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Social Choice and Welfare

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06-02-2018 | Original Paper

Revealed votes

Author: Andrei Gomberg

Published in: Social Choice and Welfare | Issue 2/2018

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Abstract

In this paper I consider choice correspondences defined on a novel domain: the decisions are assumed to be taken not by individuals, but by committees, whose membership is observable and variable. In particular, for the case of two alternatives I provide a full characterization of committee choice structures that may be rationalized with two common decision rules: unanimity with a default and weighted majority.

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These results apply when nothing is known about the group composition. When the size of the committee is known, there exist restrictions implied by various group choice rules on the minimal choice cycle length, which have been studied since Nakamura (1979).

Whether voting records of central bank decision-makers should be public has been a subject of substantial controversy and research in recent years and the international practice has varied (see, for instance, Sibert 2003, or Gersbach and Hahn 2008).

Thus, in a legislature no formal vote may be taken on an issue since the parliamentary leaders know that it would fail anyway.

See, for instance, Vargas (1996) for a discussion of the role—and lack of precedential authority—of the writ of amparo in Mexican legal system.

The details of the case assignment are described in Articles 49 and 50 of the Reglamento Interno (Internal Rules) of the Colombian Constitutional Court (1992). The author thanks Juan Bertomeu for drawing his attention to this arrangement.

The reason I am only considering choice functions, rather than choice correspondences in this section—an assumption that will be relaxed later—is that the unanimity rule, as here defined, always produces a unique choice.

That is, if $S,T\in \mathcal {E}$, then $S\cup T\in \mathcal {E}$.

This of course, covers the case when $\mathcal {E}$ has a single element.

It follows from the proof of the proposition above that, except in the trivial case where the choice is the same for all committees, the choice of the default is uniquely implied by the data.

See, for instance, Feddersen and Pesendorfer (1996).

For a recent survey of such studies see, for instance, Kugler et al. (2012). The committee overlap typically arises there from rematching committee members in order to avoid repeated interaction effects between experimental subjects.

Indeed, if we consider Example 2 above, we obtain the following system of $ K=7$ inequalities:

$$\begin{aligned} \left\{ \begin{array}{c} w_{1}+w_{2}+w_{3}>0 \\ w_{1}>0 \\ w_{2}>0 \\ w_{3}>0 \\ -w_{1}-w_{2}>0 \\ -w_{1}-w_{3}>0 \\ -w_{2}-w_{3}>0 \end{array} \right. \end{aligned}$$

which, obviously, has no solution. The dual problem solves with $ r_{1}=r_{2}=r_{3}=r_{4}=3$ and $r_{5}=r_{6}=r_{7}=2$.

On the other hand, if the choice of the committee $\left\{ 2,3\right\} $ were not observed the corresponding system would have been ($K=5$):

$$\begin{aligned} \left\{ \begin{array}{c} w_{1}+w_{2}+w_{3}>0 \\ w_{2}>0 \\ w_{3}>0 \\ -w_{1}-w_{2}>0 \\ -w_{1}-w_{3}>0 \end{array} \right. \end{aligned}$$

which would be solved, for instance, by the weight vector $w=\left( -3,2,2\right) $.

Note that the example above shows that a stronger indirect extension could be imposed here. However, reinforcement is more intuitive, so I stick to it as a necessary implication of rationalizability.

The naming suggestion for this axiom, originally introduced in Gomberg (2011), belongs to Norman Schofield.

As noted above, in the binary choice settings of Sects. 2 and 3, if individuals are merely aggregating preferences they would have no incentives for strategic voting. If, however, they share an underlying preference but aggregate information, strategic voting incentives would emerge, as in Feddersen and Pesendorfer (1996).

Apesteguia J, Ballester M, Masatlioglu Y (2014) A foundation for strategic agenda voting. Games Econ Behav 87:91–99CrossRef

Corte Constitucional de Colombia (1992) Reglamento Interno, retrieved at http://www.corteconstitucional.gov.co/lacorte/reglamento.php

Deb R (1976) On constructing generalized voting paradoxes. Rev Econ Stud 43:347–351CrossRef

Degan A, Merlo A (2009) Do voters vote ideologically. J Econ Theory 144:1868–1894CrossRef

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Fishburn P (1970) Utility theory for decision-making. Wiley, New YorkCrossRef

Fisburn P (1986) Axioms of subjective probability. Stat Sci 1:335–345CrossRef

Gersbach H, Hahn V (2008) Should the individual voting records of central bankers be published? Soc Choice Welf 30:655–683CrossRef

Gomberg A (2011) Vote revelation: empirical content of scoring rules. In: Schofield N, Caballero G (eds) Polit Econ Democr Voting. Springer, Berlin, Heidelberg, pp 411–417 2011

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Kraft C, Pratt J, Seidenberg A (1959) Intuitive probability on finite sets. Ann Stat 30:408–419CrossRef

Kugler T, Kausel E, Kocher M (2012) Are groups more rational than individuals? A review of interactive decision making in groups. Wiley Interdiscip Rev Cognit Sci 3:471–482CrossRef

McGarvey D (1953) A theorem on the construction of voting paradoxes. Econometrica 21:608–610CrossRef

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Nakamura K (1979) The vetoers in a simple game with ordinal preferences. Int J Game Theory 5:55–61CrossRef

Sibert A (2003) Monetary policy committees: individual and collective reputations. Rev Econ Stud 70:649–665CrossRef

Smith J (1973) Aggregation of preferences with variable electorate. Econometrica 41:1027–1041CrossRef

Vargas J (1996) Rebirth of the supreme court of Mexico: an appraisal of president Zedillo’s judicial reform of 1995. Am Univ J Int Law Policy 11:295–341

Young P (1975) Social choice scoring functions. SIAM J Appl Math 28:824–838CrossRef

Title: Revealed votes
Author: Andrei Gomberg
Publication date: 06-02-2018
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2/2018
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-018-1116-6

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