Some little time ago, my mother-in-law asked me if I knew of a good way for her book club to decide which book to read next. The problem is this. Usually, various members make suggestions and so there is a longish list of possible books they could choose. However, there is typically cordial disagreement among the members of the book club as to which book they should choose. Ultimately, a single choice has to be made, but how? My mother-in-law wasn’t happy with the voting procedure that her book club was using, but she was also not sure what better procedure she could suggest. So, she asked me. My response to her at the time was not very satisfying. I told her that there is something called Arrow’s impossibility theorem that states that it is impossible to come up with a good voting procedure for such problems. This didn’t help her very much. In the meantime, however, I have had the benefit of hearing (the 2007 Nobel Laureate in Economics) Eric Maskin talk about this very issue (in the 2025 Schumpeter Lectures at the University Graz), based on a recent paper. And now I know.

For the impatient reader, let me give you the answer right away. The book club should use the Borda rule or Borda count. It is very simple. Suppose there are n books on the list. Then every member of the book club declares their full preferences over all the books on the list. In other words, they each state their ranking from the top-ranked all the way down to the bottom-ranked book. This ranking translates into points: the top-ranked book gets n points, the next best-ranked book gets n-1 points, and so on down the ranking, until the least-ranked book gets 1 point. These points are then summed up over all the members’ rankings and the book with the highest point total is chosen. If two or more books have the same point total, you could flip a coin (or roll dice) to pick one among these.

The Borda count was suggested by Jean-Charles de Borda a bit more than 250 years ago and probably already earlier by others. So why do we know only now that it really should be used? This is because of a new insight provided by Eric Maskin, building on an earlier insight by Donald Saari.

We have to begin with Arrow’s impossibility theorem. We are interested in translating (or mapping) the set of individual rankings members have over the books to one collective (or societal) ranking (or at least one collective top choice) when there are at least three alternatives (three books in our case). And, because we don’t ex ante know each member’s preferences, we would like this mapping to identify a collective choice for all possible sets of individual rankings. Arrow imposed three conditions such a mapping should satisfy. It should satisfy the Pareto property: if everyone agrees that book A is ranked above book B, then the societal ranking should also rank A above B. It should also satisfy the no-dictatorship property: the societal ranking should not always coincide (regardless of what preferences the members have) with one and the same member’s ranking. If it did, this would mean that, no matter what preferences everyone has, the societal choice is decided by the one member, who we would then call the dictator. These two properties seem very non-controversial: we wouldn’t call it a collective or societal choice if only one person decides what is chosen; and why would we ever want to choose book B, when everyone prefers A over B? But there is a third property Arrow demanded, the independence of irrelevant alternatives: the societal ranking of any two books A and B should only depend on how the members rank books A versus B. It should not depend on how members evaluate those books relative to any other book C. This also seems quite reasonable. After all, we are talking about comparing A and B. So why would book C suddenly come in?

Unfortunately, though, and this is Arrow’s impossibility theorem, there is no mapping translating individual rankings to a societal ranking that satisfies all three properties. I will not give you the proof, but you can find a couple of proofs here. I will here only explain why the Borda rule does not satisfy all the properties we wanted. It clearly satisfies no-dictatorship and the Pareto property, but it does not satisfy the independence of irrelevant alternatives. To see this, we can use an example with only two book club members and three books. Suppose member 1 ranks A over B over C and member 2 ranks B over A over C. Their rankings are also given in the following table (the ones headed by 1 and 2).

Then the Borda count method would result in a tie between books A and B and a coin flip would decide which one is chosen. Now suppose person 2 had different preferences after all: say she ranks books B over C over A (as in 2-alt in the above table). In this case the Borda count makes B the unique winner, even though we did not change how the two members compare books A and B. Only the place of C has changed for one member. If you don’t like that there is a coin-toss involved you can also construct an example with four books and two members that does not involve a tie. [Member 1 ranks A over B over C over D, and member 2 either C over A over D over B or C over D over B over A. In the first case A is chosen and in the second C, even though member 2 ranks C over A in both cases.]

Donald Saari (see Theorem 6 in this article, which goes back to a book by Saari in 1995) may have been the first to realize that the Borda rule, however, satisfies a slightly less demanding version of the independence of irrelevant alternatives. Eric Maskin calls it the modified independence of irrelevant alternatives property. It states that the societal ranking of book A versus book B should only be affected by how individuals compare books A and B and by how many other books they rank in between the two. The Borda rule satisfies this modified independence of irrelevant alternatives property. If you fix the number of books that members rank between books A and B then their point difference from each member’s points is the same and, thus, their total point difference, and their societal ranking, remains the same as well.

Eric Maskin then showed that there is a sense in which the Borda rule is the only reasonable rule that satisfies the modified independence of irrelevant alternatives property. This result is a bit more complicated but let me try to, at least, state it correctly. First, we demand a bit more fairness than just the no-dictatorship property. To do so we take some properties that May has suggested (to argue that majority rule voting is best when there are only two alternatives). We demand that the societal ranking is ex-ante fair in terms of books and members. The societal ranking should not favor one book over another just because of its title. Mathematically, this can be achieved by saying that if we permute the book titles without affecting the members’ preferences, then ultimately the societal ranking should stay the same (now with the new titles). This property is called neutrality. Similarly, the societal ranking should not favor any particular member. This can be achieved by requiring that if we fix the set of members’ preferences, but change who has which of these preferences, then the societal ranking stays the same. This property is called anonymity. We then demand one last property, called positive responsiveness. If book A moves up in some members’ ranking and doesn’t move down in any member’s ranking then A should move up in the societal ranking (at least in some cases). Ok, this is a bit vague and you will have to read the paper for the full definition. Eric Maskin then showed that the Borda rule is the only rule that satisfies the modified independence of alternatives, neutrality, anonymity, positive responsiveness, and (a weaker version of) the Pareto property. So, that’s why every book club should use the Borda count.

Now, if you are still reading at this point, I have to mention a caveat to what I have just said. Unfortunately, the Borda rule is strategically manipulable. In other words, members of the book club could (in some cases) strategically misrepresent their preferences to get their most preferred book chosen, even if it would not have been chosen if everyone were truthful. To see this, consider again the book club with just two members and three books as we already had above. Consider again the case that member 1 ranks A over B over C and member 2 ranks B over A over C. The Borda count method would result in a tie between books A and B and a coin flip would decide which one is chosen. Now suppose that member 2 has a strong suspicion that member 1’s ranking is A over B over C. In that case, member 2 could make her favorite book, book B, the sole winner by pretending to rank C higher than A, that is, she submits the false ranking B over C over A. [You could do a similar exercise with the four-book example that does not involve ties.] We can also see that strategic non-manipulability has something to do with the independence of irrelevant alternatives. Indeed, there is another impossibility theorem, the Gibbard-Satterthwaite theorem, which states that if there are three or more alternatives then there is no collective choice mapping (translating individual rankings to a societal ranking) that is strategy-proof (non-manipulable) and non-dictatorial.

Because it is strategically manipulable one would not generally advocate the Borda rule for all collective decision problems, such as important political issues. [Although Eric Maskin also had a solution for those problems (which I may write about in another post).] But this is also (apart from my mother-in-law asking me about this) why I wrote about book clubs here. In a book club, one could probably appeal to the honesty and integrity of its members to refrain from strategic manipulation and then the Borda rule stands out as probably the single best rule for collectively choosing which book to read next.