This essay addresses the question: Why does the Electoral College bias campaign resource allocations in favor of large states? Using data on candidate trips as well as estimates of the time candidates spend in states, we conclude, first, that much of the apparent empirical support for Brams and Davis's 3/2's hypothesis is an artifact of the candidates' consideration of each state's relative competitiveness and the statistical relationship between size and competitiveness. There is some evidence, however, for a residual bias. That is, after controlling for each state's competitiveness, campaign allocations still appear to favor larger states—at least for the two competitive elections of 1960 and 1968. We attribute that bias to corner solutions to the candidate's maximization problem and to the effects of sequential campaign planning. Thus, while we do not dispute the existence of bias over the course of the entire campaign, the data are consistent with a modified (albeit complex) proportional rule that each candidate applies sequentially during the campaign. Our conclusion is that the unit rule feature of the Electoral College, rather than weighted voting, is the predominant cause of bias.
This research was supported by a National Science Foundation Grant to Carnegie-Mellon University. We also wish to thank Steven Brams and Morton Davis for the availability of their data, Stanley Kelley, Jr. who gave us his data on the estimated time candidates spent in each state, and Timothy McGuire and Melvin J. Hinich for their helpful suggestions.
1 Brams, Steven J. and Davis, Morton D., “The 3/2's Rule in Presidential Campaigning,” American Political Science Review, 68 (March 1974), 113.
2 In spatial models of party competition, however, strategies are not resource allocations but positions on “issues,” and the constraint should be θi1, = θi2 = … = θi51. For a discussion of this distinction and its implications see Hinich, Melvin J. and Ordeshook, Peter C., “The Electoral College: A Spatial Analysis,” Political Methodology (forthcoming).
3 See Aranson, Peter H., Hinich, Melvin J., and Ordeshook, Peter C., “Election Goals and Strategies. Equivalent and Nonequivalent Candidate Objectives,” American Political Science Review, 68 (March, 1974), 135–152.
4 Gross, O. and Wagner, R., “A Continuous Colonel Blotto Game” (Santa Monica, Calif.: RAND Memo. #408, June 17, 1950); Gross, O., “The Symmetric Blotto Game” (Santa Monica, Calif.: RAND Memo. #424, July 19, 1950); Friedman, Lawrence, “Game-Theory Models in the Allocation of Advertising Expenditures,” Operations Research, 6 (Sept.-Oct. 1958), 699–709. See also Sankoff, David and Mellos, Koula, “The Swing Ratio and Game Theory,” American Political Science Review, 66 (June, 1972), 551–554.
6 Throughout this analysis we omit Alaska and Hawaii because of the discontinuity in campaign costs associated with visits to them.
7 If we assume that the observed total number of trips taken by a candidate is not a random variable, the parameters of this model are overidentified. That is, and Σεj, = 1. We do not, however, make this assumption. Rather, we let ΣTj, be a random variable and assume that theεj's are independently distributed.
8 An alternative procedure is to let δj, = .25 for all Tj. This, however, decreases and renders it easier to secure estimates that support the proportionate hypothesis. For example, in 1960 decreases from 1.37 to 1.32 for the Democratic slate and from 1.56 to 1.48 for the Republican slate.
9 Throughout this paper, all statements concerning significance refer to the .01 confidence level. Also, the figures in parentheses denote standard errors.
10 Several of these estimates are sensitive to δj, while others are not. Letting δj vary between .1 and .5, we find that four coefficients vary by more than 5 per cent—those for both parties for the 1968 and 1972 elections (which is to be expected, of course, since in these elections the candidates bypass a great many states whereas almost all states are visited in 1960 and 1964). The coefficient for the 1972 Democratic slate is the most sensitive: = 1.72 for δ = .1 and = 1.24 for δ = .5. Thus, while there is some justification for letting δ = .25 rather than .1 or .5, this sensitivity should be kept in mind as a qualification of our conclusions.
11 The Making of the President, 1960 (New York: Atheneum, 1960), p. 338.
12 As with Brams and Davis's data, we add a δj, to those Tj,'s we observe as zero. We again choose δj = .25 since a quarter of an hour seems a sufficiently small rounding error. None of the coefficients, however, are especially sensitive to δj.
13 Because of third parties that received more votes in a state than one or both of the two major parties, we calculate two sets of coefficients in 1960 and 1968. For these elections we set Plj, equal to the relevant slate's plurality over its strongest opponent in state j.
14 The approximate adjusted R 2's for 1960 and 1968 are .10 and .22 respectively, while the adjusted R 2's for 1964 and 1972 are essentially zero.
15 Alternatively, we can assume that candidates maximize their probabilities of winning but that the variance of the random variable Ej(θ1, θ2) is approximately a constant for all θ1, and θ2, in which case expression (1) is equivalent to expression (3). One justification for this reduction, other than the obvious way in which it facilitates analysis, is the assumption that neither candidate can manipulate his strategies so as to change appreciably (from .5 say to .25) the Pij's of a great many states. Such manipulations are plausible if candidates can advocate policies that alienate large portions of the electorate. But we are considering here a different sort of strategy—resources in general and trips to states in particular. And in the absence of any strong empirical evidence that resources such as trips affect state outcomes in an overwhelming way, the assumption of a constant standard deviation is a reasonable initial approximation.
16 Letting s(θ2) denote candidate 1's prior probabilities over θ2.
17 For a more complete discussion of such possibilities see Kramer, Gerald H., “A Decision-Theoretic Analysis of a Problem in Political Campaigning,” in Mathematical Applications in Political Science, II, Bernd, Joseph L., ed. (Dallas, SMU Press, 1966), 137–160.
18 Clearly, though, if Plj, is a function of θij, in any statistically appreciable way, the appropriate methodology is simultaneous equations estimation. Our data, however, seems far too crude to apply this method and it is reasonable to asume a priori that the relationship from Pli, to θij is the stronger.
19 The adjusted R 2s with and without ∣Plj∣ are as follows:
Note, however, that these numbers are defined in terms of log Tj. To ascertain the variance accounted for in terms of Tj, it is necessary to retransform the variables and, given our estimates of , and , recalculate R 2. These new values for R 2 will be higher than those we report above but the change in R 2 owing to the inclusion of ∣Plj∣ will diminish.
20 The following caveat must necessarily precede this “rounding-off” argument. Although rounding off the optimal solution of the noninteger (i.e., continuous) problem to obtain integer value solutions is often adequate, there are pitfalls in this approach. First, after it has been rounded, the optimal noninteger solution need not be feasible. Second, there is no guarantee that a feasible rounded solution is optimal to the original integer programming problem. See, Hillier, F. S. and Lieberman, G. J., Introduction to Operations Research (San Francisco: Holden-Day, Inc.), 1967, pp. 553–570.
21 There is, however, a condition under which these estimates are not biased. Specifically, assume that a candidate confronts two decisions—the first being whether or not to visit a state—and let Qj denote the probability that he visits state j. The candidate's second decision, now, is to properly allocate his trips among those he plans to visit. From all subsequent assumptions, including that error structures are multiplicative log-normal, the distribution of Tj, becomes
then the likelihood function for our parameters becomes
and the log of the likelihood function is,
To derive the maximum likelihood estimates of α, β, and γ, we substitute the observed values of Xj, into this expression, and set the derivatives of log L with respect to α, β, and β equal to zero. Note, however, that if Qj is functionally independent of α, β, and γ, the maximum likelihood estimates of these parameters are exactly what we obtain if we reestimate expression (17) as before after deleting from the sample those states the candidates fail to visit. The assumption that Qj is functionally independent of α, β, and j, however, is inconsistent with our previous analysis, since in that analysis the decision about whether or not to visit a state depends on the magnitude of . Nevertheless, if one is willing to accept the assumption that candidates adopt simplifying heuristics such as letting their decision rest on Ej or ∣Plj∣ alone, the preceding analysis proves that the estimates in Tables 7 and 8 are unbiased.
22 We note, however, that the derivation of the 3/2's rule also admits the possibility of corner solutions, but Brams and Davis implicitly assume that such solutions do not occur. Of course, if they are admitted into their analysis then their theory would predict estimates of β in excess of 3/2's.
23 Kelley, Stanley Jr., “The Presidential Campaign,” in The Presidential Election and Transition: 1960–1961, David, Paul T., ed. (Washington, D.C.: Brookings, 1961), p. 71.
24 The data in Table 9 is taken from Runyon, John H., Verdini, Jennefer, and Runyon, Sally S., eds., Source Book of American Presidential Campaign and Election Statistics: 1948–1968 (New York: Frederick Ungar, 1971), pp. 156–173. On Johnson's exception to the general pattern see Lamb, Karl A. and Smith, Paul A., Campaign Decision-Making: The Presidential Election of 1964 (Belmont: Wadsworth, 1968), pp. 202, 205–207.
25 To forestall someone from traversing several blind alleys that we entered, we note that the conjecture that increases monotonically with the number of replanning periods is true, but only under some severely restrictive conditions—conditions that seem impossible to express in any simple, substantively meaningful way.
26 For a discussion of the relevant statistical methodology and data requirements see Farley, John U., Hinich, Melvin J., and McGuire, Timothy W., “Testing for a Shift in the Slopes of a Multivariate Linear Time Series Model,” Journal of the American Statistical Association (forthcoming). If, however, we accept the assumptions of the statistical model outlines in footnote 21, then we can conduct a comparison of estimates of for the first five weeks of the campaign as against the last three weeks (using Kelley's data). Briefly, such a comparison reveals that, as the theory predicts, for either period is always less than our estimate for the overall campaign and that only one estimate of is significantly different from 1.0 (.39 for Lyndon Johnson in 1964—all other coefficients vary between .76 and 1.22). We repeat, though, that the assumption that Q is functionally independent of α, β, and γ is at best tenuous and is not consistent with our central analysis.
27 For an analysis of reversals and indeterminate outcomes see Melvin J. Hinich, Richard Mickelsen, and Peter C. Ordeshook, “The Electoral College vs. A Direct Vote: Policy Bias, Indeterminate Outcomes, and Reversals” in Journal of Mathematical Sociology, forthcoming and Mickelsen, and Ordesbook, , “The Electoral College and the Probability of Reversals” in Modeling and Simulation, 5 (Pittsburgh: Univ. of Pittsburgh Press, forthcoming).
* This research was supported by a National Science Foundation Grant to Carnegie-Mellon University. We also wish to thank Steven Brams and Morton Davis for the availability of their data, Stanley Kelley, Jr. who gave us his data on the estimated time candidates spent in each state, and Timothy McGuire and Melvin J. Hinich for their helpful suggestions.
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