I haven't looked at all your code yet but this jumped out at me:
You should never create more than one instance of a Scanner from System.in within an ENTIRE PROJECT. Scanner has a small amount of buffering built in which could cause one Scanner to be waiting for input that another Scanner has already consumed.
You have one large method which will probably need to be broken down into smaller methods, so, rather than worrying about passing a Scanner variable around it would be simplified if you made a constant out of it.
I would start over and begin your next class look something like this:
This is just off the top of my head so there may be better ways to do it, but I think you get the point. The methods playerPicksUpSticks(), computerPicksUpSticks(), and checkForWinner() are up to you to write. Remember the DRY principle: don't repeat yourself.
All things are lawful, but not all things are profitable.