Mick Hartley

I’m not sure that I follow all this (via):

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.

What numbers will Lucy and Pete write? What number would you write?

Well – I’d write the number corresponding to the actual price of the antique, because not only would I then be telling the truth, but the chances would be reasonably high that the other person would give the same price. Everyone would be happy, and justice would be done, and be seen to be done.

Sadly, that doesn’t appear to be an option. So unlikely is it considered that anyone would be interested in telling the truth as opposed to maximising their profit, that the actual price isn’t mentioned at all. It’s irrelevant.

OK, I’m being a little obtuse here. I can see why the price is irrelevant when it comes to the logic of the game, and what it’s trying to show.

To see why 2 [$2] is the logical choice, consider a plausible line of thought that Lucy might pursue: her first idea is that she should write the largest possible number, 100, which will earn her $100 if Pete is similarly greedy. (If the antique actually cost her much less than $100, she would now be happily thinking about the foolishness of the airline manager’s scheme.)

Soon, however, it strikes her that if she wrote 99 instead, she would make a little more money, because in that case she would get $101. But surely this insight will also occur to Pete, and if both wrote 99, Lucy would get $99. If Pete wrote 99, then she could do better by writing 98, in which case she would get $100. Yet the same logic would lead Pete to choose 98 as well. In that case, she could deviate to 97 and earn $99. And so on. Continuing with this line of reasoning would take the travelers spiraling down to the smallest permissible number, namely, 2. It may seem highly implausible that Lucy would really go all the way down to 2 in this fashion. That does not matter (and is, in fact, the whole point)–this is where the logic leads us.

Yet people playing the game – and this is where the designers of the experiment get all excited – don’t in fact go right down to $2, but stick to the higher end of the range.

Well, who’d have thought?

The point is, surely, that once you get below a certain point in the backward induction, you’d see where the logic was headed, and realise that it made no sense. I don’t buy the author’s premise that by doing so you’re somehow acting irrationally. But, to be fair, neither does he, really:

If I were to play this game, I would say to myself: “Forget game-theoretic logic. I will play a large number (perhaps 95), and I know my opponent will play something similar and both of us will ignore the rational argument that the next smaller number would be better than whatever number we choose.” What is interesting is that this rejection of formal rationality and logic has a kind of meta-rationality attached to it.