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arxiv: 2606.30162 · v1 · pith:3RGCZBVDnew · submitted 2026-06-29 · 💻 cs.GT

Revenue Guarantee of Anonymous Pricing for Mixed Bidders:Bridging Value and Utility Maximizers

Zhile Jiang , Stratis Skoulakis This is my paper

Pith reviewed 2026-06-30 04:00 UTC · model grok-4.3

classification 💻 cs.GT
keywords anonymous pricingrevenue guaranteevalue maximizersutility maximizersmixed biddersmechanism designapproximation ratiofirst-price auction
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The pith

Anonymous pricing with a suitable price guarantees one over e of the optimal revenue even when bidders include both value and utility maximizers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes that anonymous pricing can extract a 1/e fraction of the optimal revenue in heterogeneous markets containing both value maximizers and utility maximizers. The key step is showing a structural equivalence in behavior between these two types, which allows the mixed-bidder problem to be reduced to a uniform type. This extends and improves upon earlier 1/2(1-1/e) guarantees that applied only to value maximizers. The work also proves a matching upper bound of roughly 1/2.62 on the guarantee and identifies cases where adding competition lowers revenue.

Core claim

By establishing a structural behavioral equivalence between value and utility maximizers, anonymous pricing with an appropriately chosen price achieves a 1/e fraction of the optimal revenue in mixed-bidder settings. This result holds while an upper bound of 1/2.62 is shown for the performance of anonymous pricing, and first-price auctions with the same reserve can underperform anonymous pricing when value maximizers are present.

What carries the argument

The structural behavioral equivalence between value maximizers and utility maximizers that reduces the mixed-type analysis to a single-type problem.

If this is right

  • Anonymous pricing guarantees at least 1/e of optimal revenue for any mix of value and utility maximizers.
  • The 1/e guarantee improves on the prior 1/2(1-1/e) bound that applied only to pure value maximizers.
  • Anonymous pricing is at most 1/2.62-competitive in the same mixed setting.
  • A first-price auction using the same reserve as the anonymous price can produce strictly lower revenue once value maximizers are present.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence reduction may simplify analysis of other simple mechanisms in mixed-type environments beyond pricing.
  • The revenue drop from added competition suggests that reserve prices in value-maximizer markets should be set more conservatively than in standard utility-maximizer settings.
  • Practical auction platforms could simulate mixed bidder populations to check whether observed revenue ratios approach the 1/e bound.
  • Similar behavioral equivalences might be identifiable in other multi-dimensional mechanism design problems involving constrained optimization.

Load-bearing premise

A structural behavioral equivalence exists between value maximizers and utility maximizers that permits reducing the mixed-type analysis to a single-type problem.

What would settle it

A concrete market instance in which no single price for anonymous pricing extracts at least 1/e of the optimal revenue, or in which the claimed behavioral equivalence between the two bidder types fails to hold.

read the original abstract

Mechanism design increasingly faces heterogeneous environments containing both traditional utility maximizers and value maximizers, the latter of whom seek to maximize acquired value subject to Return-on-Spend constraints. Designing revenue-optimal mechanisms for such multi-dimensional settings is both computationally and theoretically challenging. To address this complexity, we investigate the revenue guarantees of \textit{Anonymous Pricing} (AP), a simple and practical mechanism, in heterogeneous markets composed of both value and utility maximizers. By establishing a structural behavioral equivalence between value and utility maximizers, we show that AP, with an appropriately chosen price, achieves a \(1/e\) fraction of the optimal revenue. Our result improves upon the recent \( \frac{1}{2}(1 - 1/e) \) guarantee established by Deng et al.~(2022) for pure value maximizers, while extending it to mixed bidder types (both value and utility maximizers). We additionally establish an upper bound of \(1/2.62\) for AP. Finally, we demonstrate a counterintuitive phenomenon: competition can reduce revenue with the presence of value maximizers. In particular, running a First-Price Auction with the exact same reserve price as AP can, in the presence of value maximizers, generate lower revenue than AP itself.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that a structural behavioral equivalence between value maximizers (ROI-constrained) and utility maximizers reduces the mixed-bidder anonymous pricing problem to a single-type analysis, allowing AP with an appropriately chosen price to achieve a 1/e fraction of optimal revenue. This improves the prior 1/2(1-1/e) guarantee for pure value maximizers and extends it to mixed populations; the paper also proves an upper bound of 1/2.62 on the AP approximation ratio and exhibits a counterexample in which a first-price auction with the same reserve yields strictly less revenue than AP when value maximizers are present.

Significance. If the equivalence is shown to preserve both the optimal-revenue benchmark and the revenue of the same anonymous price under simultaneous positive mass of both types, the result supplies a clean constant-factor guarantee for a practically relevant heterogeneous setting. The technical reduction and the competition-reduces-revenue observation are both of interest to mechanism-design research on ROI-constrained bidders.

minor comments (3)
  1. The introduction should explicitly state the precise conditions on the type distributions under which the behavioral equivalence is exact (rather than approximate) when both bidder types have positive probability.
  2. Figure or table presenting the 1/2.62 upper-bound construction would benefit from an accompanying short proof sketch or reference to the relevant lemma.
  3. Notation for the mixed-type distribution (e.g., the convex combination parameter) should be introduced once and used consistently in all statements of the main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, accurate summary of our contributions, and recommendation of minor revision. The significance assessment is appreciated. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation rests on independently established equivalence

full rationale

The paper first claims to establish a structural behavioral equivalence between value maximizers and utility maximizers, then uses that equivalence to reduce the mixed-bidder setting to a single-type problem from which the 1/e AP guarantee follows. This equivalence is introduced as a derived property of bidder behavior rather than defined circularly in terms of the target revenue ratio. No equations reduce the claimed prediction to a fitted parameter or self-referential definition, the cited prior result (Deng et al. 2022) is by different authors, and the upper bound of 1/2.62 is stated separately. The central claim therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities are stated. The central step is the behavioral equivalence, whose justification is not visible.

pith-pipeline@v0.9.1-grok · 5757 in / 1037 out tokens · 46400 ms · 2026-06-30T04:00:04.578383+00:00 · methodology

discussion (0)

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Reference graph

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