In these types of situations I find that people are either dissatisfied with their job or they are simply bored or unchallenged. Either way it would be wise as the other folks also shared with you - to find out really what is going on. Otherwise you are in the dark and have no strategy!

If this person was not out looking for a job and the other company found him and made him an offer - it obviously got him thinking about what his current job is lacking - so why don't you just ask him - what IS the job lacking?

But in general - such a person has already left and has no real loyalty to your organization. And because of that, I usually prefer that person leaves. To me it's kind of like trying to stay in a relationship that really needs to break up and you're only hanging on because it's convenient to have someone to sleep with at night and go out to dinner with and you're too lazy to go through the dating process all over again. :)