In my experience, they never do (if they're a decent sized company).
> "See, I did it this way, and I ended up with this other number. What do you think causes the difference?"
This does work at times to increase your pay, so I recommend it. However, when it doesn't, they are not going to respond to your question. It will be another "We're sorry, but our data suggests otherwise. We aim to be competitive."
Of course, if you're a very senior position and they really want to, they will have a conversation. In the remaining 99% of cases, they'll adjust the salary to an extent, and then you'll hit a wall. That's when you start negotiating on non-salary stuff.
"I appreciate that you're trying to be competitive, and it is a generous offer you have presented. The problem we have is, as you know, that I had pictured something different from a company with your resources. I have shared with you how I arrived at that. If you're not willing to share your process, how can I trust you have my best interest in mind?"
Any good negotiator on the other side of the table would eventually throw you a bone if you are nice, don't relent, ask for reasonable things (if I sign up for you to own the fruits of my labour, I want to know I'm being cared for), and apply the right amounts of awkward silence.
With most companies, the issue is that the person you are dealing with is not someone you will have a relationship with. It will not be your new manager. It will be someone from HR. They'll have a salary range, and will be willing to move to near the top depending on how much the hiring manager says how good you are. At an individual level, they don't care if you "trust" them.
If you're asking for something beyond their upper bound, they'll have a few other levers to offer you (more stock, one time sign on bonus, etc). But keep in mind: If the manager said you were "good, but not great", they're not going to go much higher than their initial offer to you.
In all these scenarios, they have zero incentive to discuss how they came up with that number. If you indicate you can't trust them if they don't, they'll agree and drop you and look for another candidate. The HR person is dealing with multiple hires for multiple positions. He/she doesn't have the time to do a detailed discussion with every one. What's more, what is the point of them sharing it? Put yourself in their shoes. They've come up with a compensation strategy for the whole company. If they're offering below market and they know it, your pointing it out to them doesn't strengthen your position. They know the information your providing them. If they were trying to save money, they'll increase the offer. If their policy is to underpay, they're not going to raise the offer unless the hiring manager indicated you're well above average.
> and apply the right amounts of awkward silence.
Silence is awesome for certain scenarios, but not when dealing with an HR person on a compensation package. Remember: Your asking a question (like the one in your comment) does not throw the ball in their court. They'll either ignore you or give you a canned response. Every offer has a deadline, and they'll ping you again near the deadline. If you continue the silence, they'll withdraw the offer and move on.
If you're joining a small company/startup, your tactics will work. Not so with established companies.
There are stories here on HN about people who had been hired, but were not happy with some of the paperwork they were being asked to sign on their first day on the job. They stood their ground and that was their last day at work. If a company is willing to do that after they've gone through the headache of hiring a person, then don't assume that you have any decent leverage with notions of "it's too painful for them to withdraw an offer and hire another person - they'd rather negotiate."
Contrary to what you might think, I agree with most (if not all) of what you're saying. They might decline to give me a reason to trust them, even after I've given them several opportunities. In that case, I have a strong signal to decline, and they have to do other things very well to compensate.
I'm not saying one should be picky, I'm just saying that if the worker and the employer have very different ideas of the value the worker brings, and one side is not open to discussing their perspective of things, my experience is that doesn't turn into a fruitful partnership with long-term prospects.
At best, it's a temporary gig and then the worker leaves with a bunch of knowledge. It's not a desirable position for anyone. (But that does happen, very often in larger companies, as you say.)
In my experience, they never do (if they're a decent sized company).
> "See, I did it this way, and I ended up with this other number. What do you think causes the difference?"
This does work at times to increase your pay, so I recommend it. However, when it doesn't, they are not going to respond to your question. It will be another "We're sorry, but our data suggests otherwise. We aim to be competitive."
Of course, if you're a very senior position and they really want to, they will have a conversation. In the remaining 99% of cases, they'll adjust the salary to an extent, and then you'll hit a wall. That's when you start negotiating on non-salary stuff.