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Yvain 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.
What happens if the first bidder bids $19 (or $19.99, or in general, the amount being auctioned minus the smallest permissible increment)? Any potential second bidder can't make any money. (...Without colluding with the auctioneer -- is that allowed?)
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.
But the other person could anticipate this reasoning and then simply bid $3 knowing that his opponent has committed himself to not bidding beyond $2.
AFAICT, this is an unfortunately strong argument... Thanks.
I see two solutions to the paradox:
1) Note that auctions are usually played by more than 2 bidders. Even if the first bidder would let you have the pot for $2, the odds that you'll be allowed to have it by everyone decrease sharply as the number of participants increases. So in a real auction (say at least 5 participants), 9% probably is overconfident.
2) If we have a small number of bidders, one would have to find statistics about the distribution of winners on these auctions (10% won by first bid, 12% won on second bid, and so on...). Of course, this strategy only works if your opponents don't know (and won't catch on) that you never bid more than once. But it should work at least for a one-shot auction where you don't publish your strategy in advance.
Out of curiosity, since you argue that joining these auctions as player #2 could very well have positive EU, would you endorse the statement "it is rational to join dollar auctions as the second bidder"? If not, why not?
Against typical human opponents it is not rational to join dollar auctions either as the second player or as the first, because of the known typical behavior of humans in this game.
The equilibrium strategy however is a mixed strategy, in which you pick the maximum bid you are willing to make at random from a certain distribution that has different weights for different maximum bids. If you use a the right formula, your opponents won't have any better choice than mirroring you, and you will all have an expected payout of zero.