Benford’s Law: A curious statistical phenomenon that keeps getting curiouser

Ted Hill, one of the contributors to The Princeton Companion to Applied Mathematics, as well as the coauthor, with Arno Berger, of An Introduction to Benford’s Law, has written a post on this fascinating statistical phenomenon. You’ll be surprised at the rather unexpected places it pops up, from an analysis of Donald Trump’s finances, to earthquake detection.

Benford’s Law

The acclaimed business and technology news website Business Insider proudly offers this advice to its readers, in capital letters:


The curious statistical phenomenon known as Benford’s Law, first discovered by Newcomb in 1881 and later rediscovered and popularized by Benford in 1938, is currently experiencing an explosion of research activity, especially in fraud detection ranging from tax data and digital images to clinical trial statistics, and from voting returns to macroeconomic data. Complementing these new forensic Benford tools, recent applications also include earthquake detection, analysis of Big Data and of errors in scientific computations, and diagnostic tests for mathematical models. As is common in developing fields, the quality of this research is all over the map, from scholarly and insightful to amusing and outlandish. The most recent Benford article I have seen is an analysis of Donald Trump’s finances, and I will let interested readers have fun judging these Benford articles for themselves. Most may be found on the open access and fully searchable Benford Online Bibliography, which currently references more than 800 articles on Benford’s Law, as well as other resources (books, websites, lectures, etc.).

The First-digit Law

In its most common formulation, the special case of the first significant (i.e., first non-zero) decimal digit, Benford’s Law says that the leading decimal digit is not equally likely to be any one of the nine possible digits 1, 2, …, 9, but rather follows the logarithmic distribution

equationwhere D1 denotes the first significant decimal digit. Many numerical datasets follow this distribution, from mathematical tables like the Fibonacci numbers and powers of 2 to real-life data like the numbers appearing in newspapers, in tax returns, in eBay auctions, and in the meta-dataset of all numbers on the World Wide Web (see Figure 1).

For datasets like these that are close to being Benford, about 30% of the leading (nonzero) decimal digits are 1, about 18% are 2, and the other leading digit proportions decrease exponentially to about 5% that begin with 9.

fig 1
Figure 1. Empirical Evidence of Benford’s Law

The complete form of Benford’s Law also specifies the probabilities of occurrence of the second and higher significant digits, and more generally, the joint distribution of all the significant digits. For instance, the probability that a number has the same first three significant digits as π = 3.141… is

eqn 2(For non-decimal bases b, the analogous law simply replaces decimal logarithms with logarithms base b.)

Robustness of Benford’s Law

Benford’s Law is remarkably robust, which may help explain its ubiquity in both theory and applications. For example, it is the only distribution on significant digits that is scale invariant (e.g., converting from dollars to euros or feet to meters preserves Benford’s Law), and is the only continuous distribution on significant digits that is base-invariant.

As an example of stochastic robustness, if a random variable X satisfies Benford’s Law, then so does XY for all positive Y independent of X; thus in multiplying independent positive random variables, say to model stock prices, if you ever encounter a single Benford’s Law entry, the whole product will obey Benford’s Law. Moreover, if X follows Benford’s Law, then so do 1/X and X2, (and all other non-zero integral powers of X).

Benford’s Law is also robust under both additive and multiplicative errors: If an increasing unbounded sequence of values X obeys Benford’s Law, then so does X + E for every bounded “error” sequence E, and if X is Benford and E is any independent error with |E| < 1, then (1 + E)X is also exactly Benford.

Applications of Benford’s Law

The most widespread application of Benford’s Law currently is its use in detection of fraud. The idea here is simple: if true data of a certain type is known to be close to Benford’s Law, then chi-squared goodness-of-fit tests can be used as a simple “red flag” test for data fabrication or falsification. Whether the tested data are close to Benford’s Law or are not close proves nothing, but a poor fit raises the level of suspicion, at which time independent (non-Benford) tests or monitoring may be applied.

A similar application is being employed to detect changes in natural processes. If the significant digits are close to Benford’s Law when the process is in one particular state, but not when the process is in a different state, then comparison to Benford can help identify when changes in the state of the process occur. Recent studies have reported successful Benford’s Law tests to detect earthquakes, phase transitions in quantum many-body problems, different states of anesthesia, signal modulations in electrophysiological recordings, and output changes in interventional radiology.

Tests for goodness-of-fit to Benford are also useful as a diagnostic tool for assessing the appropriateness of mathematical models. If current and past data obey Benford’s Law, it is reasonable to expect that future data will also obey Benford’s Law. For example, the 1990, 2000, and 2010 census statistics of populations of the some three thousand counties in the United States follow Benford’s Law very closely (see Figure 1), so to evaluate a proposed mathematical model’s prediction of future populations, simply enter current values as input, and then check to see how closely the output of that model agrees with Benford’s Law (see Figure 2).

fig 2
Figure 2. Benford-in-Benford-out Diagnostic Test

The appearance of Benford’s law in real-life scientific computations is now widely accepted, both as an empirical fact (as reported in Knuth’s classic text), and as a mathematical fact (e.g., Newton’s method and related numerical algorithms have recently been shown to follow Benford’s Law). Thus, in those scientific calculations where Benford’s Law is expected to occur, knowledge of the distribution of the output of the algorithm permits better estimates of both round-off and overflow/underflow errors.

Recent Theoretical Developments

Complementing these applications are new theoretical advancements, which are useful in explaining and predicting when Benford analysis is appropriate, and which are also of independent mathematical interest. Recent results include:

  • The outputs of many numerical algorithms, including Newton’s method, obey Benford’s Law.
  • Iterations of most linear functions follow Benford’s Law exactly, and iterations of most functions close to linear, such as f(x) = 2x + ex, also follow Benford’s Law exactly.
  • Continuous functions with exponential or super-exponential growth or decay typically exhibit Benford’s Law behavior, and thus wide classes of initial value problems obey Benford’s Law exactly.
  • Powers and products of very general classes of random variables, including all random variables with densities, approach Benford’s law in the limit (see Figure 3 for the standard uniform case).
  • Many multidimensional systems such as powers of large classes of square matrices and Markov chains, obey Benford’s Law.
  • Large classes of stochastic processes, including geometric Brownian motion and many Levy processes, obey Benford’s Law.
  • If random samples from different randomly selected probability distributions are combined, the resulting meta-sample also typically converges to Benford’s Law. (This may help explain why numbers in the WWW and newspapers and combined financial data have been found to follow Benford’s Law.)


Fig 3
Figure 3. Powers of a Uniform Random Variable

The study of Benford’s Law has also at times been entertaining. I’ve been contacted about its use to support various religious philosophies (including evidence of Benford’s Law in the Bible and Quran, and its appearance in tables of the earth’s elements as evidence of Intelligent Design), as well as a website where Eastern European entrepreneurs sold Benford data to people who need it for 25 euros a pop. For me, however, the main attraction has been its wealth of fascinating and challenging mathematical questions.

Ted Hill is Professor Emeritus of Mathematics at the Georgia Institute of Technology, and currently Research Scholar in Residence at the California Polytechnic State University in San Luis Obispo. He is the co-author, with Arno Berger, of An Introduction to Benford’s Law, (Princeton University Press, 2015).

Intro to Benford's Law


Announcing the First Annual Humanities Lecture with the New York Institute for the Humanities at NYU

Humanities lecturePrinceton University Press and the New York Institute for the Humanities at New York University are pleased to announce the first Annual Humanities Lecture. With the aim of highlighting both the value and the relevance of the humanities, this new lecture will be given annually in New York by notable figures from a wide range of fields and will explore humanistic topics and themes.

The inaugural lecture will be given at New York University on October 29 by Thomas Laqueur, the Helen Fawcett Professor of History at the University of California, Berkeley and author of the forthcoming book The Work of the Dead: A Cultural History of Mortal Remains.

Laqueur’s lecture, “The Work of the Dead: How Caring for Mortal Remains has Shaped Humanity,” will speak to compelling questions: Why do we as a species care for the bodies of our dead?  What work do the dead do for the living?  How do specific ways of disposing of the dead and specific memorial practices create communities, nations, and culture more generally?

According to Princeton University Press Director Peter Dougherty: “We at PUP welcome the opportunity to collaborate with the esteemed New York Institute for the Humanities in sponsoring this new annual lecture to showcase the work of the world’s most exciting and important scholars working in the humanities, beginning next month with historian Thomas Laqueur’s fascinating study of how care for the dead has shaped humanity. We are also extremely pleased at the prospect of an annual PUP cosponsored event in New York City, a capital of global culture and intellect.”

The Annual Humanities Lecture follows in the footsteps of Princeton in Europe, a PUP-sponsored lecture that has been presented annually in London since launching at the London Book Fair in 2011.

According to New York Institute for the Humanities Director Eric Banks: “I’m delighted that Princeton University Press has decided to partner with the New York Institute for the Humanities in endowing an annual series of lectures in the humanities. For four decades, the Institute has offered public programming and weekly fellows’ luncheons that have explored a range of issues that engage the role of the humanities in our broader civic life. Princeton University Press has long published some of the most provocative and thought-provoking titles, of interest, not only to scholars but to an intellectually curious larger readership. We are excited to inaugurate our collaboration with a scholar of the caliber of Thomas Laqueur, who has combined erudition, public engagement, and a flair for style as a writer in a unique body of work. We look forward to developing our annual lecture series over the years to come to continue to highlight intriguing and far-reaching work in the humanities.”

The Work of the DeadProfessor Laqueur’s October 29 lecture is cosponsored by the College of Arts and Science at New York University and will be free and open to the public, though preregistration is required. The lecture begins at 6:30 p.m at Hemmerdinger Hall, 100 Washington Square East, New York University.

About Princeton University Press

Princeton University Press is an independent publisher with close connections, both formal and informal, to Princeton University. As such it has overlapping responsibilities to the University, the academic community, and the reading public and a fundamental mission to disseminate scholarship both within academia and to society at large. Founded in 1905, it has offices in Princeton, Oxford, and Beijing.

About the New York Institute for the Humanities

Established in 1976, the New York Institute for the Humanities at New York University is a leading forum for promoting the exchange of ideas between academics, professionals, politicians, diplomats, writers, journalists, musicians, painters, and other artists in New York City. Comprising more than two hundred distinguished fellows, the NYIH serves to facilitate conversations about the role of the humanities in public life.


Introducing the new video trailer for The Princeton Companion to Applied Mathematics

We are pleased to present the new video trailer for The Princeton Companion to Applied Mathematics. Modeled on the popular Princeton Companion to Mathematics, this is an indispensable resource for undergraduate and graduate students, researchers, and practitioners in other disciplines seeking a user-friendly reference book. Check out the video in which editor Nicholas Higham, Richardson Professor of Applied Mathematics at The University of Manchester, talks about the major ideas covered in this expansive project, which includes nearly 200 entries organized thematically and written by an international team of distinguished contributors.

(Stanley) Fish Food for Thought: Aesthetic Reflections

Welcome to Part 3 of PUP’s Stanley Fish series, Fish Food for Thought. All selections are excerpted from Fish’s new book, Think Again: Contrarian Reflections on Life, Culture, Politics, Religion, Law, and Education.


Fish Food for Thought

Part 3: Aesthetic Reflections

2.1 Why Do Writers Write?writing

February 11, 2007

Fish on the internal satisfaction when writing.

If you’ve found something you really like to do — say write beautiful sentences — not because of the possible benefits to the world of doing it, but because doing it brings you the satisfaction and sense of completeness nothing else can, then do it at the highest level of performance you are capable of, and leave the world and its problems to others, (41)

2.4 The Ten Best American Movies

January 4, 2009

Fish on one of his favorite movies, 1980’s ‘Raging Bull.’

Most boxing movies trace the classic pattern of rise, fall, and redemption…or tell a moral tale about the corruption of the sport… or detail the corruption of the protagonist. Raging Bull offers no triumph and no moral. It just exhibits the self-destructiveness of its central figure again and again…the wonder is that Scorsese was able to make something lyrical out of a polluting self-destructiveness, but that is what he did, (55)

2.6 Larger than Life: Charlton Heston

April 13, 2008

Fish on former Hollywood star Charlton Heston.

The fact is that Heston’s size, his monumentality, was an obstacle he had to overcome in order to become the actor he wanted to be… Not only was Heston capable of playing a small man; the tension between the inner smallness he was portraying and his physical mass added strength and poignancy to the performance,(63)

2.8 Little Big Men

March 1, 2010

Fish on identifying with actors.

Seeing men you know to be small playing big on the silver screen is comforting, even though the comfort depends on a very suspect transference…But you take your comfort where you can get it, and for me, comfort at the highest level would be identifying with a short, tough guy who is also Jewish, (71-72)

2.13 Country RoadsThink Again jacket

July 1, 2007

Fish on the world of country music.

But if you enter, if only vicariously, into the country music culture, you have to swallow, along with your enjoyment, some stances and attitude that might you pause (or might not, depending on who you are). It’s a man’s world… It’s a Christian world… It’s a white world… It’s a patriotic world… And it is a world that knows everything I have just said about it, revels in it, and puts it all into the songs, (88-89)


Bird Fact Friday – Where do penguins live?

From page 16 of Penguins:

A popular misconception is that all penguins live around the poles. Penguins are actually constrained to the southern hemisphere, but only four species of 18 (or 19, depending on the taxonomy used) form colonies along parts of the Antarctic coastline, remaining at least 1200 km (745 miles) from the South Pole. The entire ‘crested dynasty’ (seven species), live in slightly milder climates, mostly north of the Polar Front, nesting on subantarctic islands. Still others make their homes in Australia, New Zealand, and the Galapagos Islands. Many penguins live in areas so remote that they are rarely observed or photographed.

Penguins: The Ultimate Guide
Tui De Roy, Mark Jones & Julie Cornthwaite

PenguinsPenguins are perhaps the most beloved birds. On land, their behavior appears so humorous and expressive that we can be excused for attributing to them moods and foibles similar to our own. Few realize how complex and mysterious their private lives truly are, as most of their existence takes place far from our prying eyes, hidden beneath the ocean waves. This stunningly illustrated book provides a unique look at these extraordinary creatures and the cutting-edge science that is helping us to better understand them. Featuring more than 400 breathtaking photos, this is the ultimate guide to all 18 species of penguins, including those with retiring personalities or nocturnal habits that tend to be overlooked and rarely photographed.

A book that no bird enthusiast or armchair naturalist should do without, Penguins includes discussions of penguin conservation, informative species profiles, fascinating penguin facts, and tips on where to see penguins in the wild.

• Covers all 18 species of the world’s penguins
• Features more than 400 photos
• Explores the latest science on penguins and their conservation
• Includes informative species profiles and fascinating penguin facts

Woodrow Wilson Papers to go online with new partnership

Princeton University Press, The Woodrow Wilson Presidential Library, and the University of Virginia Press Partner To Create Digital Edition of THE PAPERS OF WOODROW WILSON

wilson portraitPRINCETON UNIVERSITY PRESS (PUP), the WOODROW WILSON PRESIDENTIAL LIBRARY(WWPL), and the UNIVERSITY OF VIRGINIA PRESS (UVaP) announced today an agreement to create THE PAPERS OF WOODROW WILSON DIGITAL EDITION (PWWDE). Edited by Arthur S. Link and published by Princeton University Press, The Papers of Woodrow Wilson will be digitized and made available online in UVaP’s Rotunda American History collection, with the permission of PUP and the generous support of friends of the WWPL.

“This partnership among two university presses and a presidential library harnesses the intellectual investment and publishing expertise represented in the great documentary editions of the last century,” said Peter Dougherty, Director of Princeton University Press, “and makes them more accessible and valuable through this century’s digital technologies.”

Princeton University Press published the print edition of the Papers of Woodrow Wilson, consisting of 69 volumes with a 5-part index, between 1966 and 1994. The edition’s editor, Arthur Stanley Link (1920–1998),Edwards Professor of History Emeritus at Princeton University, was widely considered a pioneer in the field of documentary editing as well as the foremost scholar of Woodrow Wilson, the 28th President of the United States. The Link edition includes Wilson’s personal correspondence, academic works, and speeches, minutes of the Paris Peace Conference, and diary entries of close associates Edward House, Cary Grayson, and Josephus Daniels, totaling approximately 38,400 documents from a vast range of government and academic sources. The most significant sources of Wilson material in the published volumes are stored in the Library of Congress and Princeton University.  The Journal of American History described the Papers of Woodrow Wilson as “an unprecedented illumination of Wilson’s activities and ideas.”

Woodrow Wilson is one of the most accessible presidents in American history due to the precise organization, annotation, and indexing of the Papers of Woodrow Wilson. The Rotunda digital edition will enhance discovery of Wilson’s papers by adapting the documents, annotation, and indexing created by Arthur Link and his fellow editors to a state-of-the-art electronic publishing platform. “Inclusion in Rotunda not only provides the most up-to-date digital publishing technology,” said Mark H. Saunders, Director of UVaP. “It puts the Wilson material in conversation with other important figures in American political history, from the Founding Fathers to participants in the civil rights and Vietnam eras. Comparing the view of Thomas Jefferson, Woodrow Wilson, and Lyndon Johnson on a subject such as race or presidential power can provide new scholarly insights that were hard to imagine in an age of analog information or siloed digital repositories.”

The WWPL anticipates digitizing further materials in its collection and the collection of the Library of Congress, including a selection of Wilson’s correspondence during World War I and documents from Wilson’s later public career, and making them available in the coming years. “There is a vast array of important Wilson material that could not be included due to the constraints of a print edition,” said Don W. Wilson, President of WWPL Foundation. “Those documents will now be made available to scholars, students, and the interested public.” Additional collections held at Princeton University, among them letters between Woodrow Wilson and his wives, Edith and Ellen, and his daughter Jessie Sayre, would also be added to the PWWDE.

Press contacts:

Emily Grandstaff :

Debra Liese:

New Brain & Cognitive Science Catalog

We invite you to scroll through the 2016 Brain & Cognitive Science catalog below.

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Phishing for PhoolsDon’t miss Phishing for Phools! Nobel Prize-winning economists George A. Akerlof and Robert J. Shiller challenge the traditional idea that the free market is inherently benign and show us through numerous stories how sellers can manipulate and deceive us.



ThriveIn Thrive, Richard Layard and David M. Clark argue that the economic and social advantages of investing in modern psychological therapies more than make up for the cost, and that we cannot afford to ignore an issue that affects at least 20% of people in developed countries.



future of the brainThe Future of the Brain, edited by Gary Marcus and Jeremy Freeman, is a collection of essays that explore the exciting advances that will allow us to understand the brain as we never have before.




Secret of our SuccessJoseph Henrich makes the case that our success as a species can be attributed to our ability to socially connect with each other and benefit from a collective intelligence in The Secret of Our Success.




For these and many more titles in cognitive science, see our catalog above! And be sure to subscribe to our newsletter to get 30% off on select titles through November 13, 2015.

Fall Sale

Finally, if you’ll be attending the Society for Neuroscience Annual Meeting in Chicago from October 17-21, visit us at booth 126. You can also join the conversation on Twitter using #SfN15!

Vote, or else? Jason Brennan on why moral obligations shouldn’t be enforced

Jason BrennanEthicist Jason Brennan is writing a series of posts for the PUP blog offering unique perspectives on ethics, voting, not voting, democracy, public policy and strategy. He is currently Flanagan Family Associate Professor of Strategy, Economics, Ethics, and Public Policy at the McDonough School of Business at Georgetown University, and is writing Against Politics, under contract with Princeton University Press. You can read his first post on “why smart politicians say dumb things” here–PUP Blog Editor

Turnout in American elections is low compared to some other advanced democracies. Should we force people to vote?

Brookings Institute analyst William Galston thinks so. In a recently published Op-Ed at Newsweek, Galston offers a host of arguments on behalf of compulsory voting.[1] None of the arguments are very good.

Galston’s right about one thing: Compulsory voting works. It’s clear that compulsory voting does in fact get more people to vote. But everyone agrees that alone isn’t enough to justify compulsory voting. A basic tenet of liberal democracy, or, really, fundamental human decency, is that it’s wrong to force people to do anything without a strong justification for doing so. Thus, proponents of compulsory voting bear a strong burden of proof. They must produce some reason why it’s permissible to force people to vote.

Does Compulsory Voting Lead to Moderation?

Galston argues that moderates are underrepresented. People belonging to ideological extremes are much more likely to vote than people with middle-of-the-road views. He claims that compulsory voting would thus lead to more moderate political outcomes.

He’s right that moderates vote less. Ample empirical work (e.g., see Ilya Somin’s Democracy and Political Ignorance for a review) shows that political moderates participate less than people with more extreme views. But, that same work also shows that this is because political moderates care less about politics, hold their beliefs more weakly, and also are less informed about politics.

But does compulsory voting actually lead to more moderate political outcomes? The available research (e.g., see Sarah Birch’s Full Participation for a review of the empirical literature) does not support this result. Perhaps it’s because the extremes already tend to balance each other out, and what we actually get from Congress or the president are moderate outcomes and compromise positions.

Indeed, it’s not clear compulsory voting does much of anything. It has no significant effect on individual political knowledge, individual political conservation and persuasion, individual propensity to contact politicians, the propensity to work with others to address concerns, participation in campaign activities, the likelihood of being contacted by a party or politician, the quality of representation, electoral integrity, the proportion of female members of parliament, support for small or third parties, support for the left, or support for the far right.[2]

Is Voting an Enforceable Duty?

Galston believes you have a duty to vote. I disagree,[3] but suppose he’s right and you do have a duty to vote. It doesn’t follow from the mere fact that something is a moral obligation that it’s permissible to force people to do it.

On the contrary, many moral duties—aside from duties to avoid violating others’ rights—are unenforceable. You might have moral duties to keep promises, to be nice to strangers, to buy your mom a birthday present, to be faithful to your boyfriend or girlfriend, to give to charity, to improve your moral character, to apologize for your past wrong-doing, to avoid becoming a member of the KKK, and to avoid using racist language. Nevertheless, these moral obligations are unenforceable—it would be wrong for the government to force you to fulfill these duties, even though they are (Galston and I both agree) moral duties.

So what makes voting special? Why is it an enforceable duty, rather than an unenforceable duty?

Galston says that voting is an expression of gratitude, which makes his defense of compulsory voting all the more perplexing. We often owe it to each other to express gratitude. If you buy me a present, I should say thanks. But in general, the duty is express gratitude is unenforceable. If I don’t send you a thank you note, you shouldn’t call the police and ask them to throw me in jail.

The Public Goods Argument: Are Non-Voters Free Riders?

In an earlier New York Times Op-Ed, Galston describes non-voters as free-riders: “Requiring people to vote in national elections once every two years would reinforce the principle of reciprocity at the heart of citizenship.[4] The idea here is that people who don’t vote are like people who don’t pay their taxes. Non-voters benefit from the good government provided for them by voters, but they don’t do their part in helping to provide that good government. That’s unfair. So, just as it’s permissible to force everyone to pay her fair share of taxes, maybe it’s permissible to force everyone to pay for good government by voting.

On the contrary, I think Galston has an overly narrow view of how citizens fulfill their civic obligations.

Imagine Superman were real. Now imagine Superman never votes or participates in politics. Imagine Galston said to Superman, “You’re a jerk. You free ride off of voters’ efforts. You benefit from good government but don’t do your part.” Superman could respond, “Remember all the times I saved the world? That’s how I did my part.”

Let’s take a less extreme case. Suppose there is a medical genius, Phyllis the Physician. Phyllis is such a genius that she produces new medical breakthroughs hourly. If Phyllis cares about serving the common good, helping her fellow citizens, or paying off some “debt to society”, she has little reason to vote. An hour at the voting booth is worth less than an hour at the lab. Now, imagine Galston said to Phyllis, “You’re a jerk. You free ride off of voters’ efforts.” Phyllis could respond, “No, I’ve paid voters’ back by producing my research. I don’t owe them anything more.”

Superman and Phyllis are extreme cases that illustrate a general point. Each of us in our daily lives as workers, artists, managers, parents, truckers, musicians, priests, teachers, or whatnot, does things that make distant others better off. We’re not just taking; we’re giving. We’re already doing things that make it so that the world and our fellow citizens are better off with us than without us.

There’s no obvious reason to assume that non-voters specifically owe a debt to voters, that the only way we citizens can “pay” for good government is to vote, or that the only way to avoid free-riding on voters’ efforts is to vote ourselves.  If we have a debt to society, or a duty to compensate voters for their efforts, we could instead hold that this debt can be paid, and that voters can be compensated, any number of ways. For any given citizen, given what other citizens are doing and are good at doing, there will be an optimal mix of political and non-political ways for her to pay her debt. For some citizens, this will mean heavy political engagement at the expense of other pursuits. For other citizens, it will mean complete disengagement so as to free the citizen to pursue non-political activities. For most citizens, the optimal mix will be some combination of political and non-political engagement.  Though each citizen might contribute in different ways, they can all pay their debts.

The Best Argument for Compulsory Voting

In the end, the best argument for compulsory voting begins by noting that under a voluntary voting regime, the people who choose to vote are unrepresentative of the population at large.

Voters and abstainers are systematically different. The old are more likely to vote than the young. Men are more likely to vote than women. In many countries, ethnic minorities are less likely to vote than ethnic majorities.[5] More highly educated people are more likely to vote than less highly educated people. Married people are more likely to vote than non-married people.[6] Political partisans are more likely to vote than true independents. In short, under voluntary voting, the voting public—the citizens who actually vote—are not fully representative of the voting eligible public. In general, the privileged are proportionately more likely to vote than the underprivileged. The worry, then, is that because the privileged are more likely to vote, government is likely to be unfairly responsive to their interests. Because the underprivileged are less likely to vote, governments are likely to ignore or underrepresent their interests.

As Galston summarizes the argument:

The second argument for mandatory voting is democratic. Ideally, a democracy will take into account the interests and views of all citizens. But if some regularly vote while others don’t, officials are likely to give greater weight to participants. This might not matter much if nonparticipants were evenly distributed through the population. But political scientists have long known that they aren’t. People with lower levels of income and education are less likely to vote, as are young adults and recent first-generation immigrants.[7]

Let’s put the argument in a more rigorous form. Let’s call this the Demographic Argument for Compulsory Voting:

1.     Voters tend to vote for their self-interest.

2.     Politicians tend to give large voting blocs what they ask for.

3.     When voting is voluntary, the poor, minorities, the uneducated, and young people vote less than the rich, whites, the educated, or older people.

4.     If so, then under voluntary voting, government will tend to promote the interest of the rich, of whites, and of the old, over the interests of the poor, of minorities, or of the young.

5.     Under compulsory voting, almost every demographic and socio-economic group votes at equally high rates.

6.     Thus, under compulsory voting, government will promote everyone’s interests.

7.     Therefore, compulsory voting produces more representative government.

8.     If compulsory voting produces more representative government than voluntary voting, then compulsory voting is justified.

9.     Therefore, compulsory voting is justified.

This argument appears powerful and persuasive at first glance. Nevertheless, as I’ll explain in my next post, it’s unsound. It rests on a number of false empirical assumptions.

Note, however, that Galston cannot consistently advance both the Public Goods and the Demographic Argument for Compulsory Voting. The Public Goods Argument treats voters as cooperators. One person’s vote tends to benefit others, while abstention comes at their expense. The Public Goods argument says that non-voters take advantage of voters. But the Demographic Argument treats voters as competitors. One person’s vote tends to harm other voters (by reducing the power of their vote), while abstention helps them (by strengthening the power of their vote).  The Demographic Argument assumes that non-voters advantage voters, while voters take advantage of non-voters.

At most, one of these arguments is sound. If the Public Goods Argument is sound, then when I (a privileged, upper-middle class, married, white, heterosexual, cisgendered male) abstain, most voters should be mad at me. But if the Demographic Argument is sound, then when I abstain, I do women, blacks, Latinos, the poor, the unemployed, and so on, a favor, by making it more likely the government will represent their interests rather than mine. Galston can’t have it both ways.


[2] Sarah Birch, Full Participation: 140; Benjamin Highton and Raymond Wolfinger, “The Political Implications of Higher Turnout,” British Journal of Political Science 31 (1) (2001): 179-223, 179.

[3] See Jason Brennan, The Ethics of Voting (Princeton: Princeton University Press, 2011), chapters 1 and 2.

[4] William Galston, “Telling Americans to Vote, or Else,” New York Times, 6 November 2011, SR9.

[5] In the United States, African Americans typically have a lower overall turnout than whites. However, there is some evidence that, once we control for socioeconomic status and other factors that influence voting turnout, African Americans actually vote in higher rates than whites. For instance, African Americans vote less than whites, because they are more likely to be poor, not because they are African American. However, this probably does not matter for the purposes of the Demographic Argument. See Jan E. Leighley and Jonathan Nagler, “Individual and Systematic Influences on Voter Turnout: 1984,” Journal of Politics 54 (1992): 718-40.

[6] For a review of the empirical literature establishing the claims of this paragraph, see Jocelyn Evans, Voters and Voting: An Introduction (Thousand Oaks: Sage, 2004): 152-6.

[7] Galston, “Telling Americans to Vote”: SR9.

An interview with Nicholas Higham on The Princeton Companion to Applied Mathematics

Higham jacket

We are excited to be running a series of posts on applied mathematics by Nicholas Higham over the next few weeks. Higham is editor of The Princeton Companion to Applied Mathematics, new this month. Recently he took the time to answer some questions about the book, and where the field is headed. Read his popular first post on color in mathematics here.

What is Applied Mathematics?

NH: There isn’t any generally agreed definition, but I rather like Lord Rayleigh’s comment that applied mathematics is about using mathematics to solve real-world problems “neither seeking nor avoiding mathematical difficulties”. This means that in applied mathematics we don’t go out of our way to consider special cases that will never arise in practice, but equally we do not sidestep genuine difficulties.

What is the purpose of The Companion?

NH: The Companion is intended to give an overview of the main areas of applied mathematics, to give examples of particular problems and connections with other areas, and to explain what applied mathematicians do—which includes writing, teaching, and promoting mathematics as well as studying the subject. The coverage of the book is not meant to be exhaustive, but it is certainly very broad and I hope that everyone from undergraduate students and mathematically interested lay readers to professionals in mathematics and related subjects will find it useful.

What is an example of something aspect of applied mathematics that you’ve learned while editing the book?

NH: Applied mathematics is a big subject and so there are many articles on topics outside my particular areas of expertise. A good example concerns applications of computational fluid dynamics in sport. An article by Nicola Parolini and Alfio Quarteroni describes the mathematical modeling of yachts for the America’s cup. The designer wishes to minimize water resistance on the hull and maximize the thrust produced by the sails. Numerical computations allow designs to be simulated without building and testing them. The article also describes mathematical modeling of the hi-tech swimsuits that are now banned from competition. The model enables the benefit of the suits on race times to be estimated.

The Companion is about 1000 pages. How would advise people to read the book.

NH: The book has a logical structure, with eight parts ranging from introductory material in Part I, the main areas of applied mathematics in Part IV (the longest part), through to broader essays in the final part. It is a good idea to start by reading some of the articles in Part I, especially if you are less familiar with the subject. But a perfectly sensible alternative approach is to select articles of interest from the table of contents, read them, and follow cross-references. Or, you can just choose a random article and start reading—or simply follow interesting index entries! We worked very hard on the cross-references and index so an unstructured approach to reading should lead you around the book and allow you to discover a lot of relevant material.

What was the hardest thing about editing The Companion?

NH: The hardest aspect of the project was ensuring that it was completed in a reasonable time-frame. With 165 authors it’s hard to keep track of everything and to to ensure that drafts, revisions, and corrected proofs are delivered on time.

How much of the book did you write?

NH: I wrote about 100 of the 1000 pages. This was great fun, but it was some of the hardest writing I’ve done. The reason is partly that I was sometimes writing about topics that I don’t normally write about. But it was also because Companion articles are quite different from the papers I’m used to writing: they should have a minimal number of equations and formal statements of theorems, lots of diagrams and illustrations, and no citations (just Further Reading at the end of the article).

How did you choose the cover?

NH: We considered many different ideas. But after a lot of thought we settled on the motor boat picture, which captures the important topics of fluid mechanics, waves, and ocean, all of which are covered in the book in a number of articles.

What is the most unexpected article?

NH: Perhaps the article Mediated Mathematics: Representations of Mathematics in Popular Culture and Why These Matter by sociologist of education Heather Mendick. She discusses the way mathematics is represented in numerous TV shows and films.

What would you be doing if you hadn’t become a mathematician?

NH: I’d be playing a Hammond B3 organ in a jazz or blues band. I’m a keen musician and played keyboards semi-professionally for many years, starting in my teens.

How did you go about organizing the book?

NH: I recruited five Associate Editors with expertise in different areas and we met and planned out the eight parts of the book and the articles, along with a list of authors to invite. We looked for authors who are leading international experts in their field and are at the same time excellent expositors. Signing up the 165 authors was quite a long process. We were able to find authors for almost every article, so just a very small number had to be dropped. In some cases the authors suggested changes of content or emphasis that we were happy to agree with.

What range of readers is The Companion aimed at?

NH: The target audience for The Companion is very broad. It includes mathematicians at undergraduate level or above, students, researchers, and professionals in other subjects who use mathematics, and mathematically interested lay readers. Some articles will also be accessible to students studying mathematics at pre-university level.

Why not just seek information online? Why is there a need for a book?

NH: When Princeton University Press asked me to edit The Companion they told me that reference books still have great value. Many people have trouble navigating the vast amount of information available online and so the need for carefully curated thematic reference works, written by high calibre authors, is as great as ever. So PUP’s experience is that print is definitely not dead, and indeed my own experience is that I have many books in PDF form on my computer, but if I want to read them seriously I use a hard copy.

How have you ensured that the book will not go out of date quickly?

NH: This was a major consideration. This was a five and a half year project and we wanted to make sure that the book will still be relevant 10, 20, or 50 years from now. To do that we were careful to choose articles on topics that have proven long-term value and are not likely to be of short-term interest. This is not to say that we don’t cover some relatively new, hot topics. For example, there are articles on compressed sensing (recovering sparse, high-dimensional data from a small number of indirect measurements) and on cloaking (hiding an object from an observer who is using electromagnetic, or other, forms of imaging, as in Harry Potter or Romulan space ships in Star Trek), both of which are areas that have grown tremendously in the last decade.

What sort of overview of applied mathematics does the book give?

NH: Applied mathematics is a huge subject, so we cannot cover everything in 1000 pages. We have tried to include the main areas of research as well as key underlying concepts, key equations, function and laws, as well as lots of example of applied mathematics problems. The examples range from the flight of a golf ball, to robotics, to ranking web pages. We also cover the use of applied mathematics in other disciplines such as engineering, biology, computer science, and physics. Indeed the book also has a significant mathematical physics component.

Where is the field going?

NH: Prior to the 20th century, applied mathematics was driven by problems in astronomy and mechanics. In the 20th century physics became the main driver, with other areas such as biology, chemistry, economics, engineering, and medicine also providing many challenging mathematical problems from the 1950s onwards. With the massive and still growing amounts of data available to us in today’s digital society information, in its many guises, will be an increasingly important influence on applied mathematics in the 21st century.

To what extent does The Companion discuss the history of applied mathematics?

NH: We have an excellent 25-page article in Part I titled The History of Applied Mathematics by historians of mathematics June Barrow-Green and Reinhard Siegmund-Schultze. Many articles contain historical information and anecdotes. So while The Companion looks to the future it also gives an appreciation of the history of the subject.

How do you see the connections between applied mathematics and other disciplines developing?

NH: Applied mathematics is becoming ever more interdisciplinary. Many articles in The Companion illustrate this. For example,

  • various facets of imaging feature in several articles, including those on compressed sensing, medical imaging, radar, and airport baggage screening,
  • the article on max-plus algebras shows how what may seem like an esoteric area of pure mathematics has applications to all kinds of scheduling processes,
  • the article on the spread of infectious diseases shows the value of mathematical models in epidemiology,
  • several articles show how mathematics can be used to help understand the earth’s weather and climate, focusing on topics such as weather prediction, tsunamis, and sea ice.

What are you thoughts on the role of computing in applied mathematics?

NH: Computation has been a growing aspect of applied mathematics ever since the first stored program computer was invented here in Manchester. More and more it is the case that numerical computations and simulations are used in tandem with, or even in place of, the classical analysis that relies just on pen and paper. What I find particularly interesting is that while the needs of mathematics and of science in general have, naturally, influenced the development of computers and programming languages, there have been influences in the other direction. For example, the notation for the ceiling and floor functions that map a real number to the next larger or smaller integer, respectively, was first introduced in the programming language APL.

Of course numerical computations are expressed in terms of algorithms, and algorithms are ubiquitous in applied mathematics, and strongly represented in the book.

Do you have any views on ensuring the correctness of work in applied mathematics?

NH: As the problems we solve become every more complicated, and the computations we perform run for longer and longer, questions about the correctness of our results become more important. Applied mathematicians have always been good at estimating answers, perhaps by an asymptotic analysis, so we usually know roughly what the answer should look like and we may be able to spot gross errors. Several particular aspects of treating correctness are covered in The Companion.

Uncertainty quantification is about understanding how uncertainties in the data of a problem affect the solution. It’s particularly important because often we don’t know the problem data exactly—for example, in analyzing groundwater flow we don’t know the exact structure of what lies under the ground and so have to make statistical assumptions, and we want to know how these impact the computed flows.

A different aspect of correctness concerns the reproducibility of our computations and treats issues such as whether another scientist can reproduce our results and whether a computation on a high-performance computer will produce exactly the same answer when the computation is repeated.

All of these issues are covered in multiple articles in the book.

Nicholas J. Higham is the Richardson Professor of Applied Mathematics at The University of Manchester. Mark R. Dennis is professor of theoretical physics at the University of Bristol. Paul Glendinning is professor of applied mathematics at The University of Manchester. Paul A. Martin is professor of applied mathematics at the Colorado School of Mines. Fadil Santosa is professor of mathematics at the University of Minnesota. Jared Tanner is professor of the mathematics of information at the University of Oxford.

Facebook YEAR OF BOOKS live Q&A with authors of “Portfolios of the Poor”

Collins jacketPortfolios of the Poor: How the World’s Poor Live on $2 a Day by Daryl Collins, Jonathan Morduch, Stuart Rutherford & Orlanda Ruthven is a recent choice by Mark Zuckerberg for his Year of Books project. An unusual investigation of the staggering problem of global poverty, the authors conducted year-long interviews with impoverished villagers and slum dwellers in Bangladesh, India, and South Africa. This morning the authors are taking part in a live Q&A on the Year of Books Facebook page to share the surprising and systematic methods these families used to survive on an income that is, for many, unimaginably small.

Mark Zuckerberg announced the book’s selection on his personal Facebook page with the following thoughts:

It’s mind-blowing that almost half the world — almost 3 billion people — live on $2.50 a day or less. More than one billion people live on $1 a day or less.

This book explains how these families invest their money to best support themselves.

I hope reading this provides some insight into ways we can all work to support them better as well.

You can follow the discussion here.