Civilization – When you ‘Prove the Earth is round”, do you receive any in game benefit

Summarized results:

• Happiness is calculated based on individual player's difficulty.
• Bonus vs. Barbarians uses the lowest player's difficulty
• The AI starts use the lowest player's difficulty.
• Rewards from ruins are either based on individual player difficulty or lowest player's difficulty
• Culture costs are based on individual player difficulty.
• Not sure on City State Bonuses, will probably test those tonight. I assume they will use the lowest player's difficulty.
• Not sure on AI skill, I assume they will use the lower player's difficulty. I probably won't be able to test this due to the length of Civilization games.

Onto the proof, first the two testing parameters: vs.

Happiness works as expected:

I have a starting happiness of 6 after my first city on deity. Cartecs has a happiness of 15 on Settler.

vs.

I have a happiness of 15 on settler difficulty with cartec's playing on deity.

Interestingly, it looks like it uses the lowest player's difficulty for barbarian fights:

I'm playing on deity difficulty here with a settler difficulty human in the game:

vs.

I'm playing on settler difficulty here with a deity difficulty human in the game:

vs.

Lone human at deity difficulty:

AI Starts:

Turn 9 in both cases:

Playing on settler with another human on deity results in computer's using settler starts:

vs.

This is a game with just me playing on deity difficulty.

I saw further proof in both games. While playing the multiplayer Japan managed one city by turn ~14. In the deity game alone, Egypt managed two cities and multiple warriors. This is consistent with settler vs. deity starts.

As a settler difficulty player I was able to find settlers in ruins, and get a lot more upgrades quickly. As a deity player I did not notice a particular increase in ruin rewards. This doesn't really prove much beyond the fact it either uses the lowest difficulty bonus or uses the individual bonuses.

As a settler player it cost 10 culture to get the first policy. As the deity player it cost 25 culture to get the first policy. You can see this in the happiness screenshots.

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:

1. Make a list of all the technologies you can currently research (that you have all the prerequisites for).
2. Find the cost for each of these technologies.
3. Find the median value of the above costs.
4. Multiply it by 0.5. This value gets boosted by 0.25 if you started Rationalism, 0.25 if you have the Porcelain Tower and 0.5 if you both started Rationalism and have the Porcelain Tower.

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:

1. Build the Porcelain Tower, which is a great Wonder even without this boost, on account of the free great scientist.
2. Start Rationalism. That's a great policy anyway for getting more science.
3. Make more research agreements! Remember you can make a lot of research agreements simultaneously, as long as they are not with the same civ. Also remember that the AI civs love their research agreements, so if they won't make them with you they'll make them with the other civs - meaning two other civilizations get stronger and you get nothing. It's better for you to be one of the two.