It’s kind of the opposite of a game, it’s very transparent. In reality you would accept a minimum of $X and reasonably think there’s a max of $Y. Companies don’t treat employees as purely cost, they want to give them good packages for retention and morale so they don’t just offer the lowest salaries imaginable.

The minimum you request should be a number that would actually make you happy and content to accept the job. The upper number should be a high ball that you yourself don't actually expect to get. It shouldn't be some outlandish number way outside of what someone in your field can attain. This gives the company room to bid for your expertise with other companies without having to worry too much about putting yourself in a position like the original poster. And yes, I kind of think of it as a game, but an important, open and transparent one. The range you put forth leaves room for interpretation and makes the company uncertain. If you say one minimum number and they want you, they will pay you that minimum number. If you give a range, it opens the door to the possibility that they will pay you more. The more that they like you and the more that they are worried you might accept a competing job offer the more likely they are to bid more.

There is lots of advice that says to avoid giving any number. Sure, you can try to do that, but in my experience it comes off as a more aggressive tactic that may leave a bad taste in their mouth. I'm not sure if anyone else recommends this strategy, but anecdotally it has worked well for me. You still need to do research on what your field pays and you need to understand how well you do whatever it is that you do.

edit: Over time, being assertive but not overly aggressive in salary negotiations will pay huge dividends. I've heard of far too many stories of timid engineers that accept 5% raises or salaries for new positions year after year. Each one of those small increases in salary puts you further and further behind the curve. Companies don't ask what the history of your pay raises are, just what you are currently paid. Getting more now, means getting a lot more later. Early in your career if you are good at what you do, you can be more aggressive than later because it does not impact the balance sheet of bidding companies much. I had a good 4 to 5 year stretch early in my career where I was making >50% more than I was the previous year. Switching companies usually meant bigger increases in pay, but not in all cases if you prove yourself to be a valuable asset. Doing what you do well is still extremely important and if you do it well enough, the ROI of hiring you far outpaces your salary and companies will gladly make that investment to hire you and to retain you by keeping you happy and productive.

I disagree with the above advice (as an employee, you should never ever state the number first), however I think the assumption behind this advice would be that you the company believes it's competing against other companies for you. So, they wouldn't do the lowest because they want to outbid the competition, and the higher number acts as an anchor.

But again, as a candidate, never say the number first.