Someone has probably shut you down in 4 pages but you should go and ask the police about this. Your "job" is to report it to the professionals, that is it. You have no idea who you are dealing with, you have no authority, no training and all those people decrying this are quite correct. What if this guy was armed/prepared to injure people to evade being apprehended. AFAIK good simaritan laws usually extend to violent crimes or specific situations in most countries so you could find yourself in a world of shit if it went down badly.
So, what you're telling me if that if someone just took your wallet while you had it in your hand and ran with it, you'd just call the police instead of first trying to catch him? And that you wouldn't be ok if someone else observing the scene would stop them?
Or are you telling me you wouldn't care if it was someone else's belongings?