Online Reputation and Polling Systems: Data Incest, Social Learning and Revealed Preferences

2.9 MB·English

by  Vikram Krishnamurthy

1
Online Reputation and Polling Systems: Data
Incest, Social Learning and Revealed
Preferences
Vikram Krishnamurthy, Fellow, IEEE William Hoiles, Student Member, IEEE
Abstract—This paper considers online reputation and challenges from a statistical signal processing and data
polling systems where individuals make recommendations fusion point of view. First, humans interact with and
5
based on their private observations and recommendations
1 influence other humans since ratings posted on online
of friends. Such interaction of individuals and their social
0 reputation systems strongly influence the behavior of
influenceismodelledassociallearningonadirectedacyclic
2 graph. Data incest (misinformation propagation) occurs individuals.1 Thiscanresultinnon-standardinformation

n due to unintentional re-use of identical actions in the for- patterns due to correlations introduced by the structure
a mation of public belief in social learning; the information of the underlying social network. Second, due to pri-
J gathered by each agent is mistakenly considered to be
vacy concerns, humans rarely reveal raw observations
5 independent. This results in overconfidence and bias in of the underlying state of nature. Instead, they reveal
estimates of the state. Necessary and sufficient conditions

are given on the structure of information exchange graph their decisions (ratings, recommendations, votes) which
]
I to mitigate data incest. Incest removal algorithms are can be viewed as a low resolution (quantized) function
S presented. Experimental results on human subjects are of their raw measurements and interactions with other
. presented to illustrate the effect of social influence and
s individuals.
c dataincestondecisionmaking.Theseexperimentalresults
[ indicatethatsociallearningprotocolsrequirecarefuldesign
to handle and mitigate data incest. The incest removal Motivation
1 algorithms are illustrated in an expectation polling system
v where participants in a poll respond with a summary of This paper models how data incest propagates
4 their friends’ beliefs. Finally, the principle of revealed amongst individuals in online reputation and polling
9 preferences arising in micro-economics theory is used systems. Consider the following example comprising a
9 to parse Twitter datasets to determine if social sensors
multi-agent system where agents seek to estimate an
0 are utility maximizers and then determine their utility
0 functions. underlying state of nature. An agent visits a restaurant
. and obtains a noisy private measurement of the state
1 Keywords:sociallearning,dataincest,reputationsys- (quality of food). She then rates the restaurant as excel-
0
5 tems, Bayesian estimation, expectation polling, Afriat’s lent on an online reputation website. Another agent is
1 theorem, revealed preferences influenced by this rating, visits the restaurant, and also
: gives a good rating on the online reputation website.
v
i I. INTRODUCTION The first agent visits the reputation site and notices that
X anotheragenthasalsogiventherestaurantagoodrating
Onlinereputationsystems(Yelp,Tripadvisor,etc.)are
r - this double confirms her rating and she enters another
a of increasing importance in measuring social opinion.
good rating. In a fair reputation system, such “double
They can be viewed as sensors of social opinion - they
counting” or “data incest” should have been prevented
go beyond physical sensors since user opinions/ratings
by making the first agent aware that the rating of the
(such as human evaluation of a restaurant or movie) are
second agent was influenced by her own rating. The
impossible to measure via physical sensors. Devising
information exchange between the agents is represented
a fair online reputation system involves constructing a
by the directed graph of Fig.1. The fact that there are
datafusionsystemthatcombinesestimatesofindividuals
two distinct paths (denoted in red) between Agent 1 at
to generate an unbiased estimate. This presents unique
time1andAgent1attime3impliesthattheinformation
of Agent 1 at time 1 is double counted thereby leading
V. Krishnamurthy (e-mail: [email protected]) and W. Hoiles
(email: [email protected]) are with the Department of Electrical
andComputerEngineering,UniversityofBritishColumbia,Vancouver, 181%ofhotelmanagersregularlycheckTripadvisorreviews[2].A
V6T1Z4,Canada. one-starincreaseintheYelpratingmapsto5-9%revenueincrease[3].
PartsofSec.IandIIofthispaperappearinourtutorialpaper[1].
to a data incest event. Such data incest results in a bias since in expectation polling an individual considers its
in the estimate of the underlying state. own intent together with the intents of its friends.2 If
the average degree of nodes in the network is d, then
the savings in the number of samples is by a factor of
d, since a randomly chosen node summarizes the results
form d of its friends. However, the variance and bias of
the estimate depend on the social network structure, and
data incest can strongly bias the estimate. We illustrate
howtheposteriordistributionoftheleadingcandidatein
the poll can be estimated based on incestious estimates.
Social learning assumes that agents (social sensors)
choose their actions by maximizing a utility function.
Fig.1. Exampleoftheinformationflowinasocialnetworkwithtwo
agentsandoverthreeeventepochs.Thearrowsrepresentexchangeof The final part of the paper, namely Sec.V, uses the
information. principle of revealed preferences as a constructive test
to determine: Are social sensors utility optimizers in
their response to external influence? We present a re-
Main Results and Organization
markable result arising in microeconomics theory called
Thispaperhasfourparts.Thefirstpart,namelySec.II, Afriat’s theorem [5], [6] which provides a necessary
deals with the design of a fair online reputation system. and sufficient condition for a finite dataset D to have
Sec.IIformulatesthedataincestprobleminamultiagent originated from a utility maximizer. The result is il-
system where individual agents perform social learning lustrated by tracking real-time tweets and retweets for
and exchange information on a sequence of directed reputation based review Twitter accounts. In particular,
acyclic graphs. The aim is to develop a distributed Sec.V shows that social learning associated with rep-
data fusion protocol which incorporates social influence utation agencies has the structure illustrated in Fig.2.
constraints and provides an unbiased estimate of the Thatis,thesentimentoftweetsandnumberoffollowers
state of nature at each node. Protocol 1 in Sec.II-D (reputation)constitutethepubliclyavailableinformation
givesthecompletedesigntemplateofhowincestcanbe thatdrivesthesociallearningprocess.Ifthesentimentof
avoided in the online reputation system. It is shown that the tweet published by the reputation agency improves
by choosing the costs to satisfy reasonable conditions, itsreputation,thenthisreinforcesthereputationagency’s
the recommendations made by individuals are ordinal belief that it adequately reviewed the content. This is an
functions of their private observations and monotone example of data incest.
in the prior information. This means that the Bayesian
social learning follows simple intuitive rules and is SocialLearning
Reputation sentiment
therefore, a useful idealization. Necessary and sufficient Content in
Agency
conditions for exact incest removal subject to a social Twitter
influence constraint are given.
reputation
Thesecondpartofthepaper,namelySec.III,analyzes
the data of an actual experiment that we performed on
Fig. 2. Schematic of dynamics of the reputation agency and the
humansubjectstodeterminehowsocialinfluenceaffects
Twitter network. Content (i.e. games, movies, books) is provided to
decisionmaking.Inparticular,informationflowpatterns the reputation agency which then publishes its review as a tweet.
from the experimental data indicative of social learning The Twitter network then responds to the sentiment of this tweet by
retweeting.Thismayincreaseordecreasethereputationoftheagency.
and data incest are described. The experimental results
illustrate the effect of social influence.
The third part of the paper, namely Sec.IV, describes Related work
how the data incest problem formulation and incest [7] is a seminal paper in collective human behavior.
removal algorithms can be applied to an expectation The book [8] contains a complete treatment of social
polling system. Polls seek to estimate the fraction of
a population that support a political party, executive 2[4]analyzesallUSpresidentialelectoralcollegeresultsfrom1952-
2008wherebothintentionandexpectationpollingwereconductedand
decision, etc. In intent polling, individuals are sampled
shows a remarkable result: In 77 cases where expectation and intent
and asked who they intend to vote for. In expectation pollingpointedtodifferentwinners,expectationpollingwasaccurate
polling [4] individuals are sampled and asked who 78% of the time! The dataset from the American National Election
Studiescomprisedofvoterresponsestotwoquestions:
they believe will win the election. It is intuitive that
Intention:WhowillyouvoteforintheelectionforPresident?
expectation polling is more accurate than intent polling; Expectation:WhodoyouthinkwillbeelectedPresidentinNovember?
2
learningmodels.Sociallearninghasbeenusedwidelyin Perspective
economics,marketing,politicalscienceandsociologyto In Bayesian estimation, the twin effects of so-
model the behavior of financial markets, crowds, social cial learning (information aggregation with interaction
groups and social networks; see [9], [10], [11], [8] and amongst agents) and data incest (misinformation prop-
numerous references therein. Related models have been agation) lead to non-standard information patterns in
studiedinsequentialdecisionmaking[12]andstatistical estimatingtheunderlyingstateofnature.Herdingoccurs
signalprocessing[13],[1].Atutorialexpositionofsocial whenthepublicbeliefoverridestheprivateobservations
learning in and sensing is given in our recent paper [1]. and thus actions of agents are independent of their
Online reputation systems are reviewed and studied in private observations. Data incest results in bias in the
[14],[15],[16].Informationcausedbyinfluentialagents public belief as a consequence of the unintentional re-
is investigated in [10], [17]. In [18] examples are given use of identical actions in the formation of public belief
that show that if just 10% of the population holds and insociallearning;theinformationgatheredbyeachagent
unshakable belief, their belief will be adopted by the is mistakenly considered to be independent. This results
majority of society. in overconfidence and bias in estimates of the state.
Privacy and reputation pose conflicting requirements:
Theroleofsocialinfluenceindecisionmaking(which privacy requirements result in noisier measurements or
we consider in Sec.II-C) is studied in [19]. The expec- lowerresolutionactions(sinceindividualsarenotwilling
tation polling [4] (which we consider in Sec.IV) is a to disclose private observations), while a high degree
formofsocialsampling[20]whereparticipantsinapoll of reputation requires accurate measurements. Utility
respond with a summary of their friends responses. [20] functions, noisy private measurements and quantized
analyzestheeffectofthesocialnetworkstructureonthe actions are essential ingredients of the social learning
bias and variance of expectation polls. Social sampling models presented in this paper that facilitate modeling
has interesting parallels with the so called Keynesian this tradeoff between reputation and privacy.
beauty contest, see for example https://en.wikipedia.org/
II. REPUTATIONDYNAMICSANDDATAINCEST
wiki/Keynesian beauty contest for a discussion.
A. Classical Social Learning
Consider a multi-agent system that aims to estimate
Dataincestarisesinotherareasofelectricalengineer-
the state of an underlying finite state random variable
ing. The so called the count to infinity problem in the
x ∈ X = {1,2,...,X} with known prior distribution
distance vector routing protocol in packet switched net-
π . Each agent acts once in a predetermined sequential
works[21]isatypeofmisinformationpropagation.Data 0
order indexed by k =1,2,... Assume at the beginning
incest also arises in Belief Propagation (BP) algorithms
of iteration k, all agents have access to the public belief
in computer vision and error-correcting coding theory.
π defined in Step (iv) below. The social learning
BP algorithms require passing local messages over the k−1
protocol proceeds as follows [9], [8]:
graph(Bayesiannetwork)ateachiteration.Forgraphical
(i) Private Observation: At time k, agent k records
models with loops, BP algorithms are only approximate
a private observation y ∈ Y from the observation
duetotheover-countingoflocalmessages[22]whichis k
distribution B = P(y|x = i), i ∈ X. Throughout this
similar to data incest. The algorithms presented in this iy
paper we assume that Y={1,2,...,Y} is finite.
papercanremovedataincestinBayesiansociallearning
(ii)PrivateBelief:Usingthepublicbeliefπ available
overnon-treegraphsthatsatisfyatopologicalconstraint. k−1
attimek−1(Step(iv)below),agentkupdatesitsprivate
In [17], data incest is considered in a network where
posterior belief η (i) = P(x = i|a ,...,a ,y )
agentsexchangetheirprivatebeliefstates(withoutsocial k k 1 k−1 k
using Bayes formula:
learning). Simpler versions were studied in [23], [24].
B π
η = yk , B =diag(P(y |x=i),i∈X). (1)
Regarding revealed preferences, highly influential k 1(cid:48)XByπ yk k
worksintheeconomicsliteratureincludethosebyAfriat Here 1 denotes the X-dimensional vector of ones, η
X k
[5], [25] and Varian (chief economist at Google). In is an X-dimensional probability mass function (pmf).
particular Varian’s work includes measuring the welfare (iii) Myopic Action: Agent k takes action a ∈ A =
k
effect of price discrimination [26], analysing the rela- {1,2,...,A} to minimize its expected cost
tionship between prices of broadband Internet access
a =argminE{c(x,a)|a ,...,a ,y }
and time of use service [27], and auctions for adver- k 1 k−1 k
a∈A
tisement position placement on page search results from =argmin{c(cid:48)η }. (2)
a k
Google [27], [28]. a∈A
3
Here c = (c(i,a),i ∈ X) denotes an X dimensional We refer to n as a “node” of a time dependent informa-
a
cost vector, and c(i,a) denotes the cost incurred when tion flow graph G which we now define. Let
n
the underlying state is i and the agent chooses action a.
G =(V ,E ), n=1,2,... (5)
(iv)SocialLearningFilter:Giventheactiona ofagent n n n
k
k, and the public belief πk−1, each subsequent agent denote a sequence of time-dependent directed acyclic
k(cid:48) > k performs social learning to update the public graphs (DAGs)4 of information flow in the social net-
belief πk according to the “social learning filter”: work until and including time k. Each vertex in Vn
Rππ represents an agent s(cid:48) at time k(cid:48) and each edge (n(cid:48),n(cid:48)(cid:48))
πk =T(πk−1,ak), where T(π,a)= σ(πa,a), (3) in En ⊆Vn×Vn shows that the information (action) of
node n(cid:48) (agent s(cid:48) at time k(cid:48)) reaches node n(cid:48)(cid:48) (agent s(cid:48)(cid:48)
where σ(π,a) = 1(cid:48) RπP(cid:48)π is the normalization factor at time k(cid:48)(cid:48)). It is clear that G is a sub-graph of G .
X a n n+1
of the Bayesian update. In (3), the public belief πk(i)= The Adjacency Matrix An of Gn is an n×n matrix
P(xk = i|a1,...ak) and Raπ = diag(P(a|x = i,π),i ∈ with elements An(i,j) given by
X) has elements (cid:40)
1 if (v ,v )∈E ,
(cid:88) A (i,j)= j i , A (i,i)=0. (6)
P(ak =a|xk =i,πk−1 =π)= P(a|y,π)P(y|xk =i) n 0 otherwise n
y∈Y
(cid:40) The transitive closure matrix T is the n×n matrix
1 if c(cid:48)B P(cid:48)π ≤c(cid:48)B P(cid:48)π, a˜∈A n
P(a =a|y,π)= a y a˜ y
k 0 otherwise. Tn =sgn((In−An)−1) (7)
where for matrix M, the matrix sgn(M) has elements
The following result which is well known in the
economics literature [9], [8]: (cid:40)
0 if M(i,j)=0,
Theorem 2.1 ([9]): Thesociallearningprotocolleads sgn(M)(i,j)=
1 if M(i,j)(cid:54)=0.
to an information cascade3 in finite time with probabil-
ity 1. That is, after some finite time k¯ social learning Note that A (i,j) = 1 if there is a single hop path
caegaesnetss cahnodotsheethpeubslaicmbeealicetfioπnka+k1+=1 =πka,kk, k≥≥k¯,k¯a.nd a(cid:4)ll tbheetrweeeexnisntsodanepsaitha(npdosjs,ibInlecmoumltpia-hriospo)n,beTtnw(eie,nj)i=and1jif.
The information reaching node n depends on the
B. Information Exchange Model information flow graph Gn. The following two sets will
be used to specify the incest removal algorithms below:
In comparison to the previous subsection, we now
consider social learning on a family of time dependent H ={m:A (m,n)=1} (8)
n n
directed acyclic graphs - in such cases, apart from F ={m:T (m,n)=1}. (9)
n n
herding, the phenomenon of data incest arises.
Consider an online reputation system comprised of Thus Hn denotes the set of previous nodes m that com-
agents {1,2,...,S} that aim to estimate an underlying municatewithnodeninasingle-hop.Incomparison,Fn
state of nature (a random variable). Let x ∈ X = denotes the set of previous nodes m whose information
{1,2,...,X} represent the state of nature (such as the eventually arrives at node n. Thus Fn contains all
quality of a restaurant/hotel) with known prior distribu- possible multi-hop connections by which information
tionπ .Letk =1,2,3,...depictepochsatwhichevents from a node m eventually reaches node n.
0
occur. The index k marks the historical order of events Properties of An and Tn: Due to causality with
and not absolute time. For simplicity, we refer to k as respecttothetimeindexk (informationsentbyanagent
“time”. can only arrive at another agent at a later time instant),
It is convenient also to reduce the coordinates of time the following obvious properties hold (proof omitted):
k and agent s to a single integer index n: Lemma 2.1: Consider the sequence of DAGs Gn,
n=1,2,...
n(cid:44)s+S(k−1), s∈{1,...,S}, k =1,2,3,... (4) (i) The adjacency matrices A are upper triangular.
n
A is the upper left n×n submatrix of A .
n n+1
3 Aherdofagentstakesplaceattimek¯,iftheactionsofallagents (ii)ThetransitiveclosurematricesT areuppertriangu-
after time k¯ are identical, i.e., ak = ak¯ for all time k > k¯. An lar with ones on the diagonal. Hencen, T is invertible.
informationcascadeimpliesthataherdofagentsoccur.[29]quotesthe n
followinganecdoteofuserinfluenceandherdinginasocialnetwork:
“... when a popular blogger left his blogging site for a two-week 4ADAGisadirectedgraphwithnodirectedcycles.Theorderingof
vacation, the site’s visitor tally fell, and content produced by three nodesn=1,2,...,proposedhereisaspecialcaseofthewellknown
invitedsubstitutebloggerscouldnotstemthedecline.” resultthatthenodesofaDAGarepartiallyorderable;seeSec.IV.
4
(iii) Classical social learning of Sec.II-A, is a trivial influenceconstraint(12)isimportant.H canbeviewed
n
examplewithadjacencymatrixA (i,j)=1forj =i+1 as the “social message”, i.e., personal friends of node n
n
and A (i,j)=0 elsewhere. (cid:4) since they directly communicate to node n, while the
n
The appendix contains an example of data incest that associated beliefs can be viewed as the “informational
illustrates the above notation. message”. As described in the remarkable recent paper
[19], the social message from personal friends exerts a
C. Data Incest Model and Social Influence Constraint large social influence6 – it provides significant incentive
(peer pressure) for individual n to comply with the
Each node n receives recommendations from its im-
protocolofcombiningitsestimatewithπ0 andthereby
mediate friends (one hop neighbors) according to the n−
prevent incest. [19] shows that receiving messages from
informationflowgraphdefinedabove.Thatis,itreceives
known friends has significantly more influence on an
actions {a ,m ∈ H } from nodes m ∈ H and
m n n individual than the information in the messages. This
then seeks to compute the associated public beliefs
study includes a comparison of information messages
π ,m ∈ H . If node n naively (incorrectly) assumes
m n and social messages on Facebook and their direct effect
that the public beliefs π ,m ∈ H are independent,
m n on voting behavior. To quote [19], “The effect of social
then it would fuse these as
transmission on real-world voting was greater than the
(cid:81)
π
π −= m∈Hn m . (10) direct effect of the messages themselves...” In Sec.III,
n 1(cid:48)X(cid:81)m∈Hnπm we provide results of an experiment on human subjects
that also illustrates social influence in social learning.
This naive data fusion would result in data incest.
1) Aim: The aim is to provide each node n the true
posterior distribution
D. Fair Online Reputation System: Protocol 1
π0 (i)=P(x=i|{a ,m∈F }) (11)
n− m n The procedure specified in Protocol 1 evaluates the
subject to the following social influence constraint: fair online rating by eliminating data incest in a social
There exists a fusion algorithm A such that network. The aim is to achieve (11) subject to (12).
At this stage, the public rating π computed in (13)
π0 =A(π ,m∈H ). (12) n−
n− m n ofProtocol1isnotnecessarilythefaironlineratingπ0
n−
2) Discussion. Fair Rating and Social Influence: We of (11). Without careful design of algorithm A in (13),
briefly pause to discuss (11) and (12). duetointer-dependenciesofactionsonpreviousactions,
(i) We call πn0− in (11) the true or fair online rating πn− can be substantially different from πn0−. Then ηn
available to node n since F defined in (9) denotes computed via (15) will not be the correct private belief
n
all information (multi-hop links) available to node n. andincestwillpropagateinthenetwork.Inotherwords,
By definition πn0− is incest free and is the desired ηn, πn− and πn are defined purely in terms of their
conditional probability that agent n needs. 5 computational expressions in Protocol 1; they are not
Indeed,ifnodencombinesπ0 togetherwithitsown necessarily the desired conditional probabilities, unless
n−
private observation via social learning, then clearly algorithmAisproperlydesignedtoremoveincest.Note
also the requirement that algorithm A needs to satisfy
η (i)=P(x=i|{a ,m∈F },y ), i∈X,
n m n n the social influence constraint (12).
π (i)=P(x=i|{a ,m∈F },a ), i∈X,
n m n n
are, respectively, the correct (incest free) private belief
E. Ordinal Decision Making in Protocol 1
for node n and the correct after-action public belief. If
agentndoesnotuseπ0 ,thenincestcanpropagate;for Protocol 1 assumes that each agent is a Bayesian
n−
example if agent n naively uses (10). utility optimizer. The following discussion shows that
Whyshouldanindividualnagreetouseπ0 tocom- under reasonable conditions, such a Bayesian model is
n−
bine with its private message? It is here that the social a useful idealization of agents’ behaviors.
Humans typically make monotone decisions - the
5For the reader unfamiliar with Bayesian state estimation: Com- more favorable the private observation, the higher the
puting the posterior π0 is crucial noisy observations. Then the
n−
conditional mean estimate is evaluated as E{x|{am,m ∈ Fn}} =
(cid:80)x∈Xxπn0−(x)andistheminimumvarianceestimate,i.e.,optimal 6InastudyconductedbysocialnetworkingsitemyYearbook,81%of
in the mean square error sense and more generally in the Bregmann respondentssaidtheyhadreceivedadvicefromfriendsandfollowers
loss function sense. The conditional mean is a ‘soft’ estimate and is relatingtoaproductpurchasethroughasocialsite;74percentofthose
unbiased by definition. Alternatively the maximum aposteriori ‘hard’ who received such advice found it to be influential in their decision.
estimateisevaluatedasargmax π0 (x). (ClickZ,January2010).
x n−
5
Protocol1IncestRemovalinOnlineReputationSystem that an increasing argmin is an ordinal property -
(i) Information from Social Network: that is, argmin c(cid:48)B π0 increasing in y implies
a a yn n−
1) Social Message from Friends: Node n receives argmin φ(c(cid:48)B π0 ) is also increasing in y for any
a a yn n−
socialmessageH comprisingthenamesorphotos monotone function φ(·).
n
of friends that have made recommendations. The following result gives sufficient conditions for
2) Informational Message from Friends: The reputa- each agent to give a recommendation that is monotone
tionsystemfusesrecommendations{am,m∈Hn} and ordinal in its private observation:
into the single informational message πn− and Theorem 2.2: Suppose the observation probabilities
presents this to node n. and costs satisfy the following conditions:
The reputation system computes πn− as follows: (A1) Biy are TP2 (totally positive of order 2); that is,
a) {am,m ∈ Hn} are used to compute public Bi+1,yBi,y+1 ≤Bi,yBi+1,y+1.
beliefs {πm,m∈Hn} using Step (v) below. (A2) c(x,a) is submodular. That is, c(x,a + 1) −
b) {πm,m∈Hn} are fused into πn− as c(x,a)≤c(x+1,a+1)−c(x+1,a).
Then
π =A(π ,m∈H ). (13)
n− m n 1) Under (A1) and (A2), the recommendation
In Sec.II-G, fusion algorithm A is designed as an(πn0−,yn) made by agent n is increasing and
ln−(i)= (cid:88) wn(m)lm(i), i∈X. (14) 2) hUenndceero(rAdi2n)a,lain(oπb0ser,vyati)onisynin,cfroerasainnygπin0n−b.elief
n n− n
m∈Hn πn0− with respect to the monotone likelihood ratio
Here lm(i)=logπm(i) and wn(m) are weights. (MLR) stochastic order8 for any observation yn.(cid:4)
(ii) Observation: Node n records private observation y The proof is in the appendix. We can interpret the
n
from the distribution B =P(y|x=i), i∈X. above theorem as follows. If agents makes recommen-
iy
(iii) Private Belief: Node n uses y and informational dationsthataremonotoneandordinalintheobservations
n
message π to update its private belief via Bayes rule: andmonotoneintheprior,thentheymimictheBayesian
n−
sociallearningmodel.Eveniftheagentdoesnotexactly
B π
η = yn n− . (15) follow a Bayesian social learning model, its monotone
n 1(cid:48) B π
X y n− ordinal behavior implies that such a Bayesian model is
a useful idealization.
(iv) Recommendation: Node n makes recommendation
Condition(A1)iswidelystudiedinmonotonedecision
a =argminc(cid:48)η making; see the classical book by Karlin [30] and
n a n
a [31]; numerous examples of noise distributions are TP2.
and records this on the reputation system. Indeed in the highly cited paper [32] in the economics
(v) Public Belief Update by Network Administrator: literature,observationy+1issaidtobemore“favorable
Based on recommendation an, the reputation system news” than observation y if Condition (A1) holds.
(automated algorithm) computes the public belief πn Condition (A2) is the well known submodularity
using the social learning filter (3). condition [33]. Actually (A2) is a stronger version of
the more general single-crossing condition [34], [35]
stemming from the economics literature (see appendix)
recommendation.Humansmakeordinaldecisions7 since
(c −c )(cid:48)B π0 ≥0 =⇒ (c −c )(cid:48)B π0 ≥0.
humans tend to think in symbolic ordinal terms. Under a+1 a y+1 a+1 a y
whatconditionsistherecommendationanmadebynode This single crossing condition is ordinal, since for any
nmonotoneincreasinginitsobservationynandordinal? monotone function φ, it is equivalent to
Recall from Steps (iii) and (iv) of Protocol 1 that the
recommendation of node n is φ((ca+1−ca)(cid:48)By+1π0)≥0 =⇒ φ((ca+1−ca)(cid:48)Byπ0)≥0.
a (π0 ,y )=argmin c(cid:48)B π0 (A2) also makes sense in a reputation system for the
n n− n a a yn n−
coststobewellposed.Supposetherecommendationsin
So an equivalent question is: Under what conditions action set A are arranged in increasing order and also
is the argmin increasing in observation yn? Note the states in X for the underlying state are arranged in
ascendingorder.Then(A2)says:ifrecommendationa+
7Humans typically convert numerical attributes to ordinal scales
1 is more accurate than recommendation a for state x;
before making decisions. For example, it does not matter if the cost
ofamealatarestaurantis$200or$205;anindividualwouldclassify
this cost as “high”. Also credit rating agencies use ordinal symbols 8 Given probability mass functions {pi} and {qi}, i = 1,...,X
suchasAAA,AA,A. thenpMLRdominatesq iflogpi−logpi+1≤logqi−logqi+1.
6
then recommendation a+1 is also more accurate than social unit under study (restaurant). Therefore, agents
recommendation a for state x + 1 (which is a higher minimize the penalty of writing inaccurate reviews.
quality state than x).
IntheexperimentresultsreportedinSec.III,wefound
G. Incest Removal Algorithm A
that (A1) and (A2) of Theorem 2.2 are justified.
It only remains to describe the construction of algo-
F. Discussion and Properties of Protocol 1 rithm A in Step 2b of Protocol 1 so that
Forthereader’sconvenienceweprovideanillustrative π (i)=π0 (i), i∈X
n− n−
example of data incest in the appendix. We now discuss where π0 (i)=P(x=i|{a ,m∈F }). (16)
several other properties of Protocol 1. n− m n
1) Individuals have selective memory: Protocol 1 To describe algorithm A, we make the following def-
allows for cases where each node can remember some initions: Recall π0 in (11) is the fair online rating
n−
(or all) of its past actions or none. This models cases available to node n. It is convenient to work with the
where people forget most of the past except for specific logarithmoftheun-normalizedbelief:accordinglydefine
highlights.Forexample,intheinformationflowgraphof
l (i)∝logπ (i), l (i)∝logπ (i), i∈X.
theillustrativeexampleintheappendix(Fig.13),ifnodes n n n− n−
1,3,4 and 7 are assumed to be the same individual, then Define the n−1 dimensional weight vector:
at node 7, the individual remembers what happened at
w =T−1 t . (17)
node 5 and node 1, but not node 3. n n−1 n
2) Security: Network and Data Administrator: The
Recall that t denotes the first n−1 elements of the
n
social influence constraint (12) can be viewed as a
nthcolumnofthetransitiveclosurematrixT .Thusthe
n
separation of privilege requirement for network security.
weights are purely a function of the adjacency matrix of
For example, the National Institute of Standards and
the graph and do not depend on the observed data.
Technology (NIST) of the U.S. Department of Com-
We present algorithm A in two steps: first, the actual
merce [36, Sec.2.4] recommends separating the roles
computation is given in Theorem 2.3, second, necessary
of data and systems administrator. Protocol 1 can then
and sufficient conditions on the information flow graph
be interpreted as follows: A network administrator has
for the existence of such an algorithm to achieve the
accesstothesocialnetworkgraph(butnotthedata)and
social influence constraint (12).
cancomputeweightsw (m)(13)bywhichtheestimates
n Theorem 2.3 (Fair Rating Algorithm): Consider the
are weighed. A data administrator has access to the rec-
reputation system with Protocol 1. Suppose the network
ommendationsoffriends(butnottothesocialnetwork).
administrator runs the following algorithm in (13):
Combiningtheweightswiththerecommendationsyields
the informational message πn0−1 as in (14). l (i)= n(cid:88)−1w (m)l (i) (18)
3) AutomatedRecommenderSystem: Steps(i)and(v) n− n m
of Protocol 1 can be combined into an automated rec- m=1
ommender system that maps previous actions of agents where the weights w are chosen according to (17).
n
in the social group to a single recommendation (rating) Then l (i) ∝ logπ0 (i). That is, the fair rating
n− n−
π of (13). This recommender system can operate logπ0 (i) defined in (11) is obtained via (18). (cid:4)
n− n−
completely opaquely to the actual user (node n). Node Theorem 2.3 says that the fair rating π0 can be ex-
n−
n simply uses the automated rating π as the current pressed as a linear function of the action log-likelihoods
n−
bestavailableratingfromthereputationsystem.Actually intermsofthetransitiveclosurematrixT ofgraphG .
n n
AlgorithmApresentedbelowfusesthebeliefsinalinear Achievability of Fair Rating by Protocol 1:
fashion. A human node n receiving an informational 1) Algorithm A at node n specified by (13) needs to
message comprised of a linear combination of recom- satisfy the social influence constraint (12) - that is,
mendation of friends, along with the social message has it needs to operate on beliefs lm,m∈Hn.
incentivetofollowtheprotocolasdescribedinSec.II-C. 2) On the other hand, to provide incest free estimates,
4) Agent Reputation: The cost function minimization algorithm A specified in (18) requires all previous
in Step (iv) can be interpreted in terms of the reputation beliefs l (i) that are specified by the non-zero
1:n−1
of agents in online reputation systems. If an agent con- elements of the vector w .
n
tinues to write bad reviews for high quality restaurants Theonlywaytoreconcilepoints1and2istoensurethat
on Yelp, her reputation becomes lower among the users. A (j,n) = 0 implies w (j) = 0 for j = 1,...,n−1.
n n
Consequently, other people ignore reviews of that (low- This condition means that the single hop past estimates
reputation) agent in evaluating their opinions about the l ,m ∈ H available at node n according to (13)
m n
7
in Protocol 1 provide all the information required to making his/her judgment. This mimics the social learn-
compute w(cid:48) l in (18). We formalize this condition ing Protocol 1 of Sec.II-D. The participants continued
n 1:n−1
in the following theorem. choosing actions according to this procedure until the
Theorem 2.4 (Achievability of Fair Rating): experiment terminated. The trial terminated when the
Consider the fair rating algorithm specified by (18). response of each of the two participants did not change
For Protocol 1 using the social influence constraint for three successive iterations (the two participants did
information (π ,m∈H ) to achieve the estimates l not necessarily have to agree for the trial to terminate).
m n n−
of algorithm (18), a necessary and sufficient condition Ineachtrial,theactionsofparticipantswererecorded
on the information flow graph G is alongwiththetimeintervaltakentochoosetheiraction.
n
As an example, Fig. 4 illustrates the sample path of
A (j,n)=0 =⇒ w (j)=0. (19)
n n decisions made by the two participants in one of the
1658 trials. In this specific trial, the average diameter of
ThereforeforProtocol1togenerateincestfreeestimates
for nodes n=1,2,..., condition (19) needs to hold for the left array of circles was 32.1875 and the right array
each n. (Recall w is specified in (18).) (cid:4) was30.5625(inpixels);sothegroundtruthwas0(left).
n
Summary: Algorithm (18) together with the condition
(19) ensure that incest free estimates are generated by
Protocol 1 that satisfy social influence constraint (12).
III. EXPERIMENTALRESULTSONHUMANSUBJECTS
To illustrate social learning, data incest and social
influence, this section presents an actual psychology
experiment that was conducted by our colleagues at
the Department of Psychology of University of British
Columbia in September and October, 2013, see [37]
fordetails.Theparticipantscomprised36undergraduate
students who participated in the experiment for course Fig.3. Twoarraysofcirclesweregiventoeachpairofparticipants
credit. onascreen.Theirtaskistointeractivelydeterminewhichside(either
left or right) had the larger average diameter. The partner’s previous
decisionwasdisplayedonscreenpriortothestimulus.
A. Experiment Setup
The experimental study involved 1658 individual tri-
als. Each trial comprised two participants who were
Participant1
asked to perform a perceptual task interactively. The 1.2 Participant2
pdeerncoetpetduallefttaskandwarsigahst,fowlleorwes:giTvweno taorraeyaschofpcaiirrcleosf forRight) 0.18
participants. Each participant was asked to judge which 1
array (left or right) had the larger average diameter. The Left, 0.6
participantsanswer(leftofright)constitutedtheiraction. (0for 0.4
SoTthhee accirtciolensswpaecree ipsreApa=red{0fo(lrefeta),c1h(trriigahlta)}s.follows: Action 0.02
two 4×4 grids of circles were generated by uniformly −0.20 5 10 15 20 25 30
Trial
sampling from the radii: {20,24,29,35,42} (in pixels).
The average diameter of each grid was computed, and Fig.4. Exampleofsamplepathofactionschosenbytwoparticipants
if the means differed by more than 8% or less than 4%, in a single trial of the experiment. In this trial, both participants
new grids were made. Thus in each trial, the left array eventuallychosethecorrectanswer0(left).
andrightarrayofcirclesdifferedintheaveragediameter
by 4-8%
For each trial, one of the two participants was chosen
B. Experimental Results
randomly to start the experiment by choosing an action
accordingtohis/herobservation.Thereafter,eachpartici- The results of our experimental study are as follows:
pantwasgivenaccesstotheirpartner’spreviousresponse 1) Social learning Model: As mentioned above, the
(action)andtheparticipantsownpreviousactionpriorto experiment for each pair of participants was continued
8
untilbothparticipants’responsesstabilized.Inwhatper- Fig.5 occurred (1303 out of 1658 experiments). Further,
centageoftheseexperiments,didanagreementoccurbe- in 21% of experiments, the information patterns shown
tween the two participants? The answer to this question in Fig.5 occurred and at least one participant changed
reveals whether “herding” occurred in the experiments his/her decision, i.e., the judgment of participant at time
and whether the participants performed social learning k+1differedfromhis/herjudgmentsattimek+2andk.
(influencedbytheirpartners).Theexperimentsshowthat These results show that even for experiments involving
in66%oftrials(1102among1658),participantsreached two participants, data incest information patterns occur
an agreement; that is herding occurred. Further, in 32% frequently (79%) and causes individuals to modify their
of the trials, both participants converged to the correct actions (21%). It shows that social learning protocols
decision after a few interactions. requirecarefuldesigntohandleandmitigatedataincest.
To construct a social learning model for the exper-
imental data, we consider the experiments where both
participants reached an agreement. Define the social
First Participant action: 0/1
learning success rate as
Attempt Attempt Attempt
K K+1 K+2
# expts where both participants chose correct answer
· action: 1 action: 1
# expts where both participants reached an agreement Data Incest
In the experimental study, the state space is X={0,1}
where x = 0, when the left array of circles has the action: 0/1 action: 1 action: 0/1
larger diameter and x=1, when the right array has the
Attempt Attempt Attempt
k K+1 K+2
larger diameter. The initial belief for both participants is
Second Participant
considered to be π = [0.5,0.5]. The observation space
0
is assumed to be Y={0,1}. First Participant action: 0/1
To estimate the social learning model parameters Attempt Attempt Attempt
k k+1 K+2
(observation probabilities B and costs c(i,a)), we de-
iy action: 0 action: 0
terminedtheparametersthatbestfitthelearningsuccess Data Incest
rate of the experimental data. The best fit parameters
obtained were9
action: 0/1 action: 0 action: 0/1
(cid:20) (cid:21) (cid:20) (cid:21)
0.61 0.39 0 2
Biy = 0.41 0.59 , c(i,a)= 2 0 . Attekmpt AttKe+m1pt AttKe+m2pt
Second Participant
Note that B and c(i,a) satisfy both the conditions of
iy
the Theorem 2.2, namely TP2 observation probabilities Fig.5. Twoinformationpatternsfromourexperimentalstudieswhich
andsingle-crossingcost.Thisimpliesthatthesubjectsof canresultindataincest.
this experiment made monotone and ordinal decisions.
2) Dataincest: Here,westudytheeffectofinforma-
tion patterns in the experimental study that can result IV. BELIEF-BASEDEXPECTATIONPOLLING
in data incest. Since private observations are highly Wenowmoveontothethirdpartofthepaper,namely
subjective and participants did not document these, we data incest in expectation polling. Unlike previous sec-
cannot claim with certainty if data incest changed the tions, the agents no longer are assumed to eliminate
action of an individual. However, from the experimental incest at each step. So incest propagates in the network.
data, we can localize specific information patterns that Given the incestious beliefs, the aim is to compute the
can result in incest. In particular, we focus on the two posterior of the state. We illustrate how the results of
information flow graphs depicted in Fig.5. In the two Sec.IIcanbeusedtoeliminatedataincest(andtherefore
graphsofFig.5,theactionofthefirstparticipantattime bias) in expectation polling systems. Recall from Sec.I
k influenced the action of the second participant at time thatinexpectationpolling[4],[20],individualsaresam-
k+1, and thus, could have been double counted by the pledandaskedwhotheybelievewillwintheelection;as
first participant at time k+2. We found that in 79% of opposed to intent polling where individuals are sampled
experiments, one of the information patterns shown in and asked who they intend to vote for. The bias of
the estimate from expectation polling depends strongly
9Parameterestimationinsociallearningisachallengingproblemnot on the social network structure. Our approach below is
addressedinthispaper.Duetotheformationofcascadesinfinitetime,
Bayesian: we compute the posterior and therefore the
construction of an asymptotically consistent estimator is impossible,
sinceactionsaftertheformationofacascadecontainnoinformation. conditional mean estimate (see Footnote 5).
9
We consider two formulations below: Protocol 2 Belief-based Expectation Polling Protocol
1) An expectation polling system where in addition 1) Intention from Friends: Node n receives the beliefs
to specific polled voters, the minimal number of {πm,m∈Hn} of who is leading from its immedi-
additional voters are recruited. The pollster then ate friends, namely nodes Hn.
is able to use the incestious beliefs to compute 2) Naive Data Fusion: Node n fuses the estimates of
the posterior conditioned on the observations and its friends naively (and therefore with incest) as
thereby eliminate incest (Sec.IV-A below). (cid:81)
π
2) An expectation polling system when it is not pos- π = m∈Hn m .
n− 1(cid:48) (cid:81) π
sible to recruit additional voters. The pollster then X m∈Hn m
can only compute the posterior conditioned on the 3) Observation:Nodenrecordsitsprivateobservation
incestious beliefs. (Sec.IV-B below). y from the distribution B = P(y|x = i), i ∈ X
n iy
The first approach can be termed as “exact” since of who the leading candidate is.
the estimate computed based on the incestious beliefs 4) Belief Update: Node n uses y to update its belief
n
is equivalent to the estimate computed based on the via Bayes formula:
private observations of the sampled voters. The second
B π
approach, although optimal given the available informa- π = yn n− . (20)
n 1(cid:48) B π
tion, has a higher variance (see footnote 10). X yn n−
Suppose X candidates contest an election. Let x ∈ 5) Node n sends its belief π to subsequent nodes as
n
X = {1,2,...,X} denote the candidate that is leading specified by the social network graph.
amongst the voters, i.e., x is the true state of nature.
There are N voters. These voters communicate their
expectations via a social network according to the steps the pollster needs to poll additional voters to eliminate
listed in Protocol 2. We index the voters as nodes incest. Let E denote this set of extra (additional) voters
n ∈ {1,...,N} as follows: m < n if there exists a to poll. Clearly the choice of E will depend on R and
directed path from node m to node n in the directed the structure of the social network.
acyclic graph (DAG). It is well known that such a What is the smallest set E of extra voters to poll in
labelingofnodesinaDAGconstituteapartiallyordered ordertocomputetheposteriorP(x|y ,...,y )ofstate
1 RL
set. The remaining nodes can be indexed arbitrarily, x (and therefore eliminate data incest)?
providing the above partial ordering holds. 2) IncestRemoval: AssumethatalltherecruitedL−1
Protocol 2 is similar to Protocol 1, except that voters nodes in R report to a central polling node R . We
L
exchange their expectations (beliefs) of who the leading assume that the pollster has complete knowledge of the
candidateis,insteadofrecommendations.(Sounlikethe network;e.g.,inanonlinesocialnetworklikeFacebook.
previous section, agents do not perform social learning.) Using the formulation of Sec.II-B, the pollster con-
Remark: Similar to Theorem2.2, the Bayesian update structs the directed acyclic graph Gn = (Vn,En),
(20) in Protocol 2 can be viewed as an idealized model; n ∈ {1,...,RL}. The methodology of Sec.II-G is
under assumption (A1) the belief π increases with straightforwardly used to determine the minimal set of
n
respect to the observation y (in terms of the monotone extra (additional) voters E. The procedure is as follows:
n
likelihoodratioorder).Soevenifagentsarenotactually 1) Foreachn=1,...,RL,computetheweightvector
Bayesian, if they choose their belief to be monotone in wn =Tn−−11tn; see (17).
the observation, they mimic the Bayesian update. 2) Theindicesofthenon-zeroelementsinwn thatare
not in V , constitute the minimal additional voters
n
(nodes) that need to be polled. This is due to the
A. Exact Incest Removal in Expectation Polling
necessity and sufficiency of (19) in Theorem 2.4.
AssumingthatthevotersfollowProtocol2,thepolling 3) Once the additional voters in E are polled for
agency (pollster) seeks to estimate the leading candidate their beliefs, the polling agency corrects belief π
n
(stateofnaturex)bysamplingasubsetoftheN voters. reported by node n∈R to remove incest as:
1) Setup and Aim: Let R = {R ,R ,...,R }
1 2 L (cid:88)
denote the L−1 sampled recruited voters together with ln0(i)=ln(i)+ wn(m)lm(i). (21)
node R that denotes the pollster. For example the m∈E,m<n
L
L−1 voters could be volunteers or paid recruits that
B. BayesianExpectationPollingusingIncestiousBeliefs
have already signed up for the polling agency. Since,
by Protocol 2, these voters have naively combined the Consider Protocol 2. However, unlike the previous
intentions of their friends (by ignoring dependencies), subsection, assume that the pollster cannot sample ad-
10