The Voting Power of Racial Groups In a Multiracial Political System (May 9, 1990)


Garza v. L.A. County, the lawsuit currently being brought by MALDEF, the ACLU, and the U.S. Department of Justice against the L.A. County Board of Supervisors, raises unique questions regarding the voting power of racial groups with respect to the Voting Rights Act of 1965, and its 1982 Amendments. In this paper I will focus on the sections of the Act which deal with unintentional effects on voting power, and not those dealing with electoral systems which are purposely designed to dilute minority group voting strength—admittedly an equally significant issues in the Garza suit. I hope to show that the racial demographics of L.A. County present problems which previous Voting Rights Act cases failed to consider, specifically, that the absence of an absolute racial majority among the four large racial groups in L.A. County permits in one sense a greater opportunity for political participation, but at the same time allow no totally equitable method of apportioning voting power among the racial groups.

Section 2(b) of the Voting Rights Act provides that Section 2(a) is violated where the “totality of circumstances” reveals that “the political processes leading to nomination or election…are not equally open to participation by members of a [protected class]…in that its members have less opportunity than other members of the electorate to participate in the political process and to elect representatives of their choice.” In the major Supreme Court case interpreting the Act, Thornburg v. Gingles, 478 U.S. 30 (1986), Justice Brennan for the majority cited a list of seven plus two “typical factors” which can be considered relevant to the “totality of the circumstances.” One of the factors is “(2) the extent to which voting in the elections…is racially polarized.” Such voting has been termed “racial bloc voting”. Brennan’s opinion indicated that the existence of racially polarized voting and “the extent to which minority groups have been elected to public office in the jurisdiction” are the two “most important” factors in a Section 2 claim. Gingles, 106 S.Ct. 2752, 2766 n.15.

There is a good deal of disagreement on how to determine the existence of racial bloc voting. (See, for example, “Racial Polarization in Vote Dilution Cases Under Section 2 of the Voting Rights Act. The impact of Thornburg v. Gingles”, Paul W. Jacobs, II and Timothy G. O’Rourke, Journal of Law & Politics, p. 295.) Notwithstanding such disputes, for the purposes of this paper we shall accept the plaintiff’s assertions and assume racial bloc voting exists—but for all the racial groups in L.A. County. Furthermore, we are going to assume that every citizen is a member of one and only one racial group, and that members of a racial group will all agree to vote in the same way—that is, as a voting bloc. These assumptions describe the most extreme case of racial bloc voting. Note that they should strengthen the Voting Rights Act claim. As we shall argue below, our findings will hold for less than absolute racial bloc voting as well, so these assumptions will not weaken our argument.

If racial bloc voting is assumed, the plaintiffs in Garza have a very strong case. No non-white has been elected to the L.A. County Board of Supervisors since 1875. (Elizabeth Braithwaite Burke, a black woman, was appointed to the Board by Governor Jerry Brown but lost her first election to current Board-member Dean Dana.) With Brennan’s two “most important” factors, the district court could conclude that the voting power of minority groups in L.A. County has been diluted, and that Section 2 of the Voting Right Act has been violated. But in determining the appropriate remedy, the court will have to ask again the question that Brennan attempted to avoid with his “seven plus two” factors” What is voting power, and how is it apportioned among the racial groups in L.A. County?

Voting power is the ability to determine the outcome of a vote. In theory, every individual voter has the ability to determine the outcome of the vote. That is, if all the other voters split their votes between two candidates, the individual voter will decide the election by his/her vote. In that sense, the voting power of all the individual votes in L.A. County should be equal so long as we do not assume anything about the preferences of the voters, and assuming that the districts are the same size. Each person has the opportunity to cast a potentially deciding vote for his/her supervisor. Similarly, that supervisor has voting power because he/she can cast the deciding vote on the Board of Supervisors. One individual can cast the deciding vote for a supervisor who can cast the deciding vote on an issue before the board. Therefore, each individual voter has voting power in that his/her vote can determine the outcome of a vote on the Board.

However, we have decided to assume racial bloc voting. Bloc voting implies that the individual voters are not voting independently. It may no longer be possible for one individual to cast the deciding vote, because it may not be possible for all the other voters to split their votes evenly between two candidates. Furthermore, it may be impossible for the voting bloc as a whole to decide the election. In both cases the individual is powerless, because neither he/she nor the voting bloc has the ability to decide the outcome of the election. Since this argument is at the core of our discussion, we shall explain it in more detail.

Let us take as an example a district with five voters. In theory, if we do not assume anything about voter preferences, each voter has voting power and may determine the outcome of the election if the other voters split evenly. Now let’s assume that the five voters belong to three voting blocs: A, B and C. Bloc A has three members; B and C have one member each. The members of a voting bloc will always vote for the same candidate. In the election, B and C have no voting power because they cannot affect the outcome in any way. No matter how B and C vote, A (and it’s voters) will decide the winner of the election since A controls more than half of the votes. An election which seems perfectly fair becomes one-sided when we assume bloc voting. If voting power is the ability to determine the outcome of an election, then B and C have no voting power.

Most of the cases on vote dilution have dealt with districts as composed of only two groups: whites and non-whites. In many of these cases whites composed more than fifty percent of the overall population in the jurisdiction, and the districts inside the jurisdiction were divided up so that whites also had a majority in nearly all districts. In such a case for example, a state with a 70% white population could have nearly 100% white representatives in the legislature. The Voting Rights Act sought to remedy the situation, and without guaranteeing proportional representation to minority groups in the legislature to at least give minority groups a chance to have some representatives. That usually meant creating districts where the non-white population was well over 50%. United Jewish Organizations of Williamsburgh v. Carey, 430 U.S. 144. While those solutions may have given minority voters all of the voting power within their few minority controlled districts, it did not give minority groups any voting power in the legislature where a large majority of the seats were still controlled by white-majority districts. Minority voters could elect a representative, but that representative would be powerless to affect the outcome of a vote important to his constituents. The white majority still held all of the legislative voting power.

In effect, the court confused the discussion by segregating voting power. The primary power to affect the outcome of a vote on the legislature would remain in the hands of the white majority, but a secondary voting power would be afforded to minority groups. The emasculated secondary voting power is merely the opportunity to elect a legislatively powerless representative. Certainly, that representative can serve his/her constituents by raising issues on the floor of the legislature, by serving on legislative committees, and by answering constituent letters. The secondary power is not negligible, and that is perhaps why it was desired by minority politicians, but having representation is not the whole story. Just as we showed above that the ability to vote does not guarantee the ability to affect the outcome of an election, having a representative does not guarantee power over the legislative process.

No one would disagree that the power to pass laws is the real power of a legislator. But one might argue that minority representatives do exert voting power in the legislature by forming alliances and coalitions. That is of course true, but their ability to do so is constrained by white majority bloc voting. Remember that if the white majority did not vote as a bloc, then there would be a no racially polarized voting, and likely no violation of the Voting Rights Act. If whites did not vote in a bloc, minority candidates would be able to win elections. But we have assumed racial bloc voting and the inability of minority candidates to win an election. If representatives mirror the interests of the constituents who elected them, then we can expect the representatives also to vote as a racial bloc.

It may be true that voters vote in racial blocs, but legislators do not. That would indicate that legislators are not voting with the will of the majority that elected them. But if that is true, and legislators are more enlightened and independent of their constituents, then we need not worry about racial bloc voting in the first place. The assumption of the Voting Rights Act claim was that legislators elected by all-white majorities were inconsiderate of the interests of the non-white minorities in their districts.

In truth minority voting blocs do exert some power in the legislature. But that is because the white majority vote is split into two political parties who do not vote together as a bloc. Minorities can form dominant coalitions with other interest groups within the political parties and therefore have been able to exert voting power. But this argument would indicate that minority voters should also be able to form winning coalitions to represent their interests in a district election, unless they are geographically separated from other potential allies. If the white majority does not vote as a racial bloc, then minority blocs should be able to form coalitions to win elections, and there will be likely no Voting Rights Act violation. But we began by assuming racial bloc voting and violation of the Act.

Of course the problem is that so long as there is one group with an absolute majority our political system is designed to give that group absolute power. Such is almost always the case when there are only two groups. One will have more than 50% and the other less than 50%. And the majority rules. But what happens when there are more than two groups and no one group has an absolute majority? Such is now the case for racial groups in Los Angeles County.

If no single group can control the outcome of an election by itself then a group must form a coalition to win. For example, if groups A, B and C each have one vote, then the winning coalitions are AB, AC, BC and ABC. A group has voting power if it can determine the outcome. If the group can form a coalition which is winning with the group as a member of the coalition, but losing if the group withdraws its support, then the group has power in that coalition. Group A has power in the coalitions AB and AC, but not in ABC because that coalition will still win if A changes its vote. Similarly, B has power in the coalitions AB and BC. Group C has power in the coalitions AC and BC. Since all three groups have the ability to build winning coalitions and possibly determine the outcome of an election, all three have voting power in the district.

A plurality in one district does not guarantee that every group will have voting power in the district, nor does it mean that the groups will have equal power or power proportional to their size, but at least three groups will have some power. For example, if there are four groups: A, B, C and D. A has 4 votes, B and C each have 3 votes, and D has only one vote. Any two of the groups A, B and C can form a winning coalition with 6 or 7 of the 11 votes. But group D cannot contribute anything to any coalition. Any winning coalition with D, would also be winning if D withdrew its vote. A, B and C will negotiate and form coalitions without regard for D. Therefore D has no voting power. Notice also that A’s extra vote doesn’t give the group any more power. A still must form a coalition with either B or C to win, and the extra vote shouldn’t give A any additional bargaining power.[1]

As a further example of voting power disproportionate to size, let A and B have 100 votes each. C has 2 votes. D has 1 vote. It is still true that D has no power, as above. But what about C? Extraordinarily, C has just as much power as A and B! Just as in the two previous examples, any two of the three groups A, B and C can form a winning coalition. AB has 200 votes, but AC and BC will win also with 102 votes. Each of the three groups can form two winning coalitions in which it has power. The voting power of the three groups is identical even though C has less than 1% of the total vote and A and B nearly 50%. Poor D with only one vote less than C is powerless. This example is extreme but not uncommon. In Israel the two major parties, Likud and Labor each control just under 50% of the votes. The comparably small non-aligned religious parties have an extraordinary amount of power. Until recently the two large parties formed a coalition, but now that the coalition has fallen apart the small parties have a chance to exert their power. Just last month a Labor coalition was narrowly defeated when two religious party members defected at the last minute. The power to make or break a winning coalition is what we mean by voting power and often voting power does not correspond to the size of a group.

The voting power of a group is determined by its ability to form winning coalitions. Professor John Banzhaf has proposed a model for calculating the relative voting power between voters.[2] There are five members of the Board of Supervisors. On any proposal before the Board, each of the five supervisors can vote yes or no. Therefore, there are 32(25) combinations of votes.[3] A Supervisor holds a critical vote when the four other supervisors split their votes 2-2, that is when he/she is on the winning side of a 3-2 board vote. That occurs in 12 of the 32 combinations for each supervisor.[4] Therefore, each supervisor will hold a critical vote in 12 of the 32 combinations, or 3/8 of the time. So we say that, according to the Banzhaf Index, each supervisor has voting power equal to 3/8. If we were not assuming bloc voting, we could proceed to calculate in the same manner the number of critical votes for each person voting in a Supervisorial District election. If we then multiplied the results of the calculations for the district elections with the Banzhaf index for each supervisor (3/8) we would get the voting power of each individual voter in the County of Los Angeles with respect to the Board of Supervisors. In other words, the chance that a voter will have a critical vote for his/her supervisor multiplied by the chance that that supervisor will hold a critical vote on the Board will be equal to the chance that a voter will determine the outcome of a vote on the Board of Supervisors. And the chance to determine the outcome is what we have defined as “voting power”.

Before we begin the analysis of racial bloc voting power, I would like to repeat the assumption we have made. First, voting power is the ability to determine the outcome of an election or any other voting process. Second, a representative will vote according to the will of the majority in his/her district. This second assumption lies at the center of any “one person, one vote” claim. No legislator should be free to disregard the will of the people who elected him/her. That is why legislators are forced to undergo periodic elections. If the goal of the legislator is to get reelected, he/she must serve the interests of a majority of the voting population.[5] If we were to assume that representatives do not vote in accordance with the interests of the voters who elect them, then no amount of tinkering with the electoral districts would make any difference. Unless a person’s vote can potentially determine an outcome, that is, elect a preferred representative who is preferred because he will vote a certain way, then by our first assumption there is no voting power.

Our third assumption in the Banzhaf analysis of the supervisors was that all outcomes on the Board of Supervisors are equally significant. This point has been greatly misunderstood, and has led the U. S. Supreme Court to twice reject the Banzhaf analysis. Last year in Board of Estimate of City of New York v. Morris, 109 S.Ct. 1433, Justice White writing for the majority quoted his earlier opinion in Whitcomb v. Chavis, 402 U.S. 124 (1971), stating “In [Whitcomb] we observed that the Banzhaf methodology ‘remains a theoretical one’ and is unrealistic in not taking into account any political or other facts which might affect the actual voting power of the residents, which might include party affiliation, race, previous voting characteristics or any other factors which go into the entire political voting situation.’ Id., at 145-146.” 109 S.Ct. 1433. Morris involved a New York City board which was made up of five Borough presidents with one vote each, and three at-large members (the Mayor, the Comptroller and President of the City Council) with two votes each. Residents of the larger boroughs sued because they alleged that residents of Staten Island had a disproportionate amount of voting power. Staten Island’s population was 352,121 whereas the population of the largest borough, Brooklyn, was 2,230,936. (The average borough population was 1,414,206.) The District Court and Court of Appeals made two erroneous calculations in determining the deviation in voting power among residents of the five boroughs. First, they failed to include the power of the at-large members; second they calculated voting power by dividing the member’s voting power by the voting population in his district when they should have divided by the square root of the population.[6] Because of these two mistakes, the lower courts determined a deviation of 132.9%,[7] when the more correct Banzhaf analysis would have found a deviation of only 30% in voting power. The Supreme Court agreed with Amicus Banzhaf with regard to the effect of the at-large members, but failed to adopt his rule for dividing by the square of the population. The court instead chose to follow the method elucidated in Abate v. Mundt, 403 U.S. 184, which prescribes that voting power is directly proportional to the number of voters in the district. Although the court criticized Banzhaf for not “reflect[ing] the way the board actually works in practice,” the court then went on to praise the Abate approach, because “It does not attempt to inquire whether, in terms of how the legislature actually works in practice, the districts have equal power to affect a legislative outcome.” In fact the only advantage of the Abate approach is that it preserves an aesthetic equality, because “it does assure that legislators will be elected by and represent citizens in districts of substantially equal size.”[8] The problem with the Abate test is that not only does it suffer from exactly the same criticisms as the Banzhaf method, but it also produces results which are incorrect. In Morris it distorted and exaggerated the divergence in voting power on the Board because it assumed, incorrectly, that voting power proportional to the number of voters, when in fact it is proportional to the square root.[9]

What seems to trouble the court most is that Banzhaf doesn’t take into account the “realities” of voting. Banzhaf assumes that every voter has a choice independent of the other voters. So, for example, on a Board of five members, there are 32 equally likely outcomes. The appellate court in Morris proudly pointed out that the five Borough presidents often voted as a block.[10] Banzhaf freely admits that his method did not take into account other determining factors, which might make certain votes less likely to occur. However, this is not necessarily true. Banzhaf’s method could easily accompany any predetermined set of conditions on voting, as we shall explain below. Banzhaf gave several reasons why his method failed to take into account voting patterns preferences, attributable to party affiliation and race. First, he argued that courts would have a difficult time calculating voting power disparities if it had to figure in voting patterns. (Note that much of the post-Gingles litigation has centered on proving the existence of racial bloc voting, involving reams of statistical evidence and much debate on statistical methods.) Second, he argued that many of the affiliations may be temporary, and would therefore require constant litigation and reconfiguration of districts equalized on account of those affiliations. Finally, Banzhaf argues, “the whole purpose of the ‘one man, one vote’ cases is to equalize the ‘rules of the game’—i.e., the allocation of voting power inherent in the rules for voting—and not the way the voting power is actually used in practice.”   Banzhaf, Amicus Brief. In rejecting the Banzhaf analysis in favor of the Abate test, the court chose to equalize the appearance of voting power, rather than equalize the power itself. White admits as much when he states,

The personal right to vote is a value in itself, and a citizen is, without more and without mathematically calculating his power to determine the outcomes of an election, shortchanged if he may vote for only one representative when citizens in a neighboring district, of equal population vote for two; or to put it another way, if he may vote for one representative and the voters in another district half the size also elect one representative Board of Estimate v. Morris, 109 S.Ct 1433.

The problem with the Abate approach is that it might favor a proportionally equal representation, when Banzhaf analysis would show it to be unfair. And it might strike down an equitable weighted voting regime, because it appeared to give unequal representation.[11]

The court dismissed the Banzhaf analysis because it relied on the equal likelihood of voting outcomes.[12] The question then is, would the court accept a Banzhafian analysis which did take into account the predilections of the voters. Suppose we want to find the voting power in districts in which there is racial bloc voting.[13] In our previous examples we have shown how such bloc voting will skew the power to effect the outcome of an election. Banzhaf merely directs us to calculate the number of possible outcomes in which a group has voting power. For example, suppose in one district we have four voting blocs—W, H, B, and A. W has 39%, H has 47%, B has 4%, and A has 10% of the votes. A winning coalition must have more than 50% of the votes. The winning coalitions are WH, WBA, HB, HA, WHB, WHA, HBA, and WHBA. A group has voting power in the coalition if it has a critical vote, that is, if the coalition is winning with the group, but not winning if the group withdraws. For example, all three groups have power in the coalition WBA, only H has power in WHA, and no group has power in WHBA. There are 16 different coalitions (8 winning and 8 losing). The Banzhaf index is the number of coalitions in which a group has voting power divided by the total number of possible coalitions. In our example, W has power in two coalitions (WH and WBA) out of sixteen for a Banzhaf index of 2/16 = 1/8. Group H has power in six coalitions (WH, HB, HA, WHB, WHA, and WBA) for a Banzhaf index of 6/16 = 3/8. Group B has power in two coalitions (HB and WBA) for an index of 1/8. Group A also has power in two coalitions (HA and WBA) for an index of 1/8. We write the voting power of the four groups (W,H,B,A) as a vector (1/8, 3/8, 1/8, 1/8). According to the Banzhaf index, H has three times the power of the other groups. W has the same power as the much smaller groups B and A. Once again, these calculations are based on three assumptions: (1) Voting power is the ability to determine the outcome of an election; (2) Absolute bloc voting exists; and (3) The voting blocs act independently in choosing to join or not to join a coalition. The last assumption means that any coalition of the four blocs is possible, and therefore significant.[14]

The question presented by Garza is, what is the voting power of the racial groups in L.A. County? We will use the Banzhaf index to calculate power in two ways. First, we will calculate the voting power of racial blocs within each of the five supervisorial districts. This will be equivalent to the secondary voting power, which the court has shown so much interest in, that is, the ability to elect a representative of the group’s choice. Second, we will calculate primary voting power for each group in the County as a whole, that is, the ability to determine the outcome of a vote on the Board of Supervisors. It should be clear that having secondary voting power does not guarantee having primary voting power.[15] The data we shall use is clearly inadequate, but efforts to obtain better information were unsuccessful.[16] Nevertheless, the method of our analysis would still be useful once actual statistical voting data has been determined.

Table 1 shows the population of L.A. County according to race. There are four distinct racial groups, each with a sizable population. White, Hispanic, Black and Asian. In our example above, we have already calculated the voting power of racial blocs in District 1. (W=White=39%, H=Hispanic=47%, B-Black=4%, and A=Asian=10%.) In Districts 4 and 5 it is obvious that the White voting bloc has all of the power since it constitutes more than 50% of the votes in each district, and therefore does not have to form a coalition with any other group. So we say that that in those districts Whites have voting power equal to one, and all the other groups have voting power zero. (That is, Whites will control 100% of the time and the other groups 0%.) However, in Districts 2 and 3 we again have a plurality among the racial groups, so we can proceed to calculate voting power as we did for District 1. Table 2 shows the results of these calculations. District 3 turns out to be identical to District 1, in that the Hispanic bloc can form a winning coalition with any one other group, whereas only a coalition of all three of the other blocs can defeat it. District 2 is similar to some of our previous examples, where any pair of the three larger groups can form a winning coalition, while the smallest group is powerless.

Table 1

Population of L.A. County by Race

Supervisor-ial District WHITE




1st Schabarum 732,809
















































L.A. County 3,553,695










Table 2

Secondary Voting Power in L.A. County by Race

Supervisor-ial District WHITE




1st Schabarum 0.125 0.375 0.125 0.125


0.250 0.250 0.250 0.000


0.125 0.375 0.125 0.125


1.000 0.000 0.000 0.000


1.000 0.000 0.000 0.000


At first glance, it would appear from Table 2 that the White bloc would have a great advantage when we came to calculate the primary voting power in L.A. County. After all, they control two districts outright and only three are needed to control a vote on the Board of Supervisors. But in fact, their majority in two districts does not give them any advantage. The relevant coalitions (ones in which each of the members casts a deciding vote, i.e. has voting power) which will control three or more supervisors are White-Black (Districts 2, 4 ,5), White-Hispanic (Districts 1, 2, 3, 4, 5) and Black-Hispanic (Districts 1, 2, 3). Asians can join any of the coalitions, but they will have no voting power since their votes is not necessary to make the coalition winning. Because any two of the three groups (White, Hispanic, and Black) can join to make a winning coalition, all three have equal voting power, and Asians have none. The voting power vector is (W, H, B, A) = (0.25, 0.25, 0.25, 0.0).

We might notice that the Black-Hispanic coalition is the weakest of the three winning coalitions. It wins only a narrow majority in Districts 1 and 3. If our data is not indicative of normal voting distributions, then it could very well be that the situation in L.A. County is more similar to a (3/8, 1/8, 1/8, 1/8) configuration, in which Whites can form a winning coalition with any one other racial bloc to control 3 or more districts, whereas the three other groups can only win by banding together or joining White. This is probably a more accurate description of voting power in L.A. County, since Hispanics tend not to vote in numbers proportionate to the size of their population.[17]

In the wake of the Garza suit, the court may decide to supervise the redistricting of L.A. County following the 1990 census. Our calculations above demonstrate that the configuration can have a very large impact on the voting power of racial groups. For example, Blacks may have as much power as Whites and Hispanics, but Asians, with a greater population may be left without any power. There is a potential to gerrymander the districts so that either Whites or Hispanics control three or more of the supervisors. A group only needs just over 50% in three of the five districts, or a little more than 30% of the total countywide population to control the Board of Supervisors. Both Whites and Hispanics make up well over 30% of the overall population. In fact, it has been demonstrated that the voting power of an interest group is maximized if its members are split evenly among just over half of the districts.[18]

Clearly, the most equitable solution would give some measure of voting power to all the racial groups. Since no group has more than 50% of the votes in L.A. County, such a solution would be possible. Unfortunately, Gingles and its progeny focus on the creation of majority districts (“safe seats”) and not ones which require coalitions.[19] The majority approach makes sense only when there is one group which constitutes an absolute majority, and the courts wish to afford the minority groups some secondary voting power only. But the situation in L.A. County permits distribution of primary voting power as well as secondary power so that all the racial groups have a share. Districts in which one race holds an absolute majority are the least equitable, and L.A.’s racial mix permits a solution which avoids such districts by setting up a plurality in each and every supervisorial district. Such a solution would require coalition-forming for primary and secondary voting power, and would afford all racial groups a degree of participation in the political process which previously has been impossible. As L.A. City Council member Richard Allatore predicts, “Ultimately, the success of Hispanics, of Asians, of Blacks will be through coalitions.”[20]

Although the coalition-based distribution of power allows greater participating by minority groups, it does not necessarily give groups power proportional to their size. We have demonstrated already many times in this paper how very small groups can have a relatively large amount of power, how larger groups can have no more power than much smaller groups, and how some sizable groups can be zoned out of the power distribution altogether. If we divide L.A. County into only four groups, then the power vector will probably look like (3/8, 1/8, 1/8, 1/8) or (1/4, ¼, ¼, 0). An equal distribution among all four groups is only possible if no combination of two groups has more than 50%, which will happen only if all four groups are the same size. Finally, if the two larger groups have the same population and the two smaller groups are equal too—for example, (40%, 40%, 10%, 10%)—then in that unlikely event the power vector would be (1/4, ¼, 1/8, 1/8). But that situation and the previous one are probably impossible, since there would have to be votes ending in a tie. So if we are asked to choose between (3/8, 1/8, 1/8, 1/8) and (1/4, ¼, ¼, 0), the former seems more equitable since it gives all groups a chance at achieving a winning coalition. But it also gives one group much more power than the other three. Power cannot be apportioned according to and in proportion with the population of a racial voting bloc. Perhaps the situation would appear more equitable if there were more than four voting blocs—for example, if each of the racial groups was divided into different political blocs, or if the racial groups were subdivided into smaller and more cohesive groups.

Regardless of the efficacy of the outcome, the Banzhaf method is a powerful tool in analyzing voting power. Any assumptions which the court is willing to make can be accommodated. For example, if there is not absolute racial bloc voting, but still a violation of the Voting Rights Act, the Banzhaf analysis can still determine the actual voting power of the groups assuming limited bloc voting. If only a percentage of a racial group votes as a bloc, say 70%, then assume there is a bloc of 70% and consider the other 30% as independent voters. Say there are two groups A and B with 100 voters each. Say that 70% of A and 60% of B vote as a bloc. Then our voting system has one bloc of 70 voters, one bloc of 60 voters and 70 independent voters (or 70 blocs of one voter each). The calculations are more complicated, but one can still figure out the likelihood that one of the blocs will cast a deciding vote. Banzhaf has shown that when there are a greater number of voting blocs, the members of a large voting bloc often have proportionately more power than members of small blocs.[21] But when the number of groups is small, a larger group may not have any more power, and might even have proportionately less power.

It is up to the court to decide which voting characteristics are important. There is some precedent for considering racial and political groups in determining the fairness of districting and election schemes. However, investigation into the statistical predilections of voters is extremely time-consuming, and of debatable utility. A decision must be made on the value of such inquiries. Once a standard has been established the court should not avoid mathematical models, merely because they are difficult or abstract. Indeed, mathematical models are perhaps the only method of determining the appropriate remedy. If the court has decided that the voting power of minority racial blocs may not be “diluted” then it should adopt an approach which provides for the most equitable distribution of power “to participate in the political process and to elect representatives of their choice.” (Voting Rights Act Section 2b.) Such an approach should use mathematical models to apportion both primary and secondary voting political power among the groups which the court has decided to protect.

The multiracial demographics of L.A. County present an opportunity for greater minority participation in government. My contention is that greater participation is not necessarily promoted by safe seats and supermajorities, which have been the remedies suggested by the Supreme Court—also the one sought by plaintiffs in Garza. If the Court is willing to reconsider its definition of voting power in favor of the widely accepted Banzhaf analysis, it might then achieve remedies which not only provide a fair chance of minority representation, but more importantly give all voters a piece of the ultimate political power.



[1] Group A’s extra vote could conceivably give A an advantage because it might have more resources, i.e. sources of funding, volunteers, canvassers, etc. But the availability of funding from the constituents and other resources is probably determined by the wealth or other characteristics of the group members and not their number.

[2] I first came across the Banzhaf method in an introductory game theory course at Princeton University. See Game Theory, Second Edition, Guillermo Owen (Academic Press Inc., 1982) p. 213-225.

[3] We can illustrate this by writing out the votes (Yes or No) of the five supervisors as follows:

Supervisor                    Supervisor                    Supervisor                     Supervisor

1,2,3,4,5                       1,2,3,4,5                       1,2,3,4,5                     1,2,3,4,5

(Y,Y,Y,Y,Y)                  (N,Y,Y,Y,Y)                  (Y,N,Y,Y,Y)                  (N,N,Y,Y,Y)

(Y,Y,Y,Y,N)                  (N,Y,Y,Y,N)                  (Y,N,Y,Y,N)                  (N,N,Y,Y,N)

(Y,Y,Y,N,Y)                  (N,Y,Y,N,Y)                  (Y,N,Y,N,Y)                  (N,N,Y,N,Y)

(Y,Y,Y,N,N)                  (N,Y,Y,N,N)                  (Y,N,Y,N,N)                  (N,N,Y,N,N)

(Y,Y,N,Y,Y)                  (N,Y,N,Y,Y)                  (Y,N,N,Y,Y)                  (N,N,N,Y,Y)

(Y,Y,N,Y,N)                  (N,Y,N,Y,N)                  (Y,N,N,Y,N)                  (N,N,N,Y,N)

(Y,Y,N,N,Y)                  (N,Y,N,N,Y)                  (Y,N,N,N,Y)                  (N,N,N,N,Y)

(Y,Y,N,N,N)                  (N,Y,N,N,N)                  (Y,N,N,N,N)                  (N,N,N,N,N)


[4] For each supervisor, the votes of the other four can split in the following six ways: (YYNN), (YNYN), (YNNY), (NYNY), (NYYN), and (NNYY). Since the supervisor can vote either yes or no for each of these, there are 12 combinations (of the 32 listed in footnote 3) in which the supervisor holds a critical swing vote.

[5] If an issue appeals only to a limited number of interested voters, the representative should choose the new of the majority of the interested voters. No assumptions can be made about the preferences of the disinterested voters, so the representative has to assume that they will split evenly between him/her and an opponent regarding that issue. It is therefore more correct to assume that legislator will vote for x over y if and only if the number of voters in his district who prefer x to y is greater than the number who prefer y to x.

[6] One important result of the Banzhaf analysis is that “in situations where each person has one vote his or her ability to cast a critical or decisive note doesn’t decrease linearly (proportionately) with the number of people voting.” (Banzhaf, Amicus Brief for Board of Estimate of City of New York v. Morris, 109 S.Ct. 1433.) In fact, for large numbers of voters the Banzhaf Index decreases according to the square root of the number of persons voting. For example, we have shown that the number of critical votes with five people voting is 12 out of 32 combinations or 37%. For 15 people voting, a person will cast the deciding vote in 6864 out of 32768 combinations or 21% of the time. Although the number of voters was increased by a factor of three (5 to 15), the Banzhaf Index of their voting power decreased by a factor of 1.8 (37% to 21%), which is approximately the square root of three.

[7] The total deviation is the deviation of the smallest borough from the average plus the deviation of the largest borough from the average. Staten Island was 75.1% smaller than average. Brooklyn was 57.8% larger than average. Therefore the deviation is 75.1% + 57.8% = 132.9%

[8] The appellate court held that, “the district court properly adopted the Abate theory that what the equal protection clause requires be equalized among voters is their ‘share’ of representatives on the government body in question. In the simplest cases, where each representative has an equal vote (i.e., there is no weighted voting), Abate requires only that the court look to the number of citizens in each representative’s district; equality is achieved where those numbers are equal.” 831 F.2d 384. The court’s sole reference to a voter’s share, as opposed to their more common reference to voting power, is apt. A voter’s share is proportional to the population of the district. For example, if the representative is given $1,000,000 and decides to hand it out to all 100,000 constituents in equal shares, each voter should expect to receive $10. However, whether the representative decides to hand out the money to all voters will necessarily be decided by the desires of a majority of the voters (applying the analysis above) and in that sense the voter’s power to form a majority is the essential factor. That power is best calculated by the Banzhaf Index, and corresponds to the square root of the population (footnote 6).

[9] Using the Abate test, the court has approved multi-member district schemes, which under Banzhaf analysis would appear extremely unfair. See Whitcomb v. Chavis, 403 U.S. 124.

[10] Of course, the obvious response is that if they voted as bloc, it shouldn’t make any difference that the Boroughs were different sizes. (Banzhaf, Amicus Brief, Board of Estimate v. Morris, 831 F.2d. 384.)

[11] For example, in a legislature with many multimember or weighted districts, if district A has 200 people with 2 representatives, and district B has 400 people with 4 representatives, Abate would approve although Banzhaf would not. Likewise, if district A had 200 people and a representative with 2 votes, while district B had 400 people and a representative with 2.83 votes, Banzhaf might approve, but Abate would clearly object. The former case resembles Whitcomb v. Chavis, 403 U.S. 124. See also, Banzhaf, Multi-Member Electoral Districts—Do They Violate the “One Man, One Vote” Principle, 75 Yale L.J. 1309 (1966).

[12] Banzhaf objects to the use of “likelihood” because of the court’s relentless harping on the unrealistic assumption. He instead favors “significance”. Both terms are in fact accurate. His response should be that if the court would determine which outcomes are more likely, then his method could include that into the calculations.

[13] The method we propose would also work for party bloc voting, or any other assumptions which contradict the independence of individual voters.

[14] One might argue that two of the blocs always vote together. Then they should be considered as one bloc, not two separate blocs. Remember that our example assumed that no single bloc had a majority of the votes. Having an election assumes that at some level there are independent decisions being made. If this were not true, then there would no need to have an election. The outcome would be predetermined.

[15] We have already shown that merely having a vote does not mean having voting power. Similarly, having a representative does not mean that representative has any voting power on the legislative body.

[16] Our source is the L.A, County Department of Health Services, Population Estimate and Protection System, June 21, 1989. The data shows only total population broken down by race. The data could be improved by regarding only citizens, voting-age citizens, people eligible to vote, registered citizens, actual voters. It is debatable whether actual or potential voters should be used, since the presence or absence of voting power may have an impact on the number of actual voters within a group.

[17] “Studies over the decade have shown that Asians and Latinos vote at a significantly lower rate than other citizens. A study by political scientists at Caltech found that 11% fewer Asians and 20% fewer Latinos voted in the 1984 presidential election, compared to blacks and whites who voted at virtually the same rate.” L.A. Times, May 7, 1990, p. 24.

[18] J. Snyder, “Political Geography and Interest Group Power”. Social Choice and Welfare (1989) 6:103-125. For very small groups, their power is maximized if they are spread over all the districts.

[19] The fourth Circuit rejected an innovative “limited voting” remedy because it read Gingles as requiring the acceptance of the County’s single-member district proposal which set up a few “safe” minority districts. Because the minority population was spread out among the white majority population, only a few safe districts could be created. The limited voting scheme would have afforded black voters with more power than the creation of safe seats would allow. The Court read Gingles’ admonition against a right in proportional representation as requiring only a safe seat remedy, even though some other remedy might afford greater representation. McGhee v. Granville, 860 F.2d 110.

[20] L.A. Times, May 7, 1990, p. 24.

[21] See for example, Banzhaf, 3.312 Votes, A Mathematical Analysis of the Electoral College, 13 Villanova L. Rev. 303 (1968). Presidential election voters in California have more than three times the voting power of voters in much smaller states, such Rhode Island or Washington D.C., even though smaller states have proportionately more electoral college representatives.

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