Chore division on a graph

The paper considers fair allocation of indivisible nondisposable items that generate disutility (chores). We assume that these items are placed in the vertices of a graph and each agent's share has to form a connected subgraph of this graph. Although a similar model has been investigated before for goods, we show that the goods and chores settings are inherently different. In particular, it is impossible to derive the solution of the chores instance from the solution of its naturally associated fair division instance. We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based. We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies

The Value of Knowing Your Enemy

Many auction settings implicitly or explicitly require that bidders are treated equally ex-ante. This may be because discrimination is philosophically or legally impermissible, or because it is practically difficult to implement or impossible to enforce. We study so-called {\em anonymous} auctions to understand the revenue tradeoffs and to develop simple anonymous auctions that are approximately optimal. We consider digital goods settings and show that the optimal anonymous, dominant strategy incentive compatible auction has an intuitive structure --- imagine that bidders are randomly permuted before the auction, then infer a posterior belief about bidder i's valuation from the values of other bidders and set a posted price that maximizes revenue given this posterior. We prove that no anonymous mechanism can guarantee an approximation better than O(n) to the optimal revenue in the worst case (or O(log n) for regular distributions) and that even posted price mechanisms match those guarantees. Understanding that the real power of anonymous mechanisms comes when the auctioneer can infer the bidder identities accurately, we show a tight O(k) approximation guarantee when each bidder can be confused with at most k "higher types". Moreover, we introduce a simple mechanism based on n target prices that is asymptotically optimal and build on this mechanism to extend our results to m-unit auctions and sponsored search

Chore division on a graph

Le PDF est une version non publiée datant de 2018.International audienceThe paper considers fair allocation of indivisible nondisposable items that generate disutility (chores). We assume that these items are placed in the vertices of a graph and each agent’s share has to form a connected subgraph of this graph. Although a similar model has been investigated before for goods, we show that the goods and chores settings are inherently different. In particular, it is impossible to derive the solution of the chores instance from the solution of its naturally associated fair division instance. We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based. We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies

Learning Reserve Prices in Second-Price Auctions

This paper proves the tight sample complexity of Second-Price Auction with Anonymous Reserve, up to a logarithmic factor, for all value distribution families that have been considered in the literature. Compared to Myerson Auction, whose sample complexity was settled very recently in (Guo, Huang and Zhang, STOC 2019), Anonymous Reserve requires much fewer samples for learning. We follow a similar framework as the Guo-Huang-Zhang work, but replace their information theoretical argument with a direct proof

Learning Multi-item Auctions with (or without) Samples

We provide algorithms that learn simple auctions whose revenue is approximately optimal in multi-item multi-bidder settings, for a wide range of valuations including unit-demand, additive, constrained additive, XOS, and subadditive. We obtain our learning results in two settings. The first is the commonly studied setting where sample access to the bidders' distributions over valuations is given, for both regular distributions and arbitrary distributions with bounded support. Our algorithms require polynomially many samples in the number of items and bidders. The second is a more general max-min learning setting that we introduce, where we are given "approximate distributions," and we seek to compute an auction whose revenue is approximately optimal simultaneously for all "true distributions" that are close to the given ones. These results are more general in that they imply the sample-based results, and are also applicable in settings where we have no sample access to the underlying distributions but have estimated them indirectly via market research or by observation of previously run, potentially non-truthful auctions. Our results hold for valuation distributions satisfying the standard (and necessary) independence-across-items property. They also generalize and improve upon recent works, which have provided algorithms that learn approximately optimal auctions in more restricted settings with additive, subadditive and unit-demand valuations using sample access to distributions. We generalize these results to the complete unit-demand, additive, and XOS setting, to i.i.d. subadditive bidders, and to the max-min setting. Our results are enabled by new uniform convergence bounds for hypotheses classes under product measures. Our bounds result in exponential savings in sample complexity compared to bounds derived by bounding the VC dimension, and are of independent interest.Comment: Appears in FOCS 201