Bayesian Combinatorial Auctions

We study the following simple Bayesian auction setting: m items are sold to n selfish bidders in m independent second-price auctions. Each bidder has a private valuation function that specifies his or her complex preferences over all subsets of items. Bidders only have beliefs about the valuation functions of the other bidders, in the form of probability distributions. The objective is to allocate the items to the bidders in a way that provides a good approximation to the optimal social welfare value. We show that if bidders have submodular or, more generally, fractionally subadditive (aka XOS) valuation functions, every Bayes-Nash equilibrium of the resulting game provides a 2-approximation to the optimal social welfare. Moreover, we show that in the full-information game, a pure Nash always exists and can be found in time that is polynomial in both m and n . </jats:p

Economic Efficiency Requires Interaction

We study the necessity of interaction between individuals for obtaining approximately efficient allocations. The role of interaction in markets has received significant attention in economic thinking, e.g. in Hayek's 1945 classic paper. We consider this problem in the framework of simultaneous communication complexity. We analyze the amount of simultaneous communication required for achieving an approximately efficient allocation. In particular, we consider two settings: combinatorial auctions with unit demand bidders (bipartite matching) and combinatorial auctions with subadditive bidders. For both settings we first show that non-interactive systems have enormous communication costs relative to interactive ones. On the other hand, we show that limited interaction enables us to find approximately efficient allocations

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

Draft Auctions

We introduce draft auctions, which is a sequential auction format where at each iteration players bid for the right to buy items at a fixed price. We show that draft auctions offer an exponential improvement in social welfare at equilibrium over sequential item auctions where predetermined items are auctioned at each time step. Specifically, we show that for any subadditive valuation the social welfare at equilibrium is an $O(\log^2(m))$-approximation to the optimal social welfare, where $m$ is the number of items. We also provide tighter approximation results for several subclasses. Our welfare guarantees hold for Bayes-Nash equilibria and for no-regret learning outcomes, via the smooth-mechanism framework. Of independent interest, our techniques show that in a combinatorial auction setting, efficiency guarantees of a mechanism via smoothness for a very restricted class of cardinality valuations, extend with a small degradation, to subadditive valuations, the largest complement-free class of valuations. Variants of draft auctions have been used in practice and have been experimentally shown to outperform other auctions. Our results provide a theoretical justification

Combinatorial Auctions Do Need Modest Interaction

We study the necessity of interaction for obtaining efficient allocations in subadditive combinatorial auctions. This problem was originally introduced by Dobzinski, Nisan, and Oren (STOC'14) as the following simple market scenario: $m$ items are to be allocated among $n$ bidders in a distributed setting where bidders valuations are private and hence communication is needed to obtain an efficient allocation. The communication happens in rounds: in each round, each bidder, simultaneously with others, broadcasts a message to all parties involved and the central planner computes an allocation solely based on the communicated messages. Dobzinski et.al. showed that no non-interactive ($1$-round) protocol with polynomial communication (in the number of items and bidders) can achieve approximation ratio better than $\Omega(m^{{1}/{4}})$, while for any $r \geq 1$, there exists $r$-round protocols that achieve $\widetilde{O}(r \cdot m^{{1}/{r+1}})$ approximation with polynomial communication; in particular, $O(\log{m})$ rounds of interaction suffice to obtain an (almost) efficient allocation. A natural question at this point is to identify the "right" level of interaction (i.e., number of rounds) necessary to obtain an efficient allocation. In this paper, we resolve this question by providing an almost tight round-approximation tradeoff for this problem: we show that for any $r \geq 1$, any $r$-round protocol that uses polynomial communication can only approximate the social welfare up to a factor of $\Omega(\frac{1}{r} \cdot m^{{1}/{2r+1}})$. This in particular implies that $\Omega(\frac{\log{m}}{\log\log{m}})$ rounds of interaction are necessary for obtaining any efficient allocation in these markets. Our work builds on the recent multi-party round-elimination technique of Alon, Nisan, Raz, and Weinstein (FOCS'15) and settles an open question posed by Dobzinski et.al. and Alon et. al