I was thinking more along the lines of there being no need to actually calculate anything at all because the only relevant information is the 50-50 probability of choosing the higher envelope. And since you gain no new information after choosing, there's no need to calculate anything afterward either. The "paradox" is in making one think there's something to be calculated beyond the 50-50 chance.
Let's play a game where I wager $10 on each coin toss and you're the casino. Heads, I pay you $5. Tails, you pay me my $10. Just like the envelope game, right? "The only relevant information is the 50-50 probability", right? So if I pay you $1 per toss, you'd be happy to play this game and take my money, right?
No, because you're not actually wagering anything in the envelope game, so it makes no sense to calculate probabilities as though you are.
An analogous coin toss example would be something like: I'll flip a coin, you call heads or tails. If you call it right, I'll give you $10. If you're wrong, I'll give you $5. (So either way you make $5, so I don't know why anyone would even offer this game!) After I flip the coin, I'll conceal it in my hand and give you a chance to change your choice of heads or tails. Well, of course there's no reason to change your choice, it's completely arbitrary! You've got a 50-50 chance of winning $10 regardless. The paradox makes you think you're wagering your unknown potential winnings, but you're not actually wagering anything.
It has nothing to do with calculating whether or not a specific wager is worth it given some set of probabilities. That's a different problem for a different sort of game.
> No, because you're not actually wagering anything in the envelope game, so it makes no sense to calculate probabilities as though you are.
If we consider the decision point where you can either keep the first envelope or switch it, you are effectively wagering the value of the first envelope. You're just muddying the waters when you're trying to make some kind of point about receiving the first envelope for free. Yes yes, you are only gambling your "winnings" that you won earlier, it doesn't make a difference here.
> An analogous coin toss example would be something like: [...] I don't know why anyone would even offer this game!
Look, you made the claim that expected value doesn't matter for anything at all, the only thing that matters is the probability of winning. I offered you a game where you have a 50% probability of winning on each coin toss and I offered to pay you $1 for each coin toss to incentivize you to play it. Can you please explain why you are refusing to play this game? Yes, you don't like the analogy, I get that, but you also don't like free money? I find that hard to believe. A more plausible explanation is that you don't actually believe the claim you are making. That's why you aren't willing to wager any money on it.
> Yes yes, you are only gambling your "winnings" that you won earlier, it doesn't make a difference here.
It makes all the difference because you don't know what those "winnings" are, and what you're potentially wagering them for depends on what they are. So ultimately you're not really wagering anything. (Again, I think this is the crux of the "paradox".)
> Look, you made the claim that expected value doesn't matter for anything at all
I never meant to claim it doesn't matter for "anything at all", just not for this envelope game. (Apologies if the meaning of "at all" was ambiguous in my original post.)
> I never meant to claim [expected value] doesn't matter for "anything at all", just not for this envelope game.
When you say "this envelope game", I assume you are also including minor variations of this game?
If you mean specifically this exact envelope game, where the expected value for all actions is zero, then it is strictly true that it doesn't matter how we make decisions - using expected value or not - though it's not a particularly interesting point to make.
I will proceed assuming you mean also including minor variations of this game.
You wrote this earlier:
> That is, the probability is 1/2. That's all that matters to the decision making. Calculating an "expected value" at all is completely useless, whether or not you do it "correctly".
We can make a minor variation to the game such that the probability will still remain at 1/2, but the expected value for switching can be changed. We can make the minor variation such that it will be profitable to switch, or such that it will be unprofitable to switch. We can do this while the probability remains at 1/2. According to you the probability is all that matters, and it's useless to calculate the expected value for a game like this? This would lead to making unprofitable decisions in a game with a small variation as described above.
> When you say "this envelope game", I assume you are also including minor variations of this game?
No; it seems obvious to me that the argument for switching presented in the original wikipedia article is meant to apply only to the evelope game as it's presented. Of course introducing variations could easily change the meaningfulness of the article's premise. That is, it wouldn't be considered a "problem" or a "paradox" if calculating an "expected value" were actually meaningful.
> then it is strictly true that it doesn't matter how we make decisions - using expected value or not - though it's not a particularly interesting point to make
True, but my point was more of a question: if it's obvious that calculating an "expected value" is irrelevant in this specific case (as the article says, "It may seem obvious that there is no point in switching envelopes as the situation is symmetric"), why is the argument presented in the article considered compelling? That is, either it's not actually compelling, or the "expected value" being meaningless in this case is not necessarily so obvious at first... but if so, why not? (Or, to put it another way, why is the argument in the article compelling enough to warrant such a long wikipedia page with such numerous proposed "resolutions"?)
Your question would be very easy to answer if it concerned a whole basket of "games like this". Now that you narrowed it down to this specific game, it becomes difficult to answer, because we can just throw our hands up in the air and say "it doesn't matter what we do". If we allowed slight variations in the game format, it suddenly _would_ matter what we do, and suddenly we would need to do expected value calculations to do well in these games.
Your question is akin to a driver plowing through red lights without an accident and concluding that the traffic light was useless in this specific instance. "I didn't cause a crash even though I ran a red light, so the traffic light was meaningless! In this specific instance I mean! It didn't matter if I stopped to the red light or ran across it, no accident would have happened either way in these very specific circumstances, so why are people even talking about traffic lights?"
