This problem comes as the result of using another plugin with W3TC -- Nice Search.
Nice Search changes example.com/?s=searchterm to example.com/search/searchterm
This results in cached search pages that do not purge, so W3TC serves stale results to my readers.
I have already tried adding "search" to Purge Policy: Additional Pages, but that didn't fix the problem.
Any ideas how to configure W3TC to handle this?