Here's the tweak I'd like to propose and subsequently study. I think that the distribution remains uniform despite the influence of latent karma because the kappa coefficient is constant, ie bad karma is weighted the same as good karma. If kappa increased whenever the player failed and decreased when they succeeded (inversely for AI), players would benefit from good runs while being less likely to experience bad runs, which "feels fair" to our flawed brains. This would hopefully achieve an effect similar to a GM reading the room. Calibrating kappa's rate of change (arithmetic? geometric?) requires testing; I'm curious to know if there's a way to establish a formal equation linking kappa and the new random variable we've created.
Once you accept that the game balance can be changed, we have a lot of modeling flexibility as we enter into the realm of "weighted dice". Let us try to modify the original karma dice, for the sake of simplicity. I am not against having a parameter that dynamically changes (statisticians would call it time-dependent), but instead of modifying \kappa I would include an additional parameter, say \mu_t, which plays the role of the *trend*. By modifying \kappa, we would tweak the level of determinism of the dice (entirely random vs almost deterministic) but that's not what we want. Instead, by changing \mu we get the desired effect of modifying the average success rate.
The modified karma equation, only for players and not NPCs, would look like this (you need a bit of imagination as LaTeX does not work here):
Y_t = \mu_t - \kappa_t Y_{t - 1} + \epsilon_t.
Now, if \mu_t = \mu (i.e. a constant trend that does not change over time), this is essentially adding a hidden bonus if \mu > 0, and a malus otherwise, to all the rolls. The parameter \mu could be selected by the players as part of the "game difficulty". The mathematics is particularly neat in this special case: we still get closed-form expressions for the stationary law, and therefore, we can study the resulting random variable. Hence, we could select \mu so that the expected value of the dice is, say, 12-13 (bonus, easy mode) or 8-9 (malus, hard mode).
A more elaborate idea would be to allow \mu_t to change over time and be *adaptively* selected. A complication is that the same score, say 15, is sometimes a success and, in other cases, a failure. Nonetheless, a first idea could be keeping track of the average number of the most recent successes and failures, say \pi_t, and then set \mu_t = qnorm(\pi_t, 0, sigma), where qnorm is the quantile function of a Gaussian with a variance depending on \kappa. There is a tight connection between this approach and probit regression and Kalman filtering, so I can certainly envision more sophisticated ideas based on supervised learning, but I guess I will stop here
It looks already like an overshoot!
P.S. I realize there wasn't a question in there, aside from an implicit request for comment, but I do have a stats question I can't wrap my head around involving Magic the Gathering, odds of streaks and card counting. Would you mind if I DMed some specifics?
Sure, go ahead! But I cannot guarantee I will have any smart ideas
