This competing offer thing is relatively new and restricted to the top internet and wall street kind of companies. Where there is too much money and they just want people at the top of the ladder all the time, even if they don't need them.
This kind of requires companies winning above and beyond anybody else. Those are rare.
Every where else, its work as usual and people as usual. Nobody is important. People join and leave all the time. They could care less about whatever other offers you have.
