As wedifrid said, this is approximately transparent Newcomb plus distractions. Given Eliezer’s clarifications (Omega knows the lottery numbers and is accurate even when the Omega and lottery numbers match), I’ll ask how two algorithms would perform against Omega: OneBoxBot, which always one-boxes, and ConditionalBot, which one-boxes unless the lottery and Omega numbers match, in which case it two-boxes. I’ll ignore the tiny error rates in computing payoffs.
Case 1: the lottery is going to output a prime number.
Against OneBoxBot, Omega delivers a prime number, which may or may not match the lottery number. OneBoxBot gets a payoff of $1MM.
Against ConditionalBot, Omega must deliver a prime number different from the lottery number (if it matched the lottery Omega would be handing over a prime number with a predicted response of two-boxing from Conditional). So ConditionalBot one-boxes and gets a payoff of $1MM.
Case 2: the lottery will output a composite number.
OneBoxBot will one-box, so Omega must provide it with a prime number (which will not match the lottery). So OneBoxBot gets a payoff of $3MM every time the lottery randomly outputs a composite number.
Against ConditionalBot, Omega has two choices. It could deliver a prime number which does not match the lottery. This would lead ConditionalBot to one-box and get a total payoff of $3MM, the same as OneBoxBot. Alternatively, Omega could match the composite lottery number, leading ConditionalBot to two-box and get a payoff of $2,001,000, $999,000 worse than OneBoxBot’s payoff.
So: both algorithms receive lottery wins with equal frequency, OneBoxBot always performs at least as well as ConditionalBot across the possibilities, and ConditionalBot performs worse when the lottery outputs a composite and Omega chooses to match it. Thus we should not adopt ConditionalBot over OneBoxBot and should one-box faced with the Ultimate Newcomb problem.
There is a big and implicit step that is worth explicating here, because most people who first approach Newcomb-like problems miss it completely:
TREAT HUMANS AS BOTS
By a bot I mean an algorithm, of course. An algorithm Omega can analyze for all possible combination of inputs.
That this step is valid follows from the problem’s stipulation that Omega can predict your actions. In other words, it knows your output for any combination of inputs it cares to give you.
Whether Omega does this by running your algorithm in a sandbox or by analyzing your code does not affect the answer to the puzzle, since the end result is the same. But the sandboxing version can often make it easier to find the solution, because it lets one rely on the Reflective Equilibrium of sorts: you cannot tell when deciding what to do whether you are in an Omega’s simulation of you or not, so you may as well assume that you are.
TL;DR: to Omega, you are a bot, so write down all relevant algorithms and analyze/run them before picking a winning one.
This is what I did too. One big advantage is that it changes Omega’s predictive abilities from mysterious magic to a simple process that it’s possible to completely analyse without getting confused.
As wedifrid said, this is approximately transparent Newcomb plus distractions. Given Eliezer’s clarifications (Omega knows the lottery numbers and is accurate even when the Omega and lottery numbers match), I’ll ask how two algorithms would perform against Omega: OneBoxBot, which always one-boxes, and ConditionalBot, which one-boxes unless the lottery and Omega numbers match, in which case it two-boxes. I’ll ignore the tiny error rates in computing payoffs.
Case 1: the lottery is going to output a prime number.
Against OneBoxBot, Omega delivers a prime number, which may or may not match the lottery number. OneBoxBot gets a payoff of $1MM.
Against ConditionalBot, Omega must deliver a prime number different from the lottery number (if it matched the lottery Omega would be handing over a prime number with a predicted response of two-boxing from Conditional). So ConditionalBot one-boxes and gets a payoff of $1MM.
Case 2: the lottery will output a composite number.
OneBoxBot will one-box, so Omega must provide it with a prime number (which will not match the lottery). So OneBoxBot gets a payoff of $3MM every time the lottery randomly outputs a composite number.
Against ConditionalBot, Omega has two choices. It could deliver a prime number which does not match the lottery. This would lead ConditionalBot to one-box and get a total payoff of $3MM, the same as OneBoxBot. Alternatively, Omega could match the composite lottery number, leading ConditionalBot to two-box and get a payoff of $2,001,000, $999,000 worse than OneBoxBot’s payoff.
So: both algorithms receive lottery wins with equal frequency, OneBoxBot always performs at least as well as ConditionalBot across the possibilities, and ConditionalBot performs worse when the lottery outputs a composite and Omega chooses to match it. Thus we should not adopt ConditionalBot over OneBoxBot and should one-box faced with the Ultimate Newcomb problem.
There is a big and implicit step that is worth explicating here, because most people who first approach Newcomb-like problems miss it completely:
TREAT HUMANS AS BOTS
By a bot I mean an algorithm, of course. An algorithm Omega can analyze for all possible combination of inputs.
That this step is valid follows from the problem’s stipulation that Omega can predict your actions. In other words, it knows your output for any combination of inputs it cares to give you.
Whether Omega does this by running your algorithm in a sandbox or by analyzing your code does not affect the answer to the puzzle, since the end result is the same. But the sandboxing version can often make it easier to find the solution, because it lets one rely on the Reflective Equilibrium of sorts: you cannot tell when deciding what to do whether you are in an Omega’s simulation of you or not, so you may as well assume that you are.
TL;DR: to Omega, you are a bot, so write down all relevant algorithms and analyze/run them before picking a winning one.
This is what I did too. One big advantage is that it changes Omega’s predictive abilities from mysterious magic to a simple process that it’s possible to completely analyse without getting confused.