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Optimizing political influence: a jury theorem with dynamic competence and dependence

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Abstract

The purpose of this paper is to illustrate, formally, an ambiguity in the exercise of political influence. To wit: A voter might exert influence with an eye toward maximizing the probability that the political system (1) obtains the correct (e.g. just) outcome, or (2) obtains the outcome that he judges to be correct (just). And these are two very different things. A variant of Condorcet’s Jury Theorem which incorporates the effect of influence on group competence and interdependence is developed. Analytic and numerical results are obtained, the most important of which is that it is never optimal—from the point-of-view of collective accuracy—for a voter to exert influence without limit. He ought to either refrain from influencing other voters or else exert a finite amount of influence, depending on circumstance. Philosophical lessons are drawn from the model, to include a solution to Wollheim’s “paradox in the theory of democracy”.

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Notes

  1. Some may hold a stronger view, under which (i) people have a right to participate in politics, regardless of their epistemic state; or (ii) unrestrained influence is an essential element of the deliberation which justifies a political process.

  2. Technically, dependence is bad from the point-of-view of aggregation quality only if voters are competent; that is, more likely than a coin flip to vote correctly (see §2). No assumptions about voter competence are imposed in this paper’s model.

  3. For an introduction to the epistemology of disagreement, see Christensen (2009) and Feldman (2007).

  4. On the steadfast position, see, e.g., Bergmann (2009), Kelly (2005), and Van Inwagen (2010). On conciliationism, see Christensen (2007), Elga (2007), and Feldman (2007). I offer my (generally conciliatory) thoughts in Mulligan (2021).

  5. Example: There is a three-member civil jury. Two jurors believe, with credence 0.4, that the facts and the law favor the plaintiff. The third juror believes this with credence 0.9. So they “conciliate”, and all adopt the average credence of 0.57. Before conciliating, the respondent would prevail, two votes to one; after conciliating, the plaintiff wins unanimously. It is unclear that this is an epistemic improvement, to say nothing of justice. (N.B. in American civil cases the standard of proof is a “preponderance of the evidence” and so the two possible outcomes are symmetric. For criminal cases, with the much higher “beyond a reasonable doubt” standard, this is not the case.)

  6. See also Austin-Smith (1990).

  7. This is not to say that the other assumptions may never be violated in practice. But they are often complied with. Many real-world decisions are naturally dichotomous, or they are not naturally dichotomous but are broken down into dichotomous steps for practical reasons. And simple majority rule is ubiquitous [I offer an alternative in Mulligan (2018b)]. In any case, work has been done extending Condorcet’s theorem to deal with these limitations; on the relaxation of dichotomous choice, e.g., see Hummel (2010), Lam and Suen (1996), List and Goodin (2001), Miller (1996), and Paroush (1990).

  8. In particular, the optimal decision rule may be identified by weighting each vote in proportion to the log-odds of that voter’s competence. Other relevant literature includes Grofman and Feld (1983), Nitzan and Paroush (1984, 1985), and Paroush (1998).

  9. One might think that this problem can be bypassed by shifting from a brute independence assumption to independence conditional upon common information (like evidence). Unfortunately, such a shift imperils other necessary assumptions (like the minimal competence assumption)–see Dietrich and Spiekermann (2013a, 2013b, 2020) and Ladha (1993).

  10. For other models and discussions of dependent voting, see Berend and Sapir (2005, 2007), Berg (1993, 1996), Dietrich and List (2004), Kaniovski (2010), Kaniovski and Zaigraev (2011), Ladha (1992, 1993, 1995), List and Pettit (2004), Nitzan and Paroush (1985), Peleg and Zamir (2012), Shapley and Grofman (1984), and Zaigraev and Kaniovski (2013).

  11. A natural question is whether the core results of this paper can be obtained with general functions c and \(\rho\), without choosing specific functional forms. General functions would be constrained as follows: Both at least C1 and strictly increasing in i; \(c(i)=c_{init}\) for \(i=0\), \(c(i)=1\) in the limit as \(i \rightarrow \infty\); \(\rho (i)=0\) for \(i=0\), \(\rho (i)=1\) in the limit as \(i \rightarrow \infty\). The answer is no. Some functions approximating certain fixed curves, which satisfy these constraints, may yield an infinite number of optima rather than the unique optimum obtained here. I thank Willie Wong for identifying this counterexample.

  12. As one example of the difficulties dependence may introduce, consider a simple case of three voters whose competence and pairwise correlation are known precisely. How likely is it that this electorate will choose correctly? It is impossible to know. These data fail to specify a unique joint distribution (although they do constrain potential distributions). The most common representation of the joint distribution is that given by Bahadur (1961) and Lazarsfeld (1956), which includes \(2^n-n-1\) correlation parameters. The challenge of applying this representation is that some of these parameters (often the higher-order correlations) are unavailable, and generally cannot be ignored (i.e. set to 0). To deal with these limitations, Van Der Geest (2005) uses a maximum entropy method, which can be implemented numerically, to infer higher-order correlations. Kaniovski (2008a) obtains the relevant probabilities by identifying the distribution that is “closest” (in a least squares sense) to that in which voters are assumed to be independent. This approach also may be implemented numerically, and Kaniovski obtains an analytic solution for the special case of homogeneous competence. (Numerical examples of this approach are provided in Kaniovski 2008b.)

  13. Note that we continue to assume aggregation via simple majority rule here. When an expert exists, this may no longer be optimal.

  14. See https://www.opensecrets.org/overview/topindivs.php?cycle=2016&view=fc, retrieved 6 August 2020.

  15. Technically, the game only has a prisoner’s dilemma structure if Adelson and Soros are interpreted as opinion leaders who seek justice as they see it rather than as the superior, process-optimizing type. The reason is that, so interpreted, each would prefer most of all to exert influence and have his opponent exert no influence, followed by refraining from influence in the face of his opponent’s restraint, followed by exerting influence in the face of his opponent’s exertion of influence. In any case, the dynamic described for the two relevant cases—each exerts influence and each refrains from exerting influence—holds under both interpretations.

  16. E.g. Brennan (2016), Estlund (2008), and Landemore (2013).

  17. Including Foot (2002), Schueler (2007), and Wilcox (1968).

  18. For critical commentary on Wollheim’s paradox, see Honderich (1974), Paris and Reynolds (1978), and Weiss (1973).

  19. See, e.g., Althaus (2003), Brennan (2016), Converse (1964), Delli Carpini and Keeter (1996), Neuman (1986), Shenkman (2008), and Somin (2013).

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  1. Institute for the Study of Markets and Ethics, Georgetown University, Washington DC, USA

    Thomas Mulligan

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I wish to acknowledge, with thanks, generous feedback from David Faraci, Nick Geiser, John Hasnas, Dmitrii Karp, Raphael Lehrer, Iosif Pinelis, Kirun Sankaran, Willie Wong, and Maryam Yashtini. Audiences at the 2021 International Conference on Social Choice and Voting Theory, Virginia Tech, the Georgetown Institute for the Study of Markets and Ethics Workshop, and the 2018 Philosophy, Politics, and Economics Society annual meeting (which heard an early draft of this paper) contributed as well. Two anonymous reviewers for Social Choice and Welfare provided a number of helpful comments.

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Mulligan, T. Optimizing political influence: a jury theorem with dynamic competence and dependence. Soc Choice Welf (2022). https://doi.org/10.1007/s00355-022-01407-5

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