[Paper Review] Bundling Equilibrium in Combinatorial auctions

Ron Holzman, Noa Kfir-Dahav+2 more|ArXiv.org|Jan 14, 2002

Auction Theory and Applications31 references4 citations

TL;DR

This paper introduces bundling equilibrium in combinatorial auctions, where bidders truthfully report valuations only for a predefined family of bundles Σ, achieving individually rational ex post equilibrium. It characterizes all such Σ that support this equilibrium and shows that partition-based equilibria offer a favorable tradeoff between communication complexity and economic efficiency, though quasi-fields can slightly outperform them in specific cases.

ABSTRACT

This paper analyzes individually-rational ex post equilibrium in the VC (Vickrey-Clarke) combinatorial auctions. If $Σ$ is a family of bundles of goods, the organizer may restrict the participants by requiring them to submit their bids only for bundles in $Σ$. The $Σ$-VC combinatorial auctions (multi-good auctions) obtained in this way are known to be individually-rational truth-telling mechanisms. In contrast, this paper deals with non-restricted VC auctions, in which the buyers restrict themselves to bids on bundles in $Σ$, because it is rational for them to do so. That is, it may be that when the buyers report their valuation of the bundles in $Σ$, they are in an equilibrium. We fully characterize those $Σ$ that induce individually rational equilibrium in every VC auction, and we refer to the associated equilibrium as a bundling equilibrium. The number of bundles in $Σ$ represents the communication complexity of the equilibrium. A special case of bundling equilibrium is partition-based equilibrium, in which $Σ$ is a field, that is, it is generated by a partition. We analyze the tradeoff between communication complexity and economic efficiency of bundling equilibrium, focusing in particular on partition-based equilibrium.

Motivation & Objective

• To identify all families of bundles Σ for which truthful reporting is an individually rational ex post equilibrium in Vickrey-Clarke (VC) combinatorial auctions.
• To characterize the structure of Σ that induce such equilibria, termed bundling equilibria, and analyze their properties.
• To study the tradeoff between communication complexity (size of Σ) and economic efficiency (approximation ratio rΣ) in these equilibria.
• To compare the efficiency-complexity tradeoff of general bundling equilibria with that of partition-based equilibria, where Σ is generated by a partition of goods.
• To determine whether more general quasi-fields can achieve better efficiency-complexity tradeoffs than partition-based fields.

Proposed method

• The paper defines a bundling equilibrium as a strategy profile where bidders truthfully report valuations only for bundles in a family Σ, and this constitutes an individually rational ex post equilibrium.
• It characterizes the family Σ that support such equilibria using combinatorial and set-theoretic conditions, particularly requiring Σ to be a quasi-field.
• The approximation ratio rΣ is defined as the supremum of the ratio between the maximum social surplus and the surplus achievable under Σ-based allocations.
• The paper analyzes the communication complexity as |Σ|, the number of bundles in Σ, and compares it with the efficiency of the equilibrium via rΣ.
• It uses extremal set theory arguments, including double counting and inclusion-exclusion, to derive bounds on |Σ| for given rΣ.
• It constructs specific examples, such as Σ = {D ⊆ A : |D ∩ B| = |D ∩ C|} for a balanced partition (B,C), to show that rΣ = 2 and |Σ| = C(m, m/2), demonstrating a near-optimal tradeoff.

Experimental results

Research questions

• RQ1 Which families of bundles Σ induce an individually rational ex post equilibrium in VC combinatorial auctions?
• RQ2 What is the minimal communication complexity (|Σ|) required to achieve a given level of economic efficiency (rΣ) in such equilibria?
• RQ3 Can quasi-fields (a generalization of fields) achieve better efficiency-complexity tradeoffs than partition-based equilibria?
• RQ4 Is there a fundamental limit on how efficiently a given approximation ratio rΣ can be achieved in terms of |Σ|?
• RQ5 How does the structure of Σ, particularly when it is a quasi-field, affect the existence and properties of bundling equilibria?

Key findings

• A family Σ supports a bundling equilibrium if and only if it is a quasi-field, which generalizes the concept of a field of sets.
• For partition-based equilibria, the communication complexity is at least 2^{m−2} when the approximation ratio rπ ≤ 2, and this is worse than the complexity of the quasi-field construction with rΣ = 2 and |Σ| = C(m, m/2).
• When m ≥ 10, the quasi-field Σ defined by |D ∩ B| = |D ∩ C| achieves rΣ = 2 with |Σ| = C(m, m/2) < 2^{m−2}, thus outperforming any partition-based equilibrium with the same efficiency.
• Any quasi-field Σ with rΣ ≤ m/k must contain a partition of the goods into k non-empty parts, implying |Σ| ≥ 2^k, establishing a lower bound on complexity for a given efficiency level.
• The tradeoff between communication complexity and efficiency is fundamentally constrained: even with general quasi-fields, the improvement over partition-based equilibria is limited and not asymptotically significant.
• The minimal achievable rΣ for a given |Σ| is bounded below by m/k when |Σ| ≥ 2^k, showing that high efficiency requires exponentially large Σ in the number of goods.