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VincentYu comments on Game Theory As A Dark Art - Less Wrong
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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.
Nitpick: Expected value, not utility.
It is standard to call an expected value an "expected utility" when the values in question are utilities.
Correct but irrelevant, as Yvain was discussing dollars.
You're right; I was identifying the values with utilities for the purposes of the scenario, which I only now see was precisely what VincentYu was criticizing.