (2) Each how i roken into pod, uually one- to two-minute lock of airtime reerved for 7

8 commercial placement. The numer of pod per how and their length vary from how to how. The following guarantee that the numer of commercial placement per pod doe not exceed the numer of unit availale to accommodate them, U,. p Max Availaility/Pod B U = u= x u,, U p =,...P =,...,S (3) If a particular idder doe not otain any airtime, the following contraint force all her x value to zero and thu drop her id from conideration. Buyer Selection Indicator S P U = p= u= x u,, S P = p= U y =,..., B (4) Advertiing campaign may conit of commercial of varying length. A campaign compoed of only 30, 45 or 60-econd pot mut eliminate any collection of 5-econd unit in an individual pod that will not form a deired commercial length. Buyer may alo have mixed campaign, or campaign that conit of a comination of length. N, i a et of permiile commercial length for each how upplied y uyer. Note that the permiile collection of length can vary y how facilitating a uyer' need to vary the length() of it commercial. Indutry practice dictate that no more than one commercial per uyer appear in the ame pod. An exception to thi rule allow that at mot two 5-econd unit from the ame advertier may e placed in the ame pod. To account for thi exception we modify the et N, y adding 2 a an allowale ize if the idder permit a 5-econd unit. { 2} N, if N, and 2 N, N, =. N, otherwie Below, Equation (5a) and (5) jointly control the allocation of correct length commercial. Variale I p,, i, in Equation (5a) i et to one if pod p of how for uyer ue an allowale 8

9 numer (i) of advertiing lot. Thu the equation require that the cumulative numer of unit aigned to uyer in each pod correpond to one of her allowale commercial length (lited in N, ). Ordering of aigned unit lot in a pod i not conidered here. In practice, they are poitioned within a pod jut efore airtime to handle the concatenation and equencing prolem. Campaign Length U u= xu,, = ki,, i p =,..., P =,..., S =,..., B k N, (5a) Equation (5) prevent more than one correct length commercial for uyer from appearing in pod p of how. Commercial/Pod z, p =,..., P =,..., S =,..., B (5) Variale z p,,, defined y, z p I,, i N,,, i i if uyer ha a correct length commercial in pod p of how. Placing a large numer of different commercial in the ame pod weaken the impact of all commercial meage within that pod. Thi phenomenon reult from the "clutter" exacerated y the ue of 5-econd commercial. Network eek to avoid thi prolem y allowing at mot two 5-econd ad to appear in each pod (ee Equation 6). Anti-clutter B = I,, 2 p =,..., P =,..., S (6) Controlling the numer of commercial appearing in each how allow the uyer to pread or aggregate commercial over the campaign week. K, i the numer of correct length commercial that are permitted in how y uyer. Alo, a uyer can identify a foridden how, ay how, y etting K 0. Equation (7) provide for uch contraint., = 9

10 Maximum Spot/Show P p= z, K, =,..., S =,..., B (7) Media uyer deire a pecific amount of demographic gro impreion, or numer of viewer expoed to their commercial during their campaign. There are a variety of demographic categorie upon which a how i rated. Each how' gro impreion per category form the demographic vector D and indicate the eller' etimated reach for that particular how in the upcoming eaon. In thi tudy, D i ordered y the categorie: Women 8 to 49, Women 25-54, Men 8-49, Men 25-54, Adult 8-49 and Adult T repreent the vector of demographic reach or gro impreion that the uyer need to meet the product' campaign goal in each category. D i etimated at ale time, ut later confirmed hortfall (determined from actual Nielon viewerhip rating) are made-up with urplu inventory held-ack for uch poiilitie. Alo, although the total numer of gro impreion, D, i not elieved to e a function of commercial length (i.e., a 30 econd commercial give the ame numer of gro impreion a doe a 60 econd commercial), indutry practice compute demographic expoure auming that gro impreion are proportional to the commercial length. Equation (8) aure that the minimum demographic requirement i achieved. The um of the total numer of 5-econd unit, time the pecified demographic gro impreion over all elected how mut meet or exceed the required impreion for the pecified demographic group(). Demographic Reach Required S P U p, = p= u= x u,, D T y =,..., B (8) In addition to the actual dollar amount id and demographic requirement, a uyer may pecify deired how within which they would like their commercial placed. Setting h, to indicate that uyer would prefer placement in how. She can further indicate her willingne 0

11 to deviate from her program preference y etting the upper H and lower H ound to the numer of how required. Equation (9a) and (9) determine variale j, that capture which how uyer ha een allocated. Show Allocation P p= P p= z, j, =,..., S =,..., B z, P j, =,..., S =,..., B (9a) (9) Equation (0) enure that a uyer i allotted at leat H how and no more thanh of the how requeted. Deired Show S H h j H y,...,,, = B = (0) Network routinely guarantee that competing (i.e., when Mi = Mj) advertiement do not appear in the ame pod. The group of equation ( a-d) implement thi notion of "pod protection." Pod protection i normally not given for a ingle 5-econd commercial, therefore we need only implement anti-competition when a uyer ha two or more unit in a particular pod. The deciion variale f p,, i et to if a idder ha two or more 5-econd unit in a particular pod p of how. When the numer of unit a idder ha per how i 0 there i no competition and Equation (a) force oth f p,, and z, to zero. Pod Protection A: U u= x u,, z, + f, p =,..., P =,..., S =,..., B, (a) In the cae where one unit i aigned in a particular how to uyer, pod protection i not enforced and the requirement that z p,, equal or exceed f, in Equation () et f, to zero, and z p,, to at mot one. Thi correpond with the fact that uyer ha a ingle unit in

12 any pod of how. Pod Protection B: f, z, p =,..., P =,..., S =,..., B, () Equation (c) will force z p,, to one in thi cae. When two or more unit are aigned within the ame pod of a how to a ingle idder therey generating a potential 30-econd or longer commercial, Equation (a-c) will force z p,, and f, to one. Pod Protection C: U u= x u,, z, + P p= U f, p =,..., P =,..., S =,..., B, (c) Finally, Equation (d) will keep competitor away from uyer protected pod in how. Pod Protection D: f,, i + z, j i < j, j = 2,..., B Mi = M j, p =,..., P,..., S. (d) p = 5. Heuritic Development A direct attack on olving prolem P to optimality i proaly doomed. For example, in a repreentative prolem with 325 idder competing for 587 unit in 09 pod acro 24 how, P ha approximately 278,000 inary variale and 587,000 contraint. A heuritic i clearly needed. The olution methodology employed to allocate unit in our Incompletely Specified Cominatorial Auction i fairly complicated incorporating a mixture of trategie including aggregation and linear, contraint and dynamic programming method. The auction egin with the collection of id. Each id i ujected to an initial feaiility check to enure that it meet minimum requirement for entry. Thi i accomplihed y olving an aggregate prolem defined elow. Once all id have een tendered, an initial feaile olution i generated with the ue of heuritic. An upper-ound i etalihed uing a linear relaxation of a econd type of aggregate integer program. Thi ound i ued to judge the quality of our olution. The et olution to date i then ued to tart a ranch and ound 2

13 earch. At the end of each auction round, the elected topping criterion i checked. If topping condition are not met, idder are informed of the reult and loing idder will have the opportunity to change their id commenurate with their ehavior profile. When all topping condition have een met, the current id amount are replaced with idder reervation price (ut otherwie unaltered) and the olution to thi prolem i ued to calculate the mechanim efficiency. 5. Aggregate Su-Prolem The majority of contraint involved in the auction prolem can e determined y examining the allocation of each how rather than a pod or unit level allocation. The overall driving heuritic i a greedy allocation of how advertiing unit to idder. Each id i conidered equentially, conditional on the tentative allocation made to other idder. Let δ ( x) e the normal Kronecker delta function. Alo, let x, e the numer of advertiing unit that idder purchae in how and X, e the current domain of x,. Uing contraint programming methodologie, the domain of x, will change a other variale aociated with earlier accepted id are intantiated either ecaue unit ecome unavailale or ome contraint uch a pod protection or maximum pot per how would e violated. A we will how, incorporate all the contraint given in Equation (2)-(d) except for demographic reach and the deired how contraint. Thee latter two are handled directly in the following aggregate prolem. We define the aggregated prolem, (AP ) for each idder a X, Aggregate Prolem S σ = min L x (AP ) x, X, =, 3

14 S = H D x, S = T h δ ( x, ) H The aggregate prolem formulation relie heavily on the domain X,. A imple recurive procedure, detailed y Jone [3], how how the domain are contructed. When there are no current other aignment, the ojective value of AP i laeled σ which give the minimum how cot for idder needed to atify all prolem contraint (2) (d). When there are current aignment meeting (2) (d), then AP provide an aignment for thi id (if it ha a feaile olution) that, together with the current aignment, meet all contraint (2) (d). By minimizing the cot we atify the one remaining contraint, Equation (), the eller reervation price contraint. A imple check confirm thi. If the prolem i not feaile due to a violation of the eller reervation price contraint, free-loading idder are removed until the condition i atified. If there i no feaile olution to AP, then we et the x u,, variale to zero. Otherwie, a olution can e expanded to yield x u,, value y recovering a comination yielding the correct entry in X,. There may e many uch comination. No currently protected pod will e violated y thee x u,, value. If AP ha a feaile olution, then we et y =. If not, we et y = 0. The remaining deciion variale in the original prolem P can e recovered y a imple analyi of the expanded olution x u,,. The aggregate prolem AP i olved uing dynamic programming method. The final contraint can make thi a non-linear prolem (ecaue of the Kronecker operation) if either or oth H and H are greater than zero and the h have value neceitating the conideration of 4

15 thee contraint. We utilize one of everal dynamic programming routine deigned to olve the u-prolem, the choice of which depend on the value of H and H and the idder aignment of her h. The dynamic program provide exact optimal olution to AP. However, thee can take ome time to olve ince the T value may e large. At variou point, to e dicued, we ue a heuritic aed on linear programming relaxation to give good (often optimal) olution to the aggregate prolem AP. We call thee method FatAP. Jut a we utilize one of everal dynamic programming routine deigned to olve the aggregate u-prolem, we alo have different FatAP verion. When the how election contraint aren t needed, thi i a traightforward LP Knapack prolem. A lightly more complicated verion i employed to atify how election contraint. Each olution i refined uing prolem reduction method that hrink the domain aed on imple dominance tet. 5.3 Mater Prolem An overview of the mater prolem i preented in Figure. The goal i to find a olution that maximize eller profit while atifying contraint () to (d). To etalih a good initial olution to the allocation prolem, conider the following two greedy algorithm. Aume we are given B id. Step : Repeat the following until all id have een proceed. Sort the remaining id y ome criterion of interet with the mot deirale id deignated a the top-mot id. (In our application we order the idder in uch a way that thoe which contriute the mot to maximizing eller revenue are aigned firt. We do thi y electing the idder with the highet id per thouand demographic requirement normalized y the cot of the pecific demographic category and demand within that category) Solve the aggregate u-prolem for the top-mot remaining id and make the appropriate aignment to the variale of P. Step 2: While the amount id y the elected idder i le that the eller reervation price for the collection of allocated unit, i.e. 5

16 B = a y B S < P U = = p= u= x u,, L Sort the remaining feaile id y ome criterion of interet with the leat deirale id deignated a the top-mot id. (i.e., the igget freeloader) Set the top-mot active id aggregate u-prolem olution to infeaile and remove any current allocation to thi idder. The aove procedure yield a feaile olution to the auction prolem ince Step 2 guarantee the feaiility of the reervation requirement () and contruction of the domain for each aggregate u-prolem plu the contraint of AP aure the feaiility of all the remaining contraint. 5.4 Branch and Bound Branch and ound technique are employed to invetigate the variou comination of id that will maximize eller revenue. Since total enumeration of the variou comination i impoile, we utilize heuritic to guide our ranching ehavior. At each ranch, we take the partial olution from predeceor ranche and olve AP or FatAP. INSERT FIGURE HERE The amount of time allotted to computation in each round i pre-defined y the auction rule. We ue time remaining to guide the earch proce. After preproceing i complete, an initial olution determined and an upper ound on P computed, 30% of the remaining time i pent in a Breadth Firt Search (BFS), the ret i dedicated to a Depth Firt Search (DFS). The Breadth Firt Search extend to three level. Thi mean that it look at all ordering of all comination of 3 idder a the firt three allocation time permitting. Below level three, depth firt earch i ued ut i limited to a relatively mall numer of ranching (we ue five time the numer of idder). Bid are initially ordered y the aove heuritic to rank them 6

17 o that the top-mot id contriute the mot to maximizing revenue. However, conflict etween thee id may prevent all of thee mot deirale id from achieving an allocation. The order in which the id are proceed affect the allocation o all permutation of the firt three id are explored. During the Breadth Firt phae, the FatAP heuritic i ued. A Depth Firt Search i employed during the final 70% of computational time to eek out the et comination of id. Thi earch i conducted in two tage, the firt olve (AP ) exactly uing dynamic programming and lat for 60% if the time allotted. Stage 2 utilize FatAP and run until the concluion of the computational time. 6. Experiment Our experiment conited of imulating the execution of our auction under variou condition to analyze the mechanim performance. Our imulation ue oftware agent to repreent idder for prime-time commercial. We do thi not to ugget that thee agent e ued in an actual auction ut merely a a way to tet our mechanim. A real implementation would ue real idder. Profile for our imulated idder were determined from an analyi of data received from a repreentative of a major televiion network. Baed on uch data, ditriution for idder reervation price, demographic requirement, product category, and mix of commercial length were determined (ee [3] for detail). Thee ditriution were ued to randomly generate repreentative agent for our imulated experiment. A idding trategy wa defined for each agent type ued in the experiment aed on reult due to Bapna et al. [3], who identified three pecific type of idder in current online auction etting: Participator, Evaluator and Opportunit. Participator id actively throughout the auction, Evaluator place a ingle id repreenting their reervation price early in the auction, and Opportunit enter jut efore the auction cloe eeking argain. 7

18 Opportunit were not included in our tudy ecaue the oft cloing rule and emi-ealed environment doe not provide idder the neceary ignal information to determine the auction end. Bid modification tactic are roken into ix tyle, five Participator type and one Evaluator type. Of the Participator, two modify their id amount only, two modify oth their id amount and the contraint impoed on commercial placement and one modifie only the contraint. We define BidAdjutor a thoe idder who modify the price they are willing to pay for an allocation. ContraintAdjutor modify only the demand for, or retriction to, the how included in the allocation, and AllAdjutor comine the technique of id and contraint modification. We further ditinguih BidAdjutor and AllAdjutor into max and min type. A max(min) agent id the maximum(minimum) of the minimum id increment and the recommended id increae pecified y the auction mechanim. A preliminary et of experiment wa conducted to undertand etter variou poile agent ehavior on the auction mechanim (ee the firt ection of Tale 3). Baed on thee reult and advie from an indutry expert, we generated 325 agent made up of roughly 5% Evaluator, 20% BidAdjutor-Max, 20% BidAdjutor-Min, 30% AllAdjutor-Max, 20% AllAdjutor-Min and 5% ContraintAdjutor to capture a reaonale approximation of the marketplace. Our econd imulation ue thi framework, referred to a the Generic Bidder Set, to examine our auction mechanim under variou condition (ee Tale 2). INSERT TABLE 2 HERE The meaure ued to evaluate the Incompletely Specified Cominatorial Auction performance are allocative efficiency, aignment optimality and the length of the auction. When the eller alway ell an oject to the idder with the highet realized valuation for that oject, a long a that value i greater than the eller reervation price, the auction i aid to e 8

19 efficient []. While thi meaure i fairly traightforward in a ingle unit environment, complication arie when trying to evaluate efficiency for auction involving package id. In a cominatorial auction one mut conider the impact of overlapping demand for the variou item that may limit the feaile allocation of unit when determining a mechanim efficiency. To determine the efficiency of our auction we define the final allocation of the auction mechanim a: B ( A) = arg max y a y..t. Contraint ()-(d) = Further we determine an upper ound on the value attainale in the auction a: B V * = max v y.t. Contraint ()-(d) = From thee two equation we can formulate an efficiency meaure imilar to thoe ued in prior cominatorial auction tudie [4, 8, 7]. B ( A) vy = E = V *, Note that thi i an etimation of the efficiency a it relie on the outcome of our heuritic that may not achieve an optimal olution to either prolem. However, when olving each prolem, we end with a range from our final feaile olution to a theoretical upper ound. Thi gap i often mall and i reported in our experiment. Optimal auction are thoe that maximize eller revenue. Maximum revenue, in a ingle item auction, reult from allocating efficiently then extracting a much uyer urplu a poile [22]. We report the percentage of the maximum poile revenue from the final allocation actually captured y the eller a our meaure of optimality. 9

20 7. Reult Optimality B = = B ( v a ) y ( A) = v y ( A) The reult from experiment one are hown in Tale 3. The auction imulation coniting of all ContraintAdjutor poted the et overall reult. Thi idder category wa not only ale to achieve the highet revenue ut alo had the highet numer of winning idder, had minimal or no unold inventory and 00% optimality and efficiency. Thee reult were achieved in a little a 4 round in mot cae. The data how that auction coniting of only AllAdjutor alo performed well with regard to maximizing revenue, allocating unit to a conitently high numer of idder and leaving very little inventory unold. Evaluator alo faired well poting the econd highet achieved revenue, 00% efficiency and optimality, with only one or two unit of unold inventory. BidAdjutor under-performed, leaving ignificant amount of inventory unold, achieving the leat amount of revenue, electing the fewet winner and functioning poorly in term of efficiency and optimality. INSERT TABLE 3 HERE We alo analyzed a uniform ditriution of idder type a well a the uggeted generic ditriution. A might e expected the comination of idder type had reult omewhere etween the two extreme of the flexile and inflexile idder type. Thee reult ugget that the auction perform differently with varying idder characteritic. Since auction agent typically ehave u-optimally, one hould expect etter reult outide the imulated environment. Tale 4 how the reult from our econd experiment. A Manova tet wa conducted to 20

21 determine the impact of the independent variale (minimum id increment, topping rule, and calculation time) on the dependent variale (optimality, efficiency and realized revenue). The reult indicated that the minimum id increment and topping rule had a ignificant (at a 99% level) effect on the auction performance characteritic ut the amount of calculation time given the heuritic did not. An analyi of the data in Tale 4 indicate that increaing the minimum id increment yield greater eller revenue in le time. However, when uing the activity topping rule the larget (5%) increment demontrated a decreae of the maximum overall revenue from that achieved with a 0% increment. We attriute thi to the fact that the agent are udget contrained and a high minimum increment forced udget contrained idder out of the auction ooner, thu decreaing competition y reducing the idder pool [26]. INSERT TABLE 4 HERE Uing the highet minimum id increment produced the et optimality reult. Generally, auction that were ale to extract the mot urplu from the winning idder required a 5% id increment. Experiment to determine the mot efficient auction, or the auction that mot effectively aigned unit to the idder that value them the mot, produced le conitent reult in term of the impact of the minimum id increment. When the auction wa allowed to continue for everal round, a in the cae of thoe uing the activity and maximum round (0) topping rule, the higher percentage increae forced higher efficiency. Thi could e attriuted to the fact that requiring a higher uequent id made the idder true valuation urface earlier, thu giving the mechanim the opportunity to aign unit to thoe with the highet revealed valuation. However, in the cae of the minimum revenue topping rule the reult were inconitent 2

22 etween the 0% and 5% increment. Minimum revenue wa achieved in round 3 under thee parameter and one could peculate that the mall numer of round did not give an opportunity for thoe idder with the highet true valuation to ecome the dominant idder. Due to the hort duration of the auction it wa not allowed to converge to equilirium. Another apect of the auction influenced y the minimum increment percentage i the numer of round needed to achieve competitive equilirium. Tet howed that on average, when the activity topping rule wa employed auction requiring only 5% increment lated an average of 22 round, 0% minimum increae auction averaged 20 round, while thoe requiring at leat 5% added to previou unucceful id concluded in 9 round. Figure 2 how the revenue progreion for 5-minute calculation time. Notice that the higher the required increment the fater the initial revenue gain, however the 0% increment eventually achieve the highet revenue. In thi cae udget contrained idder were ale to remain in the auction longer giving the mechanim a greater opportunity to extract uyer urplu, thu increaing overall revenue. INSERT FIGURE 2 HERE Evidence from the experiment conducted in thi tudy indicate that the choice of topping rule doe indeed influence the mechanim performance characteritic. Tale 4 detail the variaility of the performance meaure etween rule. A one might expect, the activity rule realized the et overall performance meaure a it allowed the auction to converge to a natural topping point. When the minimum revenue and maximum round (0) rule were employed the auction topped accepting id even though there may have een idder willing to participate, thu extracting le urplu and poily inefficiently allocating unit. While analyzing the data on auction uing the activity rule we noted a phenomenon that 22

23 we refer to a the Trickle Effect. In a numer of the experiment the idding continued for everal round yet there wa no improvement in revenue. The auction did not terminate ince, during each round, at leat one revied id wa umitted. A more detailed look at the data indicated that the majority of reviion were eing made to contraint, for example decreaing the numer of how required. Due to the nature of the multi-criteria id, a idder could conceivaly enter a highly retrictive low id in the initial round and continue making modification to contraint and or her id amount for an extended numer of round. There may e ome merit to thi approach; the idder could e attempting to otain ignal information from the recommended id provided y the mechanim. Thi type of idding could prolong the auction eyond a natural topping point. The final controlled variale in our experiment i the calculation time given our heuritic to determine an allocation. We experimented with four period, 5, 5, 30 and 60 minute. The preliminary reult from tet uing the activity topping rule indicated there wa no ignificant improvement from the longer 30 and 60 minute run. Therefore, we teted the remaining cloing rule uing only 5 and 5 minute calculation time. Refer to Tale 4 for a ummary of the impact on the performance meaure from altering the calculation time. It wa anticipated that the longer the mechanim wa given to determine an allocation the etter that allocation. However, at firt glance it would appear that on occaion the additional calculation time had a negative effect on the performance parameter. We were ale to attriute the occaional reduced performance to the aility of our mechanim to find earlier allocation for more idder given longer calculation time. Although the auction perform etter in initial round, improvement in later round i reduced ecaue the roader early allocation reduce the aility of the mechanim to extract uyer urplu. Additionally, reult from a Manova teting 23

24 the impact of calculation time on optimality, efficiency and realized revenue indicated that the time given to the mechanim to determine an allocation i not a ignificant factor influencing the auction performance characteritic. The olution to the integer prolem preented in Section 4 act a a enchmark againt which we compare our heuritic performance. To facilitate achieving a olution uing a commercial integer programming olver, CPLEX 6.5, the prolem had to e caled down conideraly from the ize repreented in the previou experiment. The reduced prolem conit of 30 idder vying for 04 unit ditriuted acro 3 how. We choe to compare the reult from one round of activity. The integer prolem wa generated uing the parameter of the third round of idding from an auction uing 5 minute of calculation time, 0% minimum id increment and the activity topping rule. The numer of active idder in round 3 had een reduced to 2 from the original 30. With the aove parameter the Incompletely Specified Cominatorial Auction wa ale to complete the entire auction coniting of 5 round in a total of 5 minute. The duration of the auction include time ued to generate agent, calculate upper ound, output an MPS formatted verion of the prolem for CPLEX at each auction round, output all olution file for poile CPLEX uage a well a determine the appropriate allocation of unit in each of the 5 round. The ISCA found a olution that produced revenue of $728.3 for the eller with a gap le than 0.04%. The mall gap ugget that thi anwer may e optimal. The integer programming prolem repreenting a ingle round of the auction wa entered into CPLEX. After over 43 hour the program wa unale otain a ingle feaile integer olution. We then eeded CPLEX with the olution otained y our mechanim for round three. CPLEX confirmed that the olution wa indeed feaile and after 24 hour wa unale to 24

25 improve on the olution. The initial gap reported y CPLEX for thi olution wa 43.74%, ignificantly higher than that achieved y the heuritic. All of the run were performed on the ame 600 MhZ, dual proceor, Window 2000 machine. 8. Limitation Thi reearch preent a preliminary invetigation into a new auction mechanim. The nature of the invetigation ha important limitation. Firt, in order to gain inight into the efficacy of the mechanim we have limited our experiment to imulation with artificial agent. We acknowledge the fact that all nuance inherent to human ehavior cannot e adequately programmed into agent. Additional human trial are neceary. Our model accommodate the ale of a week of airtime and we make the implifying aumption that uyer campaign are continuou and thu the purchae pattern will repeat from week to week throughout the year. In reality, many uyer will want to flight their campaign, or place advertiing only during pecific week or day of the year. In order to accommodate flighting our mechanim will need to e expanded to 52 week and accept uyer pecified airdate. Extending our imple id tructure to 52 week will e challenging. Finally, our mechanim i deigned to ait or replace a negotiated environment, yet we have not attempted to analyze the uine proce change neceary to facilitate the tranition. We anticipate, epecially in the televiion indutry, that converion from negotiation to a uine-to-uine electronic auction may e met with a great deal of reitance. 9. Concluion and Future Reearch Thi reearch ha preented evidence that computer upported on-line auction can e developed to accommodate the pecial need of a traditionally negotiated environment. The mechanim deigned in thi tudy accept rule-aed id, providing guideline to direct rather 25

26 than dictate the allocation of good. Bidder are alo given the aility to impoe retriction on the allocation through pecification of the multi-criteria package id. Experiment etalih the Incompletely Specified Cominatorial Auction mechanim i efficient and revenue maximizing. The heuritic developed accommodate large prolem dimenion inherit in thi environment and tet indicate that the olution are near optimal. A important, the time it take the heuritic to reach a atificing olution i minimal and well within the limit impoed y the real-world environment. The development and proven efficacy of the mechanim decried in thi reearch ugget effective electronic auction mechanim can e developed to upport complex environment. A indicated earlier, thi i a preliminary tudy of our mechanim and a uch there i a great deal yet to dicover aout the mechanim ue a well a new deign iue to explore. Several deign modification are of interet. For example, the mechanim could e extended to ecome a market clearinghoue y incorporating all market participant (i.e. the entire network and cale indutry). Additionally, the current deign doe not allow the idder to reject an allocation. It can e argued that ince the unit aigned are not necearily completely pecified y the uyer, due to the inexact nature of the id, that the uyer hould have the option to reject an individual allocation. A doule-ided auction would provide the neceary id-ak format to facilitate the idder rejection of an unacceptale allocation. Additionally, not all indutrie are a protective of their pricing and inventory information a the one repreented in thi tudy, therefore an oviou model modification would e to deign an open format that provide ignal information to uyer. One of the limitation of thi reearch i the ue of imulation a the ole mean of teting the mechanim performance. An experimental tudy uing human uject would upplement 26

27 our undertanding of the effectivene of the auction mechanim. Several iue, uch a human computer interaction, complexity, trut, colluion and idding trategie, can e explored empirically. Etalihing idder trut in a emi-ealed auction ha, to our knowledge, not een invetigated yet could e a defining iue on the mechanim acceptance. The fact that the mechanim ugget id to inactive uyer implie that the mechanim mut e conidered trutworthy if idder are to act on the information provided. Another quetion aout the emiealed format i it impact on colluion. Sealed-id auction are le uceptile to colluion than are open auction ecaue actual id are not diplayed allowing idder to deviate from colluive pricing agreement without fear of detection [9, 20]. It would appear that the ame reult would hold for the emi-ealed format ut thi ha yet to e invetigated. Finally, although the heuritic developed in thi tudy proved to e effective and timely for winner determination, other olution methodologie and mechanim hould e invetigated. For example, our reult how that a high percentage of idder urplu wa extracted, ut a potauction ettlement, perhap akin to the Vickrey-Grove-Clarke procedure, might produce a etter truth-revealing idding mechanim (ee for example [24]). 0. Reference [] M. Armtrong, "Optimal Multi-Oject Auction." Unpulihed working paper Nuffield College, Oxford, UK., (999). [2] J. S. Bank, J. O. Ledyard, and D. Porter, Allocating Uncertain and Unreponive Reource: An Experimental Approach. The Rand Journal of Economic 20:-25, (989). [3] R. Bapna, P. Goe, and A. Gupta, A Theoretical and Empirical Invetigation of Multi-item On-line Auction, Information Technology and Management, ():-23,(2000). [4] M. M. Bykowky, R. J. Cull, and J. O. Ledyard, Mutually Detructive Bidding: The FCC Auction Deign Prolem. Journal of Regulatory Economic, (2000). [5] S. Y. Choi and A. B. Whinton The Smart Economy: Technologie and Application. To e pulihed y Addion-Weley Longman (999). 27

28 [6] C. Y. Coleman, Two Start-Up Plan to Market Unold Radio Time on the We. Wall Street Journal, (Novemer 2,999). [7] P. C. Cramton, The PCS Spectrum Auction: An Early Aement. Journal of Economic and Management Strategy 6(3):43-495, (995). [8] C. DeMartini, A. M. Kwanica. J. O. Ledyard, and D. Porter "A New and Improved Deign for Multi-Oject Iterative Auction." Caltech Social Science Working Paper No. 054 Novemer 998, revied (Septemer 999). [9] R. Engelrecht-Wiggan, Auction and Bidding Model: A Survey. Management Science 26(2):9-42, (980). [0] M. Harri and A. Raviv, Allocation Mechanim and the Deign of Auction. Econometrica, 49: , (98). [] C. A. Holt, Jr., Competitive Bidding for Contract Under Alternative Auction Procedure. Journal of Political Economy, 88: , (979). [2] E. L. Johnon, G. L. Nemhauer, and M. W. P. Savelergh, Progre in Linear Programming-Bae Algorithm for Integer Programming: An Expoition. INFORMS Journal of Computing 2():2-23 (Winter 2000). [3] J. L. Jone, Incompletely Specified Cominatorial Auction: An Alternative Allocation Mechanim for Buine-to-Buine Negotiation. Doctoral Diertation, Univerity of Florida, (2000). [4] F. Kelly and R. Steinerg, A Cominatorial Auction with Multiple Winner for Univeral Service. Management Science (46): , (2000). [5] E. Kutanoglu and S. D. Wu, On Cominatorial Auction and Lagrangean Relaxation for Ditriuted Reource Scheduling. IIE Tranaction (3):83-826, (999). [6] A. A. Lazar and N. Semret, Deign and Analyi of the Progreive Second Price Auction for Network Bandwidth Sharing. Preented at 8 th International Sympoium on Dynamic Game and Application, Maatricht, The Netherland, (July 998) and at the DIMACS Workhop on Economic, Game Theory, and the Internet, Rutger, NJ, (April 997). [7] J. O. Ledyard, D. Porter, and A. Rangel, Experiment Teting Multioject Allocation Mechanim. Journal of Economic and Management Strategy 6(3): , (997). [8] J. McMillan, Selling Spectrum Right. Journal of Economic Perpective 8(3):45-62, (994). [9] W. J. Mead, Natural Reource Dipoal Policy Oral Auction Veru Sealed Bid. Natural Reource Planning Journal M:94-224, (967). [20] P. Milgrom, Auction Theory. In Truman F. Bewley (ed.) Advance I Economic Theory: 28

29 Fifth World Congre. Camridge: Camridge Univerity Pre, (987). [2], Auction and Bidding: A Primer. Journal of Economic Perpective 3(3):3-22, (989). [22] R. B. Myeron, Optimal Auction Deign. Mathematic of Operation Reearch 6():58-73, (98). [23] D. C Parke and L. H. Ungar, Iterative Cominatorial Auction: Theory and Practice. Proceeding of 7 th National Conference on Artificial Intelligence (AAAI-00), pp. 74-8, (2000a). [24] D. C. Parke and L. H. Ungar, Preventing Strategic Manipulation in Iterative Auction: Proxy Agent and Price Adjutment. Proceeding of 7 th National Conference on Artificial Intelligence (AAAI-00), pp , (2000). [25] S. J. Raenti, V. L. Smith, and R. L. Bulfin, A Cominatorial Mechanim for Airport Time Slot Allocation. Management Science 44(8):3-47, (982). [26] M. H. Rothkopf and R. M. Hartad, On the Role of Dicrete Bid Level in Oral Auction. European Journal of Operational Reearch. 74: , (994). [27] M. H. Rothkopf, A. Pekec, and R. M. Hartad, Computationally Manageale Cominational Auction. Management Science 44(8):3-47, (998). [28] J. Stewart, Dot.com on NAPTE Floor. Spot-n-dot: The Daily New of TV Sale Friday, (January 28, 2000). [29] J. Teich, H. Walleniu, J. Walleniu, and A. Zaitev, A Multiple Unit Auction Algorithm: Some Theory and a We Implementation. Electronic Market 9(3):-7, (999). [30] Y. Vakrat and A. Seidmann "Analyi and Deign Model for Online Auction." Preented at INFORMS 4th Conference on Information Sytem and Technology, (May 26, 998). [3] W. Vickrey, Counterpeculation, Auction, and Competitive Sealed Tender. The Journal of Finance 6:8-37, (96). [32] J. Weaver, How Much Do I Hear for that Ad?. MSNBC New Article. (Oct. 25, 999). [33] M. P. Wellman, W. E. Walh, P. R. Wurman and J. K. MacKie-Maon, Auction Protocol for Decentralized Scheduling. Preented at the Eighteenth International Conference on Ditriuted Computing Sytem, Amterdam, (May 998) [34] P. R. Wurman, M. P. Wellman and W. E. Walh. A Parameterization of the Auction Deign Space. To appear in Game and Economic Behavior. (2000). 29

30 Appendix A: Summary of Notation u,,,i Sucript =how, =uyer, p=pod, u=pod part, i= allowale commercial length (uing thi order). S Numer of how. P Numer of pod in how. L Lit price for each 5-econd unit in how. D Vector of 5-econd demographic value for how. U Numer of 5-econd portion in pod p for how. C Maximum numer of total unit in how availale to ell. B Numer of uyer. T Target Vector of deired demographic impreion for uyer. h Vector of deired how for uyer ( h, = if how i deired y uyer, 0 otherwie) Note that h can e a zero vector. N, Set of pecified commercial length() in how for uyer. N, Set of allowale commercial length() in how for uyer. H, Min/max numer of deired how that uyer mut have ( H h H ). H K, Numer of correct length commercial allowed in how y uyer. M Type of merchandie advertied y uyer. v (d) value of uyer ' advertiing given a cumulative demographic vector d. a, h, H, H, T, N,, M, K, parameter of id umitted y uyer. a i the amount id. Binary Deciion Variale f, DV: if idder ha more than 5-econd in pod p in how. y DV: 0, variale. If 0, uyer can t uy any pod p. x DV: 0, variale. If, uyer, ha pot u in pod p in how. u,, I p,, i, DV: 0, variale. If pod p of how for uyer ue an allowale numer (i) of advertiing lot., DV: 0, variale. if I,, i = otherwie 0. (Ued for notational implicity.) z p, i N, j, DV: 0, variale. If, uyer ha any unit in how. 30

31 Tale : Incompletely Specified Cominatorial Auction (ISCA) Deign Progreive Multiple round Acending (Relaxing) Increaing id amount (relaxing tight contraint) Auction Type No open diplay of id information Semi-ealed Recommended reentry id provided Cominatorial Allow package id Incompletely Specified Include price and variou high-level pecification Increae amount y minimum percentage Modification increment, uggeted new id and/or relax tight (ignalling) Bid contraint() Eligiility No retriction Jump Bidding Permitted Withdrawal Not Allowed only if non-winning id Minimum Revenue Achieve a predetermined eller revenue goal (not revealed to uyer) Cloing Rule Activity No new id or id modification received Maximum Round Attain a predetermined round (not revealed to uyer) 3

32 Start While Bid Still Unelected Compute TotT d and TotD d Select Remaining Bid having max{(a *TotT d )/ (T *TotD d )} Solve AP for thi idder uing current aignment Bid Selection Complete Overall Feaile? No Repair Solution Sort Bid Max(Sum ((L * X, ) /a )) Ye Delete Top-mot remaining active id from olution No Record Initial Feaile Solution A Ye Feaile? Figure : Heuritic Flowchart 32

33 Tale 2: Experimental deign for auction mechanim uing Generic Bidder Set Stopping Rule Minimum Bid Increment Calculation Time Activity Minimum Revenue Maximum Round (0) 5%, 0% 5% 5, 5, 30, 60 Minute 33

34 Tale 3: Experimental reult for idder ehavior ( Activity topping rule, 5-Minute calculation time, 5% minimum id increment) Bidder Ditriution Evaluator 00% /6 5% BidAdjutor Min 0 00% % 0 /6 20% Max % % 0 /6 20% AllAdjutor Min % % /6 20% Max % % /6 30% Contraint % 0 0 /6 5% Reult Set Revenue Round Achieved Efficiency Optimality Unold Inventory # Winner % Bounded % 3% 0% % 5% % % 4% 7% 5% Reult Set 2 Revenue Round Achieved Efficiency Optimality Unold Inventory # Winner % Bounded 9% 5% 8% 2% 4% % 3% 3% 9% 5% 34

35 Tale 4: Experimental reult auction performance characteritic Stopping Percent = 5% Percent = 0% Percent = 5% Rule 5 Min 5 Min 5 Min 5 Min 5 Min 5 Min Revenue (000) Set Activity MinRev MaxRound Set 2 Activity MinRev MaxRound Optimality Percent of idder urplu extracted from winning idder Set Activity MinRev MaxRound Set 2 Activity MinRev MaxRound Efficiency Ratio of the value of the winning id to the highet attainale value for a feaile olution conidering all idder competing in the final round. Set Activity MinRev MaxRound Set 2 Activity MinRev MaxRound

36 26000 Revenue Per Round Revenue Round 5% 0% 5% Figure 2: Revenue Attained and Stopping Round For Each Percentage Increment 36

37 Biographical Note Joni L. Jone i an Aitant Profeor in the Computer and Information Sytem Department at the Univerity of Michigan School of Buine. She i the recipient of the Michael R. and Mary Kay Hallman Electronic Buine Reearch Fellowhip. Dr. Jone received her doctorate in Deciion and Information Science from the Univerity of Florida Warrington College of Buine. Gary J. Koehler i the John B. Higdon Eminent Scholar and Profeor of Deciion and Information Science in the Warrington School of Buine at the Univerity of Florida. He wa recently Profeor and Area Head at the Krannert Graduate School of Management at Purdue Univerity. 37