
108 F.Supp.2d 355 (2000)
G. Oliver KOPPELL, Arnold Linhardt, Marie Morrison Plaintiffs,
v.
NEW YORK STATE BOARD OF ELECTIONS, et al., Defendants.
No. 98 Civ. 4920(SHS).
United States District Court, S.D. New York.
August 11, 2000.
Fred Taylor Isquith, Wolf, Haldenstein, Adler, Freeman & Herz LLP, New York City, for Plaintiffs.

OPINION
STEIN, District Judge.

I. INTRODUCTION
G. Oliver Koppell-a candidate for the Democratic nomination for New York State Attorney General in 1994 and 1998-and Arnold Linhardt and Marie Morrison-two New York state voters-brought this action challenging the constitutionality of New York Election Law § 7-116(3). Pursuant to Section 7-116(3), in primary elections in the 57 counties outside of New York City the order of candidates on the ballot is determined by lottery, so that the same candidate appears first on every ballot. See N.Y. Election Law § 116(3). In contrast, within New York City ballot position is rotated by election district, so that each name appears first and in each other position an equal number of times. See N.Y. Election Law § 7-116(6).
Plaintiffs argue that as a result of "position bias," which is the hypothesis that a certain number of votes are automatically *356 cast for the candidate whose name appears first on the ballot, the lottery system infringes upon their First and Fourteenth Amendment rights, specifically the right to vote, the right to freely associate, and the right to become a candidate for office.
Two years ago this week, this Court denied plaintiffs' motion for a preliminary injunction, on the grounds that plaintiffs failed to demonstrate a likelihood of success on the merits. See Koppell v. New York State Board of Elections, 8 F.Supp.2d 382 (S.D.N.Y.1998). That determination was affirmed by the U.S. Court of Appeals for the Second Circuit. See Koppell v. New York State Board of Elections, 153 F.3d 95 (2d Cir.1998). Familiarity with those two opinions is assumed.
Following discovery proceedings and the exchange of expert reports, a bench trial was held on May 22 and 23, 2000. Upon consideration of the evidence presented and the testimony adduced at the trial, this Court finds that plaintiffs have failed to demonstrate that New York Election Law § 7-116(3) infringes upon their constitutional rights. The following constitutes the Court's findings of fact and conclusions of law pursuant to Fed.R.Civ.P. 52(a).

II DISCUSSION
When entertaining challenges to state election laws, a court
must first consider the character and magnitude of the asserted injury to the rights protected by the First and Fourteenth Amendments that the plaintiff seeks to vindicate. It then must identify and evaluate the precise interests put forward by the State as justifications for the burden imposed by the rule. In passing judgment, the Court must not only determine the legitimacy and strength of each of those interests; it must also consider the extent to which those interests make it necessary to burden the plaintiff's rights.
Anderson v. Celebrezze, 460 U.S. 780, 789, 103 S.Ct. 1564, 75 L.Ed.2d 547 (1983). In Burdick v. Takushi, 504 U.S. 428, 112 S.Ct. 2059, 119 L.Ed.2d 245 (1992), the Supreme Court clarified the standard set forth in Anderson as follows:
[T]he rigorousness of our inquiry into the propriety of a state election law depends upon the extent to which a challenged regulation burdens First and Fourteenth Amendment rights. Thus, as we have recognized when those rights are subjected to `severe' restrictions, the regulation must be `narrowly drawn to advance a state interest of compelling importance.' ... But when a state election law provision imposes only `reasonable nondiscriminatory restrictions' upon the First and Fourteenth Amendment rights of voters, `the State's important regulatory interests are generally sufficient to justify' the restrictions.
Id. at 434, 112 S.Ct. 2059 (citing Anderson).
Plaintiffs' specific allegations-upon which they premise their claim that this regulation imposes a severe burden on their constitutional rights-are that position bias exists in primary elections in New York State, that position bias is of sufficient magnitude to affect the outcome of primary elections when the placement of candidates' names is not rotated on the ballot, and that defendants have not demonstrated a strong enough interest to justify the burdens imposed by the lottery system.
Defendants contend that plaintiffs' findings are flawed and do not provide conclusive evidence that position bias affects the outcome of upstate elections. Further, defendants claim that even if the Court were to accept plaintiffs' allegations, the effects of the lottery are minimal and are justified by the state interests presented.

A. The Evidence of Position Bias in New York City's Primary Elections
Dr. Henry Bain, plaintiffs' expert on the *357 issue of position bias,[1] conducted two studies for the purposes of this litigation: an analysis of the results in the 1994 Democratic party primary for Attorney General in Queens, and an analysis of the results in seventy-nine 1998 Democratic Party primaries in New York City. See Pltf's Exh. C, Bain Aff., 7/9/98, at ¶ 10; Pltf's Exh. E, Plaintiffs' Expert Report on the Existence and Magnitude of Position Bias in 1998 New York City Democratic Party Primaries [the "Bain Report"], at ¶ 14. According to Bain, the rotation of ballot positions in primary elections in New York City permitted him to isolate the effects of position bias in primary elections by comparing how candidates performed when they appeared in different positions on the ballot. Tr. 4/10/00, at 75-76; Bain Report, at ¶ 41.
Bain measures position bias in two ways. First, he compares each candidate's percentage of the total vote from the ballots on which he or she appeared in first position with the candidate's percentage of the vote in all of the election district's other ballots. Bain Report, at ¶ 16. Any excess percentage when in first position is considered evidence of position effect. Bain Report, at ¶ 16. This measure is used to state the magnitude of a single candidate's first position advantage. Bain Report, at ¶ 16. Second, he measures the overall "bounce" candidates receive from being in first position. Bain Report, at ¶ 19. "Bounce" is the term used to describe the average percentage increase that all the candidates in a given race receive from being in the first position on the ballot. Bain Report, at ¶ 19.
In his study of the 1994 Attorney General primary, Bain reported a position effect equal to "what would be found if 4.7% of the voters automatically and blindly voted for whoever was listed first, while all of the remaining voters were unaffected by any tendency to vote for the first candidate." Pltf. Exh. C, Bain Aff., at ¶ 15. In studying the 1998 primaries, he found a position effect with a reasonable statistical significance in 64 of the contests. Bain Report, at ¶ 25. In 13 of the 79 cases there was no advantage and in 2 cases there was an advantage that was not statistically significant, meaning that the odds of the results happening by random chance were greater than 1 in 10. The candidates experienced an average excess percentage of 8.5% when in first position. Bain Report, Exh. 1. Bain reported a mean position effect of 2.8% in the race for the Democratic Party nomination for Attorney General and an average mean position effect of 2.5% for all statewide offices. Bain Report Exh. 1. In statewide elections candidates received a 10.4% "bounce" when listed in first position and an average of 9.5% "bounce" over all of the elections. Bain Report, Exh. 2. These averages do not include those races for which there was no discernible position effect. Tr. 4/10/00 at 134 (Darcy).
Defendants' expert, Dr. Robert Darcy, criticized Bain's method of measuring statistical significance. Def.'s Exh. 11, Written Response to Report of Henry Bain ["Darcy Response"], at ¶¶ 6-8. In response, Bain recalculated the significance of his results by incorporating Dr. Darcy's criticism. Pltf's Exh. F, Written Response to Reports of Robert Darcy and James Chapin ["Bain Response"], at ¶ 2. He found that even assuming Darcy's criticism, 59 of the 79 races showed a statistically significant position effect. Bain Response, at ¶ 2.
Bain found that if the runner-up had been listed first in 12 of the 79 Democratic primaries, that candidate would have won the election. Pltf's Exh. J; Tr., 4/10/00, at 47. In other words, because the runner-up outpolled the ultimate winner in the precincts where the runner-up was listed first, the runner-up would have won the *358 election had the runner-up been listed first on every ballot. Bain also considered whether the outcome would have been different in the statewide primaries, by extrapolating from the mean position effect found in studying the New York City results. Pltf's Exh. I. Extrapolating from these results, Bain found three statewide elections that would have had a different outcome had the runner-up been listed first in the past 18 years: the 1992 United States Senate Democratic primary, the 1982 Democratic Comptroller primary, and the 1994 Democratic Attorney General primary. Pltf's Exh I; Tr., 4/10/00, at 43. However, his analysis also demonstrated that the outcome would have been the same had there been rotation upstate, as opposed to a lottery with a different outcome. Pltf's Exh. I; Tr., 4/10/00, at 56.

B. Is the evidence of position bias sufficient to demonstrate a constitutional violation?
Plaintiff have shown to the satisfaction of this Court that position bias affects a certain number of votes cast in primary elections in New York City. Nonetheless, they have failed to demonstrate that the lottery system employed outside of New York City is unconstitutional.

i. Plaintiffs' evidence of position bias in New York City elections cannot be extrapolated to the 57 other counties in New York State.
First and foremost, the ballots in the upstate counties-with the exception of Albany County-are printed on horizontal ballots. Def. Exh. 15, Tr. 5/23/00 DeBiase, at 68. In New York City the ballots are vertical. Def. Exh. 14. Plaintiffs have presented no evidence that position bias exists where the choices are arranged in a horizontal rather than vertical fashion. Bain admitted that he was not aware that horizontal ballots were used in elections outside of New York City. Tr. 4/10/00 at 66 (Bain). Moreover, in the 1957 monograph that he co-authored with Donald Hecock, entitled Ballot Position and Voter's Choice: The Arrangement of Names on the Ballots and its Effect on the Voter, the authors did not find a position effect when horizontal ballots were used. Def. Exh. 3 at 88 ("No one position within any one horizontal row was consistently favored."). This Court is thus unable to extrapolate the New York City results to the upstate primaries; for all but one upstate county, plaintiffs have presented no evidence that position bias will affect the outcomes of primary elections.
Second, Bain's testimony that the position bias he found in New York City can be extrapolated to the 57 upstate counties is speculative. Darcy 4/10/00 at 136-37. He did not include the upstate counties in his study and admittedly has no knowledge of how Democratic party politics is conducted in any of the 57 counties at issue or of the income and education levels of the voters. Bain 4/10/00 at 60, 62. In addition, he reports significant variations in position effect-from race to race, and within races among competing candidates for the same nomination-without any degree of predictability or the offer of a theory or explanation. Tr. 4/10/00 at 51 (Bain). For example, Bain testified that the substantial range of position effect between and among competing candidates-such as Chuck Schumer's 2.9% position effect being more than double Mark Green's 1.2% position effect-was "very interesting," but he did not attempt to account for such variations. Bain 4/10/00 at 51. As such, it is not statistically valid to generalize from his reported findings in New York City to the 57 other counties, Tr. 4/10/00, at 136-37 (Darcy), especially because he simply assumed the same effects upstate but conceded that voter behavior differs "greatly with respect to things like race and class." Tr. (Bain) 4/10/00 at 86-87.

ii Even if the Court accepts plaintiffs' allegations that position bias affects the outcomes of primary elections in New York State, the evidence does not demonstrate that the lottery system is unconstitutional.
Even if Bain's findings were accepted, the reasonable, nondiscriminatory restrictions *359 imposed by the lottery do not render it unconstitutional. As noted above, voting regulations that impose "severe" restrictions "must be narrowly drawn to advance a state interest of compelling importance." Burdick, 504 U.S. at 434, 112 S.Ct. 2059 (citations omitted). "But when a state election law provision imposes only `reasonable, nondiscriminatory restrictions' ... `the State's important regulatory interests are generally sufficient to justify' the restrictions." Id., (quoting Anderson v. Celebrezze, 460 U.S. 780, 788, 103 S.Ct. 1564, 75 L.Ed.2d 547 (1983)); see also Schulz v. Williams, 44 F.3d 48, 56 (2d Cir.1994).
The lottery system in use outside of New York City is "by definition, nondiscriminatory." Koppell, 153 F.3d at 96. "As long as a state's system of ballot placements treats all candidates in a non-discriminatory manner, there is no constitutional right to a preferred position on a ballot." Koppell, 153 F.3d at 96. No court to address ballot listing schemes has invalidated a nondiscriminatory scheme. See, e.g., McLain v. Meier, 637 F.2d 1159, 1167 (8th Cir.1980); Holtzman v. Power, 62 Misc.2d 1020, 313 N.Y.S.2d 904, 907, (N.Y.Sup.Ct.1970), aff'd 34 A.D.2d 917, 311 N.Y.S.2d 824 (1st Dept.1970), aff'd 27 N.Y.2d 628, 313 N.Y.S.2d 760, 261 N.E.2d 666 (1970) (invalidating law giving first position to incumbents). Moreover, most courts to address the issue have explicitly noted that the scheme must be discriminatory to be considered unconstitutional. See Board of Election v. Libertarian Party, 591 F.2d 22, 24-25 (7th Cir.1979) ("A successful challenge to a ballot placement procedure under the equal protection clause requires a showing of `an intentional or purposeful discrimination by authorities in which one class is favored over another'") (citation omitted); Sangmeister v. Woodard, 565 F.2d 460, 465 (7th Cir.1977) ("[T]he plaintiff must `prove the existence of an intentional or purposeful discrimination by authorities in which one class is favored over another.'"); Libertarian Party of Colorado v. Buckley, 938 F.Supp. 687 (D.Colo.1996) (denying motion for preliminary injunction against ballot placement scheme, noting that scheme "is facially neutral"); Strong v. Suffolk County Board of Elections, 872 F.Supp. 1160, 1164 (E.D.N.Y.1994) (plaintiff bringing position bias claim pursuant to Fourteenth Amendment "must prove the existence of an intentional or purposeful discrimination by authorities in which one class is favored over another").
Compared to other burdens on voting rights, the burden imposed on candidates by the lottery is minor. This is not a case where any candidate is denied access to the ballot, see, e.g. Anderson, 460 U.S. at 792, 103 S.Ct. 1564 (striking down Ohio's early filing deadline that had the effect of excluding late-filing independent candidates from the ballot), or where a regulation precludes a voter from voting, see, e.g. Dunn v. Blumstein, 405 U.S. 330, 360, 92 S.Ct. 995, 31 L.Ed.2d 274 (1972) (striking down residency requirement of one year in state and three months in county). The lottery does not infringe on First and Fourteenth Amendment rights by "freez[ing] the status quo." Jenness v. Fortson, 403 U.S. 431, 438, 91 S.Ct. 1970, 29 L.Ed.2d 554 (1971).
Nor is it a case where the one person, one vote principle is offended. Contrary to plaintiffs' arguments, votes cast in support of lottery losers are not "diluted" by the lottery in the same sense that votes cast by voters in larger-than-average election units can be diluted by vote-counting systems which, for example, weigh the votes of rural voters more heavily than those of urban voters. See, e.g., Gray v. Sanders, 372 U.S. 368, 83 S.Ct. 801, 9 L.Ed.2d 821 (1963). Position bias, unlike the dilution of votes caused by drawing of election districts or by vote-counting systems, is not the ineluctable product of the state's election system. Position bias depends on such factors as the amount of information and encouragement voters receive on how they should vote and voters' *360 motivation, and thus can largely be ameliorated by voter education concerning the candidates. Pltf's D, Supp. Bain Aff., at ¶ 4.
To the extent that plaintiffs contend that the lottery "losers" are forced to work harder to educate voters to overcome the "head start" afforded to the lottery winner, the cases addressing ballot nomination petition statutes are instructive. In Schulz v. Williams, 44 F.3d 48 (2d Cir.1994), the Second Circuit reviewed a statute requiring that ballot nomination petitions for "independent bodies" include each signer's election district, and, where applicable, assembly district or ward. The Court "recognize[d] the plaintiffs' evidence that vote canvassers spent 50% to 70% of their time processing these numbers," but concluded that "that fact alone does not make the burden `severe.' ... `[H]ard work and sacrifice by dedicated volunteers are the lifeblood of any political organization.'" Id. at 57 (quoting American Party of Texas v. White, 415 U.S. 767, 787, 94 S.Ct. 1296, 39 L.Ed.2d 744 (1974)). To conclude that position bias is insurmountable would be to abandon "faith in the ability of individual voters to inform themselves about campaign issues." Anderson, 460 U.S. at 797, 103 S.Ct. 1564. While the lottery system may not be the ideal system for limiting the effects of position bias, the form of the ballot is essentially a legislative determination and the lottery system provides every candidate with an equal chance to achieve the top ballot placement.
Defendants have presented sufficient evidence that the state has a sufficient interest in maintaining this ballot listing scheme to justify the burden imposed. The evidence was that (1) the lottery system is less costly to operate and administer than New York City's rotation system, Pltf's Exh. P (Deposition of John Howard Clifton), at 22-25; Def's Exh. 24, at ¶¶ 12-16 (DeBiase Aff.); and (2) the lottery system avoids confusion and delay, especially when last minute changes are required, Pltf's Exh. P, at 54; Tr. 5/23/00, at 54-53.
Accordingly, because the lottery system is nondiscriminatory and the burden it imposes is minimal, New York State's important regulatory interest in the efficient administration of elections is sufficient to justify the use of the lottery system in the 57 counties outside of New York city.

III CONCLUSION
For the reasons set forth above, judgment shall be entered in defendants' favor.
NOTES
[1]  This Court earlier qualified Dr. Bain as an expert regarding the existence of position bias in accordance with the standards set forth in Daubert v. Merrell Dow Pharmaceuticals, 509 U.S. 579, 113 S.Ct. 2786, 125 L.Ed.2d 469 (1993), and Kumho Tire Co. v. Carmichael, 526 U.S. 137, 119 S.Ct. 1167, 143 L.Ed.2d 238 (1999). See Koppell v. New York State Board of Elections, 97 F.Supp.2d 477 (S.D.N.Y.2000).
