Bidirectional Encryption I would like a way to hide a piece of information from Alice and Bob. Let's say this piece of information lets Alice and Bob vote me off the island. If Alice had this information she could vote me off even if Bob did not want that. Likewise if Bob had this information he could vote me off the island even if Alice did not want that. However I'd like Alice and Bob to be able to vote me off the island if they both agree that its in their interest. So I'd like to give Alice and Bob a piece of information such that they can combine that information to vote me off the island.

One very simple method I thought of was to encrypt the information. Give the encrypted information to them. And give them each half the key that I encrypted that infomation with. I wasn't however sure how secure this was or what problems there might be with it. Additionally this is (deliberately) just like some of the naively proposed methods by which back doors might be added to encryption. I'm trying to consider what is wrong with this from a mathematical stand point. So does this method have some known theoretical issue (not issues in practice)? Is there an efficient algorithm that attacks it? If this method is a bad method then is there a good method? If not then is it possible for there to be a good method?
That is called secret sharing.

The usual method for the 2-shares case is the method you thought of with the one-time pad.

Note that usual secret-sharing schemes provide no protection against fake shares.