Models of Epistemic Democracy[1]

Draft: Please Do Not Cite or Circulate Without Permission

Archon Fung

 MIT Department of Political Science

10 November 1995

1010 Massachusetts Ave., Apt 57

 Cambridge, MA 02138

 Phone: 617/876-1950

 Email: afung@mit.edu

Abstract

 This paper constructs two distinct models of epistemic democracy and then uses those models to draw several implications for the design of concrete political institutions. Epistemic democracy provides an attractive normative and descriptive understanding of majority rule institutions as mechanisms of discovery rather than as devices to aggregate competing individual interests. Most attempts to specify the mechanisms of epistemic democracy employ the Condorcet jury theorem. The first finding of this paper is that when the number of policy options in the Condorcet model is extended from 2 to N, majority-rule in a two-party system yields incorrect decisions. In the second part, I propose a model of epistemic democracy which is based on experimental learning and which does not rely on Condorcet principals. This model offers a more plausible description of democratic assemblies because it includes the important dimension of time. It also makes a stronger case for supporters of epistemic democracy because its assumptions are less demanding than those of the Condorcet theorem.

 Introduction

 In the standard account, voting is a mechanism for aggregating the preferences of citizens in order to create a single social choice from their individual values. In his article "An Epistemic Conception of Democracy," Joshua Cohen (Cohen, 1986) explores an alternative, epistemic, interpretation of voting. The epistemic account presumes that there is a consensus on the relevant values, that voters do not know how to advance those values, and that majoritarian democracy provides one "discovery" mechanism. More specifically,

An epistemic conception has three main elements: (1) an independent standard of correct decisions--that is, an account of justice or the common good that is independent of current consensus and the outcomes of votes; (2) a cognitive account of voting--that it, the view that voting expresses beliefs about what the correct polices are according to the independent standard, not personal preferences for policies; and (3) an account of decision making as a process of the adjustment of beliefs, adjustments that are taken in part in light of the evidence about the correct answers that is provided by the beliefs of others. (34)

This short paper adds nothing to the normative account of epistemic democracy above. It presumes that there is a range of values and goals which people share and which might be called a general will; that people can and should vote according to their opinion about what the general will requires rather than in their own private interest; that each knows only imperfectly what that is; and that properly constructed democratic electoral institutions provide a good mechanism for discovering the correct policy instruments with which to advance the general will. Indeed, this paper offers several innovations which aim to carry forward the project of epistemic democracy by constructing two formal models.

 The first is an extension of the Condorcet Jury Theorem (CJT) theorem to more than two policy options. Most critics have considered only the dichotomous case[2] and have taken the Jury theorem to be an argument in favor of democratic majority rule (e.g. Grofman and Feld 1986, p. 95). However, when the Condorcet Theorem is generalized to more than N policy options, it is clear that majority-rule yields incorrect decisions because the correct decision under extended CJT is the modal decision, while majority-rule selects the median. Consequently, the extended version of CJT provides an argument against, not one in favor of, two-party majority-rule.

 The second model breaks from the standard understanding of epistemic democracy as a Condorcet Jury and constructs an alternative model which is based on iterated experimentation. I propose that epistemic voters will attempt to correct their mistakes through time and thus offer learning by experimentation as a central mechanism of epistemic democracy. I argue that this model is both more descriptive and more attractive than conceptions based on the CJT. Furthermore, this experimentalist rendering of the idea of epistemic democracy is sufficiently precise to offer several practical implications for the design of democratic institutions.

Epistemic Democracy as a Condorcet Jury

 Epistemic democracy is very much a rediscovery of 18th Century political notions. The idea originates in Rousseau's account of voting in his Social Contract:

 When a law is proposed in the people's assembly, what is asked of them is not precisely whether they approve or reject, but whether or not it conforms to the general will that is theirs. Each man, in giving his vote, states his opinion on the matter, and the declaration of the general will is drawn from the counting of votes. (SC, IV.2.viii)

Contemporary scholars have used the mathematical tools of Rousseau's contemporary, the Marquis de Condorcet, to make the idea of the General Will more precise and to give it force. Specifically, the CJT states that, given any pair of alternatives and a group of individuals each of whom is somewhat more likely than not to choose the better alternative, the probability of the group making the correct choice through a majority-rule vote rises rapidly with the size of the group (Grofman and Feld, 1988). This metaphor of a democratic assembly as a jury captures some of the essential aspects of epistemic democracy: there is an objective, correct answer to the question which the body puts itself, no member of the body has access to that truth, and deliberation between each member, who is equal to all the rest, followed by collective choice is the best method available to determine the truth.

 However, all legislatures, even epistemic rather than adversarial assemblies, are different from juries in two essential ways. Whereas juries face the binary choice of determining whether a defendant is guilty or innocent, a legislative assembly must decide between a vast array of options. Furthermore, the choice of a jury is more or less a final judgment, whereas legislatures revisit the same issues time after time. Even within the normative parameters of epistemic democracy, these two differences severely limit the ability of the Jury Theorem to capture the operations of a legislature.

 On the first difference, suppose that some large group of N people must choose between M alternatives. The most straightforward translation of Jury theorem to the case of multiple alternatives utilizes plurality voting as the social choice function; each voter gives his opinion about which of the M alternatives is the best one, and the option receiving the most votes is the social choice. If each voter has a probability of choosing the correct answer which is greater than his probability of choosing any other option, then the chance of the group choosing the correct answer through plurality voting rises rapidly with N and with his competence.

 This phenomena is intuitively clear if we think of the each voter's judgment (we take all voters to be the same) as an underlying probability density function over the choices he faces: {A1, A2, ..., AM} , where A1 is the correct policy, and each voter's choice as an independent and unbiased sample of that distribution. We stipulate that each judgment is sufficiently good that his probability p1 of picking A1 is greater than his probability of picking any of the other policies (p1 > p2 >= ... >= pM); this is to say that the highest peak of the density function lies at A1. Tallying all of the votes provides an estimation of the underlying probability density function, and the point of the vote is to select its peak. Since each additional vote provides another sample, the accuracy of the total vote as an estimation of the underlying probability increases with the number of voters (Young 1988). By the law of large numbers, the probability of perfect estimation, and therefore the probability of discerning the density function's real peak, approaches unity as N grows infinite.

 Functionally, the probability of a plurality vote selecting the correct choice is given by a series of nested summations. Let N be the number of voters and M be the number of policy options from which they can choose. Let each voter have a probability p1 of choosing the correct policy, A1, probability p2 of choosing policy A2, ..., and probability pM of choosing AM. Furthermore, let p1 > p2 >= ... >= pM. (Hence p1 + p2 + ... + pM = 1). In any vote, the probability that A1 occurs k1, times, A2 occurs k2 times, ... AM occurs kM times, is equal to the multinominal expansion term (Lipschutz, 1965):

, where ( ) (1)

The total probability,PM(N), of the correct choice gaining more votes than any other choice is obtained by adding all of the multinominal terms for which (k1 > k2 >= k3 >= ... >= kM).

 For M = 2, Condorcet's case, this turns out to be (Grofman, Owen, and Feld, 1983):

(2)

For M = 3, the the chance of the correct answer winning is:

(3)

And so on.

Consider the surface formed by plotting the probability of the correct choice winning (by plurality or more) against competence (the probability of choosing the correct policy, p1, which is equal for all voters) and the number of voters for the three policy case. In the figure below, the probability of picking the correct policy (for each voter, all voters equal) ranges from 0.3334 to 0.6000, probability of picking either of the incorrect policies is equal (p2 = p3 = (1 - p1)/2), and the number of voters ranges from 3 to 100:

 Figure 1: Probability of choosing correct policy among three choices.

The function's characteristics are similar to Condorcet's binary case; the probability of the correct answer being chosen increases with the competence of each individual voter and with the number of voters.

 An interesting institutional implication of this simple extension, however, is that a two party, winner-take-all, system does not make correct decisions. On one parsimonious and trenchant understanding, a social choice rule of majority rule in a political arena of two parties yields decisions which conform to the preferences, or in this case the opinions, of the median voter. Following Duncan Black (1958, p. 4-21), we suppose that a set of M policy alternatives can be ordered {A1, A2, ..., AM} such that the preference (or opinion) curve of each voter is single-peaked. In this case, the policy preferred by the median voter can gain a simple majority over any other policy, and no other policy can do so. If we imagine, with Downs and Hotelling, that political parties operate as entrepreneurs who try to capture the greatest number of votes, then parties will converge upon the median position in a two party system; if one party begins on the left side of the political spectrum and the other on the right, each moves toward the center in order to capture more votes, until each reaches the median voter (Downs, 114-141).

 Bringing this framework to the Condorcet model, each voter does not have a preference curve, but rather a probability density function over the available alternatives which represents his competence to select the best policy. Furthermore, we retain the simplifying stipulation that the probability density function for all voters is the same. Like Black's preference model, the policy picked through voting by majority rule is the median of the density function. However, in extension of the CJT to N cases above, the correct decision is not the median opinion, but the modal one. The two are not equivalent, of course, in the case of skewed density functions. We conclude, therefore, that majority rule under the two party system selects the incorrect policy just in case the median of the "judgment" probability density function is not equal to its modal value.

 Consider the following illustration with 16 voters and 8 policy issues:

 Figure 2: Condorcet versus Majority-Rule Policy Selection

If we assume that the each voter has a greater probability of picking the correct choice than the incorrect one, then the inference from the epistemic vote above is that Policy 2 is most likely correct. However, majoritarian voting in a two party system yields Policy 4 as the social choice.

 Without offering formal proofs, there are good reasons to think that other institutional forms can better realize the model of epistemic democracy presented above. Since a detailed discussion of these alternatives would lead us too far astray, so I will only catalog some of them here. A more direct democracy, unmediated by parties, based on a winner-take all plurality rule, is the most direct translation of the extended Condorcet model. However, direct democracy has several obvious defects which might overbalance its epistemic benefits; critics have argued that there is less deliberation under direct than representative democracy, that policies are less coherent, and that information costs are higher. The proliferation of political parties under a winner take all system is another option; at the limit, a great number of political parties approximates plural voting. A reasonable number of political parties which is nevertheless greater than two might offer a wider ranging discussion of policy options and proposals.[3] However, multiple parties also encounter problems of cyclicity, and there is no persuasive reason to think that they would settle closer to the modal value than a two-party system. Finally, proportional representation offers an alternative which encourages discussion and thus the flow of information by giving those who would be loosing minorities in a winner take all system a voice. Since there are many situations in which the model policy both looses and enjoys considerable support, proportional representation as a social choice mechanism might bring the collective choice closer to the "correct" decision.

An Experimentalist Epistemic Conception of Democracy

 Having considered the effect of multiple alternatives on epistemic voting, we now turn to the more fundamental considerations of reflection and time. An epistemic democratic assembly which is true to its principles and purpose will attempt to correct its mistakes. If we presume that an assembly has repeated opportunities to correct itself and is reasonably competent to at doing so, then we can think of epistemic democracy as a continual process of experimental adjustment through time which approaches the correct policy solution. The central mechanism of this model is self correction through time rather than improvement through large numbers, and so the simplest way to capture this idea is not to modify the model above (though each iteration can certainly be thought of as a Condorcet vote), but to construct a new one from scratch.

The basic steps in this model are straightforward. We assume that voters are the equal in their ability to evaluate the performance of past policy and from this infer the position of the correct policy in relation to the (incorrect) one which they have chosen. On the first round of voting, each voter makes his best guess about the correct policy. These votes are aggregated through some rule, say majority rule, into a social judgment which is then enacted as policy. We presume that this policy is not the best possible, correct, policy. After some experience living under their chosen rule, they realize that they can do better, and each of them makes another guess about where the best policy lies, and they vote again to choose what they hope is a superior policy. They iterate this process of self-correction indefinitely.

 To illustrate, suppose that there is a one dimensional policy space over over which there is a normal distribution of opinions about the correct policy, but the median of the distribution is not "correct." After a majority rule process, the majority selects the wrong policy:

 Figure 3: An Initial Guess in Epistemic Voting

In Figure 1 above, P0 is the policy which results from the initial round of epistemic voting, Pc gives the position of the correct policy choice, and the distance between the two, which is the error, is given by[4]

d0 = Pc - P0 (4)

Suppose further, that voters learn that the policy which they enacted is incorrect, and that they have some knowledge of the relationship between the chosen policy and the correct one. Furthermore, we presume that their estimation is imperfect. To simplify, we presume that all voters are identical in their ability to evaluate the results of policy, and the estimation of the error of policy for each is given by a Voter Error Evaluation function, E(d, c)

E(d, c) = d (1 + c), (5)

where d is the distance to the correct policy, and c is the competence of the voter, with c = 0 representing perfect competence (so c > 0 represents an activist tendency to overshoot policy and c < 0 is a conservative tendency to change policy by less than the optimal amount).

 Now it may be objected that the error evaluation function in eq. (5) is unrealistically optimistic about the capacities of voters. One response to this objection is that eq. (5) is substantially less demanding than the Condorcet model. On that mechanism, voters must have a probability of picking the best policy which is greater than the probability of picking any of the others. This experimentalist model, however, is more forgiving; it does not require voters to have a high probability of picking the correct policy, but only requires voters to have a high probability of picking policies which are better, in the sense of lying closer to the correct policy, than their current position. Furthermore, the experimentalist model explicitly incorporates experience of past policies into its understanding of judgmental competence. While one of the sources of judgmental competence on the Condorcet model must certainly be experience, its one-shot character offers no way to characterize learning.

 Another objection to eq. (5) is that the functional form is arbitrary and ought to be further specified. My response to this objection is that it could certainly be further specified, but that this level of abstraction is optimal for our purposes because it makes the fewest assumptions while still capturing the most essential elements of experimental epistemic democracy. So one obvious path to further development would be to specify a functional form which maps policy choice onto policy output, and then determine the voter error evaluation function which reaches the optimal policy choice in the fewest number of iterations. This extension would bring this paper into a well traveled ground of computational search algorithms.[5] This further specification of the model, however, would detract from its realism because it would force the modeler to stipulate something which can be learned only through social experience: the function which maps policy onto policy output. As eq. (5) stands, accurate knowledge of this function and of the proper search algorithm is endogenous to the model and captured by the voter competence term c.

 On the next iteration, voters compare notes and make a judgment about how far their policy differs from the actual policy, and then vote again by adjusting their initial vote P0 according to the error function (5) above. The position of a policy chosen in any round n, then, is given by the function,

Pn = Pn-1 + E(dn-1, c) (6)

Substituting equation (5) for the error function in (6) yields

Pn = Pn-1 + dn-1 + cdn-1 (7)

Further substituting the definition of d in equation (4) for the second term on the right hand side of equation (7) yields,

Pn = Pc + cdn-1 (8)

Subtracting Pc from both sides and then applying the definition in (4) yields a recursive form of the equation for the error after the n^th vote as a function of the (n - 1)^th vote:

dn = -cdn-1 (9)

This equation can also be written non-recursively, in terms of the first error term in the sequence, d0,

dn = (-c)ⁿd0 (10)

All this is to say that if voters have perfect competence, defined as c = 0, then they reach the correct policy on the next iteration. If |c| < 1, then successive rounds of voting generate results which become asymptotically close to the correct policy. If |c| > 1, then future rounds of voting yield results which grow more distant from the correct policy. After the n^th round of voting (not including the initial vote), the distance to the correct policy decision is given by equation (7) above.

 If |c| < 1 then after (n > nthreshold) iterations of voting, the distance between the chosen policy and the correct one will be less than some epsilon, e, where

(11)

So, the number of rounds of voting, as a function of competence level, required to come within e = d0/10, is given by the inverse log function n = - 1 / log |c|:

 Figure 4: Rounds to Asymptotic Convergence (defined as d0/10)

Institutional Implications of the Experimentalist Model

 Two obvious implications for the design of institutions grow out of this this experimental version of epistemic democracy; institutions should increase the political competence of voters (c) and they should increase opportunities for experimentation (which is captured above by the variable n).

 The first point supports arguments for enhancing the policy competence of citizens through the traditional methods such as increasing political literacy and the quality of deliberation between citizens. Experimental epistemic democracy, however, is especially sensitive to voter competence in evaluating the performance of past policy. Unlike most considerations of voting and legislation which treat collective decision making prospectively, the weight of institutional performance under experimental epistemic democracy falls on the retrospective capacities of voters. The competence term c in equation (5) refers to the ability of citizens to determine how closely the policy for which they voted lies to the ideal policy.

 Another interesting implication of the rudimentary experimentalist model, as I have constructed it, comes from the absence of a requirement of voter independence. The literature on Condercet voting has been very much concerned with voter independence (Rawls 1971, p. 358; Ladha 1992; Estlund 1994) because the central mechanism of the jury theorem relies upon independent and unbiased sampling of an unknown probability density function. Independence, however, is thought to conflict with the presence and desirability of political discussion, which might contribute to the development of voter competence.[6] The core mechanism of the experimentalist model, however, relies upon learning rather than large numbers. Since considerations of political competence overshadow considerations about independent sampling, epistemic democratic institutions should, without reservation, encourage a free-wheeling discussion of past policy performance and prospective options in order to increase the overall level of judgmental competence.

Therefore, institutions ought to develop the capacity of ordinary citizens to conduct policy evaluation, a task usually reserved for trained experts. Furthermore, the institutions charged with policy implementation ought to be transparent, for citizens must be able to assess the degree to which they fell short of expectations. Finally, information about actual performance should be honest and public; this is a demanding requirement, since actors directly related to policy--its designers, executors, or subjects--standardly have self-interested motives to conceal information. In addition to these considerations of institutional structure, experimental epistemic democracy would be best served by a public discourse which substantially focuses on the honest evaluation of policy performance, rather than one which, for example, is absorbed by attempts to lay blame on particular social or political actors.

 The second implication is that democratic institutions ought to enhance opportunities for experimentation. The closeness of realized policy to correct policy is determined not only by voter competence, but by the number of rounds of voting. Experimental epistemic democracy, therefore, recommends more frequent voting because each round of voting is an opportunity to incorporate new information into the choice of policy. This pragmatic justification adds a distinctive reason to support regular elections, which are standardly justified by their capacity to discipline leaders or because they legitimize political authority by institutionalizing the idea of popular sovereignty.

Conclusion

 This article has several closely related aims. Beginning with a presumption in favor of epistemic democracy as a normative, and perhaps descriptive, notion of political institutions, I sought to draw out the implications of the Condorcet model for institutional design. The somewhat surprising conclusion of this straightforward extension is that majority-rule in a two party system generally yields incorrect decisions. A second, more fundamental, object of this paper has been to urge that Condorcet voting is not the only conception of epistemic democracy. I have offered an alternative model, and there are certainly many others, in which the truth is sought through iterated experimentation. Even in the rudimentary form presented above, this model offers a more attractive conceptualization of epistemic democracy than the Condorcet mechanism. First, it more accurately describes the operation of real world democratic assemblies by considering their actions through time. Second, demands less political competence from citizens while still yielding correct results. Finally, it captures an important characteristic of citizens and, one hopes, democratic societies--that we are capable of learning from our mistakes.

 Bibliography

Black, Duncan. The Theory of Committees and Elections, Cambridge: Cambridge University Press, 1958.

Cohen, Joshua. "An Epistemic Conception of Democracy," Ethics 97, no. 1 (Oct. 1986): 26-38.

Downs, Anthony. An Economic Theory of Democracy, New York: Harper and Rowe, 1957.

Estlund, David and Jeremy Waldron. "Democratic Theory and the Public Interest: Condercet and Rousseau Revisited," American Political Science Review 83, no. 4 (Dec 1989): 1317-1340.

Estlund, David M. "Democracy without preference.," The Philosophical Review 99, no. 3 (July 1990): 397-423.

Grofman, Bernard, Guillermo Owen, and Scott Feld. "Thirteen Theorems in Search of the Truth," Theory and Decision. 15 (1983): 261-78.

Grofman, Bernard and Owen, Guillermo "Review Essays: Condorcet Models, Avenues for Future Research" in Information Pooling and Group Decision Making, ed. Bernard Grofman and Guillermo Owen. (Greenwich, CT: JAI Press, 1986)

Grofman, Bernard and Scott Feld. "Rousseau's General Will: A Condorcetian Perspective," American Political Science Review 82, no. 2 (June 1988): 567-76.

Ladha, Krishna K. "The Condorcet Theorm, Free Speech, and Correlated Votes" American Journal of Political Science Vol. 36, No. 3 (August 1992): 617-34

Ladha, Krishna K. "Information Pooling through majority-rule voting: Condorcet's jury theorem with correlated votes," Journal of Economic Behavior and Organization 26, no. 3 (May 1995): 353-72.

Lipschutz, Seymore. Theory and Problems of Probability, New York: McGraw-Hill, 1965)

Press, William, Brian P. Flannery, Saul A. Teukolsky, and WIlliam T. Vetterling. Numerical Recipes: The Art of Scientific Computing (Cambridge: Cambridge University Press, 1986)

Rawls, John, A Theory of Justice (Cambridge: Harvard University Press, 1971)

Rousseau, Jean-Jacques. The Social Contract, in The Basic Political Writings, trans. Donald A. Cress. (Cambridge: Hackett Publishing Co., 1987)

Young, H.P. "Condorcet's theory of voting." American Political Science Review 82, no. 4 (Dec 1988): 1231-1244.