I'm taking a course on game theory and am faced with this question. What's the rational decision?

Person X should negotiate harder.  Neither option is clearly a raise for them, and if they're due a raise, they should loudly complain they're not getting one.  

As to the math, without knowing the distribution of probability over bonus amounts, this does not have a clear answer.  If $30K is above average for expected bonus next year, take option B for the guarantee.  If it's below the mean, take A and hope for a bigger bonus.  

There is a slight advantage to the raise, if it's equal to the average expected bonus.  You get the raise paid to you spread throughout the year, and the bonus only at the end of the year.  So you can invest or earn interest on the earlier payments.  Also, if you're looking for a new job that doesn't give fake raises, the increased salary is more durable in your expectations, in addition to being partly paid in the meantime while you search.

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.

This is really just a "what is your utility function and what is your prior on the bonus" question, I guess? There is no clearly correct answer with just the information given.

There is no "most rational" solution here. There are too many known unknowns as well as unknown unknowns.

But really, WTF? Neither of those are raises. Option C is to negotiate something that is actually a raise, with a failure on the company providing one meaning that person X goes somewhere else that will actually offer better.

The correct way is to calculate the expected value. Since there is little data, the expected value is equal at base +30.000. The expected volatility is zero in case A and unknown but not zero in case B so a risk averse person would gain some utility out of going A.

Note however as both options are not a raise as the expected value is equal to her current salary and if person X accepts, she will be highlighted as an example of wage discrimination against women.

So, what was the answer given in the course?

Option B, salary increases compound over the span of your career.