Computing Socially-Efficient Cake Divisions

We consider a setting in which a single divisible good ("cake") needs to be divided between n players, each with a possibly different valuation function over pieces of the cake. For this setting, we address the problem of finding divisions that maximize the social welfare, focusing on divisions where each player needs to get one contiguous piece of the cake. We show that for both the utilitarian and the egalitarian social welfare functions it is NP-hard to find the optimal division. For the utilitarian welfare, we provide a constant factor approximation algorithm, and prove that no FPTAS is possible unless P=NP. For egalitarian welfare, we prove that it is NP-hard to approximate the optimum to any factor smaller than 2. For the case where the number of players is small, we provide an FPT (fixed parameter tractable) FPTAS for both the utilitarian and the egalitarian welfare objectives

Double Auctions in Markets for Multiple Kinds of Goods

Motivated by applications such as stock exchanges and spectrum auctions, there is a growing interest in mechanisms for arranging trade in two-sided markets. Existing mechanisms are either not truthful, or do not guarantee an asymptotically-optimal gain-from-trade, or rely on a prior on the traders' valuations, or operate in limited settings such as a single kind of good. We extend the random market-halving technique used in earlier works to markets with multiple kinds of goods, where traders have gross-substitute valuations. We present MIDA: a Multi Item-kind Double-Auction mechanism. It is prior-free, truthful, strongly-budget-balanced, and guarantees near-optimal gain from trade when market sizes of all goods grow to $\infty$ at a similar rate.Comment: Full version of IJCAI-18 paper, with 2 figures. Previous names: "MIDA: A Multi Item-type Double-Auction Mechanism", "A Random-Sampling Double-Auction Mechanism". 10 page

Leximin Approximation: From Single-Objective to Multi-Objective

Leximin is a common approach to multi-objective optimization, frequently employed in fair division applications. In leximin optimization, one first aims to maximize the smallest objective value; subject to this, one maximizes the second-smallest objective; and so on. Often, even the single-objective problem of maximizing the smallest value cannot be solved accurately. What can we hope to accomplish for leximin optimization in this situation? Recently, Henzinger et al. (2022) defined a notion of \emph{approximate} leximin optimality. Their definition, however, considers only an additive approximation. In this work, we first define the notion of approximate leximin optimality, allowing both multiplicative and additive errors. We then show how to compute, in polynomial time, such an approximate leximin solution, using an oracle that finds an approximation to a single-objective problem. The approximation factors of the algorithms are closely related: an $(\alpha,\epsilon)$-approximation for the single-objective problem (where $\alpha \in (0,1]$ and $\epsilon \geq 0$ are the multiplicative and additive factors respectively) translates into an $\left(\frac{\alpha^2}{1-\alpha + \alpha^2}, \frac{\epsilon}{1-\alpha +\alpha^2}\right)$-approximation for the multi-objective leximin problem, regardless of the number of objectives. Finally, we apply our algorithm to obtain an approximate leximin solution for the problem of \emph{stochastic allocations of indivisible goods}. For this problem, assuming sub-modular objectives functions, the single-objective egalitarian welfare can be approximated, with only a multiplicative error, to an optimal $1-\frac{1}{e}\approx 0.632$ factor w.h.p. We show how to extend the approximation to leximin, over all the objective functions, to a multiplicative factor of $\frac{(e-1)^2}{e^2-e+1} \approx 0.52$ w.h.p or $\frac{1}{3}$ deterministically