## Uh-oh, here’s another “paradox” coming up

Will this be another 300 comment post? At Marginal Revolution, there’s a discussion of the Tullock Lottery. The summary from wikipedia is as follows:

The setup involves an auctioneer who volunteers to auction off a dollar bill with the following rule- the dollar goes to the highest bidder, who pays the amount he bid. The second-highest bidder also must pay the highest amount that he bid, but gets nothing in return. Suppose that the game begins with one of the players bidding 1 cent, hoping to make a 99 cent profit. He will quickly be outbid by another player bidding 2 cents, as a 98 cent profit is still desirable. However, a problem becomes evident as soon as the bidding reaches 99 cents. Supposing that the other player had bid 98 cents, they now have the choice of losing the 98 cents or bidding a dollar even, which would make their profit zero. After that, the original player has a choice of either losing 99 cents or bidding \$1.01, and only losing one cent. After this point the two players continue to bid the value up well beyond the dollar, and neither stands to profit.

Here’s my two cents. I don’t think it’s really a paradox. Some chap named Keith says the following:

If there is no equilibrium, avoiding the game altogether is
not an equilibrium, either. After all, if nobody bids at
all, then you should a penny and win.

It’s a paradox. Playing is a mistake. If nobody plays,
then not playing is a mistake, too.

But that can’t be right. If we assume complete information and rationality on behalf of the bidders, any given bidder A will know the extent of the universe of other potential bidders. So A’s utility calculuation while considering her first bid will be as follows. If she fails to bid, her expected gain is zero. If she does bid, then she must assume that everyone else has exactly the same utility calculation, and conclude that her expected gain is – all her money.

Can someone who knows more game theory than I chime in here? (Is anyone actually reading this blog yet?) I don’t think I understand how nonparticipation is not an equilibrium. The claim that there is no equilibrium seems to rely, sub rosa, on each actor’s not having access to the utility calculations undertaken by the others.

Indeed, for once it looks like wikipedia gets it right. From the wikipedia page again:

The actual expected value of bidding again is not Zero cents due to the unterminated nature of the game; the value of the bid is actually zero cents multiplied by the possibility of the other player giving up at that point, added to the value of losing two cents multiplied by the probability of the other player giving up at that point, in an infinite series with unbounded loss.

Exactly! The operative phrase there is “multiplied by the possiblity of the other player giving up at that point.” Which is zero, for the reasons given. If A has no reason to give up, then neither does the other player. It is concluded that the expected value of bidding is negative at all points. I’d suspect the experimental results to the contrary result from subjects irrationally failing to equate the opponent’s expected actions with their own and concluding that their opponent will give up at some point. I daresay it’s necessary to the bid decision to conclude that your opponent will give in sooner than you will.
Although this is all uninformed speculation. Someone clue me in? Please?

### 2 Comments to Uh-oh, here’s another “paradox” coming up

1. Brain says:

Being in full possession of my immortal soul, I can’t lay claim to having any deep knowledge of game theory. Nevertheless, it seems to me that this hardly warrants the title of paradox — as you yourself maintain, Herr Climacus.

It seems to me that intuitions are clouded by the fact that the set-up makes it sound like a simple auction — which, of course, it isn’t. In an auction you are bidding for a PARTICULAR item. That is, you are willing to pay a price for a particular outcome — a price which you pay if and only if that outcome obtains. In an ordinary auction a bid results either in a win or else in no change. In this “game”, a bid comes with an additional risk. There are, as it were, two different outcomes that may result.

Let us suppose that I bid n¢ for this Almighty Dollar. Then, if an opponent bids n+1¢, I lose n¢ and he loses (n – 99)¢. In other words, no matter what you bid, any rebid by the opponent will result in a greater loss for you than for him.

HAVING SAID ALL THAT, it seems to me that the game boils down to whether the opponent as a desire to harm you. After all, suppose that the very first bid is 99¢. Then the second bid could only be for \$1, resulting in no gain. If the other player has no interest in harming you, why would he bid? (Of course, once you both have bid, the opponent has an interest in minimising his harm — which is why he will out-bid you, even when the bids have exceeded a dollar, since each new bid will reduce his harm by 98¢. But this is covered by double-effect. Cf. Aquinas, Anscombe, etc.)

If, on the other hand, there is the slightest possibility that the other player is interested in harming you, you shouldn’t play at all.

Keith’s reasoning is specious because it relies on normal intuitions about auctions: a bid can’t expose you to harm. Whereas in this auction, any bid (unless your opponent is vindictive) exposes you to harm. It’s people like Keith who argued that invading Poland was a good idea since no one was at war with Germany AT THE TIME. Sometimes actions have bad consequences in the FUTURE.

2. This is where I have the advantage. As a lawyer, my immortal soul is already gone. Game theory is completely costless to me!

(I suppose I don’t undermine my dubious pseudonymity too much by saying “Hi Brain!”)

The bidding-as-punishment perspective makes me wonder if it isn’t rational to make just one bid and adopt a deterrent punitive strategy. If A bids 1¢ to start and credibly makes it known that he has adopted a strategy of punishing anyone else who outbids him by bidding them back up heedless of the destruction to his own position, can he run away with the prize? It would be rational for A to do that, if he could find some way to communicate an irrevocable committment to this strategy to the other parties. Then B couldn’t respond in kind because A would be un-deterrable (being irrevocably committed). Hmm. I think I might have to run this one by the marginal revolution people.

(The comment submission button for this blog apparently says “Abschicken.” I’m not sure how I feel about that. Is this supposed to be some kind of message commanding me to go to the gym? Oh, no, Google informs me that it’s German.)