There are many situations in life where one might be curious who has the most power. As an example, one might wonder about votes in the U.S. Senate and ask how much power the vicepresident (who acts as a tiebreaking vote if and only if the senate vote results in a tie) has compared to a normal senator. Similarly, does owning ten times as many shares of stock give you ten times as much power in elections? How much more power do the permanent members of the U.N. Security Council who are allowed to veto have compared to the other voting members? There are many ways one could approach these questions, but I suspect that most mathematicians’ first response would be to ask what we even mean by “power,” and how we can make it something quanitifiable.
The bestknown measure of power is probably the Banzhaf Power Index, named after John Banzhaf who wrote about it in 1965, although it actually appeared in the literature 20 years earlier in a paper by Lionel Penrose. To compute the Banzhaf power index in a given situation, one begins by listing all coalitions of the voters that would lead to a measure passing. Then in each coalition one should identify the “swing voters” who would cause the measure to fail if they alone left the winning coalition. A given voter’s power index is then defined to be the number of times they are a swing voter in a given coalition divided by the total number of swing voters in all winning coalitions. For example, in a situation with three voters Alice, Bob, and Carol so that a measure would pass if Alice and Bob agreed (whether or not Carol did) or Alice and Carol agreed (whether or not Bob did) then the winning coalitions are AB, AC, ABC, where the swing voters in each coalition are in bold. Because Alice is a swing voter in three scenarios, and Bob and Carol each are in a single case, Banzhaf would compute Alice’s power index to be 3/5, and Bob and Carol’s to each be 1/5. For more (and better illustrated) examples, this reviewer refers the reader to the Cut The Knot page on the topic.
The Banzhaf index and its close relative the ShapleyShubin Index have probably become the most commonly accepted ways of quantifying power in a given situation, in no small part due to their computational simplicity. (In fact, many textbooks for math courses for humanities majors include this topic both for its widespread interest and its relative simplicity) However both of these measures have a number of flaws and lead to some counterintuitive results, and there continues to be a growing body of literature on the subject of quantifying power. In 1982, Manfred Holler edited a volume entitled Power, Voting, and Voting Power which collected a variety of essays that considered variations on these power indices and other measurements of power. This past year, Springer published a new volume edited by Holler and Hannu Nurmi, appropriately entitled Power, Voting, and Voting Power: 30 Years After which again gathers a number of authors to write on the subject.
As is probably obvious from my description above, the material at hand lies somewhere near the intersection of mathematics, political science, statistics, and economics. And the papers collected in this volume often fall into one camp more than the others. There were some papers where the authors dwelled on topics that I found trivial as a mathematician, while there were other papers that quickly glossed over topics from political science that I had never heard before. As is to be expected in a growing field at the borders of multiple disciplines, the different authors sometimes give different (and even contradictory) definitions of certain things. There is also somewhat of a grab bag feel to the material, as the editors seemed to include articles on based on whether they found them interesting and vaguely related to the subject at hand rather than going for a cohesive narrative. The editors acknowledge all of this when they write in their introduction that “most of the chapters are more or less revised material published in the quarterly journal Homo Oeconomicus, and are, thus, accessible only for a small readership. We did not select articles that are available in leading and widely distributed journals… this of course gives an additional bias to our volume.” While the opening chapter by Holler and Nurmi does a pretty good job of giving the big picture view of the material, I would have appreciated a more consistent feel to the whole collection.
But these complaints are relatively minor and, as I say, to be expected given the nature of the book. Most readers of this book won’t be reading it cover to cover as I did and are more curious whether the individual articles are interesting. To me, they largely were. Some of the subjects covered in the thirtyeight articles collected in the volume include:

One criticism of the Banzhaf index is that it treats all possible winning coalitions as equally likely, which is not a realistic assumption. How can one adjust the index to account for the fact that certain pairs of voters are unlikely to ever vote together for ideological reasons? For example, are congressmen like Rand Paul or Dennis Kucinich more or less powerful because they seem to stick tightly to a given ideology rather than be useful as swing votes?

In a similar vein, some voters may be better able to communicate and therefore coordinate their votes. How could or should a power index take that into account?

How are power indices different in situations where we have simple YesNo votes and in situations where we have more than two options? What if people can abstain from voting?

How can we efficiently implement algorithms to allocate congressional seats to the states, as in the Method of Largest Remainders or Jefferson’s Method?

To what extent do voters take into account the competence of their elected officials rather than just their political ideologies when they make decisions about who to vote for? (Note that the chapter dealing with this issue is entitled “Gridlock or Leadership in US Electoral Politics”)

If you only have enough resources to bribe (er… influence…) one or two voters in an election, how can you decide where to exert your influence for maximum effect?

What is a fair distribution of power in a heterogeneous voting system? What do we even mean by “fair”? A result of Penrose argues that the distribution is far if the power is proportional to the square root of the population — why is this the fact?

Lest anyone think the book is overly UScentric, there are a half dozen chapters dedicated to analyzing the power structures in the European Union. As the editors write, “the analysis of the EU has become the testing field and source of inspiration for almost all questions discussed so far.” In particular, a good deal of space is dedicated to the ways that “proportional number of seats” is used as a proxy for “proportional amount of power” and the ways this does and does not work.

Several chapters are also dedicated to the socalled discursive dilemma, which Donald Saari describes as follows, in a way that may feel particularly relevant (and/or traumatic) to some readers of this review: A given Assistant Professor is up for review and is to be given tenure if they excel in both teaching and research. One member of the panel thinks they excel in both areas, one member thinks they excel in teaching but not in research and the third member feels exactly the opposite way. Therefore, whether or not they pass the review depends on whether one aggregates the preferences in the two areas separately or combined. This feels paradoxical, and Saari and others analyze why this is the case and what can be done to address it.

In the final chapter of the book, Norman Schofield applies some of these issues and models to questions related to catastrophic climate change. In particular, he considers the ways in which “Black Swan” events should be built into our models (despite the inherent difficulty in doing so) and the ways that we discount future effects of current actions.
As with any collection such as this, the quality and depth of the articles varies considerably. And given the wide range of readers who might pick up this book I suspect that tastes will vary just as widely. But I think that anyone interested in the mathematics of social choice theory could find several articles in here to be to their liking, and I could easily imagine recommending some of these articles to an advanced student who wanted to learn about the field. On a personal note, I have always enjoyed teaching the mathematics of social choice theory to lower level undergraduates, but have never found enough good source material to develop a more advanced class for math majors. This book certainly could not serve as a textbook for such a course, but enough of the articles are wellwritten enough and contain deep enough mathematics that I would certainly use it as a resource if I ever get around to developing such a course.
Darren Glass is an Associate Professor of Mathematics at Gettysburg College. He is also the department chair, but when he started to use the methods in this book to figure out how much power that actually gave him he got quite depressed. Feel free to cheer him up at [email protected].