A A Geometric Perspective on Minimal Peer Prediction

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1 A A Geometri Perspetive on Miniml Peer Predition RAFAEL FRONGILLO, CU Boulder JENS WITKOWSKI, ETH Zurih Miniml peer predition mehnisms truthfully eliit privte informtion (e.g., opinions or experienes) from rtionl gents without the requirement tht ground truth is eventully reveled. In this pper, we use geometri perspetive to prove tht miniml peer predition mehnisms re equivlent to power digrms, type of weighted Voronoi digrm. Using this hrteriztion nd results from omputtionl geometry, we show tht mny of the mehnisms in the literture re unique up to ffine trnsformtions. We lso show tht lssil peer predition is omplete in tht every miniml mehnism n e written s lssil peer predition mehnism for some soring rule. Finlly, we use our geometri hrteriztion to develop generl method for onstruting new truthful mehnisms, nd we show how to optimize for the mehnisms effort inentives nd roustness. Ctegories nd Sujet Desriptors: J.4 [Computer Applitions]: Soil nd Behviorl Sienes Eonomis; K.4 [Computers nd Soiety]: Eletroni Commere Generl Terms: Algorithms, Eonomis, Design Additionl Key Words nd Phrses: Peer Predition, Informtion Eliittion, Mehnism Design ACM Referene Formt: Frongillo, Rfel nd Witkowski, Jens A Geometri Perspetive on Miniml Peer Predition ACM Trns. Eon. Comp. V, N, Artile A (My 2017), 27 pges. DOI: 1. INTRODUCTION User-generted ontent is essentil to the effetive funtioning of mny soil omputing nd e-ommere pltforms. A prominent exmple is eliiting informtion through rowdsouring pltforms, suh s Amzon Mehnil Turk, where workers re pid smll rewrds to do so-lled humn omputtion tsks, whih re esy for humns to solve ut diffiult for omputers. For exmple, humns esily reognize elerities, wheres even stte-of-the-rt omputer vision lgorithms perform signifintly worse. While sttistil tehniques n djust for ises or identify noisy users, they re pproprite only in settings with repeted prtiiption y the sme user, nd when user inputs re informtive in the first ple. But wht if providing urte informtion is ostly for users, or if users hve inentives to lie? Consider n imge nnottion tsk (e.g. for serh engine indexing), where workers my wish to sve effort y nnotting with rndom words, or words tht re too generi (e.g. niml ). Or onsider puli helth progrm tht requires prtiipnts to report whether they hve ever used illegl drugs, nd where prtiipnts my lie out their drug use due to shme or eligiility onerns. This pper is signifintly extended version of A Geometri Method to Construt Miniml Peer Predition Mehnisms, in the Proeedings of the 30th AAAI Conferene on Artifiil Intelligene (AAAI 16). Author s ddresses: rf@olordo.edu (Rfel Frongillo) nd jensw@inf.ethz.h (Jens Witkowski). Permission to mke digitl or hrd opies of ll or prt of this work for personl or lssroom use is grnted without fee provided tht opies re not mde or distriuted for profit or ommeril dvntge nd tht opies er this notie nd the full ittion on the first pge. Copyrights for omponents of this work owned y others thn ACM must e honored. Astrting with redit is permitted. To opy otherwise, or repulish, to post on servers or to redistriute to lists, requires prior speifi permission nd/or fee. Request permissions from permissions@m.org ACM /2017/05-ARTA $15.00 DOI:

2 A:2 R. Frongillo nd J. Witkowski Peer predition mehnisms ddress these inentive prolems. They re designed to eliit truthful privte informtion from self-interested prtiipnts, suh s nswers to the question Hve you ever used illegl drugs? Cruilly, peer predition mehnisms nnot use ground truth. In the puli helth exmple this mens the progrm nnot verify whether prtiipnt hs or hs not ever used illegl drugs; it n only use the prtiipnts voluntry reports. The lssil peer predition method, introdued y Miller et l. [2005], ddresses this hllenge y ompring the reported informtion of prtiipnt (her reported signl) with tht of nother prtiipnt, nd omputing pyment rule whih ensures tht truth reveltion is strtegi equilirium. The ommon knowledge ssumptions in their nlysis, however, re too strit for prtil pplition. In prtiulr, the prtiipnts possile posterior eliefs (their elief model) re ssumed to e known nd the sme for ll prtiipnts. In the puli helth exmple, this would men tht every prtiipnt using drugs hs the sme elief tht e.g. 40% of prtiipnts use drugs, nd every prtiipnt not using drugs hs the sme elief tht e.g. 20% use drugs. Moreover, these numers need to e known y the mehnism in order to ompute the pyment rule. Sine the ssumption of perfetly known elief models is unrelisti in prtie, Jur nd Fltings [2007] pply tehniques from roust optimiztion to the lssil peer predition method nd mke it roust ginst smll vritions in the ommonly-held elief model. As we will see in Setion 7, the lssil peer predition method lredy does llow for some devitions from the ssumption of perfet knowledge. Moreover, we show how geometri perspetive on peer predition n e used to ompute mximlly-roust mehnisms for generlized notion of roustness, whih inludes the roustness of Jur nd Fltings s speil se. Byesin Truth Serum (BTS) mehnisms relx the ommon knowledge ssumptions of forementioned mehnisms in tht the elief model is still the sme for ll prtiipnts ut no longer needs to e known to the mehnism (not even pproximtely s in Jur nd Fltings [2007]). In ddition to the signl tht is to e eliited, BTS mehnisms lso require prtiipnts to report proility distriution regrding the signl reports in the popultion. Tht is, they re not miniml. A shortoming of the originl Byesin Truth Serum (BTS) [Prele 2004] is tht it is truthful only for n infinite numer of prtiipnts. The Roust Byesin Truth Serum (RBTS) [Witkowski nd Prkes 2012; Witkowski 2014] hieves truthfulness for smll popultions (ny numer of gents n 2 with inry signls nd ny n 3 for non-inry signls). The 1/p BTS [Rdnovi nd Fltings 2013] is truthful for non-inry signls nd ny numer of gents n 2 ut is not roust towrds predition reports of 0% or 100% nd hs unounded ex post pyments. 1 It lso requires different elief model onstrint thn RBTS (see lso Setion 4). All BTS mehnisms still ssume tht ll prtiipnts shre the sme elief model. Witkowski nd Prkes [2012] relx this ssumption nd llow prtiipnts to hve sujetive elief models. (In the puli helth setting, for exmple, two prtiipnts using drugs ould hve different eliefs out the prevlne of drug users in the popultion.) However, their mehnism requires the ility to eliit relevnt informtion from n gent oth efore nd fter she mkes her oservtion. In prtiulr, their mehnism is not miniml. 1 Rdnovi [2016] suggests workround for these shortomings (p.33, Footnote 5), s follows. If the peer gent ssigns 0% to signl in her predition report, then the prtiipnt is pid signl sore of 1 if nd only if the prtiipnt nd the peer gent oth report this signl, nd 0 else. However, in this miniml mehnism it is of ourse dominnt strtegy to report the 0% signl independent of the prtiipnt s true elief. In ontrst, even with 0% predition reports, the signl soring omponent of RBTS with positive delt results in non-trivil mehnism, whih is still stritly truthful for generi elief models.

3 A Geometri Perspetive on Miniml Peer Predition A:3 Multi-tsk mehnisms re pplile to settings where gents respond to severl similr questions, i.e. questions for whih n gent hs the sme elief model. This pproh llows for the design of miniml peer predition mehnisms tht do not require knowledge of the gents elief models. Witkowski nd Prkes [2013] sore prtiipnts on given tsk using signl distriutions tht were lerned from reports on other tsks. The mehnism due to Dsgupt nd Ghosh [2013], s well s its multisignl extension [Shnyder et l. 2016], rewrds greement with the peer gent on overlpping questions nd punish greement on non-overlpping questions. In the single-tsk miniml setting, line of work introduing the 1/p Mehnism [Jur nd Fltings 2008; Jur nd Fltings 2011] nd the Shdowing Method [Witkowski nd Prkes 2012; Witkowski 2014] llows for relxed ommon knowledge ssumptions when ompred to the originl work introduing the lssil peer predition method. As we show in this pper, ny miniml mehnism, regrdless of the knowledge ssumptions used to onstrut it, llows for n nlogous relxtion of ommon knowledge ssumptions. Our Results. In this pper, we provide omplete hrteriztion of the design spe of miniml, sujetive, single-tsk peer predition mehnisms, whih inludes the lssil peer predition method, Output Agreement, the 1/p Mehnism, nd the Shdowing Method s speil ses. While it ws known tht every miniml mehnism requires some onstrint on the gents elief models [Jur nd Fltings 2011], it ws unknown whih onstrints llow for truthful mehnisms nd how the onstrints of different truthful mehnisms relte to one nother. We nswer these questions in Setion 3 y dpting tehniques from the literture on property eliittion [Lmert et l. 2008; Lmert nd Shohm 2009; Frongillo nd Ksh 2014]. In prtiulr, we prove the equivlene of miniml peer predition mehnisms nd power digrms, type of weighted Voronoi digrm. In Setion 4, we use this equivlene nd results from omputtionl geometry to show tht ll forementioned mehnisms re the unique mehnisms, up to positiveffine trnsformtions, whih re truthful with respet to their elief model onstrints. One importnt orollry of this is tht mximizing effort inentives for ny of these mehnisms redues to omputing the effort-optiml positive-ffine trnsformtion. In Setion 5, we show how to onstrut new truthful mehnisms for new onditions, inluding detiled exmple of this onstrution in Setion 5.3. We ddress the reltive expressiveness of peer predition mehnisms in Setion 6. In prtiulr, we prove tht the lssil peer predition method is flexile enough to over ll possile mehnisms, in most ses even when restriting to the qudrti soring rule. In Setion 7, we show how to ompute mehnism tht is mximllyroust with respet to devitions etween the mehnism s nd the gents elief models. Finlly, in Setion 8 we give diretions for ongoing nd future work, inluding dt-driven non-miniml pproh to peer predition, where predition reports re used to trin lssifiers (miniml mehnisms) to sore signl reports. 2. PRELIMINARIES In this setion, we introdue the model, nd review onepts in peer predition nd omputtionl geometry Model There is group of n 2 rtionl, risk-neutrl, nd self-interested gents. We sometimes use shorthnd [n] := {1,..., n} to denote the set of gents. When interting with

4 A:4 R. Frongillo nd J. Witkowski the environment, eh gent i [n] oserves signl S i, 2 whih is rndom vrile with vlues [m] := {1,..., m} for m 2. The signl represents n gent s experiene or opinion. The ojetive in peer predition is to eliit n gent s signl in n inentiveomptile wy, i.e. to ompute pyments suh tht gents mximize their expeted pyment y reporting their signl to the mehnism (enter) truthfully. To hieve this, ll peer predition mehnisms require tht gent i s signl oservtion tells her something out the signl oserved y nother peer gent j i. For exmple, this ould e gent j = i + 1 (modulo n), so tht the gents form ring, where every gent is sored using the following gent. (Our results hold for ny hoie of peer gent.) Let then p i (s j s i ) = Pr i (S j = s j S i = s i ) (1) denote gent i s signl posterior elief tht gent j reeives signl s j given gent i s signl s i. We refer to p i ( ) s gent i s elief model. A ruil ssumption for the existene of stritly inentive omptile peer predition mehnisms is tht every gent s elief model stisfies stohsti relevne [Johnson et l. 1990]. Definition 2.1. Rndom vrile S i is stohstilly relevnt for rndom vrile S j if the distriution of S j onditionl on S i is different for ll possile vlues of S i. Tht is, stohsti relevne holds if nd only if p i ( s i ) p i ( s i ) for ll i [n] nd ll s i s i. Intuitively, one n think of stohsti relevne s orreltion etween different gents signl oservtions Peer Predition Mehnisms We re now redy to define peer predition mehnisms. For disussion of possile extensions to more generl mehnisms, see Setion 8. Definition 2.2. A (miniml) peer predition mehnism is funtion M : [m] [m] R, where M(x i, x j ) speifies the pyment to gent i when she reports signl x i nd her peer gent j reports signl x j. We use ex post sujetive equilirium [Witkowski nd Prkes 2012], whih is the most generl solution onept for whih truthful peer predition mehnisms re known. Definition 2.3. Mehnism M is truthful if we hve [ s i = rgmx E M ( ) ] x i, S j Si = s i, x i Sj for ll i [n] nd ll s i [m] with the expettion tken using gent i s elief model, i.e. S j p i ( s i ). The equilirium is sujetive euse it llows for eh gent to hve distint elief model, nd ex post euse it llows for (ut doesn t require) knowledge of other gents elief models. Agents only need to reson out other gents signls, not their eliefs. One implition of this equilirium onept is tht p i ( ) n e formulted fter hving seen the elief models of other gents. Note tht while it my e the se tht repeting this proess eventully yields ommon prior, ex post sujetive equilirium expliitly llows gents to gree to disgree. Ex post sujetive equilirium is thus stritly more generl thn the lssi definition of Byes-Nsh equilirium 2 We will drop the susript to denote generi signl.

5 A Geometri Perspetive on Miniml Peer Predition A:5 (BNE) [Hrsnyi 1968] s it oinides with BNE only when ll gents shre the sme elief model, i.e. if p i ( ) = p j ( ) for ll i, j [n]. 3 Definition 2.4. A mehnism M is positive-ffine trnsformtion of mehnism M if there exists f : [m] R nd α > 0 suh tht for ll x i, x j [m], M (x i, x j ) = αm(x i, x j ) + f(x j ). The importne of Definition 2.4 lies in the ft tht if M is truthful, then M is truthful s well. We stte this s Lemm 2.5, whih we prove in Appendix A. As we will see, in ertin ses these re the only possile truthful mehnisms. LEMMA 2.5. Let M e positive-ffine trnsformtion of M. Then M is truthful if nd only if M is truthful. We onlude with severl exmples of peer predition mehnisms from the literture to whih we refer throughout the pper. The first re the Output Agreement Mehnism, the 1/p Mehnism nd the Shdowing Method, whih we now define. We will use the nottion m to refer to the proility simplex, the set of proility distriutions over m outomes (see the following susetion). Definition 2.6. Output Agreement is M(x i, x j ) = 1 if x j = x i nd 0 otherwise. Definition 2.7. The 1/p Mehnism [Jur nd Fltings 2011] is given y M(x i, x j ) = 1 y(x if x i) i = x j nd 0 otherwise, for some fixed y m. Definition 2.8. The Shdowing Method [Witkowski nd Prkes 2012; Witkowski 2014] is M(x i, x j ) = R q (y, x j ), where y = ( y(1) δ m 1,..., y(x ) i) + δ,..., y(m) δ m 1 for some fixed y m, δ > 0, nd R q (y, x j ) = 2y (x j ) m k=1 y (k) 2 is the qudrti soring rule [Brier 1950]. THEOREM 2.9. The preeding mehnisms re truthful if nd only if the following onditions re stisfied for ll s [m], s s, nd for ll i [n]: (1) Output Agreement: p i (s s) > p i (s s) (2) 1/p Mehnism: p i (s s)/y(s) > p i (s s)/y(s ) (3) Shdowing Method: p i (s s) y(s) > p i (s s) y(s ). The proofs for eh of these sttements re in the respetive ppers ited in eh mehnism s definition ove. Note tht Theorem 2.9 does not sy whether or not there re other mehnisms whih re truthful under the onstrints (1), (2), nd (3). We will nswer this question in Theorem 4.1. Finlly, we will refer to the Clssil Peer Predition Method in Setion 6, whih relies on the notion of proper soring rule, tool for eliiting proilisti eliefs from gents given smple from the ground truth. In short, proper soring rule is funtion R : m [m] R suh tht Ep[R(p, X)] Ep[R(q, X)] where X hs distriution p, nd q p is non-truthful report. When the inequlity is lwys strit, we sy R is stritly proper. (One exmple of stritly proper soring rule is the qudrti soring rule due to Brier [1950], given in Definition 2.8.) As we note in Setion 6, ll proper soring rules n e given in terms of onvex funtions; we refer the reder to tht setion nd to Gneiting nd Rftery [2007] for detils. Definition Given stritly proper soring rule R nd elief model p i ( ), the Clssil Peer Predition Method [Miller et l. 2005] is M(x i, x j ) = R ( ) p i ( x i ), x j. 3 More modern definitions of BNE inlude sujetivity of eliefs, e.g. [Osorne nd Ruinstein 1994], nd our definition n e seen s speil se where gents hve n ritrry prior over their own signl.

6 A:6 R. Frongillo nd J. Witkowski THEOREM The Clssil Peer Predition Method is truthful if nd only if p i ( ) stisfies stohsti relevne for ll i [n] The Proility Simplex The intuition for our min results n e provided for m = 3 signls lredy, nd so we give suh exmples throughout the pper. For proility distriutions over only 3 signls, there is onvenient grphil representtion of the proility simplex m s n equilterl tringle, where the three orners represent the signls (see Figure 1L). The loser point is to orner (the distne from the orner s opposing side), the more proility mss of tht orner s signl is on tht point. 4 The tringulr shpe ensures tht for ny point on the tringle the vlues of the three dimensions sum up to 1. For exmple, the point p i ( ) = (0.2, 0.75, 0.05) in Figure 1L is t height 1/20 (sine the top orner represents signl ), nd one fifth wy from the right side of the tringle (euse the left orner represents signl ). Oserve tht with three signls, there re only two degrees of freedom, nd so fixing the point s position with respet to nd, the vlue for signl is fixed s well. (Confirm tht p i ( ) is three fourths wy from the left side.) 2.4. Power Digrms Our results rely on onept from omputtionl geometry known s power digrm, whih is type of weighted Voronoi digrm [Aurenhmmer 1987]. Definition A power digrm is prtitioning of m into sets lled ells, defined y olletion of points {v s R m : s [m]} lled sites with ssoited weights w(s) R, given y { } ell(v s ) = u R m : {s} = rgmin{ u v x 2 w(x)} x [m]. (2) We ll u v x 2 w(x) the power distne from u to site v x ; thus, for every point u in ell(v s ), it holds tht v s is loser to u in power distne thn ny other site v x. We hve defined power digrms for the speil se of the proility simplex, whih is the se we need in this pper. The more generl definition llows for different numer of sites thn dimensions. Also, note tht we exlude ell oundries y ensuring tht we hve unique minimizer in eq. (2); in the following we drop the set nottion round s nd just write s = rgmin{...} or s = rgmx{...}. The usul definition of Voronoi digrm follows y setting ll weights w(s) to MECHANISMS AND POWER DIAGRAMS As with previous work, we would like to mke sttements of the form, As long s the elief models stisfy ertin onstrints, the mehnism is truthful. For exmple, the Shdowing Method (Definition 2.8) is truthful if nd only if p i (s s) y(s) > p i (s s) y(s ) for ll s, s [m] : s s, ll i [n], nd some distriution y, whih is prmeter of the mehnism. When used diretly, nd not s uilding lok for more omplex mehnisms, it is often ssumed tht there is known, ommon signl prior, whih is then used s y. As we will see, oth the Shdowing Method nd the 1/p mehnism [Jur nd Fltings 2008; Jur nd Fltings 2011] re tully roust in tht they re truthful even if there is no ommon signl prior. All tht is required is tht the gents possile posteriors fll into the orret regions. While it hs een known tht the onstrints required y the Shdowing Method nd the 1/p mehnism re inomprle, 4 This is equivlent to the nturl emedding into R 3 nd viewing in the diretion ( 1, 1, 1).

7 A Geometri Perspetive on Miniml Peer Predition A:7 (L) (M) (R) p i ( ) D p i ( ) p i ( ) p i ( ) p i ( ) p i ( ) D D Fig. 1: Belief models for whih OA is not truthful (L) nd is truthful (M). The elief models for whih OA is truthful re hrterized y the ells D, D, D, where the posterior following signl s must lie in D s (R). i.e. there exist elief models for whih the Shdowing Method is truthful ut the 1/p mehnism is not, nd vie vers [Witkowski 2014], it ws not known for whih onstrints there exist truthful mehnisms. In this setion, we nswer this question, nd hrterize ll elief model onstrints for whih truthful mehnisms exist Belief Model Constrints To egin, onsider the Output Agreement mehnism (OA), with two possile elief models p i ( ) nd p i ( ) s depited in Figures 1L nd 1M, respetively: p i( ) = p i ( ) = (0.6, 0.3, 0.1), p i( ) = p i ( ) = (0.2, 0.75, 0.05), p i( ) = (0.2, 0.55, 0.25), nd p i ( ) = (0.2, 0.25, 0.55). For whih of the elief models p i( ), p i ( ) is OA truthful? Upon inspetion, one n verify tht OA is truthful with respet to p i ( ) ut not p i( ). The key point is tht upon oserving signl, p i ( ) = 0.55 > p i ( ), mening under p i ( ), the gent thinks tht the peer gent is most likely to hve seen (nd reported), so signl will e the optiml report. Clerly then, OA is not truthful with respet to every elief model, nd one my nturlly sk, for whih elief models is OA truthful? As it turns out, one n desrie ll suh elief models using very simple elief model onstrint, of the form p i ( s) D s for ll signls s, for some sets of distriutions D s. For OA, this onstrint is given y the sets D s = {p s s, p(s) > p(s )}, s depited in Figure 1R. We will revisit this onstrint in Setion 3.2. We now formlly define these onstrints on elief models tht limit whih posteriors re possile following whih signl. Definition 3.1. A elief model onstrint is olletion D = {D s m : s [m]} of disjoint sets D s of distriutions. If dditionlly we hve l( s D s ) = m, i.e. if D prtitions the simplex, we sy D is mximl. A elief model onstrint D = {D 1,..., D m } ensures tht for eh gent i, following signl oservtion S i = s i, her elief out her peer gent s signl s j is restrited to e in D si. It is esy to ome up with non-mximl elief model onstrints, suh s s p(s s) > Note tht under suh onstrint, some distriutions re not vlid posteriors for ny signl. In ontrst, mximl onstrint overs the simplex, prtitioning it into m ordering ut non-overlpping regions (e.g. Figure 1R). We n now tlk out mehnisms eing truthful with respet to elief model onstrint, whih ensures tht gents will report honestly s long s their elief model stisfies the onstrint. 5 See Figure 4M for n illustrtion, s well s the surrounding disussion.

8 A:8 R. Frongillo nd J. Witkowski Definition 3.2. A mehnism M(, ) is truthful with respet to elief model onstrint D if M is truthful whenever p i ( s) D s for ll gents i [n] nd ll signls s [m]. It diretly follows from this perspetive tht ll miniml peer predition mehnisms require elief model onstrint. 6 Consider, for exmple, posterior elief p i ( s) = (3/5, 3/20, 1/4). Without ny onstrint on the elief model, it is not ler if this is the posterior following signl 1, 2, or 3. This hoie needs to e mde sine given posterior elief n only elong to one signl (stohsti relevne, Definition 2.1), nd so every truthful miniml peer predition mehnism requires elief model onstrint. One very nturl onstrint to onsider is to tke n ritrry mehnism M nd restrit to only those elief models under whih M is truthful. It turns out tht this set n e suintly desried y elief model onstrint, whih we ll the onstrint indued y M. Moreover, the regions of this indued onstrint must tke prtiulr shpe, tht of power digrm, nd onversely, every power digrm is n indued onstrint of some mehnism. We will now oserve tht for ny mehnism M, there is elief model onstrint D M, whih extly ptures the set of elief models for whih M is truthful. In other words, not only is M truthful with respet to D M, ut under ny elief model tht does not stisfy D M, M will not e truthful. The onstrution of D M is esy: for eh signl s, Ds M is the set of distriutions p( s) under whih x i = s is the unique optiml report for M. Note tht if Ds M is empty for ny s, then M is not truthful for ny elief model. LEMMA 3.3. Let M : [m] [m] R e n ritrry mehnism, nd let D M e the elief model onstrint given y { } Ds M = p i ( s) : s = rgmx E M(x i, S j ). (3) x i S j p i( s) Then M is truthful with respet to D M, ut not truthful for elief models not stisfying D M. Moreover, if the rows of M re ll distint, D M is mximl. PROOF. Suppose p i ( s) Ds M for ll i [n], s [m]. Then y onstrution of Ds M, n gent i reeiving signl s mximizes expeted pyoff y reporting s, nd hene M is truthful. By definition then, M is truthful with respet to D M. Now suppose p i ( s) / Ds M for some i [n], s [m]. Then s rgmx xi E Sj p i( s) M(x i, S j ), nd thus M nnot e truthful. Finlly, onsider the funtion G(p) = mx x ES j p M(x, S j ), whih is onvex s pointwise mximum of liner funtions. By stndrd results in onvex nlysis (.f. [Frongillo nd Ksh 2014, Theorem 3]) G hs sugrdient M(x, ) whenever x is in the rgmx. As the rows of M re ll distint, multiple elements in the rgmx orresponds to multiple sugrdients of G, nd thus G is nondifferentile 7 t the set of indifferene points {p : rgmx x ES j p M(x, S j ) 2}. As G is onvex, it must e differentile lmost everywhere [Aliprntis nd Border 2007, Theorem 7.26], these indifferene points must hve mesure 0 in the proility simplex, nd thus D M is mximl. The onstrution ove in Eq. 3 shows even more thn the ft tht D M is mximl: the ells Ds M must e onvex sets. (One n see this diretly y tking two posteriors for whih s is the rgmx, nd oserving tht s must e the rgmx for ny mixture 6 Jur nd Fltings [2011] were the first to stte tht no miniml mehnism n e truthful for ll stohstilly relevnt elief models. However, they did not study the type of onstrints tht need to e imposed to llow for truthful mehnisms. 7 Tehnilly, we should restrit to the first m 1 oordintes of the distriution, or use the pproh of Appendix B, so tht G is defined on full-dimensionl suset of R m 1. Neither lters the rgument.

9 A Geometri Perspetive on Miniml Peer Predition A:9 of the two.) A onnetion to property eliittion in the next susetion will revel even more struture: D M must e the ells of power digrm Reltionship to Finite Property Eliittion Our results rely hevily on novel onnetion from miniml peer predition mehnisms to the literture on property eliittion, where one wishes to extrt prtiulr funtion, or property, of n gent s elief using soring rule with ess to single smple from the true distriution (unlike our setting, where no ground truth is ever oserved). Formlly, soring rule S(, ) eliits property Γ if for ll gent eliefs p, the expeted sore Ex p[s(r, x)] is mximized y the report r = Γ(p). In prtiulr, we leverge results from the finite property se, where the reports r re restrited to finite set [Lmert nd Shohm 2009]. For exmple, the mode of distriution over m possile outomes, Γ mode (p) = rgmx x p(x), hs m possile vlues, nd is eliited y the soring rule S(r, x) = 1{r = x} where 1{} is the inditor funtion. (This follows y writing out the expeted sore Ex p[s(r, x)] = Ex p[1{r = x}] = p(r), whih is mximized when r is equl to the mode of p.) In peer predition, one n view elief model onstrints s kind of finite property in the ove sense. Let D e mximl elief model onstrint, nd let us enode it in property Γ D : m [m] y tking Γ D (p) = s if p D s. (Stritly speking, this does not desrie Γ D everywhere s the ells D s re open nd disjoint; we defer the detils to the proofs in Setion 4.) Now let M e miniml mehnism. If M is truthful for D, then M is truthful with respet to ny elief model stisfying D. In prtiulr, if q( ) is gent i s posterior following some signl, then q D s for some s (gin, ignoring oundries); thus y the elief model onstrint D, q must e the posterior following s, nd y M eing truthful for D, gent i will report s. Summrizing, we hve just sid tht whenever n gent hs elief q, the report mximizing their expeted sore is the s suh tht q D s, whih is Γ D (q) y definition. Thus, M eing truthful for D implies tht M, thought of s soring rule, eliits Γ D. The onverse follows y extly the sme logi, where now M eliiting Γ implies tht M is truthful for D defined y the onstrint Γ(p i ( s)) = s, whih is extly the indued onstrint D M s one n verify from Eq. 3. Now tht we know miniml mehnisms eliit finite properties, we n sk whih properties re eliited y populr mehnisms in the literture. It follows immeditely from the ove disussion tht Output Agreement, M(x i, x j ) = 1{x i = x j }, eliits the mode Γ mode (p) = rgmx s p(s). Similrly, the Shdowing Method eliits shifted mode, Γ(p) = rgmx s p(s) y(s), tht is, the mode of the distriution q(s) = p(s) y(s). (Note tht q my lie outside the simplex m.) Finlly, the 1/p mehnism n e thought of s eliiting dmpened mode Γ(p) = rgmx p(s)/y(s), whih gin is the mode of distriution q(s) = p(s)/y(s) Equivlene to Power Digrms Leverging the onnetion to finite properties estlished ove, we n now use results in the property eliittion literture, whih show n equivlene etween eliitle finite properties nd power digrms [Lmert nd Shohm 2009; Frongillo nd Ksh 2014], nd extend this equivlene to miniml peer predition mehnisms. From there, tehniques from omputtionl geometry llow us to show further struture, s we explore in Setions 4 nd 6. To simplify exposition, we will show the orrespondene to power digrms diretly, without referene to finite properties, though properties will pper gin in the proofs. We hve seen tht every mehnism M indues some elief model onstrint D M, nd tht M is truthful with respet to D M. We now show further tht D M is power

10 A:10 R. Frongillo nd J. Witkowski digrm, nd onversely, tht every power digrm hs mehnism suh tht D M s = ell(v s ) for ll s. The onrete mpping is s follows. Given mehnism M : [m] [m] R, we onstrut sites nd weights y: v s = M(s, ), w(s) = v s 2 = M(s, ) 2. (4) Conversely, given power digrm with sites v s nd weights w(s), we onstrut the mehnism M s follows: M(x i, x j ) = v xi (x j ) 1 2 vxi w(x i), (5) where v xi (x j ) is the x j th entry of v xi. We note tht these formuls re more expliit versions of those ppering in property eliittion, s mentioned ove. With these onversions in hnd, we n now show tht they indeed estlish orrespondene etween miniml peer predition mehnisms nd power digrms. THEOREM 3.4. Given ny mehnism M : [m] [m] R, the indued elief model onstrint D M is power digrm. Conversely, for every power digrm given y sites v s nd weights w(s), there is mehnism M whose indued elief model onstrint D M stisfies Ds M = ell(v s ) for ll s. PROOF. Oserve tht if either reltion (4) or (5) holds, we hve the following for ll x, p: 2p v x + v x 2 w(x) = 2 E Sj p [M(x, S j )]. (6) To see this, note tht p M(x, ) = ES j p[m(x, S j )]. Adding p 2 to oth sides of Eq. 6 gives p v x 2 w(x) = p 2 2 E Sj p [M(x, S j )]. (7) Now pplying Eq. 7 to the definitions of power digrm nd of D M, we hve p ell(v s ) s = rgmin{ p v x 2 w(x)} x { } s = rgmin p 2 2 E [M(x, S j )] x Sj p s = rgmx x p D M s. E S j p( s) M(x, S j ) Finlly, s Eq. 4 defines power digrm for ny mehnism M, nd Eq. 5 defines mehnism for ny power digrm, we hve estlished our equivlene. COROLLARY 3.5. Let D e mximl elief model onstrint. Then there exists mehnism tht is truthful with respet to D if nd only if D is power digrm. Corollry 3.5 gives us onsiderle power in determining whether or not elief model onstrint n yield truthful mehnism. On the positive side, Figure 2L shows mximl elief model onstrint tht is power digrm, nd thus there must e truthful mehnism (in this se, the Shdowing Method). On the negtive side, we n leverge known ttriutes of power digrms, suh s the ft tht their ells re onvex, their oundries re liner/ffine, nd moreover the oundry etween two ells must e perpendiulr to the line segment onneting the orresponding sites. Figure 2M shows mximl onstrint in whih the D ell is nononvex, nd thus nnot e power digrm; we onlude there n e no mehnism truthful

11 A Geometri Perspetive on Miniml Peer Predition A:11 (L) (M) (R) D D D D y D D D D D Fig. 2: (L) The regions D s re those for whih the Shdowing Method inentivizes the gents to report the respetive signl s. For exmple, for ny posterior elief flling into D, gent i should report x i =. To see tht the depited prtitioning is indeed oming from this onstrint, first oserve tht there is indifferene t the intersetion point, i.e. when p i ( s) = y( ), nd tht the onstrints re liner, so tht the indifferene orders re lines. The only remining piee is then to determine the points for eh pir of signls, where the third (left-out) signl s weight is 0, nd drw line etween tht point nd y. For exmple, the indifferene point etween signls nd, where hs no weight is (0, 7/12, 5/12) p i ( ) 1/6 = p i ( ) 1/3; p i ( ) = 0. (M) A mximl elief model onstrint whih is not power digrm nd hene for whih there is no truthful mehnism. (R) A non-mximl elief model onstrint for whih there is still no truthful mehnism, euse there is no power digrm onsistent with the onstrints. with respet to this onstrint. 8 Finlly, Figure 2R shows onstrint whih is not mximl, ut whih is not onsistent with ny power digrm (using the linerity of ell oundries), nd hene there still nnot e truthful mehnism. These exmples illustrte the power of this geometri pproh in determining whether truthful mehnisms exist, even for non-mximl onstrints. Finlly, it is esy to see tht the onversion from mehnisms to sites nd weights of power digrms (Eq. 4) nd k (Eq. 5) re inverse opertions. In the following setion, we will leverge this tight onnetion, nd use results from omputtionl geometry to show tht severl well-known mehnisms re unique in the sense tht they re the only mehnisms, up to positive-ffine trnsformtions, tht re truthful for their respetive elief model onstrints. 4. UNIQUENESS Consider gin the the Output Agreement mehnism M(x i, x j ) = 1 if x j = x i nd 0 otherwise. It is esy to see tht the mehnism is truthful s long s eh gent ssigns the highest posterior proility p i ( s i ) to their own signl s i, yielding the onstrint p(s s) > p(s s) s s. The type of question we ddress in this setion is: re there ny other mehnisms thn M tht re gurnteed to e truthful s long s posteriors stisfy this ondition? We will show tht, up to positive-ffine trnsformtions, the nswer is no: output greement is unique. Moreover, the Shdowing Method nd the 1/p Mehnism re lso unique for their respetive onditions on posteriors. Note tht we identify mehnisms with their mtries, so tht uniqueness does not prelude tht prmeterized mehnisms, suh s the 1/p Mehnism nd the Shdowing Method, oinide for prtiulr hoies of prmeters. In ft, for uniform y, oth the 1/p Meh- 8 It is worth noting tht even if the D ws djusted to e stright (forming T intersetion with the oundry), it would still not e power digrm; even though the D ell would now e onvex, the digrm would violte the perpendiulrity ondition (this is not immeditely ovious, ut eomes ler with experimenttion).

12 A:12 R. Frongillo nd J. Witkowski v y v v Fig. 3: Construting mehnism tht is truthful under the sme onditions s output greement. nism nd the Shdowing Method oinide with Output Agreement in the sense tht their elief model onstrints re identil. (See Setion 6.) To get some intuition for this result, let us see why output greement is unique for m = 3 signls,,. From Theorem 3.4, we know tht D = D M, the indued elief model onstrint, is power digrm (whih is depited in Figure 3). In generl, there my e mny sites nd weights tht led to the sme power digrm, nd these my yield different mehnisms vi Eq. 5. In ft, it is generl result tht ny positive sling of the sites followed y trnsltion (i.e. some α > 0 nd u R m so tht ˆv s = αv s +u for ll s) will result in the sme power digrm for n pproprite hoie of weights [Aurenhmmer 1987]. As it turns out, suh slings nd trnsltions extly orrespond to positive-ffine trnsformtions when pssing to mehnism through Eq. 5. Thus, we only need to show tht the sites for the output greement power digrm re unique up to sling nd trnsltion. Here, nother useful property of sites omes into ply: the line etween sites of two djent ells must e perpendiulr to the oundry etween the ells. Exmining Figure 3, one sees tht fter fixing v, the hoie of v is onstrined to e long the lue dotted line, nd one v nd v re hosen, v is fixed s the intersetion of the red dotted lines. Thus, we n speify the sites y hoosing v ( trnsltion), nd how fr wy from v to ple v ( positive sling). We n then onlude tht output greement for three signls is unique up to positive-ffine trnsformtions. We now give the generl result. THEOREM 4.1. If there exists mehnism M tht is truthful for some mximl elief model onstrint D, nd there is some y m with y(s) > 0 s suh tht s l(d s ) = {y}, then M is the unique truthful mehnism for D up to positive-ffine trnsformtions. PROOF. As M is truthful with respet to D, we hve D = D M nd thus D is power digrm P from Theorem 3.4. By our ssumption, we oserve tht the only vertex (lso lled 0-dimensionl fe) of P must e y, the intersetion of the l(d s ), s there re m ells ut m hs dimension m 1. Throughout the proof, s efore, we impliitly work in the ffine hull A of m. We my ssume without loss of generlity tht ll sites lie in A, s we my trnslte ny site to A while djusting its weight to preserve the digrm nd resulting mehnism (see Appendix B). Thus, y definition of simple ell omplex, 9 s the vertex y is in the reltive interior of m, we see tht the extension ˆP of P onto the ffine hull of m must lso hve y s the only 9 A ell omplex P is simple if eh of P s verties is vertex of extly m ells of P, the minimum possile [Aurenhmmer 1987, p.50].

13 A Geometri Perspetive on Miniml Peer Predition A:13 vertex. (As vertex is on the oundry of every ell, nd there re m ells in m 1 dimensions with disjoint interiors, there n e only one suh point.) Now following the proof of Frongillo nd Ksh [2014, Theorem 4], we note tht Aurenhmmer [1987, Lemm 1] nd Aurenhmmer [1987, Lemm 4] together imply the following: if ˆP is represented y sites {v s } s [m] nd weights w( ), then ny other representtion of ˆP with sites {ˆv s } s [m] stisfies α > 0, u R m s.t. ˆv s = αv s + u for ll s [m]. In other words, ll sites must e trnsltion nd sling of {v s }. To omplete the proof, we oserve tht different hoies of u nd α (with suitle weights) merely yield n ffine trnsformtion of M when pssed through Eq. 5, nd s ny positive-ffine trnsformtion preserves truthfulness, the result follows. As Corollry of Theorem 4.1, we n now prove stronger version of Theorem 2.9. Not only re the three mehnisms truthful with respet to the orresponding onstrints, ut they re the only suh mehnisms, up to positive-ffine trnsformtions. COROLLARY 4.2. The following mehnisms re unique, up to positive-ffine trnsformtions, with respet to the orresponding onstrints (eh with s s nd i [n] implied): (1) Output Agreement, p i (s s) > p i (s s) (2) 1/p Mehnism, p i (s s)/y(s) > p i (s s)/y(s ) (3) Shdowing Method, p i (s s) y(s) > p i (s s) y(s ). PROOF. In ll three ses, the given mehnism is known to e truthful for its respetive elief model onstrint. Moreover, for ll three onstrints D, one n hek tht s l(d s ) = m nd s l(d s ) = {y} mening tht y is the unique distriution ordering every set D s (for Output Agreement, y is the uniform distriution). Hene, the mehnisms re unique up to positive-ffine trnsformtions y Theorem 4.1. Let us mke few remrks on Theorem 4.1. First, note tht the restrition tht y must not touh the oundry of the simplex is neessry, s uniqueness will not hold otherwise; see Figure 4L. Similrly, for onstrints D tht re not mximl, there my e mny more truthful mehnisms; Figure 4M depits two distint power digrms yielding mehnisms tht re truthful with respet to the non-mximl onstrint p(s s) > 0.6, nd thus they re not merely positive-ffine trnsformtions of eh other. Tht sid, some non-mximl onstrints D still yield unique truthful mehnism, s illustrted in Figure 4R. In Setion 6, we strengthen the results of this setion, showing tht under the sme onditions s Theorem 4.1, not only is the mehnism unique up to positive-ffine trnsformtions, ut it n e expressed s positive-ffine trnsformtion of lssil peer predition mehnism with respet to the qudrti soring rule. This is done y first showing tht in this setting, the mximl onstrint D is not just power digrm ut in ft Voronoi digrm. 5. CONSTRUCTING MECHANISMS FOR NEW CONDITIONS In this setion, we show how to ompute new mehnisms tht re truthful with respet to new onditions. Moreover, we find the positive-ffine trnsformtion tht mximizes effort inentives sujet to udget. It then follows diretly from Theorem 4.1 tht the finl mehnism s effort inentives re glolly optiml given this ondition. Tht is, there is no peer predition mehnism tht is truthful with respet to the new ondition providing etter effort inentives Computing Truthful Mehnisms To onstrut mehnisms from elief model onstrints, we turn to omputtionl geometry to find sites nd weights for the orresponding power digrm. For the lss

14 A:14 R. Frongillo nd J. Witkowski (L) (M) (R) v D D v y ˆv v D D D D Fig. 4: Illustrtions of the onditions of Theorem 4.1. (L) An exmple showing why the ondition s y(s) > 0 is neessry. In this exmple, there re two sets of sites nd weights tht represent the sme power digrm yet re not trnsltions nd slings of eh other. Here, ll sites remin the sme exept v. (M) A non-mximl onstrint, with two onsistent power digrms (red nd lue). (R) An exmple where the onstrint is non-mximl (here the grey region elongs to none of the ells) yet there is unique truthful mehnism (in this se, Output Agreement). of onstrints stisfying Theorem 4.1, where there is n intersetion point y in the interior of the simplex, whih is the most ommon se in peer predition, one n rely on the O(m) time lgorithm given y Aurenhmmer [1987]. 10 We now give version of this proedure, speilized for our setting. The first onsidertion is the form of the input to the proedure. How should we ssume the elief model onstrint D is given? One suint representtion is n m m mtrix representing truthful mehnism for the onstrint, ut of ourse speifying onstrint with mehnism defets the whole purpose of the lgorithm. To e useful, we need representtion for the elief model onstrint tht is loser to the prtitioning point of view. Here, we ssume tht one knows the onstrint oundry etween ny pir of signls (s, s ). If D is power digrm ( neessry ondition for the existene of truthful mehnism y Theorem 3.4) this oundry must e defined y hyperplne, whih in turn n e represented y its defining norml vetor u s,s. All tht remins is speifying the orienttion of u s,s so we know whih diretion is whih ell; here we ssume u s,s points in the diretion wy from D s, so tht we hve p D s u s,s p < 0 for ll s s. As n exmple, the shdowing onstrint is given y u s,s = 1 s 1 s (y(s ) y(s))1, where 1 s is the stndrd unit vetor (with 1 in oordinte s nd 0 otherwise), nd 1 is the ll-ones vetor. Hene, u s,s p < 0 p(s ) p(s) y(s ) + y(s) < 0, sine 1 p = 1. We give nother exmple in Setion 5.3. Thus, given ny mximl elief model onstrint D, we n represent D y olletion of vetors {u s,s R m : s, s [m]}. With this representtion in hnd, the following lgorithm omputes truthful mehnism for D, under the dditionl ssumption tht D stisfies the onditions of Theorem 4.1. For the orretness of Algorithm 1, first oserve tht line i of the lgorithm hs unique solution y the ssumption of Theorem 4.1. We must lso show uniqueness in line 3. This follows y the result of Theorem 4.1, whih shows tht two different olletions of sites representing these onstrints must e relted y positive-ffine trnsformtion. To see this, oserve tht one we hve fixed v s nd v s, the rest of the 10 This ondition is speil se of the so-lled simple power digrms; see footnote 9. For more omplited ses, one would use [Rynikov 1999].

15 A Geometri Perspetive on Miniml Peer Predition A:15 Algorithm 1 Compute Mehnism from Belief Model Constrint i. Solve for y s the unique solution to u s,s y = 0 s, s nd y 1 = 1. ii. Let û s,s = u s,s ( 1 m us,s 1)1 for ll s, s. # Projets u s,s onto ffine hull of m 1: Choose s [m] nd ny v s in the ffine hull of m. 2: Choose s s nd ny α > 0, nd set v s = v s + αû s,s. 3: For ll s / {s, s }, find the unique positive solution (β, γ) to αû s,s + βû s,s = γû s,s, nd set v s = v s + γû s,s. # See rgument elow 4: # The point y must hve equl power distne to ll sites: Set w(s) = 0 nd w(s ) = y v s 2 y v s 2 for ll s s. 5: Compute the mehnism y pplying Eq. 5. sites re uniquely determined. 11 Consider some positive-ffine trnsformtion of the sites tht keeps v s nd v s. As v s hs not moved, the trnsltion must e 0, nd s v s lso remins the sme, so does the length of the line segment etween v s nd v s, so the sling must e 1. We hve now determined the positive-ffine trnsformtion: the identity. Thus ll other sites re uniquely determined y the first two. Finlly, one n hek tht the projetions û s,s re orretly oriented nd remin perpendiulr to the oundry of ells D s nd D s Optimizing Effort Inentives From Setion 5.1, we know how to ompute mehnism tht is truthful with respet to given elief model onstrint. In this setion, we tke this one step further nd optimize within the spe of truthful mehnisms. As explined in Setion 1, peer predition mehnisms re espeilly useful for inentivizing effort, i.e. the ostly quisition of signls, nd we will thus ddress the following optimiztion prolem: mx effort inentives e i (M) s.t. truthfulness with respet to D udget onstrint M(x i, x j ) B non-negtive pyments M(x i, x j ) 0 Effort n e modeled in mny different wys. Following Witkowski [2014], we model effort s inry hoie: gents either exert effort or not. In ontrst to the rest of the pper, where gent i hs oserved her signl nd resons out the est report given tht signl, here the hoie of whether to invest effort lso depends on gent i s prior elief out her own signl. In this setion, we ssume tht gents re exhngele, so tht gent i s prior elief out her own signl is the sme s her prior elief out gent j s signl. Let then p i (s i ) = Pr i (S i = s i ) = p i (s j ) denote gent i s signl prior. Note tht for n gent s signl prior nd elief model, y Byes rule it holds tht p i (s j ) = m k=1 p i(s j k) p i (k). Definition 5.1. Given tht gent j invests effort nd reports truthfully, the effort inentive e i (M) tht is implemented for gent i y peer predition mehnism M is the differene in expeted utility of investing effort followed y truthful reporting nd not (8) 11 Note tht we re restriting the sites to the ffine hull of the simplex, whih is simply given y the ondition v 1 = 1; see Appendix B. Without this, there would e n extr degree of freedom in hoosing the sites, ut s tht setion shows, this is irrelevnt. 12 Letting p, q m e distint points on the oundry of these ells, nd letting z = p q, we hve û s,s z = u s,s z ( 1 m us,s 1)1 z = u s,s z s p 1 = q 1 = 1, so 1 z = 0.

16 A:16 R. Frongillo nd J. Witkowski investing effort, i.e. e i (M) = [ ] [ ] E M(S i, S j ) mx E M(x i, S j ), Si,S j x i [m] S j where x i is gent i s signl report tht mximizes her expeted utility ording to the signl prior, nd where the expettion is using gent i s sujetive elief model p i ( ). Thus, the effort inentive e i (M) is gent i s expeted gin y exerting effort. Nturlly, sling mehnism should sle the inentives; in ft, this n e generlized to positive-ffine trnsformtions. LEMMA 5.2. For ny mehnism M, nd ny positive-ffine trnsformtion M = αm + f, we hve e i (M ) = αe i (M). PROOF. This follows from simple omputtion: [ ] [ ] e i (M ) = E M (S i, S j ) mx E M (x i, S j ) Si,S j x i [m] S j [ ] [ = E αm(s i, S j ) + f(s j ) mx E Si,S j x i [m] S j ] [ ] + E f(s j ) Sj [ = α E M(S i, S j ) Si,S j = αe i (M). ] αm(x i, S j ) + f(s j ) [ α mx E M(x i, S j ) x i [m] S j ] [ ] E f(s j ) Sj From Setion 4 we know tht the spe to optimize over is restrited to positiveffine trnsformtions of ny truthful mehnism one the elief model onstrint is fixed nd given the onditions of Theorem 4.1. Using this, we n pin down the effortmximizing mehnism s given y the following theorem. THEOREM 5.3. Let mehnism M nd elief model onstrint D stisfy the onditions of Theorem 4.1. Let g(x j ) = min xi M(x i, x j ), G = mx xi,x j M(x i, x j ) g(x j ), nd α = B/G. Then mehnism M (x i, x j ) = αm(x i, x j ) αg(x j ) optimizes effort in Eq. 8. PROOF. We will show something slightly stronger, giving the full hrteriztion of mehnisms optimizing Eq. 8. By Theorem 4.1, M is the unique truthful mehnism for D up to ffine trnsformtions M α,f (x i, x j ) = αm(x i, x j ) + f(x j ), so we need only serh for the optiml α, f. By Lemm 5.2, e i (M α,f ) = αe i (M), whih sine e i (M) is positive onstnt redues the optimiztion to the following: mx s.t. α αm(x i, x j ) + f(x j ) [0, B] x i, x j f R m, α R +. Let G = mx xj ( mxxi M(x i, x j ) min xi M(x i, x j ) ) denote the mximum pyment differene within ny olumn of M. We see tht the lrgest vlue of α one ould hope for is α = B/G sine the finl mehnism should hve vlues in [0, B], nd in prtiulr the differene etween the minimum pyment nd mximum pyment of the finl mehnism must e t most B. Indeed, tking this α, one sees tht ny f in the rnge f(x j ) [ α min xi M(x i, x j ), α(mx xi M(x i, x j ) min xi M(x i, x j ))] will stisfy the onstrints. One n hek tht in prtiulr tking f(x j ) = α min xi M(x i, x j ) gives the mehnism in the theorem sttement, whih gives the lowest possile pyments mong the optiml mehnisms. As f = αg, we re done. To onfirm tht M (x i, x j ) B for ll x i, x j [m], note tht M (x i, x j ) = B ( M(x i, x j ) g(x j ) ) /G =

17 A Geometri Perspetive on Miniml Peer Predition A:17 v v y Fig. 5: The truthful mehnism with respet to the omplement 1/p ondition in power digrm form with sites v, v, nd v. Notie tht while the intersetion point y = (1/2, 1/3, 1/6) is the sme s in Figure 2L, the elief model onstrint s depited y the dshed prtitioning is now given y the new omplement 1/p ondition. (Compre to Figure 3.) B M(x i,x j) min x i M(x i,xj) mx x j ( mx x i M(x i,x j ) min x i M(x i,x j ) ) v B, s the numertor is differene of pyoff vlues within olumn of M, nd G y definition is the lrgest suh differene in ny olumn of M. Similrly, for ll x i, x j [m] we hve M (x i, x j ) = α(m(x i, x j ) min x i M(x i, x j )) 0 s α > 0 nd M(x i, x j ) min x i M(x i, x j). Note tht knowledge of only D is not suffiient to ensure tht speifi effort ost C > 0 is implemented (e.g. C = 1) sine the mgnitude of n gent s effort inentive will depend on the prtiulr vlues of their posteriors {p i ( s)} s [m]. In prtiulr, the gents posteriors ould ll e ritrrily lose to the intersetion point, in whih se the required sling α to meet effort ost C = 1 would e ritrrily high. Even knowing only D nd not the gent s tul elief model, however, Theorem 5.3 gives the highest possile sling within the udget (nd thus the highest possile effort inentives) of ny mehnism implementing D, no mtter the gent s elief model. To illustrte the power of Theorem 5.3, onsider the Shdowing Method (Definition 2.8), whih hs prmeter δ speifying how muh to pertur y in order to otin the shdow posterior y. A nturl question to sk is for whih δ the effort is mximized sujet to the onstrints in Eq. 8. (One n hek tht the resulting power digrm is the sme for ny suh δ.) A diret nlysis is quite tedious, with mny lines of lger, nd even with the optiml δ in hnd, it is not ler whether one ould do etter y perhps dding sore f(x j ) tht depends only on the peer gent s signl report x j, or y llowing for shdow posteriors tht ren t vlid distriutions followed y renormlizing the soring rule so tht the sore is gin ounded y [0, B]. Moreover, the optiml mehnism ould hve een of different form entirely. Theorem 5.3 suggests etter pproh to this prolem: tke the Shdowing Method with ny δ > 0 s lk ox nd simply find the mximum sling llowing trnsltion tht keeps the sores in the intervl [0, B] Exmple We now illustrte Algorithm 1, onstruting new mehnism tht is truthful with respet to new ondition. Moreover, we ompute the optiml suh mehnism with respet to the effort it inentivizes s explined in Setion 5.2. As intuition for the new ondition, imgine the mehnism hs n estimte of the gents signl priors p(,, ) = (0.01, 0.04, 0.95), whih it designtes s the intersetion point y( ) = p( ) of the elief model onstrint. Consider now posterior