I can see the sense of a decision like that, as Vardy will feel he has achieved more than his life's footballing ambitions at Leicester. He has won the Premier League, played (and scored) for England and is about to embark on a Champions League adventure regardless of who he is playing for
The only questions he can have are:
Will Arsenal change their game style to suit?
Will he miss the camaraderie that is so obviously in the Leicester City camp?
Does he owe Leicester (like others around him) one more season whilst they compete in the Champions League?
Will he regret not playing for a legendary (or 'big name') football club once he is retired?
The answer to all of those is open to debate, but if he stays he will have my admiration because I have to be honest and say that if I was in his position, I would be seriously looking to add a 'big name' club to my CV before I retired to write a book, star in a film and gloat about my career to my grand children!