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Addendum 3

Even better: After you get the envelope with 20D, ask how the envelopes are being filled. The answer will determine whether or not you can improve your odds beyond 50-50. So, I ask, “How are you deciding what amounts to put into these envelopes?” You say, 1) “I’m not telling you,” or don’t answer. I assert: There is no way to know what percent of the time there will be 10D or 40D in the other envelope, so there is no way to determine when to trade, 2) “I put 20D in one and then half the time I put 10D in the other and the other half of the time I put 40D in the other,” or “I put 20D in one and then I put in either half the amount or twice that amount randomly into the other envelope.” I assert that trading every time will increase the odds of winning more money, 3) “I put 20D into one and then some percent of the time I’ll put 10D into the other and some other percent of the time I’ll put 40D into the other. I’m not telling you what the percent’s are.” I assert that there is no way to determine when to trade.

The typical assumption from your language is that there will be half the amount half the time and twice the amount half the time, in which case trading will increase the average amount of money acquired. But that assumption depends on the method by which the envelopes are being filled. If the method isn’t known, then I can’t make that assumption. If I know the method, then I can form a strategy or know that there is no strategy as I discuss above.

Addendum 2

I think that the above must be taken into consideration with my prior discussion that both choices don’t exist. Here’s that discussion after seeing the amount in the chosen envelope and also using the above ideas.

Suppose I fill the envelopes with 20D and 40D. You pick one, see 20D, and then state, “the other envelope has either 10D or 40D.” I’d correctly say, “No, it doesn’t,” because it can’t have 10D because neither envelope had 10D to begin with. Just because you don’t know that doesn’t give you the choice of 10D. In other words, you are wrong if you think the other envelope can have 10D in it. You might say to yourself, “I think it can have either 10D or 40D, but it can’t. One of those choices is not available to me. I just don’t know which one it is, but I could find out by doing this a few times. Until I do that, there’s no way for me to know whether or not trading will make me more money.” It’s the “doing it a few times” part that helps me see that both choices don’t actually exist.

This is similar to my argument that the amount in the other envelope depends on whether I chose the larger or smaller amount (the hidden conditional), and until I know that, I don’t know enough to decide whether or not to trade. “I see 20D but I don’t know if this is the smaller or larger amount. Until I know whether I chose the larger or smaller amount, I can’t know what the choice is for the other envelope, so I can’t know if trading will help me or not.”

This is both fun and frustrating to try and understand and then explain. ☺

Addendum 1:

I still think that the paradox arises from ambiguity in the language. Here’s how this happens:

Odds for any experiment can be empirically estimated using repeated trials. I want to set up the actual experiment so we know what is repeated. You’ll see that after looking at one envelope’s amount, there are 2 different things that can be repeated that lead to different outcomes. Since the same language seems to be able to be interpreted in both ways, the same language can seem to result in 2 outcomes that contradict.

The problem begins the same for each case. You are given the choice of one of 2 envelopes. You know that one envelope has twice the amount of money as the other envelope. You choose one and see that it has 20D inside. It’s what happens next that matters:

1: Repeatedly give the same 2 envelopes. Then, on the next try, you get to choose from the same 2 envelopes. Maybe you get the 20D again, or maybe you get the 40D. You repeat this process multiple times, always employing some strategy to see if you can increase your odds of ending up with the 40D envelope. The strategy of changing every time doesn’t work.

That’s the way this problem is typically conceived by people who show that changing envelopes after seeing 20D doesn’t increase the odds.

2: Repeatedly see the same amount of money. Then, on all subsequent tries the envelope you choose has 20D. In this case, what changes is the amount in the other envelope: it has either 10D or 40D. If the chance of the other envelope having 10D or 40D is the same (which will depend on the exact method for filling them), then always trading will increase the odds of making more money.

That’s the way that wins when always trading.