Omega is continuing his eternal mission: To explore strange new philosophical systems... To seek out new paradoxes and new counterfactuals... To boldly go where no decision theory has gone before.
In his usual totally honest, quasi-omniscient, slightly sadistic incarnation, Omega has a new puzzle for you, and it involves the Sleeping Beauty problem as a bonus.
He will offer a similar deal to that in the counterfactual mugging: he will flip a coin, and if it comes up tails, he will come round and ask you to give him £100.
If it comes up heads, instead he will simulate you, and check whether you would give him the £100 if asked (as usual, the use of randomising device in the decision is interpreted as a refusal). From this counterfactual, if you would give him the cash, he’ll send you £260; if you wouldn’t, he’ll give you nothing.
Two things are different from the original setup, both triggered if the coin toss comes up tails: first of all, if you refuse to hand over any cash, he will give you an extra £50 compensation. Second of all, if you do give him the £100, he will force you to take a sedative and an amnesia drug, so that when you wake up the next day, you will have forgotten about the current day. He will then ask you to give him the £100 again.
To keep everything fair and balanced, he will feed you the sedative and the amnesia drug whatever happens (but will only ask you for the £100 a second time if you accepted to give it to him the first time).
Would you want to precommit to giving Omega the cash, if he explained everything to you? The odds say yes: precommitting to accepting to hand over the £100 will give you an expected return of 0.5 x £260 + 0.5 x (-£200) = £30, while precommitting to a refusal gives you an expected return of 0.5 x £0 + 0.5 x £50 = £25.
A standard way to approach these types of problems it to act as if you didn’t know whether you were the real you or the simulated you. This avoids a lot of complications and gets you to the heart of the problem. Here, if you decide to give Omega the cash, there are three situations you can be in: the simulation, reality on the first day, or reality on the second day. The Dutch book odds of being in any of these three situations is the same, 1/3. So the expected return is 1/3(£260-£100-£100) = £20, twenty of her majesty’s finest English pounds.
However, if you decide to refuse the hand-over, then you are in one of two situations: the simulation, or reality on the first day (as you will not get asked on the second day). The Dutch book odds are even, so the expected return is 1/2(£0+£50) = £25, a net profit of £5 over accepting.
So even adding ‘simulated you’ as an extra option, a hack that solves most Omega type problems, does not solve this paradox: the option you precommit to has the lower expected returns when you actually have to decide.
Note that if you depart from the Dutch book odds (what did the Dutch do to deserve to be immortalised in that way, incidentally?), then Omega can put you in situations where you lose money with certainty.
So, what do you do?