The Key Civ discovery allowed me to run some controlled tests on the numbers of beakers
required for researching new technology. I think I understand how the cost is calcuated now.
Some of this stuff is already known, but I will repeat it here just to have
the whole mechanism described in one place.

The cost of researching a new technology (the beaker count) is the product of two factors.
The first is the Tech Number which you are researching. This is the number
of Acquired Techs you have +1. Acquired Techs are all techs you have received
in gameplay from research, huts, trades, gifts, or steals.
It does not include your starting techs.

The second factor is a Base Tech Multiplier to which either a bonus or penalty
can be added. The formula would look like this:

Cost of Research = TechNumber X (Base + Penalty/Bonus)

The Penalty/Bonus is based on how far ahead or behind your research is
compared to your current Key Civ. That relationship is quantified by
comparing your TechNumber to your Key Civ's TechNumber. If you are the same
you receive the Base Tech Multiplier with no Penalty or Bonus.

If you on a lower TechNumber than your Key Civ, then you receive the Bonus.
The Bonus is the same no matter how far behind you are. On a medium map
it is either a -1 or -2 depending on the TechNumber (see chart below).

If your TechNumber is higher than your Key Civ's, you are penalized a +1
to the Base Tech Multiplier for every 3 techs which you are ahead of him.
In other words, if you are even or ahead by one or two, your penalty is 0.
If you are ahead by 3,4, or 5 techs then your penalty is +1.
Thus, the higher the TechNumber which you are researching the farther
ahead it is possible to be. If your Key Civ is stuck in the Bronze Age
while you are researching Space Flight, you will be paying an enormous
penalty in beaker costs.

Here is a chart of TechNumbers, Base Multipliers, and Bonuses for the first 20 techs.

Tech# Base Bonus

01 10 0
02 11 -2
03 11 -1
04 12 -2
05 12 -1
06 12 -1
07 12 0
08 13 -1
09 13 -1
10 14 -2
11 14 -2
12 15 -2
13 15 -2
14 15 -1
15 15 -1
16 16 -1
17 16 -1
18 17 -2
19 17 -2
20 26 -2

All TechNumbers above 20 have a Base of 26 and a Bonus of -2.
The Minimum Beaker Cost for a given TechNumber can be calculated from this chart
and that Minimum can be achieved in gameplay by lowering your TechNumber
relative to your Key Civ through tech-gifting.

One important consequence of the fact that the Penalty/Bonus is based on TechNumber,
rather than total techs, is that Starting Techs are critical in determining how much
you will pay for research in a game. If you have fewer Starting Techs than
your Key Civ, then you can only get the Bonus when that Key Civ actually
is ahead of you in researching. You can never reach the Bonus through tech-gifting alone.

On the other hand, if you have more Starting Techs than your Key Civ,
you can easily get the Bonus without even giving away all of your techs.
Starting Techs are an enormous advantage not just at the start,
but throughout the whole game as they determine your ability to reduce your science cost.

If you start with NO techs, and the other civs all have them, you may be
paying research penalties the entire game, even with aggressive tech-gifting.

samson

I missed a thanks to you for the work you put into resaerching this.

samson

Mixam,

There's still some unexplored questions, if you're interested.
Like what happens if you start with less than 7 civs? Does the missing Power rating
always correspond to the absent color? Or will there be deadzones where
you are unable to gift the missing Key Civ?
This could be especially troubling for those who play on a small map when the 7th civ
will sometimes fail to start. In gameplay, Minimum beaker tables are easier to use
than the formula, so I wouldn't throw out your work. By creating a test case,
where the human civ is locked into its own Key Civ position, you can augment
your MIN and MAX data with the NOMINAL beaker cost data.

(Seems unfair that the rotton Mongols can get Future Techs at average cost while my Romans wound up paying the maximum 32,000! And I'm the Greeks my current game)

One clarification on "Power Rankings" versus the "PowerGraph." The Power Rankings that your formula relies upon (i.e., those that may be accessed via F3,send emissary) appear to be recalibrated every other year; the "PowerGraph" represents changes every fourth year, the oedo year. This may explain some inconsistencies in terms of the timing of changes in the amount of research required.

(Since my original experiments were designed to determine what the power graph measures as opposed to what it did not, I was already thinking of the PowerGraph in terms of "snapshots" taken of the game every four years. Your formula, however, required rechecking my assumption; and thus, apparently, Power Rankings are recalibrated every other year, but illustrated on the PowerGraph every fourth year.)

Once again. Congratulations on some great work done.
(I'm post a link on the other forum.)
Once again, very good work!