I was wrong! The set from which the starting amounts are drawn is irrelevant. All that matters is the starting amounts used in any particular game.

The logic I suggested in a previous post that, for example, the highest amount included in any game will never be doubled, along with the fact that every game must end, is all that is needed to resolve the paradox, as I discuss below.

After all of these posts, I think that the following is a complete summary of all of the relevant and correct ideas already discussed:

I’ll define a trial as each time 2 envelopes are filled and one is chosen. I’ll define a game as any finite number of trials. This assumes that every game ends. No game goes on indefinitely.

The setup: An envelope is filled with some amount of money. Another envelope is then filled with either half or twice that amount, randomly. The player chooses one of those envelopes and then decides to keep the chosen envelope or trade for the other envelope. This continues with different pairs of envelopes until the game ends.

The 2 envelope paradox, which falsely claims that trading every time will increase a players winnings, is based on this logic: “When I see an amount, I can trade half the time for twice the amount and half the time for half the amount. Therefore, I should trade every time since I’ll make more on average than I’ll lose.”

In order to draw that conclusion, that logic must be true for every amount seen. However, in any actual game, the envelope that contains the highest amount used in that game cannot be traded for twice its amount. Therefore, the logic fails, and so does the conclusion that trading every time will increase the chance of winning.

In fact, in any game, 1) some of the lowest amounts will never be paired with half their amount, 2) some amounts will be paired with both half and twice their amount, and 3) some of the largest amounts will never be paired with twice their amount. When the envelopes in 1) are chosen, the player always wins by trading. When the envelopes in 2) are chosen, on average, the player wins or loses by trading according to what percent of the time the other amount is higher or lower. When the envelopes in 3) are chosen, the player always loses when trading. In any game in which the player either never trades or always trades, the sum of the amounts won or lost in each of the 3 possibilities described above equals zero.

For example, if the envelope I choose has 20D, I don’t know if any other envelope will ever have 40D in the game I’m playing. Maybe the first envelope was filled with 10D, and then, randomly, the other envelope was filled with 20D, and I picked the 20D envelope. In that case, then next time the 10D envelope is filled first, the other envelope may have 5D. The first envelope filled may never contain 20D in that game, and the 40D envelope may never appear. If a 40D envelope does exist, then I will draw it some percent of the time, and I’ll repeat the same logic – maybe an 80D envelope will never exist in this game. This logic applies to any amount drawn.

If someone says, “I see 20D. The other envelope will have 10D half the time and 40D the other half of the time.” I’ll say, “That’s not necessarily true. Maybe the 10D envelope was filled first and, randomly, the other envelope, which is the one you chose, was filled with 20D. If so, then the next time the 10D envelope is filled first, the other envelope may be filled with 5D. A 40D envelope may never exist in this game.” She may say, "But the other envelope could have 40D." I'll say, "Yes, but in the games that don't have a 40D envelope, you'll lose money every time you trade the 20D envelope. Those games are examples of when your strategy of always trading won't work." The same logic applies to any amount drawn or even considered.

I think that this exposes the illogic that leads to the paradox.

Both of your statements below were generated from my mistake. Still, I’ll address them individually.

You wrote, “What troubles me still is this: while I think it is the case that it is impossible to work with the idea of 'choosing' from infinite possibilities, this does not imply that the finite set of possibilities is fixed in the sense that there is a given finite set of pairs from which the pair the player plays is selected.”

I was wrong about this, and I agree with what you state above. I realize that the starting amounts can be drawn from an infinite set and my logic still holds as long as the game ends. It’s not the set from which the initial starting amounts were drawn that matters; it’s the actual amounts drawn that matter.

You asked, “if I say I create a pair by 'thinking of a number', then flipping a coin to put double or half into the other envelope', what is the finite set from which this pair was selected from?”

My answer is: Any set (finite or infinite) that contains whatever starting amounts were used in that particular game.