Is it "fair use" to refer to names of celebrities/cartoon characters in a game?

We have a game that requires players to refer to a list of celebrity and/or cartoon characters by name. The names simply show up in print alone, along with a short, original description of the entity.

I'm not worried about the celebrity names so much as I am the use of cartoon character and/or puppet names, as those in themselves are copyrighted works. The use in itself is a game, not a cartoon, movie, or the like, so I would think it to be both "transformative" and not in direct competition with profits of the original copyrighted characters, but I could be wrong.

From my understanding, the fair use doctrine on copyright use follows the following:

1. the purpose and character of the use, including whether such use
   is of a commercial nature or is for nonprofit educational purposes;
2. the nature of the copyrighted work;
3. the amount and substantiality of the portion used in relation to
   the copyrighted work as a whole; and
4. the effect of the use upon the potential market for or value of
   the copyrighted work.

1. It is commercial, however...
2. It is transformative in nature (not a derivative) of the copyrighted work
3. It is not very substantive, since it's only a reference to their names
4. The game would not seem to be in direct competition with the markets of the cartoon characters/celebrities/puppets

As an initial matter, the area of law that you need to be concerned about is trademark law and perhaps the right of publicity. Copyright law does not apply because you can't copyright a name. For example, the name "Mickey Mouse" is a trademark but is not copyrighted. The appearance of Micky Mouse and perhaps the sound of his voice is copyrighted, but it sounds like you are not using those things.

Beyond that, however, I'm not sure of what you can do legally. If all the characters in your game were Disney characters, then I think you would likely have a legal problem. If the characters come from all over the place, then I would think that you should probably be ok. I doubt that Trivial Pursuit got licenses for any character they mention in a question or answer.