Source Clarifications: Competitive Auctions and Digital Goods

Logarithmic domains

• Finite logarithmic bounds: use $h\geq2$ for Theorems 4.1 and 7.1–7.2, and at least two bidders for Corollary 4.2, where $h$ is the largest normalized bid. The denominators vanish at $h=1$ or bidder count $n=1$. For $1<h<2$, the logarithm is defined but the dyadic proof’s finite constant needs adjustment. The checked bounds give the source asymptotics; they do not assert the same finite coefficient throughout that smaller range.

Lemma 8.1 and Theorem 8.2

• Journal monotonicity condition: for bids $b_i\leq b_j$ and prices $x\leq b_i$, bidder $i$’s probability of winning at a price at most $x$ is no greater than bidder $j$’s. This cross-bidder condition gives Lemma 8.1’s allocation ordering and Theorem 8.2’s bound $\mathbb{E}[R]\leq F$, where $F$ is optimal fixed-price revenue. The preliminary paper infers the condition from truthfulness; the journal version, Definition 8.1 and Theorem 8.2, states it separately. Truthfulness by itself does not imply that comparison.
• Inverse-CDF notation: use the lower generalized inverse $\inf\{x:y\leq g_i(x)\}$ in the journal’s common-uniform coupling. Its printed equality-level-set formula can be undefined at a CDF jump. The lower inverse has the required marginals and preserves the CDF ordering, so the expected-revenue argument applies to arbitrary real offer laws.

Randomization and bounded supply

• Randomization model: the concrete mechanisms have finite outcome distributions, represented within a general model allowing countable distributions. Theorem 8.2 separately covers arbitrary real offer distributions. Accepted nonnegative offers are bounded by the bids, so their expected revenue is defined without an additional moment condition.
• Excess demand: when more bidders accept the price than there are items, serve them in a fixed priority order chosen independently of bids. This specifies one of the source’s permitted rejection rules; it does not claim truthfulness for every possible rule that reacts to the bid amounts.
• Theorem 6.2 count parameter: take $\alpha$ to be a natural-number lower bound on the winner count, the role it plays in the source proof. The formalization proves the displayed probability bound for uniform exact half-sampling. A separate real-$\alpha$ calculation uses independent coin flips and is not an equivalence between the two sampling schemes.
• Theorems 9.1 and 9.3 impossibility examples: for every integer $H\geq2$ and real $\alpha>0$, bids at $1$ and $H$ give $R/F\leq1/H$ and $F\geq\alpha H$. As $H$ grows, the revenue fraction tends to zero even with a large benchmark. Integer examples therefore prove the paper’s no-constant-competitive-ratio conclusion. Theorem 9.1 treats deterministic bid-independent auctions; Lemma 9.2 extends it to the truthful deterministic set-of-bids model in Theorem 9.3. The construction does not parameterize every real $H$.
• Half-sizes and parity: interpret $n/2$ and $k/2$ as floor division for odd bidder counts or capacities. Truthfulness and total supply feasibility hold for all natural inputs; even inputs give exact halves. The bounded dual-price revenue guarantee remains conditional on its sampling good-event bounds, so this convention does not by itself prove an odd-input rate.