Under typical decision theory, your decisions are a product of your beliefs and by the utilities that you assign to different outcomes. In order to argue that Jack and Jill ought to be making different decisions here, it seems that you must either:
Dispute the paper’s claim that Jack and Jill ought to assign the same probabilities in the above type of situations.
Be arguing that Jack and Jill ought to be making their decisions differently despite having identical preferences about the next round and identical beliefs about the likelihood that a ball will turn out to be red.
Are you advancing one of these claims? If (1), I think you’re directly disagreeing with the paper for reasons that don’t just come down to how to approach decision making. If (2), maybe say more about why you propose Jack and Jill make different decisions despite having identical beliefs and preferences?
I thought about it more, and I am now convinced that the paper is right (at least in the specific example I proposed).
The thing I didn’t get at first is that given a certain prior over P(extinction), and a number of iterations survived, there are “more surviving worlds” where the actual P(extinction) is low relative to your initial prior, and that this is exactly accounted for by the Bayes factor.
I also wrote a script that simulates the example I proposed, and am convinced that the naive Bayes approach does in fact give the best strategy in Jack’s case too (I haven’t proved that there isn’t a counterexample, but was convinced by fiddling with the parameters around the boundary of cases where always-option-1 dominates vs always-option-2).
Under typical decision theory, your decisions are a product of your beliefs and by the utilities that you assign to different outcomes. In order to argue that Jack and Jill ought to be making different decisions here, it seems that you must either:
Dispute the paper’s claim that Jack and Jill ought to assign the same probabilities in the above type of situations.
Be arguing that Jack and Jill ought to be making their decisions differently despite having identical preferences about the next round and identical beliefs about the likelihood that a ball will turn out to be red.
Are you advancing one of these claims? If (1), I think you’re directly disagreeing with the paper for reasons that don’t just come down to how to approach decision making. If (2), maybe say more about why you propose Jack and Jill make different decisions despite having identical beliefs and preferences?
I thought about it more, and I am now convinced that the paper is right (at least in the specific example I proposed).
The thing I didn’t get at first is that given a certain prior over P(extinction), and a number of iterations survived, there are “more surviving worlds” where the actual P(extinction) is low relative to your initial prior, and that this is exactly accounted for by the Bayes factor.
I also wrote a script that simulates the example I proposed, and am convinced that the naive Bayes approach does in fact give the best strategy in Jack’s case too (I haven’t proved that there isn’t a counterexample, but was convinced by fiddling with the parameters around the boundary of cases where always-option-1 dominates vs always-option-2).
Thanks, this has actually updated me a lot :)