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One Man, ? Votes: Mathematical Analysis of

Voting Power and Effective Representation*

JOHN F. BANZHAF III--

In order to measure the mathematical voting power... it would be neces-sary 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 SupremeCourt entered the reapportionment area amid warnings that it was gettingthe courts into a mathematical quagmire. Little did the Court know that itsdecisions would stimulate the heretofore untapped mathematical ingenuity ofpoliticians' who, in an effort to avoid simple systems of equally populated

* This article is based in large part upon studies previously reported by the authorin: 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. 1309 (1966); and One Man, 3.312 Votes, A Mathematical Analysisof the Electoral College, 13 Villanova L. Rev. 303 (%968); see also Mathematics, Voting andthe Law: The Quest for Equal Representation, prepared for presentation at the InternationalConference on Mathematical Theory of Committees and Elections, Institute for AdvancedStudies and Scientific Research, Vienna, June 26-28, 1968. Portions of these works havebeen included in this article with the kind permission of the editors. Readers who wouldlike 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 Direc-tor, Action on Smoking and Health (ASH), 777 United Nations Plaza, New YorkCity 10017. Effective Sept. %968: Associate Professor of Law, National Law Center, GeorgeWashington University.

i. Iannucci v. Board of Supervisors, 2o N.Y.2d 244, 251-53, 229 N.E.2d 195, %98-99,282 N.Y.S.ad 502, 507-09 (1967).

2. Earlier in this century the legislature of one of the more backward southern statescame very close to enacting a law which would have established 3 as the "official"state value of pi (actually about 3.1416) for the convenience of its citizens. Fortunatelyfor all of the circular structures thereafter constructed, the measure was narrowlydefeated.May 1968 Vol. 36 No. 4

HeinOnline -- 36 Geo. Wash. L. Rev. 808 1967-1968

One Man-One Vote and Local GovernmentTHE GEORGE WASHINGTON LAV REVIEW

single-member districts, would develop straight weighted voting, modifiedweighted voting, fractional voting, multi-member electoral districts, floterialdistricts and mixed plans employing two or more of these devices.

Simple as these plans may have appeared to those who enacted them, largecracks and even untenable foundations appear when they are examined withthe 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 themathematicians who must struggle to make sense of what others have un-comprehendingly created.

By using a technique which is recognized and generally accepted in thefields of mathematics 3 and political science,4 and which is gaining judicialacceptance, it is possible to measure the voting power of representatives andto determine the effects of these plans on the voting power and effectiverepresentation of the citizen-voters whose rights must be protected. In manycases the results show that the plans deprive voters of their constitutionallyprotected rights.

The Measure of Voting Power

The technique for the measurement of voting power is based on the simpleand almost self-evident proposition that the purpose of any voting system isto allow each voting member some chance, however small, to affect the de-cisions that must be made. It can be demonstrated by resort to common experi-ence that different methods of voting can vary the effectiveness of the voteof a given individual. In some cases, such as the election of a dub presidentwhere 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 BehavioralSci. 59 (1962); Shapley, Solution of Compound Games, in Advances in Games Theory267 (1964); Shapley, A Value for N-Person Games, 2 Annals of Mathematical Studies307 (1953).

For reports of mathematical applications of this technique to one aspect of theElectoral College, see Mann & Shapley, The A Priori Voting Strength of the ElectoralCollege, in Game Theory and Related Approaches to Social Behavior 151 (Shubik ed.1964); Mann & Shapley, Value of Large Games VI: Evaluating the Electoral CollegeExactly (RAND Corp. Memo. RM-315 8-PR, May %962); Mann & Shapley, Valueof Large Games IV: Evaluating the Electoral College by Monte Carlo Techniques (RANDCorp. Memo. RM-265%, Sept. i9, 196o).

4. See, e.g., David, Goldman, Bain, The Politics of National Party Conventions 174-75(196o); Schubert, Judicial Behavior (1964); Schubert, Quantitative Analysis of JudicialBehavior, 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 Associa-tion); 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. 1573 (1966); Riker, Some Ambiguities in the Notion of Power, 58 Am. Pol. Sci.Rev. 341 (1964).


to affect the outcome, and all members obviously have equal voting power.In other situations, such as a stockholders' meeting where members cast votesin proportion to their stock holdings, the ability of the voting members to af-fect 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 inwhich the different voters could vote, i.e., to imagine all possible voting com-binations. One then asks in how many of these combinations can each voter af-fect the outcome by changing his vote. Since a priori, all voting combinationsare equally likely and therefore equally significant, the number of combinationsin which each voter can change the outcome by changing his vote servesas the measure of his voting power. In other words, no one can say whichvoting combinations will occur most often, or which combinations will pre-dominate when more important issues are involved. The most a legislature orjudge can do to equalize voting power is to satisfy himself that the systemallows each voting member an opportunity to affect the outcome in an equalnumber of equally likely voting combinations.

A person's voting power, then, is measured by the fraction of the totalnumber of possible voting combinations in which he can, by changing hisvote, alter the outcome of the group's decision. To be more precise, the ratioof the voting power of voter X to the voting power of voter Y is equal to theratio of the number of possible voting combinations in which X could alterthe outcome by changing his vote (assuming that no other voters changetheir votes) to the number of possible voting combinations in which Y couldalter the outcome by changing his vote (also assuming no other voters changetheir votes).'

It is important to recognize that this technique measures the individualvoting power which is inherent in the rules governing the voting system andthe distribution of population; it does not reflect the actual ability of anygiven voter to affect the outcome of a particular election. The latter would de-pend to some extent on factors which are not inherent in the system, such asthe relative power of the political parties in different geographical areas, andconditions which may be peculiar to the voter himself; e.g., whether as a signof protest he decides to vote for a minority party candidate who has no chanceof winning. Thus, a critical distinction must be drawn between inequalitiesin voting power that are built into the system-e.g., the old county unit sys-tem in Georgia or the distribution of electoral votes in the Electoral College-and those which result either from the free choice among citizens as to howthey use their voting power-e.g., the political impotence of a Republican ina solidly Democratic district-or from factors outside of the legal rules govern-ing the process-e.g., voter intimidation, weather, the televised prediction ofelection results. Concededly, these and other external factors may affect acitizen'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 citednote * supra.


One Man-One, Vote and Local GovernmentTHE GEORGE WASHINGTON LAW REVIEW

retical voting power of an individual may not coincide with his actual abilityto affect the outcome of any particular election.

The voting power measured here is that inherent in the system and neces-sarily represents an average of a voter's effectiveness in a large number ofequally likely voting situations. However, only these inequalities, which re-sult from the rules of a particular system of voting, may properly be consideredin determining the basic "fairness" of the system itself.

The technique for measuring voting power by calculating the number ofopportunities each voter has to affect the decision has been generally acceptedin the fields of mathematics 6 and political science.7 Moreover, the technique'sacceptance has been demonstrated through its use in analyzing voting powerin Congress,' stockholders' meetings,9 the French Assembly, 10 New York'sBoard of Estimate" and weighted voting situations in general,' including theNew Jersey Senate,' 3 Nassau County, New York,' 4 and in multi-member dis-tricts.15 In several cases courts have admitted computer analyses based onthis technique and have held the weighted voting plans unconstitutional be-cause of hidden inequalities uncovered by the studies.16 Very recently, NewYork State's highest court, adopting some of the author's arguments as articu-lated in an amicus curiae brief, ruled against two weighted voting plans, andheld that such plans must be subjected to a mathematical-computer analysis.17

This decision is of particular interest for several reasons. First, it is per-suasive because it comes from one of the most highly regarded state courtsin 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 Repre-

sentatives, 56 Am. Pol. Sci. Rev. 58 (1962); Shapley & Shubik, A Method for Evaluatingthe Distribution of Power in a Committee System, 48 Am. Pol. Sci. Rev. 787, 787-90(%954)-

9. Shapley & Shubik, supra note 8, at 791.1o. Riker, supra note 3, at 122-31.%i. Krislov, The Power Index, Reapportionment and the Principle of One Man, One

Vote, 1965 Modem Uses of Law & Logic 37, 40-43.12. 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: aMathematical Analysis, %9 Rutgers L. Rev. 317 (1965); Krislov, supra note ii; Riker &Shapley, Weighted Voting: A Mathematical Analysis for Instrumental Judgments (RANDCorp. Memo. P-3 3 18, 1966).

13. Banzhaf, Weighted Voting Doesn't Work, supra note 12, at 335-38.14. Id. at 338-4 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. 6859-%965 (Sup. Ct. of West-chester 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.2d502 (1967).


experience in this specialized area through the large number of weighted vot-ing cases before New York courts. Second, the decision expressly recognizesthe validity of mathematical analysis in this area and specifically holds thatit may be the determining factor regardless of other empirical evidence. Final-ly, the decision validates this technique of measurement by equating it withthe legal meaning of voting power. It is not unlikely that other courts willnow follow the lead and likewise adopt this same technique-a technique whichhas 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 problemposed when a legislative body, state or local, is ordered to reapportion itselfand either wishes to retain existing lines or wishes to continue giving theleast-populous district a separate representative. Under weighted voting sys-tems substantially unequal districts are retained but each representative castsa number of votes proportional to the population he represents. The theory isthat by giving each legislator a number of weighted votes proportional to hispopulation he is given a voting power which is proportional to the number ofpeople he represents, and that the voting power and effective representationof all of the citizen-voters is thereby equalized. Thus a necessary (but notnecessarily a sufficient) condition for the constitutionality of such plans isthat the voting power of each legislator be approximately proportional to thenumber of votes he can cast.

By calculating the number of chances each legislator has to cast a decisivevote, it can be shown that voting power is not necessarily proportional to thenumber of votes one can cast. Usually the exact distribution of voting powerin a weighted-voting legislative body can be determined only by a complicatedcomputer analysis. Often the distribution of voting power is far from beingproportional to the population of each district. Recently New York State's high-est 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 mathematicalanalysis and that absent such an analysis the plans must be rejected. 9

The decision illustrates the application of this technique to measure the dis-tribution of voting power in a legislative body and may be applied to otherplans as well. Several paragraphs are illustrative of the application of thistechnique:

Although the small towns in a county would be separately represented onthe board, each might actually be less able to affect the passage of legislationthan if the county were divided into districts of equal population with equalrepresentation on the board and several of the smaller towns were joinedtogether 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'swork, 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. -1967).

%9. lannucci v. Board of Supervisors, 2o N.Y.2d 244, 229 N.E.2d %95, 282 N.Y.S.2d502 (-1967).


One Man-One Vote and Local GovernmentTHE GEORGE WASHINGTON LAW REVIEW

A Mathematical Analysis, i9 Rutgers L. Rev. 317.) The significant standardfor 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 apparentlydistributing this voting power in proportion to population, may actually op-erate to deprive the smaller towns of what little voting power they possess,to such an extent that some of them might be completely disenfranchisedand rendered incapable of affecting any legislative determinations at all....

Unfortunately, it is not readily apparent on its face whether either of theplans before us meets the constitutional standard. Nor will practical experi-ence in the use of such plans furnish relevant data since the sole criterionis the mathematical voting power which each legislator possesses in theory-i.e., the indicia of representation-and not the actual voting power hepossesses in fact-i.e., the indicia of influence. In order to measure the math-ematical voting power of each member of these county boards of supervisorsand compare it with the proportion of the population which he represents,it would be necessary to have the opinions of experts based on computeranalyses. The plans, then, are of doubtful constitutional validity and to es-tablish the facts one way or another would be, in all likelihood, most expen-sive. In our view, it was incumbent upon the boards to come forward withthe requisite proof that the plans were not defective.

... [A] considered judgment [of the plans] is impossible without computeranalyses and, accordingly.., there is no alternative but to require them[the boards] to come forward with such analyses and demonstrate the valid-ity of their reapportionment plans2 0

Measuring the Voting Power of Individual Citizen-Voters

The technique previously described may also be used to measure the votingpower of the individual citizen-voter who, after all, is the plaintiff in the re-apportionment cases and the one whose right to an equally effective vote isbeing protected. This application can most easily be illustrated by referenceto an analogous system, the Electoral College, which is nothing more or lessthan a large-scale, one-shot weighted voting plan. In the Electoral College thestates are analogous to the districts in local or state reapportionment and thestate's electors, who almost universally vote as a bloc, are analogous to thenumber of votes each district may cast. Although the number of electoralvotes is not directly proportional to the state's population, the difference is notimportant here. The technique of analysis applies to any electoral systememploying unequally populous districts, including multi-member district sys-tems, and demonstrates that their premises may be fallacious.

The smaller states have consistently opposed advocates of a direct presi-dential election because they feel that an election where each vote is counted

20. id. at 251, 252-53, 254, 229 N.E. 2d at 198-99, 200, 282 N.Y.S.2d at 507-09, 510.


individually could eliminate the advantages accruing to them under thepresent election system. As a practical matter, however, this supposed ad-vantage actually reduces the influence of small states; a large state, such asNew York or California, has "more than two and a half times as much chanceto affect the election of the President as a resident of a small state and morethan 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 Elec-toral College may be made as follows. First, one examines, with the aid of acomputer, all of the different possible arrangements of electoral votes anddetermines 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 thestate and determines in how many of these voting combinations a residentcould affect how that state's electoral votes would be cast. Finally, combiningthose two figures, it is possible to calculate the chance of any voter affectingthe 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 43electoral votes and Alaska 3- Since New York has approximately 74 times thepopulation of Alaska, it might be supposed that an individual Alaskan's votecarries 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 asopposed to an Alaskan's potential to affect only 3, rather than Alaska's heavilyweighted representation. Thus, the most heavily favored citizens under thepresent system are those of New York, California, Pennsylvania and Ohio.The most deprived are those in Maine, New Mexico, Nebraska, Utah and theDistrict of Columbia. The following table shows the inequities of the presentsystem:

PRESENT ELECTORAL COLLEGEPercentDevia-

Percent tion fromExcess Average

State Popula- Electoral Relative Voting VotingName tion 196o Vote Voting Power Power

(-) Census 1964 Power (2) (3) (4)

Alabama 3266740 10 1.632 63.2 - 3.0

Alaska 226167 3 %.838 83.8 9.2

Arizona "30261 5 1.281 28.% -23.9

Arkansas 1786272 6 1.315 31.5 -2.9California 15717204 40 3.%62 216.2 87.9Colorado 1753947 6 1.327 32.7 -21.1Connecticut 2535234 8 1.477 47.7 -12.2.

Delaware 446292 3 1.308 30.8 -22.3

Dist. of Columbia 763956 3 1.000 .0 -4o.6Florida 495156o 14 1.870 87.0 %1.1Georgia 3943116 12 1.789 78.9 6.3

zi. Washington Post, Dec. 31, 1967, at C-6, col. i (editorial).


One Man-One Vote and Local GovernmentTHE GEORGE WASIIINGTON LAW REVIEW

Hawaii 632772 4 %.468 46.8 -12.8Idaho 667i91 4 %.429 42.9 -15-.Illinois 10081158 26 2.491 149.A 48.0Indiana 4662498 13 1.786 78.6 6.xIowa 2757537 9 1.596 59.6 - 5.2Kansas 2178611 7 %.392 39.2 -17-3

Kentucky 3038156 9 1.52% 52.1 - 9.6Louisiana 3257022 10 1.635 63.5 - 2.9

Maine 969265 4 1.%86 18.6 -29.5

Maryland 3100689 10 %.675 67.5 - .4Massachusetts 5%48578 14 1.834 83.4 9.0Michigan 7823194 2% 2.262 %26.2 34.4Minnesota 3413864 10 %.597 59.7 - 5.%

Mississippi 2178%4% 7 %.392 39.2 -17.3Missouri 43%98%3 12 %.7%0 71.0 1.6Montana 674767 4 1.421 42.1 -15.5Nebraska 1411330 5 1.231 23.A -26.9Nevada 285278 3 1.636 63.6 - 2.8New Hampshire 606921 4 %.499 49.9 -10.9New Jersey 6066782 %7 2.063 %06.3 22.6New Mexico 951023 4 1.%97 %9.7 -28.9New York 16782304 43 3.3%2 23%.2 96.8North Carolina 4556155 13 1.807 80.7 7.4North Dakota 632446 4 1.468 46.8 -2.8Ohio 9706397 26 2.539 153.9 50.9Oklahoma 2328284 8 %.541 54.1 - 8.4Oregon 1768687 6 1.321 32.1 -21.5Pennsylvania 1x3%9366 29 2.638 163.8 56.8Rhode Island 859488 4 1.259 25.9 -25.2South Carolina 2382594 8 %.524 52-4 - 9.5South Dakota 68o514 4 1.415 4.5 -15.9Tennessee 3567089 11 1.72% 72.1 2.3Texas 9579677 25 2.452 145.2 45-7Utah 890627 4 1.237 23-7 -26.5Vermont 389881 3 %.400 40.0 -1.6.8Virginia 3966949 12 1.784 78.4 6.oWashington 2853214 9 1.569 56.9 - 6.8West Virginia 186042% 7 %.5o6 5o.6 -10.5Wisconsin 395%777 12 1.788 78.8 6.2

Wyoming 33066 3 1-521 52.A - 9.6

(%) 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 Effective Representation of IndividualCitizen-Voters

So far the only question considered is voting power, the direct effect of thecasting of a vote. A more interesting and perhaps more significant question


concerns the effect of such plans on the effective representation of individualcitizen-voters, since the electoral process is only a means to the end of equalrepresentation. The technique for studying this problem can be illustrated by aspecific analysis of a system of representation which -employs multi-memberdistricts. Speaking strictly theoretically, and within the limits of inquiry delin-eated thus far by the Supreme Court, how effectively is a citizen-voter repre-sented under a multi-member district plan?

To answer the question it is necessary to consider the role and function ofthe representative. Here one may broadly distinguish between two contrastinghypotheses. One, which is often called the Burkean (or republican) model,assumes a representative who acts for the whole area being governed withoutconsideration for the particular interests of his constituents. He decides issuesaccording to what he feels is best for the whole district, county, or divisionof local government, based either upon his assessment of their majority wishesor his own best judgment. In contrast there is the delegate (or democratic orrepresentative) model, which assumes that a representative acts as the dele-gate of his particular constituents. On each issue he is presumed to act ac-cording to what he believes to be the will of the majority in his district ortown.22 In effect the votes of all of the citizens are "funneled" into the govern-ing body through their representatives.

Certainly both theories are gross simplifications and are subject to criticismfor this and other reasons. Evidence suggests that neither model is entirelysatisfactory in general but that in many respects the representative theory pro-vides a reasonable approximation to actual situations.23 Furthermore, in termsof effective representation, only the delegate model need be considered. Tothe extent that representatives act according to the Burkean model, it is oflittle relevance what size districts they represent. Each acts for all of thepeople and does not attempt to reflect the particular wishes of his constituents.

22. See Riker & Shapley, supra note z2, at 23-31. See generally Sabine, History ofPolitical Philosophy 61o (1950); Wahlke, Eulau, Buchanan & Ferguson, The LegislativeSystem 273 (1962).

23. See, e.g., Cnudde & McCrone, The Linkage Between Constituency Attitudes andCongressional Voting Behavior: A Causal Model, 6o Am. Pol. Sci. Rev. 66, 69-70 (1966):"This analysis indicates that constituencies do not influence civil rights roll calls inthe House of Representatives by selecting Congressmen whose attitudes mirror their own.Instead, Congressmen vote their constituencies' attitudes (as they perceive them) with amind to the next election." (The authors report "influence coefficients" of 88%.); Korn-berg, 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 beregarded as a factual truth as well as a normative principle.... The taking of theBurkean-Trustee role style is apparently a luxury .... The relatively small numberwho actually took the role of Trustees suggests that empirical reality (the require-ment of being reelected) precludes the taking of this role regardless of the statusattached to it.

(Of their sample, 15% said that they acted according to the Burkean role, 49% saidthat they followed the delegate-representative model, and 36% adopted an intermediateposition); Miller & Stokes, Constituency Influence in Congress, 57 Am. Pol. Sci. Rev.45, 45-46 (1963); Froman, Congressmen and Their Constituencies (%963); Miller & Stokes,Representation in the American Congress (to be published by Prentice-Hall in %968);Wahlke, Eulau, Buchanan & Ferguson, supra note 22, at 281.


One Man-One Vote and Local GovernmentTHE 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 elec-tive system would appear to be simply to select the requisite number of wiseand able men to act together to make judgments for the people of the localgovernment unit. There is no evidence to show that the ability to select suchmen depends closely on population or that a basic constitutional right isinvolved.

On the other hand, at least part of the time a representative is supposedto act as a delegate. Certainly American voters want and expect a representa-tive who will execute their wishes on certain issues, and they repudiate thosewho deviate too far from the delegate model. While acting purely in a repre-sentative capacity, he will cast his vote as he thinks the majority of con-stituents would vote if they had the opportunity. However infrequently arepresentative may in fact try to act as a delegate and however imprecisehis estimate of his constituents' wishes, the reapportionment decisions, con-strued broadly, outlaw systems which deny voters an equal chance to havetheir wishes reflected in the votes of their representative(s). No one can saya priori which representative will be best able to perform this function norwhich will take his obligation most seriously. All that a legislative draftsmanor judge can do is to insure that the elective and representative system it-self does not tend to make it more likely that some citizens will be better ableto have their wishes reflected than others. Thus, in constructing a mathematicalmodel, which must of necessity ignore many of the real problems of the sys-tem,2 4 one may hypothesize the representative to be no more than a vehiclefor reflecting as best he can the votes of his constituents on certain issues.In such a model of the representative system, each representative would ineffect poll his district on each issue and cast his vote according to the majorityvote. For the limited purpose of establishing the outer boundaries of a fairrepresentative system, it seems reasonable to assume this type of representa-tive as an oversimplified model.2 5

24. What little is known about how legislators' votes are influenced tends to cast doubton any theory which would have a constituent's ability to affect his representative's votedepend 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 re-alities of political life. Yet, so far, the Supreme Court has looked no further than popula-tion figures in deciding reapportionment cases. Moreover, the justification offered formulti-member district systems also depends upon such a theory. If influence and repre-sentation cannot with some reasonable degree of accuracy be approximated by such atheory, 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 approximationto political reality, it is submitted that the analysis contained herein is at least mathemati-cally consistent and therefore more likely to be correct than the inverse ratio theory offeredas justification for such systems.

25. The courts can go only so far in protecting the rights of citizens to equal representa-tion. Some factors which this model necessarily ignores are no doubt beyond the compe-


With this as a model, it can be shown mathematically why in the usualreapportionment case of single-member districts the districts must be of equalpopulation to guarantee equality of representation. Consider as an example alocal legislative body governing three districts, A, B and C, each contain-ing three voters. Each district of three voters is represented by a representa-tive 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 achoice of only yea or nay, there are eight possible voting combinations. Eachrepresentative will be "critical"--able to alter the result by changing hisvote-in four of these cases. The same situation exists at the citizen-voterlevel. 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 criticalvote in his district in four of these. The true issue, however, is how well eachcitizen-voter is actually represented vis-a-vis the unit of government. In oth-er words, considering all of the combinations in which all nine of the votescould be cast, in how many combinations will any individual voter be able toaffect the outcome through the medium of his representative's vote? The an-swer is that he can cast a critical vote in his district in half of the district'svoting combinations and the resulting vote on behalf of the district, as castby its representative, is critical in half of the voting combinations. Combiningthese two figures, each voter can cast a critical vote in one-fourth of the totalvoting combinations. Put another way, if all of the other citizen-voters do notchange their original positions on any given issue, any given citizen-voterwill be able to change the ultimate decision through the medium of his repre-sentative's vote in one-fourth of the possible voting combinations by changinghis vote.

To show how this method may be applied to demonstrate the inequities in aclassic case of malapportionment in which there are single-member districtsof substantially unequal population, consider the case where the number ofconstituents in district A has grown to five. Representative A, like representa-tives B and C, can still cast a critical vote in one-half of the decisions. How-ever, at the district level, each resident of A can now cast a vote which will becritical with respect to his representative's vote in only twelve out of thirty-two possible voting combinations. Because of the increase in population, thevotes of residents of district A have less effect on their representative's voteyet his vote is no more effective in the governing body. In terms of overalleffective representation, the citizen-voters in district A can cast a vote whichwill 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 wouldbe impractical. "

This model does not assume that all legislators from a given multi-member district, be-cause they are each supposed to be responsive to the same constituency, will as a resulttend to vote as a bloc. Representatives from the same district may of course differ in theirassessment of their constituents' wishes and may tend to vote according to the delegatemodel with respect to different issues. If, however, there is some tendency for them to voteas a bloc, the result will resemble to some extent a weighted voting situation with addi-tional inherent difficulties.



of the voting combinations (one-half times twelve thirty-seconds) while resi-dents of the other two districts still may cast decisive votes in one-fourth (eightthirty-seconds) of the combinations. Thus, using this technique, it is possibleto demonstrate mathematically exactly why and by how much a citizen isdisadvantaged by being a resident of an over-populated district in the usualsingle-member district electoral system. It may likewise be used to determinewhether multi-member district systems provide equally effective representa-tion for all citizens.

Consider for purposes of illustration two districts in a local government unitusing multi-member districts of unequal size. District S has io,ooo peopleand elects one representative. District L has 40,000 voters and has four repre-sentatives elected at large. The voting power of all of the representatives inthe body is the same because each can cast only one vote. District L's fourrepresentatives, taken together, are four times as powerful as District S'ssingle representative. Proponents will argue that the voters are equally repre-sented because each of District L's legislators represents four times the num-ber of people that district S's legislator does: 4 divided by 40,0oo equals idivided by io,ooo; there is one representative for every io,ooo people. De-spite its simplistic appeal, this argument is fallacious.

There is no justification here for simple division; the resulting product hasno meaning. Although at first glance it might seem to be a logical operation,there is no mathematical theory to support it. The issue is effective representa-tion. In terms of the model and the definition of voting power, the questionis whether the relative ability, however small, of the people in districts L andS to affect the outcome of decisions through the medium of their representa-tive's votes is substantially equal. The answer is no. It can be shown mathe-matically that the ability of an individual to affect his representative(s) variesas 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. Bygiving them more representatives than the diminution in their relative effec-tiveness required, the overall effect is to give them twice as much voice inthe local government unit as residents of S. Furthermore, these inequalitiesare inherent in the system and are substantially the same as those shownby an analysis of voting power. Whether analyzed in terms of voting power orrepresentation, systems utilizing districts of substantially unequal populationcontain hidden inequalities.

Abstract mathematical analysis may suggest that the results are equallyabstract and that the conclusions are primarily of academic interest. The fol-lowing analysis of multi-member district systems in Arkansas, Georgia, Ha-waii, Texas and Wyoming - each of which has recently been involved in re-


apportionment litigation2 6 -demonstrates the extent of the inequalities whichmay exist at the statewide level today. This type of analysis is equally ap-plicable to the local unit of government. The simplified mathematical modelin each case assumes that all districts electing one legislator have the samepopulation and that the populations of the other districts are exact multiples.Because populations are not divided with mathematical precision, the actualinequalities will vary somewhat from the figures presented in each table butthese variations will be small compared with the inequalities shown in thetables. In each table minus signs (-) indicate under-representation whileplus signs (±) indicate over-representation (with respect to each of the twoaverages) .2

ARKANSAS (House)

Effective Per Cent Per CentRepresenta- Deviation in Deviation intion (as %) Effective EffectiveCompared Representation Representation

Number of Number of With With Respect With RespectLegislators Districts Single-Member to Average to AveragePer District in State District (Per District)* (Per Voter)*"

1 17 00% - 30% -46%2 15 14%% - 1% -23%3 6 173% + 22% - 6%4 3 2oo% + 41% + 9%5 2 224% + 58% +22%

13 1 361% +154% +96%

The average effective representation, computed on a per district basis, is 142% comparedwith a single-member district. Approximately 73% of the districts are below this average ineffective representation.

* The average effective representation, computed on a per voter basis, is %84% comparedwith a single-member district. Approximately 65% of the voters are below this average ineffective representation.

26. See Fortson v. Dorsey, 379 U.S. 433 (1965) (Georgia, figures cited obtained frombriefs); 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. Harrisonv. 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 anaverage per district, treating each district as the significant entity, and is obtained by multi-plying each district by its corresponding effective representation and dividing by the totalnumber of districts. The fourth column indicates deviations from this average. The secondaverage 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 votersin the state. Deviations from this second average are presented in the last column.


One Man-Ona Vote and Local GovernmentTHE GEORGE WASHINGTON LAW REVIEW

GEORGIA (Senate)

Number ofLegislatorsPer District

Number ofDistrictsin State

EffectiveRepresenta-tion (as %)Compared

WithSingle-Member

District

%00%%41%%73%264%

Per CentDeviation in

EffectiveRepresentationWith Respect

to Average(Per District)*

- 14%+ 22%

+ 49%+127%


EffectiveRepresentationWith Respectto Average

(Per Voter)**

-30%- 1%

+21%

+84%

* The average effective representation, computed on a per district basis, is 116% comparedwith a single-member district. Approximately 77% of the districts are below this average ineffective representation.

"* The average effective representation, computed on a per voter basis, is %43% comparedwith a single-member district. Approximately 70% of the voters are below this average ineffective representation.

HAWAII (Senate)




WithSingle-Member

District

100%141%%73%200%



to Average(Per District)*

-42%

-%8%0%

+16%



to Average(Per Voter)"*

-46%-24%- 6%+ 8%


** The average effective representation, computed on a per voter basis, is 185% comparedwith a single'member district. Approximately 36% of the voters are below this average ineffective representation.


TEXAS (House)




WithSingle-Member

District

1oo%

173%200%

224%

245%

264%283%316%

374%



(Per District)*

- 22%

+ 9%+ 34%+ 55%+ 74%+ 90%+105%+229%+145%+190%



(Per Voter)"*

-- 48%-27%

'10%

+ 4%+17%+28%+37%+47%+65%+95%


" The average effective representation, computed on a per voter basis, is 192% comparedwith a single-member district. Approximately 55% of the voters are below this average ineffective representation.

WYOMING (Senate)




WithSingle-Member

District

-Too%

200%273%2oo%224%



(Per District)*

-21%

+12%

+36%+57%+76%



(Per Voter)"

-34%- 7%+14%+32%+47%

The average effective representation, computed on a per district basis, is 227% comparedwith a single-member district. Approximately 59% of the districts are below this average ineffective representation.

** The average effective representation, computed on a per voter basis, is 252% comparedwith a single-member district. Approximately 6o% of the voters are below this average ineffective representation.



Conclusions

There is a technique for measuring voting power which is generally acceptedand recognized in the fields of mathematics and political science. It is basedon the assumption that the purpose of any voting system is to allow eachvoting member some chance of affecting the outcome, and it defines votingpower in terms of the number of times each voting member can change theoutcome by changing his vote.

By using this basic technique it is possible to measure the distributionof voting power in a representative body under systems of weighted voting,modified weighted voting, and fractional voting. It can be shown that thevoting power of a representative is not proportional to the number of voteshe can cast and that the resulting disparity can frequently render such plansunconstitutional-a conclusion recently acknowledged by New York's highestcourt.

Similarly, it is possible to measure the distribution of voting power amongthe individual citizen-voters under any system employing substantially un-equal electoral districts-e.g., weighted voting, multi-member district systems.Furthermore, if one considers that a representative-at least in theory-is sup-posed to represent the wishes of his constituents, it is possible to calculate theeffective representation each citizen-voter theoretically enjoys under such sys-tems. Both techniques of analysis reveal large disparities which are to thedisadvantage of residents of the least populous electoral subdivisions.

These analyses demonstrate that even the simplest reapportionment plansmay contain inherent hidden inequities which can only be discovered by acareful mathematical analysis. The analyses are complex only because theplans themselves, no matter how simple they may appear, actually introducenew and untried complications into the already difficult problem of obtainingeffective representation through the electoral and legislative process. Formu-lating a mathematical analysis of such plans forces one to examine themcritically and to reconsider heretofore unquestioned assumptions.

It also seems possible to apply similar techniques to demonstrate and docu-ment the inequalities of gerrymandering, the effects of reapportionment planson different interests and ethnic blocs and to study the workings of otherfamiliar representative systems. Who can predict what the findings might beor what new insights such research might provide for politicians, the judiciary,and scholars in every field who are interested in the problems of effectiverepresentation under a democratic system.


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