Abstract
The seminal contribution known as the Condorcet Jury Theorem observes that under a specific set of conditions an increase in the size of a group tasked with making a binary decision (“guilty or “innocent, say) leads to an improvement in the group’s ability to make the correct choice. An assumption that is not properly appreciated in the relevant literature is that the competency of the group members is assumed to be exogenous. In numerous applications, members of the group make investments to improve the accuracy of their decision making (e.g., pre-meeting efforts). We consider the collective action problem that arises. We show that if individual competence is endogenous, then increases in the group size encourages free riding. This trades off with the value of information aggregation. Thus, the value of the larger group size is muted. Extensions illustrate that if committee members are allowed to exit/not participate, the equilibrium committee size is reduced. Additionally, supermajority voting rules encourage investments and, consequently, individual competence.
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Notes
We are not the first to recognize the potential problem with endogenizing competence. Ben-Yashar and Nitzan (2001) raise the issue. They limit their analysis to a numerical example comparing a committee of three to a single decision maker, and document a reduction in aggregate competence when a social planner sets the effort investment. Persico (2003) allows uninformed agents to invest in a noisy signal before voting. Martinelli (2006) considers voters with differing preferences. He explores how investments in identifying the correct state-space are adjusted according to the size of the electorate (rather than using the common preference framework considered here). Hao and Suen (2009), in a review of committee decision-making models, also point out that the accuracy of a group’s decision can be worsened if the informational signals are acquired with a cost. They do not explore the interaction between this cost and group size. Additionally, the potential problem is hypothesized in McCannon (2011).
Thus, the literature on rationally-ignorant voters (Congleton 2001; Caplan 2007) and its moral arguments (Brennan 2011) is a useful reference. We differ, though, in that we formally explore how the quality of voter decision making adjusts with changes in the size of the relevant group. Our work is more closely related to that of Martinelli (2006), who extends the rational-ignorance framework to electorates whose investments in information decline to zero, but the decision nevertheless corresponds to the majoritys preference when the number of voters becomes arbitrarily large. We differ from these works by considering agents with common preferences.
Tullock’s main argument is that the belief that public goods problems in private markets require government intervention is incomplete. Government bureaucrats also suffer from the free-rider problem and, therefore, it is unclear whether government intervention is an effective way of dealing with public goods. As an example, he compares a designer of bumpers on Chevrolets to a politician. Both gain only a tiny fraction of the benefits they create, while incurring private costs. The primary difference between the two is that General Motors has an incentive to monitor and punish ineffective bumper designers, but that voters too suffer from free riding when holding politicians accountable. Also, see Tullock (1967) for a further discussion.
These are, of course, strong assumptions, but they are standard in the literature. For an overview discussion of the role of the interdependence in competence, see McCannon (2011). One would expect that free riding can be exacerbated if there is also positive serial correlation. This extension, though, is not considered here.
In practice, individuals differ substantially in their ability to acquire information and make good decisions. The assumption of identical competence production functions is made for convenience. Similarly, individuals may differ in the benefit they receive from a correct decision (or, relatedly, the disutility from an erroneous one), which would incentivize heterogeneous investments. The main argument (that there exists a free-riding problem in group decision making that is exacerbated in larger groups) can be expected to hold in these extended environments.
We make this assumption to minimize the difference between the model presented here and the standard framework employed in the literature. With strategic voting and abstention permitted, as in Feddersen and Pesendorfer (1996), one would expect that the decision by others to abstain would affect the effort investment. See Austen-Smith and Banks (1996) and Ben-Yashar (2006) for discussions on the impact of strategic voting.
The alternative name for this theorem is the “Two-Policemen-and-a-Drunk Theorem”: if two policemen are taking a drunk prisoner to his cell, then, regardless of the amount of wobbling that goes on, if the guards keep the prisoner between them and they make it to the cell, then the prisoner must also make it to the cell.
So long as \(\frac{dc}{de}>0\) \(\forall e\), assuming \(\underline{p}= \frac{1}{2}\) does not affect Proposition 2.
As with any theoretical model with a multiplicity of equilibria, comparative statics are difficult. Here, we deal with this difficulty by considering the addition of the new individuals to an existing cohort to retain the P partitions. Furthermore, the result presumes the existence of an equilibrium and illustrates that if an asymmetric equilibrium (defined in terms of the partition condition) exists, then the main result [Proposition 2] continues to hold.
To simplify the proof, the assumption \(\underline{p}>{1 \over 2N}+{1 \over 2}\) for a fixed e is employed, which is a sufficient (but not necessary) condition.
With a non-decisive supermajority rule, \(K_{N}(E,A)=K_{N}(E,B)=K_{N}(E)\) from (9).
This is a sufficient and not necessarily a necessary condition, since it guarantees the convexity of the function. For a simple majority (\(k=\frac{N+1}{2}\)), this requires only that \(p>\frac{1}{2}+\frac{1}{N-1}\). Also, the group size results consider increasing N by increments of d where, for common voting rules (\(\frac{1}{2}\), \(\frac{3}{5}\), \(\frac{2}{3}\), and \(\frac{3}{4}\) for example), these are small increments: 2, 5, 3, and 4, respectively.
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Acknowledgments
We would like to thank Roger Congleton, Phil Curry, Dan Grossman, Greg DeAngelo, Cliff Hawley, Nick Miller, Eric Olson, Bill Shughart, Nic Tideman, and seminar participants at West Virginia University, the Association of Private Enterprise Education, and the Public Choice Society meetings for their useful comments. Additionally, we thank Nick Miller and Jac Heckelman for their overall support on the Handbook project from which this paper derives.
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McCannon, B.C., Walker, P. Endogenous competence and a limit to the Condorcet Jury Theorem. Public Choice 169, 1–18 (2016). https://doi.org/10.1007/s11127-016-0366-z
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DOI: https://doi.org/10.1007/s11127-016-0366-z