An old mate of mine has only one rule of negotiation “he who mentions price first, loses”
Whilst there are other things that contribute to a successful outcome, he is right about the price. Once an expectation of price has been mentioned, it becomes the point from which other conversation evolves.
Typically: “I want $1,000 for it” “well, I would be willing to pay you $500”
This will evolve to a final price of $750, but had the initial price been $1,200, the end price may have been around $900.
Perhaps the most public example of this anchoring phenomenon I have seen is Alan Greenspan, former head of the US Federal Reserve, the acknowledged expert on the economy, who completely missed the GFC, simply did not see it coming despite many warnings. The preconceptions he had of the way the economy worked simply blinded him to alternatives. Later, after it became obvious that he was wrong, to the detriment of the US and ultimately world economy, he felt obliged to acknowledge his error before a congressional committee.
When negotiating anything, do not forget my mates rule 1, but recognise that it works across a wider field than just price. Any firm preconception will blind you to alternatives in a negotiation, particularly when there is an emotional investment in the result.