Whether Omega ended up being right or wrong is irrelevant to the problem, since the players only find out if it was right or wrong after all decisions have been made. It has no bearing on what decision is correct at the time; only our prior probability of whether Omega will be right or wrong matters.
the players only find out if [Omega] was right or wrong after all decisions have been made
If you observe Omega being wrong, that’s not the same thing as Omega being wrong in reality, because you might be making observations in a counterfactual. Omega is only stipulated to be a good predictor in reality, not in the counterfactuals generated by Omega’s alternative decisions about what to predict. (It might be the right decision principle to expect Omega being correct in the counterfactuals generated by your decisions, even though it’s not required by the problem statement either.)
It is extremely relevant to the original problem. The whole point is that Omega is known to always be correct. This version weakens that premise, and the whole point of the thought experiment.
In particular, note that the second decision was based on a near-certainty that Omega was wrong. There is some ordinarily strong evidence in favour of it, since the agent is apparently in possession of a million dollars with nothing to prevent getting the thousand as well. Is that evidence strong enough to cancel out the previous evidence that Omega is always right? Who knows? There is no quantitative basis given on either side.
And that’s why this thought experiment is so much weaker and less interesting than the original.
This variant is known as Transparent Newcomb’s Problem (cousin_it alluded to this in his comment). It illustrates different things, such as the need to reason so that the counterfactuals show the outcomes you want them to show because of your counterfactual behavior (or as I like to look at this, taking the possibility of being in a counterfactual seriously), and also the need to notice that Omega can be wrong in certain counterfactuals despite the stipulation of Omega always being right holding strong, with there being a question of which counterfactuals it should still be right in. Perhaps it’s not useful for illustrating some things the original variant is good at illustrating, but that doesn’t make it uninteresting in its own right.
Whether Omega ended up being right or wrong is irrelevant to the problem, since the players only find out if it was right or wrong after all decisions have been made. It has no bearing on what decision is correct at the time; only our prior probability of whether Omega will be right or wrong matters.
If you observe Omega being wrong, that’s not the same thing as Omega being wrong in reality, because you might be making observations in a counterfactual. Omega is only stipulated to be a good predictor in reality, not in the counterfactuals generated by Omega’s alternative decisions about what to predict. (It might be the right decision principle to expect Omega being correct in the counterfactuals generated by your decisions, even though it’s not required by the problem statement either.)
It is extremely relevant to the original problem. The whole point is that Omega is known to always be correct. This version weakens that premise, and the whole point of the thought experiment.
In particular, note that the second decision was based on a near-certainty that Omega was wrong. There is some ordinarily strong evidence in favour of it, since the agent is apparently in possession of a million dollars with nothing to prevent getting the thousand as well. Is that evidence strong enough to cancel out the previous evidence that Omega is always right? Who knows? There is no quantitative basis given on either side.
And that’s why this thought experiment is so much weaker and less interesting than the original.
This variant is known as Transparent Newcomb’s Problem (cousin_it alluded to this in his comment). It illustrates different things, such as the need to reason so that the counterfactuals show the outcomes you want them to show because of your counterfactual behavior (or as I like to look at this, taking the possibility of being in a counterfactual seriously), and also the need to notice that Omega can be wrong in certain counterfactuals despite the stipulation of Omega always being right holding strong, with there being a question of which counterfactuals it should still be right in. Perhaps it’s not useful for illustrating some things the original variant is good at illustrating, but that doesn’t make it uninteresting in its own right.