ProgramCrafter's Shortform

ProgramCrafter

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Anthropics based on prediction markets - Part 2

Follow-up to https://www.lesswrong.com/posts/xG98FxbAYMCsA7ubf/programcrafter-s-shortform?commentId=ySMfhW25o9LPj3EqX.

Which Questions Are Anthropic Questions? (https://www.lesswrong.com/posts/SjEFqNtYfhJMP8LzN/which-questions-are-anthropic-questions)

1. The Room Assignment Problem

You are among 100 people waiting in a hallway. The hallway leads to a hundred rooms numbered from 1 to 100. All of you are knocked out by a sleeping gas and each put into a random/unknown room. After waking up, what is the probability that you are in room No. 1?

2. The Incubator

An incubator enters the hallway. It will enter room No.1 and creat a person in it then does the same for the other 99 rooms. It turns out you are one of the people the incubator has just created. You wake up in a room and is made aware of the experiment setup. What is the probability that you are in room No.1?

3. Incubator + Room Assignment

This time the incubator creats 100 people in the hall way, you are among the 100 people created. Each person is then assigned to a random room. What is the probability that you are in Room 1?

In all of those cases, betting YES on probability 1% is coherent in the sense that it leads to zero expected profit: each of the people buys 1 "ROOM-1" share at price of 1/100, and one of them wins, getting back 1 unit of money.

1Tensor White3y

To phrase it better: You find yourself in room N, how many total rooms are there? I know UDASSA accounts for the description length of the room address, but remember that given a number of rooms, each room will have the same description length. If there are 64 rooms, then room 1 will have address "000000" and not simply "0" or "1". This way if you find yourself in a room, without knowing how many total rooms there are, and only knowing your room number, then you write it out in binary and take 2 to the bit-length of your room's address. For ex, you find yourself in room number "100111", 6 bits. So with 50% chance, there will be 64 rooms in total. Then you add an extra bit with 50% of the remaining measure (25%), 128 rooms, and repeat. If the payout doesn't scale with the number of rooms, then 64 rooms would be the most profitable bet. It's easy to test this either irl, or with a python script. python script: https://pastebin.com/b41Sa6s6 After doing so, I got unexpected results: given your room number, the most likely number of total rooms is a number whose description length is one-bit longer than the description length of your room. Weird.

The experiment is commonly phrased in non-anthropic way by statisticians: there are many items getting sequential unique numbers, starting from 1. You get to see a single item's number $n$ and have to guess how many items are there, and the answer is $2 * n$ . (Also, there are ways to guess count of items if you've seen more than one index)