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The celebrated model of auctions with interdependent valuations, introduced by Milgrom and Weber in 1982, has been studied almost exclusively under private signals $$s_1, \ldots, s_n$$ of the $$n$$ bidders and public valuation functions $$v_i(s_1, \ldots, s_n)$$. Recent work in TCS has shown that this setting admits a constant approximation to the optimal social welfare if the valuations satisfy a natural property called submodularity over signals (SOS). More recently, Eden et al. (2022) have extended the analysis of interdependent valuations to include settings with private signals and \emph{private valuations}, and established $$O(\log^2 n)$$-approximation for SOS valuations. In this paper we show that this setting admits a {\em constant} factor approximation, settling the open question raised by Eden et al. (2022).
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q-Partitioning Valuations: Exploring the Space Between Subadditive and Fractionally Subadditive Valuations
{"Abstract":["For a set M of m elements, we define a decreasing chain of classes of normalized monotone-increasing valuation functions from 2^M to ℝ_{≥ 0}, parameterized by an integer q ∈ [2,m]. For a given q, we refer to the class as q-partitioning. A valuation function is subadditive if and only if it is 2-partitioning, and fractionally subadditive if and only if it is m-partitioning. Thus, our chain establishes an interpolation between subadditive and fractionally subadditive valuations. We show that this interpolation is smooth (q-partitioning valuations are "nearly" (q-1)-partitioning in a precise sense, Theorem 6), interpretable (the definition arises by analyzing the core of a cost-sharing game, à la the Bondareva-Shapley Theorem for fractionally subadditive valuations, Section 3.1), and non-trivial (the class of q-partitioning valuations is distinct for all q, Proposition 3).\r\nFor domains where provable separations exist between subadditive and fractionally subadditive, we interpolate the stronger guarantees achievable for fractionally subadditive valuations to all q ∈ {2,…, m}. Two highlights are the following:\r\n1) An Ω ((log log q)/(log log m))-competitive posted price mechanism for q-partitioning valuations. Note that this matches asymptotically the state-of-the-art for both subadditive (q = 2) [Paul Dütting et al., 2020], and fractionally subadditive (q = m) [Feldman et al., 2015]. \r\n2) Two upper-tail concentration inequalities on 1-Lipschitz, q-partitioning valuations over independent items. One extends the state-of-the-art for q = m to q < m, the other improves the state-of-the-art for q = 2 for q > 2. Our concentration inequalities imply several corollaries that interpolate between subadditive and fractionally subadditive, for example: 𝔼[v(S)] ≤ (1 + 1/log q)Median[v(S)] + O(log q). To prove this, we develop a new isoperimetric inequality using Talagrand’s method of control by q points, which may be of independent interest.\r\nWe also discuss other probabilistic inequalities and game-theoretic applications of q-partitioning valuations, and connections to subadditive MPH-k valuations [Tomer Ezra et al., 2019]."]}
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- Award ID(s):
- 1955205
- PAR ID:
- 10658521
- Editor(s):
- Censor-Hillel, Keren; Grandoni, Fabrizio; Ouaknine, Joel; Puppis, Gabriele
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 334
- ISSN:
- 1868-8969
- Page Range / eLocation ID:
- 18:1-18:20
- Subject(s) / Keyword(s):
- Subadditive Functions Fractionally Subadditive Functions Posted Price Mechanisms Concentration Inequalities Theory of computation → Algorithmic mechanism design Mathematics of computing → Probability and statistics Theory of computation → Computational pricing and auctions
- Format(s):
- Medium: X Size: 20 pages; 992243 bytes Other: application/pdf
- Size(s):
- 20 pages 992243 bytes
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/lipics.icalp.2025.18
