Designing multi-unit multiple bid auctions: an agent-based computational model of uniform, discriminatory and generalised Vickrey Auctions Atakelty Hailu, Sophie Thoyer
To cite this version:
Atakelty Hailu, Sophie Thoyer. Designing multi-unit multiple bid auctions: an agent-based computa- tional model of uniform, discriminatory and generalised Vickrey Auctions. Economic Record, Wiley, 2007, 83 (S1), pp.S57-S72. 10.1111/j.1475-4932.2007.00410.x. hal-02664994
HAL Id: hal-02664994 https://hal.inrae.fr/hal-02664994 Submitted on 31 May 2020
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based Fax + 33 (0)4 67 54 58 05 58 54 67 (0)4 33 + Fax 18 22 61 99 (0)4 +33 Phone Email : : Email France 1 cedex Montpellier 34060 Viala place 2 –LAMETA ENSAM ** author *Corresponding
fax: +61 8 6488 1098 1098 6488 8 +61 fax: 2538 6488 8 phone:+61 [email protected] email: AUSTRALIA 6009 WA (Perth), Crawley M089 Highway, Stiriling 35 Western of Australia Univeristy Economics, Resource and Agricultural of School Designing multi-unit multiple bid auctions: An An agen bid auctions: multiple multi-unit Designing
(S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x computational model of uniform, discriminatory and and discriminatory uniform, model of computational [email protected] Comment citer cedocument:
generalized Vickreyauctions generalized Atakelty Hailu Atakelty
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Sophie Thoyer Sophie ** **
t-based Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based based modelling, reinforcement learning learning reinforcement modelling, based Key wordsKey heteroge and competition of degrees different under discriminat uniform, compare to model computational effici allocative or outcomes controve budgetary of terms in is of auctions picture complete a depict experiments multi-unit laboratory for formats pricing or lumpy supply a submit to bidders the allowing by auctions resolve auctions ag These public services. by purchase employed being are auctions Multi-unit Abstract
Designing multi-unit multiple bid auctions: An An agen bid auctions: multiple multi-unit Designing
(S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x computational model of uniform, discriminatory and and discriminatory uniform, model of computational : procurement auctions, multi-unit auctions, comput auctions, multi-unit auctions, procurement : Comment citer cedocument: generalized Vickrey auctions Vickreyauctions generalized
JEL: C900 –D440 –Q250 –D440 C900 JEL:
ency. This paper constructs an agent-based agent-based an constructs paper This ency. neity in the bidder population. population. bidder the in neity demand schedule. However, the choice of of choice the However, schedule. demand how alternative pricing formats perform perform formats pricing alternative how bid problem inherent in single-unit single-unit in inherent problem bid rsial. Neither economic theory nor nor theory economic Neither rsial. ory and generalized Vickrey formats formats Vickrey generalized and ory encies to allocate resources and to to and resources allocate to encies ational experiments, agent- experiments, ational t-based Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based 2000). Swierzbinski efficie allocative improving of hope the in formats fro switched have policy-makers which for auctions, This 2000). Swierzbinski and (Binmore States United to continues formats price uniform and pay-as-bid) re a As formats. pricing alternative of performance auctioneer. for independent private and values,payment forma all established re the which for auctions single-bid to contrast In paper. this sak the For auction. multiple-bid multi-unit the to literature the In 1999). (Tenorio, exchange foreign market electricity i wholesale including applications being are auctions bid multiple Multi-unit 1993). inh problem bid' 'lumpy the avoid helping schedules bid allow hand, other the on auctions, Multiple-bid c bid a submit to allowed is bidder individual each multiple-b or single-bid be can They good. same the aucti the which in auctions are auctions Multi-unit 1 The revenue equivalence theorem The revenue tha (RET) indicates Designing multi-unit multiple bid auctions: An An agen bid auctions: multiple multi-unit Designing I. (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x
computational model of uniform, discriminatory and and discriminatory uniform, model of computational 1 te ein f utpebd utos ufr from suffers auctions multiple-bid of design the , Introduction Introduction Comment citer cedocument: generalized Vickrey auctions Vickreyauctions generalized
ts lead to equivalent equivalent revenues lead ts expected to for the e of brevity, we will use this shorter name in in name shorter this use will we brevity, of e venue equivalence theorem (RET) has been been has (RET) theorem equivalence venue t under the hypothesis bidders' hypothesis t of the neutrality, under risk onsisting of a single price-quantity pair bid. bid. pair price-quantity single a of onsisting , the term multi-unit is mostly used to refer refer to used mostly is multi-unit term the , oneer wishes to sell or buy several units of of units several buy or sell to wishes oneer sult, the choice between discriminatory (or (or discriminatory between choice the sult, ncy and budgetary revenues (Binmore and and (Binmore revenues budgetary and ncy s as well as markets for Treasury bills and and bills Treasury for markets as well as s be controversial both in Europe and in the the in and Europe in both controversial be id auctions. Under the single-bid version, version, single-bid the Under auctions. id ders to bid with demand (or supply) supply) (or demand with bid to ders m discriminatory to uniform payment payment uniform to discriminatory m erent in single-bid auctions (Tenorio, (Tenorio, auctions single-bid in erent ncreasingly used, with well known known well with used, ncreasingly is the case for US Treasury bill bill Treasury US for case the is great uncertainty about the the about uncertainty great t-based Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based h aon o srie te wud rvd a differ at provide would they services of amount the structures cost different and capacities production private with bidders of population a from services a procurement a as cast is market auction simulated un discriminatory, auctions: multi-unit for formats exami to model agent-based an constructs paper This and experimental studies in economics in studies experimental and em are studies of number growing A designs. auction comparing for and contexts different under auctions tool research inexpensive and useful a provides ACE const complexity or cost the from less suffer hand, computatio agent-based or experiments Computational dem flat with bidders two where auctions on focused s simplified very to restricted are experiments the compl the However, 2003). Grimm and Engelmann 2001, Nous Alemsgeest, (e.g. formats payment three the of more provided have experiments field and Laboratory 407). p 2000, the is design auction multiple-bid of understanding and experimental for case The 1993). Tenorio, 1998, Empir 2002). Cramton, and (Ausubel constant not are heterogeneous is population bidder the when greater mul a in formats alternative of efficiency relative does theory economic because exists controversy The 2 information oninformation economics. researchACE in Tesfatsion's site web Tesfatsion's at (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x Comment citer cedocument: http://www.econ.iastate.edu/tesfatsi/ace.htm
2 . .
. Bidders submit supply functions indicating indicating functions supply submit Bidders . ettings. To date, most of these studies have have studies these of most date, To ettings. tiple-bid setting. The uncertainty is even even is uncertainty The setting. tiple-bid refore strong (Binmore and Swierzbinski, Swierzbinski, and (Binmore strong refore iform and the generalized Vickrey. The The Vickrey. generalized the and iform raints that limit laboratory experiments. experiments. laboratory limit that raints or when bidder marginal values (costs) (costs) values marginal bidder when or computational approaches to further our our further to approaches computational ploying ACE to complement theoretical theoretical complement to ACE ploying uction where a government agent buys buys agent government a where uction independent values reflecting different different reflecting values independent ne the performance of three alternative alternative three of performance the ne sair and Olson 1998, Kagel and Levin Levin and Kagel 1998, Olson and sair the relative performance of different different of performance relative the ent prices. Auctions are repeated and and repeated are Auctions prices. ent insights into the relative performance performance relative the into insights ical studies are also scarce (Wolfram, (Wolfram, scarce also are studies ical not provide much guidance on the the on guidance much provide not nal economics (ACE), on the other other the on (ACE), economics nal o eaiig h promne of performance the examining for and curves compete for two units. units. two for compete curves and exity of these auctions means that that means auctions these of exity is an is excellent source of Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based impact of of changes impact in number bidders. the of by supply degree aggregate bidders. reflects o the It different levels of competitionof levels different of performance The incomes. net their increasing of their update to learning reinforcement use bidders 3 antd fo 1.% o 0 o te grgt capaci aggregate the of 50% to 12.5% from magnitude summarize recommendations. we section, fifth the In increasing. are of levels all almost for performance worst the has levels heterogeneity and competition all for almost Vickre auctions. uniform the and discriminatory the the by predicted as bids truthful towards converge generali the under that demonstrate We auction. the t population, bidder the in heterogeneity of nature pattern interesting more display behaviours bidding pre analytical of confirmation some provide results for allocation of efficiency and outlays budgetary exper computational the from results the present we reinforcement (1998) Roth’s and Erev using choices boundedly of (ABM) model agent-based an develop we implied strategies bidding of properties structural briefly first We follows. as organized is paper The bidders. individual of of levels different for undertaken also is analysis The level of competitionThe level of ratio the measured here is as (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x Comment citer cedocument: 3 , with the demand level from the purchasing agency agency purchasing the from level demand the with ,
f demand rationing but it does demand butdoes the f rationing it not include heterogeneity in the size and cost structures structures cost and size the in heterogeneity by theoretical analysis. In the third section, section, third the In analysis. theoretical by of demand by the government agency the agency demand over government by of he intensity of competition, and the type of of type the and competition, of intensity he review the various auction formats and the the and formats auction various the review competition when marginal costs of supply supply of costs marginal when competition individual bid functions with the objective objective the with functions bid individual the three auction formats. The simulation simulation The formats. auction three the theory and that bid shading is the rule for for rule the is shading bid that and theory studied here. The discriminatory auction auction discriminatory The here. studied s depending on the interplay between the the between interplay the on depending s iments and compare bidding behaviours, behaviours, bidding compare and iments learning algorithm. In the fourth section, section, fourth the In algorithm. learning y auctions have the best performance performance best the have auctions y each auction format is evaluated for for evaluated is format auction each dictions. But they also indicate that that indicate also they But dictions. zed Vickrey format, simulated bids bids simulated format, Vickrey zed the paper and draw some general general some draw and paper the ty of the bidders. The comparative comparative The bidders. the of ty ainl idr rvsn ter bid their revising bidders rational ranging in in ranging Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based quantity quantity proble allocation the formats, auction three the In per-unit price per-unit price price
The residual demand facing bidder bidder facing demand residual The utoer ihs o u i fxd n eul to equal and fixed is buy to wishes auctioneer identical objects with private independent values. values. independent private with objects identical simultaneo sealed-bid of case the on concentrate We purchasing agency less the quantities supplied by a by supplied quantities the less agency purchasing schedule of bidder bidder of schedule su inverse an to equivalent are bids multiple these to willing is he price the indicating bids multiple multi- procurement a In demand. residual of concept procedur allocation the describing by start will We displ are that own) his than (other bids ofthe sum i winner each Vickrey, the Under case. auction unit generalizat a is which payment, Vickrey generalized equati price clearing the earn sold units all which actual their of sum the to equal amount an paid are (als payment discriminatory the 1) formats: payment II. (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x b
. . Q * i Multiple-bid auctions auctions Multiple-bid at which his supply schedule intersects his residua his intersects schedule supply his which at b S b D Q a b S Q max b D b * * * − . . i i i i 0 )} ( {0, ) ( = = Comment citer cedocument: i as − = − ) ( ) ( Q i = S i
( b d ) with with (2) ∑ i i j , ≠ D − Q j i ( i b the quantity that bidder bidder that quantity the )
, is defined as the total demand demand total the as defined is , (1) Q aced by his successful bids. bids. successful his by aced We assume that the number of units that the the that units of number the that assume We ng aggregate supply to demand; and 3) the the 3) and demand; to supply aggregate ng es under the three payments rules using the the using rules payments three the under es d ll other bidders bidders other ll accept for different quantities. In effect, effect, In quantities. different for accept m is solved by awarding each bidder the the bidder each awarding by solved is m winning bids; 2) the uniform payment in in payment uniform the 2) bids; winning . We compare the three most common common most three the compare We . o known as pay-as-bid) in which bidders bidders which in pay-as-bid) as known o s paid the amount corresponding to the the to corresponding amount the paid s pply function. Let’s define the supply supply the define Let’s function. pply ion of the second price payment in the the in payment price second the of ion us auctions for the procurement of of procurement the for auctions us unit auction, each bidder submits submits bidder each auction, unit l demand. demand. l j j for each level of clearing clearing of level each for i is willing to sell for a for sell to willing is Q d of the the of Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based the uniform-price auction, all units sold earn the the earn sold units all auction, uniform-price the corresponding bids. In a generalized Vickrey auctio Vickrey generalized a In bids. corresponding p receive units infra-marginal Therefore, demand. to area under the residual demand up toup demand residual underthe area discriminatory auction, each bidder is paid the are the paid is bidder each auction, discriminatory calculatio the in differ formats three the However,
(2002) demonstrate that, although it is a dominant dominant a is it although that, demonstrate (2002) Kah and Engelbrecht-Wiggans payment, uniform the In summary). a for 1 Table d take can reduction demand that but shading bid to paym uniform the that predict – auction procurement for conducted – studies Most formats. payment three the compare studies theoretical and empirical Many 2006). Chakraborty, 2002; Krishna, 1995; structura the of analysis the on focussed therefore clos as strategies equilibrium obtain to impossible t lead formats payment most Second, 2002). Cramton, Engelbrecht-Wiggan 1993; (Tenorio, case auction bid auctio the for revenues expected equivalent to lead theorem equivalence revenue the First, conclusions. leadi mechanism, this on literature rich a been has described the purchase of perfectly divisible units divisible perfectly of purchase the described auctions multiple-bid on investigations Theoretical (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x Comment citer cedocument:
Q * i . .
clearing price price clearing l properties of bidding strategies (Noussair, (Noussair, strategies bidding of properties l a under his supply schedule up to up schedule supply his under a as “auctions of shares”. Since then, there there then, Since shares”. of “auctions as n of the payments for the winners. In the the In winners. the for payments the of n n, each successful bidder is paid the entire entire the paid is bidder successful each n, neer, does not carry over to the multiple- the to over carry not does neer, formats payment all that indicates which , strategy to bid truthfully for the first unit unit first the for truthfully bid to strategy properties of bidding strategies under the the under strategies bidding of properties were initiated by Wilson (1979), who who (1979), Wilson by initiated were the case of a standard selling rather than than rather selling standard a of case the ed form expressions and authors have have authors and expressions form ed ent and the discriminatory payment lead lead payment discriminatory the and ent ng to two important and unchallenged unchallenged and important two to ng ifferent forms in the two auctions (see (see auctions two the in forms ifferent n (1998) and Ausubel and Cramton Cramton and Ausubel and (1998) n o multiple equilibria. It is therefore therefore is It equilibria. multiple o ayments that are higher than the the than higher are that ayments s and Kahn, 1998; Ausubel and and Ausubel 1998; Kahn, and s b * equating aggregate supply supply aggregate equating Q * i . In . Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based Asbl n Cramton and (Ausubel on paid price the reduce to incentive the by offset “ because offered quantities the Moreover, quantities. additional for bids their to tends quantity when case, continuous the in (or,
uto i wih rtfl idn i a eky domin weakly a is bidding truthful which in auction Ausu by designed quantities “clinched” with auction counterpart its or payment Vickrey generalized The Gr and Engelmann by experimentally confirmed is This in difference the when outcome, possible a also is to comparable shading, bid differential of strategy al (2002) Krishna equilibrium. possible a is curve, flat entirely submitting neutral, risk are bidders the for amounts large relatively by overbid bidders tha curves supply flatter submit to incentive an is setting simplified a in demonstrate, (1993) Zender more different, to leads payment discriminatory The c sincere, rarely are bids first-unit the observed, point (2000) Lucking-Reiley and List auctions, card o “incremental However, Wales. and England in auction procurement of analysis econometric an conducted as market, exchange foreign Zambian the of analysis a (2001) Levin and Kagel by experimentally verified increasing the explains which increase, quantities (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x Comment citer cedocument: , , 02 p23). 2002, the incentive to win units at any price below margi below price any at units win to incentive the
The latter becomes increasingly important when when important increasingly becomes latter The
ontrary to theoretical predicitons. predicitons. theoretical to ontrary supply curves, above the true opportunity cost cost opportunity true the above curves, supply n in a uniform price auction. In other words, words, other In auction. price uniform a in n infra-marginal units that are won anyway” anyway” won are that units infra-marginal the marginal values of the two units is high. high. is units two the of values marginal the what is observed in the uniform payment, payment, uniform the in observed is what first unit compared to subsequent units. If If units. subsequent to compared unit first with two bidders and two units, that there there that units, two and bidders two with so shows, in the case of two units, that a that units, two of case the in shows, so zero), it is efficient for bidders to shade shade to bidders for efficient is it zero), bid shading. This phenomenon has been been has phenomenon This shading. bid amount of bid shading increases with with increases shading bid of amount nd empirically by Tenorio (1993) in his his in (1993) Tenorio by empirically nd complex, bidding strategies. Back and and Back strategies. bidding complex, in the open format (i.e. the ascending ascending the (i.e. format open the in imm (2003). imm(2003). in a field experiment involving sports sports involving experiment field a in bel, 2005) is the only multiple-unit multiple-unit only the is 2005) bel, ant strategy, resulting in efficient efficient in resulting strategy, ant out that, although bid-shading is is bid-shading although that, out well as by Wolfram (1998) who who (1998) Wolfram by as well verbidding” in the electricity electricity the in verbidding” nal value is is value nal Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based the impact of these types of heterogeneity. There i There heterogeneity. of types these of impact the theore The costs. supply in heterogeneity to related su the in heterogeneity is one first The exploring. bid of sources two are There format. discriminatory g is gain performance this whether indication no is more to and behaviour strategic of reduction the to demo was compe It population. bidder the in of heterogeneity levels different for compare formats these particular, In payments. uniform and discriminatory decis help to produced be results more that crucial tw the of efficiency the compare even or coordinate st equilibrium of classes different are there Since allocation 4 formats for multi-unit auctions. auctions. multi-unit for formats strat ofequilibrium properties 1:Structural Table inefficiencies. allocation to t Decision-makers althoug auctions, payment bidders. uniform or discriminatory by understood easily not employed rarely is Vickrey generalized the However, Lucking-Reiley, and (List experimentally tested and production marginal (highest costs utility). production is when (sold) reallocation obtained aregoods bought Efficiency here social coEfficiency opportunity the refers to Generalized Vickrey Generalized Uniform-Price Discriminatory Sealed-bid format (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x 4 . It was demonstrated theoretically (Ausubel and Cr and (Ausubel theoretically demonstrated was It . Comment citer cedocument: Efficient allocation allocation Efficient dominantstrategy; weakly a is bidding Truthful allocation Inefficient equilibrium; p high ata Coordination overbidding”; “Incremental allocation Inefficient steep; is curve cost opportunity true when ov “incremental or for supply” flat “high for Scope e and strategies equilibrium of property Structural
st allocationthe of st resources.of An efficient
rategies, it is difficult to analyse how bidders bidders how analyse to difficult is it rategies, from (to) bidders with the lowest bidders from (to) marginal with the s a need, therefore, to turn to experiments to to experiments to turn to therefore, need, a s pply capacity of bidders and the second is is second the and bidders of capacity pply truthful bidding (Swinkels 1999) but there there but 1999) (Swinkels bidding truthful tical literature does not provide answers on on answers provide not does literature tical it is particularly important to assess how how assess to important particularly is it nstrated that increased competition leads leads competition increased that nstrated h it is known that overbidding may lead lead may overbidding that known is it h egies under different payment payment different under egies ion-makers to make a choice between between choice a make to ion-makers in practice because the payment rule is is rule payment the because practice in 2000; Engelmann and Grimm, 2003). 2003). Grimm, and Engelmann 2000; reater in the uniform format or in the the in or format uniform the in reater o payment formats. Therefore, it is is it Therefore, formats. payment o ders’ heterogeneity which are worth worth are which heterogeneity ders’ tition and for different types of of types different for and tition herefore prefer to implement implement to prefer herefore amton, 2002; Ausubel, 2005) 2005) Ausubel, 2002; amton, erbidding” erbidding” rice rice fficiency
Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based buyer, the government agent. The government agent h agent government The agent. government the buyer, sellin agents of population a has model auction Our model based agent of Structure strategies. bidding their update to algorithms aucti the in functions supply bid continuous submit mul of issue the tackles it because models previous mo The (2004). Schilizzi and Hailu (2001), Oliveira (2 Tesfatsion and Petrov Nicolaisen, (1995), Miller study the to ACE applying Studies 2002). (Tesfatsion theoretica in imposed normally are that through assumptions as well as learning, inductive networking, lik phenomena of incorporation explicit the through modelli where systems of study the to suited is ACE des conditions equilibrium or equations than rather specifi the is ACE in point starting the approaches, experiments. subject human from findings examine to prov and economics in experiments human-subject and w exami (2006) in Duffy applied. been areas has (ACE) economics research the surveyed 2002 Axt in and Tesfatsion (Epstein studied being system the or market env and institutions individuals, of behaviours the interac of societies artificial of study the is ACE computationa develop to chose we run,to costly are research. of area this in understanding our further
III. (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x
The modelling of bidding strategies with artificial with strategies ofbidding modelling The Comment citer cedocument:
ting autonomous agents that directly emulate emulate directly that agents autonomous ting Instead of using human experiments, which which experiments, human using of Instead cribing the system under study. Therefore, Therefore, study. under system the cribing ironmental components that make up the the up make that components ironmental g goods in a sealed-bid auction to a single single a to auction sealed-bid a in goods g l experiments. experiments. l 001), Bower and Bunn (2001), Bunn and and Bunn (2001), Bunn and Bower 001), cation of agent attributes and behaviours behaviours and attributes agent of cation e agent heterogeneity, local interactions, interactions, local heterogeneity, agent e del presented in this paper differs from from differs paper this in presented del on and employ reinforcement learning learning reinforcement employ and on ng outcomes can be gainfully enriched enriched gainfully be can outcomes ng as a fixed target or demand level. Each Each level. demand or target fixed a as ides an overview of studies using ACE ACE using studies of overview an ides tiple-bid auctions: competing bidders bidders competing auctions: tiple-bid l analysis for tractability purposes purposes tractability for analysis l the relaxation of other restrictive restrictive other of relaxation the Unlike conventional or deductive deductive or conventional Unlike of auctions include Andreoni and and Andreoni include auctions of nes the relationship between ACE ACE between relationship the nes ell 1996; and Tesfatsion 2002). 2002). Tesfatsion and 1996; ell hich agent-based computational computational agent-based hich learning agents agents learning Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based For the sake of simplicity, it is assumed that the the that assumed is it simplicity, of sake the For supply bid functions that maximize their expected n expected their maximize that bidfunctions supply l sellers time, Over functions. bid supply declared bidders different the of functions supply true the sa for has it good of amount maximum the indicating and can be written as: as: written be can and where where seller is characterized by a (true) non-decreasing non-decreasing (true) a by characterized is seller
costs as well as on the history of choices he has m hashe ofchoices history the ason as well costs a of probabilities choice strategy The round. next wh with probabilities the update to and incomes net use Sellers use. in format auction the to according payment stage, second the In demand. residual their at them of each from bought quantities equilibrium residual the calculates sellers, the from functions firs the In stages. two involves round auction Each follows: follows: curve bid learnt The curve. linear a by learnt approximated the that assumption simplifying the make We strategies choice Seller (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x ms
β β i is the capacity of bidder capacity the is i i i i i i i wt () ( with ) ( QQ bQ a Q + = Comment citer cedocument: P i = a
i 0 + i, a i, b i 0 i Q 0
is his entry price and price entry his is i t
he strategic bid of player ofplayer bid strategic he with with
true supply function of each sellers sellers each of function supply true demand for each bidder and determines the the determines and bidder each for demand supply function supply ade and rewards obtained for those choices. choices. those for obtained rewards and ade and makes selection based on submitted or or submitted on based selection makes and 0 bidder therefore depend on his opportunity opportunity his on depend therefore bidder the results of the auction to compute their their compute to auction the of results the earn to choose, in a repeated process, the the process, repeated a in choose, to earn et incomes. incomes. et ich they choose their bid strategies for the the for strategies bid their choose they ich ≤ ≤ is therefore assumed to be represented as as represented be to assumed therefore is le. The government agent does not know know not does agent government The le. the intersection of their bid supply and and supply bid their of intersection the s to individual bidders are determined determined are bidders individual to s ms Q t stage, the government collects bid bid collects government the stage, t i i bidding curve can be reasonably reasonably be can curve bidding , b , (3) (3) i 0
and a given supply capacity capacity supply given a and is the supply slope. slope. supply the is i (4) (4) i is linear linear is Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based slope parameter, for example, there is a choice of choice a is there example, for parameter, slope choi slope and Intercept bids. past of performance the maximum slope value implied by the constraints constraints the by implied value slope maximum the impose that bidders won't use extreme overbidding s overbidding extreme use won't bidders that impose reinforcement principle that is widely accepted in in accepted widely is that principle reinforcement developed was algorithm reinforcement-learning The with associated payoffs forgone about knowledgeable biddin modelling for suitable particularly is it as reinforcement-l The requirements. information their cas special are variants certain how and algorithms t shows (2003) Camerer in presented models learning over developed been have models learning Different algorithm learning The implicit an imposing to equivalent is This supplier. the of supply of cost marginal the times three than supply function parameters. parameters. function supply restricting the allowed or feasible parameter choic parameter feasible or allowed the restricting bidde the in included not therefore is and strategy functi cost true the below falling section any have strategies of choice the on imposed is constraint A seller’s the to dimensions two therefore are There
xlr dfeet obntos of combinations different explore slope choice ( choice slope and slope (b slope and (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x 0 ) parameters are included in the choice sets to all to sets choice the in included are parameters ) b i ). The learning algorithm described below will allo will below described algorithm learning The ). Comment citer cedocument:
a i
and b i
and to retain the best values based on the the on based values best the retain to and g behaviour without requiring that players be be players that requiring without behaviour g on (bidding below true costs is a dominated dominated a is costs true below (bidding on most expensive unit by the most expensive expensive most the by unit expensive most seven values equally spaced between 0 and and 0 between spaced equally values seven es (es ces are discretized into seven steps. For the the For steps. seven into discretized are ces reserve price by the auctioneer. auctioneer. the by price reserve es of others. The models differ in terms of of terms in differ models The others. of es so that the chosen bid function does not not does function bid chosen the that so the psychology literature. Erev and Roth Roth and Erev literature. psychology the earning algorithm is chosen for this study study this for chosen is algorithm earning choice strategy: intercept choice ( choice intercept strategy: choice strategies that they did not select. select. not did they that strategies trategies by restricting their bids to less less to bids their restricting by trategies discussed below. The true intercept (a intercept true The below. discussed r's choice set). This is guaranteed by by guaranteed is This set). choice r's a the last several decades. A typology of typology A decades. several last the by Roth and Erev (1995) based on the the on based (1995) Erev and Roth by he relationship between these learning learning these between relationship he and b ) as shown in Figure 1. We also also We 1. Figure in shown as ) ow for truthful revelation of of revelation truthful for ow w bidders to progressively progressively to bidders w a i ) and and 0 ) ) Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based studies mo to algorithm learning this use and extend (1998) 5 the propensity of player player of propensity the described be can algorithm the of features main The experience. gn-ae suis f lcrct acin markets auction electricity of studies agent-based Ro The predictions. theoretical outperform generally predictions model’s learning reinforcement the that propensity updating function can be written as (Erev as written be can updatingfunction propensity (or (or Experimentation prevents players from quickly being quickly from players prevents Experimentation succes chosen previously to similar strategies that (Erev and Roth 1998): 1998): Roth and (Erev princi four following the on based is algorithm The 2001). Oliveira, fact that learning curves tend to be initially stee initially be to tend curves learning that fact not. This principle implies that choice is probabili is choice that implies principle This not. ( recency. Eleven of by thesedifferent r of Eleven conducted were games reinforced forgetting (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x 5 f eetd ae wt uiu nnrva equilibri nontrivial unique with games repeated of
q The law of effect effect of law The i ab ) or weakened depending upon whether the action pro action the whether upon depending weakened or ) ( t rqie ta rcn eprec hs oe mat o impact more has experience recent that requires ) + )1 Comment citer cedocument: = 1( − φ h lw f fet te oe lw f rcie exper practice, of law power the effect, of law the ) q i o hoe taey (a,b) strategy choose to i ab set ta te edny o hoe n cin s st is action an choose to tendency the that asserts t )(
+
E cd ( a , b , R )
p. p. esearchers in the period and1960 esearchers between in period 1995. the Experimentation stic. stic. ples rooted in the psychology of learning learning of psychology the in rooted ples and Roth 1998, p. 863): 863): p. 1998, andRoth sful ones will be employed more often. often. more employed be will ones sful of the choices of experimental subjects subjects experimental of choices the of
del behaviour from twelve experimental experimental twelve from behaviour del (e.g. Nicolaisen Nicolaisen (e.g. locked into particular choices. choices. particular into locked using the following three equations. If If equations. three following the using t time at (5) The power law of practice of law power The th-Erev algorithm has been used in in used been has algorithm th-Erev a in mixed strategies. They find find They strategies. mixed in a t is denoted by by denoted is
duces favourable results or or results favourable duces ( or generalization) generalization) n behaviour than past past than behaviour n et al al et 2001; Bunn and and Bunn 2001; imentation imentation q refers to the the to refers i ab rengthened rengthened ( Recency t implies implies ) , the the , and and
Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based ial, s hw i te hr ln i euto (6) equation in line third the in shown as Finally, that a given strategy or strategy given a that proba a in made is curve bid learnt a of choice The aselected neighbouring not are and rounds previous well as the addition of new reinforcement reinforcement new of addition the as well E where: where: 6 = R(1- = has has e in line second (see experimentation of result the previ the in selected strategy a of neighbour a was If (6). equation in line first the in indicated is "neighb or "similar" with experiment to need the by eq is reinforcement additional this round, previous strategy strategy where where (equation 5). This updating includes elements of di of elements includes updating This 5). (equation previous the on update an is t+1 period in strategy attach propensity a has element strategy each Thus, 0 = function. See Erev and Roth Roth (1998, function. and See Erev 863). p. For strategy sets without linearsets For generaliz strategy order, the cd ( a ,b a n (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x egbuig taeis te xeietto param experimentation the strategies, neighbouring ε ϕ , , is an experimentation parameter and parameter experimentation an is E (c,d)
R cd is the recency parameter, parameter, recency the is ) is the following three step generalization functi generalization step three following the is ) ( a, b, Rb, a, and is the payment above true costs (net revenue) o revenue) (net costs true above payment the is and Comment citer cedocument: ) = R. ( R. = ) (a,b) value is chosen depends on that strategy’s proport strategy’s that on depends chosen is value ε ε /n) /n)
) )
R is the reward or reinforcement from previous choic previous from reinforcement or reward the is if a = c and b= d d b= and c= a if tews otherwise of(c,d) strategy neighboring b)is (a, if R j E cd n
the strategy whose propensity is being updated updated being is propensity whose strategy the ) , ( is the number of neighbours of strategy (a,b). (a,b). strategy of neighbours of number the is ation ation should function specified aas be two-step bilistic way in each round. The probability probability The round. each in way bilistic ual to the reward R achieved, discounted discounted achieved, R reward the to ual scounting by by scounting quation (6)). Note that since the mark-up mark-up the since that Note (6)). quation strategy, get no experimental spillovers. spillovers. experimental noget strategy, ous round, then the new reinforcement is is reinforcement new the then round, ous , strategies that were not selected in the the in selected not were that strategies , ouring" strategies in the next round; this this round; next the in strategies ouring" ed to it. And the propensity to choose a a choose to propensity the And it. to ed . For a strategy that was selected in the the in selected was that strategy a For . propensity of choice for that strategy strategy that for choice of propensity on eter is divided by by divided is eter 6 : : (1- btained by the seller, while while seller, the by btained ϕ ) to reflect forgetting as as forgetting reflect to n
ional share in in share ional in this line. line. this in (6) e of of e
Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based parameter (s) that determines the initial propensit initial the determines that (s) parameter used in equation (5), the experimentation parameter experimentation the (5), equation in used algorithm. learning the robu are results our that show analyses Sensitivity lea the which at speed the affect can These occurs. propensities which from scale overall the determine of significance The capacity. maximum her supplying ar harmlessly is latter The round. bidding any from t of product the to equal is value uniform this and pr choice The values. propensity 1) (period initial appear not does parameter scale The latter. the for previou the in selected strategy a to similar being o that implies of0.2 value parameter experimentation factor a by discounted get choice of propensities value The respectively. 9, and 0.2 0.1, are values R and Erev in studied games 12 the for data best the hrfr, hs erig loih hs he parame three has algorithm learning this Therefore, player player to available strategies all for sum propensity the
(S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x i uses his (c,d) (c,d) Comment citer cedocument: strategy is then given by: by: given then is strategy p i cd ( t )
= ∑∑ b a q i cd q ( t i ab ) ( t )
of 0.1 for the recency parameter implies that that implies parameter recency the for 0.1 of ies the bidder. Specifically, the probability that that probability the Specifically, bidder. the opensities are initially given uniform values values uniform given initially are opensities s round is 2.5% of the net reward achieved achieved reward net the of 2.5% is round s in the above equations; it is used to set the the set to used is it equations; above the in he scale parameter and the expected profit profit expected the and parameter scale he bitrarily set at 10% of the bidder’s cost for for cost bidder’s the of 10% at set bitrarily st to variations in the parameter values of of values parameter the in variations to st the reinforcement that a bid curve gets for for gets curve bid a that reinforcement the ( oth (1998) were used in this study. These These study. this in used were (1998) oth q get reinforced and degraded as learning learning as degraded and reinforced get es nml,te eec aaee ( parameter recency the namely, ters, ε (7) 1 ab rning converges on particular choices. choices. particular on converges rning ) used in equation (6) as well as a scale scale a as well as (6) equation in used ) f 0.9 between auction rounds. The The rounds. auction between 0.9 f . The parameter values that provided provided that values parameter The . the initial propensities is that they they that is propensities initial the ϕ ) Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based idn udr h Vcry dsrmntr ad unifo and discriminatory Vickrey, the under Bidding settings Simulation bidder populations (Table 2): 2): (Table populations bidder performan Auction available. capacity aggregate the corr These used. were 2.0, and 1.5 1.0, 0.5, namely, cons capacity supply aggregate keeping while demand level The population. bidder the of structures cost com of levels different for experiments computation
a choice a Figure 1 : Generalization (experimentation) in rei in (experimentation) Generalization : 1 Figure IV. (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x
• •
Simulation results and discussion discussion and results Simulation (c,d) Strategy costs, with a flat supply curve (constant marginal curve (constant supply flat a with costs, w bidders of population a homogeneous 1: population and 50% are large capacity bidders, all with a fla a with all bidders, capacity large are 50% and bidders the of 50% where a population 2: population Comment citer cedocument:
Neighbours of of Neighbours
c,d)(c
b choice b espond to 12.5%, 25%, 37.5% and 50% of of 50% and 37.5% 25%, 12.5%, to espond petition and heterogeneity in the size and and size the in heterogeneity and petition of competition was varied by changing changing by varied was competition of ce was simulated for the following four four following the for simulated was ce nforcement learning learning nforcement tant at 4.0. Four levels of demand, demand, of levels Four 4.0. at tant rm auctions was simulated in our our in simulated was auctions rm t supply curve (b supply t cost curve, b curve, cost are small capacity bidders bidders capacity small are ith similar capacity and capacity similar ith 0 =0) =0) 0 =0) Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based using these criteria. In 64% of the simulations, th simulations, the of 64% In criteria. these using allowe were rounds 1000 of minimum A seeds. random replica different 100 up, set experimental each For the pair (apair the ar in interested are we What bids. their adjust and u rounds of number large a over run are Simulations strategies simulated of Convergence
following two conditions hold for each bidder: bidder: each for hold conditions two following the of termination premature avoid To make. bidders fo appropriate more is it as study this in employed strategies of set the among choice of probabilities secon The selected. are that parameters of stability convergence in the choices that agents make. The fi make.The agents that choices the in convergence
1) 2) (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x ofbid populations four the of description 2: Table
Bidders Capacity Entry price aprice Entry Capacity Bidders Population 4 Population Population3 2 Population 1 Population probability of choice attached to the most likely s likely most the to attached choice of probability choice over probabilities of distribution the that that this probability is at least five times bigge times least five at is probability this that • • i ,b
i ) obtained once convergence is reached. There are a are There reached. is convergence once obtained ) along two dimensions (capacity and the steepness of andsteepness the (capacity dimensions two along wit population heterogeneous a highly 4: population (b rising with population a homogeneous 3: population 0 >0), and and >0), Comment citer cedocument: dnia idr 05 . 0.5 0 0.5 cost large-high 2 0.5 large-low cost 2 0.5 small-high cost 2 cost: small -low 2 bidders identical 8 0.5 large 4 small 4 bidders identical 8
0.75 0.75 0.75 0.25 0.25 0.75 0.25 that a bidder has. This second approach was was approach second This has. bidder a that e the bidders' convergence strategies, that is is that strategies, convergence bidders' the e r the probabilistic nature of the choices that that choices the of nature probabilistic the r e convergence criteria were met within the the within met were criteria convergence e rst is to define convergence in terms of the ofthe terms in convergence define to is rst ntil bidders have had ample time to learn learn to time ample had have bidders ntil learning process, it was required that the the that required was it process, learning r than the second highest probability probability highest second the rthan tions were generated using different different using generated were tions d is to examine the distribution of of distribution the examine to is d strategies is unimodal, i.e. the the i.e. unimodal, is strategies trategy is at least 0.5, and and 0.5, least at is trategy ders ders d before convergence was tested tested was convergence before d 0.5 0.5 0.5 0.5 0.5 0.5 t least two ways to measure measure to ways two least t h bidders differentiated differentiated bidders h the marginal cost curves). curves). cost marginal the marginal cost curves curves cost marginal i 0 0.75 0.75 0.25 0.75 0.25 0 0 Supply slope b slope Supply i 0 0 Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based before that. The 20000The that. before the if rounds 20000 of maximum a for run to allowed wit simulations the of 80% in and rounds 2000 first
auctions. Vickrey auc uniform the in demand of level the on dependent forth strategy bidding The demand. of level the of 99% of frequency a (with unit first the on truthful population this for behaviour bidding simulated The inv level price atany competition the bidders, all level the Since constant. is cost marginal of level bidd The structure. simplest the has population This with bidders of population homogenous a for Results pe ofproduction cost the by measured as allocation and unit per outlays budgetary of terms in compared perform the Then predictions. theoretical available strategie bidding the First replications. 100 these based evaluated are auctions the of performance the with replications 100 involve simulations all Since se next the in discussed are results simulation The stopped. was simulation the when wa there that indicates algorithm learning the from a highest the of examination An cases. the of 8% in (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x Comment citer cedocument: th round was reached without the convergence criteria convergence the without reached was round
olves the entire aggregate capacity. capacity. aggregate entire the olves of marginal cost is constant and identical for for identical and constant is cost marginal of or higher) in all auction formats regardless regardless formats auction all in orhigher) ction. We only analyse convergence results. results. convergence analyse only We ction. e subsequent units was found to be be to wasfound units esubsequent s observed are presented and compared to to compared and presented are observed s s a predominant choice in most of the runs runs the of most in choice predominant a s r unit. unit. r different random seeds, the strategies and and strategies the seeds, random different (Table 3) indicates that bidding is is bidding that 3)indicates (Table tion but not in the discriminatory and discriminatory the in not but tion ers are homogeneous in capacity and the the and capacity in homogeneous are ers flat supply flatsupply ance of the three auction format are are format auction three the of ance nd second highest choice probabilities probabilities choice highest second nd on the average values obtained from from obtained values average the on hin 5000 rounds. The simulation was was simulation The rounds. 5000 hin in terms of the social efficiency of of efficiency social the of terms in convergence criteria were not met met not were criteria convergence being met only only met being Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based net income by inflating prices on subsequent units. subsequent on prices inflating by income net price entry truthful with bidding by winning ensure circ such Under bids. others' in changes small with riv by undercutting price to susceptible are curves c when However, price. clearing auction the at bids there format, auction this under So unit. marginal u infra-marginal for received prices the as revenue referred (henceforth units following the on overbid selected of 96% around auctions, discriminatory In 3. Table in summarized are figures The b"). a/higher but unit first the on truthfully bid bidders cases, more is bidding truthful of frequency The strategy. bidd truthful that states which theory by predicted behaviour bidding learnt format, Vickrey the Under Uniform Discriminatory Vickrey Auction type sele for (%) frequencies strategy Bidding 3: Table
nt ( units ent higher are strategies equilibrium of properties indi 1998) Kahn and (Englebrecht-Wiggans literature (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x ih lt bidding flat high population with constant marginal costs ( a=0.5, b= a=0.5, ( costs marginal constant with population level level Demand Comment citer cedocument: 1.5 0.5 1.5 0.5 1.5 0.5 1 1 1 2 50.8 49 0.2 0 0 0 0.2 0 0.2 49 0.8 96 50.8 2 3.8 13.3 2 85.9 2
ecfrh. fatnn o te upy uv impr curve supply the of flattening A henceforth). true a/true b true a/higher b true a/higher ba/truetrue 41.2 23.6 87.8
3.5 4.3 88 85 5 5
58.9 76.4 96.5 95.7 11.9 95 95 12 15
is an incentive for bidders to organize their their organize to bidders for incentive an is
ry price and flat bidding on the following following the on bidding flat and price ry als and the bidder can easily be priced out out priced be easily can bidder the and als ing ("true a/true b") is a weakly dominant dominant weakly a is b") a/true ("true ing Thus, supply inflation is not precluded by by precluded not is inflation supply Thus, overbid on the subsequent units ("true ("true units subsequent the on overbid bidders bid truthfully on the first unit and and unit first the on truthfully bid bidders s while at the same time earning positive positive earning time same the at while s than 85%. In the remaining 15% of the the of 15% remaining the In 85%. than
umstances, bidders have an incentive to to incentive an have bidders umstances, o as to cted bidders: homogeneous homogeneous bidders: cted nits are brought closer to that of the the of that to closer brought are nits ompetition is tight, these flat supply supply flat these tight, is ompetition
higher a /true b higher a/higher a/higher higher b a bhigher /true cates that the expected structural structural expected the that cates generally conforms to the results results the to conforms generally upy inflation supply 0) 0.3 0 0 0 0 0 0 0 0
). However, the the However, ). oves bidder bidder oves 0 0 0 0 0 0 0 0 0
Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based Auction type Bidders true a/ true a/ Discriminatory Vickrey Bidders Auction type selec for (%) frequencies strategy 4:Bidding Table cas base homogenous the asin samethe remain curve supply total The 0.75). = (ms capacity high a with lo a with agents of 50% sizes: different having but ide display all who bidders of made is 2 Population more slightly a of case the for observed strategies i simulated behaviours bidding the compare we Here, suppl flat with bidders of population a for Results get. they price the impact not does it as bidd Infra-marginal increases. demand the as setter bidde each that fact the by explained be can levels truth more bid to propensity the However, auctions. behavi the therefore is It profits. greater make to setti one (the bidder marginal the allows inflation format Vickrey the in observed as bidding, truthful f whereas inflation), (supply format discriminatory is bidding the levels, demand low for competition: learnt the that format uniform the with only is It populations. homogenous this o is what exactly is it and 2002), (Krishna, theory Uniform (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x population of small and large bidders with constant with bidders large and small of population Comment citer cedocument: Small Small Small Large 3.4 96.6 0 0 0 0 0 1.4 96.6 15.2 3.4 83.4 Large Large Large 4.3 95.3 0 0 0 0 95.3 4.3 Large
true true b 88.6 94.9 5.9
higher higher b True a/ True a/ y but heterogeneous sizes sizes heterogeneous ybut behaviour changes greatly with the level of level the with greatly changes behaviour ng the clearing price) and all other winners winners other all and price) clearing the ng r has a lower probability of being the price price the being of probability lower a has r 11.4 94.1 complex population of bidders (Table 4). (Table bidders of population complex w capacity (ms = 0.25) and 50% of agents agents of 50% and 0.25) = (ms capacity w bserved in our simulations for the case of of case the for simulations our in bserved ntical constant marginal costs (flat supply) supply) (flat costs marginal constant ntical or higher demand levels, the frequency of frequency the levels, demand higher or 4.1 ers have no incentives to inflate their bids bids their inflate to incentives no have ers capacity, as well as the aggregate supply supply aggregate the as well as capacity, , increases. In a uniform auction, supply supply auction, uniform a In increases. , fully that is evident at lower completion completion lower at evident is that fully our expected by theory under uniform uniform under theory by expected our n the simplest case described above with with above described case simplest the n e (a = 0.5 and b = 0). =b and 0.5 = e(a comparable to the one observed in the the in observed one the to comparable ted bidders for demand of 1.5: of1.5: demand for bidders ted
marginal costs (a=0.5, b=0) b=0) (a=0.5, costs marginal higher a/ a/ higher true true b 0 0 0
higher a/ a/ higher higher higher b 0 0 0
Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based u we dmn lvl ices, h hg fa bidd flat high the increase, levels demand when but simul previous the in as behaviour inflation supply d low For flat. are costs marginal when observed is positively is bidding in changes major that format discriminatory production of cost marginal the When with bidders of population homogeneous for Results decline). levels competition r to tends behaviour truthful (although competition are behaviou observations these Furthermore, respectively). frequent most the are cases discriminatory Vick the in bidding a truthful strategies: Small comparable auctions. discriminatory the in and Vickrey mu have not does size case, uniform the to Contrary toadop also tend therefore and setters price be to of levels lower For units. last the on overbidding therefor They demand. of levels high for especially bidders Large follows. as is explanation The supply. goe b) a/higher (true combination the on bids their supply and truthful of mix a display bidders small infl of supply strategy a use bidders large whereas small demand, of levels high For case. significa homogeneous altering bidders, capacity high and small discrep great observe We auction. uniform the under d very a has heterogeneity size of introduction The
(S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x Comment citer cedocument:
t aggressive overbidding. overbidding. aggressive t demand, small bidders have a greater chance chance greater a have bidders small demand, ation. However, for lower levels of demand, demand, of levels lower for However, ation. fairly insensitive to the level of demand or demand of level the to insensitive fairly ations (97% on true a/higher b for D= 0.5) 0.5) D= for b a/higher true on (97% ations strategies are observed compared to what what to compared observed are strategies s up to 54% for D= 12.5% of aggregate aggregate of 12.5% D= for 54% to up s upward sloping supply supply sloping upward inflation strategies, i.e. the frequency of frequency the i.e. strategies, inflation egress slightly under the Vickrey when when Vickrey the under slightly egress ancies between the strategies adopted by by adopted strategies the between ancies ch influence on bidding strategies in the the in strategies bidding on influence ch emand levels, bidders display the same same the display bidders levels, emand bidders tend to adopt a truthful strategy strategy truthful a adopt to tend bidders e tend to inflate the clearing price by by price clearing the inflate to tend e ramatic impact on bidding behaviours behaviours bidding on impact ramatic rey case and supply inflation in the the in inflation supply and case rey ntly the conclusions drawn for the the for drawn conclusions the ntly are more likely to be price setters, setters, price be to likely more are ing becomes increasingly prevalent, prevalent, increasingly becomes ing rs (above 80% and above 94%, 94%, above and 80% (above rs nd large capacity bidders adopt adopt bidders capacity large nd sloped (Table 5), it is for the the for is it 5), (Table sloped Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based groups have two bidders each with the following cha following the with each bidders two have groups (m capacity its on depending categories four of one s the have bidders All simultaneously. slopes) cost sou two include we simulations, of set final the In heterogeneou with bidders of population for Results format. Vickrey the in stable relatively remains 1 to 0.5 of demand for 27% (from format uniform the beh this of frequency the that and inflation supply o the cases, both in that note to interesting is It th for 81% above is bids b a/true true of frequency bidding sincere formats, auction two other the With Uniform Discriminatory Vickrey Auction type sele for (%) frequencies strategy Bidding 5: Table w competitors by undercut completely ofbeing risks theoretica the to conforms behaviour latter The 50%. when b) a/zero (higher 74% of frequency a reaching
(S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x Group D: large capacity and steeper supply curves ( supply steeper and capacity large D: Group curves ( supply and steeper capacity small C: Group ( curves supply and flatter capacity large B: Group ( curves supply and flatter capacity small A: Group Homogeneous population with positively sloped margi sloped positively with population Homogeneous Comment citer cedocument: true a/ true a/ true true b 75.2 24.6 0 0 0.2 (0.2) 0.2 (0) 0 0 0 0 0 24.6 15.6 75.2 84.4 4.3 35.7 0.4 0.4 59.2 (59.2) 59.2 0.4 0.4 35.7 4.3
higher higher b true true a/
aviour decreases when demand increases in in increases demand when decreases aviour ther frequent bidding strategy observed is is observed strategy bidding frequent ther ame entry price (a = 0.5) but each falls into into falls each but 0.5) = (a price entry ame higher a/ a/ higher e Vickrey and above 72% for the uniform. uniform. the for 72% above and Vickrey e s) and true supply cost slopes (b). The four four The (b). slopes cost supply true and s) s sizes and supply slopes slopes supply and sizes s true true b is prevalent at all levels of demand. The The demand. of levels all at prevalent is hen the demand level increases. increases. level demand henthe rces of heterogeneity (size and marginal marginal and (size heterogeneity of rces racteristics: racteristics: l predictions and is explained by reduced reduced by explained is and predictions l cted bidders for demand level 1.5: level demand for bidders cted the ratio of demand to total capacity is is capacity total to demand of ratio the ms=0.75, b=0.25) b=0.25) ms=0.75, ms=0.25, b=0.25) b=0.25) ms=0.25, ms=0.75, b=0.75) b=0.75) ms=0.75, 7.9% for demand of 2) whereas it it whereas 2) of demand for 7.9% ms=0.25, b=0.75) b=0.75) ms=0.25, higher higher a/ higher b nal costs (a=0.5, b=0.5) b=0.5) (a=0.5, costs nal lower b lower (zero higher a/ a/ higher b) Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based 24% of the time, bids combining true entry prices w prices entry true combining bids time, the of 24% slopes supply true the reveal also bids the cases, 89%, respectively. Only for big capacity and steep steep and capacity big for Only respectively. 89%, f higher is behaviour This average). on (77% groups bidding truthful 0.5) of (demand levels competition varies auction uniform the under behaviour Bidding rivals. by out priced betotally to likely been inflat supply that already confirms It settings. population has what with consistent is behaviour esp strategies, inflation” “supply adopt frequently bi large are this to exceptions only The 2). Figure level demand a at 37% from increases it decreases: f increasing an with occurs which bid supply “ flat overbi predominant most The auction. discriminatory trut populations, other the for results the with As 75%, 91%, are groups four the for bidding truthful sum 1.5 of demand of case the for results frequency The curves. steeper with bidders sized equally than cur supply true shallower with bidders Furthermore, regard counterparts larger their than truthful more gro the among observed consistently are differences i bid not do groups different the However, auction. oeitn. idr peoiaty i ter re e true their bid predominantly Bidders nonexistent. decl bidders where cases auction, Vickrey the Under 7 go above 1%. go At a 0.5 ratelevel 0% of this is demand Only when the when frequency level is the o demand does Only 2 (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x Comment citer cedocument:
only 0.29% levelat and a of demand 1. resulting in truthful bidding. Between 18 and and 18 Between bidding. truthful in resulting ion is a strategy adopted by bidders who are are who bidders by adopted strategy a is ion less of the slope of their true supply curves. curves. supply true their of slope the of less hful bidding is almost non-existent with a with non-existent almost is bidding hful 86% and 56%, respectively. respectively. 56%, and 86% dders with steep true supply curves who who curves supply true steep with dders n exactly similar ways and the following following the and ways similar exactly n ves tend to bid truthfully more frequently frequently more truthfully bid to tend ves ty prices ntry se patterns can clearly be seen from the the from seen be clearly can patterns se ith higher slopes are observed under this this under observed are slopes higher ith ups. Smaller capacity bidders tend to be be to tend bidders capacity Smaller ups. of 1 to 69% at a demand level of 2 (see (see 2 of level demand a at 69% to 1 of or groups A, B, and C at 94%, 80% and and 80% 94%, at C and B, A, groups or dding is the theoretically expected “high “high expected theoretically the is dding is the most frequent behaviour for all all for behaviour frequent most the is marized in Table 6. The frequencies of of frequencies The 6. Table in marized ecially for low demand levels. This This levels. demand low for ecially requency as the level of competition competition of level the as requency cost curve bidders (group D) is this this is D) (group bidders curve cost are higher entry prices are almost almost are prices entry higher are f f entrybids prices than with higher true with competition levels. At high high At levels. competition with observed in more homogeneous homogeneous more in observed 7 . And in at least 74% of the the of 74% least at in And . Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based f counts and sizes heterogeneous with bidders of population (%) frequencies strategy Bidding 6. Table especially strategies flat” and“high “truthful” of increa demand as declines then but 1.0 to increases frequent more becomes inflation” “supply However, truthfu with bids group this as 50% below frequency
Group D 1 98 0 0 1 1 59 66 83 84 0 0 0 0 0 1 0.6 0 0 0 1 0.5 0 1.8 0 0 0 36 98 26 0 9 10 0 0 0 0 5 1 8 0 7 0.6 6 0.7 bidders All true a/ 22 43 DGroup C Group 13.7 B Group 22.7 AGroup 8.1 77 56 85.8 74.9 91.2 bidders All DGroup C Group B Group AGroup groups Bidder All bidders 67 27 0 1 5 5 0 3 14 2 1 1 0 1 0 0 1 0 1 1 27 40 13 51 6 67 58 84 33 91 bidders All DGroup C Group B Group AGroup (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x Comment citer cedocument: true a/ true a/ true a/ true a/ true true b true true b true b true b
higher b higher b Bidding strategies under Discriminatory Bidding under Discriminatory strategies higher b b higher true a/ true a/ true a/ true a/ true a/ true a/ Bidding strategies Bidding under Uniform strategies Bidding strategies under Bidding Vickreystrategies
among groups B and C. C. and B groups among higher a/ a/ higher higher a/ a/ higher higher higher a/ true true b true true b true true b ses further where we observe a mixture mixture a observe we where further ses supply slopes slopes supply for all groups as the demand level level demand the as groups all for l entry prices and inflated slopes. slopes. inflated and prices entry l or demand level 1.5 for the the for 1.5 level demand or higher higher a/ higher higher a/ higher b higher b higher a/ higher b b higher
higher a/ a/ higher higher a/ a/ higher higher higher a/ (zero (zero b) (zero (zero b) lower b lower lower b lower (zero b) lower b lower b Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based
general observations can be made. made. be can observations general t across behaviour bidding of pattern the Examining summary a strategies: Bidding 0.4 0.5 0.6 0.7 0.8 0.9 1.1 1.2 1) 2) 1 (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x (a) Learnt bid curves for bidderbiddera for with (a)curves bid Learnt
algorithm used in the simulations does lead to cohe to lead does simulations the in used algorithm at looked be also can It analysis. theoretical from biddin sincere of frequencies high to leads auction leve demand and populations bidder of types all For iiie h rs o big opeey nect y c by undercut completely being of risk the minimize an theoretical by precluded not is (which behaviour marginal the when discriminatory the under strategy slope supply higher but price levels entry (same inflation lower at predominant more becomes behaviour non-constant by characterized populations bidder of or lower and price entry (higher bidding flat high au discriminatory the under norm the is Overbidding capacity of 0.25 and slope of 0.25 of slope and0.25 of capacity sloping supply cost curves curves cost supply sloping heterogeneous of population of competition: levels a discriminatory a under behaviour Bidding 2. Figure Comment citer cedocument: D=2.0 D=1.5 D=0.5 D=1
0.4 0.5 0.6 0.7 0.8 0.9 1.1 1.2 1.3 1.4 1 (b) Learnt bid curves for bidderbiddera for with (b)curves bidLearnt he different populations, the following following the populations, different he capacity of 0.75 and slope of 0.25 of andslope 0.75 of capacity as a confirmation that the learning learning the that confirmation a as zero slope) is observed in the case case the in observed is slope) zero rent outcomes. outcomes. rent g. This conforms with predictions predictions with conforms This g. s) is the predominant overbidding overbidding predominant the is s) ction. The theoretically predicted predicted theoretically The ction. cost of supply is constant. This This constant. is supply of cost ls considered here, the Vickrey Vickrey the here, considered ls marginal cost curves. This This curves. cost marginal bidders with upward upward with bidders alysis) allows the bidder to to bidder the allows alysis) ompetitors. Therefore, this this Therefore, ompetitors. uction, for different different for uction, of competition. Supply Supply competition. of D=2.0 D=1.5 D=0.5 D=1 Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based
their overbidding when demand is high. And this coo this And high. is demand when overbidding their coor induce equilibria possible of multiplicity the tighter is competition of level the when strategies popula the of regardless low is rate this auctions, lower much is ratio this costs, marginal increasing t pass experiments the of percent 95% to 82 between bidder for levels demand all at Vickrey the in high cons that choices strategy learnt of percentage The Th one. learnt the than other bid a using income net non if (NE) equilibrium Nash a constitute strategies t bidder the allows that strategy other no is there properti reply best for tested were bids learnt The strategie learnt of tests property equilibrium Nash 4) 3) (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x
is low and/or marginal cost is constant pitting eve pitting constant is cost marginal and/or low is comp of level the when rational is strategy bidding most competitive cost structure (i.e. for group A t A group for (i.e. structure cost competitive most bid truthful of frequency the formats, auction both obtai that than lower is telling oftruth frequency behaviour bidding truthful induces context the When setters. price of being probability lower smaller among especially levels, demand in increase frequency The size. in heterogeneous is population and capacity large with bidders among predominantly for mostly observed is inflation Supply inflation. strategies: of types two induces auction uniform The Comment citer cedocument:
dination failures. Bidders learn to coordinate coordinate to learn Bidders failures. dination (demand is low). These figures confirm that that confirm figures These low). is (demand o increase its net income. The convergence convergence The income. net its increase o s s populations 1 and 2. For these populations, populations, these For 2. and 1 populations es by checking, for one bidder at a time, if if time, a at bidder one for checking, by es . And for both discriminatory and uniform uniform and discriminatory both for And . tion. Finally, more choices constitute NE NE constitute choices more Finally, tion. 7. Table in summarized are eresults e of the eight players can improve on his his on improve can players eight the of e iue Ns eulbim taey e is set strategy equilibrium Nash a titute rdination is easier when the population is is population the when easier is rdination he NE test. For the populations with with populations the For test. NE he ned under the Vickrey. Moreover, for for Moreover, Vickrey. the under ned han for group B). B). group hanfor ry unit for sale against every other. every against forsale unit ry etition is intense because demand demand because intense is etition ding is higher for the group with with group the for higher is ding of truthful bidding increases with with increases bidding truthful of high levels of competition and and competition of levels high under the uniform auction, the the auction, uniform the under capacity bidders who have a a have who bidders capacity truthful bidding and supply supply and bidding truthful steep cost curves when the the when curves cost steep Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based The Vickrey is the most efficient for all types of types all for efficient most the is Vickrey The ha what confirms efficiency auction of analysis The a 1 populations in as bidders all for identical and equally is allocation any as 4) and 3 (populations relev is criterion second The producers. cost lower efficien more are outcomes auction the perspective, socia auction's the measures latter The bidders. the measures former The costs. production total the and the using measured is auction an of performance The performance Auction (U Uniform and (D) Discriminatory (V), Vickrey (%): th strategies bidding learnt of 7:Proportion Table bidders. ofother choices coordination price the th have might bidder a Therefore, strategy. bidding t uniform, the Under fixed. bids their keep to were t with bidder the providing bid, own his on depends t In competitors. its of choices the given strategy i if down) go never would (but improve might income bene high is price clearing the that so others with for Vickrey, the With deviation. unilateral through individu as strategies reply best constitute not do fewe by set is price clearing the and heterogeneous
Demand . 86 83 97 2.0 95 1.5 1.0 0.5
(S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x
D V U D V U D V U D V U D V U D V Population 1
Comment citer cedocument: 14 14 13 91
15 17 20 30
82 82 84 90 Population 2
10 11 22 71
nd 2. 2. nd he discriminatory auction, a bidder’s revenue revenue bidder’s a auction, discriminatory he 11 60 72 8 populations. The discriminatory is the least least the is discriminatory The populations. he bidder’s revenue can depend on its own own its on depend can revenue bidder’s he e same incentives to defect or 'free ride' on on ride' 'free or defect to incentives same e bids coordinated these However, bidders. r efficient when marginal costs are constant constant are costs marginal when efficient al bidders can improve their net incomes incomes net their improve can bidders al
t if the product purchased is sourced from from sourced is purchased product the if t fits all bidders. However, a bidder's net net bidder's a However, bidders. all fits example, coordinating bidding choices choices bidding coordinating example, l cost efficiency. From a social welfare welfare social a From efficiency. cost l s been observed with bidding strategies. strategies. bidding with observed been s the monetary transfers from the buyer to to buyer the from transfers monetary the at pass the Nash equilibrium tests tests equilibrium Nash the atpass following two criteria: budgetary outlay outlay budgetary criteria: two following he incentive to deviate if other bidders bidders other if deviate to incentive he ant only for the last two experiments experiments two last the for only ant 13 15 18 30 Population 3 ) auctions auctions ) t reverts to a more truthful bidding bidding truthful more a to reverts t
25 0 0 1
10 29 8 9
11 50 Population 4 2 8
34 2 0 5
17 48 2 5
Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based
units. last the on overbidding by up price clearing l of strategies the to mainly due declines, auction ho 4), and 2 (populations populations heterogeneous demand when same the almost are uniform the in and and 3 (population bidders cost marginal increasing levels high for especially , 2) and 1 (populations bi when clear very is Vickrey the of advantage This is auction Vickrey the outlay, budgetary by Judging formats. three the under relativ the to related are results These efficient.
(S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x population of homogeneous bidders with flat supply supply flat with bidders homogeneous of population of c levels different for unit per Outlay 3. Figure
Comment citer cedocument: outlay per unit 0.50 0.55 0.60 0.65 25 50 75 50.0% 37.5% 25.0% 12.5%
demand-capacity ratio demand-capacity Uniform Discrim. Vickrey
of demand. See Figure 3. For populations of of populations For 3. Figure See demand. of See Figure 4. 4. Figure See arge bidders who tend to drive the auction auction the drive to tend who bidders arge e frequency of truthful bidding displayed displayed bidding truthful of frequency e the least expensive in almost all settings. settings. all almost in expensive least the 4), the budgetary outlays in the Vickrey Vickrey the in outlays budgetary the 4), wever, the performance of the uniform uniform the of performance the wever, dders have constant marginal costs costs marginal constant have dders is low. For higher demand levels in in levels demand higher For low. is ompetition for the the for ompetition cost curves curves cost
Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based
The discriminatory auction format is generally the the generally is format auction discriminatory The Table 8. for bidderpopulation performance Auction uniform. the than in heterogeneous are which populations with levels populati the when especially Vickrey the as well as competition high For qualifications. two following measure Performance Cost per unit 0.568 0.569 0.572 1.034 1.035 1.040 1.040 1.252 1.035 1.035 1.379 1.194 1.034 1.038 1.162 1.018 0.572 1.357 1.093 1.029 0.767 1.041 1.126 0.569 1.006 0.554 1.283 0.759 1.016 1.016 0.568 0.687 1.014 1.068 0.556 0.729 0.533 1.095 per Cost unit 0.727 1.008 0.551 0.598 Outlay per unit 1.025 0.545 0.660 0.515 per Cost unit 0.672 0.532 0.532 Outlay per unit 0.519 0.589 per Cost unit 0.560 0.516 Outlay per unit 0.539 per Cost unit Outlay per unit (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x curves curves slo upward with bidders heterogeneous of population of c levels different for unit per Outlay 4. Figure outlay per unit Comment citer cedocument: 0.50 0.55 0.60 0.65 0.70 0.75 0.80 Vickrey Vickrey Vickrey Vickrey 25 50 75 50.0% 37.5% 25.0% 12.5% Summary Performance Measures Summary Performance Measures
Discriminatory Discriminatory Discriminatory Discriminatory
ead . Demand: 0.5 Demand: 0.5 ead . Demand: 1.5 Demand: 1.5 ead Demand: 2 Demand: 1 Demand: 2 Demand: 1 demand-capacity ratiodemand-capacity
nfr Vcry Discriminatory Vickrey Uniform Vickrey Uniform Discriminatory Vickrey Uniform Uniform
size, the discriminatory can perform better better can perform discriminatory the size, on is homogeneous. For low competition competition low For homogeneous. is on levels, the discriminatory often performs performs often discriminatory the levels, most expensive auction but with the the with but auction expensive most
4 Vickrey Vickrey Summary PerformanceRelative Uniform Discrim. Vickrey ompetition for the the for ompetition Discriminatory Discriminatory ping supply cost cost supply ping Measures
Uniform Uniform Uniform Uniform
Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based different demand levels (ranging in magnitude from from magnitude in (ranging levels demand different functio bid their of parameters slope and intercept str a over learn bidders The experience. individual learn reinforcement a formulated use that agents of population then was model agent-based An discussed. studi some from Findings exist. that gaps knowledge predict theoretical discussing by started paper The auctions. Vickrey formats: pricing three for performance auction and e computational using by gap knowledge this filling objective The issue. controversial a be to continues discriminat and uniform multi-unit under strategies desc analytical an provide not does theory Economic Conclusions 5. relat are losses Efficiency truthful. completely not Vickrey the of case the in even significant be can aucti uniform the for 125% to 102% from and auction range figure This bidding. truthful under been have range learning bidder under outlay program auction; highest is distortion This formats. three the under as us help figures relative These columns). 3 (last woul what to relative measures performance presents ar 4 population for figures performance average The
(S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x Comment citer cedocument:
for program outlay under the discriminatory discriminatory the under outlay program for ively small, in a range of 0.8% to 4%. 4%. to 0.8% of arange in small, ively auction where bidding is predominantly but but predominantly is bidding where auction n. The experiments are undertaken for four four for undertaken are experiments The n. ing algorithm to update their bids based on on based bids their update to algorithm ing ory auctions. The choice of auction format format auction of choice The auctions. ory sess the relative distortion of true values values true of distortion relative the sess xperiments to simulate bidding behaviour behaviour bidding simulate to xperiments ions for the three auction types and the the and types auction three the for ions uniform, discriminatory and generalized generalized and discriminatory uniform, s from 110% to 138% of what it would would it what of 138% to 110% from s s from 102% to 116% for the Vickrey Vickrey the for 116% to 102% from s of this paper is to contribute towards towards contribute to is paper this of ategy space with two dimensions: the the dimensions: two with space ategy on. Therefore, the level of overpayment overpayment of level the Therefore, on. es using human experiments were also also were experiments human using es bidding truthful under occurred have d e presented in Table 8. The table also also table The 8. Table in presented e 12.5% to 50% of aggregate supplier supplier aggregate of 50% to 12.5% ription of the equilibrium bidding bidding equilibrium the of ription to simulate bidding among a a among bidding simulate to Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based sold are brought closer. However, this strategy inc strategy this However, closer. brought are sold bidder improve to capacity the have bids flat High from deviation this for provided be an can explanation (Engelmann studies experimental human in observed struc cost competitive less with bidders for levels compet high at bidders of types all among behaviour Our high. are levels competition when or constant fr a is auction uniform the in observed type the of comp of levels when costs, marginal increasing with bid flat high The bid). supply flat and price entry inflati supply observed: are behaviours overbidding neve hand, other the on auction, discriminatory The counterparts. expensive more or bigger their tru remain to tend they setters, price be to likely sette price the be to likely are who bidders the by c large for predominantly and competition of levels i Supply format. uniform the under occur behaviours in heterogeneous is bidders of population the When increas but price entry true (i.e. inflation supply tw induce auctions Uniform bidders. amongst size of auction uniform a in strategies bidding particular, bidder the of nature the on depends also It format. cannot behaviour bidding that indicate results Our differentiated bidders of groups four of consisting po bidder of types different four for and capacity)
(S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x Comment citer cedocument:
ding expected by theory is found for bidders bidders for found is theory by expected ding equent strategy when true marginal costs are are costs marginal true when strategy equent by size and marginal cost slopes. slopes. cost marginal size and by are extremely sensitive to the heterogeneity heterogeneity the to sensitive extremely are tures. This bidding behaviour has also been been also has behaviour bidding This tures. pulations, with the most heterogeneous one one heterogeneous most the with pulations, population and the level of competition. In In competition. of level the and population thful and "free-ride" on the risks taken by by taken risks the on "free-ride" and thful reases the risk that the bidder is completely completely is bidder the that risk the reases ing overbidding on the subsequent units). units). subsequent the on overbidding ing rs. On the contrary, when bidders are less less are bidders when contrary, the On rs. revenue as the prices received for all units units all for received prices the as revenue the high flat bidding predicted by theory. theory. by predicted bidding flat high the apacity bidders. It is the strategy adopted adopted strategy the is It bidders. apacity etition are low. However, supply inflation inflation supply However, low. are etition results provide evidence of such bidding bidding such of evidence provide results o types of strategies: truthful bidding and and bidding truthful strategies: of types o capacity, dramatic differences in bidding bidding in differences dramatic capacity, on and high flat bidding (i.e. high leant leant high (i.e. bidding flat high and on r leads to truthful bidding: two types of of types two bidding: truthful to leads r be completely characterized by auction auction by characterized completely be ition levels, and even at high demand demand high at even and levels, ition nflation is observed mostly for high high for mostly observed is nflation d Grimm 2003). An intuitive intuitive An 2003). Grimm d Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based Action Plan for Salinity and Water Quality Water and Salinity for Plan Action Ins Market-Based National The Australia. throughout con conservation allocating for Auctions Australia. mu where markets auction emerging several are There alter consider seriously to need would policymakers competiti of levels most for auction discriminatory The case. act multi-unit the in especially complexity, this use to auctioneers of reluctance The cases. lea the is Vickrey format. expensive most the cases is which auction, discriminatory The message. strong auction of performance relative the of analysis The but structure cost of terms in only not population, gran be should attention that indicates It formats. prope structural the to relation in provides theory c more is simulations these by provided picture The t it allows bidding flat high than rather inflation compe less its of because or others with similarity face bidder a when Therefore, rivals. by out priced 8 is s dniid o e polm o piae land private for problem a be to identified is bids the although auctions multi-unit been have auctions conservation trialing pilots several included have alge ad Appels and Laplagne. the and auction, discriminatory the format, payment A description of the pilot the of Aprogram description here available is (S1), S57-S72. DOI :10.1111/j.1475-4932.2007.00410.x Comment citer cedocument: (2003)). Moreover, all auction trials have focused have trials auction all Moreover, (2003)).
o avoid zero gain outcomes. outcomes. gain zero avoid o : http://www.napswq.gov.au/mbi/index.html http://www.napswq.gov.au/mbi/index.html : 8 is in its second round; and, both rounds rounds both and, round; second its in is rties of equilibrium strategies under the three three the under strategies equilibrium of rties also in terms of size. size. of terms in also uniform auction can perform better than the the than better perform can auction uniform s in terms of budget outlays also delivers a a delivers also outlays budget of terms in s natives to the discriminatory auction. auction. discriminatory the to natives st expensive procurement auction in most most in auction procurement expensive st titive cost structure, a strategy of supply supply of strategy a structure, cost titive ted to the heterogeneity of the bidding bidding the of heterogeneity the to ted on. Therefore, procuring agencies and and agencies procuring Therefore, on. commonly used in practice, is in most most in is practice, in used commonly truments Pilot Program of the National National the of Program Pilot truments omplex than the partial view that the the that view partial the than omplex tracts are currently of great interest interest great of currently are tracts s stiff competition as a result of its its of result a as competition stiff s ion might be related to its relative relative its to related be might ion auctions. However, none of these these of none However, auctions. lti-unit auctions could be applied in in applied be could auctions lti-unit re has been little consideration of of consideration little been has re holders (see, for example, example, for (see, holders lumpy bid nature of single-unit single-unit of nature bid lumpy only on one on only Chan, Chan, Version postprint computational modelof uniform,discriminatory and generalisedVickrey Auctions. EconomicRecord, 83 Hailu, A., Thoyer,S.(2007). Designing multi-unitmultiple bidauctions: anagent-based improve budgetary and allocative efficiency outcome and allocative efficiency budgetary improve m designed properly where area an also is Tasmania) environmen for rights harvest forest of buyback The o in irrigators from rights water of buyback the is appl potential Another formats. pricing alternative
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