I don’t see how one can usefully answer this question without more information. (Perhaps one point of the question is to get you to reflect on what further information might be relevant?)
To whatever extent X is an expected-utility maximizer—i.e., there’s some function U from states-of-the-world to numbers, such that X prefers outcomes where the expected value of U is higher—you can formalize the question in terms of probabilities and utilities. But the resulting decision will depend in complicated ways on X’s predictions for the likely evolution of the bonus pool and of X’s bonuses given the size of the pool. (And also on things like X’s predicted future need for money, future job opportunities, retirement date, etc., etc., etc.) And, if I am understanding the premises of the question right, X has no information to speak of on which to base those predictions.
Perhaps X has some idea of whether last year was a good year for the company or a bad one. If it was a good year, she should probably expect future bonuses to be worse. If it was a bad year, she should probably expect future bonuses to be better.
X’s predictions might be influenced by the fact that the company is offering her this choice; if this isn’t a thing they usually do, it might indicate that they hope to save money that way (which would mean that option A is likely better), or it might indicate that they are unusually anxious to keep X happy (which seems like it would mean her future bonuses are likely to be good, in which case again option A might be preferred—but if they’re giving her the option of B out of a genuine desire to do something nice for her, maybe that’s a sign that they expect future bonuses to be low and B would be better).
Without any information on previous bonus pool behaviour, though, predicting what the bonus pool will do in future seems pretty meaningless. X has (so far as we know) no useful information and her predictions are likely to be unreliable. Whatever decision she makes, she shouldn’t be at all confident that it was the right one.
I’m personally neither much convinced by the least-worst approach (extreme case: you have the option of a bet that costs you $1 and gives you $1M 99% of the time. The worst case is that you get nothing. Do you therefore pass?) nor by the “potentially infinite reward” arising from your ignorance of the bonus pool; the potential reward isn’t really infinite (the company cannot make more money than the world’s GDP[1]) and it’s monstrously unlikely to be, say, more than 10x what it was last year.
[1] Well, it could. But the scenarios where that happens are generally ones where things at the company are changing so fast that we should e.g. expect all salary arrangements to be drastically changed, and/or things in the world are changing so fast that we should expect salaries to be kinda meaningless.
I remark that if X is “due for a raise” then she might reasonably be aggrieved to be given a choice between these two options neither of which is a clear improvement on her present pay.
(Perhaps one point of the question is to get you to reflect on what further information might be relevant?)
I think this is the case. I don’t think there is a correct option, but it’s just a question aimed to see how you think about solving a problem like this and one where more information is needed.
I don’t see how one can usefully answer this question without more information. (Perhaps one point of the question is to get you to reflect on what further information might be relevant?)
To whatever extent X is an expected-utility maximizer—i.e., there’s some function U from states-of-the-world to numbers, such that X prefers outcomes where the expected value of U is higher—you can formalize the question in terms of probabilities and utilities. But the resulting decision will depend in complicated ways on X’s predictions for the likely evolution of the bonus pool and of X’s bonuses given the size of the pool. (And also on things like X’s predicted future need for money, future job opportunities, retirement date, etc., etc., etc.) And, if I am understanding the premises of the question right, X has no information to speak of on which to base those predictions.
Perhaps X has some idea of whether last year was a good year for the company or a bad one. If it was a good year, she should probably expect future bonuses to be worse. If it was a bad year, she should probably expect future bonuses to be better.
X’s predictions might be influenced by the fact that the company is offering her this choice; if this isn’t a thing they usually do, it might indicate that they hope to save money that way (which would mean that option A is likely better), or it might indicate that they are unusually anxious to keep X happy (which seems like it would mean her future bonuses are likely to be good, in which case again option A might be preferred—but if they’re giving her the option of B out of a genuine desire to do something nice for her, maybe that’s a sign that they expect future bonuses to be low and B would be better).
Without any information on previous bonus pool behaviour, though, predicting what the bonus pool will do in future seems pretty meaningless. X has (so far as we know) no useful information and her predictions are likely to be unreliable. Whatever decision she makes, she shouldn’t be at all confident that it was the right one.
I’m personally neither much convinced by the least-worst approach (extreme case: you have the option of a bet that costs you $1 and gives you $1M 99% of the time. The worst case is that you get nothing. Do you therefore pass?) nor by the “potentially infinite reward” arising from your ignorance of the bonus pool; the potential reward isn’t really infinite (the company cannot make more money than the world’s GDP[1]) and it’s monstrously unlikely to be, say, more than 10x what it was last year.
[1] Well, it could. But the scenarios where that happens are generally ones where things at the company are changing so fast that we should e.g. expect all salary arrangements to be drastically changed, and/or things in the world are changing so fast that we should expect salaries to be kinda meaningless.
I remark that if X is “due for a raise” then she might reasonably be aggrieved to be given a choice between these two options neither of which is a clear improvement on her present pay.
I think this is the case. I don’t think there is a correct option, but it’s just a question aimed to see how you think about solving a problem like this and one where more information is needed.