It is perhaps too readily supposed that candidate or party convergence is an intrinsic part of the Downsian spatial model and a necessary implication of so called "proximity" preferences. It is commonly recognized that such convergence may not occur if there are more than two candidates or if candidates do not behave in a ("Downsian") vote-maximizing manner. But most American general elections do involve two candidates, and we generally ascribe something close to vote-maximizing motives to American politicians. Thus the common and sometimes fairly striking polarization of House and Senate candidates in many districts and states, and the often abrupt change in roll-call behavior when a Democrat replaces a Republican or vice versa in a given district or state, is often viewed as anomalous in the context of Downsian spatial (or proximity) theory. But such non-convergence may be quite consistent with the theory.

        Consider a society that makes a unidimensional political choice through an (n+1)-member (American-style) legislature elected from single-member districts and without party discipline. Suppose all citizen preferences are based on proximity in the manner of the classical Downsian spatial model.

        A vacancy arises in the legislature, leaving an n-member incumbent legislature and requiring a special election in one district. In the special election there are two candidates, D and R, whose platforms (which, for the moment, we consider exogenously fixed), together with the ideal point V of a focal voter in the district, are depicted in the diagram below.
 

                                       D                                                    V     L'   L   L"              R
                                ____|___________________________|____|__|__|_________|________
 

        On the basis of proximity, voter V prefers the platform of candidate R to the platform of candidate D. Now, if the election were to chose a single executive (D or R) or to chose between two disciplined parties (D or R) one of which will exclusively control the legislature and enact policy, the voter would therefore vote for R. (This is the common "Downsian setup.")

        But in the institutional setup described above, the choice facing the voter is between adding D to the incumbent legislature or adding R to the incumbent legislature. By the Median Voter Theorem, we may suppose that any legislature (without party discipline) enacts its median position. Suppose the median of the incumbent legislature is position L lying between V and R. Then election of D will shift this median position a bit to the left, say to L'; and election of R will shift this median position a bit to the right, say to L".⁽¹⁾ Thus the question facing voter V is whether to try to nudge the legislature a bit to the left or a bit to the right. Given his (proximity) preferences, voter V wants to nudge it to the left, and the way to help accomplish this is to vote for any candidate who lies in that direction. How far a candidate lies in that direction makes no difference, since the median is (as statisticians say) "resistant to outliers." So in this case V should vote for the relatively extreme candidate D whose platform V likes less than R's.

        If V is the median voter in the district (or more generally if the median voter in the districts lies to the left of the legislative median L), the more extreme candidate D will be elected. Moreover, so long as the two candidates are on opposite sides of L, neither candidate can gain votes by moving toward the center (or will lose votes by moving to the extreme). The outcome of the election is unaffected by changes in candidate positions and is determined by the spatial relationship between the median voter in the district and the median in the incumbent legislature. (If both candidates are on the same side of L, all voters are strictly indifferent in terms of the grounds for choice focused on here.) So there is no incentive for candidate convergence, and candidate platforms may indeed be determined exogenously (on the basis of personal ideology, activist preferences, etc.), as we initially assumed. Thus while this setup does not imply that candidates are polarized, it is perfectly consistent with this result. It is also perfectly consistent with an abrupt change in the roll-call behavior of a district's representative if there is a turnover in party control of the district from one election to the next (that would result if the spatial relationship between the district and legislative medians reversed).

        Observation 1.   Focusing on a "special election" is convenient from an expository point of view, since the composition of the remaining (incumbent) legislature is literally held constant. It seems apparent that similar considerations apply in the more relevant case of a general election, where all seats are filled simultaneously, though problems of conjectural variation then arise.

        Observation 2. From a technical point of view, this setup is very closely related to the "Grofman model" (Journal of Politics, 1985), where Grofman's "status quo" is corresponds very directly to the incumbent legislative median L and Grofman's "discount factor" corresponds more roughly to the intervals from L to L' and to L". (In Grofman's model the corresponding intervals depend on the candidate positions D and R and are plausibly relatively large.)

1. Suppose incumbent legislator positions are indexed 1,...,n from left to right, and let m = (n+1)/2. Then "a bit to the left" corresponds to the interval from L_m to L_m-1, and "a bit to the right" corresponds to the interval from L_m to L_m+1. Even this is not quite right, since either n or n+1 is an even number, in which case the median is not quite a point (or some unspecified tie-breaking rule comes into play). However these technicalities are resolved, the relevant point is that these intervals are almost surely quite small (in the manner depicted in the diagram) -- though, we may note (anticipating the conclusion of this argument), not as small as they would be if there were a clear incentive for legislative candidate convergence.