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Published in: Social Choice and Welfare 2/2014

01-08-2014 | Original Paper

Inefficient committees: small elections with three alternatives

Author: J. M. M. Goertz

Published in: Social Choice and Welfare | Issue 2/2014

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Abstract

We consider small committees which have to elect one of three alternatives using the simple plurality rule. Committee members have common, state-dependent preferences and receive imprecise private signals about the state of nature prior to the election. We are interested in whether the committee decision is efficient, that is whether the probability with which the committee elects the correct alternative is higher than the probability with which one single individual alone—on behalf of the others—would. It has been shown that there exists a unique efficient equilibrium in elections with two alternatives. We show that this result does not extend to elections with more alternatives. Multiple equilibria may exist for the same committee, and there may be both efficient and inefficient ones. Informative voting may or may not be an equilibrium. Also contrary to two-alternative elections, there exist responsive equilibria in which voters vote ‘against’ their signal. As a consequence, only two alternatives receive positive expected vote shares and the outcome is inefficient.

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Appendix
Available only for authorised users
Footnotes
1
In all committee voting games, there always exist equilibria in which all voters vote for the same alternative. They are called unresponsive (e.g., Feddersen and Pesendorfer 1998) because voters do not change their strategy as a function of their private signals. In these equilibria, no voter has any influence on the voting outcome. The literature usually focuses on responsive equilibria.
 
2
An excellent overview of the literature on preference aggregation and related issues can be found in the Handbook of Social Choice and Welfare Vol. I. (2002).
 
3
The same is true for the literature on large elections, with the exception of Hummel (2010). He assumes a very particular preference structure such that there is a natural ordering of alternatives: If a voter likes alternative \(k\), then she always likes all alternatives \(1\) through \(k-1\) as well, but not alternative \(k+1\). His results only apply to large electorates. He claims that “this result [existence of a symmetric equilibrium] cannot be extended to more general preference relationships for any finite number of jurors...” (p. 72).
 
4
Duggan and Martinelli (2001) or Meirowitz (2002) assume a continuous signal space. With a continuous space, results change only for the unanimity rule, a rule with generally bad efficiency properties.
 
5
For the sake of comparison: In a large election, the definition of efficiency is quite different. As \(n \rightarrow \infty \), a limit equilibrium \(\sigma *\) in a large election is efficient if it is true that \(P_{xX}=1\) for all alternatives \(X\) and states of nature \(x\) (\(n\) is the mean of random variable \(N\)). That implies that a limit equilibrium \(\sigma *\) with \(\sigma _n\rightarrow \infty \) as \(n\rightarrow \infty \) is only efficient if the correct alternative is chosen with probability 1 in each state of nature (e.g., Goertz and Maniquet 2011b). In the literature, this definition is sometimes referred to as ‘full-information equivalence’ because it requires the outcome of the election to be the same as it would be if all private signals were public information. For a large election, the Law of Large Numbers implies that the state of nature can be inferred if all signals are publicly observable.
 
6
Playing a mixed strategy implies that several pure strategies yield the same expected utility and therefore the same \(P^{one}\), so we can focus attention on pure strategies for a single decision maker.
 
7
For comparison: A single decision maker in a two-alternative election has only two types of best replies: voting informatively or always voting for the same alternative independent of her type.
 
8
There always exist unresponsive equilibria in which all voters vote for the same alternative. However, we are not interested in those.
 
9
In Maple 12, the command to be used is fsolve \((y = 0)\). It finds the roots of the equation \(y=0\). Any other standard math software can be used as well.
 
Literature
go back to reference Bouton L, Castanheira M (2012) One person, many votes: divided majority and information aggregation. Econometrica 80(1):43–87CrossRef Bouton L, Castanheira M (2012) One person, many votes: divided majority and information aggregation. Econometrica 80(1):43–87CrossRef
go back to reference Condorcet MD (1785) Essay on the application of analysis to the probability of majority decisions. Condorcet MD (1785) Essay on the application of analysis to the probability of majority decisions.
go back to reference Duggan J, Martinelli C (2001) A Bayesian model of voting in juries. Games Econ Behav 37:259–294CrossRef Duggan J, Martinelli C (2001) A Bayesian model of voting in juries. Games Econ Behav 37:259–294CrossRef
go back to reference Feddersen T, Pesendorfer W (1998) Convicting the innocent: the inferiority of unanimous jury verdicts under strategic voting. Am Polit Sci Rev 92(1):23–35CrossRef Feddersen T, Pesendorfer W (1998) Convicting the innocent: the inferiority of unanimous jury verdicts under strategic voting. Am Polit Sci Rev 92(1):23–35CrossRef
go back to reference Feddersen TJ, Pesendorfer W (1996) The swing voter’s curse. Am Econ Rev 86(3):408–424 Feddersen TJ, Pesendorfer W (1996) The swing voter’s curse. Am Econ Rev 86(3):408–424
go back to reference Feddersen TJ, Pesendorfer W (1997) Voting behavior and information aggregation in elections with private information. Econometrica 65(5):1029–1058CrossRef Feddersen TJ, Pesendorfer W (1997) Voting behavior and information aggregation in elections with private information. Econometrica 65(5):1029–1058CrossRef
go back to reference Feddersen TJ, Pesendorfer W (1999) Abstention in elections with asymmetric information and diverse preferences. Am Polit Sci Rev 93(2):381–398CrossRef Feddersen TJ, Pesendorfer W (1999) Abstention in elections with asymmetric information and diverse preferences. Am Polit Sci Rev 93(2):381–398CrossRef
go back to reference Goertz JM, Maniquet F (2009) On the informational efficiency of simple scoring rules. CORE Discussion Paper 2009026. Goertz JM, Maniquet F (2009) On the informational efficiency of simple scoring rules. CORE Discussion Paper 2009026.
go back to reference Goertz JM, Maniquet F (2011a) On a three-alternative Condorcet jury theorem. CesIfo Working Paper No. 3457. Goertz JM, Maniquet F (2011a) On a three-alternative Condorcet jury theorem. CesIfo Working Paper No. 3457.
go back to reference Goertz JM, Maniquet F (2011b) On the informational efficiency of simple scoring rules. J Econ Theory 146(4):1464–1480CrossRef Goertz JM, Maniquet F (2011b) On the informational efficiency of simple scoring rules. J Econ Theory 146(4):1464–1480CrossRef
go back to reference Hummel P (2010) Jury theorems with multiple alternatives. Soc Choice Welf 34(1):65–103CrossRef Hummel P (2010) Jury theorems with multiple alternatives. Soc Choice Welf 34(1):65–103CrossRef
go back to reference Hummel P (2011) Information aggregation in multicandidate elections under plurality rule and runoff voting. Math Soc Sci 62(1):1–6CrossRef Hummel P (2011) Information aggregation in multicandidate elections under plurality rule and runoff voting. Math Soc Sci 62(1):1–6CrossRef
go back to reference Martinelli C (2002) Simple plurality rule versus plurality runoff with privately informed voters. Soc Choice Welf 19(4):901–919CrossRef Martinelli C (2002) Simple plurality rule versus plurality runoff with privately informed voters. Soc Choice Welf 19(4):901–919CrossRef
go back to reference McLennan A (1998) Consequences of the Condorcet jury theorem for beneficial information aggregation by rational agents. Am Polit Sci Rev 92(2):413–418CrossRef McLennan A (1998) Consequences of the Condorcet jury theorem for beneficial information aggregation by rational agents. Am Polit Sci Rev 92(2):413–418CrossRef
go back to reference Meirowitz A (2002) Informative voting and Condorcet jury theorems in models with a continuum of types. Soc Choice Welf 19:219–236CrossRef Meirowitz A (2002) Informative voting and Condorcet jury theorems in models with a continuum of types. Soc Choice Welf 19:219–236CrossRef
go back to reference Myerson RB (1998) Extended Poisson games and the Condorcet jury theorem. Games Econ Behav 25:111–131CrossRef Myerson RB (1998) Extended Poisson games and the Condorcet jury theorem. Games Econ Behav 25:111–131CrossRef
go back to reference Wit J (1998) Rational choice and the Condorcet jury theorem. Games Econ Behav 22:364–376CrossRef Wit J (1998) Rational choice and the Condorcet jury theorem. Games Econ Behav 22:364–376CrossRef
Metadata
Title
Inefficient committees: small elections with three alternatives
Author
J. M. M. Goertz
Publication date
01-08-2014
Publisher
Springer Berlin Heidelberg
Published in
Social Choice and Welfare / Issue 2/2014
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-013-0784-5

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