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A Theorem About Voting*

Published online by Cambridge University Press:  01 August 2014

Thomas W. Casstevens
Thomas W. Casstevens*
Affiliation:
Oakland University
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Extract

The objective of this essay is to present a simple decision-theoretic model of individual rational voting in a single-member district, using the simple-majority single-ballot system of election, and to derive the following theorem from the model: The rational voter votes for the candidate (party) associated with the outcome he (the voter) most prefers.

The model and theorem may interest students of voting for at least two reasons. First, the theorem contradicts the classical argument that “there is one eventuality in a multiparty system that does not arise in a two-party system: a rational voter may at times vote for a party other than the one he most prefers.” The theorem asserts, by contrast, that what is true for the two-party case is also true for the multi-party case. Thus, the model and theorem sharply differ from the classical theory of party systems. The ramifications of this conflict may affect some conventional views about the decline of third parties, the differences between two-party and multi-party systems, as well as (perhaps) other topics.

Type
Research Notes
Information
American Political Science Review , Volume 62 , Issue 1 , March 1968 , pp. 205 - 207
Copyright
Copyright © American Political Science Association 1968

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Footnotes

*

The original proof of the theorem was privately circulated in April, 1965. The penultimate version of this paper was appended to the author's doctoral dissertation, “The Decline of the British Liberal Party: A Comparative and Theoretical Analysis,” (Michigan State University, 1966), Appendix 1. The author is indebted to several persons for their private criticisms but especially to Harold T. Casstevens II for suggesting an elegant simplification of the original proof.

References

1 Downs, Anthony, An Economic Theory of Democracy (New York: Harper & Brothers, 1957), p. 47.Google Scholar See also Duverger, Maurice, Political Parties, trans. Barbara, and North, Robert (New York: John Wiley & Sons, Inc., 1955), pp. 223–226.Google Scholar

2 “In fact, use of this criterion is often cited as a necessary (if not sufficient) condition for rational choice.” Ackoff, Russell L., Scientific Method (New York: John Wiley & Sons, Inc., 1962), p. 38.Google Scholar A diffuse intellectual debt is owed to Ackoff's stimulating book.

3 The restriction to a single district is consistent with Downs' analysis of the entire nation as a single district (op. cit., pp. 23–24) and with Duverger's remarks about local bi-partism (op. cit., p. 223). The restriction fits by-elections (special elections) in Britain, Canada, the United States and elsewhere, but the general election situation is more complex. Although the theorem is derived for simple-majority single-ballot systems, an implication appears to be that (for many analytical purposes) first preference votes in preferential systems (e.g., Australia) are equivalent to votes in plurality single-ballot systems, but this conjecture is not proved in the present paper.

4 Ignoring ties is similar to ignoring the possibility that a coin will fall on its edge. This simplification is convenient since there is no voting choice uniquely associated with a tie, but it should be noted that if a tie occurs, then there is no election. (The man named Nobody is not an incumbent skin to the status quo which may reign after motions are defeated in committee voting.)

5 This postulate seems to be intuitively obvious, although its consequences (perhaps) are not. The Shapley-Shubik a priori power index is suggestive for this axiom. See Shapley, L. S. and Shubik, Martin, “A Method for Evaluating the Distribution of Power in a Committee System,” this Review, 48 (1954), 787–792.Google Scholar

6 This corollary (roughly speaking, that if any voter decides to vote for any candidate, then that candidate has some chance of winning) follows directly from the preceding postulate but, never theless, appears to conflict with the classical theory of parties.

7 This reduction postulate is not intuitively obvious, and as a result, subjective probabilities are not used in the present model. Although the postulate does not contradict the mathematical theory of probability, and although such small magnitudes cannot be empirically measured (at least at the present time), the consequences of the postulate (in a theoretical context) certainly deserve to be tested. The theorem presented in this paper is, of course, such a testable consequence.

8 Simple preferences are universally accepted as the basis for calculating utilities. The debate pivots about how to transform the non-numerical preferences into numerical utilities, but an isomorphic ordinal ranking is generally accepted as a fundamental characteristic of a suitable transformation. See Duncan, R. Luce and Raiffa, Howard, Games and Decisions (New York: John Wiley & Sons, Inc., 1957), Chapter 2.Google Scholar

9 Start by subtracting Equation (6) from Equation (4).

10 An appropriately designed survey study of a British by-election, pitting Conservative-Labour-Liberal candidates against one another, might serve as a “crucial experiment” for the model and theorem.

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