pindicator Wrote:(So, if I do understand correctly, you're saying that in order to get the most bang for your buck, you should take the ratio of beakers : gold and assign specialists according to which ever is lower? For example, if I had 7 libraries, 2 universities & 4 monestaries in my empire as well as 4 markets and 2 grocers, I would be better served running merchants. Yet if I added an additional 4 banks, then I would be better served running scientists?)
There's more to it than that. First in how you calculate the national average, and second in how you figure each city's preference.
To calculate the national aggregate multiplier for beakers and for gold, average each city's multiplier
weighted by its commerce production (including trade routes and Bureaucracy). It simplifies things to also include 100% for each city's base production in this step (so a city with only a library would be 125%.) Consider this example:
Code:
City Commerce Science Gold
Delhi 60 335% 150% (Oxford and many science buildings)
CityB 30 150% 200% (Mostly gold buildings)
CityC 10 100% 100% (No multiplier buildings)
Your national gold multiplier is (60 * 150% + 30 * 200% + 10 * 100%) / (60 + 30 + 10) = 160%. This isn't the sum of your multiplier buildings, or the straight average of each city; it's the weighted average of each city. Notice that the overall ratio is close to Delhi's (because Delhi is your commerce gorilla), but CityB pulls it above that to a greater extent than CityC pulls it down, because CityB has more weight than CityC.
Similarly, your national science average multiplier is (60 * 335% + 30 * 150% + 10 * 100%) / (60 + 30 + 10) = 256%. Your national value ratio of gold:beakers is 160:256, or exactly 1:1.6. You can derive this ratio in-game by looking at the F2 screen and clicking your slider one step. Disregarding roundoff artifacts, the change you see in each of your beaker production and gold production is your national ratio.
Suppose CityC wants to hire a specialist. This is pretty straightforward. There's no multipliers in the city, so you can produce either gold or science here on a one-to-one basis. But we should hire a merchant. Why? Because hiring a scientist will produce a flat 3 research. But the 3 gold produced by a merchant will instead enable you (via raising the science slider) to produce 3 * 1.6 = 4.8 beakers elsewhere in your empire.
Now suppose CityB wants to hire a specialist. Mr Scientist would give us 3 + 50% = 4.5 beakers. Mr Merchant would give us 3 + 100% = 6 gold, and that 6 gold translates to 6 * 1.6 = 9.6 beakers elsewhere in the empire. Both intuitively and mathematically, it's very clear-cut for the merchant.
Suppose Delhi itself wants to hire a specialist. Intuitively, we'd say a scientist with that high multiplier, and we'd be right. Mr Scientist produces 3 + 235% = 10.05 beakers. Mr Merchant produces 3 * 1.5 = 4.5 gold, which translates only to 4.5 * 1.6 = 7.2 beakers of increased science slider, less than the scientist.
Now here's a tricky example. Suppose we had a City D (producing no commerce so as not to affect the national value ratio.) And suppose this city has a library for 125% science production, and no gold multiplier buildings. Mr Scientist produces 3 * 1.25 = 3.75 beakers. But Mr Merchant produces his 3 gold which translates into 3 * 1.6 = 4.8 beakers via the slider!
How can this be? A city with a higher science multiplier than gold still came out ahead by hiring a merchant? The answer is that
each city's production is not absolute, but is relative to your national gold:science ratio. If a city's own gold:science ratio (1:1.25 for City D here) is lower than the national gold:science ratio (1:1.6) here, it should hire a merchant. If a city's own gold:science ratio is higher than the national gold:science ratio (for Delhi, compare its 1:2.17 to national 1:1.6), it should hire scientists.
As proof, consider the breakeven crossover point, which would be in a city with exactly 1:1.6 ratio. This could be 100% cash and 160% science, or 125%:200% gold:science, or 150%:240%. Let's use that middle one. Mr Scientist produces 3 * 200% = 6 beakers. Mr Merchant produces 3 * 125% = 3.75 gold, which equals 3.75 * 1.6 = 6.00 slider-beakers. Exactly the same.
Whew.
So Sulla, you may point alexman here and tell him that 1 gold may equal 1 beaker in certain specific cases, but it's not a general truth at all.
I'm glad that I had it right on an intuitive level, but I'm still trying to wrap my mind around it logically. 
This discussion is exactly why I have been lurking here for over a year. I cannot fathom figuring that merchant/scientist math, but now that I read it, it makes perfect sense. Thank you very much T-Hawk for that detailed breakdown, especially with the specific example. When I looked at this, the first thing I thought of was "I was told there would be no math" 