I'm taking a course on game theory and am faced with this prompt. What's most rational?

Person X is due for a raise at her company. Her company is offering her two options:

Option A is to continue receiving the same salary and stay within the bonus pool.

Option B is to increase her salary to match her bonus amount last year, but opt out of the bonus pool moving forward. Her bonus last year was $30,000. Therefore, her new salary would be $30,000 more than what it was.

The track rate of previous bonus pools are unknown. 

I think the most rational way to approach this is by looking at each option's worst case scenario and to take the one with the least worst. Option A's worst case scenario is there's no bonus and player X receives no extra money. Option B's worst case scenario is player X still earns $30,000 more.

The thing that's holding me up is the potentially infinite reward Option A gives because the bonus pool is not capped. How much should that option be considered, if any at all, since the past performance of the bonus pool is an unknown?


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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.

(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.

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.

That's what I'm wondering too. I don't think there is a correct answer, but I guess it's more of a "how did you think about whatever answer you gave?"

One concept you may be looking for is diminishing marginal utility of money. If I gave you a million dollars, it would give you some amount of happiness. If I then gave you another million dollars, it would again give you some happiness, but less than previously. Because the difference in quality of life between "not a millionaire" and "millionaire" is much greater than between "one-millionaire" and "two-millionaire".

(Unless you owe exactly 2 million dollars to a crime boss who will kill you tomorrow if you don't pay him the money. In such case, the second million is the one that makes you more happy. But this is a special case, where certain amount of money gives you something that a part of it couldn't give you partially. Most situations are not like this.)

As an approximation, it is said that happiness from money is proportional to the logarithm of the amount. But the logarithm does not apply to the extra money you get, but the total money you have. So the information that we lack about the person making $30K is whether they are living paycheck to paycheck, or they have savings, or possibly an extra source of income.

Generally, "$30K for sure" is better than "depending on a coinflip, $60K or nothing". But how much better it is, that depends on how much money you have otherwise. If literally zero, then the coinflip means that with 50% probability you will starve and die. On the other hand, if you already are a millionaire, then these two options are almost equal. So if you had a choice between "N for sure" and "coinflip: $60K or nothing", the millionaire would not accept N smaller than $30K, but a starving homeless guy might settle for much less.


Now, a bit more out-of-the-box:

As Dagon already said, this is not a raise. The company is just trying to distract you from the fact that they said no. Update your CV and start looking for a new job.

In the meanwhile, I would probably take the Option B. In theory, the bonus is between zero and infinity, which sounds great... but in practice, a company that refuses to give you a raise will probably be stingy about a bonus, too. (If the bonuses are so secret and unpredictable, they had an option to give you e.g. a $5K raise and then silently subtract the money from your next bonus; you would never find out.)

I think this is exactly what I was looking for, but couldn't put it quite so eloquently. Thank you.

I put Player X on a game show and imagined they had the choice to keep the $30,000 they have, or risk it for double or nothing.  As you mentioned, if that player went on the game show because they had a $30,000 debt to pay before they lost their house, they're going to keep the money and pay off that debt.

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.

Expected value is not necessarily the way to go. I would rather be paid 100k Currency Units per year than a 50% chance of 210k Currency Units and a 50% chance of zero. Expected utility is more to the point.

Note also that part of the (dis)utility of unpredictable compensation comes from things other than the actual monetary value. You might find it exciting not knowing what's coming. Or you might find it anxiety-provoking. If it's a bonus scheme, you might take some satisfaction in the feeling that if your hard work helps the company succeed then you will get a little bit of the gains; you might initially feel that way but then reflect on what a small fraction of the gains you're getting and feel grumpy about it; etc.

So, what was the answer given in the course?

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