I'm learning as I go here. — Srap Tasmaner
When you open the envelope and look at what's inside, your numéraire is dollars. When you don't open it your numéraire is 'value of contents of the first envelope', ie Y. So it's a different calculation that can validly have a different result. — andrewk
In general, it's best to avoid arguments based on 'absurdity — andrewk
Those results are both correct, and it's because each is done from the point of view of observing that envelope - two opposite points of view. It's analogous to how two spaceships travelling at half the speed of light relative to one another both measure time as going more slowly in the other spaceship. That 'feels' inconsistent but when you dig into it, trying to locate and fix the inconsistency, you find there isn't one.Then let's go with "inconsistency": given such a pair of envelopes, you can readily construct a pair of arguments that tell you each has a higher expected value than the other. I would prefer finding that such arguments are invalid. — Srap Tasmaner
That's right, in this case using either dollars or the value of Y as numéraire gives the same result - an expected gain equal to Y/4. That's why I said 'can' rather than 'does'. But using X as numéraire gives a different result - an expected gain of zero. It really is worth spending the time to come to grips with the numéraire concept. It has many more important applications than just in probability.I'm still not seeing it. We've done to death the example of finding £10 and you calculate an expected gain of £2.50. Or you can do a generic calculation and show an expected gain of £Y/4, for any value of Y. — Srap Tasmaner
You can't use Y as a value defined independently of X — Snakes Alive
Can you justify that 'since'? There is no justification provided in the sentence in which it occurs, because the words following it have no logical relation to the words before it. It sounds like you're saying that, having observed 10 in the envelope, you now believe that the other envelope might contain some amount other than 5 or 20. If that's not what you're saying, what are you saying?it is not metaphysically possible for the unopened envelope to be 'either 5 or 20' (read as an exhaustive disjunction), since neither of these is twice the other ( — Snakes Alive
We agree on that. Y and X are interdependent. That's why I define Y asYou can't use Y as a value defined independently of X and average across possibilities using that value. — Snakes Alive
It sounds like you're saying that, having observed 10 in the envelope, you now believe that the other envelope might contain some amount other than 5 or 20. — andrewk
What do you mean by 'exhaustive disjunctive possibilities'?The point is that it is not a possibility that the exhaustive disjunctive possibilities of what's contained in the other envelope are 5 and 20. — Snakes Alive
That's all correct, but is not inconsistent with my note. Nowhere does it say that 5 and 20 are X and 2X, given we have seen that Y=10. Rather, we know that X is either 5 or 10, so eitherit's known already that for some X, the amount in the other envelope is X or 2X. But 5 and 20 are not X and 2X for any value of X — Snakes Alive
What do you mean by 'exhaustive disjunctive possibilities'? — andrewk
That's all correct, but is not inconsistent with my note. Nowhere does it say that 5 and 20 are X and 2X, given we have seen that Y=10. Rather, we know that X is either 5 or 10, so either
we have X in the envelope we opened, so that X = Y = 10; OR
X is in the other envelope, so we have Y = 2X = 10 and X=5 is in the other envelope. — andrewk
Now eliminate the Y altogether, do the calculation, and you'll see that your expected value is 1.5X regardless of whether you switch or not. — Snakes Alive
Eliminating Y is making X the numéraire. That's why you need to address the numéraire issue, as explained in this post. When we use X as numéraire, the expected gain from switching is zero units of X, but when the numéraire is dollars, it is a gain.If you define Y in terms of X, then everything you write with Y can (must) be rewritten in terms of X. — Snakes Alive
That doesn't prevent us from modelling our uncertainty about it by representing it as a random variable. In Bayesian analysis we model a fixed, unknown population parameter like X as a random variable from an assume distribution we call the 'prior'. We then use new information to update that distribution to a more accurate 'posterior distribution'.You've agreed that Y must be defined in terms of X. X is fixed. — Snakes Alive
reduces the problem to the player selecting the option from which she expects the highest gain, based on the available information. — andrewk
You are playing a game for money.
I don't believe that description correctly represents the analysis.But whatever value L has, R has 5/4 of that value. So I consider R. But whatever value R has, L has 5/4 of that value. So I consider L again. — Srap Tasmaner
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