Dr. David McCune is an associate professor of mathematics at William Jewell College in Liberty, Missouri. He primarily studies problems in the field of mathematical political science, focusing on apportionment and social choice theory. He also enjoys studying games of chance, especially when Markov chains might be involved.
Episode Description
The issue of voter fraud has taken up increasing amounts of the public’s imagination since the 2020 election. Spurred in part by claims from former U. S. President Donald Trump that the election was stolen from him. On their face, some of the claims of fraud seem irrational. Others however, require a bit of statistical investigation before they can be fully debunked. That’s the focus of this episode of Stats+Stories with guest Dr. David McCune.
+Full Transcript
Rosemary Pennington The issue of voter fraud has taken up increasing amounts of the public imagination since the 2020 election, spurred in part by claims by former US President Donald Trump, that the election was stolen from him. On the surface, some of the claims of voter fraud seem irrational. Others, though, need a bit of statistical investigation before they can be fully debunked. That's the focus of this episode of stats and stories where we explore the statistics behind the stories and the stories behind the statistics. I'm Rosemary Pennington. Stats and Stories is a production of Miami University's departments of statistics and media journalism and film, as well as the American Statistical Association. Joining me is regular panelist John Bailer, Chair of Miami statistics department. Our guest today is Dr. David McCune and Associate Professor of Mathematics at William Jewell College in Liberty, Missouri. He primarily studies problems in the field of mathematical political science, focusing on apportionment and social choice theory. He also enjoys studying games of chance, especially when Markov chains might be involved. McCune is also the co-author of an article for chance about Benford's law and county level voting. David, thank you so much for joining us this afternoon.
David McCune Thank you so much for having me. It's a real honor to be on.
Rosemary Pennington
What is Benford’s law?
David McCune Yeah, the question of what Ben, first of all is, is a very good question. And the article, of course, it's a prerequisite for understanding the article. So Ben, first law is a, I consider it a statistical model that describes how certain datasets behave. And it describes the distribution of first digits from a numerical data set. So the example I always start with in my course, as I show the students a spreadsheet with the populations of every town and city in the state of Missouri, where I teach, and you ask the students, if you were to choose a town or city at random, and measure its population, what is the probability that that population starts with a number one? Like, what's the probability the leading digit of a population is one, and there are nine leading digits, so the students will usually say the probability is 1/9, so around 11%, and then when you actually show them the distribution, the percentages are usually surprised. So if you do what I described, you take all population data from Missouri cities and towns and measure the leading digit, the leading digit of one occurs approximately 30% of the time, the leading digit has to occur approximately like 17% of the time, and so on down the probabilities decrease. So the probability of seeing a population with a leading digit of nine is only around four or 5%. And students usually find this surprising.
John Bailer 2:57
So are there other other examples? I mean, before we dive into kind of the story, are there other kind of classic examples of Benford's law where this pattern of you know ones being much more frequent than twos, which are much more frequent than threes, and so on.
David McCune That, yeah, if you go to like the Wikipedia article, which is really good for Benford's law, they get a lot of the classic examples. So I don't usually show them to the students so much, because they find them. I don't want to say boring, but it is more boring than, say, population data. But, you know, if you go to Google Maps, and you grab the links, the length of every river in North America and build a spreadsheet with that data, it will follow them for as long as the physical constants and chemistry, if you fill up a spreadsheet with those, they'll follow Ben for his law. Those are sort of the classic examples. For me, I usually show population examples. So any state, if you get the town and city population data, it will follow the first law pretty well. And any numbers sort of based on population. So if you grab like, county level data, about COVID cases, cumulative COVID cases or COVID, deaths, things like that, that will follow then for his law, the classic application of it that I've seen is in financial fraud. So since 1996, or so people have known that a lot of financial data follows the law. And so you can generate examples from accounting data.
Rosemary Pennington
So your article and chances about election fraud, and particularly, you're looking at the issue of Pennsylvania. So do you want to talk us through what you and your co author were looking at and sort of why you took this particular approach to this issue?
David McCune Yes, so as a preamble, let me just say that a little bit of knowledge can be dangerous. Of course.
Rosemary Pennington John knows that.
John Bailer About me here, Rosemary.
David McCune
So I'm constantly in danger. But yeah, so when I teach this to my students, I'm always actually a little hesitant to do it because it's really cute and fun and the students find it interesting but you know, you can say things like Benford’s law has been used to detect financial fraud. And then somebody might hear that and forget willfully or not, I guess, forget the adjective financial, you know, and just think, oh, it can detect fraud? And no, not really at least, it hasn't been shown to do that in every case. So, Benford’s law got a lot of attention, actually, after the 2020 election, not in the context of this particular article. But generally speaking, I had never heard so much discussion of infers law before surrounding election data. So yeah, I mean, you can, if you are not careful, you can take a tool that has been shown effective in other contexts. And if you apply it willy nilly, you're just going to make mistakes. In terms of the election itself, the claim that I saw made about been first law that I analyzed in this paper revolves around county level votes. So trying to detect voter fraud by looking at county level data. And related to the preamble, I just gave, I would say that the people making this argument are making the error that some of my students, or myself might make, I've just forgetting the adjective, financial, and just saying, Oh, I can just use this to detect fraud when I feel like it. So that's how I found this claim, essentially, is that I was looking for arguments about fraud that were probabilistic, or statistical to discuss with my students and I stumbled across the one that that's in the article. There are other Benford's law arguments that were made that got much more into the mainstream, and other people analyze that better than I would have. So this the argument in the article itself was pretty obscure. I found it in several places, but it didn't rise up into the mainstream.
John Bailer
So what was the claim of fraud that was being asserted that was tied to Benford’s Law?
David McCune
Yes. So again, benefits law can oftentimes model population kind of data, like populations of towns, this argument was trying to say that the first law should apply to county level presidential votes for a major party candidate. So if you go to a state like Pennsylvania, or whatever, and get the vote data by county for each candidate, that should follow Ben for his law. So in Pennsylvania, in particular, if you look at Trump's county level vote data across the 67 counties, his leading digit distribution follows bench press law pretty closely like it, it matches what Benford would predict in terms of those percentages. And Biden's does not. So you could if you want to. And some people did want to, you could say, well, that's evidence of fraud, because I've detected a slight statistical irregularity in Biden's votes that were not present in Trump's votes.
John Bailer 7:49
So then how did you dive in and explore this question in some systematic way? I mean, you, I know that you started with a series of premises that are associated with county votes, a premise for whether or not things are anomalous. And another third premise about you know, whether or not the failure of the votes to follow this indicates fraudulent activity. So you have these, these three, three kinds of assertions that you're going to investigate? And then you also then generalize this beyond just a single state. So could you step us into this to your thought process about how you are going to, to formally evaluate this assertion of fraud?
David McCune Yes, so the people who made this argument that I found didn't try to formalize it in any meaningful way, as far as I can tell, which is fine. You know, people are speaking sort of colloquially, but when somebody makes an argument colloquially, and you want to analyze it, you should sort of pull it apart and see the premises that it rests on. And so if you're going to say, Trump's votes follow Ben for his law, and Biden's don't and county level votes and Pennsylvania. And that's your argument for fraud. If I want to analyze that I need to think about what kind of premises that argument rests on. So, as best as I and my co author can tell, there were three premises that these people were using to make this argument. And the first premise is that just in general, generally speaking, county level votes for major presidential candidates should follow Ben for his law in a given state. If that premise isn't true, then what are we doing here with this argument? You know, the second premise is that Biden's votes are somehow suspicious with respect to Ben for his law. If his county level votes don't raise any red flags, then again, what are we doing here? The argument falls apart. And furthermore, you would have to argue that a statistical irregularity that's in quotes, by the way, is it good to do air quotes on a podcast?
John Bailer Okay,only if you say that an air quote has occurred.
David McCune
So, air quotes, statistical irregularity, even if such a thing occurs, you have to then argue that that's evidence of fraud by certain people is not necessarily true, right? If You find some sort of irregularity, you can then point to a single person or group of people and say, fraud must have occurred there. So if you want to say that Democrats in particular, however you're using that term, for me personally, it was never clear if we were talking about the Democratic party apparatus as committing fraud, or just like, you know, random groups of Democratic voters are committing fraud. But if you want to claim such groups are committing fraud by finding statistical irregularities, you would have to make that connection. It's not just implied.
John Bailer 10:29
You know, I thought it was interesting that the way that this was framed, I mean, essentially, Benford law gives you a kind of unexpected percentage of each of these conditions, you know, a leading digit one and a vote count leading digit two and so on. And you, you then effectively just determined expected counts based upon if Benford Zod was true, and then looking for some kind of systematic deviation using some Goodness of Fit procedures. So that seems like that was a really kind of well framed way of investigating that. You didn't just restrict yourself to Pennsylvania, though. I mean, I thought that was an important part of the story is that if you look more broadly across multiple states and saw whether or not you know, they're, you know, whether it was was met or not, in terms of reasonable description of these leading digit and the votes, you saw that there were a fairly large fraction of times, it didn't apply?
David McCune Yes. So if you limit yourself to Pennsylvania, you're just sort of restricting your data set, which I don't know, I guess might be appropriate in some situations. But in this case, you want as much context as you can get. So we looked at all states with sort of enough counties where the law might apply. So some states you got to toss out like Delaware, I think only have three counties. So there's no meaningful way to talk about Benford's law there. Yeah, we looked at all states with enough counties to make meaningful conclusions from the years 1996 through 2020. So we got a pretty sizable dataset, just to get an idea of how often Ben for his law is followed in some sort of goodness of fit test multiple ways, actually. So there are multiple ways you can think about how well a distribution match has been for his law. And I should give credit where it's due, this is certainly not my framing, in any sense, the financial fraud literature on Benford's law, which as far as I know, it dates back to 1996, my co author, and I grabbed that paper by Nick Greenie. And that's their framing, you know, they talked about, they do a quick a chi square goodness of fit test, they do these individual digit tests by proportion. So we use the techniques they used.
Rosemary Pennington
You're listening to Stats and Stories, and today we're talking with David McCune of William Jewell College, you describe yourself as a mathematical political scientist. So I wonder how you frame yourself that way? And what sort of work are you doing when you're not exploring Benford’s Law?
David McCune Yes, well, the emphasis is on the word mathematical. I don't know anything about political science. And my back, my background is purely mathematical. So I have a PhD in algebra, which I've learned recently has some applications to voting theory. But that certainly wasn't what I had thought about. I like to think about the mathematics of elections and voting from a social choice standpoint. So given voter preference data, for example, you asked, you know, 1000 people to rank a certain subset of candidates, given that ranking data? How should you elect a winner? So that's one interesting question I like to think about, I like to think about apportionment problems, which involve allocating resources that can't be divided. So the classic example is how seats in the US House of Representatives, how do you give those seats out to the states in a fair, and equitable fashion? Those are the kinds of questions I usually think about.
John Bailer 13:40
So this idea of voter preference, the idea that you can vote in different ways is a surprise to people. I mean, this idea of even like the idea of transferable voting, where you where you give a preference collection, could you talk a little bit about that? And you know, you wrote a paper where you looked at one election outcome that would really freak people out. And that there was, there was sort of an argument that, you know, three, three different candidates probably have some foundation for claiming they've won.
David McCune Yeah, it's interesting. So just in general, when you think about an election, if you're an elections official or a designer of elections, you have to ask yourself, what kind of information do you want voters to tell you? Do you want them to tell you their favorite candidates or who they're willing to say their favorite candidate is right now, because of strategic considerations? Do you want them to fully rank if there are five candidates in the election? Do you want every voter to rank all five of them from most favorite to least favorite? Do you want voters to just tell you which candidates they approve of? So maybe not their favorite but like, you know, get me all the candidates you will be okay with? That all those different types of data you gather could lead to different procedures for deciding who wins. So, I mean, at base level, designing an election, you first have to decide what kind of ballot you give out. Because depending on what you want from the voters, you'll give them different ballots like different things to mark. So this election you're referencing from Minneapolis, for Minneapolis, municipal elections, like for city council, they use a method that they call rank choice voting, where the information you get from the voters, is a ranking of candidates. So in Minneapolis for a city council election, no matter how many candidates are in the election, what a voter gives to you, is their top three choices, right? So my first favorite, my second favorite, my third favorite, although voters aren't required to give even that they could just give you a single candidate if they want to. But if they're, if you're going to ask people for preference information like that, then you have to design an election procedure that can be fed that kind of information, and I'll put a winner.
John Bailer
Are there other other municipalities or other governing groups that use this, this type of ballot, this type of rank choice voting? I think that we do at the university. Does this in terms of some of the committee appointments or other components do? I've just haven't, I had not not been aware of this being used in the sort of public, you know, sort of public servants being elected in this way?
David McCune Yes, it's on the upswing, actually. So since 2004, San Francisco has used this kind of voting for local elections. So for city council, they call it the Board of Supervisors, but for city council and for like the city, da and Sheriff, Alameda County, California, which includes Oakland, Berkeley and San Leandro use it. Most recently, New York City used it in their city primary elections, the state of Maine now uses it for national elections. So their representatives to US Congress are elected using rank choice voting now, they actually use it for the presidential election and 2020. So yeah, it's being used. And that's up to as of 2020. They didn't do it in 2016. So it's being used quite a bit more now than it was like 20 years ago, it was barely used at all in the US.
Rosemary Pennington 16:57
How did you get interested in taking this sort of approach to the study of politics? Just sort of how did you? Again, you're the first person I've met, who says they take a mathematical approach to political science. So I would, I would find it interesting, but also, the work you're doing is stuff that does not seem unfamiliar to stuff that I've read. But I'm just really interested in sort of how you decided that this is what you wanted to study and wanted to go into
David McCune My PhD was in algebra, which has nothing to do with anything, really. But I found that super interesting is beautiful. But on a theoretical level, it's absolutely stunning. At my current job, though, it was difficult to find students who would do that kind of research with me for a number of reasons. So I wanted to find work that I could do with my undergraduate students at William Jewell College. Simultaneously, I was trying to improve a political science unit and one of our general education classes. So I was just sort of going through the literature over those questions. And I just thought it was super interesting. So I just decided, after a year or two of spending time with the literature to improve a course, I decided I may as well give it a shot. And that was around 2015 2016.
John Bailer
You know, I found myself thinking, I wonder how hard it would be to convince representatives in Ohio to vote in a rank choice system? You know, that they have, they'd have to think, you know, that there's some that there's some there's an advantage to that to doing so can you talk about some of the the advantages of using more information? At face value? It's pretty clear. I mean, the more information you have about your preference about not just your first choice, but maybe sort of ranking of your subsequent choices, is giving more information to support a decision that somehow is, I mean, you know, kind of brought globally best. So can you talk a little bit about what's, you know, if you've been called to Columbus, Ohio to help advise on why that is why a consideration of rank choice voting should be made? What are some of the reasons that might be that you might give?
David McCune
Well, that's a great question. I am not the kind of person that should be calling election officials in Columbus, Ohio. My background is mathematical. I mean, there are lots of reasons you can give for why you might do such a thing. Many of them are political, a lot of them are psychological, perhaps. So for things like that I have very little to contribute. From a mathematical perspective. As you said, on face value, it's clear why you would want more data from a voter perspective. The more information you are allowed to express, in some sense, the better and especially for strategic considerations. So the classic story is something like in 2016, if you really want to vote for Jill Stein, but you really don't like Donald Trump, what do you do? Right? That's the classic strategic question. So Jill Stein is going to lose. There's no getting around that and so maybe you like Jill Stein? You don't really like Hillary Clinton, but you really hate Donald Trump. What do you do if you're allowed to vote in the rank choice? selection, you can put Jill Stein as your first choice, and then put Hillary Clinton as your second choice to have that sort of preference information expressed. And then in the short term, what that probably means is Jill Stein still loses. And maybe now your support can go to Hillary Clinton and maybe get her over the hurdle. And in the long term may be enough people voting for other candidates might by creating a more robust system of competition. There are just lots of issues involved, like municipal elections are just far different than national ones. For example, like in San Francisco, the elections are all nonpartisan. So there are eight names, you don't know what party they belong to. So in a nonpartisan election, the reasoning might be different for you to want such a system than to say the presidential level. But at base level, for me, I just like more data. So if I can get preference data, there are lots of mathematical things I like to check that I cannot check without preference data of some kind. So I like to check things like in my language monotonicity issues occur. So if there's some sort of issue involving like, maybe people could strategically vote cause strategically engineered outcomes by mood shifting people up and down in their rankings, for example, I like to check for things like that. And I can't give in just like a first place ranking as we normally do.
John Bailer
So to continue kind of these other areas that you have some interest in, you've mentioned this idea of another area being apportionment and that there's the possibility of having a paradox, or more than one paradox that are associated with apportionment. So can you just talk about what it means to think about apportionment?
David McCune So the classic example of an apportionment problem is allocating House seats to states in proportion to their population. So I live in Missouri, we have eight House seats. So that means eight representatives to the US House. The question is, where does that eight come from? And how do you make sure that eight is fair, because I mean, eight is close to seven. But if you take away one of our reps, it really hurts, you know, and a state like Montana that's constantly on the verge of getting two seats. And I forget actually how the census shook out this time. But they're constantly on the border between one and two seats. If you move from one to two, that's doubling your representation. So the questions are both interesting, but also important that it's about the quality of representation you receive on some level. So that's the classic example of an apportionment problem, how seats are allocated to states based on their population. The core of the problem, by the way, is that if you just say proportional, you're going to be talking about decimals. So Missouri, if you just do the raw math, I forget what it is. But we're owed like 8.3 seats, 8.3 reps, but a representative as a human being, and you can't chop them into three tenths. So you got to do something with that decimal. And it's not as easy as it sounds, actually, what to do with all that. I personally don't study that problem. It's super well studied. So if you're like me, and you're kind of like a hack, you have to find something to do that other people haven't really tread on, you know. So my primary area of research is presidential primaries, where delegates are allocated to presidential candidates in proportion to their vote share. So mathematically, it's the same kind of problem, but politically is much different. Because with how seats you don't want to favor big seats that are expensive, the smaller vice versa. Usually, you know, you don't want to say hey, California, you got so many people will give you extra representation, because you're so big. But in an election, if I get more votes than you, it makes sense to sort of favor me, maybe. So that the context is sort of different enough that the questions you can ask are different enough for this sort of new work.
Rosemary Pennington
That's all the time we have this episode of Stats and Stories. David, thank you so much for being here today. Stats and Stories is a partnership between Miami University’s Departments of Statistics, and Media, Journalism and Film, and the American Statistical Association. You can follow us on Twitter, Apple podcasts, or other places you can find podcasts. If you’d like to share your thoughts on the program send your email to statsandstories@miamioh.edu or check us out at statsandstories.net, and be sure to listen for future editions of Stats and Stories, where we discuss the statistics behind the stories and the stories behind the statistics.