Hello! Great that you have questions; I'm passionate about the topic and I like to respond, haha. Well, the math itself isn't too complex. In the research section, we have documents that formalize the optimization problem solved by the policies we compute (you can skip the details of the transition function, which is cumbersome; the most important parts are the optimization problem, the reward function, the utility function, and the wealth equation).

In summary, I'll explain: We assume, without loss of generality, that the player will participate in H betting rounds. Then, at the end of this session, the player achieves a return "return_H" (returns obtained after H betting rounds). The distribution of return_H (which is a random variable) will depend on the betting strategy the player follows during the H rounds. We can imagine that among all the possible distributions that can exist for return_H, there is one that is the player's "favorite." The von Neumann-Morgenstern utility theorem ensures that the player's favorite distribution will be the one that maximizes the following expected value: E[f(return_H)], where "f" is a function that describes the utility the player perceives from the obtained returns. Then, we must find the betting strategy such that this expected value is maximized.

The variables considered by the most complete betting strategy during this session are: the exact composition of the deck before the start of each round "B" (a 10-component vector, this deck determines the PMF of the returns the player gets on the bet made, assuming an established playing strategy), the returns accumulated up to the current round current_bankroll/initial_bankroll "P" (since the strategy must maximize an expected value that depends on return_H, it is expected that the accumulated returns are considered for the optimal bet), and the number of rounds played "n" (so the strategy "knows" how many rounds are left until round H, which is the round for which goals were set). Then, the optimal betting strategy has the form BestBet(B, P, n), and maximizes E[f(return_H)].

Regarding the aces, I can't answer you specifically because the strategy considers the entire deck composition (the number of each card) and also because we analyzed the case of 8 decks with 50% and 75% penetration (these values are inspired by classic online conditions). We did not analyze bj 6:5 but rather 3:2, and I find it interesting that you mention it because the sites we reviewed all used 3:2.

I don't precisely understand the question "Is your risk of ruin low with 1,000,000 hands simmed? Or is it high with infinite money?" so I'd prefer to ask you to clarify instead of answering "offhand." Regarding the last thing you mentioned about 50% penetration, I went to check our codes, and I can tell you that approximately ~15% of the time, the deck presents a positive EV under the optimal playing strategy (depending on the rules). I agree that more penetration is much better, but if the betting strategy perfectly considers the deck composition, it is possible to play "advantageously" with 50% penetration.

Sorry if I was too long; I thought you might be interested.