Undominated Groves Mechanisms

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by  Mingyu Guo

JournalofArtificialIntelligenceResearch?(????)?-?? Submitted?/??;published?/??
Undominated Groves Mechanisms
MingyuGuo [email protected]
UniversityofLiverpool,UK
EvangelosMarkakis [email protected]
AthensUniversityofEconomicsandBusiness,Greece
KrzysztofR.Apt [email protected]
3 CWIandUniversityofAmsterdam,theNetherlands
1 VincentConitzer [email protected]
0
DukeUniversity,USA
2

n
a
J Abstract

4
The family of Groves mechanisms, which includes the well-known VCG mechanism (also

knownastheClarkemechanism),isafamilyofefficientandstrategy-proofmechanisms.Unfortu-
]
T nately,theGrovesmechanismsaregenerallynotbudgetbalanced.Thatis,undersuchmechanisms,
G paymentsmayflowintooroutofthesystemoftheagents,resultingindeficitsorreducedutilities
. fortheagents. We considerthefollowingproblem: withinthefamilyofGrovesmechanisms,we
s
c wanttoidentifymechanismsthatgivetheagentsthehighestutilities,undertheconstraintthatthese
[ mechanismsmustneverincurdeficits.

2 We adopt a prior-free approach. We introduce two general measures for comparing mecha-
v nismsinprior-freesettings. Wesaythatanon-deficitGrovesmechanismM individuallydominates
9 anothernon-deficitGrovesmechanismM′ifforeverytypeprofile,everyagent’sutilityunderM is
0 nolessthanthatunderM′,andthisholdswithstrictinequalityforatleastonetypeprofileandone
8
agent. Wesaythatanon-deficitGrovesmechanismM collectivelydominatesanothernon-deficit
1
. Grovesmechanism M′ if forevery type profile, the agents’ total utility under M is no less than
3 thatunderM′,andthisholdswithstrictinequalityforatleastonetypeprofile. Theabovedefini-
0
tionsinducetwopartialordersonnon-deficitGrovesmechanisms.Westudythemaximalelements
2
1 correspondingtothesetwopartialorders,whichwecalltheindividuallyundominatedmechanisms
: andthecollectivelyundominatedmechanisms,respectively.
v
i
X
1. Introduction
r
a
Mechanism design is often employed for coordinating group decision making among agents. Of-
ten, such mechanisms impose payments that agents have to pay to a central authority. Although
maximizing revenue is a desirable objective in many settings (for example, if the mechanism is an
auction designed bytheseller), itisnotdesirable insituations wherenoentity isprofiting fromthe
payments. Some examples include public project problems as well as certain resource allocation
problems without a seller (e.g., the right to use a shared good in a given time slot, or the assign-
mentoftake-off slotsamongairlinecompanies). Insuchcases, wewouldliketohavemechanisms
thatminimizepayments(or,evenbetter,achievebudgetbalance),whilemaintainingotherdesirable
properties,suchasbeingefficient,strategy-proofandnon-deficit(i.e.,themechanismdoesnotneed
tobefundedbyanexternal source).
.Someofthisworkappearedinpreliminaryformin(Apt,Conitzer,Guo,&Markakis,2008)and(Guo&Conitzer,
2008b).
(cid:13)c????AIAccessFoundation.Allrightsreserved.
GUO, MARKAKIS, APT,CONITZER
The family of Groves mechanisms, which includes the well-known VCG mechanism (also
known asthe Clarke mechanism), isa family ofefficient and strategy-proof mechanisms. Inmany
sufficientlygeneralsettings,includingthesettingsthatwewillstudyinthispaper,theGrovesmech-
anisms are the only efficient and strategy-proof mechanisms (Holmstro¨m, 1979). Unfortunately
though, the Groves mechanisms are generally not budget balanced. That is, under such mecha-
nisms, payments may flow into or out of the system of the agents, resulting in deficits or reduced
utilities for the agents. Motivated by this we consider in this paper the following problem: within
thefamilyofGrovesmechanisms, wewanttoidentify mechanisms thatgivetheagents thehighest
utilities, undertheconstraint thatthesemechanismsneverincurdeficits.1
Weadoptaprior-free approach, whereeachagent iknowsonly hisownvaluation v ,andthere
i
isnopriorprobabilitydistributionovertheotheragents’ values. Weintroducetwonaturalmeasures
forcomparingmechanismsinprior-freesettings. Givenaperformanceindicator, wesaythatmech-
anismM individuallydominatesmechanismM′ifforeverytypeprofileoftheagents, M performs
noworsethanM′fromtheperspectiveofeachindividualagent,andthisholdswithstrictinequality
foratleastonetypeprofileandoneagent. WesaythatmechanismM collectively dominatesmech-
anism M′ if for every type profile, M performs no worse than M′ from the perspective of the set
ofagents asawhole, andthisholdswithstrict inequality foratleastonetypeprofile. Inthispaper,
wefocus on maximizing the agents’ utilities. Giventhis specific performance indicator, individual
and collective dominance are determined bycomparing either individual utilities orthe sum of the
agents’utilities, respectively.
Theabovedefinitionsinducetwopartialordersonnon-deficitGrovesmechanisms. Ourgoalin
this work is to identify and study the maximal elements corresponding to these two partial orders,
which we call the individually undominated (non-deficit Groves) mechanisms and the collectively
undominated (non-deficit Groves) mechanisms, respectively. It should be noted that the partial
orderswefocusonmaybedifferentfromthepartialordersinducedbyotherperformanceindicators,
e.g.,ifthecriterion istherevenueextractedfromtheagents.
1.1 Structureofthepaper
The presentation of our results is structured as follows: In Sections 2 and 3, we formally define
the notions of individual and collective dominance, as well as the family of Groves mechanisms,
andthenprovidesomebasicobservations. Wealsoestablish somegeneralpropertiesofanonymous
Grovesmechanisms whichweuselateron,andwhichmaybeofindependent interest. Wethenbe-
ginourstudyofindividualdominanceinSection4,wherewegiveacharacterizationofindividually
undominated mechanisms. Wealso propose twotechniques for transforming anygiven non-deficit
Grovesmechanismintoonethatisindividually undominated.
In Sections 5 and 6 we study the question of finding collectively undominated mechanisms in
twosettings. Thefirst(Section 5) isauctions of multiple identical units withunit-demand bidders.
In this setting, the VCG mechanism is collectively dominated by other non-deficit Groves mecha-
nisms,suchastheBailey-Cavallo mechanism (Bailey,1997;Cavallo, 2006). Weobtainacomplete
characterization of collectively undominated mechanisms that are anonymous and linear (meaning
that the redistribution is a linear function of the ordered type profile; see Section 5 for the defini-
1.Theagents’utilitiesmaybefurtherincreasedifwealsoconsidermechanismsoutsideoftheGrovesfamily(Guo&
Conitzer, 2008a; deClippel, Naroditskiy,&Greenwald, 2009; Faltings, 2005; Guo, Naroditskiy, Conitzer, Green-
wald,&Jennings,2011),butinthispaperwetakeefficiencyasahardconstraint.
2
UNDOMINATEDGROVESMECHANISMS
tion). In particular, we show that the collectively undominated mechanisms that are anonymous
and linear are exactly the Optimal-in-Expectation Linear (OEL)redistribution mechanisms, which
includetheBailey-Cavallomechanismandwereintroducedin(Guo&Conitzer,2010). Thesecond
setting (Section 6) is public project problems, where a set of agents must decide on whether and
how to finance a project (e.g., building a bridge). We show that in the case where the agents have
identical participation costs, the VCG mechanism is collectively undominated. On the other hand,
whentheparticipation costscanbedifferentacrossagents,thereexistmechanismsthatcollectively
dominate VCG. We finally show that when the participation costs are different across agents, the
VCGmechanismremainscollectively undominated amongallpay-onlymechanisms.
1.2 Relatedwork
How to efficiently allocate resources among a group of competing agents is a well-studied topic
in economics literature. For example, the famous Myerson-Satterthwaite Theorem (Myerson &
Satterthwaite, 1983) rules out the existence of efficient, Bayes-Nash incentive compatible, budget-
balanced, and individually rational mechanisms. Cramton et al. (1987) characterized the Bayes-
Nash incentive compatible and individually rational mechanisms for dissolving a partnership, and
gavethenecessary andsufficientcondition forthepossibility ofdissolving partnership efficiently.
Themaindifferencebetweenthesepapersandoursisthatweadoptaprior-freeapproach. That
is, we do not assume that we know the prior distribution of the agents’ valuations. As a result of
this, our notion of truthfulness is strategy-proofness, which is stronger than Bayes-Nash incentive
compatibility. Inmanysufficiently general settings, including thesettings thatwewillstudy inthis
paper, the Groves mechanisms are the only efficient and strategy-proof mechanisms (Holmstro¨m,
1979). That is, the search of undominated Groves mechanisms is, in many settings, the search of
efficient,strategy-proof, andnon-deficit mechanismsthat areclosesttobudget-balance.
Recently,therehasbeenaseriesofworksonVCGredistribution mechanisms,whicharemech-
anisms that make social decisions according to the efficient and strategy-proof VCG mechanism,
and then redistribute some of the VCG payments back to the agents, under certain constraints,
such as that an agent’s redistribution should be independent of his own type (therefore ensuring
strategy-proofness), and that the total redistribution should never exceed the total VCG payment
(therefore ensuring non-deficit). Actually, anynon-deficit Grovesmechanism can beinterpreted as
suchaVCG-basedredistribution mechanism, andany(non-deficit) VCGredistribution mechanism
corresponds toanon-deficitGrovesmechanism (moredetails onthisareprovidedinSection2).
One example of a redistribution mechanism is the Bailey-Cavallo (BC) mechanism (Cavallo,
2006).2 Under the BC mechanism, every agent, besides participating in the VCG mechanism,
also receives 1 times the minimal VCG revenue that could have been obtained by changing this
n
agent’s ownbid. Insomesettings (e.g.,asingle-item auction), theBCmechanism cansuccessfully
redistribute a large portion of the VCG payments back to the agents. That is, in such settings, the
BCmechanism bothindividually andcollectively dominates theVCGmechanism.
Guo and Conitzer (Guo & Conitzer, 2009) proposed another VCG redistribution mechanism
calledtheworst-caseoptimal(WCO)redistributionmechanism,inthesettingofmulti-unitauctions
with nonincreasing marginal values. WCO is optimal in terms of the fraction of total VCG pay-
2.In settings that are revenue monotonic, the Cavallo mechanism (Cavallo, 2006) coincides with a mechanism dis-
coveredearlierbyBailey(1997). TheBailey-Cavallomechanismforasingle-itemauctionwasalsoindependently
discoveredin(Porter,Shoham,&Tennenholtz,2004).
3
GUO, MARKAKIS, APT,CONITZER
ment redistributed in the worst case.3 Moulin (Moulin, 2009) independently derived WCO under
a slightly different worst-case optimality notion (in the more restrictive setting of multi-unit auc-
tions with unit demand only). Guo and Conitzer (Guo & Conitzer, 2010) also proposed a family
of VCG redistribution mechanisms that aim to maximize the expected amount of VCG payment
redistributed, in the setting of multi-unit auctions with unit demand. The members of this family
arecalledtheOptimal-in-Expectation Linear(OEL)redistribution mechanisms.
Finally, the paper that is the closest to what we study here is an early work by Moulin on
collectivelyundominatednon-deficitGrovesmechanisms(Moulin,1986). Itdealswiththeproblem
of selecting an efficient public decision out of finitely many costless alternatives.4 Each agent
submits to the central authority his utility for each alternative. Subsequently, the central authority
makes a decision that maximizes the social welfare. Moulin showed (Lemma 2 of Moulin, 1986)
that the VCG mechanism is collectively undominated in the above setting. This result generalizes
an earlier result for the case of two public decisions by Laffont and Maskin (Laffont & Maskin,
1997).
2. Preliminaries
We first briefly review payment-based mechanisms (see, e.g., Mas-Colell, Whinston, & Green,
1995).
2.1 Payment-basedmechanisms
Assume that there is a set of possible outcomes or decisions D, a set {1,...,n} of agents where
n ≥ 2,andforeachagenti,asetoftypesΘ andan(initial)utilityfunctionv :D×Θ →R. Let
i i i
Θ := Θ ×···×Θ .
1 n
In a (direct revelation) mechanism, each agent reports a type θ ∈ Θ and based on this, the
i i
mechanism selects an outcome and a payment to be made by every agent. Hence a mechanism is
given by a pair of functions (f,t), where f is the decision function and t is the payment function
thatdetermines theagents’payments, i.e.,f : Θ→D,andt : Θ→Rn.
Weputt (θ) := (t(θ)) ,i.e.,thefunction t computes thepayment ofagenti. Foreachvector
i i i
θ of announced types, if t (θ) ≥ 0, agent i pays t (θ), and if t (θ) < 0, he receives |t (θ)|. When
i i i i
thetruetypeofagentiisθ andhisannounced typeisθ′,hisfinalutilityfunctionisdefinedby
i i
u ((f,t)(θ′,θ ),θ ) := v (f(θ′,θ ),θ )−t (θ′,θ ),
i i −i i i i −i i i i −i
whereθ isthevectoroftypesannounced bytheotheragents.
−i
2.2 Propertiesofpayment-basedmechanisms
Wesaythatapayment-based mechanism (f,t)is
• efficientifforallθ ∈ Θandd ∈D, n v (f(θ),θ ) ≥ n v (d,θ ),
i=1 i i i=1 i i
P P
• budget-balanced if n t (θ)= 0forallθ ∈ Θ,
i=1 i
P
3.Thisnotion ofworst-caseoptimalitywasalsostudiedformoregeneral settingsin(Gujar&Narahari, 2011; Guo,
2011,2012).
4.Inourpublicprojectmodel,thereisacostassociatedwithbuildingtheproject.
4
UNDOMINATEDGROVESMECHANISMS
• non-deficitif n t (θ) ≥ 0forall θ,i.e.,themechanism doesnotneedtobefunded byan
i=1 i
externalsourceP,
• pay-onlyift (θ) ≥ 0forallθ andalli∈ {1,...,n},
i
• strategy-proof ifforallθ,i∈ {1,...,n},andθ′,
i
u ((f,t)(θ ,θ ),θ ) ≥ u ((f,t)(θ′,θ ),θ ),
i i −i i i i −i i
i.e.,foreachagent i,reporting afalsetype,here θ′,isnotprofitable.
i
2.3 Individualandcollective dominance
We consider prior-free settings, where each agent i knows only his own function v , and there is
i
nobelieforpriorprobability distribution regardingtheotheragents’initialutilities. Payment-based
mechanisms can naturally be compared in terms of either the effect on each individual agent or
the global effect on the whole set of agents. We therefore introduce two measures for comparing
such mechanisms. Given a performance indicator5, we say that mechanism (f′,t′) individually
dominates mechanism (f,t) if for every type profile, (f′,t′) performs no worse than (f,t) from
the perspective of every agent, and this holds with strict inequality foratleast one type profile and
oneagent. Wesaythatmechanism(f′,t′)collectivelydominatesmechanism(f,t)ifforeverytype
profile, (f′,t′) performs no worse than (f,t) from the perspective of the whole agent system, and
this holds withstrict inequality foratleast one type profile. Inthispaper, wefocus on maximizing
theagents’utilities. Giventhisspecificperformance indicator, individual andcollectivedominance
arecapturedbythefollowingdefinitions:
Definition2.1 Given two payment-based mechanisms (f,t) and (f′,t′), we say that (f′,t′) indi-
viduallydominates(f,t)if
• forallθ ∈ Θandalli ∈{1,...,n},u ((f,t)(θ),θ )≤ u ((f′,t′)(θ),θ ),
i i i i
• forsomeθ ∈ Θandsomei ∈{1,...,n},u ((f,t)(θ),θ )< u ((f′,t′)(θ),θ ).
i i i i
Definition2.2 Giventwopayment-basedmechanisms(f,t)and(f′,t′),wesaythat(f′,t′)collec-
tivelydominates(f,t)if
• forallθ ∈ Θ, n u ((f,t)(θ),θ ) ≤ n u ((f′,t′)(θ),θ ),
i=1 i i i=1 i i
P P
• forsomeθ ∈ Θ, n u ((f,t)(θ),θ ) < n u ((f′,t′)(θ),θ ).
i=1 i i i=1 i i
P P
Fortwopayment-based mechanisms (f,t)and(f′,t′),clearlyif(f′,t′)individually dominates
(f,t), then it also collectively dominates (f,t). Theorem 3.4 shows that the reverse implication
howeverdoes not need tohold, even ifwelimitourselves tospecial types of mechanisms. That is,
thefactthat(f′,t′)collectively dominates(f,t)doesnotimplythat(f′,t′)individually dominates
(f,t).
5.Byaperformanceindicatorwemeanafunctionofthemechanism’soutcomethatservesasameasureforcomparing
mechanisms. E.g.,itcanbethefinalutilityofanagent, oranarbitraryfunctionofit,orafunctionoftheagent’s
paymentoranyotherfunctionthatdependsonthedecisionruleandthepaymentruleofthemechanism.
5
GUO, MARKAKIS, APT,CONITZER
Given a set Z of payment-based mechanisms, individual and collective dominance induce two
partial orders on Z, and we are interested in studying the maximal elements with respect to these
partialorders. Amaximalelementwithrespecttothefirstpartialorderwillbecalledanindividually
undominated mechanism, i.e., it is a mechanism that is not individually dominated by any other
mechanism in Z. A maximal element for the second partial order will be called a collectively
undominated mechanism, i.e., it is a mechanism that is not collectively dominated by any other
mechanism in Z. The maximal elements with respect to the two partial orders may differ and
in particular, the notion of collectively undominated mechanisms is generally a stronger notion.
Clearly, if (f′,t′) ∈ Z is collectively undominated, then it is also individually undominated. The
reversemaynotbetrue,examplesofwhichareprovidedinSection4.2.
If we focus on the same decision function f, then individual and collective dominance are
strictly due to the difference ofthe payment functions. Hence, (f,t′)individually dominates (f,t)
(orsimplyt′ individually dominatest)ifandonlyif
• forallθ ∈ Θandalli ∈{1,...,n},t (θ)≥ t′(θ),and
i i
• forsomeθ ∈ Θandsomei ∈{1,...,n},t (θ)> t′(θ),
i i
andt′ collectively dominates tif
• forallθ ∈ Θ, n t (θ)≥ n t′(θ),and
i=1 i i=1 i
P P
• forsomeθ ∈ Θ, n t (θ)> n t′(θ).
i=1 i i=1 i
P P
We now define two transformations on payment-based mechanisms originating from the same
decision function. Both transformations build upon the surplus-guarantee concept (Cavallo, 2006)
forthespecificcaseoftheVCGmechanism6.
Consider a payment-based mechanism (f,t). Given θ = (θ ,...,θ ), let T(θ) be the total
1 n
amountofpayments, i.e.,T(θ):= n t (θ). Foreachi ∈ {1,...,n}let
i=1 i
P
SBCGC(θ ) := inf T(θ′,θ ).
i −i i −i
θi′∈Θi
Inotherwords, SBCGC(θ )isthesurplusguarantee independent ofthereportofagent i. Wethen
i −i
definethepayment-based mechanism (f,tBCGC)bysettingfori∈ {1,...,n}
SBCGC(θ )
tBCGC(θ):= t (θ)− i −i .
i i n
Also,forafixedagent j,wedefinethepayment-based mechanism (f,tBCGC(j))bysetting for
i ∈{1,...,n}
tBCGC(j)(θ):= ti(θ)−SiBCGC(θ−i) ifi = j
i (cid:26) ti(θ) ifi 6= j
6.Thefirsttransformationwasoriginallydefinedin(Bailey,1997)and(Cavallo,2006)forthespecificcaseoftheVCG
mechanismandin(Guo&Conitzer,2008b)fornon-deficitGrovesmechanisms.WecallittheBCGCtransformation
aftertheauthorsofthesepapers(Bailey,Cavallo,Guo,Conitzer).
6
UNDOMINATEDGROVESMECHANISMS
After the first transformation (from (f,t) to (f,tBCGC)), every agent receives an additional7
amount of 1 times the surplus guarantee independent of his own type. During the second trans-
n
formation (from (f,t) to (f,tBCGC(j)), agent j is chosen to be the only agent who receives an
additional amount. This additional amount equals the entirety of the surplus guarantee indepen-
dent of j’s owntype. Forboth transformations the agents’ additional payments are independent of
their own types, thus the strategy-proofness is maintained: if (f,t) is strategy-proof, then so are
(f,tBCGC)and(f,tBCGC(j))forallj.
Thefollowingobservations generalize someoftheresultsof(Bailey,1997;Cavallo,2006).
Proposition 2.3
(i) Eachpayment-based mechanismoftheformtBCGC isnon-deficit.
(ii) If t is non-deficit, then either t and tBCGC coincide or tBCGC individually (and hence also
collectively) dominates t.
Proof. (i)Forallθ andi ∈{1,...,n}wehaveT(θ)≥ SBCGC(θ ),so
i −i
n n SBCGC(θ )
TBCGC(θ) = tBCGC(θ)= T(θ)− i −i
i n
Xi=1 Xi=1
n T(θ)−SBCGC(θ )
= i −i ≥ 0.
n
Xi=1
(ii) If t is non-deficit, then for all θ and all i ∈ {1,...,n} we have SBCGC(θ ) ≥ 0, and hence
i −i
tBCGC(θ) ≤ t (θ). 2
i i
ThesameclaimsholdfortBCGC(j) forj ∈ {1,...,n},withequally simpleproofs.
3. Grovesmechanisms
WefirstbrieflyreviewGrovesmechanisms.
3.1 Preliminaries
RecallthataGrovesmechanism(Groves,1973)isapayment-based mechanism(f,t)suchthatthe
followinghold8:
• f(θ) ∈ argmax n v (d,θ ), i.e., the chosen outcome maximizes the allocation wel-
d∈D i=1 i i
fare(theagents’totalPvaluation),
• t :Θ→Risdefinedbyt (θ):= h (θ )−g (θ),where
i i i −i i
• g (θ):= v (f(θ),θ ),
i j6=i j j
P
• h : Θ →Risanarbitrary function.
i −i
7.Receivinganadditionalpositiveamountmeanspayinglessandreceivinganadditionalnegativeamountmeanspaying
more.
8.HereandbelowP isashorthandforthesummationoverallj ∈{1,...,n}, j 6=i.
j6=i
7
GUO, MARKAKIS, APT,CONITZER
So g (θ) represents the allocation welfare from the decision f(θ)with agent i ignored. Recall
i
nowthefollowingcrucialresult(see, e.g.,Mas-Colelletal.,1995).
GrovesTheorem(Groves, 1973)EveryGrovesmechanism isefficientandstrategy-proof.
Forseveraldecisionproblems,theonlyefficientandstrategy-proofpayment-basedmechanisms
areGrovesmechanisms. Thisisimpliedbyageneralresultof(Holmstro¨m,1979),whichcoversthe
two domains that we consider in Sections 5 and 6, and explains our focus on Groves mechanisms.
Hencefromnowon,weusetheterm“mechanism”torefertoaGrovesmechanism.
Focusing on the set of non-deficit Groves mechanisms, individually (respectively, collectively)
undominated mechanisms are the mechanisms from this set that are not individually (respectively,
collectively) dominatedbyanyothernon-deficitGrovesmechanism. Asmentionedearlier, nomat-
ter which domain and which set of mechanisms we consider, collective undominance always im-
pliesindividual undominance. InSection4.2weshowtwoexamplesofsingle-item auctionscenar-
ios, wherecollective undominance isstrictly stronger thanindividual undominance, fornon-deficit
Groves mechanisms. That is, there exists an individually undominated non-deficit Groves mecha-
nismthatiscollectively dominated.
Recall thataspecial Grovesmechanism, called the VCGor Clarke mechanism (Clarke, 1971),
isobtained using9
h (θ ):= max v (d,θ ).
i −i j j
d∈D
Xj6=i
Inthiscase
t (θ):= max v (d,θ )− v (f(θ),θ ),
i j j j j
d∈D
Xj6=i Xj6=i
whichshowsthattheVCGmechanism ispay-only.
In whatfollows weintroduce aslightly different notation to describe Grovesmechanisms, that
makes the rest of our presentation more convenient. First, we denote the payment function t of
i
the VCG mechanism by VCG . Note now that each Groves mechanism (f,t) can be defined in
i
terms of the VCG mechanism by setting t (θ) := VCG (θ) − r (θ ), where r : Θ →R is
i i i −i i −i
some function of θ . Werefer then to r := (r ,...,r )as a redistribution function. Hence each
−i 1 n
Grovesmechanismcanbeidentifiedwitharedistribution functionrandcanbeviewedastheVCG
mechanism combined witharedistribution. Thatis, under rthe agents firstparticipate intheVCG
mechanism. Then,ontopofthat,agentialsoreceivesaredistribution amountequalto r (θ ). By
i −i
definition, aGrovesmechanism risnon-deficit iff n VCG (θ)≥ n r (θ )forallθ ∈ Θ.
i=1 i i=1 i −i
P P
3.2 Dominancerelations
Using the new notation above, individual and collective dominance (among non-deficit Groves
mechanisms)canbedescribed asfollows:
Definition3.1 Anon-deficitGrovesmechanismr′individuallydominatesanothernon-deficitGroves
mechanism rif
• foralliandallθ,r′(θ ) ≥ r (θ ),
i −i i −i
9.Hereandbelow,wheneverDisnotafiniteset,inordertoensurethattheconsideredmaximumexists,weassume
thatf iscontinuous,andsoisv foreachi,andalsothatthesetDandallΘ arecompactsubsetsofsomeRk.
i i
8
UNDOMINATEDGROVESMECHANISMS
• forsomeiandsomeθ,r′(θ ) > r (θ ).
i −i i −i
Definition3.2 Anon-deficitGrovesmechanismr′collectivelydominatesanothernon-deficitGroves
mechanism rif
• forallθ, r′(θ )≥ r (θ ),
i i −i i i −i
P P
• forsomeθ, r′(θ )> r (θ ).
i i −i i i −i
P P
We now consider the mechanism that results from applying the BCGC transformation to the
VCG mechanism. We refer to this as the Bailey-Cavallo mechanism or simply the BC mecha-
nism(Cavallo,2006). TheVCGmechanismischaracterizedbytheconstantredistribution function
rVCG = (0,0,...,0). AftertheBCGCtransformation, everyagent ireceivesanadditionalamount
of 1 timesthesurplusguarantee SBCGC(θ ),independent ofhisowntype. Thatis,theBCmech-
n i −i
anismisalsoaGrovesmechanism, anditsredistribution function isrBC =
(1SBCGC, 1SBCGC,..., 1SBCGC).
n 1 n 2 n n
Letθ′ := (θ ,...,θ ,θ′,θ ,...,θ ). Thenstarting fromtheVCGmechanism,wehave
1 i−1 i i+1 n
n
SiBCGC(θ−i)=θi′i∈nΘfikX=1md∈aDxXj6=kvj(d,θj′)−Xj6=kvj(f(θ′),θj′),
 
thatis,
n n
SiBCGC(θ−i)=θi′i∈nΘfikX=1md∈aDxXj6=kvj(d,θj′)−(n−1)Xk=1vk(f(θ′),θk′) (1)
 
In many settings, we have that for all θ and for all i, SBCGC(θ ) = 0, and consequently the
i −i
VCGand BCmechanisms coincide (e.g., see Proposition 6.1). Whenever they do not, by Proposi-
tion 2.3(ii), BC individually and collectively dominates VCG. This is the case for the single-item
auction, as it can be seen that there SBCGC(θ ) = [θ ] , where [θ ] is the second-highest bid
i −i −i 2 −i 2
amongbidsotherthanagent i’sownbid.
3.3 AnonymousGrovesmechanisms
Someoftheproofsofourmainresultsareobtained byarguing firstaboutaspecialclassofGroves
mechanisms,calledanonymousGrovesmechanisms. Weprovideheresomeresultsaboutthisclass
thatwewillutilizeinlatersections. Wecallafunctionf :An →B permutationindependentiffor
allpermutationsπof{1,...,n},f = f◦π. Following(Moulin,1986),wecallaGrovesmechanism
r = (r ,...,r )anonymous10 if
1 n
• alltypesetsΘ areequal,
i
• allfunctions r coincide andeachofthemispermutation independent.
i
Hence,ananonymous Grovesmechanism isuniquely determined byasinglefunction r : Θn−1 →
R.
10.Ourdefinition isslightly different than the one introduced in (Moulin, 1986) in that no conditions are put on the
utilityfunctionsandthepermutationindependencerefersonlytotheredistributionfunction.
9
GUO, MARKAKIS, APT,CONITZER
In general, the VCGmechanism isnot anonymous. Butitis anonymous when allthe type sets
areequalandalltheinitialutilityfunctions v coincide. Thisisthecaseinthetwodomainsthatwe
i
considerinlatersections.
Foranyθ ∈ Θandanypermutation π of{1,...,n}wedefineθπ ∈ Θbyletting
θiπ := θπ−1(i).
Denote by Π(k) the set of all permutations of the set {1,...,k}. Given a Groves mechanism
r := (r ,...,r )forwhich thetype set Θ isthe sameforevery agent (and equal tosomeset Θ ),
1 n i 0
weconstruct nowafunction r′ :Θn−1→R,following(Moulin, 1986),bysetting
0
n r (xπ)
r′(x) := π∈Π(n−1) j ,
P n!
Xj=1
wherexπ isdefinedanalogously toθπ.
Notethatr′ispermutationindependent, sor′isananonymousGrovesmechanism. Thefollow-
inglemma,whichcanbeofindependent interest, showsthatsomeoftheproperties of rtransferto
r′.
Lemma3.3 Consider a Groves mechanism r and the corresponding anonymous Groves mecha-
nism r′. Let VCG(θ) := n VCG (θ), and suppose that the VCG function is permutation
i=1 i
independent. Then: P
(i) Ifrisnon-deficit, soisr′.
(ii) If an anonymous Groves mechanism r0 is collectively dominated by r, then it is collectively
dominated byr′.
Proof. Forallθ ∈ Θwehave
n n n r ((θ )π)
r′(θ )= i=1 π∈Π(n−1) j=1 j −i =
i −i P P n!P
Xi=1
n r (θπ )
i=1 π∈Π(n) i −i
P P n!
wherethelastequalityholdssinceinbothtermsweaggregateoverallapplicationsofallr functions
i
toallpermutations ofn−1elementsofθ.
Lettandt′ bethepaymentfunctions ofthemechanismsrandr′,respectively. Wehave
n n
t′(θ)= VCG(θ)− r′(θ )
i i −i
Xi=1 Xi=1
andforallπ ∈ Π(n)
n n
t (θπ) = VCG(θπ)− r (θπ ).
i i −i
Xi=1 Xi=1
Hencebytheassumption aboutVCG(θ)itfollowsthat
10