I can't make any sense of your math, sorry.
My analysis is here:
http://forums.civfanatics.com/showthread.php?t=533642
If someone could tell me how to format that table properly, that would be better.
if you only look at the virtues themselves, it actually takes about 30 virtues to balance out.
However if you count the synergies gained (because you have 1 extra virtue in Knowledge and Tier 2) its ahead after ~20 virtues.
Thanks for the stuff Krikkit, that is good to know.
When you guys say "balance out", you mean "Player one, who took CV, is ahead of player two, who did not take CV, by one virtue". I.e.
The break-even point is where a player taking Cohesive Values would have 1 extra virtue over a player that doesn't. That assumes the first player going straight for Cohesive Values.
I.e. the assumption is that you bee-line for CV, then immediately leave the tree.
That isn't what 'balancing out' means to me. Balancing out means:
1) You recoup the loss of your investment (culture you spent).
2) You have spent the same amount of culture, and you have more virtues.
@Zet: Using your definition, Community Medicine, Networked Datalinks, Metaresearch Methods and Information Warfare are all not deemed to be worth it.
Irrespectively, regardless of the definition, with CV Player One gets TechnoArtisans OR Information Warfare
sooner than Player Two gets TechnoArtisans or Monomyth.
(Assuming this is a player with Madjinns approach to health)
Further, according to your own definition, the break-even point is virtue 5. Because for x number of turns at virtue 5, the player taking cohesive values has 1 extra virtue over player 2.
In fact, if you look at Zet's maths, there is an absolutely colossal flaw. This isn't about "How much you spend". It's about "how much you save".
One way of looking at it: By Virtue 9, you have saved 209 culture. That means that the REAL cost of Virtue 10 is 1220 culture with a further discount of 209, for a cost of 1011 culture.
However for player 2, the real cost of virtue 10 is 1340 culture.
Looking at Madjinn's KP stream, he has 57 culture income per turn by turn 176. I believe he got 2 free virtues, so he's about to purchase virtue 10.
For player two (Madjinn), virtue 10 is purchased on turn 176.
For player one, virtue 10 was purchased 17 turns beforehand, on turn 159.
That's the important nuance that your simulation lacks. The comparing factor isn't number of virtues, it's the number of turns used to obtain those virtues.
And by your own definition, the player who takes CV "breaks-even" every time she purchases a virtue before the other player, for the number of turns it takes the second player to catch up with her. Then there's a set amount of time, before she breaks-even again.
(Note I use the "discount" as a means of calculating the turns saved. Clearly the true calculation is the sum of culture over the last x turns, but well, I couldn't be bothered to go through the stream for that level of detail)