Final Validation Report: EOS07GSP

Updated: 2026-09-04

1. Human Verdict

The formalization covers the static GSP/VCG comparisons, the corrected finite Theorem 7 revenue result, and Theorem 8’s equilibrium and VCG-outcome claims.

Formalization gap: Theorem 8 uniqueness is proved among common symmetric continuation plans; uniqueness among arbitrary bidder-specific plans remains unproved. Details.

2. Closeout Status

3. Source and Scope

The folder tracks the NBER working-paper numbering: Remarks 1–3, Definition 4, Lemmas 5–6, and Theorems 7–8. A later public PDF renumbers analogous main results, so this report uses the NBER numbering consistently.

4. Researcher Summary of Checked Results

Result Comparison with source
First-price example; GSP/VCG two-slot example Exact: the displayed bid revisions and outcome/payment calculations.
Remarks 1–3 Exact: GSP payments weakly dominate VCG payments at the same bids; VCG is truthful and GSP is not dominant-strategy truthful.
Lemma 5; Theorem 7 Necessary strengthening: every unassigned bidder has no bottom-slot envy; otherwise the stated stability and VCG-revenue comparisons can fail. The stable-assignment bridge and seller-minimal VCG-revenue comparison use this strengthened Definition 4. Witness.
Lemma 6 Restriction for strict-bid realization: positive winner utility excludes stable assignments whose exact payments force a bid tie. This does not establish necessity for every tie-aware realization. Obstruction.
Theorem 8 Formalization gap: equilibrium and the VCG outcome are proved; uniqueness is checked only for a common symmetric continuation plan. Asymmetric-plan uniqueness remains unproved. Scope.

5. Formalization Clarifications and Additions

The checked scope is the three explicit conditions in Section 6. In particular, the strict-surplus result does not cover every stable assignment, and common-plan uniqueness leaves asymmetric-plan uniqueness unresolved.

6. Explicit Formalization Additions

The static results add no bottom-slot envy for every unassigned bidder (Definition 4) and positive realized utility for every winner (Lemma 6). Theorem 8’s uniqueness is restricted to common symmetric continuation plans. The memo supplies both static obstructions and the dynamic scope comparison.

Lemma 6 permits real-valued bids, including a potentially negative unassigned tail. A nonnegative-bid version would need a positive-tail condition.

7. Proof-Strategy Deviations

Theorem 8 first compares continuation plans pointwise for each opponent-value profile, then integrates under supported conditional beliefs. Its common-plan scope is stated in Section 6.

8. Proof Tricks Worth Reusing

9. Generalizations, Conjectures, and Extensions

The proof decomposition suggests a routine extension from one common value law to bidder-specific independent continuous positive-density laws: the conditional belief construction changes by substituting the bidder-specific marginals, while the ex-post continuation comparison remains pointwise. A second reusable extension is to expose the clock-clamped effective-action semantics as a generic interface for continuous ascending auctions. Neither extension is needed for the paper’s result, and neither is claimed here without a dedicated reviewed theorem.

10. Mathematical Typos or Other Fixes Suggested in the Source Paper

See Section 6 and the memo for the static additions and common-plan restriction.

11. Source Corrections and Formalization Clarifications

No wider static or dynamic conclusion is inferred beyond Section 6. The below-clock threshold uses the stated immediate-drop interpretation of a stopped auction.

12. Detailed Formalization Evidence

The formalization is organized around a compact paper interface and an exhaustive audit ledger. PaperInterface.lean exposes source-facing rows for the definitions, examples, lemmas, and main theorems. PostPaperAudit.lean keeps the longer implementation ledger, including support lemmas and alternate route endpoints that are useful for audit but too detailed for human review.

The current machine-readable audit artifacts report:

13. Paper Assumption Provenance

No paper-facing assumption declaration is used. Source model conditions such as strictly decreasing positive click rates and continuous independent values with positive density occur directly as visible theorem parameters or constructed law fields, not as conclusion-bearing certificates.

Current Canonical Evidence

Generated from the configured source-condition surface and exact current statement digests in the canonical assumption-provenance sidecar. Model, agent, and automated checks are identified as such; no human review is inferred.

Assumption declaration Lean declaration Source location / statement Assumption validators Comments
None none None None No paper-facing assumption declarations are configured.

14. Displayed Formula Provenance

The main displayed formulas used in the paper-facing results are exposed in PaperInterface.lean and checked by row-local statement judgments.

Current Canonical Evidence

Rows are selected from semantic formula and equation source-kind metadata within the configured semantic source-coverage scope, never from declaration names. Coverage and statement checks remain separate evidence lanes. Scope metadata needs repair, so the full inventory is shown: paper_statement_map.json must explicitly set source_coverage_mode before a source-coverage closeout

Paper formula / subclaim Lean declaration Provenance Validators Comments
Theorem 8 states the dropout price formula p_i(k,h,s_i)=s_i-alpha_k/alpha_{k-1}(s_i-b_{k+1}); the appendix defines q by the same indifference equation. theorem8_dropout_formula_eq_bstar_threshold covered. theorem8_dropout_formula_eq_bstar_threshold: no completed statement check covered; Agent check by Codex EOS07 source-first v10 full-formalization audit; 2026-07-18 After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
The q formula is defined by indifference between position k at price b_{k+1} and position k-1 at price q, placing q between the previous dropout price and the bidder value under the weak source conditions. theorem8_q_mem_interval_review covered. theorem8_q_mem_interval_review: no completed statement check covered; Agent check by Codex EOS07 source-first v10 full-formalization audit; 2026-07-18 After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Under the paper’s strict descending click-through-rate convention and positive values, q lies strictly between the previous dropout price and the bidder value. theorem8_q_strict_mem_interval_review covered. theorem8_q_strict_mem_interval_review: no completed statement check covered; Agent check by Codex EOS07 source-first v10 full-formalization audit; 2026-07-18 After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.

15. Library Lift Pass

Reusable position-auction, GSP, VCG-payment, locally-envy-free, stable- assignment, and generalized-English payoff-game primitives live under EconCSLib/MechanismDesign/Auctions. No additional library extraction is required for closeout; future cleanup can continue reducing EOS-specific proof aliases in PostPaperAudit.lean.

16. DAG Audit

17. Validation Checks

LLM-as-Judge Results

The closeout validation path for this report is:

The final command is the targeted repository audit command required for this paper’s closeout gate.

18. Paper Definitions Checked

19. Named Theorem Statements Checked

Remarks 1–3

Paper statement. Same-bid GSP payments weakly dominate VCG payments, truth-telling is dominant under VCG, and truth-telling is not dominant under GSP.

Lean interface statement. - remark1_gsp_payments_weakly_dominate_vcg - remark2_vcg_truthful - remark3_gsp_not_truthful

Status. formalized.

Running Example

Paper statement. The two-slot, three-bidder first-price example has profitable bid revisions; the GSP/VCG example has truthful GSP bids as a Nash equilibrium and GSP revenue higher than VCG revenue.

Lean interface statement. - first_price_running_example_profitable_revision_chain - running_example_truthful_gsp_nash - running_example_truthful_gsp_revenue_comparison

Status. formalized.

Lemma 5

Paper statement. The outcome of any locally envy-free equilibrium is a stable assignment.

Lean interface statement. - corrected_lemma5_ranked_more_bidders_stable

Status. archival statement not claimed; the corrected full K>N ordinary-GSP bridge is proved.

Lemma 6

Paper statement. Any stable assignment is an outcome of a locally envy-free GSP equilibrium.

Lean interface statement. - corrected_lemma6_ranked_more_bidders_constructs_tiebreak_equilibrium

Status. archival statement not claimed. The explicit strict-surplus amended domain excludes the zero-surplus counterexample, and its direct full K>N construction/equilibrium proof is complete.

Theorem 7

Paper statement. The constructed B* equilibrium yields the same positions and payments as VCG and is revenue-minimal among locally envy-free equilibria.

Lean interface statement. - theorem7_finite_static_bstar_full_corrected

Status. archival statement not claimed. The corrected finite target is proved: B-star is locally envy-free with the VCG ledger, and its revenue is weakly minimal over corrected strict ranked-GSP comparisons.

Theorem 8

Paper statement. The generalized-English auction has a unique continuous- strategy perfect Bayesian equilibrium with the displayed dropout-price formula; the equilibrium is ex post and yields VCG-equivalent positions and payoffs.

Lean interface statement. - theorem8_dropout_formula_eq_bstar_threshold - theorem8_q_step1_dropping_after_q_review - theorem8_q_step2_waiting_before_q_review - theorem8_continuous_full_history_bayes_ex_post_review

The finite source-event VCG outcome is supplied by the checked support declaration theorem8_source_event_strict_values_unique_pbe_formula_conclusion together with the Theorem 7 B* outcome and the dropout-formula bridge.

Status. formalized. The selected finite endpoint proves the full-history ex-post PBE with actual conditional Bayes beliefs, arbitrary complete common-plan deviations, effective-action uniqueness, and the displayed dropout formula. The common-plan scope is the explicitly stated formalization clarification, while the archival bidder-indexed wording remains visible in the source notes.

20. Statement Review Evidence

The current source-to-statement assessment and exact review targets are recorded in the following artifacts. Mathematical scope and qualifications are explained with their named results above.

21. Source-Coverage Audit Ledger

Current Canonical Evidence

Source statement Linked Lean review rows Coverage judgment Row-local statement checks Comments
Definition 4 defines a locally envy-free equilibrium of the static GSP game as an equilibrium in which no player can improve by exchanging bids with the player ranked one position above; formally, for each allocated adjacent position the adjacent no-envy inequality holds. definition4_locally_envy_free covered definition4_locally_envy_free: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
The paper maps GSP outcomes to the Shapley-Shubik assignment game, where advertiser-position pair values are alpha_k s_i and stability means no profitable advertiser-position rematch. stable_assignment covered stable_assignment: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
In the Section 2.2 first-price running example with two slots, click rates 200 and 100, and bidder values 10, 4, and 2, bidder 1 prefers the minimal top bid 2.02 to paying more for the same top slot, bidder 2 can profitably raise from 2.01 to 2.03 to take the top slot, and bidder 1 can then profitably raise to 2.04 to take the top slot back. first_price_running_example_profitable_revision_chain covered first_price_running_example_profitable_revision_chain: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Remark 1 states that if advertisers bid the same amounts under GSP and VCG, each advertiser payment is at least as large under GSP as under VCG, with the recursive VCG-payment induction displayed immediately after the remark. remark1_gsp_payments_weakly_dominate_vcg covered remark1_gsp_payments_weakly_dominate_vcg: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Remark 2 states that truth-telling is a dominant strategy under VCG, citing the standard VCG property. remark2_vcg_truthful covered remark2_vcg_truthful: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Remark 3 states that truth-telling is not a dominant strategy under GSP and gives the two-slot, three-bidder deviation where bidder 1 shades to obtain the second position and higher payoff. remark3_gsp_not_truthful covered remark3_gsp_not_truthful: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
The running two-slot, three-bidder example has click rates 200 and 100 and values 10, 4, and 2; the text states truthful GSP bids are an equilibrium in this example. running_example_truthful_gsp_nash covered running_example_truthful_gsp_nash: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
The running example computes GSP payments of 4 and 2 per click, VCG payments of 600 and 200 total, and states that VCG revenues are lower than GSP revenues in the example. running_example_truthful_gsp_revenue_comparison covered running_example_truthful_gsp_revenue_comparison: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Lemma 5 states that the outcome of any locally envy-free equilibrium of auction Gamma is a stable assignment; the appendix proof derives assortativity and telescopes adjacent no-envy inequalities. lemma5_locally_envy_free_stable covered lemma5_locally_envy_free_stable: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Lemma 6 states that, when the number of bidders exceeds the number of positions, any stable assignment is an outcome of a locally envy-free GSP equilibrium; the appendix constructs bids from assignment payments. lemma6_tiebreak_ranked_gsp_stable_assignment_locally_envy_free covered lemma6_tiebreak_ranked_gsp_stable_assignment_locally_envy_free: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Theorem 7 constructs the B* profile, proves bid order is preserved, and shows each bidder position and payment coincide with VCG in the constructed equilibrium. theorem7_ranked_gsp_bstar_mechanism_realizes_bstar_outcome covered theorem7_ranked_gsp_bstar_mechanism_realizes_bstar_outcome: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
The B* construction sets bid b_i* from the previous VCG payment divided by the previous click-through rate; the proof checks that the resulting next-price payments coincide with VCG payments. theorem7_bstar_payment_identity covered theorem7_bstar_payment_identity: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Theorem 7 states that the B* strategy profile is locally envy-free; the proof checks that each bidder is indifferent between remaining in position and swapping with the bidder one position above. theorem7_bstar_locally_envy_free covered theorem7_bstar_locally_envy_free: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Theorem 7 concludes that the B* equilibrium has VCG-equivalent payments and that any other locally envy-free equilibrium yields seller revenue at least as high; the proof uses stable-assignment payment lower bounds. theorem7_no_positive_transfer_conclusion covered theorem7_no_positive_transfer_conclusion: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Theorem 7 comparison component links locally envy-free GSP outcomes to stable assignments and VCG payment lower bounds, under the paper’s ranked strict-order conventions. theorem7_strict_tiebreak_gsp_comparison_conclusion covered theorem7_strict_tiebreak_gsp_comparison_conclusion: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Theorem 8 states the dropout price formula p_i(k,h,s_i)=s_i-alpha_k/alpha_{k-1}(s_i-b_{k+1}); the appendix defines q by the same indifference equation. theorem8_dropout_formula_eq_bstar_threshold covered theorem8_dropout_formula_eq_bstar_threshold: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Step 2 argues that if a bidder drops before q, waiting until q gives a strictly better payoff because another dropout occurs with positive probability and q is the indifference threshold. theorem8_q_step2_waiting_before_q_review covered theorem8_q_step2_waiting_before_q_review: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Step 1 argues that dropping after q cannot be optimal; above q, the bidder strictly benefits by dropping slightly earlier or at the indifference price. theorem8_q_step1_dropping_after_q_review covered theorem8_q_step1_dropping_after_q_review: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
The q formula is defined by indifference between position k at price b_{k+1} and position k-1 at price q, placing q between the previous dropout price and the bidder value under the weak source conditions. theorem8_q_mem_interval_review covered theorem8_q_mem_interval_review: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Under the paper’s strict descending click-through-rate convention and positive values, q lies strictly between the previous dropout price and the bidder value. theorem8_q_strict_mem_interval_review covered theorem8_q_strict_mem_interval_review: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Theorem 8 restricts attention to strategies continuous in valuation, and the displayed q formula is affine, hence continuous, in the bidder value. theorem8_q_continuous_value_review covered theorem8_q_continuous_value_review: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Theorem 8’s proof identifies the continuous dropout-price formula by excluding deviations above and below q, so continuous one-step best responses agree with q on the relevant support. theorem8_continuous_source_local_best_response_support_unique_review covered theorem8_continuous_source_local_best_response_support_unique_review: no completed statement check After unfolding the reviewed predicates, the linked conclusion preserves the complete finite-auction domain, exact real formulas, quantifier order, strategic scope, and conclusion of this cited source item without relying on a result-bearing input package.
Theorem 8 is ex post: the prescribed strategy is a best response regardless of realized values; the strict ranked-value source event gives the finite ex-post local-deviation condition. theorem8_continuous_full_history_bayes_ex_post_review covered theorem8_continuous_full_history_bayes_ex_post_review: no completed statement check The legal-history row proves a strictly stronger ex-post statement: every complete history-dependent continuation plan is weakly dominated for every realized ordered opponent profile, and hence under every supported posterior.
Theorem 8 states that the generalized English auction has a unique perfect Bayesian equilibrium with strategies continuous in valuations: an advertiser with value s_i drops out at price p_i(k,h,s_i)=s_i-(alpha_k/alpha_{k-1})(s_i-b_{k+1}); in this equilibrium each advertiser’s position and payoff coincide with VCG, and the equilibrium is ex post. theorem8_continuous_full_history_bayes_ex_post_review
theorem8_dropout_formula_eq_bstar_threshold
theorem7_ranked_gsp_bstar_mechanism_realizes_bstar_outcome		 covered theorem8_continuous_full_history_bayes_ex_post_review: no completed statement check
theorem8_dropout_formula_eq_bstar_threshold: no completed statement check
theorem7_ranked_gsp_bstar_mechanism_realizes_bstar_outcome: no completed statement check		 The full-history row proves the source-intended legal-history ex-post PBE, actual Bayes consistency, arbitrary-continuation sequential rationality, and effective uniqueness. The dropout bridge identifies the displayed threshold with the B-star payment threshold; the Theorem 7 outcome row and checked finite source-event support theorem derive the VCG-equiv…