A Simplified Analysis of the Ascending Auction to Sell a Matroid Base

Abstract

We give a simpler analysis of the ascending auction of Bikhchandani, de Vries, Schummer, and Vohra to sell a welfare-maximizing base of a matroid at Vickrey prices. The new proofs for economic efficiency and the charge of Vickrey prices only require a few matroid folklore theorems, therefore shortening the analysis of the design goals of the auction significantly.

Figures (4)

• Figure 1: Example iteration on a graphic matroid with four buyers 1 (black, solid), 2 (red, dashed), 3 (blue, dotted), 4 (green, dotdashed): For $p=0$ and $p=1$ there are no critical items. When $p$ increases to $2$ multiple items get critical and trigger some sales, including some of critical items.
• Figure 2: Sketch of the proxy auction.
• Figure 3: At price $p = 2$ two items are (truthfully announced) as critical: one by buyer 1 (black, solid), one by buyer 2 (red, dashed). If buyer 2's item gets deleted first by the auctioneer, player 1 has a monopsony and could now announce her critical item as best, which would be inconsistent with the information that her other item is not critical.
• Figure 4: Graphic matroid with two buyers 1 (red, dashed) and 2 (blue, dotted) where signaling truthfully is not a dominant strategy for buyer 1.

Theorems & Definitions (14)

• Example
• Lemma 0: Cocircuits after deletion
• Lemma 0: Cocircuits before deletion
• Lemma 0: Augmentation by cocircuit, dawson1980optimal
• Theorem 1: Theorem 13 in bikhchandani2011ascending
• proof
• Theorem 2: Theorem 17 in bikhchandani2011ascending
• Lemma 2: Cocircuits after deletion
• proof
• Lemma 2: Cocircuits before deletion