Bayesian Truth Serum and Information Theory: Game of Duels

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1 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < Bayesian Tuth Seu and Infoation Theoy: Gae of Duels Jaša Cvitanić, Dažen Pelec 2, Sonja Radas 3, and Hvoje Šiić 4 Abstact This pape ais to develop insights into Bayesian Tuth Seu algoith fo the pespective of Shannon theoy. We postulate a natual sequence of seven axios that poduce Bayesian Tuth Seu scoing ule in such a way that it eflects quality of infoation. This aes it possible to egad Bayesian Tuth Seu as a easue of cobined infoation-pediction quality in situations whee espondents ae ased to choose an altenative fo a finite set and povide pedictions of thei pees popensities to choose. This is possible fo finite and infinite sets of espondents. Index Tes Bayesian Tuth Seu, infoation entopy, Bayesian gae, Shannon theoy B I. INTRODUCTION ayesian Tuth Seu algoith was developed in []. The Bayesian Tuth Seu equies a single ultiple-choice question. The espondents ae ased, in addition to poviding thei pesonal answe, to pedict the pecentage distibution of answes in the entie saple. The algoith has applications in studies of public o expet opinions: voting intentions, poduct atings, expet foecasting, and vaious cowdsoucing applications. Published studies of Bayesian Tuth Seu include application to assess the degee of leaning in design couses [2], validity of deteence in ciinology [3], incentivizing digital piates confession [4], and new poduct adoption in phaaceutical setting [5]. The algoith has been studied fo the econoics point of view, its scoing ule aspects have been exploed, but as fa as we now it has not been appoached fo the Shannon theoy point of view. The pupose of this pape is to ephasize that thee is a connection between the Bayesian Tuth Seu algoith and the Shannon Theoy and that it lies at the vey essence of the algoith itself. Thee ae two fundaental featues of the Bayesian Tuth Seu algoith. One is that it is (tuth) incentive-copatible (i.e. assuing that all othe espondents tell the tuth, it is in expectation ost pofitable fo the selected espondent to tell the tuth as well). Although highly desiable and ipotant fo the geneal point of view, this aspect is not of inteest hee. The second fundaental featue of the Bayesian Tuth Seu algoith is that its scoing ule ans playes accoding to thei posteio pobability on the actual pecentage distibution of answes in the saple []. If that distibution is egaded as the tue state of natue, this aning ay be intepeted as a aning accoding to doain expetise [6, 7]. The Bayesian Tuth Seu algoith is not the only incentive-copatible algoith with this aning popety, as shown in [6]. In this pape we focus on anings, but appoach the poble without using Bayesian gae theoy. Instead of having one gae whee any playes povide thei one-tie answes and pedictions, we view this situation in tes of paiwise duels between playes. These duels esult in tansfe of points between the playes accoding to soe function P. We use siple siultaneous duels to show that unde natual set of axios copatible with Shannon theoy we can deive the sae scoing ule as []. In the est of the pape we pesent the Bayesian Tuth Seu algoith foally in Section II. Ou axioatic syste and ain esults ae pesented in Section III. We conclude the pape with the discussion in Section IV. This pape was fist subitted fo eview on Febuay 25, 206. Division of the Huanities and Social Sciences, Caltech. E-ail: cvitanic@hss.caltech.edu. Reseach suppoted in pat by NSF gant DMS MIT, Sloan School of Manageent, Depatent of Econoics, Depatent of Bain and Cognitive Sciences. E-ail: dpelec@it.edu. 3 The Institute of Econoics, Zageb. and MIT, Sloan School of Manageent. E-ail: sadas@it.edu This eseach was suppoted by a Maie Cuie Intenational Outgoing Fellowship within the 7 theuopean Counity Faewo Pogae, PIOF-GA BayInno. 4 Univesity of Zageb, Faculty of Science, Depatent of Matheatics. E- ail: hsiic@ath.h. Reseach suppoted in pat by the MZOS gant of the Republic of Coatia an in pat by Coatian Science Foundation unde the poject 3526 II. BAYESIAN TRUTH SERUM ALGORITHM By R we denote the set of playes (espondents). We assue that R is not epty, not a singleton, and at ost countable (i.e. the cadinal nube of the set R satisfies 2 cad(r) ℵ 0 ). Suppose that the playes ae pesented with a ultiple choice question, offeing a choice of N {} answes (we use the standad atheatical notation whee N is the set of natual nubes, R is the set of eal nubes, and R = R { } {+ } ). Each playe pics a siple answe

2 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 2 (the one s/he thins is the coect one) and gives a pediction in tes of pobabilities on the distibution of answes within R. Moe pecisely, we pesent the answe of a playe R as a pai of odeed -tuples ((x,, x ); (y,, y )) () whee x,, x {0,}, and y,, y [0,] such that = x = and = y =. Exactly one of x is equal to one (the non-zeo te which coesponds to the selected answe), while (y,, y ) is a pobability distibution on {,2,, }. As a consequence, a coplete data containing the answes of all playes can be pesented as a (finite o infinite) atix (X; Y); it is of the ode cad(r) 2 and its th ow, R, is given by (). Based on (X; Y) we want to assign a nueical scoe fo each playe, say u = u (X; Y) (2) fo playe R. Eventually we expect ou scoes to be ealvalued, but hee at the outset we shall not estict ouselves and in pinciple we allow even fo infinite values, i.e. u (X, Y) R (3) A. Classic definition of the scoe in Bayesian Tuth Seu Befoe stating ou axios we pesent how the scoe is defined by the Bayesian Tuth Seu. We shall use the notation in both finite and infinite case. If R is finite, then has its usual eaning of the su ove all eleents of R. If R is infinite, then we conside R = n N R n, whee cad(r n ) = n, and the eaning of is in the sense of li n n ; the notation coes togethe with an assuption that the liit exists within. R Siilaly, the eaning of (av) is in the cad(r) case of a finite R, while in the case of an infinite R it is li n n n. With this notation in ind, we conside x : = (x,, x ) whee, fo =,, x : = (av) x i.e. aithetic eans of X-coluns, and y = (y,, y ) whee, fo =,, s ln(y ) (av) ln (y ) i.e. geoetic eans of Y-coluns. The latte question is usually ased in the following way: please estiate the pecentage of you pees who will choose answe, the question is epeated fo each =,..,. Using the notation above, the algoith in [] is given as u (X, Y) = = x ln x + x = ln y whee R. The fist pat of the su is the infoation scoe, while the second one is the pediction scoe []. y III. AXIOMATIC SYSTEM Ou goal in this pape is develop an axioatic syste fo (4) that is copatible with Shannon theoy. In ou appoach playes choose an expet aong theselves via siultaneous conceptual duels. Each duel has a challenge, say playe R, and an offende, say playe s R. 2 We denote such duel as s. Each espondent plays a duel with evey othe espondent, including oneself. Each duel s ends with a tansfe of points fo playe to playe s. We denote the nube of tansfeed points by x (4) T s = T s (X; Y) R. (5) We can thin of positive T s as the winning case fo the offende, while negative T s eans that the challenge pevails. All the possible duels ae to be pefoed (including the duel with oneself) in ode to deteine scoes u fo all espondents R. In paticula, if R is finite, thee will be [cad(r)] 2 duels. Let us intoduce the basic ule fo a duel. Fo evey R the scoe u equals the nube of eceived points inus the nube of given points, i.e. u = u (X, Y) = T s (X; Y) - T s (X; Y) (6) Thee ae two iediate ipotant consequences of (6). Fist, assuing that all the sus ae finite-valued (which is the only inteesting case), the duel is a zeo-su gae, R u = R T s R T s = 0 (7) The second consequence of (6) is that the desciption of u educes to the desciption of T s. Hence, we pesent a set of axios about T s that geneate the Bayesian Tuth Seu algoith (4). Fo each axio we give an intuitive justification (which ay include soe ideas fo statistics) and a foal stateent (which is always going to be deteinistic). Ou fist axio is vey uch in the spiit of edieval duels. We can intepet it as the offende chooses the playgound fo the duel. Axio. The challenge will tansfe points to the offende s, based on the x answe of the offende s. Moe 2 We use taditional duel teinology, whee one playe (offende) offends the othe (challenge), who in tun challenges the fist playe to a duel

3 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 3 pecisely, fo evey, s R and fo evey {,, } thee is P s (X; Y) R such that T s s (X; Y) = = x P s (X; Y) (8) Obseve that, aong othes, Axio educes ou analysis fo T s to P s. Obseve also that, fo evey s R, thee is exactly one {,, } such that x s =. Hence, we can thin of that as being the function of s, i.e. = (s). It follows then that (8) becoes T s (X; Y) = P s (s) (X; Y) (9) In ode to undestand the second axio, we intoduce the following patition of R R = {s R x s = }, =,, (0) Obviously, the patition R = R. R is a function of X. Fix fo a oent and conside R, which is a subset of playes who choose the sae answe. In geneal, the nube of points P s ay vay as s changes within R. The pupose of ou second axio is to pevent this fo happening, i.e. that axio can be thought of as the egalitaian pinciple within R. " Axio 2. Given R and {,, } we have (s, s R P s (X; Y) = P s (X; Y) ). Axio 2 says that if the offendes s, s R choose the sae answe, then in a duel with evey challenge they will eceive the sae nube of points. Obseve that Axio 2 includes even the cases when fo soe the set R ay be an epty set; in this case the iplication in Axio 2 is tue, since the peise of the iplication is neve tue. Using a slight abuse of notation (thin of = (s)), Axio 2 iplies that P s (X; Y) = P (X; Y) () In ode to undestand the thid axio, obseve that by choosing the answe, the offende s decides (given that is nown) on a type of function P that will be used in the duel s. Howeve, the P will in geneal still depend on (X; Y). Ou next axio can be thought of as stengthening the Axio. The offende s chooses the playgound, and in doing so it educes the vaiable dependence accodingly. Axio 3. Fo evey R and fo evey {,, }, P (X; Y) = P ((x q ) q R ; (y q ) q R ) Next we tun to Axio 4 which has deteinistic fo, and which can be justified using soe ideas fo statistics. One of the ain pobles in statistical analysis is to ae infeence about soe unnown paaete θ. The infeence is based on the infoation given in a saple X,, X n. In ost cases the obseved saple x,, x n is just a long list of nubes, which can be difficult to intepet diectly. Theefoe, we eploy the pinciple of data eduction to obtain soe statistic t = T(X,, X n ) which siplifies ou analysis. Thee ae vaious data eduction pinciples: aong othes these ae sufficiency pinciple, the equivaiance pinciple and the lielihood pinciple. In this pape we ae paticulaly inteested in the sufficiency pinciple, which elies on the notion of sufficient statistic fo θ. If t is such a statistic, then wheneve we have two saple points x = (x,, x n ) and x = (x,, x n ) with the popety T(x) = T(x ), then the infeence about θ is the sae egadless whethe x o x is obseved. Typical exaple is a Benoulli saple fo which we infe about the pobability of success, p. It is well nown and easy to undestand intuitively that X = n X n i= i is an exaple of a sufficient statistic egading p. We would ague hee that the X-pat of ou data is ain to the Benoulli saple set-up. We ae inteested in ω = (ω,, ω n ), whee ω gives the actual faction of the population that thins is the coect answe to the oiginal question. Hence, since we ae inteested in ω, then the aveage value gives as uch infoation about ω as the entie -th colun of the atix X, i.e. (x q ) q R. Theefoe, we te ou fouth axio the data eduction pinciple fo X. Axio 4. Fo evey R and fo evey {,, }, P ((x q ) q R ; (y q ) q R ) = P (x ; (y q ) q R ) Ou second data eduction pinciple deals with Y. Ou axios so fa povided the offende s with the advantage to choose the playgound. In the next axio we give an advantage to the challenge by giving hi/he an option to choose the weapon. We can thin of it as allowing the challenge to select soe infoation fo the th colun of Y in ode to pedict ω. We assue that the challenge is vey self-confident and always eeps with his/he own choice i.e. y. This gives us the data eduction pinciple fo Y. Axio 5. Fo evey R and fo evey {,, }, P (x ; (y q ) q R ) = P (x ; y ) Obseve that ou axios have educed a function defined on a atix (X; Y) to a function defined on a pai of nubes (x ; y ) which ae between 0 and. Howeve, on this level of geneality we still allow the fo of the function to change with o with (i.e. the function can vay with the choice of diffeent playes o answes). A syste that would allow fo such level of geneality would not be vey pactical, as fo evey and evey we would have a diffeent function P. Hence we opt fo a oe obust selection and intoduce the following univesality axio.

4 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 4 Axio 6. Thee exists a function P: [0,] [0,] R such that fo evey R and fo evey {,, } we have P = P. In othe wods, Axio 6 insues that function P is the sae fo evey playe and fo evey answe. We believe that the fist six axios ae natual and easy to accept. Thei cobined effect is that, fo evey, s R T s (X; Y) = s = x P(x ; y ) (2) Let us now tun ou attention to the last and the ost deanding axio. In ode to justify it, we boow ideas fo infoation theoy 3. We eploy again a copaison with the Benoulli pobabilistic odel; we thin of both x and y as estiates of the pobability of success, say p. Intuitively speaing, let us thin of P(x ; y ) as an estiate of soe function h(p ), whee h awads points with the intention of easuing the uncetainty of the associated Benoulli odel. If we have two independent Benoulli ando vaiables U (with success pobability p) and V (with success pobability q), then the pobability of joint success is pq. Following the sae aguent as in [8], page 6, it is natual to equie h(pq) = h(p) + h(q). By tanslating this equieent to the language of P, we obtain P(x(p) x(q); y(p) y(q)) = P(x(p); y(p)) + P(x(q); y(q)) As in [8] we exclude the case of zeo and teat it sepaately (see also []). Hence, we intoduce the additivity popety axio in the following fo. Axio 7. The estiction P 0,] 0,] of the function P given in (2) is a continuous function such that, fo evey u 0, ], P(u, u) = 0, and fo evey u, u 2, v, v 2 0, ], P(u u 2, v v 2 )= P(u, v ) + P(u 2, v 2 ). Obseve that if the selected playgound infoation of the offende esults in the x which is exactly equal to the challenge infoation, then the natual outcoe is a daw, i.e. P(u, u) = 0. Obviously, as in the Shannon theoy, the consequence of the Axio 7 is that one can ely on one of the well-nown functional equations fo atheatics. Moe pecisely, the following esult is standad and well-nown. Lea. If h: 0, ] R is continuous and such that, fo evey u, v 0, ], h(uv) = h(u) + h(v), then h(u) = a ln(u), whee a = h(e ). 3 In paticula, one ay consult a chapte on a easue of infoation in [8] with the ephasis on section.2. Recall that the additivity popety is vey stong. The conclusion of the Lea follows even with uch ilde equieents than continuity on function h; fo exaple it is sufficient to equie onotonicity o easuability. Although this would allow us to educe the equieent on continuity given in Axio 7, in ode to avoid unnecessay atheatical inticacies we pesented the Axio 7 in the above fo. Naely, to constuct non-easuable additive functions one needs to go into details about the axio of choice and we thin that fo the point of view of vaious applications it is pehaps not necessay to go into such axio iniization issues futhe. Using Lea it is not difficult to see that Axio 7 educes function P to a paticula fo. Coollay. If a function P: 0, ] 0, ] R satisfies Axio 7, then thee exists a R such that, fo evey u, v 0, ], P(u, v) = a ln (u/v) Poof. Tae u = u, u 2 =, v = v, v 2 = in Axio 7. We obtain P(u, v) = P(u, ) + P(, v). We stat with the function u P(u, ). If we apply Axio 7 with v = v 2 =, we obtain P(u u 2, ) = P(u, ) + P(u 2, ). Hence, u P(u, ) satisfies the equieent of the Lea. We conclude that thee exists a R such that P(u, ) = a ln (u). Conside now the function v P(, v). If we apply Axio 7 with u = u 2 =, we obtain P(, v v 2 ) = P(, v ) + P(, v 2 ). Again, using Lea, we conclude that thee exists b R such that P(, v) = b ln (v). Finally, using P(u, u) = 0 and P(u, u) = P(u, ) + P(, u) = a ln (u)+ b ln (u), we obtain b = a. Hence, fo evey u, v 0, ], it follows P(u, v) = a l n ( u v ). Q.E.D. Rea. We need to decide on a paticula choice of the noalizing constant a R fo the pevious Coollay. Suppose fo the oent that the challenge has selected y =, fo soe. This iplies y l = 0 fo all l, i.e. the challenge has put his entie tust on. If, in this case, the playgound chosen by the offende is indeed, then it is the challenge who should ean points in this duel. Moe pecisely, if 0 < u <, then P(u, ) < 0, and it follows that a > 0 (3)

5 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 5 What is then the natual choice fo the constant a? This is now just the atte of noalization. Suppose fo the oent that all offendes have chosen playgound. In that case the challenge would eceive in total 4 a cad(r) P(x ; ) points in the finite case, and li a(r n ) cad(r n ) n P(x ; ) points in the infinite case. It is natual to noalize so that the total is P(x ; ) points. Hence we define the constant a to be a = in finite case, o a(r n ) = cad(r) cad(r n ) in the infinite case. (4) Theoe. If the scoing syste satisfies Axios -7 and condition (4), then the esulting syste is the Bayesian Tuth Seu algoith, i.e. u satisfies (4). Poof. Without loss of geneality we pesent the poof fo the finite case. In the infinite case we can use exactly the sae poof unde the liit sign li n n n, so we do not conside the infinite case sepaately. We poceed with the poof in the finite case. Using (2) and the Coollay, we obtain u = u (X, Y) = T s (X; Y) - T s (X; Y) = = x cad(r) (ln x s y ) x s cad(r) (ln x y ) = The fist su becoes = x cad(r) (ln(x ) ln(y s )) = = = x [ cad(r) ln(x ) cad(r) ln(y s )] = Since the choice of depends on (not on s), we obtain cad(r) ln(x ) = ln(x ) On the othe hand, cad(r) ln(y s ) = (av) ln(y s ) = ln(y ) It follows that the fist su equals = x ln ( x ), i.e. y equals the infoation scoe in (4). Fo the second su we obtain s x cad(r) (ln x y ) = x s cad(r) ln y = = x = = ln y ( cad(r) x s ) = x ln y = = This is equal to pediction scoe in (4). IV. CONCLUSION = x x Q.E.D. Foula (6) descibes a lage class of aning systes which ae all based on totality of the points playes eceive afte the sequence of duels is pefoed. Evey playe pefos two duels against evey othe playe: one tie as an offende and one tie as a challenge. These duels ae deteinistic and in pinciple they wo in the sae way fo the finite and the infinite nube of playes. As in the Shannon theoy, the ey function involved is the logaith. Hence, it sees plausible that the Bayesian Tuth Seu ay be soewhat connected to the notion of entopy. The connection is athe subtle, and it exists only in the infinite case whee we tae into account full Bayesian stochastic appoach togethe with the exchangeability assuption on the set of playes (espondents). The de Finetti theoe guaantees that thee is an undelying govening ando vaiable (we call the outcoes of this vaiable states of natue ), which epesents vaious possible belief systes within the population. Fo each {,, } and fo evey state of natue, say i, we conside conditional pobabilities z i = Pob(state i esponse ), so called posteios. It is shown in [] 5 that fo the tue state of natue, assuing that x =, we have that the Bayesian Tuth Seu scoing ule is given by u = ln(z i ) + A The te A does not eflect aning and ensues that the zeosu popety is satisfied. Let us denote Z = {z i i state of natue}. Hence, the conditional expectation can be expessed as E(u esponse ) = entopy(z ) + A Obseve howeve that the u aning is not based on entopy, but athe on z i itself. Thus it oe esebles the axiu lielihood estiato based on posteios. 4 In total hee eans fo all the offendes. 5 Fo oe details see also [6]

6 > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 6 Obseve that an essential featue of the BTS algoith is that it siultaneously easues the quality of infoation and pediction. We ephasize that the syste of axios in this pape does not use the zeo-su popety as the distinguishing one aong vaious algoiths of the fo (6) (copae this to the chaacteization of BTS n [6]). So fa Bayesian Tuth Seu bas been successfully tested fo huan espondents, hence the ipotance of incentivecopatibility. By setting aside incentive-copatibility and focusing on the aning syste, we believe that the algoith can be applied in the context whee playes ae achines instead of huans. One application would be easuing infoation-pediction capability in such vaious fields as eteoology, finance, edicine, etc. REFERENCES [] D. Pelec, A Bayesian Tuth Seu fo Subjective Data. Science 306, , [2] S.R. Mille, B.P. Bailey, and A. Kili, Exploing the Utility of Bayesian Tuth Seu fo Assessing Design K nowledge, Huan-Copute Inteaction, vol.29, no.5-6, pp , 204. [3] T. Loughan; R. Patenoste; and K. Thoas, Incentivizing Responses to Self-epot Questions in Peceptual Deteence Studies: An Investigation of the Validity of Deteence Theoy Using Bayesian Tuth Seu. Jounal of Quantitative Ciinology, Vol. 30 Issue 4, p , Dec 204. [4] A. Kula-Gyz, J. Tyowicz, M. Kawczy, K. Siwinsi, We All Do It, but Ae We Willing to Adit? Incentivizing Digital Piates' Confessions, Applied Econoics Lettes, v. 22, iss. -3, pp , Febuay 205. [5] Howie, Patic J.; Wang, Ying; Tsai, Joanne. Pedicting new poduct adoption using Bayesian tuth seu, Jounal of Medical Maeting. Vol. Issue, p6-6. p, Feb20. [6] J. Cvitanić, D. Pelec, S. Radas, and H. Šiić, Mechanis design fo an agnostic planne: univesal echaniss, logaithic equilibiu payoffs and ipleentation, subitted fo publication, Octobe 205. [7] D. Pelec,, H.S. Seung, and J. McCoy, A supisingly popula solution to the single question cowd wisdo poble. Woing pape, MIT, 205. [8] R. B. Ash, Infoation theoy, Dove Publications,990. :