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A Theory of the Calculus of Voting*

  • William H. Riker (a1) and Peter C. Ordeshook (a1)


Much recent theorizing about the utility of voting concludes that voting is an irrational act in that it usually costs more to vote than one can expect to get in return.1 This conclusion is doubtless disconcerting ideologically to democrats; but ideological embarrassment is not our interest here. Rather we are concerned with an apparent paradox in the theory. The writers who constructed these analyses were engaged in an endeavor to explain political behavior with a calculus of rational choice; yet they were led by their argument to the conclusion that voting, the fundamental political act, is typically irrational. We find this conflict between purpose and conclusion bizarre but not nearly so bizarre as a non-explanatory theory: The function of theory is to explain behavior and it is certainly no explanation to assign a sizeable part of politics to the mysterious and inexplicable world of the irrational.2 This essay is, therefore, an effort to reinterpret the voting calculus so that it can fit comfortably into a rationalistic theory of political behavior. We describe a calculus of voting from which one infers that it is reasonable for those who vote to do so and also that it is equally reasonable for those who do not vote not to do so. Furthermore we present empirical evidence that citizens actually behave as if they employed this calculus.3



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We wish to thank a number of colleagues who have helpfully contributed improvements to our argument, especially Professors Richard Rosett, Richard Niemi, and Steven Brams. William Riker wishes to thank the National Science Foundation for a grant under which this essay was written.



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1 The two most interesting presentations of this analysis are Downs, Anthony, An Economic Theory of Democracy (New York: Harper, 1957), pp. 3650 , 260–276 and Tullock, Gordon, Toward a Mathematics of Politics (University of Michigan Press, 1968), Chapter 7.

2 In Downs' theory, which he characterizes as “positive, but not descriptive,” there is no reason to expect descriptive accuracy, although in science one would expect to discard positive theories that are inadequate as descriptions. Tullock's theory is, however, intended to be descriptive.

3 There is continuing interest in political science in the question of whether or not the substantive decisions in voting are rational: e.g., Berelson, B. R., Lazarsfeld, P.F., and McPhee, W. N., Voting (Chicago, 1954); and Key, V. O., The Responsible Electorate (Cambridge, Mass., 1966), the former of which takes the position that substantive decisions are irrational while the latter argues that they are rational. See also Riker, W. H., Democracy in the United States (2nd ed., New York, 1965), pp. 4749. This essay may be taken as supporting the position of Riker and Key on the procedural level, i.e., that voters are rational in deciding whether or not to cast a substantive ballot.

4 When we say “customarily,” we are referring not only to the work of Downs and Tullock, but also to the numerous unpublished analyses we have read or heard from political scientists and economists over the past twenty years.

5 Bureau, U. S. of the Census, Current Population Reports Series P-20 #143, “Voter Participation in the National Election: November 1964,” (Washington, 1965).

6 Downs refines (5) considerably and we refer the reader to his work cited, pp. 38–47, for the full presentation.

7 Downs, of course, recognizes that his is an incomplete analysis of the act of voting. Because of the particular theory of rationality he adopted, however, he was unable to include other considerations. Rationality may be interpreted broadly as the ability to order preferences and to choose the more preferred action over the less preferred. In this sense, almost all behavior is rational and the assumption of rationality is close to tautological. Because he thinks the tautology is sterile, Downs rejects this definition of rationality, which is the one customarily used by economists, for a narrower interpretation in which rational behavior is only that behavior directed toward the goals that the theorist postulates as appropriate for a particular realm of action. As a result Downs cannot include “non-political” benefits (i.e. those calculated in (7) but not in (1)) in his analysis, although many of the benefits he cannot include are clearly political. It seems to us that he is unduly bothered by the tautology, which after all does affirm something, namely that people calculate about their actions and that their calculations can be understood by others. It seems to us also that in a descriptive theory it is unwise for the theorist to impose his own interpretation of goals on the observed behavior. By so doing, he falls into the trap (that all the natural law theorists fall into) of saying that one goal is rational and another is not. Because it is not possible to judge the “rationality” of goals—unless one adopts some sort of natural law theory—we will adopt here the broader interpretation of rationality, recognizing its tautological character, in order to develop a theory that may more adequately describe behavior.

8 Shapley, L. S., “A Value for n-Person Games,” Annals of Mathematics Study #28 (Princeton, 1953), pp. 307317. See also Shapley, L. S. and Shubik, M., “A Method for Evaluating the Distribution of Power in a Committee System,” this Review, 48 (1954), 787792.

9 It might be argued that & C 1>C 2 owing to the fact that in an area of strong partisanship, the party label reduces the cost of decision in the general election to zero. If this is true then the argument in the text is even more persuasive because B 1 must then be even greater than B 2 for ν 1 to be greater than ν 2.

10 In the general case, since we are dealing with subjective probabilities, we may write pr [1; x]=A, where A is any positive number.

11 The variance is a function of certainty and certainty is in turn a highly complicated function of information such that increases in information may either increase or decrease certainty. For example, information about opinion surveys taken closer and closer to the election and showing a large and sustained majority for one candidate might increase certainty while information about opinion surveys showing an increase in undecided voters as the election approached might decrease certainty.

12 The data utilized in this study were made available by the Inter-University Consortium for Political Research and were originally collected by the Survey Research Center of the University of Michigan.

13 An operationally equivalent assumption is that the variance of the probability density function is randomly distributed, independently of x 0. Hence those citizens with a subjective x 0 “close” to ν/2 are expected to have a higher P than those whose x 0 is “far” from ν/2.

14 It is conceivable that some people who have a high P by this operationalization actually view the probability density as continuous so that for them P = 0 rather than P = ½ pr [1; ν/2]. If some “low” P persons are by this error of operationalization included in the “high” P category, our predictions are of course rendered more difficult to verify.

15 Unfortunately the SRC did not ask the appropriate questions in 1964 so that we cannot use other data from that year in our analysis. The sense of citizen duty scale is fully described in Campbell, A., Gurin, G. and Miller, W. E., The Voter Decides (Evanston, 1954), pp. 194199.

16 The constancy of D is open to some technical question. The SRC ceased in 1964 to collect the necessary information, first, because the scale of citizen duty turned out to be very similar to the scale of the sense of political efficacy, which was more useful for SRC purposes, and, second, because it seemed possible that the four questions on the sense of citizen duty scale might be contaminated by a positive response set. Their first reason for the rejection of the scale is not immediately relevant to our concerns. Their second reason, if true, does, however, bias the data in favor of our predictions. Suppose some persons with an actually low D are included (by reason of response set) in our categories I and II of high and medium D. They will then lower the percentage of voters in these categories. And if low D is associated with not caring about the outcome of the election so that these low D's add disproportionately to the non-carers, their presence among the non-carers in category I or II will make it easier to verify our hypothesis that those who care about the outcome are more likely to vote than those who do not care. If, however, this bias, for our purposes, was significant it would tend to wipe out differences among like a i.'s between categories. Since, as will be shown later, this does not occur, we may suppose that the response-set contamination of the scale of citizen duty is minimal.

17 This is the necessary confirmation of assumption, mentioned earlier, that the effect of response set is minimal. (See footnote 16.)

18 If we drop from our calculations in Table 3 all those persons whom the SRC coded “not appropriate,” “don't know,” or “no answer” to the question about the closeness of the outcome (whom we have actually included in Table 3 in the “high” P category), then the results are somewhat improved. Two of the three bad predictions in Table 5 are eliminated and the level of significance increases.

19 If the vote for the citizen's preferred party is ν/2, ν even, before he votes, then this outcome is worth B utiles since his vote gives his party ν/2+1 votes, or enough to win. If, however, ν is odd, then, assuming the voter does not value a tie, only until an outcome of (ν+1)/2+1 votes is assured can the voter realize the desired outcome of his preferred candidate winning when he votes.

20 It is assumed that g(x-x0) goes to zero sufficiently rapidly so that small changes in x 0 do not violate

Also it is assumed that μ(x) does not change when one more voter is added to the electorate.

21 Integration by parts is defined as follows:

In this particular case we let x = μ and g′(xx 0)dx = dv.

22 An n-order polynomial is any function of the following form:

23 This result differs by a factor of ½ from our original formulation in equation (21) because we have implicitly assumed ν even. Our original analysis showed however that when ν is odd the voter cannot affect the outcome so as to bring about the victory of his preferred candidate. The gain associated with increasing his favored candidate's chance of winning is offset by increasing the chance of a tie (from 0 to prν, [1; ν+ 1/2]). Given an equal likelihood for even and odd ν we would once again let P = ½prν [l; ν/2]. If we assumed the voter equally valued creating a tie to having his preferred candidate win then we would drop the ½ once again.

24 Or if one wishes ½pr ν [1; ν/2].

* We wish to thank a number of colleagues who have helpfully contributed improvements to our argument, especially Professors Richard Rosett, Richard Niemi, and Steven Brams. William Riker wishes to thank the National Science Foundation for a grant under which this essay was written.


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