WW, I don't want to disappoint you, but I have been sort of stuck.
What the analysis says, pretty much, is that Harems versus Golds yield the same result. However, this is caused by a few nonsensical assumptions:
1) You always draw an "average" hand.
2) Cards you buy mix instantaneously
3) The game is continuous (specifically, you can use "half-a-turn" to buy "half-a-province", etc.)
4) The PPP does not apply.
2 is a unharmful assumption since no strategy benefits from it. However, clearly 1, 3, and 4 are assumptions that greatly hurt the Harem strategy for three reasons:
1) You never draw a "random" hand. Buying extra Harems for a slightly longer period of time induces less variance in the deck whereas buying Golds for a shorter amount of time induces more variance.
3) The continuousness of the game allows a Gold player to use "one-third-of-a-turn" to buy "one-third-of-a-province" which would make up one Harem. Specifically, the all or nothing affair makes catching up harder
4) The Harem player has an advantage if going into the PPP phase of the game by the nature that he or she "starts" with more points.
Things that I will consider, but they will take me a lot of time:
1) How to model the variance of a hand (or shuffle) and how that affects what one can buy. Specifically, having an average above $1.6 does not guarantee the ability to buy a province, and nor does $1.8 or $2.0. However, the higher, the more likely you are able to buy a province... based upon the variance of your deck.
3) Using a differential equation (e.g., continuous model), makes things easier. Using a discrete recurrence relation might be better, but they are often much more painful to solve and manipulate.
4) I don't know. I think the next simulation challenge should be encoding a successful PPP. The PPP ultimately decides many games, that will ultimately be the key for a good simulator.
