My first choice in cases like this is a simple 'double the one, half the other and if the two numbers come together(ish), there's your ballpark number'.
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While this heuristics, when it works out that way, may provide you with a first idea that may be in the right ball parc, I think it would be unwise to stick with a "factor 2" rule. There is no propaganda handbook that instructs propagandists to double good-news numbers and half bad-news numbers. Some numbers are made up whole cloth were reality is zero / factor is infinity.
And even if the other side goes by a factor-two rule: Once you let them know you divide or mutiply by 2, they may take that into account and change to a factor-four rule.
I have a real-world anecdote for this:
I used to work in IT support for the national sales office (150 employees) of some chemical company. One collegue in our team of three (let me call him "Hans") had a knack for working out details and was usually the first to come up with estimated figures for cost and duration when we were thinking about some major change to the IT landscape or other such projects. However, after a while, we noticed that Hans' estimates were consistently too low by a factor of 4. How long would it take till we establish an outsourced first level phone support? Hans said 3 months; it took a year. What will it cost to move network and workstations to a new office? Hans said 50,000 Deutsche Mark, we ended up paying 200,000. So when a new purchase, change, project rolled in and we asked Hans for an estimate, and he said half a year, quarter million DM, we multiplied by 4 without telling him, and it would work out nicely, we'd finish within 2 years at half a million Marks.
Then, someone made a grave mistake and told Hans that we multiplied his estimates by 4.
From that point on, Hans divided his estimates by four, thinking we'd multiply by four to arrive at his actual estimate, but we adapted quickly and learned to multiply by 16... (before the next round of escalation and us using a factor of 256, a company merger dispersed the team).