Since you can do only things which can happen, you actions are unable to change the set of things which will happen to multi-you across all timelines.
That's the predestination argument, isn't it? Whatever the choices available to multi-me are, it's impossible for me to compute them all, which provides sufficient uncertainty for something resembling free will to apply. I don't know whether any given future-me will even have the option to buy a lottery ticket, let alone what the consequences of making that choice one way or the other might be; and so I might as well treat any given timeline as one in which that version of me can make decisions which affect his particular future.
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?