Efficient voter distributions under the electoral college are those that allow a presidential voting coalition to capture an electoral majority with less popular support than its opposition. Efficient distributions for particular parties are produced by the impact of the electoral apportionment combined with the impact of asymmetric partisan pluralities. The latter is the basis of gerrymandering and will be referred to as the gerrymander bias, although the bias obviously cannot be attributed to legislative manipulation. Figure (3) represents the most extremely efficient distribution of a coalition based on the gerrymander bias with no apportionment bias. The voter coalition defined by the shaded area has a cross-state appeal that just gains the electoral majority line and a within-state

appeal that just gains the necessary segment of the popular majority line. With no malapportionment, this winning voter coalition forms a square segment embracing only a quarter of the nation's voters. Figure (3) clearly represents the most extreme gerrymander with no apportionment bias, because of the extreme asymmetry of one party carrying its states by virtually zero percent pluralities, while the other party carries its states by 100 percent pluralities. Although the gerrymander bias is precisely the aspect of "efficiency" that Best argues deserves special electoral weight, the electoral rules obviously do not favor the more broadly based coalition in figure (3).

Although no voter coalition could plausibly achieve the extremely efficient asymetrical distribution depicted in figure (3), it is now apparent that a winning coalition under the present system need only embrace one of the four sections of the electorate defined by the electoral and popular majority lines in two-coalition competition. Votes not necessary to carry a section are "wasted" and undermine the efficiency of the coalition's conversion of popular support into electoral strength. Coalitions cannot expect to neatly embrace an essential section of the electorate because the underlying preference cleavages of the electorates must limit the alliances that political leaders can consruct. In terms of the geometric scheme, coalitions cannot formulate appeals that bend sharply around the intersection of the popular and electoral majority lines to achieve maximum advantage of asymmetrical distributions.

A less efficient but more plausible distribution may be defined by an arc that contains an essential segment of the electorate. In figure (4), this are has a radius extending from the lower left corner to the intersection of the two bisectors. This radius is about .7 by the Pythagorean theorem. Because the arc defines a winning coalition occupying a quarter of a circle, simple geometry shows that the quadrant represents slightly under 40 percent of the electorate." This indicates that the favored coalition can win the Presidency against a coalition of 60 percent of the nation's voters. The favored coalition represented by the shaded area in Figure (4) is clearly more broadly based than the favored coalition in Figure (3). It has some support in states with 70 percent of the nation's voters and is supported by about 70 percent of the state voters in its strongest states. However, the losing coalition in figure (3) is more broadly based in terms of both its greater vertical and horizontal reach. The crucial standoff in the reach of the two coalitions appears along the secondary diagonal (lower left to top right).

Because a winning coalition must extend its support beyond the intersection of the two bisectors, the vote in the surrounding state segment(s) would appear pivotal. This accords with the notion that because the competitive states “decide” presidential elections, campaigns generally concentrate their attention on these "doubtful" states. This principle is often cited without full appreciation that the basic relative appeals of tickets determine the distribution of popular support that makes state(s) pivotal. One the basic structure of competitive coalitions is established, campaign managers may seek marginal increments of support in the doubtful states. Thus one would predict that candidate itineraries, special parochial appeals, vote-buying, and election irregularities would be concentrated disproportionately in likely pivotal states where marginal gains might be decisive." Carrying pivotal states by narrow margins makes a coalition more representative only in the limited sense of representing popular majorities, however slim, in states containing a majority of the voters, assuming no apportionment biases. With apportionment biases, the winning coalition can only claim to

The square of the hypotenuse of a right triangle equals the sum of the squares of the two short sides; thus if C is the hypotenuse and A and B the short sides, then the radius, C=(A2+B2) = ( (.5)2 + (.5)2)%=(.5)%, which is about .7.

10 The area of a quadrant is wr2/4=(3.14) (.5) (.25)=.39.

11 This analysis differs from that of James MacGregor Burns and others who suggest that the voters in the doubtful states force presidential aspirants to shape their policy commitments to appeal disproportionately to the doubful state voters. Credence is lent to this proposition by the observations that presidential campaigns devote disproportional attention to doubtful states, and a coalition would gain strength to the extent that it could augment is support in critical states even at the expense of nonessential support elsewhere. However, the underlying structure of preferences among the nation's voters limits the ability of coalitions to tailor their appeals to create a favorable distribution of support. Moreover, the concentration of campaigning in critical states is in pursuit of incremental strength, precisely because the competitive states can not be identified until the basic appeals of the tickets are known. Thus, my analysis assumes that basic appeals of the tickets shape the distribution of the coalitions they mobilize, rather than placing special weight on the policy preferences of voters in competitive states. James MacGregor Burns, The Deadlock of Democracy (Englewood Cliffs, NJ: Prentice-Hall, 1963), pp. 302306.

represent majorities in states containing a majority of the electoral votes, which may not be representative of the nation in terms of either the voters or the populace.

Defenders of the electoral college do not envision biases so extreme as to allow 40 percent of the voters to defeat a coalition of 60 percent. Indeed, they justify the electoral college's deviation from the majoritarian principle by citing Robert Dahl's proposition that "the closer a group approaches to an equal division, the less valid the majority principle becomes." " Thus, although efficiency is said to deserve special electoral weight, it is assumed that efficiency will not get out of hand.

Less extreme gerrymander distributions for electorally competitive coalitions may be represented by flatter arcs that reach the intersection of the two bisectors. The flatter the arc, the more it approaches the principal diagonal, one of the lines that represents voter coalitions reaching electoral majorities just when they reach popular majorities. The shaded area depicted in figure (5) clearly represents the voters throughout the nation better than the shaded areas in the previous two figures where electoral majorities were reached with far less than popular majorities. Significantly, figure (5) displays a distribution of partisan pluralities with wasteful pluralities for one coalition off-set by equally wasteful pluralities for the opposition. If the more symmetrical partisan distributions that balance out each party's wasteful vote margins produce better representation of the nation's voters, then "efficient" distributions based on asymmetrical pluralities would not seem to deserve extra electoral weight.

This geometric analysis also serves to show that the electoral college rules do not foster the bridging of regional or class cleavages better than direct election. Without apportionment or gerrymander biases, both the electoral college and direct election are equally responsive to class and regional cleavages between two coalitions. Consider the partisan division in figure (5) which cuts diagonally across the electorate, indicating neither a sectional (potentially regional) nor a transnational (potentially class) cleavage. Rotating the line of cleavage in figure (5) on its midpoint describes other coalitions equally competitive under either the electoral or popular systems, but more sectional or transnational in their compositions. When the line of cleavage is rotated toward the vertical axis, it approaches perfect homogeneity of the state's support for their preferred ticket as depicted in figure (6). When the line is rotated toward the horizontal axis, it approaches the perfect intra-state competition as depicted in figure (7).

The slope of the line demarking the competing coalitions identifies the relative sectional, as opposed to transnational, voting. Because the cleavage line in these cases displays each coalition as including both the half of the voters essential to a popular election and a quarter segment essential to an electoral victory, both systems are equally susceptible to either sectional or transnational class voting in two-party competition when no distributional biases are present. In the absence of symmetrical pluralities, however, a winning electoral coalition can be based on a majority class within a majority section, with little or no support outside that section and class. Direct election insures that when a coalition's supporters are largely limited to one section, it must have extraordinary crossclass support within its section, or, alternatively, when its supporters are largely limited to one class it must have broad, cross-sectional support to reach a popular majority. Therefore, direct election is more constrained to be broadly representative than is the electoral college system.

The electoral college will tend to magnify the size of a winning coalition's electoral majority according to the extent of transnational as opposed to sectional voting, although apportionment and gerrymander biases exert confounding influences. With the extreme sectional voting depicted in figure (6), the coalitions will win electoral votes proportional to their share of the nation's popular votes. But as the voting approaches the transnational pattern of figure (7), the leading popular coalition will control a disproportional share of the electoral votes, and in the extreme case, control of a bare popular majority will win all the electoral votes. Thus, the recognized tendency of the electoral college to magnify the winning coalition's national popular votes into proportionally greater electoral strength depends on the voting cleavage being predomi

12 Robert Dahl, A Preface to Democratic Theory (Chicago: Chicago University Press. 1956), p. 41. Dahl's pronositions about democracy are cited in defense of the electoral college by Best, pp. 51, 77, and by Max Power, "Logic and Legitimacy: On UnderstandIng the Electoral College Controversy," in Donald R. Matthews (editor). Prespectives on Presidential Selection (Washington, D.C.: Brookings Institution. 193). pp. 209-211. Bickel also asserts that defeat of majority coalitions by narrow margins is not of consequence, Reform and Continuity, p. 31.

nantly transnational." Winning the presidency, however, does not depend on the magnitude of electoral majorities, but on winning a majority by mobilizing a coalition embracing an essential segment of the electorate. The electoral college is not a deliberative body, and therefore numbers of electors won by a coalition beyond a majority fail to contribute anything to that coalition's political power. Because the magnification effect confers no additional power on the coalition with an electoral majority, it cannot encourage transnational coalition formation beyond securing the essential segment of the voters.

The conclusion that the electoral college fails to outperform direct election in encouraging coalitions across either class or sectional divisions is further strengthened by considering apportionment biases. In figure (8), the electoral majority line is off-center from the vertical bisector of the electorate so that states with 40 percent of the voters reach the electoral majority line." This means that one coalition can win with a very narrow popular vote base because its essential segment is only 20 percent of the voters, as depicted by the shaded area of Figure (8), while the opposition requires 30 percent. When there is no gerrymandering bias permitting a coalition to win with anything approaching these narrow bases of support, the coalition favored by the apportionment will need between 40 and 50 percent of the voters to control the electoral majority. The favored coalition will need about 40 percent of the electorate when the coalitions approach the solidly sectional voting pattern depicted in figure (9), while the favored coalition will need about 50 percent of the votes when the coalitions approach the purely transnational pattern depicted in figure (10). Direct election would eliminate the electoral apportionment which can favor one coalition according to the extent of sectional voting. Some political observers might argue that states with small populations or low voter turnout rates deserve special electoral weight to offset their small portion of the electorate, but these observers should note that the impact of the apportionment bias is diminished to the extent of transnational as opposed to sectional voting.

Even with sectional voting, the equal electoral vote line will not be far off center when there is no strong tendency for small or low voting states to associate with one particular party. Most defenders of the electoral college, moreover, cannot rest their case on possible benefits for the small states because they appear to believe that the rules actually create a large state bias," and few would

13 The relative class versus sectional voting is of tremendous political importance. The extent of transnational as opposed to sectional voting appears to fit into the mathematical logic of Rein Taagepera's "n-power law of elections." The more heterogeneous the elec toral districts, the more competitive the transnational coalitions within the districts, and the larger the magnification of popular pluralities in the electoral count. Malapportionment and gerrymandering would exert disturbing influences on this relationship. Rein Taagepera, "Seats and Votes: A Generalization of the Cube Law of Elections," Social Science Research, v. 2 (September, 1973), pp. 257–275.

14 The representation of a coalition of states with 40 percent of the nation's voters carrying an electoral majority, although arbitrary for purposes of illustration, is not at all incredible. If states with popular election of presidential electors in 1860 were ordered according to the ratio of their popular to electoral vote strengths, the 12 states with the most popular votes per elector would barely control a majority of the popularly chosen electors while casting about 68 percent of the popular ballots. The coalition of states with the least popular votes per elector could control an electoral majority with 35 percent of the popular votes. This bias had some political relevance because none of the 12 disadvantaged states supported secession in 1861, all but three voted for the Republican ticket, and all but one cast their electoral votes for either Lincoln or Douglas, the northern based candidates. 15 The faith in a large state bias developed because the large states containing most of the nation's voters tend to be competitive in presidential elections when the national contest is competitive. The outcome of the competitive contest for large blocs of electoral votes is obviously critical. Although the logical casual order would be that the competitiveness of states depends on the appeal of partisan tickets, campaign managers rationally seek marginal popular vote gains in the large states where the outcome of the contest for large blocs of electors is doubtful. Scholars infer that the large states are disproportionally powerful under the electoral college. All that was lacking was "scientific" demonstration of the large state bias. Many scholars assume this was supplied by the electoral power calculations associated with John F. Banzhaf. These calculations rest on the "heroic" game theory assumption that voter coalitions are randomly formed, with each voter's choice between two tickets constituting a statistically independent event. This would predict extremely competitive "voting" within states, making all sates doubtful. Based on these assumptions, Banzhaf calculates the relative probabilities from state to state that an individual "citizen voter" would cast a decisive ballot, swaying not only his state but the national election. These calculations show that the individual voters in the large states have a statistically greater chance to cast a decisive vote than voters in the small states. Students of politics are intrigued by these calculations without adequate recognition of how restrictive the assumptions are. The bias identified by Banzhaf does not appear in the geometric scheme, because I have specified coalitions based on possible underlying social cleavages in the electorate rather than the random voting assumption which is irrelevant to the conscious construction of voter coalitions. Nevertheless, the leading defenders of the electoral college who assume a large state bias are precluded from arguing that the electoral college is useful for the small states. John F. Banzhaf III, "One Man. 3.312 Votes." Villanova Law Review, v. 13 (Winter. 1968), p. 303. The Banzhaf effect is also at the heart of the analyses presented by Guillermo Owen. "Evaluating of a Presidential Game." American Political Science Review, v. 69 (September 1975), pp. 947-953. and Stephen J. Brams and Morton D. Davis. "The 3/2 Rule of Presidential Campaigning." American Political Science Review, v. 68 (March 1974), 113-134.