a caught "dropped" third strike

[GarthB]

Bob:

Dave said the pitch was caught for third strike. The runner took off anyway either accidentally or perhaps purposely attempting to draw a throw if, let's say, there is a runner on second and they want to get him over to third on a defensive mistake.

The question is, do you announce loudly and overtly the out, thus alerting the defense to the scheme or let nature take its course and allow the defense to pay for being asleep at the wheel?

There are good arguments to be made on both sides. This isn't so much a rules situation as it is one of game management.