Calculating net value after a series of transactions — family of problems Is there a generic name for the type of problem which involves transactions between multiple containers with ownership of the containers and ownership of transactions by multiple individuals?
Here is a word description of the problem with a simple image:
Bob and Alice have bank accounts with 100% ownership, and a third account with joint ownership at 50% each.

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*Initially, Alice has 110 in her account, and Bob has 90 in his account.


*Alice contributes 110 to the joint account, and Bob contributes 90 to the joint account. Bob and Alice have a net value of 100 each.


*Bob makes a transaction of 50 from Bob's account (container 2) to another person (container 4). Bob has a net value of -50 + 50% * (200) = -50 + 100 = 50


*A transaction of 50 is made from the joint account (container 1) to Bob's account (container 2). Bob and Alice now both have a net value of 75 each.
Suppose Alice "owns" 100% of the 50 transaction out of Bob's account in step 3, even though it passed through Bob's container 2. Thus, Alice agrees her net value should be 50, while Bob's should be 100. How should accounts be settled given that we know how the transaction occurred in step 1-4, and that the transaction history cannot be changed, and that Bob's final net value should match the net value of Bob and Alice from an arbitrary earlier step (step 2 in this case, but it could be step 1 if we wish).
In this example, 50 should be transferred from container 1 to container 2, which gives a net value of { Alice : 50, Bob : 100 }. But what if there were more containers, and more players? What if ownership of the transaction in any of the steps were jointly owned by multiple people (many to one)?

 A: Let state $x_i$ denote the amount in the $i$-th container. Hence, after all transactions, the state vector is
$$ {\bf x} = \begin{bmatrix} 0 \\ 90 \\ 110 \\ 0 \end{bmatrix} + 90 \begin{bmatrix} +1 \\ -1 \\ 0 \\ 0 \end{bmatrix} + 110 \begin{bmatrix} +1 \\ 0 \\ -1 \\ 0 \end{bmatrix} + 50 \begin{bmatrix} -1 \\ +1 \\ 0 \\ 0 \end{bmatrix} + 50 \begin{bmatrix} 0 \\ -1 \\ 0 \\ +1 \end{bmatrix} = \begin{bmatrix} 150 \\ 0 \\ 0 \\ 50 \end{bmatrix} $$
After all these transactions, the amounts owned by Alice and Bob are a linear function of the state vector
$$ {\bf W} {\bf x} = \begin{bmatrix} \frac12 & 0 & 1 & 0 \\ \frac12 & 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} 150 \\ 0 \\ 0 \\ 50 \end{bmatrix} = \begin{bmatrix} 75 \\ 75 \end{bmatrix} $$
If transactions between any two containers are allowed, then we have a directed complete graph on $4$ nodes. This digraph has $\frac{4!}{2!} = 12$ (directed) edges. Let $\bf C$ denote the corresponding $12 \times 4$ signed incidence matrix. Hence, a transaction corresponds to updating the state vector as follows
$$ {\bf x}_{k+1} = {\bf x}_k + u_k {\bf C}^\top {\bf e}_j$$
where $u_k$ is the amount transferred and $j$ is the chosen edge (which can be translated to a pair of nodes $(i,j)$ where $i$'s account is debited and $j$'s account is credited).
