One Man,? Votes: Mathematical Analysis of Voting Power and Effective Representation*

Transcription

1 One Man,? Votes: Mathematical Analysis of Voting Power and * JOHN F. BANZHAF III-- In order to measure the mathematical voting power... it would be necessary to have the opinions of experts based upon computer analyses.' To paraphrase a popular thought, "figures don't lie, but lawyers don't figure." Unfortunately this is all too true in the area of reapportionment. The Supreme Court entered the reapportionment area amid warnings that it was getting the courts into a mathematical quagmire. Little did the Court know that its decisions would stimulate the heretofore untapped mathematical ingenuity of politicians' who, in an effort to avoid simple systems of equally populated * This article is based in large part upon studies previously reported by the author in: Weighted Voting Doesn't Work: A Mathematical Analysis, %9 Rutgers L. Rev. 317 (1965); Multi-Member Electoral Districts-Do They Violate the "One Man, One Vote" Principle, 75 Yale L.J (1966); and One Man, Votes, A Mathematical Analysis of the Electoral College, 13 Villanova L. Rev. 303 (%968); see also Mathematics, Voting and the Law: The Quest for Equal, prepared for presentation at the International Conference on Mathematical Theory of Committees and Elections, Institute for Advanced Studies and Scientific Research, Vienna, June 26-28, Portions of these works have been included in this article with the kind permission of the editors. Readers who would like additional information or a fuller exposition are referred to the originals. B.S.E.E., M.I.T.; LL.B., Columbia Law School. Currently serving as Executive Director, Action on Smoking and Health (ASH), 777 United Nations Plaza, New York City Sept. %968: Associate Professor of Law, National Law Center, George Washington University. i. Iannucci v. Board of Supervisors, 2o N.Y.2d 244, , 229 N.E.2d 195, %98-99, 282 N.Y.S.ad 502, (1967). 2. Earlier in this century the legislature of one of the more backward southern states came very close to enacting a law which would have established 3 as the "official" state value of pi (actually about ) for the convenience of its citizens. Fortunately for all of the circular structures thereafter constructed, the measure was narrowly defeated. May 1968 Vol. 36 No. 4 HeinOnline Geo. Wash. L. Rev

2 One Man-One Vote and Local Government THE GEORGE WASHINGTON LAV REVIEW single-member districts, would develop straight weighted voting, modified weighted voting, fractional voting, multi-member electoral districts, floterial districts and mixed plans employing two or more of these devices. Simple as these plans may have appeared to those who enacted them, large cracks and even untenable foundations appear when they are examined with the cold light of mathematical analysis. If the analyses appear complicated, it is the fault of the politicians who refuse to enact simpler plans, not of the mathematicians who must struggle to make sense of what others have uncomprehendingly created. By using a technique which is recognized and generally accepted in the fields of mathematics 3 and political science, 4 and which is gaining judicial acceptance, it is possible to measure the voting power of representatives and to determine the effects of these plans on the voting power and effective representation of the citizen-voters whose rights must be protected. In many cases the results show that the plans deprive voters of their constitutionally protected rights. The Measure of Voting Power The technique for the measurement of voting power is based on the simple and almost self-evident proposition that the purpose of any voting system is to allow each voting member some chance, however small, to affect the decisions that must be made. It can be demonstrated by resort to common experience that different methods of voting can vary the effectiveness of the vote of a given individual. In some cases, such as the election of a dub president where each member may cast one vote, each member has the same chance 3. See, e.g., Kemeny, Snell & Thompson, Introduction to Finite Mathematics 74-76, ios-ro (1957); Riker, A Test of the Adequacy of the Power Index, 4 Behavioral Sci. 120 (1959); Shapley, Simple Games: An Outline of the Descriptive Theory, 7 Behavioral Sci. 59 (1962); Shapley, Solution of Compound Games, in Advances in Games Theory 267 (1964); Shapley, A Value for N-Person Games, 2 Annals of Mathematical Studies 307 (1953). For reports of mathematical applications of this technique to one aspect of the Electoral College, see Mann & Shapley, The A Priori Voting Strength of the Electoral College, in Game Theory and Related Approaches to Social Behavior 151 (Shubik ed. 1964); Mann & Shapley, Value of Large Games VI: Evaluating the Electoral College Exactly (RAND Corp. Memo. RM PR, May %962); Mann & Shapley, Value of Large Games IV: Evaluating the Electoral College by Monte Carlo Techniques (RAND Corp. Memo. RM-265%, Sept. i9, 196o). 4. See, e.g., David, Goldman, Bain, The Politics of National Party Conventions (196o); Schubert, Judicial Behavior (1964); Schubert, Quantitative Analysis of Judicial Behavior, ch. 4 (1959); Riker, Bargaining in a Three-Person Game, Sept. 6-1o, 1966 (paper delivered at 1966 Annual Meeting of the American Political Science Association); Krislov, Power and Coalition in a Nine-Man Body, 6 Am. Behavioral Sci. 24 (April 1963); Krislov, Theoretical Attempts at Predicting Judicial Behavior, 79 Harv. L. Rev (1966); Riker, Some Ambiguities in the Notion of Power, 58 Am. Pol. Sci. Rev. 341 (1964). HeinOnline Geo. Wash. L. Rev

3 to affect the outcome, and all members obviously have equal voting power. In other situations, such as a stockholders' meeting where members cast votes in proportion to their stock holdings, the ability of the voting members to affect the outcome is not equal and they do not have equal voting power. In any voting situation it is possible to consider all the possible ways in which the different voters could vote, i.e., to imagine all possible voting combinations. One then asks in how many of these combinations can each voter affect the outcome by changing his vote. Since a priori, all voting combinations are equally likely and therefore equally significant, the number of combinations in which each voter can change the outcome by changing his vote serves as the measure of his voting power. In other words, no one can say which voting combinations will occur most often, or which combinations will predominate when more important issues are involved. The most a legislature or judge can do to equalize voting power is to satisfy himself that the system allows each voting member an opportunity to affect the outcome in an equal number of equally likely voting combinations. A person's voting power, then, is measured by the fraction of the total number of possible voting combinations in which he can, by changing his vote, alter the outcome of the group's decision. To be more precise, the ratio of the voting power of voter X to the voting power of voter Y is equal to the ratio of the number of possible voting combinations in which X could alter the outcome by changing his vote (assuming that no other voters change their votes) to the number of possible voting combinations in which Y could alter the outcome by changing his vote (also assuming no other voters change their votes).' It is important to recognize that this technique measures the individual voting power which is inherent in the rules governing the voting system and the distribution of population; it does not reflect the actual ability of any given voter to affect the outcome of a particular election. The latter would depend to some extent on factors which are not inherent in the system, such as the relative power of the political parties in different geographical areas, and conditions which may be peculiar to the voter himself; e.g., whether as a sign of protest he decides to vote for a minority party candidate who has no chance of winning. Thus, a critical distinction must be drawn between inequalities in voting power that are built into the system-e.g., the old county unit system in Georgia or the distribution of electoral votes in the Electoral Collegeand those which result either from the free choice among citizens as to how they use their voting power-e.g., the political impotence of a Republican in a solidly Democratic district-or from factors outside of the legal rules governing the process-e.g., voter intimidation, weather, the televised prediction of election results. Concededly, these and other external factors may affect a citizen's ability to affect the outcome of an election, and, therefore, the theo- 5. This method of measuring voting power is discussed more fully in articles cited note * supra. HeinOnline Geo. Wash. L. Rev

4 One Man-One, Vote and Local Government THE GEORGE WASHINGTON LAW REVIEW retical voting power of an individual may not coincide with his actual ability to affect the outcome of any particular election. The voting power measured here is that inherent in the system and necessarily represents an average of a voter's effectiveness in a large number of equally likely voting situations. However, only these inequalities, which result from the rules of a particular system of voting, may properly be considered in determining the basic "fairness" of the system itself. The technique for measuring voting power by calculating the number of opportunities each voter has to affect the decision has been generally accepted in the fields of mathematics 6 and political science. 7 Moreover, the technique's acceptance has been demonstrated through its use in analyzing voting power in Congress,' stockholders' meetings, 9 the French Assembly, 10 New York's Board of Estimate" and weighted voting situations in general,' including the New Jersey Senate,' 3 Nassau County, New York,' 4 and in multi-member districts. 15 In several cases courts have admitted computer analyses based on this technique and have held the weighted voting plans unconstitutional because of hidden inequalities uncovered by the studies. 16 Very recently, New York State's highest court, adopting some of the author's arguments as articulated in an amicus curiae brief, ruled against two weighted voting plans, and held that such plans must be subjected to a mathematical-computer analysis. 17 This decision is of particular interest for several reasons. First, it is persuasive because it comes from one of the most highly regarded state courts in the country. It is all the more persuasive because the court has gained 6. See note 3 supra. 7. See note 4 supra. 8. Riker & Niemi, The Stability of Coalitions on Roll Calls in the House of Representatives, 56 Am. Pol. Sci. Rev. 58 (1962); Shapley & Shubik, A Method for Evaluating the Distribution of Power in a Committee System, 48 Am. Pol. Sci. Rev. 787, (%954)- 9. Shapley & Shubik, supra note 8, at o. Riker, supra note 3, at %i. Krislov, The Power Index, Reapportionment and the Principle of One Man, One Vote, 1965 Modem Uses of Law & Logic 37, Banzhaf, Multi-Member Electoral Districts--Do They Violate the "One Man, One Vote" Principle, 75 Yale L.J. 13o9 (1966); Banzhaf, Weighted Voting Doesn't Work: a Mathematical Analysis, %9 Rutgers L. Rev. 317 (1965); Krislov, supra note ii; Riker & Shapley, Weighted Voting: A Mathematical Analysis for Instrumental Judgments (RAND Corp. Memo. P , 1966). 13. Banzhaf, Weighted Voting Doesn't Work, supra note 12, at Id. at o. 15. Banzhaf, Multi-Member Electoral Districts, supra note 12. %6. Dobish v. Board of Supervisors, 53 Misc. 2d 732, 279 N.Y.S.2d 565 (Sup. Ct. 1967); Morris v. Board of Supervisors, 50 Misc. 2d 929, 273 N.Y.S.2d 453 (Sup. Ct. 1966); Town of Greenburgh v. Board of Supervisors, Index No %965 (Sup. Ct. of Westchester County, May 21, 1968) (Dillon, J.). 17. Iannucci v. Board of Supervisors, 2o N.Y.2d 244, 229 N.E.2d 195, 282 N.Y.S.2d 502 (1967). HeinOnline Geo. Wash. L. Rev

5 experience in this specialized area through the large number of weighted voting cases before New York courts. Second, the decision expressly recognizes the validity of mathematical analysis in this area and specifically holds that it may be the determining factor regardless of other empirical evidence. Finally, the decision validates this technique of measurement by equating it with the legal meaning of voting power. It is not unlikely that other courts will now follow the lead and likewise adopt this same technique-a technique which has been noted with interest by the United States Supreme Court.", The Distribution of Voting Power in a Weighted Voting Legislature Weighted voting has been widely suggested as an answer to the problem posed when a legislative body, state or local, is ordered to reapportion itself and either wishes to retain existing lines or wishes to continue giving the least-populous district a separate representative. Under weighted voting systems substantially unequal districts are retained but each representative casts a number of votes proportional to the population he represents. The theory is that by giving each legislator a number of weighted votes proportional to his population he is given a voting power which is proportional to the number of people he represents, and that the voting power and effective representation of all of the citizen-voters is thereby equalized. Thus a necessary (but not necessarily a sufficient) condition for the constitutionality of such plans is that the voting power of each legislator be approximately proportional to the number of votes he can cast. By calculating the number of chances each legislator has to cast a decisive vote, it can be shown that voting power is not necessarily proportional to the number of votes one can cast. Usually the exact distribution of voting power in a weighted-voting legislative body can be determined only by a complicated computer analysis. Often the distribution of voting power is far from being proportional to the population of each district. Recently New York State's highest court was asked to rule on the constitutionality of two weighted voting plans. It held that such a decision was impossible in the absence of a mathematical analysis and that absent such an analysis the plans must be rejected. 9 The decision illustrates the application of this technique to measure the distribution of voting power in a legislative body and may be applied to other plans as well. Several paragraphs are illustrative of the application of this technique: Although the small towns in a county would be separately represented on the board, each might actually be less able to affect the passage of legislation than if the county were divided into districts of equal population with equal representation on the board and several of the smaller towns were joined together in a single district. (See Banzhaf, Weighted Voting Doesn't Work: %8. Kilgarlin v. Hill, 386 U.S. 12o, 125 (1966). For other courts citing the author's work, see, e.g., WMCA, Inc. v. Lomenzo, 246 F. Supp. 953, 959 (S.D.N.Y. 1965) (Levet, J., dissenting); Dobish v. Board of Supervisors, 53 Misc. 2d 732, 734, 279 N.Y.S.2d 565, 568 (Sup. Ct ). %9. lannucci v. Board of Supervisors, 2o N.Y.2d 244, 229 N.E.2d %95, 282 N.Y.S.2d 502 (-1967). HeinOnline Geo. Wash. L. Rev

6 One Man-One Vote and Local Government THE GEORGE WASHINGTON LAW REVIEW A Mathematical Analysis, i9 Rutgers L. Rev. 317.) The significant standard for measuring a legislator's voting power, as Mr. Banzhaf points out,... is not the number or fraction of votes which he may cast but, rather, his "ability, by his vote, to affect the passage or defeat of a measure."... And he goes on to demonstrate that a weighted voting plan, while apparently distributing this voting power in proportion to population, may actually operate to deprive the smaller towns of what little voting power they possess, to such an extent that some of them might be completely disenfranchised and rendered incapable of affecting any legislative determinations at all... Unfortunately, it is not readily apparent on its face whether either of the plans before us meets the constitutional standard. Nor will practical experience in the use of such plans furnish relevant data since the sole criterion is the mathematical voting power which each legislator possesses in theory -i.e., the indicia of representation-and not the actual voting power he possesses in fact-i.e., the indicia of influence. In order to measure the mathematical voting power of each member of these county boards of supervisors and compare it with the proportion of the population which he represents, it would be necessary to have the opinions of experts based on computer analyses. The plans, then, are of doubtful constitutional validity and to establish the facts one way or another would be, in all likelihood, most expensive. In our view, it was incumbent upon the boards to come forward with the requisite proof that the plans were not defective.... [A] considered judgment [of the plans] is impossible without computer analyses and, accordingly.., there is no alternative but to require them [the boards] to come forward with such analyses and demonstrate the validity of their reapportionment plans 2 0 Measuring the Voting Power of Individual Citizen-Voters The technique previously described may also be used to measure the voting power of the individual citizen-voter who, after all, is the plaintiff in the reapportionment cases and the one whose right to an equally effective vote is being protected. This application can most easily be illustrated by reference to an analogous system, the Electoral College, which is nothing more or less than a large-scale, one-shot weighted voting plan. In the Electoral College the states are analogous to the districts in local or state reapportionment and the state's electors, who almost universally vote as a bloc, are analogous to the number of votes each district may cast. Although the number of electoral votes is not directly proportional to the state's population, the difference is not important here. The technique of analysis applies to any electoral system employing unequally populous districts, including multi-member district systems, and demonstrates that their premises may be fallacious. The smaller states have consistently opposed advocates of a direct presidential election because they feel that an election where each vote is counted 20. id. at 251, , 254, 229 N.E. 2d at , 200, 282 N.Y.S.2d at , 510. HeinOnline Geo. Wash. L. Rev

7 individually could eliminate the advantages accruing to them under the present election system. As a practical matter, however, this supposed advantage actually reduces the influence of small states; a large state, such as New York or California, has "more than two and a half times as much chance to affect the election of the President as a resident of a small state and more than three times as much chance as a resident of the District of Columbia." 21 The analysis of the voting power of individual citizen-voters under the Electoral College may be made as follows. First, one examines, with the aid of a computer, all of the different possible arrangements of electoral votes and determines those in which any given state, by a change in its electoral vote, could change the outcome of the -election. One then looks to the people of the state and determines in how many of these voting combinations a resident could affect how that state's electoral votes would be cast. Finally, combining those two figures, it is possible to calculate the chance of any voter affecting the election of the President through the medium of his state's electoral votes; in other words, his chance to effectively participate in the presidential election. This conclusion may be illustrated by a specific example. New York has 43 electoral votes and Alaska 3- Since New York has approximately 74 times the population of Alaska, it might be supposed that an individual Alaskan's vote carries much more weight than a New Yorker's. But the computers disagree; the significant factor is a New Yorker's potential to affect 43 electoral votes as opposed to an Alaskan's potential to affect only 3, rather than Alaska's heavily weighted representation. Thus, the most heavily favored citizens under the present system are those of New York, California, Pennsylvania and Ohio. The most deprived are those in Maine, New Mexico, Nebraska, Utah and the District of Columbia. The following table shows the inequities of the present system: PRESENT ELECTORAL COLLEGE Percent Devia- Percent tion from Excess Average State Popula- Electoral Relative Voting Voting Name tion 196o Vote Voting Power Power (-) Census 1964 Power (2) (3) (4) Alabama Alaska % Arizona " % Arkansas California % Colorado Connecticut Delaware Dist. of Columbia o.6 Florida o %1.1 Georgia zi. Washington Post, Dec. 31, 1967, at C-6, col. i (editorial). HeinOnline Geo. Wash. L. Rev

8 One Man-One Vote and Local Government THE GEORGE WASIIINGTON LAW REVIEW Hawaii % Idaho 667i91 4 % Illinois A 48.0 Indiana x Iowa Kansas % Kentucky % Louisiana Maine % Maryland % Massachusetts 5% Michigan % % Minnesota % % Mississippi 2178%4% 7 % Missouri 43%98%3 12 %.7% Montana Nebraska A Nevada New Hampshire % New Jersey % % New Mexico %97 % New York %2 23% North Carolina North Dakota Ohio Oklahoma % Oregon Pennsylvania 1x3% Rhode Island South Carolina % South Dakota 68o Tennessee % Texas Utah Vermont % Virginia o Washington West Virginia % 7 %.5o6 5o Wisconsin 395% Wyoming A (%) Includes the District of Columbia. (2) Ratio of voting power of citizens of state compared with voters of the most deprived state. (3) Percent by which voting power exceeds that of the most deprived voters. (4) Percent by which voting power deviates from the average of the figures in column 4. Minus signs (-) indicate less than average voting power. Measuring the of Individual Citizen-Voters So far the only question considered is voting power, the direct effect of the casting of a vote. A more interesting and perhaps more significant question HeinOnline Geo. Wash. L. Rev

9 concerns the effect of such plans on the effective representation of individual citizen-voters, since the electoral process is only a means to the end of equal representation. The technique for studying this problem can be illustrated by a specific analysis of a system of representation which -employs multi-member districts. Speaking strictly theoretically, and within the limits of inquiry delineated thus far by the Supreme Court, how effectively is a citizen-voter represented under a multi-member district plan? To answer the question it is necessary to consider the role and function of the representative. Here one may broadly distinguish between two contrasting hypotheses. One, which is often called the Burkean (or republican) model, assumes a representative who acts for the whole area being governed without consideration for the particular interests of his constituents. He decides issues according to what he feels is best for the whole district, county, or division of local government, based either upon his assessment of their majority wishes or his own best judgment. In contrast there is the delegate (or democratic or representative) model, which assumes that a representative acts as the delegate of his particular constituents. On each issue he is presumed to act according to what he believes to be the will of the majority in his district or town. 22 In effect the votes of all of the citizens are "funneled" into the governing body through their representatives. Certainly both theories are gross simplifications and are subject to criticism for this and other reasons. Evidence suggests that neither model is entirely satisfactory in general but that in many respects the representative theory provides a reasonable approximation to actual situations. 23 Furthermore, in terms of effective representation, only the delegate model need be considered. To the extent that representatives act according to the Burkean model, it is of little relevance what size districts they represent. Each acts for all of the people and does not attempt to reflect the particular wishes of his constituents. 22. See Riker & Shapley, supra note z2, at See generally Sabine, History of Political Philosophy 61o (1950); Wahlke, Eulau, Buchanan & Ferguson, The Legislative System 273 (1962). 23. See, e.g., Cnudde & McCrone, The Linkage Between Constituency Attitudes and Congressional Voting Behavior: A Causal Model, 6o Am. Pol. Sci. Rev. 66, (1966): "This analysis indicates that constituencies do not influence civil rights roll calls in the House of Representatives by selecting Congressmen whose attitudes mirror their own. Instead, Congressmen vote their constituencies' attitudes (as they perceive them) with a mind to the next election." (The authors report "influence coefficients" of 88%.); Kornberg, Perception and Constituency Influence on Legislative Behavior, 19 West. Pol. Q. 285, 291 (1966): [Slubstantial constituency control over legislative leaders has by now come to be regarded as a factual truth as well as a normative principle... The taking of the Burkean-Trustee role style is apparently a luxury... The relatively small number who actually took the role of Trustees suggests that empirical reality (the requirement of being reelected) precludes the taking of this role regardless of the status attached to it. (Of their sample, 15% said that they acted according to the Burkean role, 49% said that they followed the delegate-representative model, and 36% adopted an intermediate position); Miller & Stokes, Constituency Influence in Congress, 57 Am. Pol. Sci. Rev. 45, (1963); Froman, Congressmen and Their Constituencies (%963); Miller & Stokes, in the American Congress (to be published by Prentice-Hall in %968); Wahlke, Eulau, Buchanan & Ferguson, supra note 22, at 281. HeinOnline Geo. Wash. L. Rev

10 One Man-One Vote and Local Government THE GEORGE WASHINGTON LAW REVIEV If one representative has more or less than the average number of constituents, how is anyone advantaged or harmed? To this extent the purpose of the elective system would appear to be simply to select the requisite number of wise and able men to act together to make judgments for the people of the local government unit. There is no evidence to show that the ability to select such men depends closely on population or that a basic constitutional right is involved. On the other hand, at least part of the time a representative is supposed to act as a delegate. Certainly American voters want and expect a representative who will execute their wishes on certain issues, and they repudiate those who deviate too far from the delegate model. While acting purely in a representative capacity, he will cast his vote as he thinks the majority of constituents would vote if they had the opportunity. However infrequently a representative may in fact try to act as a delegate and however imprecise his estimate of his constituents' wishes, the reapportionment decisions, construed broadly, outlaw systems which deny voters an equal chance to have their wishes reflected in the votes of their representative(s). No one can say a priori which representative will be best able to perform this function nor which will take his obligation most seriously. All that a legislative draftsman or judge can do is to insure that the elective and representative system itself does not tend to make it more likely that some citizens will be better able to have their wishes reflected than others. Thus, in constructing a mathematical model, which must of necessity ignore many of the real problems of the system, 2 4 one may hypothesize the representative to be no more than a vehicle for reflecting as best he can the votes of his constituents on certain issues. In such a model of the representative system, each representative would in effect poll his district on each issue and cast his vote according to the majority vote. For the limited purpose of establishing the outer boundaries of a fair representative system, it seems reasonable to assume this type of representative as an oversimplified model What little is known about how legislators' votes are influenced tends to cast doubt on any theory which would have a constituent's ability to affect his representative's vote depend solely on the population of the district. Such a theory would ignore party alliances, ethnic blocs, regional differences and interests, lobbying, influence peddling, and other realities of political life. Yet, so far, the Supreme Court has looked no further than population figures in deciding reapportionment cases. Moreover, the justification offered for multi-member district systems also depends upon such a theory. If influence and representation cannot with some reasonable degree of accuracy be approximated by such a theory, then the justification fails and multi-member district systems should be abandoned. On the other hand, if any such numerical theory can give even a reasonable approximation to political reality, it is submitted that the analysis contained herein is at least mathematically consistent and therefore more likely to be correct than the inverse ratio theory offered as justification for such systems. 25. The courts can go only so far in protecting the rights of citizens to equal representation. Some factors which this model necessarily ignores are no doubt beyond the compe- HeinOnline Geo. Wash. L. Rev

11 With this as a model, it can be shown mathematically why in the usual reapportionment case of single-member districts the districts must be of equal population to guarantee equality of representation. Consider as an example a local legislative body governing three districts, A, B and C, each containing three voters. Each district of three voters is represented by a representative who casts his vote in accordance with the majority vote of his constituents. Each of the three representatives has one vote in the local government body. Looking first at the local government body, with three persons voting and a choice of only yea or nay, there are eight possible voting combinations. Each representative will be "critical"--able to alter the result by changing his vote-in four of these cases. The same situation exists at the citizen-voter level. The citizens of each district can cast their votes in eight different ways (voting combinations) and each citizen-voter will be able to cast a critical vote in his district in four of these. The true issue, however, is how well each citizen-voter is actually represented vis-a-vis the unit of government. In other words, considering all of the combinations in which all nine of the votes could be cast, in how many combinations will any individual voter be able to affect the outcome through the medium of his representative's vote? The answer is that he can cast a critical vote in his district in half of the district's voting combinations and the resulting vote on behalf of the district, as cast by its representative, is critical in half of the voting combinations. Combining these two figures, each voter can cast a critical vote in one-fourth of the total voting combinations. Put another way, if all of the other citizen-voters do not change their original positions on any given issue, any given citizen-voter will be able to change the ultimate decision through the medium of his representative's vote in one-fourth of the possible voting combinations by changing his vote. To show how this method may be applied to demonstrate the inequities in a classic case of malapportionment in which there are single-member districts of substantially unequal population, consider the case where the number of constituents in district A has grown to five. Representative A, like representatives B and C, can still cast a critical vote in one-half of the decisions. However, at the district level, each resident of A can now cast a vote which will be critical with respect to his representative's vote in only twelve out of thirtytwo possible voting combinations. Because of the increase in population, the votes of residents of district A have less effect on their representative's vote yet his vote is no more effective in the governing body. In terms of overall effective representation, the citizen-voters in district A can cast a vote which will be critical with respect to the ultimate decision in only six thirty-seconds tence of the courts to correct. Others are so imperfectly understood that theorizing would be impractical. " This model does not assume that all legislators from a given multi-member district, because they are each supposed to be responsive to the same constituency, will as a result tend to vote as a bloc. Representatives from the same district may of course differ in their assessment of their constituents' wishes and may tend to vote according to the delegate model with respect to different issues. If, however, there is some tendency for them to vote as a bloc, the result will resemble to some extent a weighted voting situation with additional inherent difficulties. HeinOnline Geo. Wash. L. Rev

12 One Man-One Vote and Local Government THE GEORGE WASHINGTON LAW REVIEW of the voting combinations (one-half times twelve thirty-seconds) while residents of the other two districts still may cast decisive votes in one-fourth (eight thirty-seconds) of the combinations. Thus, using this technique, it is possible to demonstrate mathematically exactly why and by how much a citizen is disadvantaged by being a resident of an over-populated district in the usual single-member district electoral system. It may likewise be used to determine whether multi-member district systems provide equally effective representation for all citizens. Consider for purposes of illustration two districts in a local government unit using multi-member districts of unequal size. District S has io,ooo people and elects one representative. District L has 40,000 voters and has four representatives elected at large. The voting power of all of the representatives in the body is the same because each can cast only one vote. District L's four representatives, taken together, are four times as powerful as District S's single representative. Proponents will argue that the voters are equally represented because each of District L's legislators represents four times the number of people that district S's legislator does: 4 divided by 40,0oo equals i divided by io,ooo; there is one representative for every io,ooo people. Despite its simplistic appeal, this argument is fallacious. There is no justification here for simple division; the resulting product has no meaning. Although at first glance it might seem to be a logical operation, there is no mathematical theory to support it. The issue is In terms of the model and the definition of voting power, the question is whether the relative ability, however small, of the people in districts L and S to affect the outcome of decisions through the medium of their representative's votes is substantially equal. The answer is no. It can be shown mathematically that the ability of an individual to affect his representative(s) varies as the inverse of (i divided by) the square root of the district population, rather than as the simple inverse of the population as many have expected. People in District L have more representation than they are entitled to. By giving them more representatives than the diminution in their relative effectiveness required, the overall effect is to give them twice as much voice in the local government unit as residents of S. Furthermore, these inequalities are inherent in the system and are substantially the same as those shown by an analysis of voting power. Whether analyzed in terms of voting power or representation, systems utilizing districts of substantially unequal population contain hidden inequalities. Abstract mathematical analysis may suggest that the results are equally abstract and that the conclusions are primarily of academic interest. The following analysis of multi-member district systems in Arkansas, Georgia, Hawaii, Texas and Wyoming - each of which has recently been involved in re- HeinOnline Geo. Wash. L. Rev

13 apportionment litigation 2 6 -demonstrates the extent of the inequalities which may exist at the statewide level today. This type of analysis is equally applicable to the local unit of government. The simplified mathematical model in each case assumes that all districts electing one legislator have the same population and that the populations of the other districts are exact multiples. Because populations are not divided with mathematical precision, the actual inequalities will vary somewhat from the figures presented in each table but these variations will be small compared with the inequalities shown in the tables. In each table minus signs (-) indicate under-representation while plus signs (±) indicate over-representation (with respect to each of the two averages).2 ARKANSAS (House) Per Cent Per Cent Representa- Deviation in Deviation in tion (as %) Compared Number of Number of With With Respect With Respect Legislators Districts Single-Member to Average to Average Per District in State District (Per District)* (Per Voter)*" % - 30% -46% %% - 1% -23% % + 22% - 6% 4 3 2oo% + 41% + 9% % + 58% +22% % +154% +96% The average effective representation, computed on a per district basis, is 142% compared with a single-member district. Approximately 73% of the districts are below this average in * The average effective representation, computed on a per voter basis, is %84% compared with a single-member district. Approximately 65% of the voters are below this average in 26. See Fortson v. Dorsey, 379 U.S. 433 (1965) (Georgia, figures cited obtained from briefs); Kilgarlin v. Martin, 252 F. Supp. 404 (S.D. Tex. 1966); Yancy v. Faubus, 25% F. Supp. 998 (E.D. Ark. %965), aff'd sub nom. Crawford County Bar Ass'n v. Faubus, 383 U.S. 271 (%966); Schaefer v. Thomson, 251 F. Supp. 450 (D. Wyo. m965), aff'd sub nom. Harrison v. Schaefer, 383 U.S. 269 (1966); Holt v. Richardson, 240 F. Supp. 724 (D. Hawaii 1965), vacated and remanded sub. nom. Bums v. Richardson, 384 U.S. 73 (1966). 27. Two different averages and deviations therefrom have been computed. The first is an average per district, treating each district as the significant entity, and is obtained by multiplying each district by its corresponding effective representation and dividing by the total number of districts. The fourth column indicates deviations from this average. The second average is an average per citizen-voter, treating each citizen-voter as the significant entity, and is obtained by multiplying the number of voters in each district (distributed as assumed) by his corresponding effective representation and dividing by the total number of voters in the state. Deviations from this second average are presented in the last column. HeinOnline Geo. Wash. L. Rev

14 One Man-Ona Vote and Local Government THE GEORGE WASHINGTON LAW REVIEW Number of Legislators Per District Number of Districts in State GEORGIA (Senate) (as %) Compared With Single-Member District %00% %41% %73% 264% Per Cent Deviation in With Respect to Average (Per District)* - 14% + 22% + 49% +127% Per Cent Deviation in With Respect to Average (Per Voter)** -30% - 1% +21% +84% * The average effective representation, computed on a per district basis, is 116% compared with a single-member district. Approximately 77% of the districts are below this average in "* The average effective representation, computed on a per voter basis, is %43% compared with a single-member district. Approximately 70% of the voters are below this average in Number of Legislators Per District Number of Districts in State HAWAII (Senate) (as %) Compared With Single-Member District 100% 141% %73% 200% Per Cent Deviation in With Respect to Average (Per District)* -42% -%8% 0% +16% Per Cent Deviation in With Respect to Average (Per Voter)"* -46% -24% - 6% + 8% * The average effective representation, computed on a per district basis, is 173% compared with a single-member district. Approximately 25% of the districts are below this average in ** The average effective representation, computed on a per voter basis, is 185% compared with a single'member district. Approximately 36% of the voters are below this average in HeinOnline Geo. Wash. L. Rev

15 TEXAS (House) Number of Legislators Per District Number of Districts in State (as %) Compared With Single-Member District Per Cent Deviation in With Respect to Average (Per District)* Per Cent Deviation in With Respect to Average (Per Voter)"* 1oo% 173% 200% 224% 245% 264% 283% 316% 374% - 22% + 9% + 34% + 55% + 74% + 90% +105% +229% +145% +190% -- 48% -27% '10% + 4% +17% +28% +37% +47% +65% +95% * The average effective representation, computed on a per district basis, is 229% compared with a single-member district. Approximately 69% of the districts are below this average in " The average effective representation, computed on a per voter basis, is 192% compared with a single-member district. Approximately 55% of the voters are below this average in WYOMING (Senate) Number of Legislators Per District Number of Districts in State (as %) Compared With Single-Member District Per Cent Deviation in With Respect to Average (Per District)* Per Cent Deviation in With Respect to Average (Per Voter)" -Too% 200% 273% 2oo% 224% -21% +12% +36% +57% +76% -34% - 7% +14% +32% +47% The average effective representation, computed on a per district basis, is 227% compared with a single-member district. Approximately 59% of the districts are below this average in ** The average effective representation, computed on a per voter basis, is 252% compared with a single-member district. Approximately 6o% of the voters are below this average in HeinOnline Geo. Wash. L. Rev

16 One Man-One Vote and Local Government THE GEORGE WASHINGTON LAW REVIEW Conclusions There is a technique for measuring voting power which is generally accepted and recognized in the fields of mathematics and political science. It is based on the assumption that the purpose of any voting system is to allow each voting member some chance of affecting the outcome, and it defines voting power in terms of the number of times each voting member can change the outcome by changing his vote. By using this basic technique it is possible to measure the distribution of voting power in a representative body under systems of weighted voting, modified weighted voting, and fractional voting. It can be shown that the voting power of a representative is not proportional to the number of votes he can cast and that the resulting disparity can frequently render such plans unconstitutional-a conclusion recently acknowledged by New York's highest court. Similarly, it is possible to measure the distribution of voting power among the individual citizen-voters under any system employing substantially unequal electoral districts-e.g., weighted voting, multi-member district systems. Furthermore, if one considers that a representative-at least in theory-is supposed to represent the wishes of his constituents, it is possible to calculate the effective representation each citizen-voter theoretically enjoys under such systems. Both techniques of analysis reveal large disparities which are to the disadvantage of residents of the least populous electoral subdivisions. These analyses demonstrate that even the simplest reapportionment plans may contain inherent hidden inequities which can only be discovered by a careful mathematical analysis. The analyses are complex only because the plans themselves, no matter how simple they may appear, actually introduce new and untried complications into the already difficult problem of obtaining effective representation through the electoral and legislative process. Formulating a mathematical analysis of such plans forces one to examine them critically and to reconsider heretofore unquestioned assumptions. It also seems possible to apply similar techniques to demonstrate and document the inequalities of gerrymandering, the effects of reapportionment plans on different interests and ethnic blocs and to study the workings of other familiar representative systems. Who can predict what the findings might be or what new insights such research might provide for politicians, the judiciary, and scholars in every field who are interested in the problems of effective representation under a democratic system. HeinOnline Geo. Wash. L. Rev