General Truthfulness Characterizations Via Convex Analysis

0.19 MB·English

by  Rafael M. Frongillo

General Truthfulness Characterizations Via Convex
Analysis
Rafael M. Frongillo Ian A. Kash
UCBerkeley MSRCambridge
[email protected] [email protected]
2
1
ABSTRACT rule. Moregenerally,thisremainstrueinhigherdimensions
0
(see [1]). Note that here the restriction that S(t′) is affine
2 Wepresentamodeloftruthfulelicitation whichgeneralizes
iswithout lossof generality,sincewemayconsidertypesas
and extends both mechanisms and scoring rules. Our main
v functionsmappinganoutcometotheagent’sutilityforthat
resultisacharacterizationtheoremusingthetoolsofconvex
o outcome, and the evaluation of a type on an outcome (or a
analysis,ofwhichcharacterizationsofmechanismsandscor-
N distribution over outcomes) is affinein that type.
ingrulesrepresentspecial cases. Moreover,wedemonstrate
3 thatavarietyofexistingresultsinthemechanismdesignlit- Another special case is a scoring rule, where an agent is
asked to predict the distribution of a random variable and
1 eratureareoftensimplerandmoredirectwhenfirstphrased
givenascorebasedon theobservedrealization ofthatvari-
intermsofconvexity. Wethengeneralizeourmaintheorem
able. In this setting, typesare distributions over outcomes,
] to settings where agents report some alternate representa-
T and S(t′)(t) is the agent’s expected score for a report that
tion of their private information rather than reporting it
G directly, which gives new results about both scoring rules the distribution is t′ when he believes the distribution is t.
As an expectation, this score is linear in the agent’s type.3
. and mechanism design.
s GneitingandRaftery[13]unifiedandgeneralizedexistingre-
c sultsinthescoringrulesliteraturebycharacterizing proper
[ 1. INTRODUCTION
scoring rules in terms of convex functions (the negative of

Inthispaper,weexamineageneralmodelofinformation theBayesriskfromdecisiontheory)andtheirsubgradients.
1
v elicitation where a single agent is endowed with some type The similarities between these two models were noted by
3 t that is private information and is asked to reveal it. Af- Fiat et al. [11], who gave a construction to convert mecha-
4 ter doing so, he receives a score that depends on both his nisms into scoring rules and vice versa. In this paper, we
0 report t′ and his true type t. For reasons that will become proveageneralcharacterization,ofwhichthesecharacteriza-
3 clear, we represent thisas a function S(t′)(t) that maps his tionsofscoringrulesandmechanismsarespecialcases. Our
. reported type to a function that maps types to real num- proofisessentially acombination ofGneitingandRaftery’s
1
bers, with his score being this function applied to his true scoring rule construction [13] with a techniquefrom Archer
1
type (equivalently his reported type selects from a param- andKleinberg[1]forhandlingmechanismswithnon-convex
2
eterized family of functions with the result applied to his typespaces. Ourcharacterizationnotonlyshowshowmech-
1
true type). We allow S to be quite general, with the main anisms and scoring rules relate to each other, but also pro-
:
v requirement being that S(t′)(·) is an affine1 function of the vides an understanding of how results about mechanisms
i true type t, and seek to understand when it is optimal for relate to results about scoring rules and vice versa. In par-
X
the agent to truthfully report his type. Given this restric- ticular,manyresultsineachliteratureareessentiallyresults
r tion, it is immediately clear why convexity plays a central in convex analysis, and by phrasing them as such it is im-
a
role – when an agent’s type is t, the score for telling the mediately clear how they applyin theotherdomain.
truth is S(t)(t)=supt′S(t′)(t), which is convex function of
t as thepointwise supremum of affine functions.
Implementable
Onespecialcaseofourmodelismechanismdesign,where
S(t′)canbethoughtofastheallocationandpaymentgiven
Rochet [21] Myerson [18]
areportoft′,whichcombinetodeterminetheutilityofthe
agent as a function of his type2. In this context, S(t)(t) CMON Subgradient
is the consumer surplus function, and Myerson’s [18] well-
knowncharacterization statesthat,in single-parameterset- Mu¨ller et al. [17] Archer,Kleinberg [1]
tings, a mechanism is truthful if and only if the consumer
WMON + PI LWMON + VF
surplus function is convex and its derivative (or subgradi-
(for convex T) (for convex T)
ent at points where it is not differentiable) is theallocation
1Amappingbetweentwovectorspacesisaffineifitconsists
Figure 1: Proof structure of existingmechanismde-
of a linear transformation followed bya translation.
2It suffices to consider a single agent because notions of sign literature
truthfulness such as dominant strategies and Bayes-Nash
arephrasedintermsofholdingthebehaviorofotheragents 3Sincedistributionslieonanaffinespace,anyaffinefunction
constant. See [10, 1] for additional discussion. can beimplemented as well.
Implementable Notation..
We define R = R ∪{−∞,∞} to be the extended real
Thm 1 numbers. Given a set of measures M on X, a function f :
X → R is M-quasi-integrable if f(x)dµ(x) ∈ R for all
X
WMON + PI Thm 3 µ ∈ M. Let ∆(X) be the set ofRall probability measures
Subgradient
(for convex T) on X. . We denote by Aff(X → Y) and Lin(X → Y) the
Thm 2
set of functions from X to Y which are affine and linear,
Cor 6 Thm 4 CMON respectively. WewriteConv(X)todenotetheconvexhullof
vector space X, the set of all (finite) convex combinations
of elements X.
LWMON + VF
Weak Local Subgradient
(for convex T)
(for convex T) 2. MODEL ANDMAIN RESULT
Weconsider a very general model with an agent who has
Figure 2: Our proof structure a given type t∈T and reports some possibly distinct type
t′ ∈ T, at which point the agent is rewarded according to
After proving our main result, we examine a number of somescoreS whichisaffineinthetruetypet. Thisreward
characterizationsfromthemechanismdesignliteraturethat we call an affine score. We wish to characterize all truthful
generally focus on when there exist payments that make a affinescores,thosewhichincentivizetheagenttoreporther
givenallocationruletruthful. Figure1illustratesthesechar- truetypet.
acterizationsandhowtheywereproved. Asitshows,several
Definition 1. Any function S : T → A, where T ⊆ V
ofthemrelyonshowingequivalencetoaconditionknownas
for some vector space V and A ⊆ Aff(T → R), is a affine
cyclic monotonicity. Instead,wetranslatetheseresultsinto
convex analysis terms and prove them by showing equiva- score. We say S is truthful if for all t,t′∈T,
lence to the property of being a family of subgradients of S(t′)(t)≤S(t)(t). (1)
a convex function (see Figure 2). This has two main bene-
fits. First,sincecyclicmonotonicityisadifficultconditionto Ifthisinequalityisstrictforallt,t′,thenS is strictlytruth-
workwith,weareabletogreatlysimplifytheproofsofthese ful.
results. Second, our proofs generally proceed by explicitly
Before stating our characterization result, we note two
constructingtheconvexfunction,whichgivesanaturalchar-
important applications of, and indeed motivations for, this
acterization of the payments rather than just showing that
framework: mechanism design, and proper scoring rules.
they exist. This approach also illuminates how a result by
Carroll[7]abouttruthfulmechanismsisessentiallyasimilar Definition 2. Given outcome space O and a type space
characterization (see Theorem 4). T ⊆(O→R), consisting of functions mapping outcomes to
Next,weturnanalyzingsituationswhere,ratherthanask- reals, a mechanism is a pair (f,p) where f : T → O is an
ing for an agent’s type, we wish to elicit a simpler repre- allocation rule and p : T → R is a payment. The utility
sentation of it. In the scoring rules context, this has been of the agent with type t and report t′ to the mechanism is
studiedastheelicitationofpropertiesofadistributionsuch U(t′,t) = t(f(t′))−p(t′); we say the mechanism (f,p) is
as the mean or median [12, 16]. In mechanism design, this truthful if U(t′,t)≤U(t,t) for all t,t′ ∈T.
isimplicitinsettingssuchasmatchingwherearankingover
potential matches is elicited rather than the agent’s utility Definition 3. Given outcome space O and set of prob-
for them. We generalize our main result to this setting, ability measures P ⊆ ∆(O), a scoring rule is a function
which allows us to prove several new results. Perhaps the S :P×O→R. We say S is proper if for all p,q∈P,
most notable is that if the set of distributions has positive
E [S(q,o)]≤ E [S(p,o)]. (2)
measureinitsconvexhullthenthenthereisauniquevalue
o∼p o∼p
of thepropertyalmost everywhere.
If the inequality in (2) is strict, then S is strictly proper.
Inadditiontoourmoresignificant results,we getanum-
berofsmallernewresultsalongthewayessentially forfree.
These two models fit comfortable within our framework.
These include:
Forthemechanism,viewingT asavectorspaceoffunctions,
thetermt(f(t′))islinearint. Hence,U(·,t′)isaffine. Thus,
• the fact that in mechanisms which select among a fi-
the mechanism is simply the affine score with A = {t 7→
nitesetof(allocation,payment)pairs,thesetofsetsof
t(o)+c|o∈O,c∈R} defined by S(t′)(t):=U(t,t′). Simi-
typesthat select each outcome forms not just a set of
larly,ascoringruleS isanaffinescoreS :T →AwithT =
polyhedrabut a power diagram;
P, and A⊆{p7→ f(o)dp(o)|f is P-quasi-integrable}.
O
• a relaxation of an outcome compactness assumption Our characterizaRtion relies heavily on convex analysis, a
neededbyArcherandKleinberg[1]forallocationrules central concept of which is thesubgradient of a function.
on non-convextypespaces;
Definition 4. Given some function G:T →R, a func-
• acharacterizationofproperscoringrulesfornon-convex tion dGt ∈Lin(T →R) is a subgradient to G at t if for all
sets of distributions; and t′ ∈T,
G(t′)≥G(t)+dGt(t′−t). (3)
• a characterization that a scoring rule is proper if and
only if it is locally proper We denote by ∂Gt the set of subgradients to G at t, and
∂G=∪t∈T∂Gt.
Formechanismdesign,itistypicaltoassumethatutilities ofT isaconvexfunctionwhichhasasubgradientconsistent
are always real valued. However, the log scoring rule (one withf onT. Thisisaknowncharacterizationforthecaseof
of the most popular scoring rules) has the property that if convex T (as well as non-convexT that satisfy an assump-
an agent reports that an outcome will occur with probabil- tionknownasoutcomecompactness)[1],butinpracticethe
ity 0 and then that outcome does occur, the agent receives consumer surplus function is not always the most natural
a score of −∞. Essentially solely to accommodate this, we representation of a mechanism. In this section, we exam-
allow affine scores and subgradients to take on values from ine two other approaches to characterizing truthful mecha-
the extended reals. It is standard (see [13]) to restrict con- nisms that have been explored in the literature and show
sideration to the regular case, where essentially only things that theyhaveinsightful interpretationsin convexanalysis.
like the log score are permitted to be infinite. In particu- This interpretation has two benefits. First, by focusing on
lar, an affine score S is regular if for all t ∈ T S(t)(t) ∈ R theessentialconvexanalysisquestionsweareabletogreatly
and for t′ 6=t S(t′)(t)∈R∪{−∞}. Similarly, a parameter- simplify many of the proofs. Second, our proofs are con-
izedfamilyoflinearfunctions(e.g. afamilyofsubgradients) structive; in many cases we explicitly construct a consumer
{dGt ∈Lin(T →R)}t∈T isregularifforallt∈T dGt(t)∈R surplusfunctionG,whichwhenthemechanismisbeingrep-
andfort′6=tdGt′(t)∈R∪{−∞}. Fortheremainederofthe resented byitsallocation rulegivesthenecessary payments
paperweassumeallaffinescoresandparameterizedfamilies rather thansimply providinga proof that paymentsexist.
of linear functions are regular. Note that certain results in
3.1 Subgradient characterizations
Section 3 require a stronger assumption that the relevant
parameterized families are in fact real valued rather than From an algorithmic perspective,it may be more natural
simplyregular. Areadernotinterestedinthedetailsofhow tofocusonthedesignoftheallocationrulef ratherthanthe
ourframework incorporates thelogscoringrulecanassume specific payments. There is a large literature that focuses
thatallaffinescoresandfamiliesarerealvaluedthroughout on when thereexists achoice of paymentsp tomakef into
thepaper with little loss. a truthful mechanism (e.g. [24, 2]). Since such payments
We now state, and prove, our characterization theorem. existifandonlyifthereisaconvexfunctionforwhichf isa
Note that the proof draws techniques and insights from subgradientatpointsinT,thisisessentially averynatural
GneitingandRaftery[13]andArcherandKleinberg[1],but convexanalysisquestion: whenisafunctionf asubgradient
in such a way as to simplify theargument considerably. of a convex function? Unsurprisingly, the central result in
this literature is closely connected to convexanalysis.
Theorem 1. Let an affine score S : T → A be given.
SConisv(tTru)th→fulRifwaitnhdGo(nTly) ⊆if tRh,eraendexsiostmsesosmeleectcioonnveoxf sGub-: fiesDceyfcilnicitmioonno5to.nAiciftaym(CilyM{OdNG)ti∈ffoLrina(lTlfi→niteRs)e}tts∈T{ts,a.t.is.-,t }⊆
0 k
gradients {dGt}t∈T, such that T,
S(t′)(t)=G(t′)+dGt′(t−t′). (4) k
Proof. It is trivial from the subgradient inequality (3) dGti(ti+1−ti)≤0, (6)
X
thattheproposedformisinfacttruthful. Fortheconverse, i=0
we are given some truthful S : T → A. Note first that for whereindicesaretakenmodulok+1. Werefertotheweaker
any tˆ∈ Conv(T) we may write tˆas a finite convex combi- condition that (6) hold for all pairs {t0,t1} as weak mono-
nation tˆ= mi=1αiti where ti ∈T. Now, as the range of S tonicity (WMON).
isaffine,wePmaynaturallyextendS(t)toall ofConv(T)by
defining Awellknowncharacterizationfromconvexanalysisisthat
a function f defined on a convex set is a subgradient of
m
S(t)(tˆ)= αiS(t)(ti). (5) aRoccohnevte’sx[2fu1]npctriooonftohnattahastucshetpaiyffmietnstasteixsfiisetssoCnMaOpNoss[i2b2l]y.
X
i=1
non-convex T iff f satisfies CMON is effectively a proof of
One easily checks that this definition coincides with the thefollowing generalization of thistheorem.
given S on T.
NowweletG(tˆ):=supt∈T S(t)(tˆ),whichisconvexasthe Theorem 2. A family {dGt ∈Lin(T → R)}t∈T satisfies
pointwisesupremumofconvex(inourcaseaffine)functions. CMONifandonlyifthereexistsaconvex G:Conv(T)→R
SinceS istruthful,weinparticularhaveG(t)=S(t)(t)∈R such that dGt is a subgradient of G at t for all t∈T.
for all t ∈ T by our regularity assumption. Also by truth-
fulness, we havefor all t,t′ ∈T, Rochetnotesthathisproofisadaptedfromtheonegiven
in Rockafellar’s text [22] of the weaker theorem where T is
G(t)=S(t)(t)≥S(t′)(t)=S(t′)(t′)+S (t′)(t−t′)
ℓ restricted to be convex. We adapt Rochet’s proof to high-
=G(t′)+Sℓ(t′)(t−t′), light how its core is a construction of G.
where Sℓ(t′) is the linear part of S(t′). Hence, Sℓ(t′) satis- Proof. GivensuchaG,by(3)wehavedGti(ti+1−ti)≤
fies (3) for Gat t′ and so S is of theform (4). G(ti+1)−G(ti). Summing gives (6). Given such a family
{dGt}t∈T, fixsome t0∈T and define
3. CONVEXITYIN MECHANISMDESIGN
k
Interpreted in the general mechanism design framework G(t)= sup dGti(ti+1−ti), (7)
given by Definition 2, Theorem 1 says that a mechanism {t1,...,tk+1}⊆T,Xi=0
(f,p) is truthful if and only if the consumer surplus func- tk+1=t
tion t7→U(t,t) that it implicitly defineson theconvexhull where{t ,...,t }denotesanyfinitesequence(kisnotfixed).
1 k
By CMON, for t ∈ T this sum is upper bounded by so G is convex. Similarly, for x,y ∈ T, dGt satisfies (3)
−dGt(t0 −t). Thus, the supremum is finite on T. G is because
apointwisesupremumofconvexfunctions,soisconvex. By
convexity,G is also finiteon Conv(T). dGx(y−x)≤Z dGt(y−x)dt=G(y)−G(x).
For any t∈T and t′∈Conv(T), Lxy
3.2 Local convexity
k
G(t)+dGt(t′−t)=dGt(t′−t)+ sup dGti(ti+1−ti)In many settings, it is natural to specify mechanisms in
{t1,...,tk+1}⊆T,Xi=0 termsofanalgorithmthatcomputestheallocationandpay-
tk+1=t
ment. As it is often easier to reason about the behavior of
k
algorithmsgivensmallchangestotheirinputratherthanar-
= sup dGti(ti+1−ti) bitrarychanges,severalauthorshavesoughttocharacterize
{tt1k,=..t.,,ttkk++11}=⊆tT′,Xi=0 truthfulmechanisms usinglocal conditions [1, 5, 7].
We show in this section how many of these results are
k
≤{t1,.t.k.,s+tuk1+p=1t}′⊆T,Xi=0dGti(ti+1−ti) itinnhaettshsceeontncwveiecxaeitcdyoinffisseearqenuneitninahcbeelreoenfctaalsymelooitcreaclafupnnrobdpeaemvrteeyrn.itfiaFelodsrtbeayxteadmmeeptnelert-,,
=G(t′), mining whether the Hessian is positive semidefinite at each
point. We start with a local convexity result, and use it to
so dGt satisfies (3). showthatanaffinescoreistruthfulifandonlyifitsatisfies
a very weak local truthfulness property introduced by Car-
A number of papers have sought simpler and more natu- roll[7]. Afterwardsweturntoasimilar characterization by
ralconditionsthanCMONthatarenecessaryandsufficient Archerand Kleinberg [1].
in special cases, e.g. [24, 1, 2]. These results are typically
provenbyshowingtheyareequivalenttoCMON.However, Definition 6. GivenafunctionG,afamily{dGt ∈Lin(T →
itismuchmorenaturaltodirectlyconstructtherelevantG. R)}t∈T is a weak local subgradient with respect to G (G-
Thisalso often hastheadvantageofprovidingacharacteri- WLSG)ifforallt∈T thereexistsanopenneighborhoodUt
zationofthepaymentsthatismoreintuitivethanthesupre- of t such that for all t′ ∈Ut,
msuucmh riensuRltochhaest’ascsoimnsptlreupctriooonf.uAsisnagnoeuxrafmrapmlee,wwoerks.howone G(t)≥G(t′)+dGt′(t−t′) and G(t′)≥G(t)+dGt(t′−t).
(8)
AsinMyerson’s[18]constructionforthesingle-parameter
case, we construct a G by integrating over dGt. In partic- We now show that G-WLSG is a sufficient condition for
ular, for any two typesx and y our construction makes use afamily offunctionstobeasubgradientofG. Theproofis
of theline integral heavily inspired byCarroll [7].
ZLxydGt(y−x)dt=Z01dG(1−t)x+ty(y−x)dt. R}Tt∈hTeoisreamsu4bg.rLadetieTntboefcaongvievxe.nAfufnacmtiiolyn{GdGifta∈nLdino(nTly→if
it satisfies G-WLSG.
AsBergeretal.[5]andAshlagietal.[2]observed,if{dGt}t∈T
Proof (Adapted from [7]). As usual, the forward di-
satisfiesWMONandT isconvex,this(Riemann)integralis
rection is trivial. For the other, let t,t′ ∈ T be given;
welldefinedbecauseitistheintegralofamonotonefunction.
If these line integrals vanish around all triangles (equiva- we show that the subgradient inequality for dGt′ holds at
t. By compactness of Conv({t,t′}), we have a finite set
xle)ndttl)y)RwLexysadyG{t(dyG−t}x)sdatti+sfiReLsypzadtGhti(nzd−epyen)ddten=ceR.LxzdGt(z− stiuc=h tαhiatt′ +G-(W1−LSαGi)ht,olwdshebreetw0e=enαea0ch≤t·i·a·n≤d tαi+k+1.1 (=Th1e,
Theorem 3 (adapted from [17]). ForconvexT,afam- cover {Us|s ∈ Conv({t,t′}) has a finite subcover. Take t2i
ily {dGt ∈ Lin(T → R)}t∈T is a subgradient of a convex fcroonmditthioens(u8b)c,owveerhaanvdetf2oir+e1a∈chUit,2i∩Ut2i+2.) BytheWLSG
function if and only if {dGt}t∈T satisfies WMON and path
independence. 0≥G(ti+1)−G(ti)+dGti+1(ti−ti+1) (9)
Proof. Given a convex function G and {dGt}, {dGt} 0≥G(ti)−G(ti+1)+dGti(ti+1−ti). (10)
satisfies CMON and thus WMON. Path independence also Now using the identity ti+1−ti = (αi+1−αi)(t′−t) and
follows from convexity(Rockafellar [22] p. 232). Nowgiven addingαi/(αi+1−αi)times(9)toαi+1/(αi+1−αi)times(10),
a {dGt} that satisfies WMON and path independence, fix we have
daetfiynpeedtb0y∈WTMaOndNdaesfitnheeGin(tte′)gr=alRoLft0at′mdoGnto(tto′n−etf0u)ndctt(iowne)ll. 0≥G(ti)−G(ti+1)+αidGti(t′−t)−αi+1dGti+1(t′−t). (11)
Given x,y,z ∈ T such that z = λx+(1−λ)y, by path Summing (11) over0≤i≤k gives
independenceand thelinearity of dGz we have 0≥G(t0)−G(tk+1)+α0dGt0(t′−t)−αk+1dGtk+1(t′−t),
λG(x)+(1−λ)G(y)
whichwhen recallingourdefinitionsforαi andti yieldsthe
result.
=G(z)+λZ dGt(x−z)dt+(1−λ)Z dGt(y−z)dt
Lzx Lzy TheWLSGconditiontranslatestoananalogousnotionin
≥G(z)+λdGz(x−z)+(1−λ)dGz(y−z)=G(z), terms of truthfulness, weak local truthfulness.
Definition 7. An affine score is weakly locally truthful reports from a different (intuitively,smaller) space than T.
if for all t ∈ T there exists some open neighborhood Ut of Our results shed light on the structure of properties and
t, such that truthfulness holds between t and every t′ ∈ Ut, their affinescores.
and vice versa. That is,
4.1 General setting
∀t∈T, ∀t′∈Ut, S(t′)(t)≤S(t)(t) and S(t)(t′)≤S(t′)(t′). We wish to generalize the notion of an affine score to ac-
(12)
cept reports from a space R which is different from T. To
evendiscusstruthfulnessinthissetting,weneedanotionof
Corollary 5 (Generalization of Carroll [7]). An
a truthful report r for a given type t. We encapsulate this
affine score S :T →A for convex T is truthful if and only
notionbyageneralmultivaluedfunctionwhichspecifiesonly
if it is weakly locally truthful.
and all of thecorrect values for t.
Proof. DefiningG(t):=S(t)(t),byweak local truthful-
Definition 8. LetRbesomegivenreportspace. Aprop-
ness we may write erty is a multivalued map Γ : T ⇒ R which associates a
G(t)=S(t)(t)≥S(t′)(t)=G(t′)+S (t′)(t−t′) nonempty set of correct report values to each type. We let
ℓ
G(t′)=S(t′)(t′)≥S(t)(t′)=G(t)+S (t)(t′−t), Γr :={t∈T |r∈Γ(t)}denotethesetoftypestcorrespond-
ℓ ing to report value r.
where Sℓ(·) is the linear part of S(·). This says that dGt = Onecan thinkofΓr asthe“levelset”of Γcorrespondingto
S (t) satisfies G-WLSG; the rest follows from Theorem 4
ℓ valuer. Thisconceptwillbeusefulmainlywhenweconsider
and Theorem 1.
finite-valuedproperties in Section 4.2.
We extend the notion of an affine score to this setting,
Finally, in the spirit of Section 3.1, Archer and Klein-
where thereport space is R instead of T itself.
berg [1] characterized local conditions under which an allo-
cationrulecanbemadetruthful. Akeyconditionfromtheir Definition 9. An affine score S :R→A elicits a prop-
paperisvortex-freeness,whichisaconditiontheyshowtobe erty Γ:T ⇒R if for all t,
equivalenttolocalpathindependence. Theothercondition,
Γ(t)=argsupS(r)(t). (13)
local WMON, means that WMON holds in some neighbor- r∈R
hood around each type. Their result follows directly from Aproperty Γ:T ⇒Ris elicitableifthereexistssomeaffine
an analogous characterization of subgradients.
score S :R→A which elicits Γ.
Corollary 6. LetT beconvex. Afamily{dGt ∈Lin(T → As before, we need a notion of regularity here as well –
R)}t∈T is a subgradient of a convex function if and only if an affine score S is Γ-regular if S(r)(t) < ∞ always and
it satisfies local WMON and is vortex-free. S(r)(t)∈R wheneverr∈Γ(t).
The simplest way to come up with an elicitable property
Proof. Weprovethereversedirection;suppose{dGt}t∈T is to induce one from an affine score. For any S : R →
satisfies local WMON and is vortex-free. From Lemma 3.5 Aff(T →R), theproperty
of[1]wehavethatvortex-freenessisequivalenttopathinde-
pendence, so by Theorem 3 for all t there exists some open ΓS :t→argsupS(r)(t) (14)
Ut such that {dGt′}t′∈Ut is thesubgradient of some convex r∈R
function G(t) :Ut →R. Weneedonlyshowtheexistenceof is trivially elicited by S.
someGsuchthat{dGt}t∈T isthesubgradientofGoneach Observe also that any affine score S eliciting Γ gives rise
Ut; therest follows from Theorem 4. to a truthful affine score in the original sense – in fact, this
isFwiexllsdoemfienetd0 b∈yTcoamnpdacdtenfienses oGf(Ct)on=v(R{Ltt0,tt}d)Gatn′ddt′t,hwehfaiccht liestarvte∈rsiΓo(nt)ofbethaerseop-coarltlecdhoreicveelafotriotn; pthrienncitphlee.aFffionreeasccohret
that a locally increasing real-valued funct0ion is increasing. ST(t′)(t) := S(rt′)(t) is truthful. Moreover, by our choices
But for each t′ and t ∈ Ut′ we can also write G(t′)(t) = of {rt}, we have
RdLent′ctedwGet′′sdete′′tbhyat[2G2,anpd. G23(t2′]),dainffdernboywabcyonpsattahnti.ndHepenecne- G(t):=ts′u∈pT ST(t′)(t)=rs∈uRpS(r)(t). (15)
Of course, in general, ST will not be strictly truthful,since
{t′d∈GtT}t.∈T must bea subgradientof Gon Ut′ as well, for all by definition, any reports t′,t′′ with rt′ = rt′′ (which can
happen when t′,t′′ ∈ Γr for some r) will have ST(t′) ≡
ST(t′′). Thus we may think of a property as refining the
4. EXTENSION TO PROPERTIES
notion of strictness for a truthful affine score. The connec-
In many settings, it is difficult, or even impossible, to tion we will draw in Theorem 8 is that, in light of (15), a
haveagentsreportanentiretypet∈T. Forexample,when property Γ therefore specifies theportions of thedomain of
O = R, then a mechanism with T = (O → R) requires T where Gmust be“flat”.
agentstosubmitaninfinite-dimensionaltype,andeventype Togetattheconnectionbetweenpropertiesand“flatness”,
spaces which are exponential in size can be problematic. It westartwithatechnicallemmawhichshowsthathavingthe
is therefore natural to consider a model of truthful report- same subgradient at two different points implies that G is
ingwhereagentsprovidesomesortofsummaryinformation flat in between.
abouttheirtype. Suchamodelhasbeenstudiedinthescor-
Lemma 7. For convex T, let G : T → R be convex, and
ingrulesliterature,whereonewishestoelicitsomestatistic,
or property, of a distribution [12, 16]. We follow this line of let d∈∂Gt for some t∈T. Then for all t′ ∈T,
research, and extend our affine score framework to accept d∈∂Gt′ ⇐⇒ G(t)−G(t′)=d(t−t′).
Proof. First, the forward direction. Applying the sub- Bydefinitionofaproperty,wehaveforallt∈T thatΓ(t)6=
gradient inequality(3)att′ fordGt =dandat tfordGt′ = ∅, so by the above G(t) = S(r)(t) for some r. Combining
d, we have this with the fact that G dominates S, we have G(t) =
sup S(r)(t) for all t ∈ T. Putting this together with (17)
G(t′)≥G(t)+d(t′−t) r
and our assumption that r ∈ Γ(t) ⇐⇒ ϕ(r) ∈ ∂Gt, we
G(t)≥G(t′)+d(t−t′), have
from which the result follows. For the converse, assume r∈Γ(t) ⇐⇒ G(t)=S(r)(t) ⇐⇒ r∈argsupS(r′)(t)
G(t) = G(t′)+d(t−t′) and let t′′ ∈ T be arbitrary. Then r′
since d∈∂Gt we have Returningto theabovediscussion, byfocusing on ST in-
G(t′)+d(t′′−t′)=G(t′)+d(t′′−t)+d(t−t′) stead of S, we now see how properties are essentially a re-
finement of strictness. In fact, one can can formalize this
=G(t)+d(t′′−t)
by considering that, up to a remapping of R, ST is strictly
≤G(t′′), truthfulifandonlyifS elicitsΓ:t7→{t},andST istruth-
fulifandonlyifS elicitssomeΓsuchthatt∈Γ(t)forallT.
We are now ready to state our characterization in this
Hence, Theorem 8 is actually a generalization of our main
setting, which in essence says that eliciting a property Γ is
result, Theorem 1.
equivalent toeliciting subgradientsof a convexfunction G.
ReturningourfocustoR,ourcharacterizationshedslight
Theorem 8. LetpropertyΓ:T ⇒RandΓ-regularaffine on the structure of elicitable properties. For example, we
may apply the fact that convex functions are differentiable
score S : R → A be given. Then S elicits Γ if and only if
there exists some convex G: Conv(T) → R with G(T) ⊆ R almost everywhere to yield a very strong result about the
structureof elicitable properties.
and ϕ:R→∂G satisfying r ∈Γ(t) ⇐⇒ ϕ(r)∈∂Gt, such
that Corollary 9. If Γ:T ⇒R is an elicitable property, T
S(r)(t) = G(tr)+ϕ(r)(t−tr), (16) is of positive measure in Conv(T), and no property value is
where {tr}r∈R satisfies r∈Γ(tr) for all r. reevdeurynwdahnerte.(∀r1,r2 ∈ R,Γr1 6⊆ Γr2) then |Γ(t)| = 1 almost
Proof. Fortheforwarddirection,assumethataffinescore
4.2 Finitecase
S elicits Γ. For each r, we may extend S(r) to all tˆ ∈
Conv(T) by linearity as in Theorem 1, whence we may de- WenowturntothecasewhereRisfinite. Inthisscoring
fineG(tˆ):=sup S(r)(tˆ), which is finitefor tˆ∈T as S is rules literature, this case has been thought of as eliciting
r∈R
Γ-regular. We wish to show that the choice ϕ : r 7→ S (r) answers to multiple-choice questions [15]. There are also
ℓ
suffices, where S denotesthelinear part of S. naturalapplicationstomechanismdesign,whichwemention
ℓ
By standard arguments(cf. Proposition 8.12 of [23]),the below.
subgradients to a convex function F :X →R at x∈X are AssumethroughoutthatRisfiniteandthatT isasubset
precisely thegradientsof itsaffinesupportsat x. Morefor- of avectorspace V endowed with an innerproduct,so that
mally, let AM(F)={a∈Aff(X →R)|∀xF(x)≥a(x)} be we may write ht,t′i and in particular ktk2 = ht,ti. In this
the set of affine functions minorizing F, and let AS(F,x)= moregeometricalsetting,wewillusetheconceptofapower
{a ∈AM(F)|a(x)= F(x)} be the set of affine supports of diagram from computational geometry.
F Batyxd.efiTnhietnionweofhGav,ew∂eFhxa=ve{S∂a(r|)a∈∈AAMS((FG,)x.)N}.oting that Definition 10. Given points pi ∈ V, called sites, and
∂S(r)t =Sℓ(r),we havefor all t∈T and r∈R, wcoelilgehcttisonw(cpeil)ls∈ceRll(fpoir)i⊆∈T{1d,e.fi..n,emd}b,ya power diagram is a
r∈Γ(t) ⇐⇒ S(r)(t)= sup S(r′)(t)
r′∈R cell(pi)={t∈T |i=argmin{kpi−tk2−w(pj)}. (18)
j
⇐⇒ S(r)(t)=G(t)
The following result is a generalization of Theorem 4.1
⇐⇒ S(r)(t)∈AS(G,t)
of [15],and isessentially adirectapplication ofProposition
⇐⇒ Sℓ(r)∈∂Gt, 1 from [3].
where we use elicitation in the first biconditional. Finally, Theorem 10. A property Γ:T ⇒R for R finite is elic-
using Γ-regularity and the definition of tr, we show the itable if and only if the level sets {Γr}r∈R form a power
form (16): diagram. Moreover, every S :R→A can be written
G(tr)+Sℓ(r)(t−tr)=S(r)(tr)+Sℓ(r)(t−tr)=S(r)(t). S(r)(t)=ktk2−kpr−tk2+w(pr), (19)
For the converse, let G, ϕ, and tr be given, and assume forsomechoiceofsites{pr ∈V}r∈R andweightsw(pr)∈R.
S has the form (16). First note that by definition of {tr},
andbyassumptiononϕ,wehaveϕ(r)∈∂Gtr forallr. We thePfroormof.(1G9)i.veTnhseintetsa{kpinrg}Γan(td):w=eiagrhgtssupw(pSr()r,)(let)t,Swetaskeee
claim that G dominates S, meaning for all r ∈ R and all r
t ∈ T, G(t) ≥ S(r)(t); this follows from the definition of S Γr =cell(pr).
Given affine score S eliciting Γ, note that since we are in
and thesubgradient inequality (3) applied to ϕ(r) at t.
Now byLemma 7,we have an innerproduct space, we may write S(r)(t)=hxr,ti+cr
for xr ∈ V and cr ∈ R. Now take pr = xr/2 and w(pr) =
ϕ(r)∈∂Gt ⇐⇒ G(t)−G(tr)=ϕ(r)(t−tr) kprk2+cr; then
⇐⇒ G(t)=S(r)(t). (17) ktk2−kpr−tk2+w(pr)=2hpr,ti−kprk2+cr+kprk2 =S(r)(t),
which simultaneously proves the form (19) and that Γr = where Gp :q 7→ OGp(o)dq(o) is a subgradient of G for all
cell(pr). p∈P. R
Proof. Thegivenformistruthfulbythesubgradientin-
We can now combine Theorem 10 and Corollary 5 to ob-
equality. Let S : T → A be a given truthful affine score.
tain the following simple truthfulness check for the finite
case. Since S(p)∈A, we have some fp ∈F generating S(p). We
can therefore use Gp : q 7→ Ofp(o)dq(o) as the subgradi-
Theorem 11. For finite R, an affine score S : R → A ents in the proof of TheoremR 1, thus giving us the desired
elicitsΓ ifandonly ifforall r,r′ such that Γr∩Γr′ 6=∅, we form.
have S(r′)(t)<S(r)(t) for all t∈Γr\Γr′. Corollary 12 immediately generalizes their characteriza-
We now apply our results to mechanism design with a tion to the case where P is not convex. While this case is
finite set of allocations. One example of such a setting is notofcentralimportancetothetheoryofscoringrules,itis
ordinal utilitiesoverafiniteset ofoutcomes. Let T =O→ sometimesanaturalcasetoconsider. Forexample,Babaioff
RandletRbethesetofpreferenceorderingsoverO,which et al. [4] examine when proper scoring rules can have the
werepresentaspermutationsπofO(higherpreferencebeing additionalpropertythatuninformedexpertsdonotwishto
first). Then anatural property is make a report (have a negative expected utility), while in-
formed experts do wish to make one. They show that this
Γ(t)={π∈R|∀i<j, t(πi)≤t(πj)}, (20) ispossibleinsomesettingswherethespaceofreportsisnot
the set of rankings consistent with type t. More generally, convex. Thus our characterization shows that, despite not
truncatedrankingsor anymechanism with afinitemessage needing to ensure properness on on reports outside P, es-
space yields such a property. sentially the only possible scoring rules are still those that
Carroll[7]pointsoutthatitisknownthatforallfinitesets are proper on all of ∆(O).
ofallocations, thesetoftypesforwhich aparticularalloca- While local truthfulness is a natural property to exam-
tion is optimal forms a polyhedronand then shows that for ine in mechanism design, it is not obvious how it is as use-
allconvexsetoftypesitissufficienttocheckincentivecom- ful for scoring rules since scoring rule designers tend to op-
patibility constraints between polyhedra that intersect at a erate under significantly fewer constraints than mechanism
face. Our results in this section show something stronger: designers. Nevertheless, to results from Section 3.2 apply,
the types form not just an arbitrary set of polyhedra, but so Corollary 5 shows that local properness is equivalent to
apowerdiagram. However,thesufficiencycondition weget global propernessfor scoring rules on convexP.
is slightly weaker, as it requires checking incentive compat-
Corollary 13. For a convex set P ⊆∆(O) of probabil-
ibility between typesthat intersect at avertexas well as at
ity measures, a scoring rule S :P×O→R is proper if and
a face.
only if it is (weakly) locally proper.
5. ADDITIONALAPPLICATIONS 5.2 DecisionRules
In this section, we demonstrate the power of our frame- Theorem 1 also generalizes Gneiting and Raftery’s [13]
work with several additional applications drawn from the characterization to settings beyond eliciting a single distri-
scoring rules literature. bution. Forexample,alineofworkhasconsideredasetting
whereadecisionmakerneedstoselectfromafinitesetDof
5.1 Two Results About Scoring Rules
decisions and so desires to elicit the distribution over out-
The role of convexityin the scoring rules literature dates comes conditional on selecting each alternative [20, 9, 8].
back to 1971 when Savage showed how to construct a scor- Sinceonlyonedecisionwillbemadeandsoonlyonecondi-
ing rule from a convex function [25]. The general, mod- tional distribution can be sampled, simply applying a stan-
ern characterization of scoring rules is due to Gneiting and dardproperscoringrulegenerallydoesnotresultintruthful
Raftery [13], who used a more nuanced convex analysis ap- behavior. ApplyingTheorem 1 to this setting characterizes
proachtoclarifyandgeneralizeanumberofpreviouscharac- what expected scores must be, from which many of the re-
terizations,includingSavage[25]andSchervish[26]. Inthis sultsinthesepapersfollow. Asitisnotourmainfocus,we
section, we show that the Gneiting and Raftery characteri- refrain from introducing the model necessary to explicitly
zation is a simple special case of Theorem 1, and moreover state a characterization result similar to Corollary 12.
thatourresultsenableseveralgeneralizationsinthescoring
rule setting. 6. DISCUSSION
Letting F be the set of P-quasi-integrable functions f :
O → R, we apply our characterization with T = P and We have presented a model of truthful elicitation which
generalizesandextendsbothmechanismsandscoringrules.
A={p7→ f(o)dp(o)|f ∈F}.
O Onthemechanismdesignside,wehaveseenhowourframe-
R
Corollary 12. For an arbitrary set P ⊆∆(O) of prob- work provides simpler and more constructive proofs of a
ability measures, a regular4 scoring rule S : P ×O → R numberof known results, some of which (as we saw in Sec-
is proper if and only if there exists a convex function G : tion 5.1) lead to new results about scoring rules. We then
Conv(P)→R with functions Gp ∈F such that generalized our model to allow reports of a succinct rep-
resentation of an agent’s private information rather than
S(p,o)=G(p)+Gp(o)−ZOGp(o)dp(o), (21) tshcoartiningforurmlesatliiotenraittsuerlef,aabotouptieclitchitaitnghapsrobpeeerntisetsuodfieddisitnribthue-
4Thisisthesameconceptaswithaffinescores: scorescannot tions. This led us to new results in both mechanism design
be ∞ and only incorrect reports can yield −∞. and scoring rules.
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