This book consists of expository lectures by Sen and Maskin on Arrow’s work, commentary by Arrow himself, and supplementary materials especially an introduction by Pattanaik and mathematical papers by Sen and Maskin. However the result would have been far more successful if the organizers had invited a talk by someone like Donald Saari who could have given a far deeper view of the famous Arrow theorem.

At one point Sen even talks about the need to broaden “social choice” theory to include richer sources of information (pp. 76 – 80) – more than just the preferential or rank orderings assumed by the Arrow theorem and used in common voting methods like the Borda Count and Instant Runoff Voting. While a laudable goal, my feeling is that this needs a much different approach, one involving nonlinear agents or dynamics. Yet incredibly Sen fails to note that Arrow’s impossibility result is, in fact, a direct consequence of an assumption that drastically restricts the use of information that is already presents in the voters’ rankings of candidates. This is, of course, the axiom of “Independence of Irrelevant Alternatives”, which I think might be better called the assumption of “Forbidden Intensity” (FI), as it forbids the voting algorithm from using the intensity of a voter’s ranking of one candidate over another.

To illustrate this concept, suppose I am a voter in an election with 4 candidates {a,b,x,y} and I feel strongly that candidate x is by far the best and that candidate y is by far the worst, while I am not enthusiastic about either candidate a or b but would pick a over b if forced to. Thus I would rank these 4 candidates x > a > b > y. However FI says that the electoral ranking, or “social ordering”, of these 4 candidates, as produced by the voting algorithm under consideration, should ignore this difference. That is, the electoral ranking of x and y should not be affected by how I rank a or b – that such rankings are “irrelevant” even though the way I rank a and b in between x and y is the way I express the strong intensity of my opinion, or the weakness of this intensity when I rank no candidates between a and b.

In other words, FI forces the voting algorithm to ignore very important information provided by the voters. The intensity of a ranking of x over y is easily quantified, using a variation on Saari’s concept (pp. 189 - 191," Decisions and Elections", 2001), by defining Iv(x,y) = k if x > a1>… ak-1>y when voter v ranks k-1 candidates (a1,…,ak-1) between x and y. In addition set Iv(x,x) = 0 and Iv(x,y)= -Iv(y,x) if x < y. Then Saari shows that the Borda Count satisfies all the Arrow axioms if FI is modified to permit the voting algorithm to use this intensity function (which implies voter transitivity). This modification Saari calls “Intensity of Binary Independence” but I’d prefer to call it the assumption of “Permitted Intensity” (PI).

In fact, Arrow’s theorem more properly applies to pairwise comparisons (showing the severe limitations of that approach to voting, in line with the Condorcet paradox) rather than rank ordering, where Saari’s theorem hints at the fundamental role of the Borda Count. Arrow’s theorem is also a demonstration of the limitations of the axiomatic approach to voting algorithms, in that some properties which may seem natural from one point of view may have hidden consequences which require a much broader perspective. That perspective should focus more on the practical aspects of voting than on theoretically undesirable properties which may rarely come into play or which can be anticipated and guarded against by appropriate procedures.

For example, as an activist mathematician I often recommend that the Borda Count be used in the form “rank your top 3 choices” (or sometimes 4 or 5), to reduce the cognitive workload of the voter, the vote counting work load, and tactical voting. This seems to be by far the best method in most informal or low key voting situations, such as when a group wants to rank action plans in a way that builds consensus. Combining it with Peter Emerson’s “modified Borda Count” is a good way to discourage single choice voting. In more partisan or large scale situations, a danger is that a faction may run “clone” candidates or superfluous alternatives to subvert the ranking of true alternatives. In this case, a primary election may eliminate the clones, or in some cases proportional representation may be used to identify different factions and their top candidates (I’ve developed my own clustering algorithm for this purpose).

Instead of this kind of practical work, Maskin tries to rescue FI by identifying restrictions to the rankings which will satisfy FI. He cites Black’s “single peakedness” criterion, which guarantees a Condorcet winner. Then he defines simple, but impossible to enforce, criteria for Borda and Plurality voting to satisfy FI. However I see FI as the problem, not the solution, because of the way it refuses to admit important information. And how would you restrict how people rank candidates in any kind of legal, let alone moral, way?

There are also some issues with the mathematics. First of all, Sen’s definition of decisiveness on p. 35 is not clear. His wording says that for a particular voting algorithm a set of voters G is decisive for a pair of candidates {x,y} if whenever all voters in G rank x > y, then x > y in the electoral ranking. The problem here is that {x,y} uses set notation (is not ordered) but the notation x > y specifies an ordering. So should the set {x,y} be replaced by the ordered pair (x,y)?

The answer is “No” because then the proof of the Spread of Decisiveness Lemma fails. That is, the way to prove an ordered pair version of the Lemma, following Sen’s sketch, would be to assume that (a,b) is an arbitrary ordered pair such that all voters in G rank a > b. Then show that the electoral ranking must specify a > b, given that G is decisive for some ordered pair (x,y).

However consider the case (a,b) = (y,x). Then the decisiveness of (x,y) tells us nothing about (a,b) since the hypothesis fails; that is, x > y for all voters in G whereas we need y > x. If {a,b,x,y} constitutes 3 voters with a = y, we encounter a similar problem. We’d want to show a > b given x > y by decisiveness, but the method of proof gives us instead b > x > y=a.

Hence the correct definition of decisiveness is that G is decisive for {x,y} as a set if it decisive for both orderings. That is, if x > y for all voters in G, then x > y electorally, or if y > x or all voters in G, then y > x electorally. This concept is easily generalized to any set X of size 2 or more by saying that G is decisive for X as a set if G is decisive for each ordering of X.

Then Sen’s proof works if {a,b} does not intersect {x,y}. For all voters ranking a > b, which includes all voters in G, we can use FI to transform any ranking of {a,b,x,y} into the canonical ranking a > x > y > b first by switching the ranking of x, then the ranking of y, without affecting the ranking a > b. For all remaining voters we can switch x until a > x and y until y > b, getting y > b > a > x. In both cases a > x and y > b for all voters, so a > x > y > b electorally using Pareto for a > x and y > b and decisiveness for x > y. Hence a > b electorally by transitivity. Likewise we get b > a electorally if b > a for all voters in G using the decisiveness of (y,x) for G, transforming all b > a orderings to the canonical ordering b > y > x > a and the remaining orderings to x > a > b > y.

However, in the case of one of a or b = x or y, separate proof is needed. Take a = x for example and first consider the case that a > b, including all voters in G. Then we can switch a voter’s ranking of a=x > b > y or of y > a=x > b into the canonical ranking a=x > y > b by FI. For the remaining voters we could switch a ranking of b > a=x > y or of b > y > a=x into the canonical ranking y > b > a=x. Thus we get y > b for all voters, so that y > electorally by Pareto, which when combined with x > y electorally by decisiveness yields a > b electorally by the electoral transitivity of a=x > y > b. If instead we want to prove b > a electorally, we’d end of up using the decisiveness of y > x by reduction to the canonical ordering b > y > a=x.

This lemma demonstrates the incredible power of the assumption of Forbidden Intensity, in fact how it is way too powerful for its own good. It’s too bad that these economists have not been talking to mathematicians like Saari, who have moved far ahead in their analysis.