Private Interdependent Valuations: New Bounds for Single-Item Auctions and Matroids

TL;DR

The paper advances algorithmic mechanism design for interdependent valuations with private valuation functions. It delivers a $5$-approximate truthful mechanism for SOS valuations in single-item auctions via a novel eating mechanism analyzed with LP-duality, and a tight $(d+1)$-approximate truthful mechanism for matroid-feasible environments with private $d$-critical valuations, using shadow values and a matroid-partition approach. The results extend interdependent-valuation guarantees beyond single-item settings and include an extension to heterogeneous and unknown $d$. Together, these contributions narrow the gap to optimal welfare in private interdependent valuation models and provide tools that apply to broader combinatorial feasibility constraints, with potential impact on auction design in information-influenced environments.

Abstract

We study auction design within the widely acclaimed model of interdependent values, introduced by Milgrom and Weber [1982]. In this model, every bidder $i$ has a private signal $s_i$ for the item for sale, and a public valuation function $v_i(s_1,\ldots,s_n)$ which maps every vector of private signals (of all bidders) into a real value. A recent line of work established the existence of approximately-optimal mechanisms within this framework, even in the more challenging scenario where each bidder's valuation function $v_i$ is also private. This body of work has primarily focused on single-item auctions with two natural classes of valuations: those exhibiting submodularity over signals (SOS) and $d$-critical valuations. In this work we advance the state of the art on interdependent values with private valuation functions, with respect to both SOS and $d$-critical valuations. For SOS valuations, we devise a new mechanism that gives an improved approximation bound of $5$ for single-item auctions. This mechanism employs a novel variant of an "eating mechanism", leveraging LP-duality to achieve feasibility with reduced welfare loss. For $d$-critical valuations, we broaden the scope of existing results beyond single-item auctions, introducing a mechanism that gives a $(d+1)$-approximation for any environment with matroid feasibility constraints on the set of agents that can be simultaneously served. Notably, this approximation bound is tight, even with respect to single-item auctions.

Figures (2)

• Figure 1: The transitions from $\sigma$ to $\hat{\sigma}$ used in the proof of \ref{['lemma:before_still_selected']}.
• Figure 2: The eating process used to determine bidder $i$'s allocation. bidder $i$ starts eating at time $-\ln v_i(\mathbf{{s}})$ and all other bidders $j\neq i$ start eating at time $-\ln {v}_{j}(\mathbf{{s}}_{-i},0_{i})$. At each point in time, all bidders eating at that time eat at the same speed. The solid blue line denotes bidder $i$'s share and the dashed blue line denotes $j$'s (pretend) share. The red square denotes the time $t$ when the sum of the blue lines adds up to $1$, thus halting the eating process.

Theorems & Definitions (29)

• Definition 2.1: EPIC-IR
• Proposition 2.1: EdenGZ22
• Definition 2.2: SOS Valuations
• Lemma 2.1: SOS functions are self-bounding EdenGZ22LuSZ22
• Definition 2.3: $d$-critical Valuations
• Proposition 2.2: EdenGZ22
• Definition 2.4: Matroid
• Definition 2.5: Rank Function
• Theorem 2.1: Greedy is Optimal (Edmonds71)
• Lemma 2.2