Algorithmic Pricing for the Partial Assignment

A seller has an items set \(M=\left\{ 1,2,\dots ,m\right\} \) and the amount of each item is 1. In the buyer set \(N=\{1, 2, ..., n\}\), each buyer i has an interested bundle \(B_i\subseteq M\), a valuation \(v_i\) on \(B_i\), and a set of budgets \(e_{ij}\) on each item j. The seller knows the whole information of the buyers and decides the price for each item so as to maximize the social welfare. Buyers come in an arbitrary order. When a buyer comes, each item can be only sold integrally. In previous works, if a buyer’s interested bundle does not exist, he will buy nothing and cannot be satisfied. In this paper, we consider the buyer can be partially satisfied. If some items in \(B_i\) are sold out on the arrival of buyer i, he might be partially satisfied by buying a subset of \(B_i\) without violating the budget condition. We first show that in this new model, achieving the maximum social welfare is NP-hard. Moreover, an optimal assignment oracle can be used to achieve the maximum social welfare. We then analyze two pricing schemes to approximate the optimal social welfare. In the trade-off pricing scheme, an O(k)-approximation algorithm can be achieved if considering the subset of the bundle, where k is the maximal cardinality of the assigned bundle among all buyers. In the item-set pricing scheme, when \(d=\dfrac{1}{2}|x_{iB}^*|\), the social welfare can be approximated within the factor 2, moreover, compared with the traditional (non-partial assignment) setting, the social welfare can be increased by at least \(\sum _{S_{i}}\dfrac{d}{\left| x_{iB}^*\right| }\sum _{j\in S_{i}}e_{ij}^{'}\), where \(S_i\) is the buyer subset containing item j, \(x_{iB}^*\) represents the ith optimal assignment subset’s complete set, i.e, if the optimal assignment item set is one buyer bundle \(B_{i}\)’s subset, then \(x_{iB}^*=B_{i}\), \(x_{iB}\) is the intersection of \(B_{i}\) and the optimal assignment item set S, d is the minimum of \(\left| x_{iB}\right| \) and \(\left| x_{iB}^*\right| -\left| x_{iB}\right| \), and \(e_{ij}^{'}\) is the optimal assignment item’s budget.