¹ It is not difficult, in fact, to imagine a plethora of stochastic models that give rise to this relationship (and that also promise to be more consistent with the data). For example, suppose state j consists of R_j perfectly homogeneous regions of equal size that are populated wholly by Democrats, Republicans, or uncommitted voters. Let the character of each region be determined by a random draw from an infinite population, where p_d (p_r) denotes the probability that the region is Democratic (Republican). If uncommitted voters are as likely to vote for one candidate as the other, it follows that the Democrat's proportionate expected plurality equals while the standard deviation of equals . Thus, if R_j is proportional to the state's size, σ again decreases with the square root of N_j. We are not, of course, proposing this model to explain observed variations in ∣Pl_j∣—we require a more sophisticated probability model dealing, perhaps, with mobility patterns and socialization. What we seek to show, however, is that an inverse relationship between ∣Pl_j∣ and N_j is a likely consequence of most simple (and perhaps more complex) probability models, and, hence, its confirmation should not be particularly exciting. An adequate model of campaigning must account, additionally, for the magnitudes of ∣Pl_j∣ and σ.

² Alternatively, suppose that we attribute y per cent of the variation in Pl_j to violations of the matching hypothesis. Then even if n_j = N_j/2, a value for y as low as 10 (per cent) presupposes that the candidates shifted their resources out of the local equilibrium region of the 3/2's rule in those states with ∣Pl_j ≥ .05 (a majority). That is, to accommodate ten per cent of .05 or more, it must be the case that ∣2p_j − 1∣/2 > .05(.1), i.e., either p_j < .495 or p_j > .505. It is straightforward to show, however, that the small deviations from matching that raise p_j to .505 in general lie outside the local equilibrium region.

³ Brams, Steven J. and Davis, Morton D., “The 3/2's Rule in Presidential Campaigning,” American Political Science Review, 68 (March, 1974), 126CrossRefGoogle Scholar.

⁴ Ibid., p. 132.

⁵ Ibid., p. 131.

⁶ Admittedly, our measure of competitiveness is ex post, owing to the unavailability of ex ante data. We do not regard this as serious, however, since it seems reasonable to assume that resources such as trips affect the biases of states less than those biases affect trips (see our note #18) . Ideally, of course, we would prefer an ex ante measure such as the opinions of the candidates themselves as their campaigns commence.

⁷ To see this we use Brams and Davis's original method of proof: Brams, Steven J. and Davis, Morton D., “Resource-Allocation Models in Presidential Campaigning: Implications for Democratic Representation” Annals of the New York Academy of Sciences, 219 (November, 1973), 105–123CrossRefGoogle ScholarPubMed. Evaluating, as they do, the rate of change in the Republican's expected electoral vote with respect to r_i (dW_r/dr_i) at the point.

and letting

we obtain (ignoring all higher order terms in ε),

.

Thus, (a, b) is a local maximum for the Republican only if K_i equals K_j, i.e., only if dW_r/dr_i = 0 for ε = 0 (note that if α_i = α_j, = 1, expressions (1) and (2) reduce to those seen in the original derivation of the 3/2's rule). The second condition for a local maximum at (a, b) is that dW_r/dr_i < 0 for ε > 0 and dW_r/dr_i > 0 for ε < 0. Setting K_i = K_j, this condition is satisfied (after some rearranging of terms) if and only if

.

Repeating the analysis for the Democrat by letting

.

we find that (α_i,a, α_jb) is a local maximum for him if and only if K_i = K_j, and

.

Thus, a sufficient condition for [d = (α_i,a, α_jb), r = (a, b)], to be a local equilibrium in the gametheoretic sense is,

To show now that this is, for all practical purposes, a necessary condition, note that if 1 + 2/n_i < α_i, or α_i </(n_i + 2) for two or more states, a local equilibrium cannot exist. There remains, then, two possibilities. First, condition (5) is satisfied for all states but one; second, condition (5) is satisfied for all but two states, with α > 1 + 2/n in one state and α < n/(n + 1) in the other. Suppose state j satisfies the condition and, in particular (to avoid excessive algebra) let α_j = n_j/(n_j, + 2) so that, from condition (4), α_i must satisfy α_i ≥ n_i/(n_i + 2). Using the fact that K_i = K_j, expression (3) now requires that

This expression is readily reduced by noting that the maximum value of

is one and that

equal in this instance to

increases as n_j increases and closely approximates 1 for relatively small values of n_j (e.g., for n_j = 12, it equals .968). Adding without proof the condition for α_j = 1 + 2/n_j, we conclude that in the one or two remaining states, the maximum admissable bounds on α_i are,

where . Note, however, that the bounds on α_i, must hold for all j ≠ i, in which case the maximum value presently for is about 2. Clearly, then, the second term in the expression for Γ is, for reasonable values of n_i, several orders of magnitude smaller than one. Thus, we can set Γ = 1 and approximate condition (6) simply by condition (5). To ascertain, then, the maximum admissable discrepancy between R and D, assume that D > R and α_i = 1 + 2/n_i for all i. Since D = Σr_i[1 + 2/n_i] = R + 2Σ(r_i/n_i) and , the necessary and sufficient condition for a local equilibrium is,

For example, then, if all n_i,'s are on the order of 10,000, a local equilibrium exists if and only if 0.9998 ≤ D/R ≤ 1.0002!

⁸ One objection to this example, perhaps, is that we choose n_j too large and that if we let n_j = .01N_j or .001N_j, the 3/2's allocation rule is less unstable (although a 1 per cent increase in Missouri's budget at the expense of Vermont remains advantageous even if there are as few as 4,000 uncommitted voters in Missouri). To repeat our earlier argument, however, the problem is that for smaller values of n_j, the model is less able to accommodate the apparent biases of states. Thus, the model is either irrelevant or incomplete from a practical perspective or the instability of a 3/2's rule vitiates its function as a “reference point.”

⁹ Brams and Davis's objection to the statement (in our abstract) that “the unit rule … is the predominant cause of bias” is legitimate. We perhaps overstated the case there, however, in satisfying the restrictions on an abstract's length. We suggest that they read the conclusion to our essay for a more complete account of our argument, which is simply that “even without weighted voting we should anticipate some variation in resource allocations” across units.