Source Clarifications: Competitive Auctions and Digital Goods

Logarithmic domains

Unqualified logarithmic revenue bounds → $h\geq2$ in Theorem 4.1, Theorem 7.1, and Theorem 7.2’s lower bound; at least two bidders in Corollary 4.2. Here $h$ is the largest normalized bid. At $h=1$ or bidder count $n=1$ , the relevant denominator $\log_2h$ or $\log_2n$ is zero. The $h\geq2$ scope also excludes $1<h<2$ , where the logarithm is defined; this proof restriction is not shown necessary for every revenue bound.

Preliminary-version inference from truthfulness → the journal version’s monotone-offer condition: if bids satisfy $b_i\leq b_j$ and $x\leq b_i$ , bidder $i$ ’s probability of receiving and accepting an offer at price at most $x$ is no greater than bidder $j$ ’s. Truthfulness controls changes in one’s own report and does not alone supply this cross-bidder comparison. The Lemma 8.1 and Theorem 8.2 proofs use the journal condition.

General randomized offers → countably supported outcome laws, and finite offer support in Lemma 8.1 and Theorem 8.2. These restrict the current formalization’s randomization domain; continuous-offer extensions remain unproved.
Arbitrary excess-demand rejection → one report-independent priority among accepted bidders. This fixes the source’s unspecified choice; the current result does not quantify over every rejection rule.
Theorem 6.2 count parameter → natural-number $\alpha$ in the current probability bound; arbitrary real parameters would need a rounding argument.
Theorems 9.1 and 9.3 high-value constructions → integer $H\geq2$ , with bids $1$ or $H$ and at least one high bid. Its revenue/benchmark ratio is at most $1/H$ ; no arbitrary-real- $H$ construction is asserted.
Bounded dual-price mechanism → an exact equal-half partition and even total supply $2q$ , assigning capacity $q$ to each side. Other partition or odd-supply cases are outside that checked mechanism.