Philosophy

A Twist In A Solution To Newcomb’s Paradox

Let’s take a break from the insanity and wade into cool waters to start the weekend. I’ll take it that you’ve watched this video, which purports to give a solution to Newcomb’s paradox, or that you otherwise know about both.

The idea is simple. There are two boxes. A variable one has either a million or nothing, and the other always had a thousand. You can (A) choose the first variable box or (B) pick both. But there is an entity that can, with some unspecified degree of accuracy, predict what you will do, choose A or B. If the entity thinks you’ll pick A, he’ll stick the million in first box; if he thinks you’ll pick B (both boxes), he’ll leave the first box empty.

Obviously, you’d like to max out your purse. So, which choice should you make?

Here is the twist: all probability, really all information, is conditional. Keep it in mind. Not so easy to do; I know I always didn’t.

Wildberger’s solution uses “expected values”, of which I am not a universal fan. Not that they are without value, but there are deep suspicions about them in odd situations.

Skip that for now, and accept the idea of expected value (which is explained in the video). Wildberger says that, somehow, we know

     p_A = Pr( Entity guess you pick A | you pick A, I),

and

     p_B = Pr( Entity guess you pick B | you pick B, I).

The “I” is the information that lets you deduce the numbers. We do not (and few do) discuss where this I comes from. But see below.

Note that some versions of this paradox have p_A = p_B = 1; i.e. the entity is an infallible oracle.

Lastly, it’s convenient to let m = money in the fixed box, and mK = money in the variable box. Above, m = 1,000 and K = 1,000. Then we can do the expected value calculation.

     E_A = p_A * (mK) + (1-p_A) * (0),

     E_B = p_B * (m) + (1-p_B) * (m + mK).

If you choose A, only the variable box, you get mK if the entity guesses correctly, which has probability (we say) of p_A; if the entity, however, guesses you’ll pick B (even though you pick A), you get nada, with probability (1-p_A). The sum of both possibilities is the expected value when you pick A.

A similar argument holds for picking B, as you see.

Expected value logic says pick the strategy with the larger value, E_A or E_B. For example, you should pick A when E_A ≥ E_B, or when, as Wildberger shows us,

     p_A + p_B ≥ 1 + 1/K.

Assuming the entity is an oracle, i.e. p_A + p_B = 1, you’d choose A when K ≥ 1. If K < 1, you’d pick B (both boxes). If either p_A or p_B is less than 1, which happens when the entity can make mistakes, you’d have to plug in what’s what and solve the simple calculation. So says expected value calculus.

Now the twist, which involves all that stuff about quantum mechanics and free will and whatnot Wildberger was referencing.

For ease, let’s stick with the entity-as-oracle, so that p_A = p_B = 1. Fix a K, which is known, both by you and the entity. Doesn’t matter which K you pick; it only matters if K ≥ 1 or K < 1. Above we said that pick A (only variable box) if K ≥ 1, else pick B (both boxes).

Since the entity is an oracle, this means the entity is never wrong about what you’d pick—no matter when you pick (saying the oracle never guesses wrong makes it irrelevant when he guesses; and even though somebody can predict what you will choose, you still have free will). That means the entity knows you know about expected value. Since we all know K ≥ 1, the entity knows you’d pick A. Which means he’d put the mK in the variable box. So far, it’s seems there is no problem and no twist. (Obviously as K goes to infinity, there is less incentive to choose B anyway.)

But, since you know the entity can’t guess wrong, because you started with expected value, and expected value says pick only the variable box K ≥ 1, why not pull a fast one and pick B (both boxes), since the entity assumes you’ll only pick the variable box?

Yet the entity can’t guess wrong! He’ll know you’ll try a fast one, and so leave zippo in the variable box. You only get m. But, we just figured out the entity would know this “second iteration”, as it were, and that we’d have to pick A, so we have the opportunity for another sly move. Yet the entity would have figured this, too, since he always guesses right. Meaning if you try to get clever, you end up with only m.

That works if you go to infinity in the chain of reasoning, too. If K ≥ 1, pick A and win mK. Any tricks brings you only m.

Now if K < 1, expected value says to pick B, both boxes. But the entity knows you’d do this, which means he’d—again—leave the variable box empty, meaning you ought not to pick both boxes, since you’d only get m.

Hold up. Why would you pick only the variable box, which only has mK < m in it? You’re better off picking both boxes (option B)! True, the entity would know this, and he’d leave the variable box empty, but you’d at least come away with m, and not less than m if you chose A.

That’s the twist. If you rely on expected value, the entity would know this. If K > 1, pick A and come away with mK > m; if K = 1, pick A or B, or if K < 1, choose B, and in both cases you come away with m. So it seems the size of the reward drives the decision. And there is no way to reward your greed. At least with oracles, expected value is not needed.

This is for the case of an omniscient entity. Playing this game instead only with a merely intelligent one (and finding a new “I”) and the answer can change. If there is enough interest, I’ll write up that solution, too. The theological aspects of this one are, I think, more or less obvious.

Addendum

I couldn’t wait and did it anyway, since it’s too juicy to leave sit. Suppose first of all, for ease, that the intelligent entity guesses right about the same amount, i.e. p_A = p_B = p (this is your new I). Then you should choose A (only the variable box) when

     p ≥ 1/2 + 1/(2K),

else choose B (both). This can be easily pictured:

For any “large” K, choose the variable box as long as you think the entity is at least as good as just guessing. For “small” K, or for inaccurate entities (when you think you have him fooled), choose B.

Here’s a blowup of the K < 1.

When you’re dealing with a dummy, be greedy. The case where the oracle always guesses wrong, i.e. p_A = p_B = 0, regardless of the value of K, says grab both (B).

Why is this fun? Because the p is deduced from your I, the information you are using to guess the guessing ability of your opponent. Your opponent does not “have” a probability of guessing your guess correctly. He has an ability, perhaps imperfect, perhaps flawed, but no probability.

This becomes a bit more obvious if you imagine playing this game iteratively, switching sides about who takes the pot. I’m doing a dice version of this to further annoy my friends and relatives.

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Categories: Philosophy, Statistics

34 replies »

  1. This is obviously O.T. …

    Great opportunity to increase your knowledge of the threat Judaism poses to The Catholic Church
    https://www.bitchute.com/emichaeljones/

    The link will take you to Jones’ Bitchute account and on the top left you can view the vid about The Revolution and take advantage of his free offer to get information the Shadow Church does not want you to know about – the danger the Jews present to the Catholic Church as explicated by Civilta Cattolica (Catholic Civilisation) lo these many years ago.

    Remember, what was true in the past of Catholic-Jewish relations is true today despite the Fiddy plus years of The Vatican’s political accommodation with our enemies.

    The Shadow Church will not speak this truth and so you will have to learn the information you own self.

    Get the free download…

  2. ABS

    If ABS knows who God is
    If ABS knows where God is

    Then GO to HIM!

    ABS all but insists and ABS intone that Jesus Christ is NOT in ABS’ church

    Jesus Christ is ABS’ intercessor! WHY does ABS insist on go through anybody or anything else?

    ABS’ warped vinyl disk has been crackling for a long time and is now in full skip
    Slapping the console isn’t going to help

    https://www.catholicamericanthinker.com/Catholic-Shadow-Government.html

    I pray for ABS

  3. Dear L. Ron.

    Did you Baptise your own self?

    Why does an Angel tell those he is appearing to to not be afraid? Because of their light and power.

    If you think you could be in the presence of an Angel – a creature – and not be awed and fearful of his presence and power, what makes you think you could stand before Jesus now without His appointed ministers?

  4. Just choose B, pocket the 1000 from one box and sell the other one on E-Bay for half a million.

  5. Wildberger defines K and derives a condition, under which one would choose A based on the expected value criterion. One can easily generalize the solution by using an expected utility criterion, where the utility function depends on a player’s risk aversion. Not sure if it can be called a solution to Newcomb’s paradox as if the paradox is a paradox no more.

    Assuming the entity is an oracle, i.e. p_A + p_B = 1, you’d choose A when K ? 1.

    p_A + p_B = 2.

  6. The nature of Newcomb’s Paradox does not lie in the problem of what do you choose (assuming that you somehow know with certainty what P sub A, P sub B, and the two potential box contents are), but rather lies in the stark contradiction between the two equally plausible justifications for choosing one box or two. That is the contradiction that makes it a paradox instead of a simple mathematical puzzle. The argument for choosing one box makes sense. The argument for choosing two boxes makes sense also — but why does it make sense? It makes sense because we KNOW that whatever is in the big box NOW cannot be changed. “What’s done is done!” Why do we believe that? Because in everyday life causality always goes from the present to the future. That understanding is so deep in us that we do not normally even consider the possibility of backward causality, i.e., causality which goes from the future to the present. And yet, if we take as a given (as we do in Newcomb’s Paradox) that an entity can foretell the future with high and repeatable accuracy, that backward causality is exactly what we are positing. We are saying that a future event is causing a present action. Once you understand that we are speaking of backward causality, Newcomb’s Paradox is resolved. You pick the single box because picking that box CAUSES it to have had (in the your past) the million dollars placed into it.

  7. Matt says that “the theological aspects” of playing the game with an omniscient entity are “more or less obvious.” I would say instead:

    Pr(“obvious” | J )

    In probability [ e.g., Pr(A | I) ], Matt rarely forgets the ‘I’, and tries hard not to.

    In theology, Matt always forgets the ‘J’, because he implicitly asserts that his ‘J’ — a welter of cosmological, Greek-y, and scholastic assumptions and processes — is itself “obvious,” even “universal.”

    At times, Matt will go so far as to say that his ‘J’ is and must be obvious and universal, otherwise we commit the Deadly Sin of Relativism.

    But what intrigued me today was the reference to the notorious “free will of the gaps” involving “quantum mechanics” and “whatnot” that is occasionally trotted out in these more enlightened times in which there is such a thing as ‘pure nature’.

    What difference would it make if we had this “free will”? Suddenly there are no real world consequences to our actions? Hitting ourselves over the head with a hammer won’t hurt, unless we have free will? Other people will never hold us accountable, unless we have free will?

  8. Yes and then there’s that squeaky clean Swiss Guard protecting the Vatican and Pope,
    and those squeaky clean Swiss Banks hoarding the money. It’s a full circle set in a square.

  9. Why not just repeatedly tell the Oracle: “I’m taking B” – and then do so ? It’s a classic situation in which both sides win if they trust each other…

  10. Hey Fredo! You say, “there’s that squeaky clean Swiss Guard protecting the Vatican and Pope,
    and those squeaky clean Swiss Banks hoarding the money.”

    And isn’t it odd that the US President is guarded by the Secret Service, a branch of the US Treasury which is always headed by someone from the biggest US banks?

  11. Dear John ABS was baptized by a church ABS doesn’t trust or respect – that works for you?

    Now you are beginning to sound like Joe Biden. A Priest Baptised ABS.

    A person, not a Church, Baptised you; that is, a minister, one standing between Jesus and you but you claim to reject that.

  12. All thanks be to God, acricketchirps!! Beautifully true!!!

    “If He’s there and the light’s not burning that’s on us.”

    Love this!! God bless, C-Marie

  13. Interesting question.
    So, if we assume the oracle is infallible, the question then is: about what?

    IF it knows which box you will pick, then you maximise your return by picking A – oracle says “million goes in only if you pick A”, so pick A and win.

    IF, instead, the oracle knows your intent (greedy vs modest) and decides based on that, then you maximise your return (because you are greedy!) by picking B, even though you don’t get the million – oracle says “million goes in only if you aren’t trying to obtain best outcome”, so you can’t get the million if you want to maximise your result, by definition!

  14. @Sheri — two thumbs up.

    There is a step before this. Am I forced to make a choice? If someone presents this choice to me, my answer is “back up slowly, and move on”. This is similar to my answer for getting the vaccine. “Yep, I will get the vaccine, as soon as everyone else has gotten it. I will sacrifice my safety to make sure others get the limited safe juice. I will always be at the back of the line. Don’t let anyone suggest that I am NOT in line.” If someone is trying to get me to make this choice against the machine, there is a scam in play. I am not saying the vaccine is a scam, but I am going to let everyone else partake in the scam.

    Sheri and I may fight to be the last person in the line. Hopefully she will accept the biscotti I offer with the coffee so we can have something to nibble and sip as we watch everyone else getting poked.

    PS. I just caught myself wearing my mask in the car. Please send nasty thoughts in my direction for forgetting to take it off as soon as the nice lady handed me my tags for my car. She distracted the part of my brain worried about my mask.

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