Antisocial Agents and Vickrey Auctions

13 Pages·2012·0.13 MB·English

Antisocial Agents and Vickrey Auctions
FelixBrandt and GerhardWeiß
Institutfu¨rInformatik,TechnischeUniversita¨tMu¨nchen
80290Mu¨nchen,Germany
Tel.[phone]
{brandtf,weissg}@in.tum.de
Abstract. In recent years auctions have become more and more important in
thefieldofmultiagentsystemsasusefulmechanismsforresourceallocationand
task assignment. In many cases the Vickrey (second-price sealed-bid) auction
is usedas a protocolthat prescribeshow the individual agents have to interact
in order to come to an agreement. We show that the Vickrey auction, despite
its theoretical benefits, is inappropriate if “antisocial” agents participate in the
auctionprocess.Morespecifically,anantisocialattitudeforeconomicagentsthat
makesreducingtheprofitofcompetitorstheirmaingoalbesidesmaximizingtheir
ownprofitisintroduced.Underthisnovelcondition,agentsneedtodeviatefrom
thedominanttruth-tellingstrategy.Thispaperpresentsastrategyforbiddersin
repeatedVickreyauctionswhoareintendingtoinflictlossestofellowagentsin
ordertobemoresuccessful,notinabsolutemeasures,butrelativelytothegroup
ofbidders.Thestrategyisevaluatedinasimpletaskallocationscenario.
1 Introduction
The area of multiagent systems [20], which is concerned with systems composed of
technicalentitiescalled agentsthatinteractandinsomesensecanbesaid tobe intel-
ligentandautonomous,hasachievedsteadilygrowinginterestinthepastdecade.Two
key problems to be addressed in this area are automated resource allocation and task
assignmentamongtheindividualagents.Asasolutiontotheseproblemsithasbecome
commonpracticetoapplywellknownresultsandinsightsfromauctiontheory(e.g.,[9,
10])andwellunderstoodauctionprotocolsliketheEnglishauction,theDutchauction,
andtheVickreyauction.Amongthedifferentprotocols,theVickreyauction[18](also
knownassecond-pricesealed-bidauction)hasreceivedparticularattentionwithinthe
multiagentcommunityandhasbeenappliedinavarietyofcontextslikee-commerce,
operating systems, and computer networks (e.g., [17,6,5,19,8,4]). The Vickrey auc-
tionisfavoredbecauseofthreemaincharacteristics:
– itrequireslowbandwidthandtimeconsumption
– itpossessesadominantstrategy,namely,tobidone’struevaluation(seeSection2)
– itisasealed-bidauction(bids(expressingprivatevalues)remainsecret)
ThesecharacteristicsmaketheVickreyauctionprotocolparticularlyappealingfromthe
pointofviewofautomation.TheVickreyauction,initsoriginalformulationandasit
isusedforsellinggoodsorresourceallocation,worksasfollows:eachbiddermakesa
sealedbidexpressingtheamountheiswillingtopay,andthebidderwhosubmitsthe
highestbidwinstheauction;thepricetobepayedbythewinnerisequaltothesecond
highestbid.Intask assignmentscenariostheVickreyauctionworksexactlytheother
way round (and for that reason is often referred to as reverse Vickrey auction): each
bidderwillingtoexecuteataskmakesabidexpressingtheamounthewantstobepayed
fortaskexecution,andthebiddersubmittingthelowestbidwinstheauction;thewinner
receivesan amount equaling the second lowest bid (and his payoffthus is the second
lowestbidminushisprimecostsforexecution).Ifthereismorethanonewinningbid,
thewinnerispickedrandomlyfromthehighestbidders.Thispaperconcentratesonthe
useoftheVickreyauctionfortaskassignmentscenarios;itshouldbenoted,however,
thatallpresentedconsiderationsandresultsdoalso hold forVickreyauctioninginits
originalformulation.Itisanimportantfact,thatbidsaresealed.Biddersthatdidnotwin
an auction do not necessarily get to knowwho placed the lowest bid and particularly
howlowthisbidwas.Theonlyinformationthatisrevealedtoallbiddersistheselling
price(see [2] for a more detaileddiscussionofprivacyandsecurityissues in Vickrey
auctions).
A general assumption often made in applications of multiagent systems is that an
individualagentintendstomaximizehisprofitwithoutcaringfortheprofitsofothers.
This, however,is by no means the case in real-world settings. What can be often ob-
servedhereisthatacompany,besidesmaximizingitsownprofit,deliberativelyinflicts
lossestorivalingcompaniesbyminimizingtheirprofits.Infact,itisreal-worldpractice
that a companyaccepts a lower profit or is even willing to sell goods at a loss if this
financiallydamages a competingcompanyoratleasthelpstobind availableandgain
newcostumers.Suchacompanyobviouslyexhibitsanantisocialbehaviorandattitude
inthesensethatitconsiders(atleastforsomeperiodoftimeorinsomecases)itsabso-
luteprofitnotasimportantasitsprofitrelativetoothercompanies’profits.(Itisworth
topointoutthatan“antisocialcompany”behavesrationallygivenitsobjective,evenif
it couldbe said to act irrationallyunderthe conditionthat its objectivewereto maxi-
mizeitsabsoluteprofit.)ThispaperinvestigatestheperformanceofVickreyauctioning
inmultiagentsystemsinwhichthepresenceofantisocialagentscannotbeexcluded.
Arelatedproblemthathasbeendiscussedintheeconomicliteratureare“externalities”
betweenbidders [7]. Externalitiesdescribepreferencesofa bidder specifyingwho he
wantstowinanauction(besideshimself),e.g.,whensellingpatentsornuclearweapons.
However,ourapproachissolelyprofit-oriented.Antisocialagentsdonotcarewhobuys
a good or who is awarded a task contract. They are just interestedin decreasing their
fellows’profitstostrengthentheirownmarketposition.Thisisamajordifference.
Thepaperisstructuredasfollows.Section2explainswhyagentsaimingatamax-
imizationoftheirabsoluteprofitsshouldbidtheirtruevaluationsinVickreyauctions.
Section3formallycapturestheantisocialattitudesketchedaboveandSection4shows
itsimpactonthebiddingstrategy.Section5introducesandanalyzesanantisocialstrat-
egy for repeated Vickrey auctions. Section 6 presents experimental results that illus-
tratetheimplicationsofthisstrategy.Finally,Section7concludesthepaperwithabrief
overviewofadvantagesanddisadvantagesoftheagents’“badattitude”.
2 Thedominant Strategy
TheVickreyauctionhasadominantstrategy,whichmeansthatifanagentappliesthis
strategyhereceivesthehighestpossiblepayoff,nomatterwhichstrategiesareusedby
theotherbidders.Thedominantstrategyistobidone’struevaluationofthetask.Even
ifanagentknowsalltheotherbidsinadvance,hestilldoesbestbybiddinghisprivate
valuation.Theconditionsfortheexistenceofthedominantstrategyequilibriumarethat
the bidders are symmetric (i.e., they can not be distinguished) and have independent
valuationsofthetasks1.Thisimpliesthattaskscannotberecontracted.
GiventwoagentsAandB,theircorrespondingprivatevaluesv andv ,andtheir
a b
submittedbidsb andb ,theprofitforagentAisdefinedbythefollowingequation.
a b
b −v ifb ≤b
b a a b
profit (b ,b )= (1)
a a b
(cid:40)0 ifba ≥bb
It can easily be seen why bidding the task value is the optimal strategy. We consider
agentAandinvestigatethepossibleprofitshewould makebynotbiddinghisprivate
value. It sufficesto model only one opposing agent B representing the entire compe-
tition because A only cares whether he wins or loses. He does not draw distinctions
betweenhisfellowbidders.
IfAbidsmorethanhisprimecosts(b > v )therearethreesubcasesconditional
a a
onagentB’sbidb :
b
i) b <v <b :BwinstheauctionandreceivesmoremoneythanifAhadbidv .
b a a a
ii) v <b <b :Alosestheauction,insteadofwinningitbybiddingv .
a b a a
iii) v <b <b :Astillwinstheauction,butdoesnotgainanything,becausethetask
a a b
priceremainsb andhispayoffisstillb −v .
b b a
Ifhebidsb <v thefollowingcasesdescribetheresultingsituations:
a a
iv) b <v <b :Awins,butispaidthesameamountofmoney(b )asifhebidv .
a a b b a
v) b < b < v : A wins, but gets less money than his own prime costs, i.e. he is
a b a
losingv −b .
a b
vi) b <b <v :AstilllosesandreducesB’spayoffbyv −b .
b a a a a
Concluding, bidding anything else than v cannot yield more profit than bidding
a
the true valuation v . Obviously,this extremely simplifies the bid preparation, due to
a
theabsenceofwastefulcounter-speculation,thatisneedede.g.,infirst-priceorDutch
auctions.
3 TheAntisocial Attitude
Inmostmultiagentapplicationsitisassumedthattheobjectiveofanagent(orofateam
ofagents)istomaximizehisabsoluteprofitwithoutcaringfortheprofitsmadebythe
1Ifthebiddersdon’thavetoestimatetheirprivatevalues,thedominantequilibriumexistsinde-
pendentlyofriskneutrality[11].
other agents. However,in many real-world applications it is more realistic to assume
thatagentsmaybepresentthattrytogainasmuchmoneyaspossiblerelativetoothers
(theircompetitors).Inother words,in manyscenariositiswiseto takeinto consider-
ation the availability of “antisocial agents,” that is, agents who accept small losses if
they can inflict great losses to other agents. To make this more precise, we developa
formaldescriptionofthisantisocialattitude.Asastartingpointforthisformalization,
it appears to be reasonable to assume that an antisocial agent wants to maximize the
differencebetweenhisprofitandthegainofhiscompetitors;thismeansthat theown
profitontheonehandandtheotheragents’lossesontheotherhandareconsideredto
beofequalimportancefromthepointofviewofthisantisocialagent.Inatwo-player
scenario, this view capturesthe antisocial agent’sintention to be better than his rival.
Toachieveahigherdegreeofflexibilityindescribingandanalyzingantisocialagents,
itisusefultothinkofdifferentdegreesofanti-socialitylike“aggressiveanti-sociality”
(where itisanagent’sobjectivetoharm competitorsatanycost)and“moderateanti-
sociality”(whereanagentputssomewhatmoreemphasisonhisownprofitratherthan
the loss of other agents). These considerations lead to our formal specification of an
antisocial agent (or an agent’s antisocial attitude) as an agent who tries to maximize
theweighteddifferenceofhisownprofitandtheprofitofhiscompetitors.Precisely,an
antisocialagentiintendstomaximizehispayoff2thatisgivenby
payoff = (1−d )profit − d profit , (2)
i i i i j
j(cid:54)=i
(cid:88)
whered ∈ [0,1]isaparametercalledderogationrate.Thederogationrateiscru-
i
cialbecauseitformallycaptures,andallowstomodify,anagent’sdegreeofantisocial
behavior. It is obvious that this formula covers “regular” agents by setting d = 0. If
dishigherthan0.5,hurtingothershasgreaterprioritythanhelpingyourself.Apurely
destructiveagentisdefinedbyd=1.Wesayanagentisbalancedantisocialifd=0.5,
e.g.,hisownprofitandtheprofitofhiscompetitorsareofequalimportance.3
Pleasenotethatthedescribedantisocialattitude,whichisfeasibleinallkindsofcom-
petitivemarkets,isrationalastheagentsintendtobenefitfromtheirrivals’lossesata
laterdate(e.g.,byshorteningacompetitor’sbudgetforfuturenegotiationsorbyputting
arivalcompletelyoutofbusiness).
4 Antisocialityand Vickrey Auctions
TheimplicationsofthisformalizednotionofanantisocialattitudeontheVickreyauc-
tionisenormous. Intheremainingofthissection,theseimplicationsaretheoretically
investigated.Bycombining equations1 and2we get the (antisocial)payofffor agent
2Thepayoffforanon-antisocialagentissimplyhisprofit.
3Whenconsideringothertypesofauctionslikeuniform-priceauctions,itmightbewisetoscale
thesuminequation2.InaVickreyauction,however,allprofitsbutonearezerowhichmakes
thescalingredundant.
Aasafunctionofthebidsb andb .
a b
(1−d )(b −v ) ifb ≤b
a b a a b
payoff (b ,b )= (3)
a a b
(cid:40)−da(ba−vb) ifba ≥bb
Considercasevi)ofSection2.AgentAis notableto effectivelywin theauction,
butthepriceagentBreceivescompletelydependsonA’sbid.So,ifAcarefullyadjusts
hisbiddownwards,heiscapableofreducingB’sprofit.SupposingthatAknowsB’s
private value v , his optimal strategy would be to bid v + (cid:15) (see Figure 1), which
b b
reducesB’sprofittotheabsoluteminimumof(cid:15).
v v
b a
Fig.1.AreducesB’sprofittoaminimum
However,ifBknowsA’sprimecostsv aswellandhealsoprefersinflictinggreat
a
losses to A over gaining small profits (i.e., his derogation rate is positive) he can bid
v +2(cid:15),rejectingapossiblegainof(cid:15)andmakingAlosev −v +2(cid:15).
b a b
Asaresult,ifA’sderogationrated is0.5,Ashouldonlybidv +va−vb +(cid:15)tobesafe
a b 2
frombeingoverbidbyB (seeFigure2).IfB stilltopsA’sbid,heisrenouncingmore
moneythanAloses.
v v
b a
Fig.2.Careful,aggressivebidding
Ifd =0.5aswell,B’sbeststrategyistobidv + va−vb.
b b 2
Concluding, the following bidding strategy seems to be “safe” for an antisocial
agent i (still under the unrealistic assumption that an agent knows private values of
otheragents:v isthelowestprivatevalue,v thesecondlowest):
1 2
v −d (v −v )+(cid:15) ifv >v
b = i i i 1 i 1 (4)
i
(cid:40)vi+di(v2−vi) else
Itispossibletoomitthemargin(cid:15).Weincludedittoavoidrandomizedexperimental
resultsthatoccurwhentwoormorebidderssharethesameminimumbid.However,we
willusethestrict(cid:15)-lessversioninthetheoremsstatedbelow.
Theorem:InVickreyauctionswithbalancedantisocialbidders(∀i:d =0.5),the
i
strategydefinedbyequation4isinNashequilibrium.
Proof: The assumption states that under the supposition that all agents apply
thisstrategy,thereisnoreasonforasingleagenttodeviatefromit.
WeconsiderthepayoffofagentA.Ifv isthelowestprivatevalue,letBbethesecond
a
lowest bidder. Otherwise, B represents the lowest bidder. It suffices to consider B as
A cannot differentiate between the individual bidders and the Vickrey auction has a
solevictor.Thereareonlytwopossibilitiesforanantisocialagenttocauseharm.The
cheapestprovidercanhurtthesecondhighestbidderandaninferioragentcanreduce
theprofitofthelowestbidder.AssumeBappliestheantisocialstrategydefinedin4.
v − 1(v −v ) ifv ≤v v +v
b = b 2 b a a b = a b
b (cid:40)vb+ 21(va−vb) ifva ≥vb 2
v +v vb−va ifb ≤ va+vb
⇒payoff (b ,b )=payoff b , a b = 4 a 2
a a b a(cid:18) a 2 (cid:19) (cid:40)−ba−2vb ifba ≥ va+2vb
v < v v > v
a b a b
payoff payoff
a a
v −v
b a v +v
a b
4
2
b b
a a
va va +vb vb vb −va vb va
2 4
v +v v −v v +v
a b b a a b
maxpayoff b , = ⇒b ≤
a a a
ba (cid:18) 2 (cid:19) 4 2
Concluding, if A bids less than va+vb, he only receives equal payoff; if he bids
2
more,hispayoffisdiminishing.(cid:3)
Theorem:Thestrategydefinedbyequation4isinMaximinequilibriumindepen-
dentlyofthederogationratesofthebidders.
Proof:It isclaimedthatthestrategyisanoptimalstrategyto reducethepos-
sible losses that occur in worst case encounters. “Worst-case” means that the other
bidders(representedbyasingleagentBagain)trytoreduceagentA’spayoffasmuch
aspossible.
(1−d )(b −v )ifb ≥b
minpayoff (b ,b )=min a b a b a
bb a a b bb (cid:18) −da(ba−vb) ifbb ≤ba(cid:19)
=min{(1−d )(b −v ),−d (b −v )}
a a a a a b
f(ba) g(ba)
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
f yieldstheminimumprofitifAwinsandg yieldstheminimumprofitifhelosesthe
auction.Inthefollowing,weconsiderthemaximumoftheseminima.
maxminpayoff (b ,b )=maxmin{f(b ),g(b )}
a a b a a
ba bb ba
v < v v > v
a b a b
min payoff min payoff
a a
f
g vb va −d a (va −v b ) va
b b
a a
v v +d (v −v ) v g
a a a b a b
f
Due to the fact that f is increasing and g is decreasing, the Maximin equilibrium
pointcanbecomputedbysettingf(b )=g(b ).
a a
f(b )=g(b )⇔(1−d )(b −v )=−d (b −v )
a a a a a a a b
⇔b −v −d b +d v =−d b +d v
a a a a a a a a a b
⇔b =v +d (v −v ) (cid:3)
a a a b a
5 Antisocial BiddinginRepeated Auctions
OnthebasisofthetheoreticalfoundationsofthepreviousSection,wenowdevelopan
antisocialbiddingstrategythatcanactuallybeusedinrealisticenvironments.
Inthegeneralcase,anagentdoesnotknowtheprivatevalueofotherbidders.However,
in principle an agent has several possibilities to figure out that value, for instance by
means of espionage, careful estimation (e.g., by assuming a uniform distribution of
private values), colluding with the auctioneer (e.g., bribing him) or by learning from
previousauctions.Thispaperdealswiththelattertechnique.
5.1 RevealingPrivateValuesbyUnderbidding
Weconsidertheauctioningofafixednumberoftasks,thatrepeatsforseveralrounds.
Now,supposeabalancedantisocialagentlosesanauctioninthefirstround.Whenthe
sametaskisauctionedforthesecondtime,hebidszero.Asaconsequence,hewinsthe
auction4,andreceivesanamountequalingthesecondlowestbid,whichisthe private
valueofthecheapestagent(supposingthisagentappliedthedominantstrategy).Thus,
heisabletofigureouttheneededprivatevalueandcanplacehisnextbidrightinthe
middle between the two private values. Using this technique, he loses the difference
betweenbothvaluesonce,butcansafelycutoff50%ofthecompetitor’sprofitforall
followingauctionrounds.Ifthetotalnumberofroundsishighenough,theinvestment
pays.
Inascenariowhereallotheragentsdefinitelyfollowthedominantbiddingstrategyand
no counter-speculation is needed, an effective, antisocial bidding strategy looks like
this:
1. Bid0(p=receivedprice)
v +d (p−v ) ifv <p
i i i i
2. Bid
(cid:40)vi−di(vi−p)+(cid:15) else
Bidding zero is elegant but dangerous, especially if more than one agent is applying
thisstrategy.Inthiscase, oneof thezero-bidding agentswinsthe auction, butis paid
nomoneyatall(becausethesecondlowestbidiszeroaswell),thusproducingahuge
deficit.
5.2 StepbyStepApproach
It’ssafer,butpossiblynotasefficient,toreduceabidfromroundtoroundbyasmall
marginsuntilthebestagent’sbidisreached.Figure3displaysthemodifiedstrategy.If
thestepsizesequalstheprivatevalue(s = v),thisalgorithmemulatestheaggressive
zero-biddingstrategy.Thealgorithmworkssomewhatstableindynamicenvironments
whereagentscanvanishandnewonesappearfromtimetotime.However,thestrategy
isnotperfect,e.g.,iftwobalancedantisocialagentsapplythisstrategy,themoreexpen-
siveagentisonlyabletoreducethewinningagent’sprofitby25%becauseheisusually
4unlesssomeotheragentbidszeroaswell.
start here
bid v
won lost
(p=price)
lost
bid v+d(p−v) bid last_bid−s
won
won
(p=price) lost
bid v−d(v−p)+e
p<v
won (p=price)
lost
p>=v
Fig.3.AntisocialstrategyforrepeatedVickreyauctions
notabletofigureouttherealprivatevalueofthecheaperagentintime.Nevertheless,
thisisstilltolerable,becauseinagroupofantisocialagentsabiddernevertakesmore
riskthanhisderogationrateprescribes.
Generally, a careful agent should use a small step size s in order to be safe that
the competitor already suffered huge losses before he makes negative profit himself.
Areasonablesettingofsdependsonthenumberofrounds,thedistributionofprivate
valuesandhisderogationrate(anupperboundforsisspecifiedin[1]).
5.3 LeveledCommitmentContracting
Ifthetaskexecutioncontractsarenotbindingandcanbebreachedbypayingapenalty
(leveledcommitmentcontracting[15,16,3]),theunavoidablelossanagentproducesby
underbiddingthecheapestcompetitorcanbereducedbybreakingthenegativecontract.
Duetothefactthattheonlyreasonforclosingthatdealistofigureouttheprivatevalue
of another agent, the agent has no incentive to really accomplish the task. Therefore,
a contractee will break the contract if the loss he makes by accepting the contract is
greaterthanthepenaltyhepaysbybreakingthedeal.Supposingthecommondefinition
ofapenaltyasafractionofthecontractvalue,agentiisbetteroffbreachingthecontract
if
v
p≤ i (5)
pr+1
withpbeingtheactualtaskpriceandpr ∈ [0;1]thepenaltyrate.Togiveanexample,
undertheassumptionthatpr =0.25,anagentshouldbreakacontractifthetaskprice
is less or equal than 4 of his private value. When the distribution of prime costs is
5
uniform,thisistruein80%ofallpossiblecases.
6 ExperimentalResults
Theexperimentalsettinginvestigatedinthispaperissimilartotheoneusedin[3].Due
toreasonsoflimitedspace,wecanonlypresentsomeoftheexperimentsweconducted.
Pleasesee[1]formoredetailedresultsincludingarandomizedcosttable.
Thereisanumberofbuyersorcontractees(CE )whoarewillingtoexecutetasks.
i
Contracteesassociateprimecostswithtaskexecutionandareinterestedintaskswhose
pricesarehigherthantheirowncosts.Allpricesandbidsareintegervalues((cid:15)=1).
Wheneverthesellingofataskisannounced,eachinterestedcontracteecalculatesand
submitsonesealedbid.Thecontracteewhosubmittedthelowestbidisdeclaredasthe
winneroftheauction,andthesecondlowestbidistakenasthepriceoftheannounced
task;thecontracteeispaidthispriceandexecutesthetask.Iftherearetwoormoreequal
winningbids,thewinnerispickedrandomly.Asacontracteewantstoearnmoneyfor
handlingtasks,hisprivatevalueofataskishisprimecostsplus(cid:15).Itisassumedthateach
contracteecanexecuteasmanytasksashewants duringoneround(fullcommitment
contracting).
Table1containstheprimecostsofthreecontracteeswithexactlyidenticalabilities.
Eachcontracteehasonetask,thathecanhandleforthecheapestprice.Ifallthreetruly
bidtheirprivatevaluesfor100rounds,eachonewouldgain((50−30)+1)×100=
2100.Figure 4 showsthe profits accumulated by the contractees in 100 rounds. CE
1
Task1Task2Task3
CE1 70 50 30
CE2 50 30 70
CE3 30 70 50
Table1.Faircosttable
andCE applythedominantstrategyandbidtheirprimecostsplusone.CE ,however,
2 3
isantisocialandtriestoharmhiscompetitorsbyreducingtheirprofitstoaminimum.
AsCE istheonlyantisocialagentandbecausehisderogationrateis1,hecouldusea
3
verylargestepsize,e.g.,s =v .Wechoseacarefulstepsizesetting(s =(cid:15))fortwo
3 3 3
reasons.Firstofall,CE maynotknowheistheonlyantisocialbidder,andsecondly,
3
thissettingsuperiorlyvisualizeshowtheantisocialstrategyworks.
Despitethedominantstrategy,CE andCE areincapableofgainingtheirregular
1 2
profit(2100).CE outperformshisrivalsbylosingonly60.Thesummedupprofitof
3
theentiregroupofcontracteesisreducedbymorethan50%byasingleagentusingan
aggressivestrategy.ThisemphasizestheparticularvulnerabilityofVickreyauctionsto
“irrational” bidding. There is no counter-strategyavailable for CE or CE . As they
1 2
apply a dominant strategy, they already make the highest possible payoff (which is
almostnil)accordingtotheirdefinitionofpayoff.However,iftheydecidedtobecome
antisocialaswell,theywouldbeabletoincreasetheirpayoff.Incertainscenarios,this