What you're saying would be true if companies were navigating the process as blindly as candidates are. Then a high anchor could lead the employer to negotiating from a position of weakness. But, armed with data, companies don't do that.
1 - With no asymmetry both peers would have a value anchored before negotiations start.
Two stupid things on their part...I got headcount, not a budget. So when negotiating salary, it wasn't in my best interest to haggle. I'd rather have happy relationships with my direct reports than know that I screwed them out of a few dollars on behalf of the company. Had they made it my duty to make the numbers add up, I might have been a bit more frugal. Oh, and they didn't filter the data to only show me positions below my title. So I went into my reviews armed with the exact market salary data that the company used to negotiate with me.
>>In negotiations, anchoring refers to the concept of setting a boundary that outlines the basic constraints for a negotiation; subsequently, the anchoring effect is the phenomenon in which we set our estimation for the true value of the item at hand.[5] In addition to the initial research conducted by Tversky and Kahneman, multiple other studies have shown that anchoring can greatly influence the estimated value of an object. [0]
Do you or your Roman generals (who probably wouldn't be fighting each other anyway, but that's besides the point) have any hard data/sources you can show that demonstrate a link between anchoring and coming off worse in a negotiation?
You can hurt yourself by giving a number, but you're more likely to help yourself if you choose a realistic but slightly high number and indicate that you are willing to negotiate based on non-monetary factors.
[0] : https://en.wikipedia.org/wiki/Anchoring#In_negotiations