Selfish Knapsack

Abstract

We consider a strategic variant of the knapsack problem: the items are owned by agents, and agents can misrepresent their sets of items---either by hiding items (understating), or by reporting fake ones (overstating). Each agent's utility equals the total value of her items included in the knapsack. We wish to maximize social welfare, and attempt to design mechanisms that lead to small worst-case approximation ratios at equilibrium. We provide a randomized mechanism with attractive strategic properties: it has a price of anarchy of 2 for Bayes-Nash and coarse correlated equilibria. For overstating-only agents, it becomes strategyproof, and has a matching lower bound. For the case of two understating-only agents, we provide a specialized randomized strategyproof 1.522-approximate mechanism, and a lower bound of 1.09. When all agents but one are honest, we provide a deterministic strategyproof 1.618-approximate mechanism with a matching lower bound. The latter two mechanisms are also useful in problems beyond the one in consideration.