A simple direction model of electoral competition

Summary

The direction model of the electoral process allows limits to candidate mobility or voter perception and cognition. It is applicable (1) if only issue outcomes near the status quo are associated with candidates; (2) if only directional information is transmitted to voters; (3) if voter preferences are only well-defined near the status quo or are only defined for directions in which it can shift; or (4) if the outcome space is curved so that it can be modeled as a hypersphere.

Assuming that a voter will vote for the candidate who campaigns for a direction closest to his own preferred direction, plurality equilibria were shown to be undominated. Equilibrium and undominated directions were shown to be indentical if nobody is totally indifferent. Then a necessary and sufficient condition for the existence of undominated directions was determined. The first part of the condition, stating that any hyperplane containing the undominated direction vector and the origin bisects the distribution of preferred directions, is analogous to the total median condition in the simple Euclidean models. The remainder of the condition in Theorem 2, stating that a majority of the electorate's preferred direction vectors lie on the same side as the undominated direction vector of any hyperplane containing the origin, is not implied by the median-like property in this model because of the ‘curved’ nature of the directional domain space. The second part of the condition is what allows a candidate to diverge from a fixed direction chosen by an extremist opponent, where at least half the feasible directions are defined to be extremist for every distribution of the electorates' preferred directions.

Although the addition of a second part to the characterizing condition for equilibrium seems to further decrease the likelihood of its occurrence, it was shown that in situations where the assumptions of the simple Euclidean model are met, point equilibria exist only if corresponding undominated directions also exist. But the converse of this theorem is false — some distributions of voter preferences yield direction but not point equilibria. In situations where both types of equilibria exist, contradictory predictions will not occur since equilibrium direction vectors point in the direction of existing equilibrium points.

Finally, it was argued that a candidate has no incentive to adopt a type of strategy different from the type he knows his opponent will choose. This result can be interpreted as an internal stability property for each model. However, it was suggested that when a candidate's uncertainties about voters' preferences away from the status quo and about the extent of his opponent's information is considered, only the direction model may exhibit this internal stability.

This is a preview of subscription content, access via your institution.

References

  1. Campbell, Angus, Converse, Philip E., Miller, Warren E., and Stokes, Donald E. The American Voter. New York: John Wiley and Sons, 1960.

    Google Scholar 

  2. Cohen, Linda, and Matthews, Steven. ‘Constrained Plott Equilibria, Directional Equilibria, and Global Cycling Sets.’ Social Science Working Paper 178. Pasadena: California Institute of Technology, September 1977 (fortcoming in Review of Economic Studies).

    Google Scholar 

  3. Davis, Otto A., DeGroot, Morris H., and Hinich, Melvin J. ‘Social Preference Orderings and Majority Rule.’ Econometrica 40, no. 1 (1972), pp. 147–157.

    Google Scholar 

  4. Davis, Otto A., Hinich, Melvin J., and Ordeshook, Peter C. ‘An Expository Development of a Mathematical Model of the Electoral Process.’ American Political Science Review 64 (1970), pp. 426–428.

    Google Scholar 

  5. Downs, Anthony. An Economic Theory of Democracy. New York: Harper and Row, 1957.

    Google Scholar 

  6. Fiorina, Morris P., and Plott, Charles R. ‘Committee Decisions under Majority Rule: An Experimental Study.’ American Political Science Review 72 (1978), pp. 575–598.

    Google Scholar 

  7. Hinich, Melvin J., and Ordeshook, Peter C. ‘Abstention and Equilibrium in the Electoral Process.’ Public Choice 8 (Fall 1968), pp. 97–98.

    Google Scholar 

  8. Hoyer, Robert W., and Mayer, Lawrence S. ‘Comparing Strategies in a Spatial Model of Electoral Competition.’ American Journal of Political Science (August 1974), pp. 501–523.

  9. —, and — ‘Social Preference Orderings Under Majority Rule.’ Econometrica 43, no. 4 (July 1975), pp. 803–806.

    Google Scholar 

  10. Kramer, Gerald H. ‘On a Class of Equilibrium Conditions for Majority Rule.’ Econometrica 41, no. 2 (1973), pp. 285–297.

    Google Scholar 

  11. Matthews, Steven A. ‘Undominated Directions in Simple Dynamic Games.’ Social Science Working Paper No. 169. Pasadena: California Institute of Technology, June 1977.

    Google Scholar 

  12. McKelvey, Richard D. ‘Policy Related Voting and Electoral Equilibrium.’ Econometrica 43, no. 5–6 (1975), pp. 815–843.

    Google Scholar 

  13. Page, Benjamin I. ‘Elections and Social Choice.’ Paper delivered at the Research Conference on Social Choice Theory and Democratic Theory, December 1975.

  14. Plott, Charles R. ‘A Notion of Equilibrium and its Possibility under Majority Rule.’ American Economic Review 57 (1967), pp. 787–806.

    Google Scholar 

  15. Rabinowitz, George B. ‘On the Nature of Political Issues: Insights from a Spatial Analysis.’ American Journal of Political Science (November 1978), pp. 793–817.

  16. Riker, William H., and Ordeshook, Peter C. An Introduction to Positive Political Theory. Englewood Cliffs: Prentice-Hall, 1973.

    Google Scholar 

  17. Sloss, Judith. ‘Stable Outcomes in Majority Rule Voting Games.’ Public Choice (Summer 1973), pp. 19–48.

  18. Weisberg, Herbert F. ‘Dimensionland: An Excursion into Spaces.’ American Journal of Political Science 1974, pp. 743–776.

Download references

Authors

Additional information

California Institute of Technology and University of Illinois at Urbana-Champaign. The author wishes to thank John Ferejohn, Morris Fiorina, Melvin Hinich, and Charles Plott for their valuable criticism, comments, and time.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Matthews, S.A. A simple direction model of electoral competition. Public Choice 34, 141–156 (1979). https://doi.org/10.1007/BF00129523

Download citation

Keywords

  • Prefer Direction
  • Direction Vector
  • Direction Model
  • Internal Stability
  • Yield Direction