It begins by addressing a disagreement over the probability spaces associated with feasible payments and the likelihoods for minimum cost flows, clarifying that the original assertion—that for a payment to be feasible, its MCF probability must be lower than its feasibility likelihood—might not hold under a broader consideration of probability spaces.

An error in counting feasible wealth distributions is corrected, showing that instead of eight, there are ten such distributions, which are detailed in a provided table. This correction leads to a revised calculation of the probability that a specific node can make payments to two other nodes, adjusting the figure to 4 out of 10, contrary to an earlier estimate of 3 out of 8. The discussion further examines these distributions to identify how many states allow for a certain payment configuration, concluding that four distributions meet the criteria.

A significant part of the analysis is dedicated to comparing different flows that could fulfill a payment scenario, specifically assessing their feasibility based on the number of states in which they succeed. It challenges an initial example with a counter-argument supported by diagrams and theoretical models from a research paper, which assumes liquidity is uniformly and independently distributed across each channel (link to the paper). Two potential flows are considered, and their probabilities are calculated based on this assumption, leading to the conclusion that the initially described flow indeed represents the minimum cost flow but with a success probability that deviates from the previously calculated likelihoods.

This discussion underscores the complexity of modeling payment systems in networked environments, highlighting the importance of accurately accounting for all possible states and considering various assumptions about the distribution of resources. It also exemplifies the application of theoretical models to practical scenarios, providing insights into the dynamics of financial networks through the lens of probability and cost optimization.