Network assessment fee is a US-flavoured name for the charges covered under scheme fees (assessment fees), which explains how these fees are structured and how they reach merchants. This page explains the wording.
A network assessment fee is the fee a card network charges its member banks for access to its brand, rules and processing systems. On the acquiring side, the network "assesses" the acquirer a small percentage of the card volume it processes, often alongside fixed per-transaction fees for authorisation, clearing and cross-border activity. Issuers pay their own network fees too. Unlike interchange, which flows from the acquirer to the card issuer, assessment fees are revenue for the network itself, such as Visa, Mastercard or a domestic scheme.
In the United States, merchant statements traditionally list these charges as "assessments", so "network assessment fee" is common in American pricing discussions, while Europeans usually say "scheme fees". Merchants on interchange-plus pricing see them as a separate pass-through line; merchants on blended pricing see them folded into one rate, the merchant discount rate. A breakdown of how the pieces fit together is given under merchant discount rate components.