Making Democracy Count: How Mathematics Improves Voting, Electoral Maps, and Representation

by Ismar Volic

How we can repair our democracy by rebuilding the mechanisms that power it

What’s the best way to determine what most voters want when multiple candidates are running? What’s the fairest way to allocate legislative seats to different constituencies? What’s the least distorted way to draw voting districts? Not the way we do things now. Democracy is mathematical to its very foundations. Yet most of the methods in use are a historical grab bag of the shortsighted, the cynical, the innumerate, and the outright discriminatory. Making Democracy Count sheds new light on our electoral systems, revealing how a deeper understanding of their mathematics is the key to creating civic infrastructure that works for everyone.

In this timely guide, Ismar Volić empowers us to use mathematical thinking as an objective, nonpartisan framework that rises above the noise and rancor of today’s divided public square. Examining our representative democracy using powerful clarifying concepts, Volić shows why our current voting system stifles political diversity, why the size of the House of Representatives contributes to its paralysis, why gerrymandering is a sinister instrument that entrenches partisanship and disenfranchisement, why the Electoral College must be rethought, and what can work better and why. Volić also discusses the legal and constitutional practicalities involved and proposes a road map for repairing the mathematical structures that undergird representative government.

Making Democracy Count gives us the concrete knowledge and the confidence to advocate for a more just, equitable, and inclusive democracy.

Genres: Politics, Nonfiction, Mathematics, History

394 pages, Kindle Edition

Published April 2, 2024

Reviews

[Delaney · Rating 5 out of 5]

You should read this book because it’s important and funny and also I’m thanked in the acknowledgements

[Lydia]

Counting as read because I still have a few chapters but I’ve read so many similar papers in the past two weeks I’ve basically read it

[Suzanne · Rating 4 out of 5]

Familiarity can blind people to their own environments. It often takes a trip away from home to recognize and analyze aspects of our homelands. In Making Democracy Count: How Mathematics Improves Voting, Electoral Maps, and Representation Ismar Volić brings our attention to our governmental environment. He has written a valuable book that clearly elucidates the often-overlooked subjects of evaluating voting methods and assessing modes of democratic representation.

Volić makes a bold claim: “Everything you will read here is grounded in the quantitative.” While the book is indeed well-grounded in mathematics, it is important to acknowledge that not all valuable insights can be quantified, even when the topic under discussion involves quantities. For instance, math cannot tell us that democracy is preferable to autocracy; philosophy is where such questions are debated. At times, Making Democracy Count blurs the lines between mathematical analysis and philosophical inquiry.

Math can, however, shine at describing the outcomes of given scenarios under various rules. Volić does this exceptionally well: he uses math to illustrate the difference among voting schemes and institutional rules. Mathematics informs our understanding of the tradeoffs involved, yet whether these tradeoffs compel change lies outside the field of mathematics.

The book succeeds in explaining the advantages and disadvantages of different voting schemes. Anyone who has wondered about the nuts and bolts of group decision-making will find their curiosity sated. Volić states the assumptions being used in his clear and well-developed examples. He explains his essential criteria for voting methods, including anonymity, neutrality, and monotonicity, but in a way that remains accessible for readers who are neither mathematicians nor policy geeks.

While the book’s first section, on voting methods, is particularly strong, the second, on representation, starts to drift into the normative questions of how democracy should work and away from the mathematical foundations of how rules function. For example, Volić thoroughly explains and illustrates how gerrymandering(manipulation of political district boundaries to affect a given electoral outcome)works along with the multiple ways to gerrymander. However, the discussion limits itself to static outcomes- party identity is taken as given permanently rather than as a fluid state that can change. Although the evolution of party platforms and affiliation lies outside this book’s scope, the possibility of such fluidity would have enhanced the discussion.

Similarly, while Making Democracy Count justly criticizes arbitrariness in voting schemes(including the Electoral College), it overreaches by positing that institutions that appear arbitrary are necessarily flawed. We certainly do not want to be members of a legally capricious society. Yet, it is worth recognizing that just because something began arbitrarily does not mean its persistence is arbitrary. Institutions can and do survive for reasons beyond mere inertia, even when their original justifications have faded, and such reasons are not always clear nor communicable.
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Making Democracy Count makes a mathematically convincing argument that plurality(winner-take-all) voting is problematic. However, it sometimes veers into proposing solutions that lie outside the scope of mathematics. The book is well worth reading for its insights into how math can guide understanding of electoral procesess; I wish it did a better job differentiating normative questions from mathematical ones.
Overall, Making Democracy Count provides a comprehensive exploration of electoral systems. It is an informative and thought-provoking work for non-specialists that makes a significant contribution to the discourse on how societies structure their electoral systems and methods.

[Tom · Rating 4 out of 5]

For the most part I really enjoyed this book. I especially enjoyed how voting history was interwoven throughout to describe how we ended up with our flawed voting systems. For example, just the "simple" act of allocating representative after a census went through several iterations.

But I have one small complaint about how he arrived at IRV as the "best" voting system. It seemed a bit of hand-waving after describing several voting methods the book basically saying, "trust me, I've studied a lot of voting methods and IRV works pretty well". My counterpoint to that is that the book describes how the 2022 special house election in Alaska, using IRV, chose Peltola as the winner even though Begich was the Condorcet winner. This feels like it will be the downfall of IRV in public sentiment when a candidate beats all other candidates head-to-head but doesn't win the election.

I would love to hear the author's thoughts on some other IRV methods that DO pass the Condorcet criterion such as BTR-IRV or Condorcet-IRV. BTR-IRV in particular is very easy to implement and easy to explain to voters but it still picks the Condorcet winner if there is one. Shouldn't a method that guarantees a Condorcet winner be a pretty big deal?

But don't let my small complaint sway you from reading the book. It is very good and thought-provoking and I'm glad that I read it. You will learn things you never realized and have a better appreciation for how flawed our voting systems really are. But there's hope and math can help point the way!

[Jovan Tay · Rating 4 out of 5]

A good read about different kinds of voting systems. It is to be expected that many examples will originate from the political system but he does bring in unique examples from other fields. He does a good job walking the reader through simpler hypothetical examples before the factual empirical, and also to confess when certain axioms/lessons are to be taken as a starting point (rather than jump through rabbits holes of history). Any reader will walk away from this book quite easily with jargon of voting systems and verbalize their flaws/benefits (with examples).

I did not entirely buy into the idea of switching into instant run-off ... sure the only benefit of plurality voting is simplicity. But, I wished more was said about the value of simplicity. Time spent on retrieving information about various candidate solutions, how to conduct switch in voting systems (going beyond how do people switch). I also wished he separated value judgement questions from the mathematical questions more clearly in a future chapter and direction (or to suggest places to figure those answers).

Nonetheless, I will be returning to this book if I had to conduct a voting system - great.

[Maggie]

Really interesting book about our broken voting systems. Yes, the electoral college is busted, but so is congressional representation and standard issue “vote for one” voting. And yeah, there’s math involved but it’s not hard math. Also, he’s witty. Read it.

Here’s his epilogue action list of mathematically informed recommendations for what should change:
• Use instant runoff voting in single-winner elections
• Eliminate the Electoral College
• Use popular vote with instant runoff for presidential elections
• Use multimember districts with single transferable voting
• Increase the size of the House of Representatives
• Use the Webster apportionment method
• Use independent districting commissions
• Introduce more political quantitative literacy into school curricula

[Sekar Writes · Rating 5 out of 5]

I really enjoyed how this book starts by showing how most democracies work, highlighting that the majority vote doesn’t always reflect what most people want. Just because something gets the most votes doesn't mean it represents the true majority. This book is well-organized and thoughtful, helping readers avoid being blindsided by politicians. It empowers you with knowledge, built on clear and objective math, giving you the confidence to make informed decisions. Such an insightful book!

Thank you BookSirens for an advance review copy for free, and I am leaving this review voluntarily.

[Aaron Lee · Rating 5 out of 5]

Fascinating book that’s well written and easy to understand practically given the amount of information and mathematical knowledge needed. The approach of laying out first a concept or method and then explaining which axioms or tests it satisfies or fails is really helpful, and makes you feel like you are part of the scientific investigation. I am reading this as part of a brother set of books which look at the political system, and I think this one has really helped solidify mathematically what is right or wrong about the current setup

[Abby Carlson · Rating 5 out of 5]

The timing of finishing this book, on the eve of the most important presidential election of my lifetime (so far) is no coincidence. That said, it is a very good book. It covers the topic deeply, without getting bogged down. The examples are visual and help to really understand the flaws in our democratic processes.

Fingers crossed for tomorrow and that maybe just maybe we can starting incorporating some of the needed changes to our electoral system soon.

[Christopher · Rating 4 out of 5]

A fun and engaging read - both for its accessible and robust engagement with the mathematics of voting systems and its wide variety of hypothetical and actual examples. The book overall makes a very strong and well-supported case for improving the electoral process in terms of both voting structures and civic literacy and engagement, and shows how these measures would be intertwined and mutually reinforcing

[Chris · Rating 5 out of 5]

So much good information. Lots of history. The math examples are simplified to present the concept without needing all the inputs of actual elections, etc.
Mathematical proofs for why some solutions really are better than others.

[Tiffany · Rating 4 out of 5]

I learned so much from this book, partcularly about gerrymandering, which is absolutely abhorrent! I wish I understood all of the math in the book, but I caught enough to consider it a fascinating read.

[Emily Gorman · Rating 5 out of 5]

A book for everyone who wants to understand why and how our system makes it feel so difficult to have your voice heard, and wants to know what can be done about it.

[Josef Myers · Rating 5 out of 5]

Very eye-opening look at the democratic process in the US and its severe shortcomings (including math-based alternative options).

[Jeff · Rating 5 out of 5]

[B: Making Democracy Count] is one of the most vital popular math books I have ever read. It is essential reading for any American that cares about our democracy. Volic lays out in clearly defined terms the underlying mathematics of voting and representation in a way that makes a compelling argument for how we might make our systems better align with our shared values of representing the people.

In a balanced fashion, Volic lays out the pros and cons of voting methods and why our plurality, winner-takes-all approach is so damaging. He then provides a variety of alternatives with their pluses and minuses. He allows the reader to consider these but still makes a compelling case for rank-choice voting with instant run-off as probably the best option.

Then he tackles representation including how seats are allotted to states for the House of Representatives (and consequently the Electoral College); the creation of Congressional districts including the evils of Gerrymandering; and finishes with arguments for multimember districts and abolishing the Electoral College.

The brilliance of this work is that Volic doesn’t take a particularly partisan perspective on these issues. He simply stands on values of one-person-one-vote and creating space for all voices to be heard and represented including minority voices. Then he lays out what makes the most mathematical sense to best live up to those values.

[Eleanor Latimer · Rating 5 out of 5]

What I really appreciated from reading this book, was a much greater understanding of both the ways we currently hold elections and the options for changing them. Prof. Volic clearly believes in the concept of "one person, one vote" and that belief runs through all his descriptions and explanations. Given that belief, he provides clear, concise and easy to understand mathematical models to show how our elections currently run and what would change if a different model was applied.
He also provides history lessons on how we got to where we are. I laughed at his description of the founding fathers decision-making process for the electoral college while at the same time being horrified at the result. He uses humor throughout the book, making his mathematical models easier to access.
I highly recommend it to anyone either interested in the history of our election process or interested in exploring ways to move toward "one person, one vote".