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The 3/2's Rule in Presidential Campaigning*

  • Steven J. Brams (a1) and Morton D. Davis (a2)

Abstract

The purpose of this article is to assess the effect of the winner-take-all feature of the Electoral College on the allocation of resources by candidates to the states in a presidential campaign. Conceptualizing the campaign as a two-person zero-sum infinite game, it is found that the main effect of this feature is to induce candidates to allocate campaign resources roughly in proportion to the 3/2's power of the electoral votes of each state, which creates a peculiar bias that makes voters living in the largest states as much as three times as attractive campaign targets as voters living in the smallest states. Empirically, it is shown that the 3/2's rule explains quite well the time allocations of presidential and vice-presidential candidates in the 1960, 1964, 1968, and 1972 campaigns; for presidential campaigns in 1976 and 1980, optimal allocations are indicated for all fifty states and the District of Columbia. A comparison with optimal allocations under a system of direct popular-vote election of the president reveals that such a system would be less susceptible to manipulative strategies than the Electoral College as well as being compatible with the egalitarian principle of “one man, one vote.”

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*

This article is a condensation of a paper presented at the 1972 Annual Meeting of the American Political Science Association, Washington, D.C., September 5–9, 1972. The original paper includes, among other things, mathematical derivations and proofs, and a description of a sequential-strategy model, not included in this article. This back-up material has been published in Steven J. Brams and Morton D. Davis, “Resource-Allocation Models in Presidential Campaigning: Implications for Democratic Representation,” Annals of the New York Academy of Science (Proceedings of the Conference on Democratic Representation and Apportionment: Quantitative Methods, Measures and Criteria), Vol. 219, ed. Lee F. Papayanopoulos (New York: New York Academy of Science, 1973). We wish to acknowledge the very helpful comments made on an early formulation of the models by John C. Harsanyi, Anatol Rapoport, and Lloyd S. Shapley, and on the original paper by Walter Dean Burnham and Norman Nie, none of whom should be held responsible for the present analysis. We are also grateful to Roman Frydman, Elizabeth Gidengil, and Jeffrey Richelson for valuable research assistance.

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1 Congressional Record, September 8, 1970, p. 30813 .

2 In the Senate, the most recent hearings have been Electoral College Reform, Hearings before the Committee on the Judiciary, United States Senate, 91st Congress, Second Session (1970); in the House, Electoral College Reform, Hearings before the Committee on the Judiciary, United States House of Representatives, 91st Congress, First Session (1969).

3 Bickel, Alexander M., Reform and Continuity: The Electoral College, the Convention, and the Party System (New York: Harper and Row, 1971), p. 35 .

4 From a critical review of recent literature on campaigning, Gerald Pomper concludes that “in future research, more attention needs to be directed to the effects, rather than the characteristics, of campaigns.” See Pomper, Gerald M., “Campaigning: The Art and Science of Politics,” Polity, 2 (Summer 1970), 533539, at p. 539. Toward this end, mathematical models of the campaign process are developed in Ferejohn, John and Noll, Roger, “A Theory of Political Campaigning” (Paper delivered at the 1972 Annual Meeting of the American Political Science Association, Washington, D.C., September 5–9, 1972); and Kramer, Gerald H., “A Decision-Theoretic Analysis of a Problem in Political Campaigning,” in Mathematical Applications in Political Science, II, ed., Bernd, Joseph L. (Dallas: Southern Methodist University Press, 1966), pp. 137160 . In the Kramer article, a resource-allocation model is used to analyze the effects of different canvassing techniques on turnout and voting from the vantage point of one candidate—and not his opponent(s) directly, whose possible strategies our later game-theoretic models explicitly take into account; for an empirical test of the effects of canvassing in recent elections, see Kramer, Gerald H., “The Effects of Precinct-Level Canvassing on Voter Behavior,” Public Opinion Quarterly, 34 (Winter, 19701971), 560572 . The most explicit treatment of different strategic factors in a campaign, developed from a coalition-theoretic perspective, is Kessel, John H., The Goldwater Coalition: Republican Strategies in 1964 (Indianapolis: Bobbs-Merrill Co., 1968). A useful compilation of material on techniques of campaign management and communication and their effects on the electorate can be found in Nimmo, Dan, The Political Persuaders: The Techniques of Modern Election Campaigns (Englewood Cliffs, N.J.: Prentice-Hall, 1970); and for a collection of articles on new campaign methods, see Agranoff, Robert, ed., The New Style in Election Campaigns (Boston: Holbrook Press, 1972). A study of the use of electoral propaganda in Great Britain, whose practices are compared with those of the United States, is given in Rose, Richard, Influencing Voters: A Study of Campaign Rationality (London: Faber & Faber, 1967).

5 A recent summary of different proposals and a biased assessment (in favor of the present Electoral College) of their likely impact, considered especially in light of the three-way presidential contest in 1968, is given in Sayre, Wallace S. and Parris, Judith H., Voting for President: The Electoral College and the American Political System (Washington, D.C.: Brookings Institution, 1970). We find it difficult to accept Sayre and Parris's belief (p. 43) that “the political effects of the electoral college system are as clear as any in the nonexact science of American politics,” their “non-exact science” qualification notwithstanding. Other summaries that reflect a similar bias in favor of the Electoral College include Bickel, op. cit., pp. 4–36; and Polsby, Nelson W. and Wildavsky, Aaron, Presidential Elections: Strategies of American Electoral Politics, 3d ed. (New York: Charles Scribner's Sons, 1971), pp. 258271 . On the other side, a report by the American Bar Association's Commission on Electoral College Reform has called the popular-vote plan “the most direct and democratic way of electing a President.” See American Bar Association (ABA), Electing the President: A Report of the Commission of Electoral College Reform (Chicago: ABA, 1967). Also supportive of the direct-vote plan is Pierce, Neal R., The People's President: The Electoral College in American History and the Direct-Vote Alternative (New York: Simon and Schuster, 1968); Longley, Lawrence D. and Braun, Alan G., The Politics of Electoral College Reform (New Haven, Conn.: Yale University Press, 1972); and Zeidenstein, Harvey, Direct Election of the President (Lexington, Mass.: Lexington Books, 1973). See also John H. Yunker and Lawrence D. Longley, “The Biases of the Electoral College: Who Is Really Advantaged?” and Power, Max S., “Logic and Legitimacy: On Understanding the Electoral College Controversy,” both in Perspectives on Presidential Selection, ed. Matthews, Donald R. (Washington, D.C.: Brookings Institution, 1973).

6 Congressional Record, September 18, 1969, pp. 2600726008 .

7 Congressional Quarterly Almanac, Vol. 26, 91st Congress, 2nd Session (Washington, D.C.: Congres-sional Quarterly, 1971), p. 840 . For further details, see Sindler, Allan P., “Basic Change Aborted: The Failure to Secure Direct Popular Election of the President, 1969–1970” in Policy and Politics in America: Six Case Studies, ed. Sindler, Alan P. (Boston: Little, Brown, and Co., 1973), pp. 3080 .

8 Sayre and Parris, p. 15.

9 Flanigan, William H., Political Behavior of the American Electorate, 2d ed. (Boston: Allyn and Bacon, 1972), p. 109 .

10 Key, V. O. Jr., with the assistance of Cummings, Milton C. Jr., The Responsible Electorate: Rationality in Presidential Voting, 1936–1960 (Cambridge, Mass.: Belknap Press, 1966). For a general discussion of the role of issues in presidential elections, see the articles, comments, and rejoinders by Pomper, Gerald M., Boyd, Richard W., Brody, Richard A., Page, Benjamin I., and Kessel, John H., American Political Science Review, 66 (June, 1972), 415470 . The voting decisions of ticket-splitters, in particular, who comprised 54 percent of the American electorate in the 1968 election, are influenced primarily by the candidates and issues, and only secondarily by party identification and other group affiliations; see De Vries, Walter and Tarrance, Lance Jr., The Ticket-Splitter: A New Force in American Politics (Grand Rapids, Michigan: William B. Eerdmans Publishing Co., 1972).

11 Downs, Anthony, An Economic Theory of Democracy (New York: Harper and Row, 1957). For a review of the more recent literature on party-competition models, see Riker, William H. and Ordeshook, Peter C., An Introduction to Positive Political Theory (Englewood Cliffs, N.J.: Prentice-Hall, 1973), chaps. 11 and 12; and Shepsle, Kenneth A., “Theories of Collective Choice,” in Political Science Annual V: An International Review, ed. Cotter, Cornelius P. (Indianapolis: Bobbs-Merrill Co., forthcoming, 1974).

12 For a related application of a game-theoretical model to the analysis and evaluation of election procedures in the American Political Science Association, see Brams, Steven J., “The APSA and Minority Representation,” PS, 3 (Summer 1970), 321335 .

13 As another possible goal, Stanley Kelley, Jr. has suggested that “… at least some of the Goldwater inner circle set control of the Republican party—not winning the Presidency—as their principal objective in 1964. That is the implication, certainly, of Senator Goldwater's statement that the conservative cause would be strengthened if he could win as much as 45 percent of the vote.” Kelley, Stanley Jr., “The Presidential Campaign,” in The National Election of 1964, ed. Cummings, Milton C. Jr. (Washington, D.C.: Brookings Institution, 1966), p. 58 . Further, there is evidence that Goldwater, who was well aware of his impending defeat from polls commissioned by the Republican National Committee, did little to try to stem its magnitude in the latter half of his campaign (which would be consistent with the goal of winning) but instead tried to rationalize the conduct of his campaign and the anticipated action of the voters. See Shadegg, Stephen C., What Happened to Goldwater? (New York: Holt, Rinehart and Winston, 1965), p. 241 .

14 Senate Hearings on Electoral College Reform (1970), p. 82.

15 Senate Hearings (1970), pp. 82–83.

16 This model leads to the same results as one in which the quantity maximized is the expected plurality of uncommitted voters:

See Epstein, Richard A., The Theory of Gambling and Statistical Logic (New York: Academic Press, 1967), pp. 121123 . Note that the summations which follow range only over the fifty states, though in presidential elections beginning in 1964 the District of Columbia must also be included.

17 In Colonel Blotto games—where the candidate who outspends his opponent in a state by whatever amount wins that state with certainty, in contrast to the probabilistic relationship that we assume between expenditures and winnings—a minimax solution in mixed strategies has been found when all states have the same number of electoral votes. See Sankoff, David and Mellos, Koula, “The Swing Ratio and Game Theory,” American Political Science Review, 56 (June, 1972), 551554 ; and for a discussion of related Colonel Blotto games, see Friedman, Lawrence, “Game Theory Models in the Allocation of Advertising Expenditures,” Operations Research, 6 (Sept.-Oct., 1958), 699709 . For a non-mathematical description of pure and mixed strategies, see Davis, Morton D., Game Theory: A Nontechnical Introduction (New York: Basic Books, 1970), pp. 1948 .

18 Kelley, , in Cummings, , The National Election of 1964, pp. 50–51, 75 .

19 Whenever we use the term “unstable,” we mean “globally unstable,” for this point is stable locally (i.e., is impervious to small deviations).

20 If the total resources of the candidates are not equal (i.e., R ≠ D), it is not difficult to show that a “proportionate matching” of expenditures, wherein the candidates spend the same proportion (or percentage) of their total resources in each state, yields a somewhat more complicated expression for a local maximum.

21 James S. Coleman has developed a measure of power based on this term, which here gives the probability of an exactly even split of the uncommitted voters in a state. (Note that “π” in the expressions in this section refers to the number 3.14159, not the probability we defined earlier.) See Coleman, James S., “Loss of Power,” American Sociological Review, 38 (February 1973), 117 .

22 Cf. the similar findings in Banzhaf, John P. III, “One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College,” Villanova Law Review, 13 (Winter 1968), 304332 . Banzhaf's analysis of power is similar to that of Coleman, op. cit., but the correctness of Banzhaf's numerical calculations of voting power in the Electoral College has recently been contested in Longley, Lawrence D. and Yunker, John H., “The Changing Biases of the Electoral College” (Paper prepared for delivery at the 1973 Annual Meeting of the American Political Science Association, New Orleans, Louisiana, September 4–8, 1973). While Banzhaf bases his analysis on the concept of a “critical vote,” which can occur only if there is a 50–50 split, our concept of “decisiveness” takes into account other possible divisions of the vote.

23 See Lamb, Karl A. and Smith, Paul A., Campaign Decision-Making: The Presidential Election of 1964 (Belmont, Calif.: Wadsworth Publishing Co., 1968).

24 The population of a state, of course, is not an exact reflection of the proportion of the voting-age population who are registered and actually do vote in a presidential election. Since it is not at all clear whether and how the proportion of uncommitted voters in a state is related to differences in registration and turn-out among the states, we have taken the simplest course of using population as a first-approximation estimate of the proportion of voters likely to be uncommitted at the start of a campaign. This assumption can, of course, be modified at a later time.

25 The New York Times, January 31, 1972, p. 48 .

26 The New York Times, November 19, 1972, p. 1 .

27 Computed from figures given in Dunn, Delmer D., Financing Presidential Campaigns (Washington, D.C.: Brookings Institution, 1972), p. 33 .

28 The New York Times, April 9, 1972, p. 44 .

29 Federal Communications Commission (FCC), Survey of Political Broadcasting: 1960, 1964, 1968 (Washington, D.C.: FCC, April 1961, July 1965, August 1969), which include data on both primary and general election campaigns.

30 The most comprehensive source of this information for the 1968 election is Alexander, Herbert E., Financing the 1968 Election (Lexington, Mass.: D. C. Heath and Co., 1971).

31 After collecting our data we learned of the data on the itineraries of presidential candidates published in 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 Publishing Co., 1971), pp. 139173 . This work contains no data on the itineraries of vice-presidential candidates, however, whose campaign appearances we have combined with the appearances of presidential candidates in the data analyzed below.

32 We are grateful to Stanley Kelley, Jr. for furnishing us with his data, which he has analyzed in his articles entitled The Presidential Campaisn,” in The Presidential Election and Transition, ed. David, Paul T. (Washington, D.C.: Brookines Institution, 1961), pp. 5787 ; and in Cummings, , The National Election of 1964, pp. 4281 .

33 Computed from data given in Kelley, , in David, , The Presidential Election and Transition, p. 72 .

34 We must also note that the 3/2's rule works less well than the proportionate rule (correlation coefficients of 0.84 and 0.87, respectively) in the case of the only major third-party presidential bid since 1948, that by George Wallace's American Independence Party in 1968, which captured 14 per cent of the popular vote and 9 percent of the electoral vote. However, Wallace and his running mate, General Curtis LeMay, did not campaign in nearly one-third of the fifty states, which throws doubt on the substantive significance of correlations based on all fifty states.

35 In the context of voting in the Electoral College, Mann and Shapley demonstrated that the Shapley-Shubik index of pivotalness for each state produces only a slight bias in favor of the largest states. See Mann, Irwin and Shapley, L. S., “The A Priori Voting Strength of the Electoral College,” in Game Theory and Related Approaches to Social Behavior, ed. Shubik, Martin (New York: John Wiley & Sons, 1964), pp. 151164 . For an elementary discussion of the Shapley value on which the Shapley-Shubik index is based, see Davis,

36 The Democrats reportedly wrote off twenty-four states in 1968, though our data indicate that one or both of the Democratic candidates made appearances in more than half of these states. See Napolitan, Joseph, The Election Game and How to Win It (Garden City, N.Y.: Doubleday and Co., 1970), p. 62 .

37 Key, V. O. Jr., Southern Politics in State and Nation (New York: Alfred A. Knopf, 1949), p. 122 . We are indebted for this citation to Hinich, Melvin J. and Ordeshook, Peter C., “The Electoral College: A Spatial Analysis” (Paper, Carnegie-Mellon University, February, 1973), in which the effect of the Electoral College on equilibrium strategies, given different distributions -of policy preferences of the electorate, are examined.

38 Scammon, Richard M. and Wattenberg, Ben J., The Real Majority: An Extraordinary Examination of the American Electorate (New York: Coward-McCann, 1970), p. 213 .

39 According to a series of Gallup polls, less than 6 per cent of all people saw any one of the presidential candidates in person in 1968. See Davidson, Sara, “Advancing Backward with Muskie,” Harper's, June 1972, p. 61 .

40 Scammon, and Wattenberg, , The Real Majority, pp. 214216 . Moreover, television news shows, documentaries, and other specials, which are generally beyond the direct control of candidates, rank far above television advertising as the most important media influences, at least for the split-ticket voter. See De Vries, and Tarrance, , The Ticket-Splitter, p. 78 .

41 Scammon, and Wattenberg, , The Real Majority, p. 217 . Italics in original.

42 At the local level, party organization activities do appear to influence election results favorably, including the vote for president. See Crotty, William J., “Party Effort and Its Impact on the Vote,” American Political Science Review, 65 (June 1971), 439450 , and references cited therein; also, Huckshorn, Robert J. and Spencer, Robert C., in The Politics of Defeat: Campaigning for Congress (University of Massachusetts Press, 1971), stress the importance of campaign organization on election results. At the national level, on the other hand, there is no clear-cut relationship between the campaign spending of each party and the outcome of presidential elections. See Twentieth Century Fund, Voters' Time: Report of the Twentieth Century Fund Commission on Campaign Costs in the Electronic Era (New York: Twentieth Century Fund, 1969), pp. 1113 ; Congressional Quarterly, Dollar Politics: The Issue of Campaign Spending (Washington, D.C.: Congressional Quarterly, 1971), pp. 14, 1931 ; and Dunn, , Financing Presidential Campaigns, p. 9 . More opaque, still, is the relationship between major strategic decisions made by candidates, especially during the heat of a close race, and their effect on the vote. See Polsby, and Wildavsky, , Presidential Elections, pp. 199206 .

43 That the organization and planning of presidential campaigns is critical to a candidate's eventual success is stressed in Bruno, Jerry and Greenfield, Jeff, The Advance Man (New York: William Morrow & Co., 1971).

44 Polsby, and Wildavsky, , Presidential Elections, p. 183 . If the steeliness of a candidate's nerves can be judged by this standard, Richard Nixon's nerves progressively hardened in his successive bids for the presidency: as an incumbent president in the 1972 campaign, he made only 18 campaign appearances, while as a nonincumbent in 1968 and 1960 he made 140 and 228 appearances, respectively. Yet, though he cut down his number of appearances in successive campaigns, in all three races Nixon made disproportionately greater allocations to the large states than to the medium and small states, as did the entire Republican state in each of these campaigns (see Table 3).

45 Kingdon, John W., Candidates for Office: Beliefs and Strategies (New York: Random House, 1968), pp. 109114 . See also Schoenberger, Robert A., “Campaign Strategy and Party Loyalty: The Electoral Relevance of Candidate Decision-Making in the 1964 Congressional Elections,” American Political Science Review, 63 (June, 1969), 515520 .

46 Brown, Richard H., “Theory of Combat: The Probability of Winning,” Operations Research, 11 (May-June 1963), 418425 .

47 For a discussion of literature in this area, see Brams, Steven J., “Positive Coalition Theory: The Relationship between Postulated Goals and Derived Behavior,” in Political Science Annual IV: An International Review, ed. Cotter, Cornelius P. (Indianapolis: Bobbs-Merrill Co., 1973), pp. 340 .

* This article is a condensation of a paper presented at the 1972 Annual Meeting of the American Political Science Association, Washington, D.C., September 5–9, 1972. The original paper includes, among other things, mathematical derivations and proofs, and a description of a sequential-strategy model, not included in this article. This back-up material has been published in Steven J. Brams and Morton D. Davis, “Resource-Allocation Models in Presidential Campaigning: Implications for Democratic Representation,” Annals of the New York Academy of Science (Proceedings of the Conference on Democratic Representation and Apportionment: Quantitative Methods, Measures and Criteria), Vol. 219, ed. Lee F. Papayanopoulos (New York: New York Academy of Science, 1973). We wish to acknowledge the very helpful comments made on an early formulation of the models by John C. Harsanyi, Anatol Rapoport, and Lloyd S. Shapley, and on the original paper by Walter Dean Burnham and Norman Nie, none of whom should be held responsible for the present analysis. We are also grateful to Roman Frydman, Elizabeth Gidengil, and Jeffrey Richelson for valuable research assistance.

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