which astonishingly (to me) converges quite rapidly on 1/e
To make it less surprising that (1-1/n)^n converges, here are two arguments that may help.
First: take logs. You get n log (1-1/n). Now for small x, log(1+x) = x + lower-order terms, so n log (1-1/n) = n (-1/n + lower-order terms) which obviously -> -1.
(Is it obvious enough that log(1+x) = x + lower-order terms? The easiest way to prove that might be to say that log x = integral from 1 to x of 1/t, and for x close to 1 this is roughly the integral from 1 to x of 1, or x-1.)
Second: use the binomial theorem. (1-1/n)^n = sum {k from 0 to n} of (-1)^k (n choose k) n^-k. Now (n choose k) = n(n-1)...(n-k+1) / k!, and for small k this is roughly n^k/k!. So for large n, the "early" terms are approximately +- 1/k!. And for large n, the "late" terms are relatively small because of that factor of n^-k. So (handwave handwave) you have roughly sum (-1)^k 1/k! which is the series for exp(-1).
I haven't been able to find the source of the idea, but I've recently been reminded of:
This is, of course, based on the Multiple Worlds Interpretation: if the lottery has one-in-a-million odds, then for every million timelines in which you buy a lottery ticket, in one timeline you'll win it. There's a certain amount of friction - it's not a perfect wealth transfer - based on the lottery's odds. But, looked at from this perspective, the question of "should I buy a lottery ticket?" seems like it might be slightly more complicated than "it's a tax on idiots".
But I'm reminded of my current .sig: "Then again, I could be wrong." And even if this is, in fact, a valid viewpoint, it brings up further questions, such as: how can the friction be minimized, and the efficiency of the transfer be maximized? Does deliberately introducing randomness at any point in the process ensure that at least some of your MWI-selves gain a benefit, as opposed to buying a ticket after the numbers have been chosen but before they've been revealed?
How interesting can this idea be made to be?