Democratic Deficit and the Majority Principle

Hannu Nurmi ∗

Department of Political Science and Contemporary HistoryUniversity of Turku

Finland

Abstract

The concept of democratic deficit refers to the discrepancy betweenthe decision outcomes of the representative institutions and the opin-ions of the electorate. Whenever these two coincide, the democraticdeficit vanishes.

We shall first deal with the simplest settings that involve only twodecision alternatives and two levels of decision making. It will be seenthat here we may encounter the democratic deficit in its plainest form:the clear majority of voters preferring alternative a to alternative b,while a vast majority of representatives having the opposite preferenceeven though every representative correctly votes according to the opin-ion of the majority of his/her electors. The culprit responsible for thedemocratic deficit is the majority rule, simple or qualified.

With three or more alternatives the will of the people can becomeinherently ambiguous, i.e. procedure-dependent. It turns out that amajor culprit is again the majority principle. The avoidance of demo-cratic deficit in representative systems without sacrificing the majorityprinciple seems impossible, pace those deeming the majority principleas a necessary ingredient of democratic governance.

Do the new advances in electoral system design provide ways toameliorate democratic deficit? With appropriate (cryptographic) meth-ods that guarantee ballot secrecy, correct counting of ballots, randomsampling of voters and speedy processing of ballots, we could gradu-ally replace representative systems with direct forms of participation.We shall discuss these innovations and argue that the problems relatedto the agenda control make these innovations of rather limited impor-tance in democratic governance. At the same time it is likely thatthe role of direct participation – possibly augmented with deliberativeprocesses – will increase in the future largely as a result of the electoralinnovations.

∗The author wishes to acknowledge Andranik Tangian’s many useful comments on anearlier version.

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1 Introduction

Abraham Lincoln’s well-known definition of democracy as the governmentof the people, by the people and for the people contains the basic goalscustomarily associated with democracy. The first part, ‘of the people’, doesnot require further commentary since it merely delineates the subjects ofgovernment. The second and third parts are less obvious. Does the secondgoal, ‘by the people’, mean that only direct forms of government will bedemocratic or can indirect, representative forms also be included? Does ‘forthe people’, in turn, mean that only Pareto optimal political (i.e. those thatcan be achieved without diminishing anyone’s welfare from the level of thestatus quo) outcomes are justifiable? Upon strict literal reading no presentday democracy works according to the latter two desiderata; all systemsinvolve a significant, even dominant, degree of representative governance.Moreover, very often the benefits accruing to a group or stratum of thesociety are achieved at the loss of other strata or groups. This does not, ofcourse, in any way diminish the value of Lincoln’s definition as a normativegoal.

Very often the ‘by the people’ is translated into ‘by a majority of thepeople’ and somewhat less often ‘for the people’ into ‘for a majority of thepeople’. The reason for these translations is typically practical: there isno unanimity or even near unanimity about policies or candidates in theelectorate. Hence, the requirement of consensus regarding the courses ofaction collectively pursued would leave us completely empty-handed. Thus,by diminishing the number of voters whose consent is required for collectivedecisions one has better chances of getting motions passed in legislaturesand persons elected to public offices. An inevitable consequence is that inmost cases there will be voters whose opinions do not coincide with thevictorious policy options or candidates. Therefore, a democratic deficit issimply a consequence of the lack of consensus among voters. What majorityseems to guarantee, however, is that in dichotomous choice situations thedeficit always victimizes strictly less than a half of the electorate. If higherthan majority thresholds, qualified majorities, were in use, the proportionof frustrated voters could be larger. This would be the case if the status quois regarded worse than a given policy option by a larger than majority butsmaller than the required qualified majority of voters.

In this paper we shall dwell on the relationships between the majorityrule and democratic deficit. Our approach is purely theoretical, but at thesame time applied. To wit, we shall look at the implications of several well-known results of social choice theory in an effort to trace their consequencesfor the design of democratic decision making institutions, especially votingsystems. Surely, democratic deficit can also pertain to situations involvingno voting at all. Instead, other ways of achieving collectively binding de-cision are resorted to, e.g. bargaining among interest organizations. These

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are beyond the scope of the present paper. In what follows we shall firstlook at the main justifications of the majority rule. Thereafter, we discussa couple of majority voting paradoxes that may be encountered in dichoto-mous settings. We then move on to more general k-option choice situationsand see that the majority principle becomes inherently ambiguous there.This ambiguity seriously undermines efforts to use the majority preferencesas tokens of the will of the people. In particular, the majority should not beconsidered as an actor like in ‘by the people’ nor an object like in ‘for thepeople’. We shall then consider the role of some modern innovations in re-ducing the democratic deficit. Our focus is on some versions of e-voting anddeliberative mechanisms. While a completely satisfactory e-voting mecha-nism is yet to be found (Galois 2015), some shortcomings of the existingsystems may be improved in several respects (e.g. verifiability of the ballotassignments). From the democratic deficit viewpoint more promising areattempts to combine deliberative mechanisms with agenda-formation in ref-erenda. The work of List et al. suggests that deliberation may bring aboutpreference domain restrictions that are conducive to stable voting outcomes(List et al. 2013). Hence, some types of preference aggregation paradoxesmay be avoidable through deliberative agenda-formation.

2 Why majority rule?

When a group of people has to make a choice between two options, say,to join a trade treaty or not, it seems natural to give each voter one vote– ‘yes’ or ‘no’ – and count the ballots. Whichever alternative gets morevotes is then declared the winner. Should each alternative get the samenumber of votes, then the outcome is a tie. This setting describes the simplemajority rule. More than half a century ago Kenneth May gave an axiomaticcharacterization of the majority rule (May 1952). To remind ourselves aboutthis result, we first define the majority rule in precise terms. There are twoalternatives – x and y – and n voters. We assume that each voter i’s opinionDi concerning x and y is one and only one of the following: ‘x is strictlybetter than y’, ‘x and y are equally good’ or ‘y is strictly better than x’.These opinions are denoted by Di = 1, 0,−1, respectively. The collectivedecision D can have each of these values as well. A group decision functionis then

D = f(D1, D2, . . . , Dn) (1)

Now, denote by N(1) the number of 1’s in the decision function (i.e. thenumber of individuals strictly preferring x to y). Similarly, let N(0) andN(−1) be the number of 0’s and −1’s, respectively, in the decision function.The simple majority rule can now be defined as follows:

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Definition 1 Simple majority rule is a decision function that has the valuesD = 1, 0 or −1 according to whether N(1) −N(−1) > 0,= 0 or < 0.

May’s characterization involves the following properties:

1. Decisiveness: the domain of f consists of the Cartesian product D1 ×D2 × . . . × Dn. In other words, the function yields a value for anycombination of individual preferences over x and y.

2. Anonymity: any permutation of the individuals leaves the value of Dunchanged. That is, only the number of 1’s, 0’s and −1’s, not howthey are attached to specific individuals, determines the value of f .

3. Neutrality: f(−D1,−D2, . . . ,−Dn) = −f(D1, D2, . . . , Dn). In words,if everyone changes his/her mind so that those strictly preferring x toy now strictly prefer y to x and vice versa and, moreover, those whoare indifferent between x and y, remain indifferent, then the outcomeshould change from 1 to −1, from −1 to 1 or remain unchanged if itwas a tie.

4. Positive responsiveness: only one voter’s change of mind is requiredto break a tie. More formally, if D = f(D1, D2, . . . , Dn) = 0 or 1 andif a new profile is formed so that all individuals except i keep theiropinions unchanged and i changes his opinion from −1 to 0, from−1 to 1 or from 0 to 1, then x is chosen under the new profile, i.e.D′ = f(D′1, D

′2, . . . , D

′n) = 1.

May’s characterization theorem states that a group decision function isthe simple majority decision if and only if it satisfies decisiveness, anonymity,neutrality and positive responsiveness. Arguably these properties are quitenatural and plausible. The first property pertains to the general applica-bility of the function in guaranteeing that under all opinion distributions acollective decision can be found. The second and third properties, in turn,exclude discriminating decision function. The fourth property, finally, statesthat additional support, ceteris paribus, never harms a candidate and thatties can be broken by the change of mind of a single individual.

The theorem gives quite a strong theoretical reason for adopting thesimple majority rule. Similar, but more conjectural justifications have beenpresented by Buchanan, Tullock and Rae (Buchanan and Tullock 1962; Rae1969). The upshot of all these efforts is that in two-alternative settings thesimple majority rule seems quite plausible. As is well-known, difficulties arisewhen more than two alternatives are being considered. Some of these willbe discussed later in this paper. Before that, however, it is worth pointingout that anomalies can well arise already in the two-alternative settings.

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policy areavoter welfare foreign affairs culture the voter votes for

voter A X X Y Xvoter B X Y X Xvoter C Y X X Xvoter D Y Y Y Yvoter E Y Y Y Y

winner Y Y Y ?

Table 1: Ostrogorski’s paradox

3 Majority paradoxes in two-alternative settings

Most political decisions involve comparisons of alternatives (candidates, poli-cies) along several criteria. It is quite typical that one alternative is preferredto another on one criterion, but the preference is inverted on another crite-rion. When voting for a presidential candidate, we may have a large numberof criteria in mind ranging from purely personal characteristics to stands onvarious political issues. In these kinds of situations, the majority may re-sult in an ambiguous outcome. The case in point is Ostrogorski’s paradoxintroduced and elaborated by Daudt and Rae (Rae and Daudt 1976; Daudtand Rae 1978). Table 1 presents a instance of the paradox.

There are two candidates, X and Y, running for presidency and fivevoters, A - E, voting.1 Each voter evaluates candidates using three criteria,say welfare policy, foreign policy and cultural policy. Table 1 indicates foreach voter which candidate is closer to the voter’s policy ideal. Thus e.g.voter B thinks that X is closer to his/her (hereafter his) ideal position onwelfare and cultural policy, while Y is closer on foreign policy. Let us assumethat each voter deems each criterion of roughly equal importance and castshis vote for whichever candidate is closer to his position on a majority ofcriteria. The right-most column lists the candidates to be voted for by eachvoter on the basis of this principle. We see that voters A, B and C would votefor X since X is closer to their positions on a majority of criteria. SimilarlyD and E vote for Y since they prefer Y on every criterion. The outcomethen is that X receives 3 and Y 2 votes. Hence X wins.

Looking at Table 1 from another angle, we see that on the welfare crite-rion a majority of voters deems Y closer than X. The same is true on the twoother policy areas. Hence, one could argue that the overall winner ought tobe Y as it is the majority winner on all three policy areas. Thus, if the votewas taken as in a direct democracy – i.e. each policy area being voted upon

1The number of voters representing each preference rankings can be multiplied with afixed constant without changing anything. Thus e.g. instead five we could consider fivemillion voters with voters A - E standing for one million voters each.

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issuevoter issue 1 issue 2 issue 3

voter 1 Y Y Xvoter 2 X X Xvoter 3 X Y Yvoter 4 Y X Yvoter 5 Y X Y

Table 2: Anscombe’s paradox

separately – the winner would be Y. This outcome would, however, leave amajority of voters – viz. A, B and C – frustrated since their candidate inthe indirect election is X. As a consequence of the ambiguity, democraticdeficit will emerge: the issue by issue voting winner Y will be defeated bythe ‘representative’ voting winner X.

Anscombe’s paradox demonstrates a seemingly similar discrepancy be-tween various methods of aggregating votes (Anscombe 1976). The paradoxcan be summarized by stating that it is possible that a majority of voters willbe on the losing side in dichotomous voting on a majority of issues. Table 2illustrates. There two alternatives X and Y are again being voted upon.Like above, they may be presidential candidates. There are three relevantissues on which the voters are able to locate the candidates vis-a-vis theirown ideal positions at least to the extent that they can say which candidateis closer to their own ideal position. The closest candidates for each voteron each issue are indicated in Table 2. A glance at the table reveals thatthis is not an instance of Ostrogorski’s paradox: Y wins on a majority ofissues and by a majority of voters. Instead, we observe that voter 1 is onthe losing side on issues 2 and 3, voter 2 on issues 1 and 3 and voter 3 onissues 1 and 2. Hence, three voters out of five is in a minority on a majority(two issues out of three) of issues. Clearly now a majority of voters becomesa victim of democratic deficit.

The setting of Table 2 invokes incentives for collusion among voters 1-3. By virtue of constituting a majority these voters may impose whicheveroutcome they choose. On the other hand, none of them is likely to agree ona joint voting strategy that would involve deviating from his true opinionon two or more issues. However, by coordinating their votes to X,Y,X (i.e.X on issue 1, Y on issue 2 and X on issue 3) they can bring about thisoutcome. This would involve only one change in each of the three voters’true opinions. Hence, it would make sense for them to coordinate in thismanner.

One could conjecture that qualified majority rules – i.e. rules that requirelarger than simple majorities to change the status quo – could eliminateAnscombe’s paradox and the possibility that the democratic deficit afflicts

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opinion district 1 . . . district 9 district 10 total

yes 45000 . . . 45000 100000 505000no 55000 . . . 55000 0 495000

Table 3: Referendum paradox

the majority of voters. It turns out, however, that this is not the case (Nurmiand Saari 2010, Th 6): even if the qualified majority calls for majorities oneshy of unanimity, i.e. n− 1 out of n votes are required for a motion to pass,it is possible that the majority of voters does not win on any issue.

Both Ostrogorski’s and Anscombe’s paradoxes are related to aggrega-tion. A similar but conceptually distinct paradox that also involves just twoalternatives is called the referendum paradox (Nurmi 1998). It occurs whenthe same issue is being voted upon directly by voters and indirectly by theirrepresentatives and when these two votes result in different outcomes. Thewider the margin of victory in the two votes, the more dramatic instance ofthe paradox we are dealing with. Table 3 gives an instance of the paradox.A country of 1 million voters is partitioned into 10 districts of equal popula-tions; each district has 100000 voters. Each district sends one representativeto the parliament. The issue to be voted upon is dichotomous, e.g. joiningor not joining a multilateral trade agreement. Table 3 presents a fictitiousdistribution of ‘yes’ and ‘no’ voters in the districts. In the first nine districtsa majority of voters supports the ‘no’ option, while district 10 is unani-mously behinds the ‘yes’ alternative. In a referendum the ‘yes’ alternativewins. Now, suppose that the same issue is subjected to a vote in the par-liament. Then 9 MP’s out of 10 have a plausible reason to vote ‘no’ as themajority of their supporters prefer this alternative. Hence, a clear majorityopinion in the popular vote can – quite plausibly – be contradicted in theparliament with a large margin. This reminds us of the famous dictum ofthe current president of European Commission Jean-Claude Juncker: ‘we allknow what to do, but we don’t know how to get re-elected once we have doneit’ (Juncker 2007). If the opinion of the whole population is ‘what shouldbe done’, then the MP’s elected from the 9 first districts might have hardtime justifying a ‘yes’ vote the majority of their electors and might, thus,fail to be re-elected. More importantly, the paradox shows that should thedecisive vote be the one taken in the parliament (as is the case in countrieswith consultative referenda), the democratic deficit may afflict a majorityof population even though the margin of majority in the parliament for theopposite outcome is quite overwhelming.

The referendum paradox shows that in ordinary legislation even largemargins in parliamentary support for legislative proposals cannot excludethe possibility of the opposite proposals enjoying a large majority supportin the electorate at large. In fact, very little can be inferred about the latter

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4 voters 3 voters 2 voters

A B CC C BB A A

Table 4: Ambiguous majority principle

support on the basis of parliamentary vote margins.

4 More than two alternatives: Condorcet’s prin-ciple

In the preceding we have focused on situations involving only two alterna-tives for the simple reason that it is in these situations that the the simplemajority decision has an unambiguous meaning. With the advent of a thirdalternative, the majority decision becomes ambiguous. Consider the follow-ing example (Table 4). All three alternatives can be considered winnersunder three different procedures each resorting to some majority-relatedprinciple. Firstly, A is the plurality winner, i.e. it is ranked first by morevoters than any other alternative. Secondly, B is the plurality runoff win-ner since no alternative gets the support of at least half the electorate in aone-person-one-vote- election. Hence a runoff between A and B – the twolargest vote-getters – is required. In this contest, B wins with 5 votes to 4.Thirdly, C is the Condorcet winner, i.e. it defeats its two competitors witha majority of votes in pairwise comparisons (C beats A with 5 votes to 4, Cbeats B with 6 votes to 3).

So, the ambiguity between three majority-related principles is maximalin this setting. Let us see what the arguments against the selection ofeach alternative might look like. Firstly, one could object the choice of A bypointing out that it considered the worst alternative by a majority of voters.Secondly, against the choice of B one could argue that a majority of votersprefers another alternative, C, to it. Thirdly, those opposing the choice ofC could point out that C is the favorite alternative of the smallest numberof voters. So, each choice can be objected to with at least a modicumof plausibility. It is worth observing that all three rules collapse into thesame outcome in all profiles where one alternative is ranked first by morethan half of the electorate since obviously the plurality and plurality runoffmethods coincide as no second round contest is required. At the same time,the candidate ranked first by most voters becomes the (strong) Condorcetwinner. Hence, no discrepancy between rules emerges.

Despite the discrepancy exhibited by Table 4 and similar settings, theCondorcet winner is often considered a particularly plausible criterion of

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4 voters 3 voters 2 voters

A B DC C CD A BB D A

Table 5: Condorcet winner ranked first by nobody

winning. Hence, the methods that result in a Condorcet winner when oneexists – the Condorcet extensions – are often deemed superior to the othermain class of procedures, the positional methods (Felsenthal and Machover1992; McLean 1991; Risse 2001). And indeed, the Condorcet winner cri-terion is clearly majoritarian in spirit. Additional advantages have beendiscovered by Campbell and Kelly (Campbell and Kelly 2015). Using theirterminology, let us call the method that always chooses the Condorcet win-ner the Condorcet rule. An important result of Campbell and Kelly statesthat the Condorcet rule is the only anonymous, neutral and strategy-proofrule in Condorcet domains (Campbell and Kelly 2003; Campbell and Kelly2015; Merrill 2011). A rule is strategy-proof if and only if there is no situ-ation where it is manipulable by an individual voter. A rule is manipulableby voter i in the preference profile P = (P1, . . . , Pn) when by changing hispreference ranking from Pi to P∗i, ceteris paribus, the ensuing outcome ispreferable by i to the original outcome. Thus, strategy-proof rules are notmanipulable by any individual under any profile. Note, however, that theresult is restricted to Condorcet domains, i.e. domains where a Condorcetwinner exists.2 Indeed, by a result of Gardenfors, all Condorcet extensionsthat are anonymous and neutral are manipulable (Gardenfors 1976). Theresult of Campbell and Kelly rests on a restriction of the domain of socialchoice functions, viz. to the Condorcet domains.

Despite its prima facie plausibility it is not difficult to see that Condorcetextensions may lead to severe problems related to the democratic deficit. Towit, as Table 4 shows the Condorcet winner may be considered best by asmaller group of voters than any other alternative. In fact, one may envisionsettings where the Condorcet winner is not ranked first by a single voter.Table 5 is an example of this kind of profile. Here C is the Condorcet winner,but is ranked first by no voter. Thus, if the Condorcet winner is elected, thedemocratic deficit afflicts every voter, not just a majority.

2There are a couple of minor restrictions. Firstly, it does not hold when the number ofalternatives is 2 and the number of voters is even. Secondly, it is not known if the resultholds when the number of voters is a multiple of 4 and the number of alternatives is 3.

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2 voters 2 voters 2 voters 1 voter

D A B DC D A CB C D BA B C A

Table 6: Subset choices by Borda count

5 Is the Condorcet winner criterion worth holdingon to?

In the debate concerning the relative plausibility of Condorcet winner vs.Borda winner, an often stated claim is that the Borda winner is cruciallydependent on the alternative set under consideration. More importantly, aremoval of an alternative may dramatically change the Borda ranking be-tween the remaining alternatives. Similarly, adding an alternative may es-sentially change the Borda ranking among the rest of the alternatives. Thesefindings were made by Fishburn in his early book on social choice theory(Fishburn 1973). Consider the 7-voter, 4-alternative profile of Table 6. Thealternatives might be the candidates in the athlete of the year contest whereprominent sport journalists vote on 4 main candidates by indicating theirranking over these sportspersons.

Borda count results in the ranking D � A � B � C. Before the resultsare made known, some evidence turns out suggesting that D is guilty ofusing illegal performance-enhancing drugs. D is, therefore, found ineligiblein the contest at hand. Since nothing else has changed in the circumstances,it is decided that the submitted rankings be used in determining the Bordaranking over the remaining three candidates. Upon computing the newscores it is found that the new ranking is C � B � A, i.e. a completereversal of collective preference over A, B and C. Fishburn’s result statesthat if alternative x is the Borda winner in set X, there are such profiles thatx wins in only one proper subset of X. Clearly then widening or narrowingthe alternative set opens avenues for outcome control. Could one then findsome other positional procedure that provides more stable outcomes undervariations of the alternative set? Saari’s answer is a resounding: no (Saari2001). Table 7 shows the extreme instability of the plurality procedure(Saari 2001, 70).

In Table 7 the collective ranking in terms of the plurality votes is: A �B � C � D. Strike now the last alternative D out and recompute theplurality scores for the remaining three alternatives to get C � B � A,which reverses the previous ranking among A, B and C. Let us now elimatethe lowest-ranked alternative A and recompute the plurality scores to get

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voters3 6 3 5 2 5 2 4

A A B B C C D DC D C D B D B CD B D C D B C BB C A A A A A A

Table 7: Instability of plurality procedure

B � C, which again reverses the the previous ranking over B and C.These examples come nowhere near in describing the profound instability

possibilities underlying the positional procedures. These are captured inSaari’s theorem (Saari 2001, 72).

Theorem 1 Saari 1984. Consider a setting with at least three candidates.Then proceed as follows:

• rank the candidates in any desired way and choose procedure to be usedin this set of candidates,

• eliminate one candidate, place the remaining ones into any ranking(independently of the preceding ranking) and choose again a positionalprocedure to be applied to this set of candidates,

• continue in this way until just two candidates are left and rank thesetwo using the majority rule.

There exists a profile which produces exactly the outcomes described abovewhen the voters vote in each subset using the designated positional procedure.

These profiles are completely chaotic: the result in the superset of alter-natives in no way enables one to predict the collective choices in the subsets.There is no systematic connection between winning in subsets and in theoutcome rankings in the supersets – or subsets, for that matter.

In defense of the Condorcet extensions one could, however, maintainthat removing alternatives from consideration because of ineligibility doesnot change Condorcet winners. If an alternative wins all the others in binarycontests by a majority of votes in a set of alternatives, surely it will alsowin all the remaining ones if an ineligible alternative is removed. So, Con-dorcet extensions would seem to be immune to removing alternatives fromthe alternative set, while this is clearly not always the case with positionalprocedures. There are, however, modifications in the choice setting thatchange the Condorcet winner in a implausible way – while not changing theBorda winner. Consider the profile of Table 8.

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voters7 4 4 4 4

A B A C BB C C B AC A B A C

Table 8: Condorcet instability

Ignoring the vertical line for a moment, we have a 23-voter profile overthree alternatives. There is a Condorcet winner, viz. B which is also theBorda winner. Observe that the 12-voter sub-profile on the right-hand sideof the vertical line constitutes a Condorcet-paradox profile: A defeats C,C defeats B and B defeats A, all with an 8 to 4 margin. Focusing on thissub-profile only, there is no reason – based on the ranking information only– to put one alternative ahead of another since each alternative is rankedfirst, second and third equally many (4) times. A perfect tie, then, in thissub-profile. Focus now on the left side sub-profile of 11 voters. There A is astrong Condorcet winner, but B is the Borda winner. Now, if the 12-votersub-profile is a perfect tie, its addition to some profile should, intuitively.make no difference to the outcome in the latter: if there was a winner,it should not change by the addition of a perfectly tied sub-profile. And,indeed, this is the case if the Borda count is applied; B wins both in the11- and 23-voter profiles. For Condorcet extensions, on the other hand, theaddition of the tied sub-profile changes the Condorcet winner from A to B.

So, both Condorcet extensions and the Borda count are subject to in-stabilities as the result of various modifications in the choice settings. Whatmakes the former methods more questionable, however, is the existence ofseveral results showing the incompatibility of the Condorcet extensions withsome other social choice desiderata. Of particular interest in the democraticdeficit discussion are results showing that all Condorcet extensions are vul-nerable to the no-show paradox (Moulin 1988). A no-show paradox occurswhen a group of voters with identical opinions is better off (in the senseof their own preferences) by not voting at all than by voting according totheir preferences (Fishburn and Brams 1983). In determining whether theparadox can occur one has to compare two profiles: (i) one where everyonesubmits a ranking and the result is determined on the basis of this, and (ii)one which is otherwise the same as in (i), but a group of identically mindedvoters abstains. If the outcome in (ii) is ranked higher in the preferenceof the abstainers than the outcome of (i), then an instance of the no-showparadox has occurred. The no-show paradox comes in two versions: the‘plain’ one, just defined, and the strong version. The latter occurs when theoutcome in (ii) consists of the alternative ranked first by the abstainers. In

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1 voter 1 voter 1 voter 1 voter 1 voterD E C D EE A D E BA C E B AB B A C DC D B A C

Table 9: Black’s method and the no-show paradox

other words, the result of abstaining is not only required to be preferableto the one in (i), but the best one for the abstainers. Table 9 presents aninstance of the strong version under a specific Condorcet extension, Black’smethod. The method is a combination of two principles: (i) if a Condorcetwinner exists, it is elected, otherwise (ii) the Borda winner is chosen.

In Table 9 alternative D is the Condorcet winner and is, therefore, thewinner of Black’s method. Now, consider the same profile modified so thatthe right-most voter abstains. This is now the profile (ii) in the abovedefinition. In this (ii) profile there is no Condorcet winner. Accordingly, theBorda winner E is elected by Black’s method. A glance at Table 9 revealsthat E is the first ranked alternative of the abstaining voter. We thereforehave an instance the strong no-show paradox.

Obviously the strong no-show paradox is more dramatic failure of re-sponsiveness of a voting system than the ‘plain’ version of the paradox.It is therefore worth asking what kind of procedures are vulnerable to thisstronger version. Perez gives an answer to this question (Perez 2001): nearlyall Condorcet extensions are vulnerable to the strong no-show paradox. Theonly commonly known exception is the max-min rule (Kramer 1977).

From the vantage point of minimizing democratic deficit, the positionalprocedures would seem preferable to Condorcet extensions. At least situa-tions resembling that depicted in Table 5 can be avoided by plurality-relatedsystems. This is, admittedly, not a conclusive argument in favour of posi-tional systems, but in the restricted domain of minimizing democratic deficitit should have some bearing. By adopting Condorcet extensions one runsthe risk of encountering no-show paradoxes and these undermine the veryrationality of ‘going to the people’, i.e. turning to the electorate for advice.The Borda count, on the other hand, can be directly related to the mini-mization of democratic deficit (Nitzan 1981). Consider a profile of individualpreferences over a set of alternatives. Take now any alternative, say x, anda voter, say i, into consideration and determine the number of pairwise pref-erence switches that are needed to make x the first ranked by i. Obviouslythis is the same as counting the number of alternatives ranked higher thanx by i. Considering all voters gives us the sum measure of how far from

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the observed profile is one where everybody ranks x the first. Comparingthese sum measures of all alternatives suggests a reasonable way of electingthe winner, viz. the alternative which has the smallest sum. It has beenshown by Nitzan that is precisely the Borda winner. This gives us a prettystrong case for using Borda count as a method: it minimizes the democraticdeficit when the latter is measured as the distance from consensus. Admit-tedly, this argument against Condorcet extensions rests to some extent onthe definition of the democratic deficit as the difference between collectiveoutcomes and the individual preference rankings. Should one adopt a dif-ferent approach to describing voter opinions, the conclusion might also bedifferent.

We now turn to some ideas, approaches and techniques that have beendeveloped over the past few decades to overcome the difficulties that areassociated with democratic deficit. Is it likely that the theoretical andmethodological innovations will improve the performance of existing – pri-marily representative – democracies specifically by reducing the degree ofdemocratic deficit? Our focus will be on innovations in the two target areas:deliberative mechanisms and secret balloting in computer networks.

6 Technical innovations and democratic deficit

The first forms of democratic decision making were very different from thoseobserved in contemporary political systems. Especially, the city states ofancient Greece – notably Athens – made extensive use of direct democracy.Moreover, executive offices were typically filled by lot rather than election(Tangian 2014). Over time, however, representative, election-based systemshave become the predominant forms of the democratic decision making. Inthese the population elects a set of representatives who then deal with theissue of public policy, including constitutional development. Several reasonsfor the emergence of representative systems can be envisioned: with theincreasing division of labor and the ensuing complexity of governance, theamount of time and resources required for handling legislative issues as wellas the very act of balloting have made it impossible for large segments of thepopulation to engage in direct democracy on a weekly, let alone daily ba-sis. With the advent of modern information processing and communicationtechnology, this practical impossibility is about to vanish. In other words,in developed polities basically all citizens could in principle take part in themaking of decisions on a number of issues on a daily basis. Thus, one of theclassic reasons for representative decision making has lost some of its sig-nificance. Could one then envision a system where all adult citizens would– with the aid of modern IC technology – directly participate in legislationand public policy choice, in general? While no one is seriously proposing asystem of governance based solely on direct democracy, it seems clear that

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such a utopian system would go along way in minimizing the democraticdeficit – under the important proviso that the system applied in aggregat-ing the individual ballots is reasonable. This proviso has been discussedabove. What we focus on in this section is what kind of opportunities themodern IC technology provides for reducing the democratic deficit and whatare the main thus far unsolved challenges.

6.1 Agenda control

Regardless of their type (consultative, binding, constitutionally mandatory,etc) the referenda are the main avenue of direct democracy in modern so-cieties (Setala 1999). The extent of their use varies a great deal from onecountry to another. It is fair to say that they are nowhere the predomi-nant method of making public decisions, but at best auxiliary devices ofpopular control over policies and other legislation. Of crucial importance inreferenda is the question that is subjected to a vote, i.e. the agenda. Un-fortunately, there are no direct ways of assessing the impact of the phrasingof referendum questions to the outcomes, but the rich experimental litera-ture on framing effects on individual choice behavior suggests that this is anissue of considerable importance (Kahneman and Tversky 1979; Quattroneand Tversky 1988; Shafir et al 1989). The order of voting effects, on theother hand, have been given considerable attention in the theoretical socialchoice literature. The pioneering result in this field is McKelvey’s theoremon intransitivities of majority preference relations in spatial models of voting(McKelvey 1979). As will be recalled, it states that in the absence of a core(a majority undominated alternative), the trajectory of pairwise comparisonwinners can lead from any point in the policy space to any other point in it,i.e. under myopic voting the majority outcome depends solely on the agendaof pairwise contests. The result assumes that the voter preferences have acontinuous utility presentation in the policy space. The conditions underwhich the spatial voting game core is empty have been studied by Banks,Saari and Schofield (Banks 1995; Saari 1997; Schofield 2008). The main im-plication of McKelvey’s theorem is thoroughly negative: in the absence of aCondorcet winner-like alternative, the outcomes of pairwise majority votingare basically arbitrary. In particular, they may not even be ‘close to the pre-ferred alternatives of the individuals. In other words, with myopic votinga fully informed agenda-controller essentially determines the outcomes ofpairwise majority voting. And yet, at each stage of the process, the winningalternative is majority-preferred to the losing one, i.e. a ‘democratic’ choicein the majoritarian sense.

A customary objection raised against the use of referenda is that thevoters allegedly use the opportunity to voice their opinions on other mat-ters than the referendum issue, e.g. their opinion about the government orthe head of state. While this may, indeed, be the case, the plausibility of

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this objection is questionable. Firstly, the objection may equally well beused to support referenda so that the voters could signal their opinions onmore matters and, therefore, need not take the opportunity to ‘punish’ theirgovernment, but rather its policies. Secondly, it is always possible for thevoters to cast their votes having something else in mind than the referendumissue. In tactical voting (or – in more technical parlance – manipulation)this happens most of the time in voting by representatives in parliaments.Tactical considerations most certainly enter into the debates (and voting)in parliaments when the referendum issues are being dealt with. In thisphase approaches stemming from the deliberative democracy might be use-ful. After all, deliberative democracy is more about discussing the issuesthan settling them once and for all (LeDuc 2015, 140). It would thus seemthat deliberative bodies would have a useful role to play in the preparationspreceding referenda.

Another perhaps more technical and conjectural role could be envisagedto deliberative bodies in the process of translating popular opinions into col-lective decisions bypassing the parliaments: the restriction of the domain ofopinions. More than half a century ago Duncan Black introduced the notionof single-peaked preferences (Black 1948) and showed that if the preferencesof voters can be represented by single-peaked utility functions, the pairwisemajority voting could not end in a cyclic majority preference relation. Thisresult has subsequently been specified and extended in many directions.E.g. Shepsle and Weingast argued that the legislative committee systemcan under some circumstances act as a mechanism whereby legislative out-comes emerge as structure-induced equilibria as a consequence of the processwhereby each committee handles the issues from a single-dimensional per-spective (Shepsle and Weingast 1981). Perhaps deliberative bodies couldperform similar dimensionality reducing activities as legislative committeesthus making the decision alternatives more transparent and manageable.Some evidence to this effect already exists (List et al. 2013). Perhaps theycould even transform some opinion distributions into the Condorcet domain(cf. the discussion on Campbell and Kelly’s theorem above). In any eventall mechanisms that might curtail the possibilities of agenda manipulationare also devices that make referenda meaningful. As such deliberative insti-tutions are well worth pondering about.

6.2 Elections in computer networks

The almost universally declining voting turnouts have prompted ideas toresort to ICT in governance, in general, and elections, in particular. Indeed,some countries – notably Estonia – already resort to electronic voting in com-puter networks (see www.vvk.ee/voting-methods-in-estonia/engindex/reports-about-internet-voting-in estonia). Advantages of this are several: the con-venience of voters, the speedy determination of election results, the inter-

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pretation of the voter’s intention expressed in balloting, the reduced costs inadministering the election locales, to name a few. At the same time, thereare several weaknesses: the possible exclusion or alienation of voters notused to computers, possibilities of failures due to technical problems (cutsin electricity supply, hardware failures), possibilities of large-scale electoralfraud by hacking the voting system, the possibility of breaches in ballot se-crecy. These caveats notwithstanding, it is reasonable to ask if ICT basedvoting would ameliorate concerns of democratic deficit.

Obviously, internet voting makes the act of balloting a lot easier – atleast to the computer-savvy voters – than the traditional voting in des-ignated locales at publicly announced times using the equipment at handthere. By the same token, elections and referenda could, technically, be usedmuch more frequently than currently. Hence, the possibilities for resortingto direct democracy would be improved. At the same time, however, newproblems arise. To wit, how to combine the results of frequently held ref-erenda into a coherent and consistent policy? The aggregation paradoxesdealt with above can also be viewed as paradoxes pertaining to amalgama-tion votes on separate issues into public policy. In representative systemsit is the task and duty of political parties to present coherent and consis-tent policy alternatives to voters to be voted upon. The introduction offrequently held policy related referenda would seem to diminish the role ofparties as organizers of interests or as agents of interest aggregation, in thepreferred parlance of scholars of the structure-functionalist persuasion.

So, ICT voting is certainly not a panacea, but it can rectify some short-comings of existing voting systems. Of particular interest is the possibilityfor voters to ascertain that their vote has been correctly counted. Withmodern public-key cryptography methods have been developed to make thispossible without sacrificing ballot-secrecy, one of the corner-stones of democ-racy (Chaum 2004; Nurmi,Salomaa and Santean 1991). Similarly, cancellingand re-casting one’s ballot can be made possible with use of cryptographicprotocols. The main obstacle in the way of large-scale adoption of ICT-reliant balloting methods is the fact that with internet voting the conditionsunder which the ballots are cast cannot be supervised by the voting author-ities. Hence, there is no way of ascertaining whether the ballots have beencast under morally acceptable circumstances (and not, e.g. at gun-point).The Estonian voting protocols mix traditional voting (paper balloting at su-pervised locales) with the internet one in the sense that a voter can alwayscancel his electronic ballot, show up at an election locale and re-cast hisballot. Thereby the incentives for buying votes or forcing voters to vote ina certain way are expected to be eliminated.

So, the ICT-based approaches to voting can be of considerable benefitshould one wish to engage the electorates in frequent referenda. As suchthey do not, however, provide any means to counteract one of the mostserious drawbacks of referenda, viz. agenda manipulation. It seems that

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deliberative mechanisms have some potential in curtailing the possibilitiesof agenda manipulation. The extent that deliberative democracy can beimplemented using ICT remains to be seen. Anyway, it is at the intersectionof modern ICT and deliberative mechanisms that the most important stepsin direct democracy are likely to be taken.

6.3 Elections by sampling

Frequent, large-scale opinion polls and surveys are nowadays a common prac-tice in all developed democracies. As far as the election polling is concernedthe opinion polls are also becoming increasingly accurate tools to predictthe election results.3 So, why not use a random sample of voters insteadof the population of active voters as the proxy of ‘demos’? This would ina way continue the ancient Greek tradition of selecting the rulers by lot, atradition that has largely been forgotten over the past two millennia.4 Somerecent work aims at reviving the lottery as a method of democratic gover-nance and using a truly random sample of the electorate to determine theelection result (Chaum 2015). The main advantage of random sample voting(RSV) is the dramatic reduction of costs of a single election. This enablesthe arrangement of a large number of elections and/or referenda withoutincreasing the overall expenditure on democratic decision making. Severalother advantages are also attributed to RSV. To wit, they provide moreincentives to participation than the traditional mass voting. This is due tothe fact that the likelihood of an individual voter in determining the elec-tion outcome is much higher in a random sample than in the population atlarge. Similarly, the fact that a random – possibly small – sample of votersdetermines the outcome of balloting changes the circumstances of election

3Two recent quite high-profile failures in accuracy should, however, be born in mind:the result of the UK parliamentary election of 2015 and the Greek referendum on thefinancial package offered by the ‘institutions’ in July of 2015. In the former case, mostpollsters predicted a nearly even contest between the Conservative and Labour parties,while the Conservatives in fact defeated Labour hands down. In the latter, the ‘yes’ and‘no’ sides were estimated to be of roughly equal strength with ‘yes’ the slightly morelikely outcome, but in fact ‘no’ gained more than 60% of the votes. The vice-president ofWorld Association of Public Opinion Research, Claire Durant, reports in the context of theScottish 2014 referendum that there seems to be a tendency that the polls systematicallyunderestimate the support of ‘no’ side due to the fact that persons who do not disclosetheir opinion are predominantly ‘no’ supporters (Durant 2014). The present author isgrateful to Andranik Tangian for reference to Durant’s work.

4The ancient methods of election and governance are described in detail in (Tangian2014). In fact, Aristotle considered both lottery and election as democratic methods,provided that the body consisting of all citizens appoint from the set of all citizens byusing either of these methods. When some citizens – i.e. a proper subset of all citizens –appoint by lot or by election from the set of all citizens, then the method is oligarchic inAristotle’s terminology (Aristotle, 100a31). Obviously this terminology has not survivedto the present day. Note, however, that Aristotle was not a proponent of democracy, but‘mixed’ forms of government.

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campaigning and advertising. To the extent that differences in resourcesallocated to these activities creates a bias in the results in mass elections,the higher relative cost of a single vote can diminish this bias. RSV alsoenables the voters to ascertain the integrity of the ballot counting system ina mathematically proven and yet relatively easily understandable manner.Buying votes is made much more costly per vote than in mass election sincethe voters whose ballots will be counted is not known to the buyers. By thesame token the coercion of voters is made far more difficult than in masselections. These are the main advantages of the RSV when compared withmass elections. For details, see (Chaum 2015).

To an extent RSV only extends the practice of opinion polls and surveysto a new level, viz. the collectively binding decisions. The main hurdle in itsadoption will no doubt be the perceived legitimacy. Is the electorate willingto accept as legitimate the outcomes determined on the basis of finding outthe opinion of a small subset of it? Are people willing to play a part inelections where it is not certain that their vote will be taken into account indetermining the outcome? This remains to be seen. For the purposes of thispaper RSV has a lot of potential in reducing the democratic deficit. Thisis mainly due the possibility of conducting a large number of elections thatreliably reflect the opinions of the electorate on a large number of issues.At the same time, the consistency and coherence of policies chosen is notguaranteed. Also the possibilities of agenda control remain.

7 Conclusion

Democratic deficit is a somewhat loosely defined concept, but in this paper itrefers to a discrepancy between collective decisions and individual opinions.In the preceding we have argued that equating the will of the people with thewill of the majority of people aggravates the democratic deficit. As is knownfrom Condorcet’s paradox, majority may be endowed with a ‘will’ that is noteven structurally similar to the will of any individual in the group. Majorityrule is, nonetheless, a clearly definable and often used method for teasing outthe will of the people. We have seen that – in contrast to a common opinion– it may lead to paradoxes already in two-alternative contexts. From theviewpoint of democratic deficit minimization it is important to note that itsextension to multi-alternative settings, the Condorcet extension methods,may end up with alternatives that maximize the democratic deficit in thesense of producing choices that are most preferred by no one in the electorate.This suggests that perhaps positional methods could be more useful in aneffort to minimize the democratic deficit of outcomes. The distance-baseddefinition of the Borda count suggests a solution to the deficit minimizationproblem: select as the collective choice the alternative that can be madeunanimously first-ranked with a smallest number of individual preference

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switches of adjacent alternatives. If the manipulability of the Borda countis viewed unacceptable, one could suggest Kemeny’s rule which – given thereported individual preference ranking – determines the closest unanimousranking obtainable from the reported rankings with the smallest number ofpreference switches between adjacent alternatives. As the result of Camp-bell and Kelly shows, this would guarantee non-manipulability in Condorcetdomains. As a drawback this method is afflicted by all those undesirableproperties that characterize Condorcet extensions (including the vulnerabil-ity to the no-show paradox). So, our search for a unique democratic deficitminimizing method turned out to be inconclusive.

We also touched upon issues that may affect democratic deficit from theoutside of voting procedures. We argued that deliberative mechanisms maybe helpful in phrasing the referendum questions and restricting the domainsof preference profiles. The internet voting techniques can also be usefulin expanding the possibilities of direct democracy which presumably is thebest way of finding out the will of the people. The problems of aggregatingseparate referendum outcomes into consistent policies, however, loom large.Random-sample voting is a new approach to democratic governance. Itsbasic principles are well in line with the minimization of democratic deficit,but it still faces hurdles related to legitimacy and consistency.

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