UPDATE The formula has changed as of the March 2011 patch, it is now
(city population * 1.1) + (capital population * .15) - 1
for each city connected to the capital, not including the capital.
- The Machu Pichu wonder increases the modifier by 20%, to
(city population * 1.3) + (capital population * .15) - 1
.
- Arabia increases the constant by 1, to
(city population * 1.1) + (capital population * .15)
(thanks WillfulWizard).
The economic overview also gives a far better explanation of how it is calculated - in other words, this question is now trivially solved by simply looking at that economic overview :)
Original answer below.
Okay, I did a bit more testing, and bwarner's answer is almost accurate:
Each city, excluding the capital, provides (city's population * 1.25) + 0.01
gold per turn. Owning the Machu Pichu wonder increases the modifier by 20%, to (population * 1.5) + 0.01
, for all the cities.
The capital does not provide any gold.
The 1.25 and 0.01 can be seen in the Assets\Gameplay\XML\GlobalDefines.xml
file:
<Row Name="TRADE_ROUTE_BASE_GOLD">
<Value>1</Value>
</Row>
<Row Name="TRADE_ROUTE_CAPITAL_POP_GOLD_MULTIPLIER">
<Value>0</Value>
</Row>
<Row Name="TRADE_ROUTE_CITY_POP_GOLD_MULTIPLIER">
<Value>125</Value>
</Row>
The economic overview also demonstrates these values.
I couldn't find any other factor which affects these values, and I checked different difficulty levels, different distances between cities, different city route type (road vs railroad, road/railroad vs harbor), different city health and whether it is occupied. Looks like it's solely the population.
Correct, they are not necessarily the same. You will receive a random technology from those that are available to you, and they will receive a random one for them. This does imply that if you are making use of research agreements, you should try not to "skip" techs that are cheap, since you want the research agreement to give you something as "expensive" as possible.
Best Answer
How acquiring technologies works
First of all, it's important to understand that you do not get technologies by the number of turns you invest, but by the number of science points you have, just like it works for culture policies. It's just hard to see when looking at the technology graph because it only lists the amount of turns it will take; but each technology does, in fact, have a cost in science points. You can see that cost by opening the Technology's Civilopedia page (this can be quickly done by right-clicking the technology in the graph).
This is the reason, by the way, that you sometimes get technologies earlier or later than what is listed in the graph - because the estimation is done based on your current amount of science per turn, and that can change.
Also, be aware science points overflow - if you completed a technology, all left-over points will be immediately put to use on your next technology being researched. This applies regardless of the source of science points, so it will also work with a science boost (see below).
How the science boost works
So, a science boost is given in science points, not in number of turns. If I understand correctly, a Science Boost works this way:
This number is the number of science points you get.
For example, say you can currently research Steam Power, Biology and Fertilizer. Steam Power and Biology cost 2350 Science, while Fertilizer costs 1600 science. The median of the set {1600, 2350, 2350} is 2350. Let's also assume you have the Porcelain Tower. The number of Science points you will get is
2350 * 0.75 = 1762.5
.By the way, on a personal note, this is a much better system than the previous one, in which you just got the cheapest technology available for free... although admittedly this is also much more tedious to calculate. As a rule of thumb, this will usually advance you by "half a circle" in your current research.
How to maximize the effects of a science boost
As for maximizing the effects, there are 3 basic ways to make it better: