Auctioning a cake: Truthful auctions of heterogeneous divisible goods

We consider the problem of auctioning a one-dimensional continuously-divisible heterogeneous good (a.k.a. "the cake") among multiple agents. Applications include auctioning of time intervals, e.g. auctioning time for usage of a shared device, auctioning TV commercial slots, and more. Different agents may have different valuations for the different possible intervals, and 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 which slice. The associated computational problem is NP-hard even under very restrictive assumptions. 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. For this setting we provide a truthful auctioning mechanism that approximates the optimal welfare to within a log m factor. The mechanism works for arbitrary monotone valuations. In the continuous setting we are given a continuous, infinitely divisible interval, and the auction must allocate each agent a sub-interval. The agents' valuations are non-atomic measures on the interval. For this setting we provide a truthful auctioning mechanism that approximates the optimal welfare to within a O(logn) factor (where n is the number of agents). Additionally, we provide a truthful 2-approximation mechanism for the case that all slices must be of some fixed size.