Proximity: I head toward the closer [person].
Bible: I open my ship's bible and count up the number of words on the first page of Genesis--if it is odd then I will head east, if it is even then I will head west.
Algorithm: I turn on my ship's computer and run the "random number" generator (which outputs a number between 0 and 1, following an algorithm that is entirely unknown to me). If it gives me a number >0.5 then I will head east, if it gives me a number <0.5 then I will head west.
Coin: I spin a coin--heads and I will head east, tails and I will head west.
Photons: I fire a small batch of photons southwards through a narrow slit--if more deflect east then I will head east, if more deflect west then I will head west.
If you don't know the identity of the closer (eastward) person, then even Proximity yields equal epistemic chances. (Though, personally, I'm not sure why more knowledge should make the choice any more unfair: after all, there remains a sense in which the locations are arbitrary and hence it's a priori equally likely for either person to have ended up being the closer one. If self-interested agents could agree to the method from behind a veil of ignorance, I can't make any sense of residual "unfairness" complaints.)
Requiring strict physical indeterminacy (or the epistemic state of an ideal observer) would rule out everything up to Photons -- and, as Hare points out, even that could be ruled out by a supernatural ideal observer. Since Coin is uncontroversially fair, something less stringent is required. But what could it possibly be?
One interesting possibility that Hare doesn't consider (though my thought here is inspired by his discussion of counterfactual open-ness in chapter 13) is that the relevant sense of chanciness concerns whether the choice of method leaves open who will be saved. We can draw a line between Bible and Algorithm. The result of Algorithm depends upon the details of how you implement the method -- at what precise time do you run the computer's random number generator? Exactly how much pressure do you put behind the coin flip? Even if determinism is true, these methods may be counterfactually open in the sense that if you don't actually perform them, then there's no fact of the matter as to which precise way of performing them you would have done, had you done them. Clearly, Proximity and Bible are not like this: the same result will occur no longer how long you take to open the Bible, or (say) whether you are inhaling or exhaling at the moment you turn towards the closer person.
My proposal, then: Perhaps a method is sufficiently "robustly" chancy just if, the result of the method, had you not used it, would have been counterfactually open. (Forgive the nested counterfactuals. They offer a clunky, but precise, way to spell out the intuitive idea of some methods being more "open" than others, regardless of determinism.)
One implication to note: While Bible is "unfair" according to this criterion, a closely related method Bible* (just like Bible but where you open the book to an arbitrary page, rather than to the first page) is "fair". Is this really a difference that should make a difference? You be the judge.