But it isn't logically consistent. With anything. That's what I keep trying to say over and over.That's fine with me. In that case, one must be open to embracing both horns of the dilemma, and realize that there being an unconditional expectation 1.25v for switching, whatever value v one might find in the first envelope, isn't logically inconsistent with ... — Pierre-Normand
We're given a choice between envelopes valued unequally at a and b. We won't know which one we picked. The expected value of switching is
(1/2)(a−b)+(1/2)(b−a)=0
...
Isn't...this true whichever of a and b is larger — Srap Tasmaner
But it isn't logically consistent. With anything. That's what I keep trying to say over and over.
1.25v is based on the demonstrably-false assumption that Pr(X=v/2)=Pr(X=v) regardless of what v is. It's like saying that the hypotenuse of every right triangle is 5 because, if the legs were 3 and 4, the hypotenuse would be 5. — JeffJo
Exp(other) = (v/2)*Pr(picked higher) + (2v)*Pr(picked lower) is a mathematically incorrect formula, because it uses the probabilities of the wrong events.
Exp(other) = (v/2)*Pr(V=v|picked higher) + (2v)*Pr(V=v|picked lower) is the mathematically correct formula, because it uses the probabilities of the correct events.
Isn't all of this true whichever of a and b is larger, and whatever their ratio? — Srap Tasmaner
So we’re assuming that the other envelope is equally likely to contain either £20 or £5, and that’s a reason to switch. We either lose £5 or gain £10. That, to me, is a reasonable gamble. — Michael
A B 1 x 2x 2 2x x
If there's no reason to believe that we're more likely to lose than win on switching, i.e. if there's no reason to prefer sticking, and if we can afford to lose, then switching is a good gamble for a single game, even if not a winning strategy over many games. I either lose £5 or I gain £10. That's a bet worth making for me, and so if I were to play this game and find £10 in my envelope then I would switch. — Michael
I am simply assuming that the player's only goal is to act such as to maximize expected value — Pierre-Normand
But you're also not saying that sticking is a winning strategy. If sticking isn't preferred then I am going to switch, because I am willing to risk losing £5 for the chance to win £10. I have more to gain than I have to lose. That neither strategy gains over the other after repeated games doesn't change this. — Michael
Suppose I offer you to play a game with two dice. You throw them once and sum them up. If you roll any value from 1 to 11, you must give me £5. If you roll 12 then I must give you £10. Let us assume that we only are going to play this game once. Would you also say, in this case, that you are willing to risk losing £5 for the chance to win £10? — Pierre-Normand
No, because I know the probabilities aren't in my favour. If I know that they're not in my favour then I won't play. If I know that they're in my favour then I will play. If I don't know the odds then I will play. — Michael
Suppose I offer you to play a game with two dice. You throw them once and sum them up. If you roll any value from 1 to 11, you must give me £5. If you roll 12 then I must give you £10. Let us assume that we only are going to play this game once. Would you also say, in this case, that you are willing to risk losing £5 for the chance to win £10? — Pierre-Normand
In the two envelope case, you don't know the odds of winning. But you do know (or ought to be able to deduce) that the odds aren't either in your favor, neither in your disfavor — Pierre-Normand
How is this any different to saying that I'm equally likely to win as lose? — Michael
It is obviously different since on the assumption that you are equally likely to win as lose it follows that the expected value of switching is 0.25*v whereas saying that the odds neither are nor aren't in your favor is equivalent to saying that your average expected value from switching is v. — Pierre-Normand
... if a) I have no reason to believe that losing is more likely (specifically >= 2/3 chance), b) I gain twice as much if I win than I lose if I lose, and c) I can afford to lose, then I have a good reason to switch when playing a single game.
I don't see how the expected value of either always switching or always sticking over repeated games has any bearing on this.
It's not necessarily equally likely — Pierre-Normand
I know it's not necessarily equally likely. — Michael
You are only willing to make that traded because it is only 10 bucks. That is not much to lose in the first place. It is a subjective criteria and it does not say anything about the actual probability.
What would you do if you opened envelope A and saw 1000 bucks? — Jeremiah
My point is that your personal cretria is something that we cannot objectivly measure — Jeremiah
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