I assume that if you have m positions open, you extend offers to at most m people, not n > m? Do you ever try to get a sense of how many other companies the potential hire is looking at?
That’s always a question you try to get answered during the introduction and qualification process. It also depends on where the candidate came from: job posting response, friend referral, Monster resume bank, etc. Sometimes with a candidate that comes from a referral, the chances of there being high competition are lower. If it’s a hot skill set - always assume they have other activity and potential offers in the works.