I want to solve a optimization problem, which mimics the actions between a seller and several buyers. A seller has several goods, 1, 2, ... J, with prices $p_j$ and quantity $q_j$. A buyer can only buy one merchandise and he wants to buy the merchandise which he considers to be the best among all of them. If we use variable $x_{ij}=1$ to indicate whether buyer i wants to buy merchandise j, and $x_{ij}=0$ otherwise. The strategy for buyer i is simple:
$j^* = arg~max~(b_j-p_j)$, if $max_j~(b_j-p_j) \geq 0$, where $b_j$ is the evaluation of product $j$.
However, the seller wants to maximize his revenue by solving:
max $\sum_i \sum_j x_{ij}p_j$
subject to: $\sum_i x_{ij} \leq q_j$
Suppose that the seller doesn't know $b_j$ of each buyer. How can the seller dynamically adjust the price $p_j$ so that his total revenue can be maximized?