Geico advertises that "People who switch to Geico save 15% or more". Notice that people who would lose money by switching to Geico should rarely switch. Accordingly, this metric will usually be positive.
Consider that people moving to a new city for a job may need more incentive to move.
Same basic problem as an employee who has worked at the same company for 10 years, and gotten their 3% raise every year - the market may have changed radically, but looking is a burden (emotionally, and otherwise) so they are unaware of that.
Enter a tremendous flood of car insurance advertising... to get people to actually evaluate if they are paying too much.
For what it's worth, when I had to make claims with Geico it was fine. But I was clearly not at fault in either case so it was not a difficult process.
I also think part of their ability to offer lower premiums is generally having lower human involvement -- no independent agents, claims routed through Web site or call centers, claims adjuster on site in body shop instead of visiting you, etc.
"Fifteen minutes could save you 15 percent or more on car insurance."
https://web.archive.org/web/20111201164345/http://www.the-dm...
While I agree with the others calling this a type of sampling bias, I've seen this type of error frequently enough that I think it deserves a more specific name.