Voter registration laws can devalue a vote on average

The right to vote is among the most treasured privileges for Americans.  From time to time, it would seem reasonable that the Texas Legislature close loopholes that invite election fraud, which devalues a legal vote.  Matt Mackowiak, 27 Jan. 2001

Statements like this are the standard justification given for new legislation that requires higher standards of identification before a person can vote.  We could interpret what is not said (e.g., the treasured privilege or right—which is it?—is not so treasured that everyone should be issued a national ID card and be automatically eligible to vote), but let us focus on the phrase “devalues a legal vote.”

Actually, it is possible to devalue a vote more through restrictive voter ID laws.   Some algebra (described below) shows,  if I am not mistaken, that the value of a vote decreases on average if

  • the proportion of falsely rejected votes > (originally fraudulent fraction * accepted fraction after the law is implemented) / (originally legit fraction * rejected fraction after the law is implemented)

For example, if 10 votes of a 1000 would have been fraudulent and the new law leads to 20 votes not being cast, of which 10 would not have been fraudulent, then the vote is devalued because .5 > (.01 * .98)/(.99 * .02).

The way this result comes about is that under the restrictive laws, for the people who get to cast their vote legitimately, the value of their vote goes up, but for those who do not get to cast a vote that would have been legitimate, their vote value goes to 0.

Moreover, if the value of a vote is measured in terms of whether the candidate who would have had the most legitimate votes is elected, then the situation can be even more skewed.  However, to show that requires detail about a specific election.

Even if we put aside the obvious political motivation in disenfranchising likely voters for one’s opponent, there’s still a conceptual question about why the standard justification given for new legislation seems plausible, even unimpeachable, to most listeners—No-one wants to be on the side of allowing fraudulent votes, right?  But why don’t people automatically say: “Yes, but how many legitimate votes will not happen using your laws.  After all, the right to vote is treasured by American citizens.”  (At least, by the fraction who register and then vote…)


If N= no. of voters before law is implemented; fN = no. of fraudulent voters before law is implemented; (1-p)N = no. of accepted votes after law is implemented; and ipN = no. of votes not accepted or not cast after law is implemented, then

True value of a vote is 1 for eligible voters N(1-f) and 0 for ineligible voters fN.

Value before law is implemented is (1-f) for eligible and ineligible voters.

Value after law is implemented is 1 for (1-p)N (which may still include some ineligible voters) and 0 for pN, which includes ipN eligible voters.  The average of this is (1-p)N / [(1-p)N +ipN) = (1-p) / [(1-p)+ip].  Compare this quantity with (1-f) and you find that it’s less if

  • i> [f(1-p)] / [(1-f)p]

To use the language of medical screening, this is a matter of how many true positives there are—that is, how many fraudulent votes (or attempts to vote) are detected and rejected by a given ID requirement—compared to false positives (rejected votes from eligible citizens), true negatives (accepted votes from eligible citizens), and false negatives (accepted votes from non-eligible citizens).

About Peter J. Taylor
Peter Taylor is a Professor at the University of Massachusetts Boston where he teaches and directs undergraduate and graduate programs on critical thinking, reflective practice, and science-in-society. His research and writing focuses on the complexity of environmental and health sciences in their social context, incl. Unruly Complexity: Ecology, Interpretation, Engagement (U. Chicago Press, 2005) and Nature-nurture? No (2014, On reflective practice, see Taking Yourself Seriously: Processes of Research & Engagement (with J. Szteiter, 2012,

2 Responses to Voter registration laws can devalue a vote on average

  1. Another way to present this is to say fraudulent voters devalue a legitimate vote from 1 down to 1-f
    Eliminating them revalues the vote for those who are able to cast one up to 1, which sounds right until you see that it comes at the expense of devaluing the vote for legitimate voters who are prevented from voting down to 0. What does a citizen’s right to vote mean if it comes at the expense of some other people’s right to vote?

  2. Pingback: Epidemiological thinking in public discourse: Voter registration laws can devalue a vote on average | Innovation in Epidemiological Research

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