# Motivation

Several years ago, I was looking for speakers on Amazon. Prices varied widely, from $25 to$250 or more. As I browsed, I noticed an interesting pattern: the most expensive speakers had the lowest average rating. More recently, I discovered the study “Expensive Running Shoes Are Not Better Than More Affordable Running Shoes.” The study finds that shoes with higher prices tended to get lower ratings. The 10 most expensive shoes in the study had an average cost of $191 and an average rating of 79, while the 10 cheapest had an average cost of$60 and an average rating of 86. At a brand level, Sketchers had the lowest average cost, and the highest average rating.

# Possible Explanations

What could explain this pattern?

The study’s title assumes that average rating corresponds to true quality. It implies that if people were awarded a free pair of shoes of their choosing (and could not resell these shoes), many would choose a cheap pair.

An alternative explanation is that ratings reflect value, rather than quality. Someone might rate cheap shoes more highly than expensive ones, even if they thought the quality was slightly lower. If we believe this hypothesis, we might title our study “Expensive Running Shoes Are Penalized for their Price.”

This post will explore a third explanation: the shoes are being rated by different groups of people. People who buy Sketchers tend to be different from those who buy high-end Adidas. This hypothesis suggests that a more accurate title for the study might be “Expensive Running Shoes Are Purchased by Demanding Customers.”

# A Simple Model

Let’s illustrate with a simple example. There are two types of shoe: basic for $60 and premium for$180. There are also two types of runners: competitive and casual. Casual runners are not very discerning: they would rate the basic sneakers 90 and the premium ones 95. Meanwhile competitive runners are aware of every imperfection: they rate the basic sneakers 50 and the premium ones 75.

Of course, runners don’t choose which shoe to rate at random. Most casual runners buy basic shoes, though there are some exceptions (Silicon Valley venture capitalists come to mind). Meanwhile, competitive runners almost always buy premium shoes. To make things concrete, let’s suppose that there are 130 runners of each type. 90 of the casual runners buy basic shoes, and 40 choose premium. Meanwhile, only 10 of the competitive runners buy basic shoes – the remaining 120 choose premium.

The end result? The basic shoes get $$90+10 = 100$$ reviews, and an average rating of $$(90 \times 90 + 10 \times 50)/100 = 85$$. The premium shoes get $$40+120 = 160$$ reviews, and an average rating of $$(40\times 95 + 120 \times 75)/160 = 80$$.

Note that something strange has happened: even though every individual customer would rate premium shoes above basic ones, the premium shoes end up with a lower rating! This could never happen if everyone rated both shoes (or chose which shoes to rate randomly). Instead, it arises because of selection.