I recently learned about the Hungarian algorithm for maximum-weight matching in bipartite graphs and the "market" interpretation of the primal and dual LPs. (See also these notes.)
The setup: We have a bipartite graph $G = (L,R,E)$ with nonnegative edge-weights, and we want to find a matching $M$ maximizing $\sum_{i \sim j \in M} v_{ij}$, where $v_{ij}$ is the edge weight. We can write this as an LP, which turns out to be integral, and its dual is to find $\vec{p} \in \mathbb{R}_{\geq 0}^L$ minimizing $\sum_i p_i + \sum_{j} u_j(\vec{p})$ where we define $u_j(\vec{p}) = \min_i (v_{ij} - p_i)$.
These both have a "market" interpretation. We can view $L$ as items and $R$ as buyers. In the primal problem, we want to match each buyer to a unique item; $v_{ij}$ represents how much buyer $j$ wants item $i$; and the objective $\sum_{i\sim j\in M} v_{ij}$ is the "social welfare", representing how "collectively happy" the buyers are with the matching. In the dual problem, we view $p_i$ as a "price" and $u_j(\vec{p})$ as the "utility" of buyer $j$ when buying their most preferred item.
By duality, any setting of $\vec{p}$ witnesses an upper bound on the best possible matching. But from the market perspective, I don't understand why this should be the case. It means that we can prove that the buyers cannot collectively be very happy (in the primal) by imagining a "hypothetical scenario" where the items are assigned prices (in the dual). In this hypothetical scenario, the buyers can behave selfishly (i.e., they take their most preferred item), and the objective contains a $\sum_i p_i$ term, which seems to correspond to a "seller" who pays the price $p_i$ exactly once per item, regardless of which the buyers actually want to buy.
Is there an example (maybe even an "everyday life" one), or any sort of reasoning, that shows why this makes any sense? My only intuition is that in the primal buyers may have to be selfless, while in the dual they can be selfish.