Preference Aggregation

Adding Numbers, Rules, or Actions Is Different from Adding Preferences

I'll take advantage of the electoral situation in several Latin American countries to emphasize the edges of an idea that intuitively arises every four years, generating skepticism, but in whose principle organizations of all kinds trust when making decisions.

Paradoxes

Nicolas de Caritat, Marquis de Condorcet, was a mathematician and economist from that French city, who lived in the 18th century. Since the rest is covered on Wikipedia, I'll go straight to talking about the paradox that bears his name, also known as the voting paradox.

The Condorcet paradox is interesting because it shows how, from a set of rational decisions, irrational behaviors can emerge. In political elections, it reveals how the majority's decision can conflict with individual preferences. In short, it demonstrates that even if individual preferences are transitive, collective ones may not be.

Okay, let’s break this down with the help of our candidates:

These days, exercises like "If the second round were between X and Y, who would you vote for?" are common.

Different scenarios are posed, replacing X and Y with various candidates' names, and conclusions are drawn.

The plausibility of these scenarios is determined in light of another infamous electoral figure: the opinion poll. When results don't match the public's expectations, that’s when the trouble and accusations start.

It's about sample size, universes, municipalities, representativeness, the impartiality of the pollster... all valid objections that often hide something simpler: the set does not always preserve the characteristics of its parts, and therefore the results of adding them together may seem counterintuitive.

This happens in many scenarios, but especially, as the marquis pointed out over 200 years ago, with the collective set of preferences. (Be careful when launching/changing that product after asking people about their preferences!)

Ok, but what does it mean that individual preferences are transitive and collective ones aren't?

In other words, collective preferences can be cyclical even if those of a rational individual are not. In this context, being rational means adhering to statements like "if A is preferred to B and B is preferred to C, THEN A is preferred to C."

It wouldn't make sense (it would be irrational) for A to be preferred to B, B to C, and yet C to A.

Of course, not all individual preferences are rational; you could prefer apples to oranges, oranges to bananas, but end up choosing a banana over an apple.

The problem arises when quantitative methods are applied to qualitative elements. Often, surveys or market studies are not adequate vehicles to convey all the information behind a choice involving a set of possibilities.

(Ever been unsure of what to say when asked "why did you rate us that way?" after a purchase? Your customers have, too, and yet many organizations treat that information as disconnected from other data sources. Your NPS should connect with other key customer metrics and business data, otherwise its usefulness is, at best, limited.)

Returning to our candidates, let’s simulate an election with three voters who, for our example, we will assume are rational and have transitive preferences. The results are as follows:

The results are computed as follows:

Since all our voters are rational, we conclude that Wolverine beats Batman, and Ironman beats Wolverine, so surely Ironman beats Batman… Or perhaps not:

Let's look at the aggregated preference to understand what happens:

In the image above, we clearly see what the Marquis meant back in the 1700s when he said that collective preferences can be cyclical (see the green arrows) even if the individuals making up that collective are not. The transitive property is lost in the result of adding up preferences.

The key element here is that this type of scenario creates room for the propagation of uncertainty and leads to concepts like strategic voting or the distortion of preferences by voters (and candidates, in political elections).

Besides the blatant exercise of including an election-related example to improve this post's traffic, the goal is to emphasize the importance of considering how user, customer, or prospect preferences are collected.

Even if the instrument is correctly designed, there are dangers when interpreting results and drawing direct conclusions without integrating other data sources (especially behavioral data). Doing so can lead to suboptimal decision-making.

The matter of transitivity is not trivial. In an example like the one with fruit, given three options, an individual has eight possible preferences (preferring oranges to bananas or vice versa; preferring oranges to apples or vice versa; preferring apples to bananas or vice versa). 2*2*2 = 8 (Note that we're not talking about the number of possible combinations).

In a transitive scenario like that of our candidates, there's an implicit order (A > B > C) that imposes a restriction on the set of possibilities with the same options: at each step of the decision-making process, there's one less option (if I say I prefer A, in the second step I can only choose between B and C, and in the third step only C remains). 3*2*1 = 6.

Imagine that instead of voters 1, 2, and 3 from our example, these are target audiences or the "marketing personas" your marketing agency suggested to categorize your customers, and instead of candidates A, B, and C, you have key functions or products to prioritize in your catalog. What would be the cost of a misinterpretation?