The fundamental flaw in this game is the separation of 'me' from the 'many possible worlds' that I may occupy. Abstracting the self from the world is done as a matter of convenience, but there are just as many possible "Me's" as there are possible future states in which 'I' exist. In Quantum Roullette, what if the new 'me' in the new state doesn't care about being rich, or what if inflation has devalued my wealth, or what if it's a possible world without the money-form?
The advice presented above about behaving as you would were you not feeling nihilistic, while you are, is strikingly similar to Aristotle's advice about virtue: 'we become just by doing just acts; good by doing good deeds. virtue is the result of habit'. For Aristotle, virtue was a means to the end he defined as the 'good', which in turn was pleasure. Not hedonistic pleasure, but intellectual pleasure.
The biological commentary is indeed accurate, but I question its relevance in the context of the question, which seems to be one in favor of a utilitarian ethical discourse without the biological considerations. It might be better to assume the biological factors involved are compatible, or assume all other factors are equal, and disregard the biology.
The first answer that comes to mind for most I'm sure is that 10 is greater than 1, and that such a sacrifice would return a net gain in lives saved. However, this question is complicated by what it is about saving lives at all that is good. If you can save a life of a dying patient without risk then you should. We assume this is true - even from a utilitarian position - that life has intrinsic value and therefore saving lives when possible is the correct decision. Thus, as a strictly quantitative normative comparison, the decision to kill one healthy person to save 10 dying patients is right. But qualitative interests should be accounted for too: treating the act of killing a healthy individual as a separate act, it stands in direct contradiction to the supposed ethical duty we are to uphold, mainly, saving life, by taking the life of one who is not sick. Consider it in these terms: is it ethically correct, then, to take the lives of the healthy to save those of the sick? This amounts to a zero-sum game and a clear logical aporia.