# 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.

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## References

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Authors

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.

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