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An Expository Development of a Mathematical Model of the Electoral Process*

Published online by Cambridge University Press:  01 August 2014

Otto A. Davis
Carnegie-Mellon University
Melvin J. Hinich
Carnegie-Mellon University
Peter C. Ordeshook
Carnegie-Mellon University


The fundamental process of politics is the aggregation of citizens' preferences into a collective—a social—choice. We develop, interpret, and explain non-technically in this expository essay the definitions, assumptions, and theorems of a mathematical model of one aggregative mechanism—the electoral process. This mechanism is conceptualized here as a multidimensional model of spatial competition in which competition consists of candidates affecting turnout and the electorate's perception of each candidate's positions, and in which the social choice is a policy package which the victorious candidate advocates.

This approach, inaugurated by Downs's An Economic Theory of Democracy, and falling under the general rubric “spatial models of party competition,” has been scrutinized, criticized, and reformulated. To clarify the accomplishments of this formulation we identify and discuss in section 2 the general democratic problem of ascertaining a social preference. We review critically in section 3 the definitions and assumptions of our model. We consider in sections 4 and 5 the logic of a competitive electoral equilibrium. We assume in section 4 that the electorate's preferences can be summarized and represented by a single function; the analysis in section 5 pertains to competition between two organizational structures or two opposed ideologies (i.e., when two functions are required to summarize and represent the electorate's preference). Finally, we suggest in section 6 a conceptualization of electoral processes which facilitates extending and empirically testing our model.

Research Article
Copyright © American Political Science Association 1970

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This research was supported by a grant from Resources for the Future, Inc., to Carnegie-Mellon University, and a National Science Foundation Grant to the University of Rochester. The authors are indebted to many persons for comments and criticism and wish especially to thank Professors Peter H. Aranson and William H. Riker, University of Rochester, Howard Rosenthal, Carnegie-Mellon University, and Michael J. Shapiro, University of California, Berkeley.


1 See the following: Davis, Otto A., and Hinich, Melvin J., “A Mathematical Model of Policy Formation in a Democratic Society,” Mathematical Applications in Political Science II, Bernd, J. L., ed. (Dallas: Arnold Foundation, SMU Press, 1966)Google Scholar; Some Results Related to a Mathematical Model of Policy Formation in a Democratic Society,” Mathematical Applications in Political Science III, Bernd, J. L., ed. (Charlottesville: University of Virginia Press, 1967)Google Scholar; On the Power and Importance of the Mean Preference in a Mathematical Model of Democratic Choice,” Public Choice, 5 (Fall, 1968), 5972CrossRefGoogle Scholar; Some Extensions to a Mathematical Model of Democratic Choice,” forthcoming in Social Choice, Lieberman, B., ed. (New York: Gordon and Breach)Google Scholar; Hinich, Melvin J. and Ordeshook, Peter C., “Abstentions and Equilibrium in the Electoral Process,” Public Choice, 7 (Fall, 1969)Google Scholar; Social Welfare and Electoral Choice in Democratic Societies,” (unpublished, Carnegie-Mellon University, 1969)Google Scholar; Ordeshook, Peter C., “Some Extensions to a Mathematical Model of Electoral Competition, and Implications for the Theory of Responsible Parties,” Midwest Journal of Political Science, (02 1970)CrossRefGoogle Scholar; Theory of the Electoral Process (unpublished Ph.D. dissertation, University of Rochester, 1969)Google Scholar.

2 Downs, Anthony, An Economic Theory of Democracy (New York: Harper and Row, 1957)Google Scholar. For additional theoretical developments see: Garvey, Gerald: “The Theory of Party Equilibrium,” this REVIEW, LX (1966), 2938Google Scholar; Chapman, David E., “Models of the Working of a Two-Party Electoral System,” Papers on Non-Market Decision Making III (Fall, 1967)Google Scholar, and Public Choice (Fall, 1968)

3 Arrow, Kenneth J., Social Choice and Individual Values (New York: Cowles Commission Monograph #12, Wiley, 1951)Google Scholar. See also: Black, Duncan, The Theory of Committees and Elections, (Cambridge: Cambridge University Press, 1968)Google Scholar; and with Newing, R. A., Committee Decisions with Complementary Valuation (London: W. Hodge, 1951)Google Scholar. A general exposition of the paradox and its implications is given by Riker, William H., “Voting and the Summation of Preferences: An Interpretive Bibliographical Review of Selected Developments During the Last Decade,” this Review, LV (12, 1961), 900911Google Scholar.

4 Duncan Black, op. cit., pp. 21–25.

5 Duncan Black and R. A. Newing, loc. cit.

6 Such situations are examined closely by Buchanan, James M., Public Finance in Democratic Process (Chapel Hill: University of North Carolina Press, 1967)Google Scholar.

7 Tullock, Gordon, “The General Irrelevance of the General Impossibility Theorem,” Quarterly Journal of Economics (05, 1967)CrossRefGoogle Scholar. Richard G. Niemi presents an excellent formal treatment and interpretation of the probability of a paradox occurring in Majority Decision-Making with Partial Unidimensionality,” this Review, LXIII (06, 1969), 488497Google Scholar.

8 Key, V. O., Public Opinion and American Democracy (New York: Knopf, 1963), Ch. 7Google Scholar; Converse, Phillip E., “The Nature of Belief Systems in Mass Publics,” in Apter, David E. (ed.), Ideology and Discontent (New York: Free Press, 1964), pp. 206261Google Scholar; The Problem of Party Distances in Models of Voting Change,” in Jennings, M. Kent, and Zeigler, L. Harmon (eds.), The Electoral Process (Englewood Cliffs: Prentice-Hall, 1966), 175207Google Scholar; Stokes, Donald E., “Spatial Models of Party Competition,” this Review, LVII (06, 1963), 368377Google Scholar.

9 The relative importance of issues, compared to image and partisan bias, as causal determinants of voting behavior remains an open question. Aggregate analyses of cross-sectional survey data demonstrate clearly the predictive dominance of partisan identification. Key, V. O., however, concludes in The Responsible Electorate (Cambridge: The Belknap Press of Harvard University Press, 1966)CrossRefGoogle Scholar that policy counts heavily. Arthur S. Goldberg, moreover, demonstrates “that there is a rational component to party identification rooted in group norms” (p. 21) with the suggestion that these norms are related to issues, in Social Determinism and Rationality as Bases of Party Identification,” this Review, LXIII (03, 1969), 525Google Scholar.

10 See Goldberg, ibid.

11 This interpretation of rationality is equivalent to the as if principle of rational behavior as presented by Friedman, Milton in “The Methodology of Positive Economics,” Essays in Positive Economics (Chicago: University of Chicago Press, 1963)Google Scholar. See also Riker, William H., and Zavoina, William, “Rational Behavior in Politics, this Review, LXIV (03, 1970)Google Scholar.

12 Op. cit. For a spatial analysis of discrete dimensions see Chapman, op. cit.

13 Ibid.

14 The densities illustrated in Figures 3 and 4 are represented as discrete although the scales are assumed to be continuous because electorates are finite populations. Nevertheless, our analysis is facilitated by assuming that f(x) is continuous, which is not a serious distortion of any significance if the electorate is large. Hence, in all subsequent illustrations we represent f(x) as a continuous density.

15 For a discussion of the role of cognitive balance see: Stokes, Donald E., “Some Dynamic Elements of Contests for the Presidency, this Review, LX (03, 1966), 1928Google Scholar; Berelson, Bernard R., Lazarsfeld, Paul F., and McPhee, William N., Voting (Chicago: University of Chicago Press, 1954), ch. 10Google Scholar; Shapiro, Michael J., “Rational Political Man: A Synthesis of Economic and Social-Psychological Perspectives,” this Review, LXIII (12, 1969)Google Scholar. Cognitive balance poses a problem for our theoretical analysis, but it also reduces the validity of much cross-sectional survey research about attidudes and voting behavior. Briefly, the causal link between attitude (i.e., preference) and vote is bidirectional for many issues. Simply regressing attitude on vote does not reveal the importance of an issue for a citizen's choice—a significant regression coefficient may indicate only that the attitude has been made consistent with a predetermined preference because it is unimportant. Multiple regression analysis with many attitudinal variables, moreover, is not a satisfactory solution either. A statistically insignificant regression coefficient may indicate only that that variable is related to some other independent variable in the analysis although it may in fact be an important determinant of candidate preference. Because of such difficulties Gerald Kramer analyzes the relationship between policy preference and voting with variables which are more objectively measurable than attitudes in An Empirical Analysis of Some Aggregative Hypotheses About U.S. Voting Behavior, 1896–1964,” (unpublished, Yale University, 1968)Google Scholar.

16 In some of our papers individual loss functions are symbolically represented by the function ϕ (x − θ) to indicate that loss is a function of the difference between x and θ.

17 In matrix notation, expression (6) becomes

(xi - θj)' A (xi - θj

where (xi - θj)' is the transpose of (xi - θj), i.e.,

and where A is the n×n matrix of weights, i.e.,

18 There exists, moreover, a linear transformation on the axes so that any quadratic of the form (x − θ)′ A (x − θ) can be reduced to (x − θ)′ (x − θ) without loss of generality (i.e., A becomes the identity matrix I). Thus, without loss of generality, we can assume that

so that the indifference contours for a citizen's loss function are concentric circles.

19 The assumption of a common A matrix does not imply an interpersonal comparison of utility. It implies that, when the loss functions for citizens are ascertained independently, there exists a monotonic transformation on each loss function such that all loss functions have a common A matrix in one coordinate system.

20 When nonvoting (which is discussed later) is caused by alienation, we must assume that variations in level of concern are independent of preference—a somewhat stronger assumption.

21 Mathematically, we may assume that Lij) is a function, ϕ, of the quadratic form. Thus

Lij) = ϕ ((xi − θj))′ A (xi − θj))

where ϕ is any monotonically increasing function of its argument. Note that if A = I, the citizen's indifference contours remain concentric circles under the transformation ϕ.

22 Op. cit. p. 908. See also Niemi, loc. cit., and Coombs, Clyde H., A Theory of Data (New York: Wiley, 1964), Cps. 5–7Google Scholar.

23 Theory of Voting (New Haven: Yale University Press, 1969)Google Scholar. For examples of the occurrence of paradoxes in legislatures and possible occuranees of contrived paradoxes see Riker, William H., “The Paradox of Voting and Congressional Rules for Voting on Amendments,” this Review, LII (1958), 349366Google Scholar, and Arrow's Theorem and Some Examples of the Paradox of Voting,” in Claunch, J. M. (ed.), Mathematical Applications in Political Science (Dallas: Arnold Foundation, SMU Press, 1965)Google Scholar.

24 Andrews, William G., “American Voting Participation,” The Western Political Quarterly, (12, 1966), 639652Google Scholar.

25 A Theory of the Calculus of Voting,” this Review, LXII (03, 1968), 2542Google Scholar.

26 Our assumptions about nonvoting conform closely to the two factors Garvey (op. cit.) identifies.

27 See Ordeshook, “Some Extensions …,” op. cit.

28 See Hinich and Ordeshook, “Abstentions and Equilibrium …,” op. cit.

29 For an analysis of electoral strategies when vote maximization is the posited goal see Hinich, and Ordeshook, , “Plurality Maximization vs. Vote Maximization: A Spatial Analysis with Variable Participation,” this Review (forthcoming, 09 1970)Google Scholar.

30 A counter example to dominance is presented in Ordeshook, “Some Extensions …,” Op. cit.

31 Tullock, Gordon, Toward a Mathematics of Politics (Ann Arbor: University of Michigan Press, 1968), p. 52Google Scholar.

32 Ordeshook, op cit.

33 See, for example, McCosky, Herbert, Hoffman, Paul J., and O'Hara, Rosemary, “Issue Conflict and Concensus Among Party Leaders and Followers,” this Review, LIV (06 1960), 406427Google Scholar, and; Eldersveld, Samuel J., Political Parties (Chicago: Rand McNally, 1964), ch. 8Google Scholar.

34 Miller, Warren E., “Majority Rule and the Representative System of Government,” in Allardt, E., and Littunen, Y. (eds.), Cleavages, Ideologies, and Party Systems: Contributions to Comparative Political Sociology (Helsinki: Transactions of the Westermarck Society, 1964), 343376Google Scholar.

35 See Peter H. Aranson and Peter C. Ordeshook, “Spatial Strategies for Sequential Elections,” (forthcoming); and R. G. Niemi and H. P. Weisberg, Probability Models in Political Science.

36 For our use of the word theory see Hempel, Carl G., Philosophy of Natural Science (Englewood Cliffs: Prentice-Hall, 1966)Google Scholar. See also, Davis, Otto A., “Notes on Strategy and Methodology for a Scientific Political Science,” Bernd, J. (ed.), Mathematical Applications in Political Science, IV (Charlottesville: University of Virginia Press, 1969)Google Scholar.

37 Pool, Ithiel de Sola, Abelson, Robert P., and Popkin, Samuel, Candidates, Issues, and Strategies (Cambridge: M.I.T. Press, 1964)Google Scholar.

38 Some practitioners of simulation methods might object to our removing simulations from the class of deductive scientific theories. We agree with Hayward R. Alker's observation that the logical operations of computer simulations are deductive (Computer Simulation, Conceptual Frameworks and Coalition Behavior,” in Groennings, S., et. al. (eds.), The Study of Coalition Behavior, forthcomingGoogle Scholar). But Alker's assertion constitutes a serious confusion of the use of deduction as a method of science with the notion of a deductive theory. In the first usage a deduction is the process of inference from general statements to concrete instances. Thus, one infers that, if all α are β, then a particular α is a β. In the second usage, deduction is the process of inference from general sentence to general sentence. Thus, one infers that if all α are β and if all β are α, then α is equivalent to β. The first kind of deduction is used in fitting models to reality, and, where analysis is complex, is the proper function of stimulations; the second constitutes finding necessary and sufficient relations (i.e., cause) and is the proper domain of abstract mathematics. See, for example, Waltz, Kenneth, “Realities, Assumptions, and Simulations,” in Coplin, William D. (ed.), Simulation in the Study of Politics (Chicago: Markham, 1968)Google Scholar, and Charles A. Powell's review of Coplin's book, this Review, LXIII (September 1969), p. 937. Perhaps the most fruitful attempt at applying in concert simulation and the generalizations of game theory and coalition theory, and the one which comports with our understanding of the proper uses of simulation—is presented in Coplin's volume by Howard Rosenthal in “Voting and Coalition Models in Election Simulations.”

39 See Kramer, Gerald, “A Decision-Theoretic Analysis of a Problem in Political Campaigning,” in Bernd, J. L. (ed.), Mathematical Applications in Political Science II (Dallas: Arnold Foundation, SMU Press, 1966)Google Scholar.

40 For a unidimensional spatial analysis of uncertainty see Shepsle, Kenneth, “Essays on Risky Choice in Electoral Competition,” (unpublished Doctoral dissertation, University of Rochester, 1970)Google Scholar.

41 McPhee, William N., Formal Theories of Mass Behavior (New York: The Free Press, 1963), p. 40Google Scholar.