NOT Metcalfe’s Paradox – A Puzzle

Bob Metcalfe posed this puzzle at an unconference we were recently at. He says, however, he didn't invent it. Most people, including me, get the answer wrong; in fact, I got the answer wrong twice – once by not listening and once by not thinking.

Here's the puzzle:

You are a guest on a game show. There are three closed doors; behind one of them is a car you want to own; behind the other two are goats you don't want despite the fact that they don't burn gas.

You have to pick a door. After you do that, the host will pick a door behind which there is a goat (he knows what's where and has to follow the rules). You then get to decide whether you should be awarded what's behind the door you picked initially or what's behind the door that neither of you picked.

The questions, smart reader, are:

1. Does it matter which strategy you pick?
2. If so, which strategy is favored?
3. What is the quantitative advantage, if any, of the favored strategy?
4. For extra credit: why?