|Journal||The British Journal for the Philosophy of Science|
The authors do not assume backward causation, merely an extremely effective predictor:
You have enormous confidence in the Being's ability to predict your actions, and you know that he has correctly predicted all choices of all players who have played this game with him to date (in fact, he has predicted correctly all choices that you have made in this game in some previous 'warming up' trials, played for points, say, rather than money. (Bar-Hillel1972, 295)
They suggest that the dominance principle should not be applied.
Consider, for instance, the following example: Israel must decide whether to withdraw from its occupied territories or not, and Egypt must decide whether to declare war on Israel or not.
Clearly, remaining in the occupied territories is the dominant strategy […]. Suppose, however, that you believe that with a high probability withdrawal will be conducive to peace while remaining in the territories will eventually lead to war. (Bar-Hillel1972, 296–297)
This example makes clear that the dominance principle loses its appeal when applied to situations where the states of the world (or the opponent's moves) are affected by the decision maker's actions, and its logic is overriding only when the states are independent of the actions, i.e. when the probability distribution over states of the world (matrix columns) is the same for all actions (matrix rows). (Bar-Hillel1972, 297)
They do notice the problem:
In other words, choosing A1 rather than A2 seems essentially to be justified by the fact that thereby a very high,rather than very low, probability can be assigned to a certain desired event, namely that the Being put a million dollars into the covered box. But to go about assigning probabilities to past events,unaffected by present events, in what appears to be a completely arbitrary, ad hoc, and wilful fashion is, to say the least, highly unorthodox and more than a little unsettling. (Bar-Hillel1972, 299)
They proceed to argue that the problems seems confusing because it's very difficult for us to believe in such a reliable predictor–we're more inclined to think it's cheating, somehow. Otherwise, it seems to violate free will.
Suppose you are playing the following game: Someone is tossing a fair die, and after each toss you guess whether it came up on 6 or not. You receive a penny for each correct guess. Clearly, you will be maximising your expected gains by guessing not-6 on every toss. Now suppose that the die was tossed the previous day, the outcomes were noted down on a list, and you are now guessing, entry by entry, whether the number is 6 or not. Had you been able to obtain a copy of the list you could change your previous strategy to one that would ensure that you get a penny on each trial. But without such a list, you can still do no better than to guess not-6 on each trial! The fact that the order of 6's and not-6's is predetermined, given that you do not know what it is, does not affect your strategy. What this example serves to point out is that the mere knowledge that things are not what they seem to be does not necessarily supply you with an alternative strategy for dealing with them. (Bar-Hillel1972, 302)