# Voting, mathematics, critical thinking

I recently attended a popular mathematics lecture on voting, which got me thinking. Many people, with some justification, have criticised the role of mathematics in our economic troubles since 2007/8. As a mathematician I suppose the mathematics, as mathematics, to be innocent: any problem must be with the interpretation of the mathematics, so the cure would be to better understand the mathematics. I got embroiled in some of the debates about voting methods in support of the UK Jenkins’ commission. Technically, the mathematics in the lecture provided some of the key insights into the single constituency case (it didn’t really address issues of proportionality for a whole parliament). But having realised that some people really did manage to seriously misinterpret the mathematics of finance, I came away with the impression that – far from being enlightened – most of the audience would have been confused or mislead by it.

In the economic case, many had colleagues suggested that what we needed was more critical thinking, so I consulted a guide intended for undergraduates. One can immediately see some problems with the popular lecture, if not how to fix them. As in finance and economics, the problems are with the presentation of the mathematics, not the actual mathematics. But, clearly, one has to rely on mathematicians to ensure that the mathematics is not being misrepresented or misinterpreted. This is the role that I have often tried to fill, but which in recent years seems to be under-appreciated, to our cost. So I thought I’d have a go at re-presenting the key result of the lecture, Arrow’s Impossibility Theorem, based on the account in Wikipedia.

Mathematically, the theorem has some axioms, including this one, known as ‘IIA’: *or?

For two preference profiles (*R*_{1}, … *R _{N}*) and (

*S*

_{1}, …

*S*) such that for all individuals

_{N}*i*, alternatives

**a**and

**b**have the same order in

*R*

_{i}as in

*S*

_{i}, alternatives

**a**and

**b**have the same order in

*F*(

*R*

_{1}, …

*R*) as in

_{N}*F*(

*S*

_{1}, …

*S*).

_{N}It states – correctly, that no ‘preference aggregation rule’, *F*, can satisfy all these axioms simultaneously. On an historical note, Condorcet had highlighted the issue earlier. He reasoned as follows.

Suppose that *N*=3 and the alternatives are **a**, **b**, **c**. If the preference profiles are (**abc**, **bca**, **cab**) then (apply axioms of fairness like Arrow’s) there must be a draw. But if we demote **c** to yield (**abc**, **bac**, **abc**), **a** would have a definite majority for any sensible method, yet our criterion above requires that we still have a draw. So (Condorcet argued), the criterion is not sensible.

Arrow called the above axiom ‘independence from irrelevant alternatives’. From a critical thinking point of view, this is ‘question begging’. The choice of the sequence of characters used to label an axiom is not a part of the mathematics as such, but I got the impression that people really were drawing conclusions from the name of the axiom, as distinct from the mathematical content. In this case, Condorcet is arguing that alternatives that are included in a cycle with **a** and **b** are ‘relevant’ to the order of **a** and **b**, despite Arrow’s intuition. A variant of Arrow’s axiom is:

Independence totally irrelevant alternatives:

For two preference profiles (*R*_{1}, … *R _{N}*) and (

*S*

_{1}, …

*S*) suppose that the alternatives can be divided into two sets, A and B such that:

_{N}- all members of A are ordered the same in both profiles
- for all
**a**∈ A, all**b**∈ B, the majority prefer**a**to**b**.

Then all members of A are ranked the same in *F*(*R*_{1}, … *R _{N}*) as in

*F*(

*S*

_{1}, …

*S*).

_{N}For example, if **a** and **b** have a clear majority over the others, then the relative ranking of them is independent of the ordering of the lower-ranked alternatives. But, as we have seen, re-ordering a comparably ranked alternative may change things.

The mathematical challenge is to see whether some preference aggregation rule satisfies this axiom, and – if there are possible variations – how we should choose between them. (For example, under the conditions of the above axiom, is it necessary to require that all members of A are ranked above all members of B? Or to consider other changes to profiles?)

## See Also

Wikipedia notes that no method that always respects majority rule can respect IIA. It discusses some variations, including Schulze, which looks pretty good to me. If we regard a candidate with an unambiguous winner as ‘a clear winner’, then when we have no clear winner it seems reasonable to ask how far each candidate is from winning, in some sense. The minimal change to a ballot is to reverse the order of two adjacently ordered alternatives. A natural measure of order is the number of reversals required to reach a clear win. It then seems natural to select the alternative that is nearest such a win. One has the ‘participation paradox’, but that is another story.