The epistemic account on voting

For Condorcet, as for all epistemic approaches to democracy, the object of voting is not merely to balance subjective opinions or satisficing self-interested preferences; rather, it is a collective quest for truth, or knowledge (^Young 1986, 1995; ^Cohen 1986; ^Schwartzberg 2015). Although it is controversial if truth aptness is attributable to all political decisions, we are going to put that question aside. For the present we are going to grant -in accordance with Young (1988)- Condorcet’s hypothesis that the notion of a “correct” decision also applies to many political decisions, and that all voters have equal epistemic competence higher than 0.5. Thus, voters evaluate candidates in terms of their competence for the position.

The properties of social choice functions on two alternatives are quite well understood, and in general we may say that simple majority is the most natural rule in this case. In particular, May (1952) has given the standard axiomatization of simple majority. But there is no natural extension of simple majority rule to three or more alternatives, because the application of simply majority rule to all pairs of alternatives (tournaments or pairwise comparisons) may lead to cycles (i.e., A beats B, B beats C, C beats A: the famous voting paradox). However, Condorcet proposed that if there is some alternative that obtains a simple majority over every other, then that alter- native should be chosen as the winner. This principle is known as the “Con dorcet’s rule”, and any such alternative is known as a “Condorcet winner”.

What voting rules are most likely to yield good outcomes in cases with more than two alternatives? Ideally, we would like to choose the al ternative that has a simple majority over every other (i.e. the Condorcet winner), but such an alternative may not exist. Moreover, invoking any other method based on pairwise comparisons (v.g. Copeland, that selects as the winner(s) the alternative(s) that has/have more victories in pairwise com- parisons) is not helpful in order to solve this problem from an epistemic point of view, just because majoritarian victories in pairwise comparisons are not epistemic sources in their own, and epistemic credentials are only derivative from the epistemic reliability of individuals.

Assuming an equal epistemic competence and an individual proba- bility higher than 0.5 of getting a true vote in any situation, Young (1988, 1995) showed that the best way to interpret Condorcet’s original purpose with more than two alternatives, is to try to find the ranking most likely to be correct. With that goal in mind, the most likely best ranking turns out to be the one that has máximum pairwise vote support, which is the ranking selected by Kemeny’s rule (Kemeny 1959).

Kemeny’s rule avoids the problem of voting cycles in the social output (not because voting cycles disappear, but because by definition Kemeny’s rule always brings about transitive rankings), although it may lead to ties between rankings. Inherent in that rule is the idea of being “close” to majority rule. Intuitively, ranking the alternatives according to Kemeny’s rule can be seen as the best compromise in the sense that, on average, gives the “closest” social preference to all possible individual rankings given individual voter profiles, and also minimizes the inconsistencies between adjacent alternatives in the social ranking with individual rankings.² Young and Levenglick (1978) also proved that Kemeny’s rule is the unique preference function that is, at the same time, neutral³, consistent⁴ and Condorcet efficient (it picks, as the top alternative in the best ranking, the Condorcet winner when there is one). It also shows certain stability when irrelevant alternatives are withdrawn or added (Young 1986): the ranking on a subset of alternatives is stable if we remove from consideration a block of bottom ranked alternatives (or a block of top ranked alternatives).

However, all these properties are met at certain price: the price of “weakening the crucial assumption about the individual rationality of the voters” (Saari and Merlin 2000: 404). It can be argued that Kemeny’s rule misinterprets certain voters’ real preferences, as coming from a group of voters that do not exist, or that they exist but support individual cyclical preferences.

Remember, from an epistemic point of view, individual preference orderings are the outcome of judgments about the common good. Though it might be disturbing to interpret a particular hierarchical relation between adjacent alternatives inside the social ranking as coming from cyclical individual preferences, it should not cause too much worry if it is interpreted as coming from non-existent voters. Perhaps, although it remains to be studied, it might be argued that some judgmental relations between adjacent alternatives are the supervening judgmental output of the wisdom of crowds. Or, more weakly, it might be argued that if, as it happens, those particular “inver- sions” or “swaps” between adjacent alternatives in the social ranking -with respect to real individual majoritarian rankings- are always placed below the top, it is the epistemic price to pay for getting it right -about facts- at the top. Can these interpretations be traced back to Rousseau’s insights? We have not found any reason to infer from Rousseau’s writings that the general will steams from non-existent or irrational voters, so if we are seeking for a materialization of Rousseau’s thought it is worth considering existent rational voters.

From an epistemic point of view, the problem with Kemeny’s rule is not just that it could end in a draw between so many rankings, or that it could partially violate the rational consistency (transitivity) of individual rankings. The problem is that we can question the very goal of finding the best overall ranking.Why not just try to find the best alternative, irrespective of its relative position in a given ranking?

Consider Borda count. Since the Borda winner is not necessarily the alternative placed in the highest position in the Kemeny ranking, one is bound to infer that selecting the best ranking is a different task to that of se- lecting the best alternative (Balinski and Laraki 2010: xii). And what is more disturbing from an epistemic point of view, we don’t have a single privileged method, since Kemeny and Borda are prone to bring about different social transitive rankings with the same profile of individual preferences. It is well known that the Borda count does not guarantee a Condorcet winner, if it exists. Conversely, when the Condorcet winner and loser are defined, they are, respectively, top and bottom ranked by Kemeny’s rule (Saari and Merlin 2000: 418). However, the Borda count only ensures that the Condorcet winner is strictly ranked above the Condorcet loser. It is also proved that the Borda count always ranks the Kemeny top-ranked candidate strictly above the Kemeny bottom-ranked candidate. Conversely, Kemeny’s rule ranks the Borda top-ranked candidate (or Borda winner) strictly above the Borda count bottom-ranked candidate or Borda loser (Saari and Merlin 2000: 418). The stability of both methods are also quite different: while Borda count is extremely vulnerable to the addition or withdrawal of irrelevant alternatives (sometimes bringing about the complete reversion of the social ranking after the withdrawal of an irrelevant alternative), it was proved that the withdrawal of a block of top ranked adjacent alternatives or a block of bottom ranked adjacent alternatives -placed in the Kemeny ranking- leaves the order of the remaining alternatives untouched (Young 1988).

In addition to those formal relations between Borda count, on one side, and the methods of Condorcet and Kemeny, on the other, we also want to focus on a substantial, ontological difference. Condorcet’s method uses votes as a mean to build a tournament from which counting victories is a derivative calculation. In fact, there exists a vast literature on tournament so- lutions as a basically qualitative problem totally independent of how tourna- ments are built (^{Brandt 2010}).⁵ In Borda count, instead, the number of votes is an essential quantitative information for the whole process that is not con- tained in a tournament. And that quantitative information can tell us in what extent or proportionality a candidate or ranking gets majoritarian support. Young’s maximum likelihood estimation and, by equivalence, also Kemeny’s rule, build a bridge between both ontologies. Still, doubts about whether Ke meny’s rule extends Condorcet method in accordance with Rousseau’s ge neral will persist. Insofar as minority votes are counted, we obtain a different proportionality than if we counted only majority votes. So, the question we raise is: Is there any overcoming way to synthesize the epistemic aptness of the Condorcet method with a deference commitment to majority votes? Next, we reconsider Rousseau’s theory of voting seeking support for an affirmative answer to that question and guidelines for a novel approach.

Reconsidering Rousseau’s theory of voting

Fousseau has been declared by many scholars as the patron saint of epistemic democrats (Cohen 1986, Young 1988, Schwartzberg 2015, Waldron 1990, Estlund 2008, Pettit 2016). Although there is some controversy about him being really an epistemic democrat (who asks voters to follow judgments about facts on the best policy) or, instead, a non-epis- temic, agonistic democrat (who ask voters simply to follow the general nude “will” on the preferred policy; see Tuck 2019), he is, in our opinion, a clear enthusiast of epistemic democracy, properly defined along the canonical lines settled by Cohen (1986).

Curiously, what seems to be undisputed in the literature is that Condorcet’s theory of voting is, indeed, the best operative incarnation of Rousseau’s theory of the general will along the epistemic lines (^{Barry 1967}, Cohen 1986, Grofman and Feld 1988, Young 1988, Schwartzberg 2015, Estlund 2008). On the contrary, we are going to argue that Condorcet’s theory of voting must be considered fundamentally distinct from Rousseau’s theory: the basic epistemological premises that Rousseau defended in the Social Contract are not properly captured by Condorcet’s statistical approach (i.e. one that gives operative justification, as we explained, to Kemeny’s rule). Instead, we believe that Rousseau’s basic premises provide justification for a very different operative statistical estimation account, the mechanics of which we will explain in detail below.

Let’s begin with the tenets of Rousseau’s theory of the general will. In addition to the passage cited in the introduction, the following important paragraph comes from Book II, chapter 3. In outlining the difference be- tween the will of all and the general will, Rousseau says:

The latter [ie. the general will] looks only to the common interest, the former [i.e. the will of all] looks to private interest, and is nothing but a sum of particular wills; but if, from these same wills, one takes away the pluses and the minuses which cancel each other out, what is left as the sum of the differences is the general will (emphasis added).

There are many possible interpretations of these and other important paragraphs on the notion of general will from the Social Contract, and our aim is not to dismiss the validity of other readings. However, in our understanding, the general will of Rousseau is compatible with the will of an un-factionalized assembly where each voter searches for the truth about a matter of fact. That is why “when the opinion contrary to my own prevails, it proves nothing more than that I made a mistake and that what I took to be the general will was not”. It can, therefore, be interpreted as an epistemic enterprise. Thus, the general will may be expected to track, through the aggregation of votes sus- tained in judgments (perhaps preceded by deliberation), the common interest of citizens, not the interests of this or that individual or subgroup: “it is con cerned with their common preservation, and the general welfare”, as it is said.

Now, voters may defend diverse judgments over what is best for the common interest, and some may vote for A, some others may find reasonable to vote for B, others for C, and so on. These disagreements may lead to contingent and variable splits between majorities and minorities on agenda voting issues, with no permanent identifiable minority on each and every issue (otherwise, that fact would be evidence of a factionalized assembly). Democracy involves some ruling others, notwithstanding the disagreements that might continue to exist between sequences of majorities and minorities (of different size and people across issues). Roughly, on standard liberal accounts, the majority on any decision rules over the minority. However, Rousseau tries to show, to the contrary, that in a proper democracy, each “obeys no one but himself”. How could this be achieved? Rousseau gives a provocative answer: by asking minority voters to surrender judgment to the majority judgment. A problem with Rousseau’s idea of deference comes when we try to justify in what conditions minority voters should surrender judgment, and what does it mean. Does it mean only that they should give less credence to their beliefs (perhaps to the extent that they must suspend judgment), does it mean that they have epistemic reasons not only to suspend judgment but also to uphold the majority belief, or does it mean to follow blindly the majority belief?

From an epistemological point of view, in political matters where normative principles and values are always at stake, the idea of feeling obliged to surrender judgment, just because we are in a minority, is intriguing to say the least. If we are convinced that a certain person is clearly superior compared to us on many disagreement factors when it comes to answering the question “Is A good, or better, compared to B, C, or D?” then we’ll probably say she is more likely to answer the question correctly. However, we would suspend judgment not because she has more followers than we have, but just because we acknowledge she is an expert on the dimensions that are relevant to decide the issue at stake. But in political matters where com- petent experts usually are in both sides, or where there are many dimen- sions involved pushing to divergent directions, the reasons for surrendering judgment are less clear. Citizens in a democracy have a moral duty to obey a binding rule coming from democratic voting. As Schwartzberg argues, the outcome under majority vote “possess moral force insofar as [it] induced outvoted minorities to recognize that they had likely erred” (Schwartzberg 2008: 406). But they do not have an epistemic duty to submit their judgment to it (cf. Estlund 2008: 103 ss.; Rawls 1950: 319).⁶

This is the problem of deference faced by Rousseau’s epistemic approach to voting, a problem that -we argue- doesn’t fit well with Condorcet’s theory of voting and its byproduct, Kemeny’s voting rule, as long as Kemeny’s rule aggregates -while searching for the “average” ranking- not only majority votes, but also minority votes. Remember that when the voting agenda comprises more than two alternatives, Condorcet’s theory leads to the application of Kemeny’s rule, which stands as a reasonable rule for estimating the most probable correct ranking. But Kemeny’s rule counts minority votes, not only majoritarian votes. This counting of minoritarian votes, we argue, is in stark contrast with Rousseau’s theory of voting.

To be honest with Rousseau’s thought, any statistical account on the probability of getting it right must portray in some way this theory of de- ference. And we think that that deference commitment could be perfectly introduced in an algorithm, by setting a simple condition: minority votes on pairwise comparisons should not be counted, insofar they are expressions of a wrong epistemic judgment.

Let us go back to Young (1988, 1995), whose interpretation of Rous- seau’s thought as the philosophical source of Condorcet’s epistemic theory of voting has become canonical in the literature. Young argued that, when interpreting the phrase “from these same wills, one takes away the pluses and the minuses which cancel each other out, what is left as the sum of the differences is the general will” (emphasis ours), the only mathematical sense that we can make from that is to look up for the workings of something like the jury theorem elaborated by Condorcet. Young thinks that the way in which that theorem involves error (which is particular to each individual and randomly distributed among them) cancelling, and leaving truth (which is common to all) as the remainder, is thus the best operative embodiment of Rousseau’s theory of voting. We want to bring up for discussion another interpretation of Rousseau’s minority deference commitment. Error in Condorcet’s theory comes from an a priori definition of individual competence. In contrast, error in Rousseau’s theory comes from an a priori definition of competence and from the ex post awareness of being in the minority side. Sources of error are philosophical different, and that divergence should be consequential.

Proportional majoritarian support

How can we formulate an algorithm compatible with Rousseau’s idea about the minorities’ deference commitment? There surely exist many ways to carry out the task. But we also want the method to satisfy the other reasonable conditions that we have been discussing in accordance with Rousseau’s thought. First, the epistemic virtues of the Condorcetian method should be respected, in the sense that if a Condorcet winner exists, then it should be chosen. Moreover, “taking away the pluses and minuses that cancel each other” supposes, in principle, that the choice should take into account the quantities of the votes, and not just how the candidates are beaten in a tournament. This is an important difference with Condorcet’s method (similarly, Copeland’s, and others) where votes are only a mean to reach the tournament in which victories are counted. So, how votes are distributed in the preference profile is crucial. In addition, we should assume that voters are rational agents that always posit transitive rankings of alternatives. Once the majorities are identified in the pairwise comparison matrix, we calculate, for each alternative, the probability of that alternative being elected by the majority against any other alternative. Finally, the result should be such “what is left as the sum of the differences is the general will”, i.e. the alternative with greatest proportional majoritarian support is chosen.

Tablas 1-2 

Let us see how our proposal works with an example. Consider a setting with eleven voters and alternatives A, B, C, D, E, F, and G. Adhesions to rankings are distributed as shown in Table 1, with alternatives ranked in top-down order:

We proceed as follows. First, in each row we delete all the minority votes, which leaves us only with the numbers in bold. This intends to model the deference commitment of the minority in favor of the majority. Then, we calculate a coefficient for each alternative, call it the proportional ma- joritarian support, which is the sum of all the majoritarian votes in favor of the alternative (bold numbers in the alternative’s row) divided by the sum of all the majoritarian votes both in favor and against that alternative (bold numbers in the alternative’s row and column). This puts the majority support in relation to all majority opinions. For instance, the coefficient of alternative A is 10x5/((10x5)+6) = 50/56 = 0.892. Doing the same with the remaining alternatives, we find that B is the chosen alternative, having the maximum proportional majoritarian support: 36/36 = 1.

It is clear that the more majoritarian “pro” votes an alternative gets in the pairwise comparison (i.e. more bold numbers in its row) the less majoritarian “con” votes it gets (i.e. less bold numbers in its column). In con- sequence, the greater this difference between majority votes for and against, the closer the proportional majoritarian support will be to 1. This remark leads us to conjecture that our method guarantees that the winner alter native (or alternatives, since ties are possible) will never be covered⁷ by other alternatives, provided that voters are rational agents⁸. It follows that the Con- dorcet winner, if it exists, will have the maximum proportional majoritarian support. In the above example, there is no Condorcet winner, but we still get a best alternative, B. This is also the alternative with the greatest difference between the number of victories and the number of defeats, which makes it also the Copeland winner (Copeland 1951). Even more important, our method could be more selective than Copeland’s as the example of Tables 3 and 4 show.

There are two Copeland winners, B and E, both with a difference of 5 points between victories and defeats, while our method chooses B, whose proportional majoritarian support is 0.877. Once the Copeland’s rule focuses on the resulting tournament to solve the outcome, leaving behind the voting tally, an important source of quantitative information is lost. And that information makes the difference to find more accurate results.

Tablas 3-4 

Another important relation among Copeland winners and alternatives with maximum proportional majoritarian support involves the uncovered set of alternatives (Miller 1980). It includes all the alternatives that are not covered by others. In formal terms, let C={(x, y): x covers y} (see footnote 7). Then, the uncovered set of a set of alternatives X is UC(X) = {xeX: for all ye X, (y, x)¿C}. If a Condorcet winner exists, it is the unique element of UC(X). Otherwise, UC(X) includes at least three alternatives (i.e. the set of all the alternatives that are not covered by others). Miller observed that Copeland winners always belong to the uncovered set. A fortiori, the approach based on the proportional majoritarian support enables in some cases -as the last example shows- a refinement of the uncovered set.

In the next section, we make a case for the importance of the un covered set as a limit to instability introduced by strategic voting.

Setting bounds to self-interested strategic voting

The view of voting as based not on judgment but on naked self-interest is attributable to Bentham, but was later assumed by most standard utilitarian approaches to voting. Bentham was a psychological egoist, and assumed that people always act to further their own satisfactions (Waldron 1990). Provided that people may vote pursuing their self-interested preferences, they may find appropriate -after being aware of other voter preferences- to express insincere individual rankings in their votes, if that strategy leads to a better satisfaction of their true -concealed- preferences. Considered from an utilitarian legislator’s point of view, if all participants vote this way, the political choice tends to represent the correct answer, since by definition the outcome should maximize the aggregated satisfaction of individual utilities. This view of voting is non-epistemic by definition: voters do not vote following their judgments on the best choice, but their naked selfish preferences. Needless to say, the outcome of the procedure may be regarded as true or valid (in the sense that -according to utilitarianism- it might fit with a true or valid principle ofjustice: the utility principle), although the nature of voting as such, as satisfaction of selfish preferences, remains non epistemic. In this view, correct answers (given by the utility principle) can be derived from majority selfish preferences, and the “general consent” that is brought about by majority rule provides “the surest visible sign and immediate evidence of general utility” (Harrison 1983: 214; Cohen 1986: 28).⁹ The important point about this model is that individual votes represent individual satisfactions, and majority vote-counting approximates a social welfare function with individual satisfactions as its non-epistemic support.

According to this model, voters may behave strategically to pursue their self-interests. That would imply that they might occasionally find in their interest to vote first for a candidate or alternative they do not place at the highest level in their preference order, if that strategy has a higher probability of achieving -given other’s exhibited preferences- his first true preference.

This theory is implicit in most positive approaches with which political economy and political science study the strategic behavior of voters and its outcomes, either in elections or legislatures. However, from a phi- losophical perspective, their difficulties are plain enough. Some are internal to the model (such as that voting cannot possibly be made to reflect the intensity of the satisfaction or dissatisfaction anticipated by individuals with respect to some law), some are external, this latter on which we would like to pay attention. As a predictive theory is almost certainly false. People, whether they are voters or politicians, do not make decisions purely on the basis of self-interest. They are occasionally (we think, often) motivated by their sympathies for others, their own judgments -based on imperfect knowledge- of what would be conducive to the general good, or adherence to some moral ideals.

Now, let’s take for granted that people vote pursuing their self-inte- rested preferences, and that the aggregation of votes “as mere preferences”, trough majority rule, should be considered the maximization of aggregated utility or preference satisfaction, and therefore that the social outcome would be -according to utilitarian thought- the true or correct social welfare function. Well, it happens that there could be situations in which there is no single outcome, and in which the social outcome violates basic principles of consistency, stability of preferences, and rationality. These problems are sum- marized in the celebrated Impossibility Theorem demonstrated by Kenneth ^{Arrow (Arrow 1951}), the details of which are widely known and we have no space here to explain. According to Riker (1982), Arrows Theorem under- mines the very pretense of finding always a majoritarian “will” with more than two alternatives (and remember, in politics there are always more than two alternatives available, even if all of them are not submitted to a vote).

When a Condorcet winner exists, according to a certain distribution of preference profiles among voters, and the method picks the Condorcet winner, perhaps that outcome might be considered the true utilitarian outcome. But that very outcome may come at the price of introducing or withdrawing irrelevant alternatives (with the aim of manipulating the result), or at the price of restricting the menu of alternatives that are submitted to a vote (what is called “universal domain”). Arrow demonstrated that there is no single method that could meet the three conditions altogether: uni versal domain, independence from irrelevant alternatives, and transitivity (no cyclic majorities). But things don’t end there. It was later shown that cyclic majorities are infrequent in unidimensional spaces with few alternatives, but abound in multidimensional and continuous policy spaces¹⁰, and that, when majority cycles exist, they might cover -under certain condi tions elabo- rated by McKelvey (1976, 1979)- the entire space of political alternatives. A strategic agenda setter with full information of voter preferences can reach -through successive binary elections- to any point in the policy space, pro- vided voters are sincere and shortsighted. Intuitively, the widespread possi- bility of cyclical majorities in these settings implies that the outcomes pro- duced by voting may not have the consistency we require of a “common will”, along utilitarian lines. Riker (1982) concludes that we should reject the goal of finding a common or collective preference “not because it is morally wrong, but merely because it is empty”. With insincere, self-in- terested voters, the whole adventure of finding a utilitarian social function collapses in the maelstrom of McKelvey’s chaos theorem.

This instability, however, has limits. Miller (1980) and later McKelvey (1986) showed that, if we drop sincerity and assume sophisticated voters (that is, voters that behave strategically and often vote insincerely for an alternative they don’t prefer as the best, or against an alternative that they prefer in first place) with complete information on the alternatives, the set of sophisticated equilibrium outcomes -through voting by successive elim- ination- must lie inside a central bunch of alternatives, the uncovered set. Thus, the uncovered set puts some bounds on the manipulability of the agenda, while introduces a relative limit to the general instability of possible outcomes in multidimensional spaces. With sophisticated voters with com plete information, voting cycles may emerge but the uncertainty is confined in a central area identified by the uncovered set. To put it clear: outcomes guaranteed within the uncovered set do not imply the strategy-proofness of the method, but manipulability can only replace one uncovered alter- native with other uncovered alternative. In this sense, manipulability with the method proposed here is limited in such a way that an outcome obtained by manipulation cannot be worse (inasmuch it is not covered) that other alternatives.

The uncovered set might be the political equilibrium to which sophisticated voters concur with full information of other’s preference profiles, so while it might be unstable inside it, it could be regarded as the set of alter- natives that loosely define -in an indecisive way- the social choice outcome that maximizes satisfaction of self-interested preferences under the utilitarian model. It’s worth exploring whether, departing from the epistemic model, in accordance with Rousseau, it is possible or not to always guarantee the selection of an alternative that, under certain preference profiles, are placed inside the uncovered set. Intuitively, there is no reason to think that any epistemic method, however designed, can assure the choice of an unco- vered alternative. For the meantime, it is enough to observe that epistemic methods could display divergent outcomes from the ones that could be predicted under strategic behavior by self-interested voters, an intuition that only would confirm that the rationality of the strategy is essentially different than the rationality of judgment about the common good.

In any case, the important point is that when an epistemic voter is ex- pressing an opinion based on a judgment about what justice, or the common good, requires, on the overall utilitarian model, however, an individual’s vote only expresses self-interested preferences. The utilitarian approach attempts to sum votes as utilitarianism sums satisfactions, but with strategic voters and more than three alternatives, the predicted outcome of that method is unstable and indecisive, however confined to the uncovered set. Instead, the epistemic model of voting tries to estimate the most probable correct alternative or ranking, assuming certain conditions (of which equal competence higher than 0.5 and sincerity are common) and while they are challenged from several angles, from the standpoint of judgmental rationality and sta- bility, their potential outcomes go in predictable different directions than the utilitarian model, even if we put in brackets the internal quality (self-inte- rested or judgmental) of preferences.