A Unifying Hierarchy of Valuations with Complements and Substitutes

We introduce a new hierarchy over monotone set functions, that we refer to as $\mathcal{MPH}$ (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, $\mathcal{MPH}$-$m$ (where $m$ is the total number of items) captures all monotone functions. The lowest level, $\mathcal{MPH}$-$1$, captures all monotone submodular functions, and more generally, the class of functions known as $\mathcal{XOS}$. Every monotone function that has a positive hypergraph representation of rank $k$ (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in $\mathcal{MPH}$-$k$. Every monotone function that has supermodular degree $k$ (in the sense defined by Feige and Izsak [ITCS 2013]) is in $\mathcal{MPH}$-$(k+1)$. In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of $\mathcal{MPH}$-$k$. One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the $\mathcal{MPH}$ hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of $k+1$ if all players hold valuation functions in $\mathcal{MPH}$-$k$. The other is an upper bound of $2k$ on the price of anarchy of simultaneous first price auctions. Being in $\mathcal{MPH}$-$k$ can be shown to involve two requirements -- one is monotonicity and the other is a certain requirement that we refer to as $\mathcal{PLE}$ (Positive Lower Envelope). Removing the monotonicity requirement, one obtains the $\mathcal{PLE}$ hierarchy over all non-negative set functions (whether monotone or not), which can be fertile ground for further research

Simple and Approximately Optimal Pricing for Proportional Complementarities

We study a new model of complementary valuations, which we call "proportional complementarities." In contrast to common models, such as hypergraphic valuations, in our model, we do not assume that the extra value derived from owning a set of items is independent of the buyer's base valuations for the items. Instead, we model the complementarities as proportional to the buyer's base valuations, and these proportionalities are known market parameters. Our goal is to design a simple pricing scheme that, for a single buyer with proportional complementarities, yields approximately optimal revenue. We define a new class of mechanisms where some number of items are given away for free, and the remaining items are sold separately at inflated prices. We find that the better of such a mechanism and selling the grand bundle earns a 12-approximation to the optimal revenue for pairwise proportional complementarities. This confirms the intuition that items should not be sold completely separately in the presence of complementarities. In the more general case, a buyer has a maximum of proportional positive hypergraphic valuations, where a hyperedge in a given hypergraph describes the boost to the buyer's value for item $i$ given by owning any set of items $T$ in addition. The maximum-out-degree of such a hypergraph is $d$, and $k$ is the positive rank of the hypergraph. For valuations given by these parameters, our simple pricing scheme is an $O(\min\{d,k\})$-approximation.Comment: Appeared in the 2019 ACM Conference on Economics and Computation (ACM EC '19

Improved Prophet Inequalities for Combinatorial Welfare Maximization with (Approximately) Subadditive Agents

m-1} measures the maximum number of items that complement each other, and (3) as a byproduct, an O(1)-competitive prophet inequality for submodular or fractionally subadditive (a.k.a. XOS) agents, matching the optimal ratio asymptotically. Our framework is computationally efficient given sample access to the prior and demand queries

Capturing Complementarity in Set Functions by Going Beyond Submodularity/Subadditivity

We introduce two new "degree of complementarity" measures: supermodular width and superadditive width. Both are formulated based on natural witnesses of complementarity. We show that both measures are robust by proving that they, respectively, characterize the gap of monotone set functions from being submodular and subadditive. Thus, they define two new hierarchies over monotone set functions, which we will refer to as Supermodular Width (SMW) hierarchy and Superadditive Width (SAW) hierarchy, with foundations - i.e. level 0 of the hierarchies - resting exactly on submodular and subadditive functions, respectively. We present a comprehensive comparative analysis of the SMW hierarchy and the Supermodular Degree (SD) hierarchy, defined by Feige and Izsak. We prove that the SMW hierarchy is strictly more expressive than the SD hierarchy: Every monotone set function of supermodular degree d has supermodular width at most d, and there exists a supermodular-width-1 function over a ground set of m elements whose supermodular degree is m-1. We show that previous results regarding approximation guarantees for welfare and constrained maximization as well as regarding the Price of Anarchy (PoA) of simple auctions can be extended without any loss from the supermodular degree to the supermodular width. We also establish almost matching information-theoretical lower bounds for these two well-studied fundamental maximization problems over set functions. The combination of these approximation and hardness results illustrate that the SMW hierarchy provides not only a natural notion of complementarity, but also an accurate characterization of "near submodularity" needed for maximization approximation. While SD and SMW hierarchies support nontrivial bounds on the PoA of simple auctions, we show that our SAW hierarchy seems to capture more intrinsic properties needed to realize the efficiency of simple auctions. So far, the SAW hierarchy provides the best dependency for the PoA of Single-bid Auction, and is nearly as competitive as the Maximum over Positive Hypergraphs (MPH) hierarchy for Simultaneous Item First Price Auction (SIA). We also provide almost tight lower bounds for the PoA of both auctions with respect to the SAW hierarchy

When Are Welfare Guarantees Robust?

Computational and economic results suggest that social welfare maximization and combinatorial auction design are much easier when bidders\u27 valuations satisfy the "gross substitutes" condition. The goal of this paper is to evaluate rigorously the folklore belief that the main take-aways from these results remain valid in settings where the gross substitutes condition holds only approximately. We show that for valuations that pointwise approximate a gross substitutes valuation (in fact even a linear valuation), optimal social welfare cannot be approximated to within a subpolynomial factor and demand oracles cannot be simulated using a subexponential number of value queries. We then provide several positive results by imposing additional structure on the valuations (beyond gross substitutes), using a more stringent notion of approximation, and/or using more powerful oracle access to the valuations. For example, we prove that the performance of the greedy algorithm degrades gracefully for near-linear valuations with approximately decreasing marginal values; that with demand queries, approximate welfare guarantees for XOS valuations degrade gracefully for valuations that are pointwise close to XOS; and that the performance of the Kelso-Crawford auction degrades gracefully for valuations that are close to various subclasses of gross substitutes valuations

Simple Mechanisms For Agents With Complements

We study the efficiency of simple auctions in the presence of complements. [DMSW15] introduced the single-bid auction, and showed that it has a price of anarchy (PoA) of $O(\log m)$ for complement-free (i.e., subadditive) valuations. Prior to our work, no non-trivial upper bound on the PoA of single bid auctions was known for valuations exhibiting complements. We introduce a hierarchy over valuations, where levels of the hierarchy correspond to the degree of complementarity, and the PoA of the single bid auction degrades gracefully with the level of the hierarchy. This hierarchy is a refinement of the Maximum over Positive Hypergraphs (MPH) hierarchy [FFIILS15], where the degree of complementarity $d$ is captured by the maximum number of neighbors of a node in the positive hypergraph representation. We show that the price of anarchy of the single bid auction for valuations of level $d$ of the hierarchy is $O(d^2 \log(m/d))$, where $m$ is the number of items. We also establish an improved upper bound of $O(d \log m)$ for a subclass where every hyperedge in the positive hypergraph representation is of size at most 2 (but the degree is still $d$). Finally, we show that randomizing between the single bid auction and the grand bundle auction has a price of anarchy of at most $O(\sqrt{m})$ for general valuations. All of our results are derived via the smoothness framework, thus extend to coarse-correlated equilibria and to Bayes Nash equilibria.Comment: Proceedings of the 2016 ACM Conference on Economics and Computatio

Weakly Submodular Functions

Submodular functions are well-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call {\em weakly submodular functions}. Our extension includes some (mildly) supermodular functions. We show that several natural functions belong to this class and relate our class to some other recent submodular function extensions. We consider the optimization problem of maximizing a weakly submodular function subject to uniform and general matroid constraints. For a uniform matroid constraint, the "standard greedy algorithm" achieves a constant approximation ratio where the constant (experimentally) converges to 5.95 as the cardinality constraint increases. For a general matroid constraint, a simple local search algorithm achieves a constant approximation ratio where the constant (analytically) converges to 10.22 as the rank of the matroid increases