Abstract
The present paper introduces a new approach to the theory of voting in the context of binary collective choice, which seeks to define a dynamic optimal voting rule by using insights derived from the mathematical theory of information. In order to define such a voting rule, a method of defining a real-valued measure of the weight of independent opinion of an arbitrary set of voters is suggested, which is value free to the extent that it depends only on probabilistic information extracted from previous patterns of voting, but does not require for its definition any direct information concerning either the correctness or incorrectness of previous voting decisions, or the content of those decisions. The approach to the definition of such a measure, which I call gravitas, is axiomatic. The voting rule is then defined by comparing the gravitas of the set of those voters who vote for a given motion with the gravitas of the set of those who vote against that motion.
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Notes
- 1.
The present section arose from my reflections on a conversation in 1968 between John Bell and my late father, recorded in John’s autobiography Perpetual Motion.
- 2.
… provided, of course, that the results of such a calculation were likely to be consistent with their own assessments of the correct decision.
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- 4.
- 5.
It may be noted that the philosophical idea of a separate notion of probabilistic correctness relative to limited information makes sense even in the case when we suppose that there exists an “objectively true” answer. For example, in a jury trial, the criterion for conviction is typically that guilt is proved “beyond reasonable doubt.” If therefore we make the reasonable assumption that all judgments in such trials are de facto made on a probabilistic basis, then, given that the information which can be made available to a jury is of necessity limited, a jury (or indeed an individual jury member) may in fact make a decision which is probabilistically correct on the basis of the evidence available, but which is nonetheless incorrect in an absolute sense. Our restriction of the admissible information available to the decision rule to σ together with the actual division of the voters, may in this case be interpreted as a uniform (or fair) method of reifying the information contained in the accumulated subjective judgments of jury members on the evidence available to them. (Of course this presupposes that an estimate for σ is actually available, which would not be the case for a one-time only jury).
- 6.
See (Cohen, 1986) for an account of the latter.
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- 8.
We may note here that types of information other than that encoded in σ might in principle also be recorded and used in the calculations of a decision rule; for example, normalised information about the strength of conviction which individual voters attach to individual judgments could be recorded and used in some way. A closely related point is made in Dummett’s discussion of Arrow’s theorem in (Dummett, 1984).
- 9.
I do not intend by this statement to minimise the difficulty of the computational problems involved, which I have not addressed here, and which would certainly be substantial in the case of a large electorate.
- 10.
We confine ourselves here to noting that the first of these questions can reasonably be considered an analogue in collective choice theory of the problem in uncertain reasoning (by a single agent) of choosing a canonical probability distribution from a set of possible distributions constrained by certain data. The method of choice for solving the latter problem is the use of the maximum entropy principle (see, e.g., (Paris, 1994)), but without additional insight maximum entropy appears powerless to help in solving the former problem. The author believes however that the notion of gravitas can be used to provide the appropriate missing idea necessary to partially solve this problem.
- 11.
The importance of invariance properties of the notion of margin in a classical Condorcetian analysis of voting has been emphasized by List (2004).
- 12.
That is, if π is a permutation of \(D(M)\) and σ π is defined by \(\sigma^{\pi}(\pi(\alpha))\) = \(\sigma(\alpha)\) for each \(\alpha \in D(M)\), then \(G^{\sigma^{\pi}}(M) = G^{\sigma}(M)\). Note that whereas a permutation of M always induces a permutation of \(D(M)\) the converse is not true.
- 13.
See (Laruelle et al., 2006) for an analysis of the concept of success in the context of voting systems.
- 14.
- 15.
The original French text of the first sentence, which is difficult to translate exactly, is as follows: “Si, quand le peuple suffisamment informé délibère, les citoyens n’avaient aucune communication entre eux, du grand nombre de petites différences résulterait toujours la volonté générale, et la délibération serait toujours bonne.”
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Acknowledgement
I wish to express my thanks to Moshé Machover, Alena Vencovska, Hykel Hosni, Luc Bovens and Greg Holland for helpful comments on earlier versions of some of the ideas presented in this paper.
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Wilmers, G. (2011). Collective Choice as Information Theory: Towards a Theory of Gravitas. In: DeVidi, D., Hallett, M., Clarke, P. (eds) Logic, Mathematics, Philosophy, Vintage Enthusiasms. The Western Ontario Series in Philosophy of Science, vol 75. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0214-1_23
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