If you're doing something like selling your startup, you want to set the anchor price, yes. There is such a huge range of outcomes, it's best for you to shoot high (as long as you're comfortable with the risk of scaring the buyer off) as the seller. (Or to start and shoot low as the buyer; with the walkaway risk reversed.)

For something like salary negotiation where results are more well-bounded and there is a lot more commoditization (ie replacements are relatively easy to find, or are perceived to be relatively easy to find), you generally don't want to anchor. An anchor that's way out of whack is easy to shoot down (show comp data). Anchoring slightly high/low doesn't buy you much as there is no real anchor... you're just in the established/known range... and so it's more valuable to see where the counterparty starts off an dplay from there.

There are unique situations / exceptions of course. For example, you might have uniquely valuable skills for a given company (or be able to convince them you do) and therefore move into a less-bounded zone of negotiation, in which making the ask/establishing the anchor would again be a good idea. But for a normal, well-established position, there is literally no benefit in "anchoring" as the bounded range is already fairly well established and that is the anchor.

Still, coming up with that first number is very tricky. A application of mine once reached the salary negotiation phase, and the company asked me what I was expecting. I didn't want to go so high they were insulted, so I took what I thought was a reasonable number and bumped it by 40%.

They said "okay". No negotiation, no fireworks, nothing. I've always wondered what I could have gotten out of them.

I guess the fear is that you'll anchor it "lower" than they would have. Example if they are intending to pay 120, and you say "I'm hoping to make 110" they might just come down to 110. But I suppose if you start at your high range that might be in your favor. Some of us (myself included) just don't know what we're worth so it's hard to know how to start the deal...

That sounds similar to the idea the Overton window[0]. I've thought for a while that I hope I get to say what I want before they say what they're offering. The hope is that if I say a high number then we will be negotiating based on that high number rather than some unknown, possibly lower one.

[0] https://en.wikipedia.org/wiki/Overton_window

This works iff you shoot very high/low and if it's a very fungible range of possible outcomes. In something that is fairly well-bounded like job compensation (or most residential real estate transactions, as another example) it's usually better to let the counterparty turn the first card.