This is really cool puzzle. By accepting the £10, you're in a conditional "Alpha never sent you the money", but by refusing you're in conditional "Alpha sent you the money". However, that choice doesn't actually affect Alpha sending or not sending you the money. This is unlike the Newcomb's problem, where you can truly choose, acausally, what the opaque box will contain.
What gets me is the peculiarly elaborate pitfall into which I, at least, fell.
Suppose you said: "Invent a thought-experiment which could trick people who know to one-box in the classic Newcomb's paradox, into thinking that here was a higher-order analogue; the source of the error to be, that people who reason wrongly do experience a higher payoff in this case."
Perhaps it should be called Armstrong's trap. But did he make it by design, or did he just fall into it first?
This is a variant built on Gary Drescher's xor problem for timeless decision theory.
You get an envelope from your good friend Alpha, and are about to open it, when Omega appears in a puff of logic.
Being completely trustworthy as usual (don't you just hate that?), he explains that Alpha flipped a coin (or looked at the parity of a sufficiently high digit of pi), to decide whether to put £1000 000 in your envelope, or put nothing.
He, Omega, knows what Alpha decided, has also predicted your own actions, and you know these facts. He hands you a £10 note and says:
"(I predicted that you will refuse this £10) if and only if (there is £1000 000 in Alpha's envelope)."
What to do?
EDIT: to clarify, Alpha will send you the envelope anyway, and Omega may choose to appear or not appear as he and his logic deem fit. Nor is Omega stating a mathematical theorem: that one can deduce from the first premise the truth of the second. He is using XNOR, but using 'if and only if' seems a more understandable formulation. You get to keep the envelope whatever happens, in case that wasn't clear.