Mathematical Social Sciences 64 (2012) 100–102

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Mathematical Social Sciences

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Short communication

Should social network structure be taken into account in elections?✩

Vincent ConitzerDuke University, United States

a r t i c l e i n f o

Article history:Available online 13 December 2011

a b s t r a c t

If the social network structure among the voters in an election is known, how should this be taken intoaccount by the voting rule? In this brief article, I argue, via the maximum likelihood approach to voting,that it is optimal to ignore the social network structure altogether—one person, one vote.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

The maximum likelihood framework provides a natural ap-proach to determining an ‘‘optimal’’ voting rule. Under this ap-proach, we suppose that there exists a ‘‘correct’’ outcome, whichcannot be directly observed; however, voters’ votes are noisy esti-mates of this correct outcome. For example, in an election betweentwo alternatives, we may suppose that one of the alternatives isbetter in some objective sense, and each voter ismore likely to votefor the correct (better) alternative than for the incorrect alterna-tive. To make this concrete, suppose that there is a fixed p > 1/2such that each voter, independently, votes for the correct alterna-tive with probability p, and for the incorrect one with probability1 − p.

Under such a model, it makes sense to choose the maximumlikelihood estimate of the correct outcome. For example, supposethat we have a specific profile in which there are na votes foralternative a, and nb votes for alternative b. The likelihood of thisparticular profile happening given that a is the correct outcome ispna(1−p)nb , and given that b is the correct outcome is (1−p)napnb .If na > nb, then the former is larger, so a is themaximum likelihoodestimate and vice versa. That is, the maximum likelihood estimatehere is obtained by applying themajority rule: the alternativewithmore votes wins.

The study of this maximum likelihood approach to voting datesback to Condorcet (Condorcet, 1785), who proposed a noise modelthat generalizes the one above to more than two alternatives. Twocenturies later, Young (1988, 1995) showed that the maximumlikelihood solution for this model coincides with the Kemeny rule(Kemeny, 1959). A recent body of work has been devoted to the

✩ The technically rather simple observation presented in this short article waspresented in full in a short rump session talk at the 2010 Dagstuhl seminar onthe Computational Foundations of Social Choice. This work has been supported byan Alfred P. Sloan Fellowship and by NSF under award numbers IIS-0812113 andCAREER-0953756.

E-mail address: conitzer@cs.duke.edu.

0165-4896/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.mathsocsci.2011.03.006

investigation of how alternative noise models lead to differentrules (Drissi-Bakhkhat and Truchon, 2004; Conitzer and Sandholm,2005; Truchon, 2008; Conitzer et al., 2009). There is also a literatureon strategic voting in such contexts – e.g., Austen-Smith and Banks(1996) and Feddersen and Pesendorfer (1996) – but we will notconsider such issues here. Another recent paper by Elkind et al.(2010) explores the relationship between themaximum likelihoodframework and the distance rationalizability framework, where wefind the closest ‘‘consensus’’ election that has a clear winner, forsome definition of consensus and some distance function, andchoose its winner.

A common assumption in the existing work on the maximumlikelihood approach is that votes are drawn independently(conditional on the correct outcome). It is easy to dispute thisassumption. Voters do not make up their minds in a vacuum: theygenerally discuss the matter with others, and this is likely to affecttheir votes. Specifically, voters that talk to each other are morelikely to reach the same conclusion, whether right or wrong.

This brief article considers elections in which there is somesocial network structure among the voters, representing whichvoters interact with which other voters (as well as how frequentlyor intensively they do so) and, moreover, this social networkstructure is known to the election designer. Hence, the voting rulecan take the social network into account. This is not always asimpractical as it may sound. For example, Facebook, Inc. decidedto allow its users to vote on its future terms of use (Zuckerberg,2009). Naturally, Facebook knows the social network structureamong its users very accurately. But, in order for the general ideato make sense, it is not necessary to have this level of detailedknowledge of the social network structure among the voters. Forexample, we may simply observe that the residents of the stateof Hawaii are in some ways disconnected from the continentalUnited States, so that there is perhaps less interaction between thetwo groups and perhaps it makes sense to somehow take this intoaccount in the design of a voting rule. Making such a statementcan hardly be considered a taboo, since a voter’s state of residenceis already taken into account via the electoral college. In fact, onemay even go so far as to wonder whether a version of the electoral

V. Conitzer / Mathematical Social Sciences 64 (2012) 100–102 101

college is perhaps themaximum likelihood solution for a particularprobabilistic model.

So, given that we know the social network structure (or someestimate thereof), how should this affect the voting rule? We canstate various suggestive intuitive arguments, such as the following.1. Well-connected voters benefit from the insight of others so they are

more likely to get the answer right. They should be weighed moreheavily.

2. Well-connected voters do not give the issue much independentthought; the reasons for their votes are already reflected in theirneighbors’ votes. They should be weighed less heavily.

3. It is oversimplifying matters to discuss the matter merely in termsof voters’ weights. We can do better by considering richer options,such as an electoral college.In this brief article, Iwill argue that all of these three suggestions

are wrong—or, arguably, 1 and 2 are both correct, but cancel eachother out. That is, optimal rules ignore the social network structurealtogether, affirming the familiar principle of one person, one vote.The mathematical result is quite trivial once the model has beenset up.

2. Model

In this section, I give a concrete probabilistic model that takesthe social network structure into account; I will discuss thebenefits and drawbacks of this model in the final section. Let vdenote a vertex in the social network graph, corresponding to avoter. Let N(v) denote v’s neighbors. Let Vv denote voter v’s vote,and VN(v) the subprofile consisting of the votes of all the neighborsof v. Let c denote the ‘‘correct’’ outcome. For ease of reading, thereader is encouraged to keep the simple example of an electionbetween two alternatives in mind, so that there are two outcomesand two possible votes. However, this is not at all necessary for themain result: there can be arbitrarily many alternatives, and in factwe will not require any restriction on the outcome space, on thespace of possible votes, or on the relationship between the two. Forexample, an outcome could consist of an individual alternative, asubset of the alternatives, or a ranking of the alternatives. So coulda vote, independently of what an outcome consists of.

For the maximum likelihood approach, we are interested inthe probability of the full profile of votes, given c. I will assumethat this probability can be factored as

v fv(Vv, VN(v)|c), where

fv is a function associated with vertex v. (The dependence of f onVN(v) captures that voter v’s vote can be affected by her neighbors’opinions (votes)—as opposed to more standard models that takethe form

v fv(Vv|c).) This type of factorization assumption is

commonly made in undirected graphical models of probabilitydistributions (Markov random fields).1 It is worth noting that, inthe special case where the social network graph is a clique, thisassumption is not restrictive at all. I now make the followingassumption which does introduce a restriction that is crucial forthe result.

Assumption 1. For every v, there exist functions gv and hv so thatfv can be factored as fv(Vv, VN(v)|c) = gv(Vv|c) · hv(Vv, VN(v)).

Here, gv is intended to capture the effect that a voter is morelikely to vote for the correct outcome, and hv to capture the effectthat voters are likely to vote similarly to their neighbors. The keyassumption is that gv does not depend on VN(v), and hv does notdepend on c. That is, the tendency to vote for the correct outcomeis in a sense independent from the tendency to agree with one’sneighbors.

1 In this literature, factors can only be associated with cliques of the graph. Incontrast, here, a factor can be associatedwith the neighborhood of any vertex. Sinceany clique is contained in the neighborhood of any one of its vertices, the modelstudied here is in a sense less restrictive—though this is in any case not the mainassumption of this article, which follows next.

For example, in the two-alternative case, Assumption 1 pre-cludes models where, given that all of a vertex’s neighbors vote forthe correct outcome, this vertex also does sowith probability 1; butgiven that all of the neighbors vote for the incorrect outcome, thisvertex still has a positive probability of choosing the correct out-come. If this were the noise model, then if we found a vertex vot-ing differently from all its neighbors, we should conclude that thatvertex has identified the correct outcome, no matter what every-one else votes. So, in that case, the social network structure wouldmatter.

3. Detailed example

For concreteness, let us consider a simple example with twovertices (voters) connected by an edge, voting over two alterna-tives (so a vote corresponds simply to one of the alternatives). Let−c denote the alternative other than c and let−v denote the voterother than v. For each of the two vertices v, we let gv(Vv = c|c) =

0.7, gv(Vv = −c|c) = 0.3, indicating that votes for the correctoutcome are more likely. We also let hv(Vv = c, V−v = c) =

1.142, hv(Vv = c, V−v = −c) = 0.762, indicating that voters aremore likely to agree with each other. We can now calculate thatP(Vv = c|c) = P(Vv = c, V−v = c|c) + P(Vv = c, V−v = −c|c) =

0.7 · 1.142 · 0.7 · 1.142 + 0.7 · 0.762 · 0.3 · 0.762 = 0.761.In contrast, a model in which the two voters do not interact

can be obtained by setting h = 1 everywhere; in this model,P(Vv = c|c) = 0.7. Thus, there is some truth to the first argumentin the introduction: well-connected voters benefit from the insightof others so they are more likely to get the answer right. However,it does not yet follow from this that they should be weighed moreheavily, and as we will see this is in fact not the case.

4. Main result

We now arrive at the main result, which states that (underAssumption 1) the social network structure should not be takeninto account by the voting rule.

Proposition 1. Under Assumption 1, the maximum likelihood esti-mate does not depend on the social network structure—more precisely,it does not depend on the functions hv (assuming they are positive ev-erywhere). Specifically, under these conditions, the maximum likeli-hood estimate of the correct outcome is argmaxc

v gv(Vv|c).

As noted before, the proof is quite trivial.Proof. By Assumption 1, we can write the likelihood asv

fv(Vv, VN(v)|c) =

v

gv(Vv|c) · hv(Vv, VN(v))

=

v

gv(Vv|c)

·

v

hv(Vv, VN(v))

.

Therefore, the maximum likelihood estimate of the correct out-come is

argmaxc

v

fv(Vv, VN(v)|c) = argmaxc

v

gv(Vv|c)

·

v

hv(Vv, VN(v))

= argmax

c

v

gv(Vv|c)

which does not depend on the h functions. �

Because of Proposition 1, results derived in themaximum likeli-hood framework with the standard independence assumption stillhold in our setting. For example, we have the following corollary.

102 V. Conitzer / Mathematical Social Sciences 64 (2012) 100–102

Corollary 1. In a two-alternative election where Assumption 1 holds,if gv is the same for all voters v (so gv = g), and g(c|c) > g(−c|c),then the majority rule gives the maximum likelihood estimate of thecorrect winner.

Proof. By Proposition 1, the maximum likelihood estimate ofthe correct winner is argmaxc

v g(Vv|c) (regardless of the hv

functions). The remainder of the proof is as described in theintroduction, letting p = g(c|c). �

This also provides justification for the strong-seeming assump-tion of independence across voters in themaximum likelihood vot-ing literature, showing thatmodeling interaction among the votersdoes not necessarily change the solution.

5. Discussion

As noted earlier, while the result in this brief article ispresumably easiest to illustrate in the context of an electionbetween two alternatives, everything in the article applies justas well to settings with more alternatives. Also, there is norequirement that votes have to take anyparticular form—avote canbe a ranking of the alternatives, a subset of approved alternatives,or anything else. Similarly, the correct ‘‘outcome’’ can be analternative, a ranking of the alternatives, or anything else. Finally,no particular structure on the social network graph is requiredfor the result—for example, all voters may be directly connectedto each other. The crucial assumption is Assumption 1, whichseparates the correct-outcome effect from the interaction effect.An argument for voting rules that take social network structureinto account could be based on models that violate Assumption 1.

In this article, I have not proposed any concrete mechanism2

by which voters influence their neighbors’ opinions (for example,voters could sequentially send each other particular kinds ofmessages before voting); all that is needed is Assumption 1 onthe distribution. On the one hand, it can be argued that it is anadvantage of the model that it does not rely on such a concretemechanism, because any analytically tractable mechanism willpresumably be at best a very rough approximation of the truth. Onthe other hand, it still seems desirable to propose and analyze suchconcretemechanisms, to seewhether they violate the assumptionsmade here, and, if so, what their effect is on the maximumlikelihood approach. In this sense, this brief article is potentiallya first step in a richer body of research.

It is instructive to compare the result in this article to a resultby Nitzan and Paroush (1982). They consider a model with twoalternatives where some voters are more likely to identify the

2 ‘‘Mechanism’’ here is in the common sense of the word, not a reference tomechanism design.

correct alternative than others, and find that the optimal ruleis a weighted majority rule, where more skillful voters receive alarger weight. In contrast, in the framework of this article, better-connected voters are more likely to vote correctly (as we sawin Section 3), but are not given a correspondingly larger weight(Corollary 1). There is no contradiction here: in theNitzan–Paroushmodel, voters vote independently—there are no edges in the graph.Hence, more skillful voters really do contribute more valuableinformation, and should be weighed more heavily. In contrast,in the framework of this article, better-connected voters are notreally contributing more valuable information, because some oftheir information is already reflected in their neighbors’ votes. Infact, the Nitzan–Paroush framework can easily be accommodatedin the framework of this article, by reflecting skill in the g functions.If we do so, then there are two reasons that a voter may be morelikely to vote correctly than others: one is being more skillful, andthe other is being better connected. By Proposition 1, the formerresults in a larger weight (exactly as in the result by Nitzan andParoush), but the latter does not—a generalization of Corollary 1.

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