This is the sort of thing that I hesitate to share, because the rule in MM for coin purses/bags is so elegant, but...

This is the sort of thing that I hesitate to share, because the rule in MM for coin purses/bags is so elegant, but math wise...I think there may be an issue for consideration. Its not an issue with the dice themselves, its an issue with the splitting/combining rule. (Also, forgive me if someone else has already talked about this I couldn't find a post about it).

Thinking about the probabilities associated with risk dice, we can apply the negative binomial theorem (https://stattrek.com/probability-distributions/negative-binomial.aspx) to determine the expected number of purchases you can get with a die before it steps down (or in the case of the d4, disappears). The formula for the expected number of purchases BEFORE the step down is

(1-Prob of Step down)/Prob of Step Down

so the expected number of purchases is

(1-prob of step down)/prob of step down + 1

This gives us the following:

d12 - 4 purchases - 13.33 total before exhaustion
d10 - 3.33 purchases - 9.33 total before exhaustion
d8 - 2.66 purchases - 6 total before exhaustion
d6 - 2 purchases - 3.33 total before exhaustion
d4 - 1.33 purchases

(Total before exhaustion is simply the sum of the die plus all lower dice expected purchases).

The rule for coins is that you can split a bag into 2 bags by stepping down the die of both bags. However, if you look at the table, in every case the expected number of purchases you can make with 2 bags of a step lower is more than the expected number of purchases you can make with the original bag. This leads to the seemingly weird situation where you can buy more stuff if you split up your pile of cash into smaller piles, and less if you horde it into a big pile.

In fact, assuming you start with a d12 bag of coins, here are the expected number of purchases you can make if you...

Leave it as a d12 - 13.33 purchases
Split into two d10s - 18.66 purchases
Split into four d8s - 24 purchases
Split into 8 d6s - 26.66 purchases
split into 16 d4s - 21.33 purchases

I could have a major problem in the logic or math above, in which case please call me out on it.

I suspect that maybe the idea is that by combining bags you reduce the encumbrance (if I am following the rule correctly) but reduce purchases, but clever players will be inclined to try to split the bags up immediately before purchase. 8 d6s buys TWICE as much stuff, on average, as a single d12.

I'm not sure if there is any "solution" to this, because as I stated before, the splitting/combining rule is very elegant at the moment, and all the potential changes I can think of make it more complicated in ways that don't seem worth the cost to me. But its worth remembering that, the way the rule is written at the moment there is a strong incentive for the players to combine bags as much as possible while outside of town (to reduce encumbrance) and then to split bags as much as possible while in town on a shopping spree. Maybe force the players to combine up all of their party/personal cash into the highest levels possible prior to the shopping spree? At a minimum, I think GM's should be cautious about allowing splitting of coin bags when there is not a clear in-game reason for the split.

Again, this only applies to coins, because only coins can be freely split and combined. You can't split/combine your beef jerky rations or your armor or your ammo.