¹ Brams, Steven J. and Davis, Morton D., “The 3/2's Rule in Presidential Campaigning,” American Political Science Review, 68 (March 1974), 113CrossRefGoogle Scholar.

² In spatial models of party competition, however, strategies are not resource allocations but positions on “issues,” and the constraint should be θ_i1, = θ_i2 = … = θ_i51. 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)Google Scholar.

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), 135–152CrossRefGoogle Scholar.

⁴ Gross, O. and Wagner, R., “A Continuous Colonel Blotto Game” (Santa Monica, Calif.: RAND Memo. #408, June 17, 1950)Google Scholar; Gross, O., “The Symmetric Blotto Game” (Santa Monica, Calif.: RAND Memo. #424, July 19, 1950)Google Scholar; Friedman, Lawrence, “Game-Theory Models in the Allocation of Advertising Expenditures,” Operations Research, 6 (Sept.-Oct. 1958), 699–709CrossRefGoogle Scholar. See also Sankoff, David and Mellos, Koula, “The Swing Ratio and Game Theory,” American Political Science Review, 66 (June, 1972), 551–554CrossRefGoogle Scholar.

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 Σ_εj, = 1. We do not, however, make this assumption. Rather, we let ΣT_j, be a random variable and assume that theε_j's are independently distributed.

⁸ An alternative procedure is to let δ_j, = .25 for all T_j. 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 δ_j, while others are not. Letting δ_j 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. 338Google Scholar.

¹² As with Brams and Davis's data, we add a δ_j, to those T_j,'s we observe as zero. We again choose δ_j = .25 since a quarter of an hour seems a sufficiently small rounding error. None of the coefficients, however, are especially sensitive to δ_j.

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 Pl_j, 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 E_j(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 P_ij'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), 137–160Google Scholar.

¹⁸ Clearly, though, if Pl_j, is a function of θ_ij, 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 Pl_i, to θ_ij is the stronger.

¹⁹ The adjusted R ²s with and without ∣Pl_j∣ are as follows:

Note, however, that these numbers are defined in terms of log T_j. To ascertain the variance accounted for in terms of T_j, 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 ∣Pl_j∣ 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. 553–570Google Scholar.

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 Q_j 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 T_j, becomes

where

and

Letting

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 X_j, into this expression, and set the derivatives of log L with respect to α, β, and β equal to zero. Note, however, that if Q_j 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 Q_j 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 E_j or ∣Pl_j∣ 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. 71Google Scholar.

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. 156–173Google Scholar. 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, 205–207Google Scholar.

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)Google Scholar. 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)Google Scholar.