Auctioning time: Truthful auctions of heterogeneous divisible goods

We consider the problem of auctioning time - a one-dimensional continuously-divisible heterogeneous good - among multiple agents. Applications include auctioning time for using a shared device, auctioning TV commercial slots, and more. Different agents may have different valuations for the different possible intervals; the goal is to maximize the aggregate utility. Agents are self-interested and may misrepresent their true valuation functions if this benefits them. Thus, we seek auctions that are truthful. Considering the case that each agent may obtain a single interval, the challenge is twofold, as we need to determine both where to slice the interval, and who gets what slice. We consider two settings: discrete and continuous. In the discrete setting, we are given a sequence of m indivisible elements (e₁, ⋯, e_m), and the auction must allocate each agent a consecutive subsequence of the elements. In the continuous setting, we are given a continuous, infinitely divisible interval, and the auction must allocate each agent a subinterval. The agents' valuations are nonatomic measures on the interval. We show that, for both settings, the associated computational problem is NP-complete even under very restrictive assumptions. Hence, we provide approximation algorithms. For the discrete case, we provide a truthful auctioning mechanism that approximates the optimal welfare to within a logmfactor. The mechanism works for arbitrary monotone valuations. For the continuous setting, we provide a truthful auctioning mechanism that approximates the optimal welfare to within an O(log n) factor (where n is the number of agents). Additionally, we provide a truthful 2-approximation mechanism for the case that all pieces must be of some fixed size.