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Dreamed 1998/10/13 by Chris Wayan


Analog magazine has a strange series of letters from physicists and Mensa people with nothing better to do, arguing over a conundrum that goes as follows:

You're on a gameshow. Three doors: one hides a treasure, two hide goats. You pick one, but they don't open it yet. First, the host opens one of the other doors to reveal a goat (no matter what you picked, there's at least one goat to show you, right?) Now you can change your choice, if you want. Should you?
You chose your door when it was 1 of 3, but now it's 1 of 2. Some writers argue it's the same door it always was and the odds are 50% for both; and the host will always show you a goat no matter where the treasure is; so he tells you nothing.

One letter argues "If a stranger walks in at this midpoint and sees two doors, they have a 50% chance of finding treasure. But how's it different for you?" Yet you can't choose among three doors and end up always with a 50% chance of being right!

One writer makes a chart, shows four possibilities: if you guessed right, the talkshow host has a choice of TWO goats to show you; in either of those cases, you lose if you switch. But if you initially guess either goat-door, when the host shows you a goat he gives you VITAL data--the other door hides treasure! Two ways to win by switching. Equal! So why switch?

That chart's deceptive--but it leads me toward the truth. The two cases where switching wins seem parallel to the switch-and-lose pair, where you've picked the treasure, and the show host has a choice of two goat-doors to open. But they're NOT parallel. If you were right to start with, the host has two doors to open, yes, but you're initially right just 1/3 of the time--each choice by the host happens only 1/6 of the time. But you'll initially choose each of the two goat-doors 1/3 of the time, and in BOTH these cases the host DOES help you. So 2/3 of the time switching wins, and 2/6 of the time it loses. You should switch.

And even if I'm wrong and the odds are always even, you have nothing to lose by switching. So you should!

It's easy to see that the host does help you if we expand it to 100 doors. You pick one, the host opens 98, leaving yours and one other, one of which hides treasure. Wouldn't you switch?

This debate between articulate amateur mathematicians convinces me that even simple problems involving uncertainty and partial information are inherently difficult for human brains--it's tempting to dive into calculation and defend your first plausible analogy before you've mulled over a problem enough to be sure you're framing it optimally. And it's not clear what "optimal", or even "right" means. The charts of possible states and outcomes in all those passionate letters were SO convincing!

I wish I could sum up the lesson as "If two choices look equal, favor the one based on fresher data." Sort of the reverse of "new friends are silver, old friends are gold." But if this puzzle teaches anything, it's the dangers of generalizing.


Map of a handlike mesa. Hobbits hold the valley between pointer and index fingers, with the only creek. Men are on the wrist.

War in a desert. A set of ridges like fingers stretch out from a hand-plateau. The valley below the longest finger has a creek; the other two valleys between fingers are dry.

I'm a hobbit. A human tribe wants the area. Their warriors seem tall as giraffes to us, slender black figures in bright robes, carrying spears and guns (we have guns too). I worry the war's due to ignorance or mutual racism--bet we could end it if we'd just talk. But no one tries. Battle's already engaged.

The hobbits hold the stream in Valley 1. Men hold the high ground of the wrist and are streaming along the back of the hand toward Finger 1 and Valley 1, joining the fight there. I'm one of a few hobbit scouts in the short dry valleys 2 and 3, checking that they're being ignored. The Men are out to burn the valleys (stupid goal--they're mostly rock and sand, little to burn). The only one with vegetation has water--we doused the tree trunks round the spring pretty thoroughly just before the fight. Both sides clearly want Valley 1--it's the only usable land.

I hate to see whole squads of Men joining the fight without challenge. I consider sniping at them from my hiding place on Finger 2, or better still, from a small knoll on the far side of their path--from there I could kill quite a few and pin them down a while, block all their reinforcements. But I'd be traceable and they'd kill me in the end. I don't want to die. I feel angry that the hobbit strategy was to cling to the waterhole ONLY--we should have hidden half our force as snipers and guard lines all along the first finger and part of the hand, to keep them out of the valley we want.

But we ceded them all the high ground! Stupid, stupid, stupid.



The dream went to a lot of trouble to create a situation where simple assertiveness, courage or open expression of anger wouldn't work--but sniping would. At a cost. It isn't so simple as "Be yourself" or "Sarcasm sucks." My life right now requires care, skill, and ruthless strategy. Both retreat and a frontal assault will fail; I need to snip away at my problems cautiously, even furtively. But in my head are hippie/hobbit values that devalue that sort of wary planning for the worst. It's how I burned my hands. If I don't want to get burned far worse, I'd better quit trusting--and think things through.

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