Improved approximation algorithms for a bilevel knapsack problem

We study the Stackelberg/bilevel knapsack problem as proposed by Chen and Zhang [1]: Consider two agents, a leader and a follower. Each has his own knapsack. (Knapsack capacities are possibly different.) As usual, there is a set of items $i=1,...,n$ of given weights $w_i$ and profits $p_i$. It is allowed to pack item $i$ into both knapsacks, but in this case the corresponding profit for each player becomes $p_i+a_i$, where $a_i$ is a given (positive or negative) number. The objective is to find a packing for the leader such that the total profit of the two knapsacks is maximized, assuming that the follower acts selfishly. We present tight approximation algorithms for all settings considered in [1].