If you can go through that chain of reasoning, so can the other person - therefore, it doesn't seem entirely ridiculous to me to bid $2 to the other person's $1 in the hope that they won't want to enter a bidding war and you'll win $18.

Let's say there's an X% chance you expect the other person to surrender and let you have the $20 for $2 rather than enter the bidding war, and let's also say you don't intend to ever make a bid after your first bid of $2. Then expected value is (X)(20) - (1-X)(2) = 20x - (2 - 2X) = 22X - 2. If X is greater than 1/11 or about 9%, then it's profitable to enter the auction. So unless you're greater than 91% sure that the other person will start a bidding war instead of sacrificing their $1 and letting you have the money, it's positive expected value to enter the auction.