This month’s Scientific American has an interesting story about the Travelers’ Dilemma, which is simply described as follows.

Lucy and Pete, returning from a remote Pacific island, find that the airline has damaged the identical antiques that each had purchased. An airline manager says that he is happy to compensate them but is handicapped by being clueless about the value of these strange objects. Simply asking the travelers for the price is hopeless, he figures, for they will inflate it.

Instead he devises a more complicated scheme. He asks each of them to write down the price of the antique as any dollar integer between 2 and 100 without conferring together. If both write the same number, he will take that to be the true price, and he will pay each of them that amount. But if they write different numbers, he will assume that the lower one is the actual price and that the person writing the higher number is cheating. In that case, he will pay both of them the lower number along with a bonus and a penalty–the person who wrote the lower number will get $2 more as a reward for honesty and the one who wrote the higher number will get $2 less as a punishment. For instance, if Lucy writes 46 and Pete writes 100, Lucy will get $48 and Pete will get $44.

The author then goes on to talk about how the “rational” strategy will drive both players’ bids down to the minimum, but I’m reminded of a Princess Bride quote.

You keep on using that word. I do not think it means what you think it means.

The “rational” strategy supposedly goes like this. If we both bid $100, we each get $100, but I can improve on that by bidding $99 so I get $101 and you get $97. If you know, or suspect that I’m going to do that you can do better by bidding $98. I can beat that by bidding $97, and so on all the way down to $2. The problem is that people don’t really try to guess what **number** the other person is going to use until they try to guess what **strategy** they’re going to use. If you’re smart enough to follow the above train of thought to it’s $2 conclusion, then I’ll guess that you’re also smart enough to realize that the conclusion is highly undesirable to both of us. The only obvious alternative to the “try to maximize my return even if it’s unfair” strategy is the “fair share of the highest total return” strategy, which means a $100 bid. Not coincidentally, that’s the same result we’d almost certainly get if we met and made an enforceable deal ahead of time. Knowing that one strategy leads to a bad result for both of us and the other strategy leads to a very good (though not quite perfect) result for both of us, I’d say the only **truly** rational conclusion would be that my partner/opponent will choose the latter, so I do the same. Besides, the difference between being wrong with the cooperative strategy and being right with the competitive one is only $4 – not much of an inducement to change, and it’s the same even if the minimum bid is raised. A more interesting experiment might be one that shares some of the same dynamics, but where the penalty for guessing wrong about strategies – not bids – is larger.

Hi Jeff, I know it’s not the point of your post, but I couldn’t resist commenting (even though it’s obvious):

The airline manager’s logic is severely flawed:

statement : “they will inflate it”

conclusion : I’ll trust the one who quotes the lowest price

if the statement is right, this means the airline always pays too much.

It depends how much you trust your partner.

[1] You conclude that bidding down to $2 is a bad deal for both. You assume your partner has reached the same conclusion.

[2] You assume your partner will therefore adopt the â€œfair share of the highest total returnâ€ strategy and bid $100.

Having both reasoned as above – what’s to stop you screwing your parner by bidding $99? But you can safely assume your partner will think along the same lines so we have reached a circular reasoning scenario – so where does it end?

True I would not actually care if my partner bid $99 against my $100 since I still end up with $97, but if the penalties were harsher as you suggested then we would end up with a circular situation.

Even more interesting would be to repeat the experiment multiple times so each traveller can take into account the number of times he has been screwed by his partner. I believe in that case both travellers would reason that it *wouldn’t* pay to screw the other guy (given the less-then-optimal cash receivable when he tries to get his revenge in subsequent bids) and bid $100 every time.