This is the first part of a three post sequence on a problem that is similar to Newcomb's problem but is posed in terms of probabilities and limited knowledge.

Part 1 - stating the problem

Part 2 - some mathematics

Part 3 - towards a solution

Omega is an AI, living in a society of AIs, who wishes to enhance his reputation in that society for being successfully able to predict human actions. Given some exchange rate between money and reputation, you could think of that as a bet between him and another AI, let's call it Alpha. And since there is also a human involved, for the sake of clarity, to avoid using "you" all the time, I'm going to sometimes refer to the human using the name "Fred".

Omega tells Fred:

I'd like you to pick between two options, and I'm going to try to predict which option you're going to pick.

Option "one box" is to open only box A, and take any money inside it

Option "two box" is to open both box A and box B, and take any money inside them

but, before you pick your option, declare it, then open the box or boxes, there are three things you need to know.

Firstly, you need to know the terms of my bet with Alpha.

If Fred picks option "one box" then:

If box A contains $1,000,000 and box B contains $1,000 then Alpha pays Omega $1,000,000,000

If box A contains $0 and box B contains $1,000 then Omega pays Alpha $10,000,000,000

If anything else, then both Alpha and Omega pay Fred $1,000,000,000,000

If Fred picks option "two box" then:

If box A contains $1,000,000 and box B contains $1,000 then Omega pays Alpha $10,000,000,000

If box A contains $0 and box B contains $1,000 then Alpha pays Omega $1,000,000,000

If anything else, then both Alpha and Omega pay Fred $1,000,000,000,000

Secondly, you should know that I've already placed all the money in the boxes that I'm going to, and I can't change the contents of the boxes between now and when you do the opening, because Alpha is monitoring everything. I've already made my prediction, using a model I've constructed of your likely reactions based upon your past actions.

You can use any method you like to choose between the two options, short of contacting another AI, but be warned that if my model predicted that you'll use a method which introduces too large a random element (such as tossing a coin) then, while I may lose my bet with Alpha, I'll certainly have made sure you won't win the $1,000,000. Similarly, if my model predicted that you'd make an outside bet with another human (let's call him George) to alter the value of winning $1,001,000 from me I'd have also taken that into account. (I say "human" by the way, because my bet with Alpha is about my ability to predict humans so if you contact another AI, such as trying to lay a side bet with Alpha to skim some of his winnings, that invalidates not only my game with you, but also my bet with Alpha and there are no winning to skim.)

And, third and finally, you need to know my track record in previous similar situations.

I've played this game 3,924 times over the past 100 years (ie since the game started), with humans picked at random from the full variety of the population. The outcomes were:

3000 times players picked option "one box" and walked away with $1,000,000

900 times players picked option "two box" and walked away with $1,000

24 times players flipped a coin and or were otherwise too random. Of those players:

12 players picked option "one box" and walked away with $0

12 players picked option "two box" and walked away with $1,000

Never has anyone ever ended up walking away with $1,001,000 by picking option "two box".

Omega stops talking. You are standing in a room containing two boxes, labelled "A" and "B", which are both currently closed. Everything Omega said matches what you expected him to say, as the conditions of the game are always the same and are well known - you've talked with other human players (who confirmed it is legit) and listened to their advice. You've not contacted any AIs, though you have read the published statement from Alpha that also confirms the terms of the bet and details of the monitoring. You've not made any bets with other humans, even though your dad did offer to bet you a bottle of whiskey that you'd be one of them too smart alecky fools who walked away with only $1,000. You responded by pre-committing to keep any winnings you make between you and your banker, and to never let him know.

The only relevant physical object you've brought along is a radioactive decay based random number generator, that Omega would have been unable to predict the result of in advance, just in case you decide to use it as a factor in your choice. It isn't a coin, giving only a 50% chance of "one box" and a 50% chance of "two box". You can set arbitrary odds (tell it to generate a random integer between 0 and any positive integer you give it, up to 10 to the power of 100). Omega said in his spiel the phrase "too large a random element" but didn't specify where that boundary was.

What do you do? Or, given that such a situation doesn't exist yet, and we're talking about a Fred in a possible future, what advice would you give to Fred on how to choose, were he to ever end up in such a situation?

Pick "one box"? Pick "two box"? Or pick randomly between those two choices and, if so, at what odds?

And why?

Part 1 - stating the problem

next Part 2 - some mathematics

Part 3 - towards a solution