Assume we have some currency where $n$ is the value of the smallest bill and all the coins have a value in $C\subseteq[n]:=\{0,\dots,n-1\}$. Assume further that for each $a\in\mathbf{N}$ there is a unique decomposition into a minimal amount of coins, so e.g. not $1,2,3,4\in C$.
Now assume that the change I receive from a shop always assumes this smallest decomposition, e.g. I'd never receive 57 single pennies in the US or cents in the EU. I ignore bill transactions and assume modulo $n$ arithmetic, i.e. I have to pay the price $a \in [n]:=\{0,\dots,n-1\}$, for which I can pay any amount $b\in [n]$ in any coin configuration and receive $b-a$ if $b\ge a$ and $n+b-a$ else.
In an alternative formulation of the problem I can pay any amount I want (even larger than $n$) but then the question is, if $n=100$, $a=1$ and $b=500$, do I get $499$ or $99$ back? It seems to make the math more complicated in any way.
So a coin selection strategy is a map from multisets $m_C$ over $C$ and $a\in [n]$ to submultisets of $m_C$, i.e. I can only select coins which are already in my wallet. If $a$ in uniquely distributed in $[n]$ then for every fixed coin selection strategy I get a Markov chain where the nodes are multisets over $C$ (or wallet configurations).
My questions:
Can I always limit my reachable wallet configurations to a finite set if I choose the right strategy (e.g. never having more than 100 coins)?
If this is the case, what strategy minimizes the expected amount of coins in my wallet in the limit when I start with $0$ coins?
Is there a case where selecting a greedy strategy minimizing the amount of coins after each transaction does not yield the best overall strategy?
For fixed $n$, how should $C$ look like to minimize the expected amount of coins in the wallet under the optimal coin selection strategy?