Since we rejected independence, we must now consider the lotteries when taken as a whole, rather than just seeing them individually. When considered as a whole, "reasonable" lotteries are more tightly bunched around their total mean than they are individually. Hence the more lotteries we consider, the more we should treat them as if only their mean mattered.

You are absolutely correct, and it pains me because this issue should have been settled a long time ago.

When Eliezer Yudkowsky first brought up the breakdown of independence in humans, way, way back during the discussion of the Allais Paradox, the poster "Gray Area" explained why people aren't being money-pumped, even though they violate independence. He/she came to the same conclusion in the quote above.

Here's what Gray Area said back then:

Finally, the 'money pump' argument fails because you are changing the rules of the game. The original question was, I assume, asking whether you would play the game once, whereas you would presumably iterate the money pump until the pennies turn into millions. The problem, though, is if you asked people to make the original choices a million times, they would, correctly, maximize expectations. Because when you are talking about a million tries, expectations are the appropriate framework. When you are talking about 1 try, they are not. [bold added]

I didn't see anyone even reply to Gray Area anywhere in that series, or anytime since.

So I bring up essentially the same point whenever Eliezer uses the Allais result, always concluding with a zinger like: If getting lottery tickets is being exploited, I don't want to be empowered.

Please, folks, stop equating a hypothetical money pump with the actual scenario.