A Solution to Newcomb’s Paradox?

Source: Julia Galef

Newcomb’s paradox/problem is a simple thought experiment that has implications for decision theory, causality, free will, logical fatalism, and even machine consciousness. But I’m going to try to avoid most of that and instead define Newcomb’s problem in a specific enough way to where the “rational” solution is more or less obvious.

First, the Wikipedia outline, a criticism of the original formulation, then my version.

There is a reliable predictor, a player, and two boxes designated A and B. The player is given a choice between taking only box B, or taking both boxes A and B.

The player knows the following:

1. Box A is clear, and always contains a visible $1,000.

2. Box B is opaque, and its content has already been set by the predictor:

3. If the predictor has predicted the player will take both boxes A and B, then box B contains nothing.

4. If the predictor has predicted that the player will take only box B, then box B contains $1,000,000.

5. The player does not know what the predictor predicted or what box B contains while making the choice.

At first glance, the solution seems obvious. Why wouldn’t you just go for box B and get a million dollars? Well, the predictor could be wrong of course. But let’s assume the predictor is above 99% accuracy.

The paradox arises due to an apparent conflict between the expected utility principle and the strategic dominance principle in game theory.

Considering the expected utility when the probability of the predictor being right is almost certain or certain, the player should choose box B. This choice statistically maximizes the player’s winnings, setting them at about $1,000,000 per game.

Under the dominance principle, the player should choose the strategy that is always better; choosing both boxes A and B will always yield $1,000 more than only choosing B. However, the expected utility of “always $1,000 more than B” depends on the statistical payout of the game; when the predictor’s prediction is almost certain or certain, choosing both A and B sets player’s winnings at about $1,000 per game.

— Wikipedia

Here, I disagree. Insofar as the predictor predicts that you will choose both A and B, B will have nothing in it, whereas if the predictor predicts that you will choose only B, the box will have a million dollars in it.

So choosing both boxes A and will not always have $1000 more dollars. In this case, it results in $999,000 dollars more if you pick only box B. In my opinion, there is no dominance strategy in this case.

It seems like there is a fallacy based on thinking in a wrong sequence of events. In our mind, we step up to the counter, decide we are only going to pick Box B, but at the last second also get box A because given the state of the world and the fact that we were only going to pick box B, we can now get a little extra cash by picking both.

But in reality, the predictor would obviously predict that. If this predictor played the game frequently enough, they would know that most people have an internal debate as to whether or not they should go for the extra box.

As an added side note, there is way too much of a discrepancy between the amount of money contained in each box which makes this a paradox only to self interested economists who are surprised to find out that people aren’t willing to get $1 if that means someone else gets $20.

The reason this is a paradox goes as follows. If you were predicted to pick only box B, then box B will have $1,000,000. Yet, box A also has $1000 right in front of you, so why not just pick both boxes?!

As Julia Galef puts it:

“Well, I sure hope that first box has a million dollars in it. But whether it does or not, that’s fixed now. The box is already on the table. There’s nothing I can do to change that, so I might as well take both boxes.”

“The other way of looking at it is to say, well, everyone in my position who takes both boxes only ends up with $1000. Whereas everyone in my position who takes only the first box ends up with $1 million.”

My version:

  1. You are on a game show in the year 2500. This means that you have seen others play this game, perhaps hundreds of times. And obviously you have pretty advanced science, like AI, brain scanning, and psychological algorithms.
  2. Box A has $1000 and Box B has $2000, instead of one box having 100x the value of the other box. This is more symmetric, which makes it more interesting. The reason is that only an extremely greedy person would risk $1 million for an extra $1000, whereas there is legitimacy in trying to earn an extra 50%.
  3. “The Predictor” is notoriously the most contentious aspect of Newcomb’s paradox. We could imagine that God is the predictor (99%+ accuracy), a private detective (75%+ accuracy), carnival barker (50/50), a superintelligent AI (95%+), Descartes Demon (100%?).
  4. If the predictor predicts that you choose only Box B and you pick only Box B, then you get $2000. If the predictor predicts that you choose both, then Box B contains nothing and you get only $1000. If the predictor predicts that you only pick box B, but you pick both, then you get the extra $1000.

Now that I have outlined a clearer version of the paradox, it is clear there is no paradox. The predictor is the crucial variable. How accurate is the predictor? If the predictor is 100% accurate, then there is no point in ever going for anything but box B.

Really, there is no point to playing this game, unless you get free money out of it.

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Thoughts on Economics, Politics, Philosophy, Ethics, and Computing by Adam Smith Reincarnated

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