David Kurokawa

The Unreasonable Fairness of Maximum Nash Welfare

The maximum Nash welfare (MNW) solution --- which selects an allocation that maximizes the product of utilities --- is known to provide outstanding fairness guarantees when allocating divisible goods. And while it seems to lose its luster when applied to indivisible goods, we show that, in fact, the MNW solution is unexpectedly, strikingly fair even in that setting. In particular, we prove that it selects allocations that are envy free up to one good --- a compelling notion that is quite elusive when coupled with economic efficiency. We also establish that the MNW solution provides a good approximation to another popular (yet possibly infeasible) fairness property, the maximin share guarantee, in theory and --- even more so --- in practice. While finding the MNW solution is computationally hard, we develop a nontrivial implementation, and demonstrate that it scales well on real data. These results lead us to believe that MNW is the ultimate solution for allocating indivisible goods, and underlie its deployment on a popular fair division website.

Optimising trade-offs among stakeholders in ad auctions

We examine trade-offs among stakeholders in ad auctions. Our metrics are the revenue for the utility of the auctioneer, the number of clicks for the utility of the users and the welfare for the utility of the advertisers. We show how to optimize linear combinations of the stakeholder utilities, showing that these can be tackled through a GSP auction with a per-click reserve price. We then examine constrained optimization of stakeholder utilities. We use simulations and analysis of real-world sponsored search auction data to demonstrate the feasible trade-offs, examining the effect of changing the allowed number of ads on the utilities of the stakeholders. We investigate both short term effects, when the players do not have the time to modify their behavior, and long term equilibrium conditions. Finally, we examine a combinatorially richer constrained optimization problem, where there are several possible allowed configurations (templates) of ad formats. This model captures richer ad formats, which allow using the available screen real estate in various ways. We show that two natural generalizations of the GSP auction rules to this domain are poorly behaved, resulting in not having a symmetric Nash equilibrium or having one with poor welfare. We also provide positive results for restricted cases.

Leximin Allocations in the Real World

As part of a collaboration with a major California school district, we study the problem of fairly allocating unused classrooms in public schools to charter schools. Our approach revolves around the randomized leximin mechanism. We extend previous work to the classroom allocation setting, showing that the leximin mechanism is proportional, envy-free, efficient, and group strategyproof. We also prove that the leximin mechanism provides a (worst-case) 4-approximation to the maximum number of classrooms that can possibly be allocated. Our experiments, which are based on real data, show that a nontrivial implementation of the leximin mechanism scales gracefully in terms of running time (even though the problem is intractable in theory), and performs extremely well with respect to a number of efficiency objectives. We take great pains to establish the practicability of our approach, and discuss issues related to its deployment.

A Note on the Convergence of SOR for the PageRank Problem

A curious phenomenon when it comes to solving the linear system formulation of the PageRank problem is that while the convergence rate of Gauss-Seidel shows an improvement over Jacobi by a factor of approximately two, successive overrelaxation (SOR) does not seem to offer a meaningful improvement over Gauss-Seidel. This has been observed experimentally and noted in the literature, but to the best of our knowledge there has been no analytical explanation for this thus far. This convergence behavior is surprising because there are classes of matrices for which Gauss-Seidel is faster than Jacobi by a similar factor of two, and SOR accelerates convergence by an order of magnitude compared to Gauss-Seidel. In this short paper we prove analytically that the PageRank model has the unique feature that there exist PageRank linear systems for which SOR does not converge outside a very narrow interval depending on the damping factor, and that in such situations Gauss-Seidel may be the best choice among the relaxation parameters. Conversely, we show that within that narrow interval, there exists no PageRank problem for which SOR does not converge. Our result may give an analytical justification for the popularity of Gauss-Seidel as a solver for the linear system formulation of PageRank.

Fair Enough: Guaranteeing Approximate Maximin Shares

We consider the problem of fairly allocating indivisible goods, focusing on a recently introduced notion of fairness called maximin share guarantee: each player’s value for his allocation should be at least as high as what he can guarantee by dividing the items into as many bundles as there are players and receiving his least desirable bundle. Assuming additive valuation functions, we show that such allocations may not exist, but allocations guaranteeing each player 2/3 of the above value always exist. These theoretical results have direct practical implications.