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Beyond Independent Values: Optimal Auctions with Correlated Valuations for LLM-Native Advertising
Shiman Sheng, Mingyu Guo
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
Last extracted: 10/14/2026, 10:20:33 AM
Summary
The paper introduces AffMyerson, an optimal auction mechanism for LLM-native advertising that accounts for correlated valuations among advertisers. Unlike existing methods that assume independent private values, AffMyerson models advertiser values as a multivariate normal distribution conditioned on the conversation context, ensuring the precision matrix satisfies the M-matrix condition for affiliation. This approach allows for truthful, ex-post individually rational auctions with O(n^2) pricing complexity. Empirical results with LLM-agent bidders show AffMyerson outperforms second-price, Ronen's lookahead, and independent Myerson auctions in revenue.
Entities (10)
Relation Signals (8)
AffMyerson → provides → Truthfulness
confidence 95% · We prove that every prior in this family yields a truthful, ex-post individually rational auction
AffMyerson → uses → Multivariate Normal Distribution
confidence 95% · We model the advertisers’ values given the conversation as a multivariate normal
AffMyerson → enforces → M-matrix
confidence 92% · turn the economic premise of positive dependence (affiliation) into a sign condition on the precision matrix... Statisticians call this the M-matrix condition
M-matrix → ensures → Affiliation
confidence 90% · for a Gaussian this one sign condition is all that affiliation requires
AffMyerson → hascomplexity → O(N^2)
confidence 90% · pricing an auction needs no matrix inversion and takes O(n^2) time
AffMyerson → outperforms → Ronen's Lookahead Auction
confidence 90% · 6.1% more than Ronen’s lookahead auction with the same prior
AffMyerson → outperforms → Myerson's Auction
confidence 90% · 9.2% more than Myerson's auction run as if values were independent.
AffMyerson → outperforms → Second-Price Auction
confidence 90% · AffMyerson earns 6.2% more than a second-price auction with a tuned reserve
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Abstract
Abstract:Conversational assistants are becoming the place where people ask what to buy, and sponsored content is moving into their answers. Several advertisers then compete for one sponsored slot, and all of them read the same conversation, so their willingness to pay usually moves together. Existing auctions for LLM advertising treat these values as independent. We therefore present AffMyerson, an auction that captures this correlation and turns it into revenue. We model the advertisers' values given the conversation as a multivariate normal and turn the economic premise of positive dependence (affiliation) into a sign condition on the precision matrix. We design a neural parameterization whose every output satisfies it. We prove that every prior in this family yields a truthful, ex-post individually rational auction, so estimation error can cost revenue but never truthfulness. We derive the conditional virtual values in closed form. By the design of the prior, pricing an auction needs no matrix inversion and takes O(n^2) time, fast enough to run at query time. We also work out the log-normal family used in empirical ad-auction work. There we prove that the auction is truthful for every prior and revenue-optimal when the prior is correct, and that the virtual value is regular everywhere whenever the conditional spread is below a loose threshold, which our fitted priors stay well under. With LLM-agent bidders, AffMyerson earns 6.2% more than a second-price auction with a tuned reserve, 6.1% more than Ronen's lookahead auction with the same prior, and 9.2% more than Myerson's auction run as if values were independent. This gain is still 4.5% on real keyword search queries where willingness to pay is hard to read. The correlation comes from common demand. Agents that judge the conversation bid together and agents told to calculate do not. The gain from conditioning moves with the correlation.
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- Source: https://arxiv.org/abs/2610.14646v1
- Canonical: https://arxiv.org/abs/2610.14646v1
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Beyond Independent Values: Optimal Auctions with Correlated Valuations for LLM-Native Advertising Shiman Sheng Affiliation: The University of Adelaide, Adelaide, Australia email: shiman.sheng@adelaide.edu.au and Mingyu Guo Affiliation: The University of Adelaide, Adelaide, Australia email: mingyu.guo@adelaide.edu.au Abstract. Conversational assistants are becoming the place where people ask what to buy, and sponsored content is moving into their answers. Several advertisers then compete for one sponsored slot, and all of them read the same conversation, so their willingness to pay usually moves together. Existing auctions for LLM advertising treat these values as independent. We therefore present AffMyerson, an auction that captures this correlation and turns it into revenue. We model the advertisers’ values given the conversation as a multivariate normal, the standard model for correlated quantities, and turn the economic premise of positive dependence (affiliation) into a sign condition on the precision matrix. We design a neural parameterization whose every output satisfies it. We prove that every prior in this family yields a truthful, ex-post individually rational auction, so estimation error can cost revenue but never truthfulness. We derive the conditional virtual values in closed form. By the design of the prior, pricing an auction needs no matrix inversion and takes O(n2)O(n^2) time, fast enough to run at query time. We also work out the log-normal family used in empirical ad-auction work. There we prove that the auction is truthful for every prior and revenue-optimal when the prior is correct, and that the virtual value is regular everywhere whenever the conditional spread is below a loose threshold, which our fitted priors stay well under. With LLM-agent bidders, AffMyerson earns 6.2%6.2\% more than a second-price auction with a tuned reserve, 6.1%6.1\% more than Ronen’s lookahead auction with the same prior, and 9.2%9.2\% more than Myerson’s auction run as if values were independent. This gain is still 4.5%4.5\% on real keyword search queries where willingness to pay is hard to read. The correlation comes from common demand. Agents that judge the conversation bid together and agents told to calculate do not. The gain from conditioning moves with the correlation. Keywords: LLM advertising; optimal auction design; correlated values; affiliation; M-matrix priors; truthful mechanisms 1. Introduction Conversational assistants such as ChatGPT, Perplexity, Google AI Overviews, and Bing Copilot are becoming the place where people ask what to buy, and sponsored suggestions are moving into their answers (22, 23). Consider a user who asks for “the best hiking boots for a trip to the Alps in October.” Several outdoor brands would pay to have their boot mentioned in the answer. All of them read the same conversation, so they can all see that this user is ready to buy and that October is peak season for the category. If the conversation is worth more than usual to one brand, it is likely to be worth more to the others as well. Milgrom and Weber’s linkage principle says that such correlation is worth money to the platform (2). A rival’s high bid is evidence that the winner’s own value is high, so the platform can charge the winner more without losing the sale. The auctions proposed so far for LLM advertising treat advertisers’ values as independent, as in Myerson’s independent-private-values (IPV) model (1). They include segment auctions in retrieval-augmented generation (22), token auctions inside decoding (23), alignment-based allocation (24), and Myersonian payments in two-stage pipelines (25, 26). None of them uses what one advertiser’s bid says about another’s value, and that is the information this paper builds into the auction. Two-stage pipeline: a network maps the conversation to a joint prior, and the auction prices each opportunity from that prior. Figure 1. Two stages. Per conversation (top): a network trained offline on past bids maps the conversation to a joint prior over the advertisers’ values. Every prior it can output is in the family of Section 3, so the auction below is truthful whatever the fit (Theorem 5.7). Per auction (bottom): the winner is the bidder with the highest conditional virtual value and pays the lowest bid at which it would still win (Section 4).Two-stage pipeline: a network maps the conversation to a joint prior, and the auction prices each opportunity from that prior. The prior the platform needs, and how we build it. The prior must do three things: carry correlation among n values, be fast to fit and to condition on the rivals’ bids at auction time, and keep the auction truthful when the estimate is inaccurate, since conversations vary too much for any fitted prior to be exact. We use the multivariate normal, the standard model for several correlated quantities, and build the economic premise of positive dependence (affiliation) into it through one condition on its shape: the precision matrix (the inverse covariance) has nonpositive off-diagonal entries. Statisticians call this the M-matrix condition, and for a Gaussian this one sign condition is all that affiliation requires (3). It also lets the platform read one bidder’s conditional distribution from a single row of the matrix, with no inversion. The auction is then the natural one: rank bidders by their conditional virtual value (Myerson’s virtual value (1) computed under the distribution of the bidder’s value given the rivals’ bids), give the slot to the top-ranked bidder if its virtual value is nonnegative, and charge it the lowest bid at which it would still win. Under the true prior this is the revenue-optimal truthful auction (6). Truthfulness does not need the true prior. For every prior in the family, fitted well or badly, the auction is dominant-strategy truthful and never leaves a bidder worse off (Theorem 5.7), and revenue degrades continuously in the estimation error (Proposition 5.8). A learned prior is therefore safe to deploy, since the only thing a bad estimate can cost is revenue. A second prior family meets the same three requirements. Advertising values are commonly modeled as log-normal, in distributions fitted to bid data (16), in simulation (17), and in field experiments (18), so we work out that family as well. The auction stays truthful and optimal there, and its virtual value is regular everywhere (so no ironing is needed) whenever the conditional spread of log values is below a threshold we compute, σ⋆=1.5176σ =1.5176. The threshold is loose in practice: at σ⋆σ , one standard deviation in log scale multiplies a value by 4.64.6, our fitted priors never exceed 0.640.64, and every prior the parameterization of Section 6 outputs satisfies it by construction (Appendix S4). The fitting method is chosen the same way. We use a small network that reads a sentence embedding of the conversation and can only output priors inside the family, so nothing has to be checked at auction time. On our designed queries, the network can draw on the text of the conversation. On the short search queries we draw from Amazon ESCI and MS MARCO, a few words say little about how much an advertiser values the query, so the network has nothing to learn from the text, and a convex maximum-likelihood fit with no network earns the same revenue. Either way the auction stays truthful, so the platform can choose the fitting method by what its data support. Contributions. (1) Two tractable prior families with incentive and optimality guarantees. Roughgarden and Talgam-Cohen (6) verify the conditions of their optimal auction for correlated bidders on one symmetric normal. We verify them for every normal prior with an M-matrix precision, for any means and any number of bidders, which needs cross-monotonicity through the sign of the precision matrix and a revenue identity on the unbounded support (Theorem 5.5). For the log-normal family we give an exact regularity condition: the virtual value is globally regular if and only if the conditional log-scale is at most σ⋆=1.5176σ =1.5176, a threshold that is loose in practice (our fitted priors stay below 0.640.64) and that the parameterization keeps by design (Theorem 5.9). (2) Truthful whatever the fit. For every prior in the family, accurate or not, truthful bidding is a dominant strategy and no bidder ever pays more than its value (Theorem 5.7). Learned mechanisms such as CAN (21) also get truthfulness from their structure. Here it comes with exact optimality when the prior is correct, and estimation error costs only revenue, by an amount we bound in the size of the error (Proposition 5.8). (3) Fast enough to run inside a conversation. Because we parameterize the prior by its precision matrix rather than its covariance, pricing an auction needs no matrix inversion and takes O(n2)O(n^2) time, microseconds in practice (Section 4). (4) How LLM agents turn common demand into correlated bids. When LLM agents bid on behalf of advertisers, given a brand profile, the conversation, and a demand signal shared by all brands, they reproduce two features of real sponsored-search markets: bids rise with purchase intent, and rival bids move together. The shared signal does not make the correlation automatic. Across seven prompt schemes and two models, agents that judge the conversation bid together (within-query correlation +0.78+0.78), agents told to calculate do not (+0.06+0.06), and the gain from conditioning rises and falls with the correlation. To our knowledge, this is the first study of how LLM bidding agents turn a common demand shock into correlated bids, a setting we expect to become common (Section 7.7). (5) Revenue. With LLM-agent advertisers and hidden demand shocks, AffMyerson earns 6.2%6.2\% more than a second-price auction with a tuned reserve, 6.1%6.1\% more than Ronen’s lookahead auction (8) with the same prior, 9.2%9.2\% more than Myerson’s auction run as if values were independent, and 1.1%1.1\% more than CAN, a neural mechanism trained on the same bids (21). We call the gap over the unconditioned Myerson auction the affiliation rent. The rent vanishes when correlation is removed from the data, grows monotonically as it is added back, survives twelve deliberate corruptions of the prior, and is still +4.5%+4.5\% on 580580 real keyword search queries where willingness to pay is hard to read (Section 7). In a market where the conversation matters to one advertiser and not to the other, AffMyerson earns 24.8%24.8\% more than CAN, and the whole gain comes from the part of our design that reads the conversation (Section 7.3). 2. Related Work LLM-native advertising. Segment auctions (22) auction discourse segments in a retrieval-augmented generation (RAG) pipeline; token auctions (23) embed second-price logic into decoding; LLM-Auction (24) casts ad allocation as preference alignment; LERA (25) and quality-preserving mechanisms (26) apply critical-value (Myersonian) payments in two-stage RAG systems. None of them models the correlation among advertisers’ values, so none of them can price it. Optimal auction theory, affiliation, and M-matrices. Myerson (1), Milgrom and Weber (2), and Roughgarden and Talgam-Cohen (6) are the theory we build on. Our mechanism is the Gaussian case of the latter’s Theorem 5.3, and Section 5 states the exact relation, including to Crémer and McLean (7). For general correlated priors, the optimal deterministic auction is NP-hard to compute with three or more bidders (9), and the optimal randomized auction is a linear program whose size grows with the support (11), which is one reason to work inside a family where it has a closed form. Two lines of work learn the auction itself rather than the prior. For a given correlated prior, Guo et al. (13) interpolate Myerson’s greedy rule, which need not be monotone, and certify monotonicity afterwards by integer programming. In our family the greedy rule is already monotone (Lemmas 5.1 and 5.2). Huo et al. (21) learn rank-score functions whose architecture enforces the single-crossing conditions, so truthfulness holds without any prior, and train them on about 100,000100,000 samples of one market. Both learn the allocation, and neither learns the value distribution. We learn the prior instead and keep the auction in closed form, so the platform gets a distribution it can inspect and that varies with the conversation, exact optimality within the family, and pricing in O(n2)O(n^2) time. What we give up is a distributional assumption, which Section 7.3 tests. Learning the prior from samples is studied in (12) and, for correlated buyers, in (10), and (18) gives field evidence for prior-based reserves. To our knowledge, we are the first to use Karlin and Rinott’s characterization of affiliated Gaussians by an M-matrix precision (3) in computational advertising or mechanism design. It is standard in graphical models (5), and estimation under it is well studied: the constrained Gaussian maximum-likelihood fit is convex and has a solution under mild conditions (14, 15), which Section 7.6 uses. 3. Model and Preliminaries This section sets up the environment, states the prior, and derives the two things the mechanism needs from it: how to condition one bidder’s value on the rivals’ bids, and why the prior’s shape guarantees affiliation. 3.1. Environment An LLM system is generating an answer to a user query x. At certain positions in the generation process the system identifies an ad opportunity: a location where sponsored content can be inserted in a contextually relevant way. For each opportunity, a set =1,…,nN=\1,…,n\ of candidate advertisers competes for a single slot. Each opportunity is a single-item auction with no repeated interaction. Each advertiser i holds a private valuation signal si∈ℝs_i , its willingness to pay for appearing at this position for query x, and its value is its own signal, vi=siv_i=s_i. Three settings differ in how the sis_i relate. Under independent private values (IPV) the signals are drawn independently. Under correlated private values (our setting, the “correlated values” cell of (6)) each bidder still knows its own value, but the signals are jointly distributed, so a rival’s report tells the seller something about i’s value. Under interdependent values a bidder’s value itself depends on others’ signals, which we do not treat. We assume the dependence is positive (affiliation, defined below). Negative correlation, as between close substitutes, is outside the model (Section 8). Two facts about sponsored-search values shape the prior. First, values depend on the query, since a conversation with strong commercial intent is worth more to every advertiser than a navigational one, so the prior’s mean must move with the context. Second, values move together, because a demand shock the platform does not observe, such as a product launch or a seasonal rise in demand, raises every advertiser’s value at once, and this co-movement is affiliation. We capture both with one model, a multivariate normal whose mean is a function of the context and whose precision matrix (the inverse covariance) is restricted to the class that affiliation requires. Three terms recur below. Affiliation is the economic premise: when one advertiser’s value is high, every other advertiser’s value is more likely to be high too. Its formal definition is MTP2MTP_2, that the joint density is log-supermodular. An M-matrix is a positive definite matrix whose off-diagonal entries are nonpositive, and it is what MTP2MTP_2 reduces to for a Gaussian: the precision matrix must be an M-matrix, which says that every pair of bidders is nonnegatively partially correlated (Proposition 3.1). Assumption 1 (Affiliated valuations). Given a context embedding e (derived from the query and the response generated so far), the valuation vector =(s1,…,sn)s=(s_1,…,s_n) is drawn from a multivariate normal (MVN) distribution (1) |∼((),Q()−1),s ( μ(e),\,Q(e)^-1 ), where the precision matrix Q()∈ℝn×nQ(e) ^n× n satisfies (i) Q≻0Q 0 (symmetric positive definite) and (i) Qij≤0Q_ij≤ 0 for all i≠ji≠ j. A matrix satisfying (i)–(i) is a symmetric positive definite M-matrix. Condition (i) is the one that matters, since for a Gaussian it is equivalent to affiliation (Proposition 3.1). Section 3.3 explains why we choose this distribution, after the quantities it must provide have been introduced. Section 6 describes how we parameterize and fit this class. Remark 1 (Negative signals). A Gaussian puts some mass on negative values, and so do our fitted priors (Appendix S1). This has no formal consequence, since the incentive results hold on all of ℝnR^n, a bidder with a negative signal has a negative conditional virtual value (it is always below the signal itself, Definition 4.1) and so is never allocated and pays zero, and every signal in our data is an observed bid in cents. Proposition 3.1 (Affiliation ⇔ M-matrix). A multivariate normal (,Q−1)N( μ,Q^-1) is affiliated (MTP2MTP_2) if and only if Q≻0Q 0 and Qij≤0Q_ij≤ 0 for every pair i≠ji≠ j. The characterization is Karlin and Rinott’s (3). Why affiliation pays. Under IPV the winner’s payment depends on competitors’ bids only through the ranking. Under affiliation, a high bid from one advertiser is evidence that the others’ values are higher than usual as well, so when a rival bids high the seller can raise its estimate of the winner’s value and link the winner’s payment to the rivals’ reports. The winner’s own value does not change, but what the seller knows about it does, and this is what reduces the winner’s information rent, the surplus a bidder keeps because the seller does not know its value (the linkage principle (2)). 3.2. Conditioning from the Precision Matrix At auction time the mechanism needs, for each bidder i, the mean and standard deviation of sis_i given the rivals’ reports −is_-i (here and below, −i-i denotes all bidders other than i, and a subscript i|−i -i a quantity of bidder i conditional on them), which we call the conditional moments. Parameterizing the precision matrix Q directly, rather than the covariance Σ=Q−1 =Q^-1, means they can be computed without inverting a matrix: Lemma 3.2 (Precision-based conditioning). For (,Q−1)N( μ,Q^-1), the conditional law of sis_i given −is_-i is (2) si|−i s_i _-i ∼(μi|−i,σi|−i2),σi|−i2=1Qii, ( _i -i,\, _i -i^2 ), _i -i^2= 1Q_i, μi|−i _i -i =μi−1Qii∑j≠iQij(sj−μj). = _i- 1Q_i _j≠ iQ_ij(s_j- _j). The lemma is standard in Gaussian graphical models (5). Both conditional moments come from a single row of Q, so the auction in Section 4 is fast to price. 3.3. Why a Context-Dependent Gaussian? Our prior is a Gaussian whose mean moves with the conversation and whose precision matrix is an M-matrix. We choose it for three reasons. (1) Theoretical support. Roughgarden and Talgam-Cohen (6) verify the conditions of their ex-post optimality framework for one example, a symmetric multivariate normal. We verify them for the whole M-matrix class, with arbitrary context-dependent means and covariances (Proposition 3.1, Section 5), so the optimal auction for correlated values extends from one example to the whole class, and through a log transformation to the log-normal family (Section 5.3). (2) Everything the auction needs has a closed form. Per-query pricing needs three things: the conditional distribution (Lemma 3.2), because the price is set after the rivals report; the conditional virtual value (Section 4), because the optimal allocation ranks bidders by it; and a checkable criterion for affiliation (Proposition 3.1), because the incentive results need it to hold for every prior the fitting procedure can output. The Gaussian gives all three in closed form. The first two are one-line formulas in the Mills ratio, and the third is a sign condition on the precision matrix that every output of the parameterization satisfies (Section 6). (3) Truthful for every member of the family. The fitted prior is an estimate, and real bids follow the Gaussian only approximately. Because every prior in the family gives a truthful auction (Theorem 5.7), estimation error can cost revenue (Proposition 5.8) but not incentives. 4. The AffMyerson Mechanism Figure 2. How conditioning moves money (illustrative two-bidder prior with means 4545 and 3535 cents, standard deviations 1313 and 1212, ρ=0.7ρ=0.7; derived quantities Appendix S1). Left: allocation regions of AffMyerson (solid) vs. IPV-Myerson (dashed). The winner’s threshold moves with the opponent’s report. Center: the winner’s payment t0t_0, which rises with the opponent’s bid (∂t0/∂sj>0∂ t_0/∂ s_j>0 in this bivariate case, Section 4.3). Right: the revenue difference at each bid pair (AffMyerson minus IPV-Myerson). The ellipses enclose 1σ1σ and 2σ2σ of the prior, that is, where most bid pairs fall. Averaged over the prior the difference is +1.9+1.9 cents (+5.4%+5.4\%).Two-bidder example: the conditional distribution, the winner's payment surface, and the revenue gain over IPV-Myerson. We now define the mechanism, first the quantity it ranks, then the allocation and payment rule, and finally its comparison with IPV-Myerson (Section 4.3). 4.1. Conditional Virtual Values Myerson’s optimal auction ranks bidders by virtual value, the bidder’s value minus the information rent it can extract, si−(1−F(si))/f(si)s_i-(1-F(s_i))/f(s_i) (1). Milgrom and Weber (2) and Roughgarden and Talgam-Cohen (6) extend the notion to correlated values by replacing the marginal distribution with the distribution of sis_i conditional on the other bids. Under our prior that conditional distribution is normal with the moments of Lemma 3.2, and the virtual value has a closed form in one standard function, the Mills ratio R(z)=(1−Φ(z))/ϕ(z)R(z)=(1- (z))/φ(z), which is the tail probability of a standard normal divided by its density. R is decreasing and tabulated, so the virtual value below costs one function evaluation per bidder. Definition 4.1 (Conditional virtual value). ψi(si∣−i)=si−σi|−iR(zi) _i(s_i _-i)=s_i- _i -i\,R(z_i), where zi=(si−μi|−i)/σi|−iz_i=(s_i- _i -i)/ _i -i is the standardized residual. 4.2. Allocation and Payment The mechanism allocates to the bidder with the highest nonnegative conditional virtual value and charges the threshold bid, the lowest report at which the winner still wins. The threshold t0t_0 solves the implicit equation (3) ψi∗(t0∣−i∗)=max(0,maxj≠i∗ψj(sj∣t0,−i∗,j)), _i^*(t_0 _-i^*)= (0,\ _j≠ i^* _j(s_j t_0,s_-\i^*,j\) ), and the payment is pi∗=t0p_i^*=t_0. Since vi∗=si∗v_i^*=s_i^*, the critical payment equals the threshold signal, and Lemma 5.3 shows that t0t_0 is unique in (0,si∗](0,s_i^*], so we find it by bisection on that bracket (numerical tolerance: Appendix S2). If all ψi<0 _i<0 the item is unallocated. Each virtual value needs one row of Q, so pricing costs O(n2)O(n^2) plus a constant number of O(n)O(n) bisection steps, microseconds in pure Python (Appendix S14). 4.3. Comparison with IPV-Myerson: the Affiliation Rent Under independent private values, Myerson’s virtual value uses the marginal distribution of sis_i. AffMyerson uses the conditional one, which differs in two ways. Its spread is smaller, since the rivals’ bids resolve part of the uncertainty about sis_i (the law of total variance), and a smaller spread leaves the bidder a smaller information rent σi|−iR(zi) _i -iR(z_i). Its mean moves with the rivals’ bids, so a high rival bid raises the threshold t0t_0 the winner must clear (∂t0/∂sj>0∂ t_0/∂ s_j>0 for n=2n=2; Figure 2). In the two-brand prior of Figure 2, a rival bid of 6161 cents moves bidder 1’s conditional mean from 4545 to 64.764.7 cents and shrinks its conditional deviation from 1313 to 9.39.3. For n≥3n≥ 3 a high rival bid also lowers the other rivals’ virtual values, so the net effect on t0t_0 can take either sign (Appendix S2). Truthfulness does not depend on this, only on monotonicity in a bidder’s own report (Lemmas 5.1 and 5.2). We call the revenue gap between the two the affiliation rent. Whether it survives an estimated prior is the question of the next section. 5. Theoretical Analysis We first establish the two monotonicity properties the incentive argument needs, then the incentive and optimality results, and finally what survives when the prior is only an estimate. 5.1. Regularity and Cross-Monotonicity Lemma 5.1 (Unconditional regularity). For every −is_-i, ψi(si∣−i) _i(s_i _-i) is strictly increasing in sis_i. The derivative exceeds one, so regularity holds and no ironing (Myerson’s repair for non-monotone virtual values) is needed (4). Because the conditionals stay Gaussian with a variance that does not depend on −is_-i, regularity holds at every −is_-i. Lemma 5.2 (Cross-monotonicity). For all j≠ij≠ i, ψj(sj∣−j) _j(s_j _-j) is nonincreasing in sis_i, strictly decreasing when Qji<0Q_ji<0. Proofs: Appendix S1. Together the two lemmas give the two-sided monotonicity that ex-post incentive compatibility requires under correlated values (6). Regularity needs only a Gaussian conditional. Cross-monotonicity is where the M-matrix condition enters, through the sign of QjiQ_ji, and it holds for every M-matrix Q, which Theorem 5.7 builds on. 5.2. Incentive Compatibility and Optimality Lemma 5.3 (Threshold well-posedness). For the provisional winner i∗i^*, Equation (3) has a unique solution t0∈(0,si∗]t_0∈(0,s_i^*] (proof: Appendix S2). The incentive guarantee is the standard ex-post characterization for correlated private values ((6), Prop. 5.1) applied to our allocation rule. A bidder cannot gain by misreporting when winning is monotone in its own report and the price is the lowest report that still wins. Lemmas 5.1 and 5.2 supply the monotonicity and Lemma 5.3 the price. A winner pays t0≤si∗t_0≤ s_i^* and losers pay zero (full proof: Appendix S2). Theorem 5.4 (EPIC and EPIR). The mechanism of Section 4 is ex-post incentive compatible (EPIC) and ex-post individually rational (EPIR); with private values (vi=siv_i=s_i), EPIC is equivalent to dominant-strategy IC (6). Theorem 5.5 (Revenue optimality). If the learned distribution equals the true distribution, the mechanism maximizes expected revenue among all EPIC and EPIR mechanisms with no positive transfers and finite expected payments (two conditions the unbounded Gaussian support requires). The theorem is Theorem 5.3 of (6). This paper verifies its hypotheses for every prior in the family and supplies the revenue identity on the unbounded support (Theorem S3). Condition-by-condition verification. (6), Thm. 5.3 requires (i) correlated private values (Section 3), (i) affiliation (Proposition 3.1), (i) regularity at every −is_-i (Lemma 5.1, so ironing never arises), (iv) cross-monotonicity (Lemma 5.2), and (v) single-item feasibility (rank-one matroid). Under (i)–(v), the revenue-maximal EPIC/EPIR mechanism in that characterization is exactly “highest nonnegative conditional virtual value wins at the threshold payment”; boundary and tie-breaking cases are Appendix S3. ∎ Corollary 5.6 (Strict advantage over IPV-Myerson). Suppose the mechanism uses the true distribution and the IPV baseline uses its true regular marginals. The weak inequality [RevAffMyerson]≥[RevIPV]E[Rev_ AffMyerson] [Rev_IPV] is immediate from Theorem 5.5, since IPV-Myerson is itself EPIC and EPIR; the inequality is strict whenever some Qij<0Q_ij<0, because the two allocations then disagree on a set of positive probability (proof Appendix S3). 5.3. Truthfulness under an Estimated Prior The results so far use the true prior. We now let the mechanism run on any prior in the family. Theorem 5.7 (Truthfulness for every prior in the family). Let the mechanism run with any M-matrix parameters (^,Q^)( μ, Q) satisfying Assumption 1, not necessarily equal to the true ones. Then it remains EPIC, hence dominant-strategy IC, and EPIR. Proof. EPIC and EPIR are structural properties of the allocation and payment rules ((6), Prop. 5.1): monotone allocation and threshold payments, neither of which references the true distribution. Lemmas 5.1 and 5.2 hold for every M-matrix Q Q, so the argument for Theorem 5.4 goes through with Q Q in place of Q. ∎ The M-matrix parameterization therefore separates learning the prior from incentives. Crémer and McLean (7) extract the full surplus under correlation but need the exact prior. We give up full extraction, and in exchange truthfulness holds under any prior error. Proposition 5.8 bounds what an error in the prior can cost, measured in the quantities the mechanism uses, each bidder’s conditional mean and conditional scale. Proposition 5.8 (Revenue stability). For any two priors in the family, the revenue difference is at most an explicit constant times the average error in the conditional means and scales, plus a term for the profiles where the two priors disagree about who wins or whether to sell. Revenue is therefore continuous in the fitted prior. Formal statement, constant, and proof: Appendix S6. The second family: log-normal values. The same guarantees hold for a second family, the log-normal, the long-tailed model commonly used in empirical work on advertising values (16, 17, 18). Under the transformation yi=logsiy_i= s_i affiliation is preserved, but the Jacobian puts a factor sis_i in front of the virtual value, and regularity is no longer automatic. Theorem 5.9 (The log-normal family: optimality and exact regularity). Let =logy= follow the M-matrix Gaussian of Assumption 1, with conditional moments μi|−i _i -i and σi|−i=Qii−1/2 _i -i=Q_i^-1/2. Then (i) the conditional virtual value is ψi(si∣−i)=si[1−σi|−iR(zi)] _i(s_i _-i)=s_i [1- _i -iR(z_i) ] with zi=(logsi−μi|−i)/σi|−iz_i=( s_i- _i -i)/ _i -i; (i) allocating to the highest nonnegative conditional virtual value at the threshold payment is EPIC and EPIR for every such prior, and when the prior is correct it maximizes expected revenue among EPIC and EPIR mechanisms with no positive transfers and finite expected payments; (i) ψi _i is increasing on its entire support if and only if σi|−i≤σ⋆=minz2/R(z)−z=1.5176 _i -i≤σ = _z\2/R(z)-z\=1.5176 The threshold characterizes global regularity. Neither truthfulness nor optimality needs it, because virtual values increase throughout their nonnegative region and negative virtual values never receive the item. It is loose (one standard deviation in log scale multiplies a value by 4.64.6, while the largest conditional log-scale in our fitted priors is 0.640.64), and the parameterization keeps every output globally regular by a floor of 0.45>1/(σ⋆)20.45>1/(σ )^2 on the precision diagonal, a design choice that the guarantees do not need. The experiments use the normal family as their main line. 6. Context-Dependent Neural Parameterization The theory above holds for any M-matrix prior (Figure 1). This section is about learning one from data. A network reads the embedding of the conversation and outputs, for each ad opportunity, a mean vector and a precision matrix, and the difficulty is to keep every output inside the M-matrix class. We give two constructions that do so by design: a one-factor form, which is small and matches the common-demand story, and a spectral form, which covers the whole class. We then describe two alternatives that we use as controls and a convex fit that needs no network. A frozen sentence encoder (all-MiniLM-L6-v2 (27, 28)) embeds the query, and shared MLP (multilayer perceptron) heads output, for each context, the means μi=Softplus(⋅) _i=Softplus(·) and the precision parameters below. Training is by maximum likelihood in precision form on historical bids and never forms the covariance matrix. Default construction (diagonal plus rank one). The economic premise of Section 1 is a common demand factor, and a prior with one common factor is automatically an M-matrix prior: Proposition 6.1 (Single factor ⇒ M-matrix). If Σ=D+λvv⊤ =D+λ v with D positive diagonal, λ>0λ>0, and v≥0v≥ 0 entrywise, then Q=Σ−1Q= ^-1 is a symmetric positive definite M-matrix. The proof is a Sherman–Morrison computation (Appendix S5). Proposition 6.1 gives a sufficient condition. Every one-factor prior is in the class, but the class is larger, and the spectral construction below covers the rest. Under the one-factor construction the network outputs only d∈ℝ>0nd ^n_>0 and v∈ℝ≥0nv ^n_≥ 0 (2n2n parameters), and viv_i is brand i’s sensitivity to the category demand shock. On real queries this default matches the full class to 0.020.02 cents (Section 7.6). General construction: spectral M-matrix. When correlation beyond a common factor is needed, we want a network head that can output any matrix in the class and no matrix outside it. The head outputs a positive diagonal D, a symmetric nonnegative matrix C0C_0 with zero diagonal, and a scalar g, and the prior’s precision is Q=D1/2(I−αC0)D1/2Q=D^1/2(I-α C_0)D^1/2 with α=sigmoid(g)/λmax(C0)α=sigmoid(g)/ _ (C_0). Dividing by the largest eigenvalue of C0C_0 keeps I−αC0I-α C_0 positive definite, multiplying by D1/2D^1/2 on both sides preserves positive definiteness, and the off-diagonal entries of Q are nonpositive because those of C0C_0 are nonnegative. Every output is therefore an M-matrix, and every M-matrix can be written this way, so the head covers the whole family except for two numerical-stability margins, a small gap below α’s upper limit and a floor on D (Appendix S5). The main-line experiments use this construction. Two alternative constructions. Strict diagonal dominance also yields M-matrices but caps the correlation the prior can express, and at n=4n=4 the cap binds (Section 7.2). Unconstrained learning leaves 2424–27%27\% of the off-diagonal entries outside the class and would need a repair step at every auction (Appendix S5). Both stay truthful by Theorem 5.7, so only revenue is at stake. The affiliation-constrained Gaussian maximum-likelihood fit needs no network at all (14, 15), and Section 7.6 shows it matches the network on real queries. The network extends that convex fit to conversation-dependent priors. 7. Experiments We evaluate AffMyerson, run with the normal prior family, on a semi-synthetic LLM-advertising benchmark where queries and bids are produced by LLMs, a hidden demand shock the platform cannot see induces correlation, and all revenue is settled on held-out LLM-agent bids. The prior is therefore estimated, never exact, which is the regime Theorem 5.7 is designed for. Section 7.1 describes the benchmark, Sections 7.2 and 7.3 the fitted priors and the revenue comparison, and Sections 7.4–7.8 test where the rent comes from and when it fails. 7.1. Benchmark Queries. We use gpt-4o-mini to generate 600600 designed conversational queries in the outdoor category over a “subcategory × purchase intent” grid with four intent tiers (transactional, comparison, informational, adjacent), plus 180180 special queries (niche and brand-specific) for the n=4n=4 study. Advertisers. Four price-stratified brands (flagship, premium, value, clearance) each bid independently per (query, demand level) through an LLM agent call (62,40062,400 calls; prompts and brand profiles Appendix S7). Because each brand bids without seeing the others, any subset of brands is a valid smaller market. The main line is n=2n=2 (premium++value, 12,00012,000 auctions). Hidden demand shock. A topic-independent demand signal u enters only the agents’ prompts and not the platform’s context encoding. All brands in a scenario read the same u, which creates the correlation the mechanism prices. Within-query correlation is +0.66+0.66, and all pairwise within-query correlations are positive (Appendix S7). Protocol. Queries are split 6:1:36:1:3 into train/validation/test, so no test conversation is seen in training. Learned quantities are means over 55 training seeds (seed deviations in the tables), and the bidders are gpt-4o-mini (31). Full prompts, brand profiles, and hyperparameters are in Appendix S7, and the code will be made publicly available. 7.2. Fitting the Conditional Distribution We first ask how well the parameterization fits. Of four priors fitted on the full n=4n=4 data (table Appendix S5), strict diagonal dominance learns an informative context mean (R2R^2 0.600.60 on unseen queries) but allows little correlation (mean fitted correlation ρ¯ ρ 0.220.22), and an unconstrained precision fit learns neither. The deployed spectral fit reads the context mean poorly (R2≈0R^2≈ 0) but captures the correlation (ρ¯ ρ 0.470.47), and revenue depends on that channel (Section 7.6). Per-cell likelihoods favor the log-normal overall but the normal in the high-value region where revenue concentrates (Appendix S10). 7.3. Revenue on Held-Out LLM-Agent Bids Table 1. Revenue on held-out LLM-agent bids (n=2n=2, cents, mean± deviation over 55 training seeds where a prior is learned; percentages use unrounded means). The last column is AffMyerson’s lead over that row. Five-seed and five-split reruns: Appendix S14. Mechanism Revenue Lead Second price (SPA) 28.3928.39 +33.3%+33.3\% Contextual-reserve SPA 32.98±0.1832.98± 0.18 +14.7%+14.7\% IPV-Myerson (same prior) 34.66±0.1934.66± 0.19 +9.2%+9.2\% Tuned-reserve SPA (r∗=32r^*=32) 35.6235.62 +6.2%+6.2\% Ronen lookahead (8) 35.67±0.0835.67± 0.08 +6.1%+6.1\% CAN, rank scores, no context (21) 37.43±0.0737.43± 0.07 +1.1%+1.1\% AffMyerson 37.84±0.0637.84± 0.06 Baselines. Besides the second-price auction (SPA, here equal to VCG) and SPA with the reserve tuned on the training set (r∗=32r^*=32), we run a contextual-reserve SPA (ctxSPA), which sets per-advertiser reserves from the same learned marginals; IPV-Myerson, which uses AffMyerson’s exact means and marginal deviations but does not condition; Ronen’s lookahead auction (8) with the same learned prior (LA), which gives the item to the highest bidder at the larger of the second bid and its conditional monopoly price; and CAN (21), a neural mechanism whose rank-score networks enforce the single-crossing conditions, trained on the same 7,2007,200 training bids with no prior and no context (implementation Appendix S7). In Table 1, AffMyerson earns 37.8437.84 cents. The strongest baseline is CAN at 37.4337.43, and the strongest closed-form baselines are LA at 35.6735.67 and the tuned-reserve SPA at 35.6235.62. The affiliation rent over IPV-Myerson is +9.2%+9.2\% and the margin over the tuned reserve is +6.2%+6.2\% (seed deviations in the table, reruns Appendix S14). Two comparisons show where the margin comes from. The ctxSPA comparison shows that the margin does not come from contextual reserves, since reserves set on the noisy context model earn 7.4%7.4\% less than the tuned constant. LA shows that conditional pricing alone is not enough. Pricing the highest bidder conditionally earns +2.9%+2.9\% over IPV but only ties the tuned reserve. AffMyerson also ranks bidders by conditional virtual value (its winner differs from the highest bidder in 5.6%5.6\% of auctions) and earns +6.1%+6.1\% over LA, so the margin comes from conditional allocation. CAN, at 37.4337.43, recovers most of the rent from the bids alone. AffMyerson earns more, +1.1%+1.1\%, and every AffMyerson seed exceeds every CAN seed. Beyond revenue, AffMyerson is a closed-form rule with a prior the platform can inspect, priced in O(n2)O(n^2) time, and its incentive compatibility and the robustness guarantee of Theorem 5.7 hold by construction. Revenue does not come at the cost of efficiency, since against IPV-Myerson AffMyerson raises both welfare, the winner’s value (53.853.8 vs. 48.248.2 cents), and the trade rate (89.8%89.8\% vs. 69.7%69.7\%). What the conversation is worth. In a market where the conversation matters to one advertiser and not to the other, AffMyerson earns 24.8%24.8\% more than CAN, and the gap is the value of the information that bids alone do not carry. Markets like this exist. A marketplace that sells everything bids about the same on any outdoor query, while a specialist brand’s value depends on what the conversation says. We write 400400 conversations in 200200 pairs, where the same product words carry either a high-value cue (an upcoming trip, a broken item) or a low-value one (a school project, an item already bought). The marketplace’s value is 3030 cents in every auction, and the specialist’s is 200200 on a high cue and 100100 on a low one, each with a spread of 2525. Values are drawn from the Gaussian family, so the prior is fitted as on the main line. The two advertisers’ values are independent here, so the whole gain comes from the part of our design that reads the conversation, the context-dependent prior. On 2,4002,400 held-out auctions AffMyerson earns 104.8104.8 cents and CAN 84.184.1. CAN earns as much as the bids allow. Without the context, the most any mechanism can earn is what Myerson’s auction on the marginal bid distribution earns, 84.484.4, and CAN reaches it. The rest is earned on the low-cue conversations, where a mechanism that sees only bids sells to the marketplace for 24.724.7 cents while the fitted prior places the specialist’s value near 100100 and sells to it for 66.866.8. On the main line every advertiser reads the same conversation, the bids move together, and CAN reads the conversation through the bids. Here the marketplace’s bid does not reveal the cue, so only a prior that reads the conversation can price it, at no cost in truthfulness (Theorem 5.7; design and full table Appendix S7). 7.4. Rent versus Affiliation Strength The rent rises and falls with the correlation in the data (Figure 3). To check that the rent comes from the correlation, we remove the correlation without changing the marginals. Within each query we re-pair the second brand’s bids across scenarios, so that full shuffling zeroes the within-query correlation, partial shuffling interpolates, and each brand’s marginal is unchanged element by element. The prior is refit and the reserve re-tuned at every level. Figure 3 shows the rent over IPV-Myerson and the margin over the tuned-reserve SPA along this curve. We observe that the rent rises monotonically with ρ, is statistically zero at ρ≈0ρ≈ 0, and tracks the fitted ρ¯ ρ, while the margin over the tuned reserve becomes significant only above ρ≈0.3ρ≈ 0.3. These thresholds are specific to this benchmark, so a platform should measure its own before it adopts conditioning. Revenue lift over IPV-Myerson and over the tuned-reserve second-price auction as the within-query correlation is varied. Figure 3. Shuffling scan (prior refit and reserve re-tuned at every level): lift over IPV-Myerson (blue) and over the tuned-reserve SPA (orange) against within-query ρ. The panel that tracks the fitted ρ¯ ρ is Appendix S9.Revenue lift over IPV-Myerson and over the tuned-reserve second-price auction as the within-query correlation is varied. 7.5. n=4n=4 Stratified Markets We next ask whether the margin survives more bidders. On the full four-brand data the revenue is 49.0549.05 cents, +4.2%+4.2\% over the tuned-reserve SPA and +6.2%+6.2\% over matched IPV. Across the 1111 brand submarkets (table Appendix S14) we observe that the fitted ρ¯ ρ alone does not rank the markets. The three lowest-rent submarkets each pair a premium brand with a clearance brand whose near-constant bids carry no usable signal, so the rent depends on how informative competitors’ reports are as well as on correlation. The median rent across submarkets is flat in n, so added competitors bring conditioning information that offsets dilution. 7.6. Real Search Queries To test whether the findings depend on simulated queries, we build a real-query benchmark of 580580 genuine user queries from Amazon ESCI (29) (274274) and MS MARCO (30) (306306), filtered by embedding ranking and two LLM gates and passed unchanged through the same bidding pipeline (Appendix S16). The rent persists on real queries. The premium brand bids 4.604.60 times more on transactional than on informational queries and the two brands’ bids correlate at 0.7220.722 within a query, so both market features of Section 7.7 reappear on queries we did not write. A short search query gives the prior little to read (Appendix S16). Even so, the rent is +4.5%+4.5\% and the margin over the tuned reserve is +4.7%+4.7\% (per-source and channel diagnostics: Appendix S16). What the network adds. On short search queries, where the text gives the prior little to read, a convex MTP2MTP_2-constrained Gaussian fit with no network (14, 15) earns the same revenue as the embedding network (43.6543.65 vs. 43.4843.48 cents, within seed noise) and a one-factor prior (Proposition 6.1) is within 0.020.02 of it, so the rent does not depend on the fitting method. On the designed conversational queries, where the text does predict values, an unconstrained network repaired into the class earns 2.0%2.0\% more than the best convex fit (49.2949.29 vs. 48.3348.33 cents; Appendix S11). 7.7. How Bidding Agents Turn Common Demand into Bids We now ask where the correlation the mechanism prices comes from. Our bidders are LLM agents acting as advertisers. Each receives a brand profile of behavioral traits, the conversation, and the shared demand signal, and returns a bid. We check their behavior against two features of real sponsored-search markets, that bids rise with commercial intent (19) and that rivals’ values move with common demand shocks (2), and the agents reproduce both. On the main line the premium brand bids 5.765.76 times more on transactional than on informational queries, and the two brands’ bids within a query correlate at +0.66+0.66. Within a query, the shared demand shock is the only input the agents have in common (Appendix S7), so the question is how differently prompted agents translate it into bids. With queries and shocks fixed, within-query correlation is +0.78+0.78 when the agents judge the conversation and +0.06+0.06 when they are prompted to calculate (seven schemes, two models; table Appendix S7), and the calculating agents also concentrate their bids near zero. This variation matters for auction design. On the full-scale calculation-type bids AffMyerson still earns +2.5%+2.5\% over IPV-Myerson but less than the tuned reserve, so what conditioning is worth depends on how agents express demand in their bids. 7.8. Robustness to Prior Perturbation Last, we corrupt the learned prior deliberately, in twelve controlled ways (n=4n=4, seed 00), and find that revenue falls gradually while truthfulness is unaffected. We observe three things. Mean noise ε reduces the rent monotonically, from +6.4%+6.4\% at ε=0 =0 to −4.3%-4.3\% at ε=20 =20, consistent with Proposition 5.8, which bounds the loss from above only. Setting the correlation to zero gives the revenue of IPV-Myerson. No corruption produces an incentive violation (00 of 28,80028,800 probes), as Theorem 5.7 requires. At ε=20 =20 AffMyerson falls below the tuned-reserve SPA, so a platform with an unusable prior should prefer prior-free second pricing. 8. Scope and Limitations Our LLM-agent benchmarks show strong positive within-query correlation, consistent with affiliation, and IPV mechanisms give up the rent on such data. Five limitations remain. (i) The guarantee covers nonnegative dependence only, so negatively correlated substitutes are outside it. (i) The values are correlated private values and not interdependent values. (i) The auction sells one item, and multi-slot allocation needs a combinatorial extension. (iv) Bids are LLM-agent outputs, since no public dataset offers jointly observed real bids on one opportunity. (v) Learned priors invite history shading (20), and the channel that earns the rent also transmits a rival’s misreport. Averaging the prior across periods and capping the conditional-mean shift limit both, at little or no revenue cost (Appendices S12 and S13). References (1) R. B. Myerson. Optimal auction design. Math. Oper. Res., 6(1):58–73, 1981. (2) P. R. Milgrom and R. J. Weber. 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MS MARCO: A human generated machine reading comprehension dataset. In CoCo@NeurIPS, 2016. (31) OpenAI. GPT-4o mini: advancing cost-efficient intelligence. OpenAI blog, July 2024. https://openai.com/index/gpt-4o-mini-advancing-cost-efficient-intelligence/ Supplementary Material Sections S1–S16 below are the supplementary material. “Main text” refers to Sections 1–8. Appendix S1 Complete Gaussian derivations S1.1. Affiliation and the M-matrix sign condition (main text, Proposition 3.1) The log-density of (,Q−1)N( μ,Q^-1) is lnf()=const−12(−)⊤Q(−) f(s)=const- 12(s- μ) Q(s- μ), whose cross-partials ∂2lnf/∂si∂sj=−Qij∂^2 f/∂ s_i∂ s_j=-Q_ij are constant in s. MTP2MTP_2 requires them to be nonnegative. If some Qij>0Q_ij>0 the requirement fails; if all Qij≤0Q_ij≤ 0 then lnf f is supermodular on ℝnR^n, which implies MTP2MTP_2 by Karlin and Rinott (1980, J. Multivariate Anal. 10, Theorem 3.1). S1.2. Conditioning from the precision matrix Partition the quadratic form for bidder i: (−)⊤Q(−)=Qii(si−μi)2+2(si−μi)∑j≠iQij(sj−μj)+C(−i).(s- μ) Q(s- μ)=Q_i(s_i- _i)^2+2(s_i- _i) _j≠ iQ_ij(s_j- _j)+C(s_-i). Completing the square gives si|−i∼(μi−1Qii∑j≠iQij(sj−μj),1Qii).s_i _-i \! ( _i- 1Q_i _j≠ iQ_ij(s_j- _j), 1Q_i ). Only row i of Q is needed. S1.3. Regularity and cross-monotonicity Let R(z)=(1−Φ(z))/ϕ(z)R(z)=(1- (z))/φ(z). Direct differentiation gives R′(z)=zR(z)−1R (z)=zR(z)-1. Hence, for ψi(si∣−i)=si−σiR(zi) _i(s_i _-i)=s_i- _iR(z_i), ∂ψi∂si=2−ziR(zi)>1. ∂ _i∂ s_i=2-z_iR(z_i)>1. For zi>0z_i>0, the standard Mills bound gives R(zi)<1/ziR(z_i)<1/z_i; for zi≤0z_i≤ 0, ziR(zi)≤0z_iR(z_i)≤ 0. For j≠ij≠ i, ∂μi|−i∂sj=−QijQii≥0. ∂ _i -i∂ s_j=- Q_ijQ_i≥ 0. Because R is strictly decreasing, raising sjs_j weakly lowers ψi _i, strictly when Qij<0Q_ij<0. Mass below zero of the fitted priors (main text, Remark 1). A prior whose marginal mean satisfies μi()>3(Q()−1)ii _i(e)>3 (Q(e)^-1)_i puts negligible mass (<0.14%<0.14\%) below zero. The fitted spectral priors do not satisfy this on most contexts: on the seed-00 prior and test set the ratio μi/Σii _i/ _i has minimum 0.370.37 and exceeds 33 on only 14%14\% of context–bidder pairs, so on many contexts the prior puts non-negligible mass below zero (kappa_box.py). Nothing formal depends on it, and in the replay every signal is an observed bid in [2,185][2,185] cents. Parameters of the main text’s Figure 2 and the example of Section 4.3. The illustrative two-bidder prior has =(45,35) μ=(45,35) cents, marginal deviations σ=(13,12)σ=(13,12) cents and ρ=0.7ρ=0.7, i.e. Σ=(169109.2109.2144) = ( smallmatrix169&109.2\\ 109.2&144 smallmatrix ) and precision Q=Σ−1=(0.01160−0.00880−0.008800.01362)Q= ^-1= ( smallmatrix0.01160&-0.00880\\ -0.00880&0.01362 smallmatrix ), an M-matrix. Conditional scales σi|−i=Qii−1/2=(9.28, 8.57) _i -i=Q_i^-1/2=(9.28,\,8.57) against marginal (13,12)(13,12); conditional-mean slopes β12=−Q12/Q11=0.758 _12=-Q_12/Q_11=0.758 and β21=0.646 _21=0.646, so a bid of 6161 by bidder 2 moves bidder 1’s conditional mean to 45+0.758⋅26=64.745+0.758· 26=64.7. IPV-Myerson reserves (the dashed lines of the left panel) solve φiIPV(r)=0 ^IPV_i(r)=0: r1=34.8r_1=34.8, r2=27.3r_2=27.3. Expected revenue over the prior, by quadrature on an 8012801^2 grid over [−60,200]2[-60,200]^2 with 6060-step bisection payments: AffMyerson 36.8936.89, IPV-Myerson 35.0035.00, difference +1.90+1.90 cents (+5.4%+5.4\%). Script draw_fig2.py (with fig2_core.py) regenerates the figure and prints these numbers. Appendix S2 Incentives and payments Numerical tolerance of the payment (main text, Section 4.2). Bisection with 6060 steps on the bracket [0,si∗][0,s_i^*] locates t0t_0 to 2−60si∗<2×10−162^-60s_i^*<2× 10^-16 cents for bids capped at 200200. The mechanism is exactly EPIC; only the payment is evaluated numerically, to this tolerance plus the floating-point error of the Mills ratio (same order). Theorem S1 (EPIC and EPIR). For any mean vector and any SPD M-matrix used by the mechanism, allocation to the highest nonnegative conditional virtual value with critical payment is ex-post incentive compatible and ex-post individually rational. Proof. Fix −is_-i. Raising sis_i strictly raises bidder i’s score and weakly lowers every competing score, so the binary allocation xi(si,−i)x_i(s_i,s_-i) is monotone. A monotone single-parameter allocation with its critical payment is dominant-strategy incentive compatible. If bidder i wins truthfully, its report is at least the critical threshold, so payment is at most sis_i. Losers pay zero. Thus utility is nonnegative for every report profile. ∎ Theorem S2 (Threshold existence and uniqueness). For a provisional winner i with report sis_i, the score gap in the payment equation has a unique zero t0∈(0,si]t_0∈(0,s_i]. Proof. Let D(t)=ψi(t∣−i)−max0,maxj≠iψj(sj∣t,−i,j).D(t)= _i(t _-i)- \0, _j≠ i _j(s_j t,s_-\i,j\)\. The first term is strictly increasing in t and every term inside the maximum is nonincreasing, so D is strictly increasing and continuous. Since ψi(0∣−i)<0 _i(0 _-i)<0, D(0)<0D(0)<0. Since i is the selected provisional winner, D(si)≥0D(s_i)≥ 0. The intermediate value theorem therefore gives a zero in (0,si](0,s_i], and strict monotonicity gives uniqueness. Hence the bracket [0,si][0,s_i] contains exactly one root; the implementation bisects it for 6060 steps (run_auctions4.py), reaching resolution 2−60si<2×10−162^-60s_i<2× 10^-16 cents for bids capped at 200200. ∎ Payments need not increase in every competitor’s report when n≥3n≥ 3. Cross-monotonicity concerns the allocation. The winner’s threshold t0t_0 solves ψw(t0∣−w)=max0,maxj≠wψj(sj∣t0,−w,j) _w(t_0 _-w)= \0, _j≠ w _j(s_j t_0,s_-\w,j\)\; a third bidder’s report enters both sides. For n=2n=2 both channels push t0t_0 up. For n≥3n≥ 3 they can conflict: with Q=(10001−0.80−0.81)Q= ( smallmatrix1&0&0\\ 0&1&-0.8\\ 0&-0.8&1 smallmatrix ) (eigenvalues 0.2,1,1.80.2,1,1.8, an SPD M-matrix), =(5,5,0) μ=(5,5,0), s1=20s_1=20, s2=10s_2=10, raising s3s_3 from 00 to 11 leaves bidder 11 the winner but lowers its payment from 10.00010.000 to 9.9689.968: bidder 11 is uncoupled from 33, while 33’s higher report lowers ψ2 _2 and with it the competing threshold. The main text’s “∂t0/∂sj>0∂ t_0/∂ s_j>0” is therefore stated for the bivariate case only. Appendix S3 Revenue identity on unbounded Gaussian support Theorem S3 (Gaussian-tail extension). When the learned prior equals the true Gaussian prior, AffMyerson maximizes expected revenue among EPIC, EPIR, no-positive-transfer mechanisms with finite expected payments. Proof. Fix −is_-i and abbreviate the conditional CDF and density by F and f. For a monotone allocation x(t)x(t), the envelope theorem gives u(t)=u(a)+∫atx(z)z.u(t)=u(a)+ _a^tx(z)\,dz. EPIR and no positive transfers imply x(t)=0x(t)=0 for t<0t<0 and u(0)=0u(0)=0. Therefore, for t≥0t≥ 0, p(t)=tx(t)−∫0tx(z)z,p(t)=tx(t)- _0^tx(z)\,dz, while p(t)=x(t)=0p(t)=x(t)=0 for t<0t<0. Since the Gaussian has a finite first moment, Tonelli’s theorem applies: [p∣−i] [p _-i] =∫0∞tx(t)f(t)t−∫0∞f(t)∫0tx(z)zt = _0^∞tx(t)f(t)\,dt- _0^∞f(t) _0^tx(z)\,dz\,dt =∫0∞x(t)tf(t)−(1−F(t))t = _0^∞x(t)\tf(t)-(1-F(t))\\,dt =[x(si,−i)(si−1−F(si)f(si))|−i]. =E\! [x(s_i,s_-i) (s_i- 1-F(s_i)f(s_i) ) |s_-i ]. Summing over bidders converts revenue to conditional virtual surplus. Pointwise, single-item feasibility is maximized by allocating to the highest nonnegative conditional virtual value. The preceding monotonicity proof makes this allocation implementable, and critical payments attain the identity. Finally ψi<si _i<s_i, so any allocated type is positive and EPIR holds. ∎ Boundary and ties (main text, Theorem 5.5). Two events need a convention: a tie for the highest conditional virtual value, and a maximum exactly equal to zero. Under any nondegenerate Gaussian prior both have probability zero, so every fixed convention yields the same expected revenue and the same incentive properties (the allocation remains monotone and the payment remains the threshold). The implementation allocates when the maximum is ≥0≥ 0 and breaks exact ties by the lowest bidder index (argmax); the threshold bracket [0,si∗][0,s_i^*] of the main text’s Lemma 5.3 is unaffected. Dominance over IPV: full proof of the main text’s Corollary 5.6. The main text states the corollary without a sketch; the argument is as follows. IPV-Myerson is a feasible EPIC/EPIR mechanism whose revenue equals its expected true conditional virtual surplus (Roughgarden and Talgam-Cohen 2016, Prop. 5.2), while AffMyerson maximizes that integrand pointwise, which gives the weak inequality. For strictness: when Qij<0Q_ij<0 the moving threshold of AffMyerson differs from the fixed IPV reserve for all but one value of sjs_j, and on the open interval between them exactly one mechanism allocates to i (the one with the larger integrand) on a set of positive MVN measure. Regular IPV-Myerson is itself EPIC and EPIR, hence the theorem gives weak revenue dominance under the true prior. The inequality is strict exactly when the IPV allocation fails to maximize true conditional virtual surplus on a positive-probability set; when Qij<0Q_ij<0 for some pair, such a set can be exhibited constructively, so strictness there is derived, not assumed. Constructive proof of strictness. IPV-Myerson is a feasible EPIC/EPIR mechanism: its allocation is monotone in each own signal (regular marginals) with threshold payments. By the revenue identity above (equivalently, Prop. 5.2 of Roughgarden and Talgam-Cohen), any EPIC/EPIR mechanism’s expected revenue equals its expected conditional virtual surplus, [∑ixi()ψi(si∣−i)]E_s[ _ix_i(s) _i(s_i _-i)] with the true ψi _i common to both mechanisms. AffMyerson maximizes the integrand pointwise, so its revenue weakly dominates. Strictness: pick i,ji,j with Qij<0Q_ij<0 and fix the other coordinates. Let rir_i solve φiIPV(ri)=0 ^IPV_i(r_i)=0 (independent of −is_-i) and ρi(−i) _i(s_-i) solve ψi(ρi∣−i)=0 _i( _i _-i)=0 (unique by the unconditional-regularity lemma). By cross-monotonicity, sj↦ψi(ri∣−i)s_j _i(r_i _-i) is strictly decreasing, so ρi(−i)≠ri _i(s_-i)≠ r_i except at most one sjs_j. Choose sjs_j low enough that competitor virtual values are negative on the relevant compact set; then for sis_i in the nonempty open interval between rir_i and ρi _i, exactly one mechanism allocates to i: if IPV allocates where AffMyerson does not (ρi>ri _i>r_i), the IPV integrand is ψi<0 _i<0 versus 00; in the opposite case AffMyerson’s integrand is ψi>0 _i>0 versus 00. Either way IPV’s integrand is strictly smaller on an open set of positive MVN measure; integrating gives the strict inequality. ∎ Appendix S4 Log-link regularity: the complete characterization Under the identity link (main text), ∂ψi/∂si=2−ziR(zi)>1∂ _i/∂ s_i=2-z_iR(z_i)>1 by the standard bound zR(z)<1zR(z)<1, so regularity is automatic. This section gives the complete characterization promised in the main text for the multiplicative (log-normal) specification. Lemma S1 (Conditional virtual value under the log link). Let yi=logsiy_i= s_i follow the M-matrix MVN of the main text’s assumption. Then si|−is_i _-i is log-normal and ψi(si∣−i)=si[1−σi|−iR(zi)],zi=logsi−μi|−iσi|−i,σi|−i=Qii−1/2. _i(s_i _-i)=s_i [1- _i -iR(z_i) ], z_i= s_i- _i -i _i -i, _i -i=Q_i^-1/2. Proof. The conditional law of yiy_i is (μi|−i,σi|−i2)N( _i -i,σ^2_i -i). Under si=eyis_i=e^y_i, Fi|−i(s)=Φ(z)F_i -i(s)= (z) and fi|−i(s)=ϕ(z)/(sσi|−i)f_i -i(s)=φ(z)/(s\, _i -i) (the 1/s1/s is the Jacobian), so (1−F)/f=sσi|−iR(z)(1-F)/f=s\, _i -iR(z), and the general conditional virtual value s−(1−F)/fs-(1-F)/f gives the display. ∎ Theorem S4 (Regularity threshold σ⋆σ ). In the setting above, ∂ψi∂si=2−(zi+σi|−i)R(zi), ∂ _i∂ s_i=2- (z_i+ _i -i )R(z_i), so ψi _i is strictly increasing in sis_i iff supz(z+σi|−i)R(z)≤2 _z(z+ _i -i)R(z)≤ 2. Writing gσ(z)=(z+σ)R(z)g_σ(z)=(z+σ)R(z), the map σ↦supzgσ(z)σ _zg_σ(z) is strictly increasing and crosses 22 at a unique σ⋆=1.517619…,σ =1.517619…, so regularity holds iff σi|−i≤σ⋆ _i -i≤σ , equivalently Qii≥1/(σ⋆)2=0.434184…Q_i≥ 1/(σ )^2=0.434184… For σi|−i<σ⋆ _i -i<σ the slope is bounded away from zero; at σi|−i=σ⋆ _i -i=σ it vanishes at exactly one point (z⋆=−0.551z =-0.551) and ψi _i is still strictly increasing; for σi|−i>σ⋆ _i -i>σ it is negative on an open interval and regularity fails. Proof. Differentiating, with ∂zi/∂si=1/(siσi|−i)∂ z_i/∂ s_i=1/(s_i _i -i) and R′(z)=zR(z)−1R (z)=zR(z)-1 (from ϕ′=−zϕφ =-zφ), ∂ψi∂si=(1−σi|−iR(zi))−R′(zi)=2−(zi+σi|−i)R(zi). ∂ _i∂ s_i= (1- _i -iR(z_i) )-R (z_i)=2-(z_i+ _i -i)R(z_i). Finiteness of the supremum: as z→+∞z→+∞, R(z)=z−1−z−3+O(z−5)R(z)=z^-1-z^-3+O(z^-5) gives gσ(z)→1g_σ(z)→ 1; as z→−∞z→-∞, R(z)∼2πez2/2R(z) 2πe^z^2/2 while (z+σ)→−∞(z+σ)→-∞, so gσ→−∞g_σ→-∞; hence the supremum is attained at an interior point. Monotonicity in σ: ∂gσ/∂σ=R(z)>0∂ g_σ/∂σ=R(z)>0 pointwise, so the supremum is strictly increasing in σ; at σ=0σ=0, supzzR(z)=1<2 _zzR(z)=1<2, and the supremum diverges as σ→∞σ→∞, so continuity gives a unique crossing σ⋆σ . The stationarity condition R(z)[1+z(z+σ)]=z+σR(z)[1+z(z+σ)]=z+σ is transcendental, so σ⋆σ is determined numerically: σ⋆=1.517619…σ =1.517619… ∎ Constructive guarantee (design choice). In the spectral parameterization Qii=DiiQ_i=D_i (the coupling matrix has zero diagonal), so raising the diagonal floor of D to c0≥0.45>0.434184c_0≥ 0.45>0.434184 makes σi|−i<σ⋆ _i -i<σ hold for every network output. Under the log link, global regularity becomes an architectural property exactly as affiliation is, with no inference-time checks; by the proposition above this is a design choice, since neither truthfulness nor optimality depends on it. The identity-link deployment in the main text needs no floor at all. What σ⋆σ does and does not govern. The theorem characterizes textbook regularity, monotonicity of ψi _i on the whole support. Neither truthfulness nor optimality needs it in the single-item setting, as the next proposition (main text, Theorem 5.9(i)) shows. Proposition S1 (Truthfulness and optimality for every conditional log-scale). Under the log link with any SPD M-matrix Q, the allocation “highest nonnegative conditional virtual value wins” with threshold payments is EPIC and EPIR. If the prior is correct, it maximizes expected revenue among all EPIC and EPIR mechanisms with no positive transfers and finite expected payments, for every value of σi|−i _i -i. Proof. Step 1 (the nonnegative region is an up-set). ψi/si=1−σi|−iR(zi) _i/s_i=1- _i -iR(z_i) is strictly increasing in sis_i, since ziz_i increases in sis_i and R is decreasing; hence ψi≥0=[si⋆,∞)\ _i≥ 0\=[s_i ,∞) for some si⋆∈(0,∞)s_i ∈(0,∞). Step 2 (monotone on that region). On ψi≥0\ _i≥ 0\ one has σR(zi)≤1σ R(z_i)≤ 1, hence ∂ψi∂si=2−ziR(zi)−σR(zi)≥ 1−ziR(zi)> 0 ∂ _i∂ s_i=2-z_iR(z_i)-σ R(z_i)\ ≥\ 1-z_iR(z_i)\ >\ 0 for every σ>0σ>0, so ψi _i is strictly increasing wherever it is nonnegative. Cross-monotonicity carries over under the monotone link (∂μj|−j/∂yi=−Qji/Qjj≥0∂ _j -j/∂ y_i=-Q_ji/Q_j≥ 0 and ψj _j is decreasing in μj|−j _j -j). Therefore, as sis_i rises, bidder i moves from “never wins” (ψi<0 _i<0) to a region where its own virtual value rises and every rival’s falls: the allocation is monotone in sis_i for every −is_-i, the threshold t0t_0 is unique, and EPIC/EPIR follow exactly as in the main text’s Theorem 5.4. Step 3 (revenue identity without regularity). Fix −is_-i and let xi(⋅∣−i)x_i(· _-i) be any allocation rule that is nondecreasing in sis_i, with the EPIR-binding payment pi(si)=xi(si)si−∫0sixi(t)tp_i(s_i)=x_i(s_i)s_i- _0^s_ix_i(t)\,dt. Integrating by parts against the conditional density fi|−if_i -i gives [pi∣−i]=[xiψi∣−i]E[p_i _-i]=E[x_i\, _i _-i] provided the boundary term xi(s)s(1−Fi|−i(s))→0x_i(s)\,s\,(1-F_i -i(s))→ 0 as s→∞s→∞ and [si∣−i]<∞E[s_i _-i]<∞. Both hold for the log-normal: [si∣−i]=eμ+σ2/2<∞E[s_i _-i]=e^μ+σ^2/2<∞ and s(1−Φ(z))≤sϕ(z)/z→0s(1- (z))≤ s\,φ(z)/z→ 0. The identity uses only monotonicity and the IR normalization, not regularity. Summing over i and taking expectations, every EPIC/EPIR mechanism with no positive transfers and finite expected payments has Rev=[∑ixiψi]Rev=E[ _ix_i _i]. Step 4 (pointwise bound and attainment). Since ∑ixi≤1 _ix_i≤ 1 and xi≥0x_i≥ 0, ∑ixiψi≤max0,ψ1,…,ψn _ix_i _i≤ \0, _1,…, _n\ pointwise, so Rev≤[max0,maxiψi]Rev [ \0, _i _i\]. The mechanism of Step 2 attains this bound pointwise, is EPIC/EPIR by Step 2, and has finite expected payments (payments are bounded by the winner’s value, which has finite mean). Hence it is optimal. ∎ Global regularity therefore matters only for the un-ironed rule to coincide with Myerson’s ironed rule on the whole support; where it fails, on part of ψi<0\ _i<0\, the item is never sold, so nothing changes. We keep σ≤σ⋆σ≤σ as the condition under which the virtual value is a textbook regular one, and enforce it in the deployed parameterization as a design choice (floor c0=0.45c_0=0.45), not as a requirement of the guarantees. Appendix S5 Learned hypothesis class Proof of the main text’s Proposition 6.1 (single factor ⇒ M-matrix). Let Σ=D+λvv⊤ =D+λ v with D positive diagonal, λ>0λ>0 and v≥0v≥ 0 entrywise. By the Sherman–Morrison formula, with c=1+λv⊤D−1v>0c=1+λ v D^-1v>0, Q=Σ−1=D−1−λc(D−1v)(D−1v)⊤,Q= ^-1=D^-1- λc\,(D^-1v)(D^-1v) , Qij=−λvivjcdidj≤0(i≠j),Q_ij=- λ\,v_iv_jc\,d_id_j≤ 0 (i≠ j), so the off-diagonal entries are nonpositive, and Σ≻0 0 gives Q≻0Q 0. Hence Q is a symmetric positive definite M-matrix. For every context, the deployed network emits a positive diagonal D, a symmetric nonnegative zero-diagonal matrix C0C_0, and a scalar g. It sets α=(1−ϵ)sigmoid(g)λmax(C0),Q=D1/2(I−αC0)D1/2.α= (1-ε)sigmoid(g) _ (C_0), Q=D^1/2(I-α C_0)D^1/2. For a symmetric nonnegative matrix, Perron–Frobenius gives spectral radius λmax(C0) _ (C_0). Hence all eigenvalues of αC0α C_0 have magnitude strictly below one, so I−αC0≻0I-α C_0 0. Congruence by D1/2D^1/2 preserves positive definiteness, and the off-diagonals of Q are nonpositive. Thus Q is an M-matrix for every network weight vector and context. A stricter diagonally dominant construction is retained as an ablation. Figure 4. Deployment pipeline. Offline (top lane): historical bids and query embeddings train the context-to-prior map; the spectral parameterization makes every network output an M-matrix, so truthfulness never depends on fit quality (main text, misspecification-robustness theorem). Online (bottom lane): one O(n2)O(n^2) row-read of the precision matrix plus a 6060-step bisection prices each auction in ∼ 12μ12\, at n=2n=2 (pricing only; p50 0.30.3 ms end-to-end with the network forward pass, Section S14). Theorem S5 (Completeness of the spectral representation). Let Q be any symmetric positive-definite Z-matrix. Define D=diag(Q),B=I−D−1/2QD−1/2.D=diag(Q), B=I-D^-1/2QD^-1/2. Then D≻0D 0, B is symmetric, nonnegative and zero-diagonal, ρ(B)<1ρ(B)<1, and Q=D1/2(I−B)D1/2Q=D^1/2(I-B)D^1/2. Conversely, any positive diagonal D and symmetric nonnegative zero-diagonal B with ρ(B)<1ρ(B)<1 produce an SPD Z-matrix through this formula. Proof. Positive definiteness gives Qii>0Q_i>0. Directly, Bii=0B_i=0 and Bij=−Qij/QiiQjj≥0B_ij=-Q_ij/ Q_iQ_j≥ 0 for i≠ji≠ j. Moreover, I−B=D−1/2QD−1/2≻0I-B=D^-1/2QD^-1/2 0, hence every eigenvalue of B is strictly below one. Since B is nonnegative, Perron–Frobenius identifies its spectral radius with its largest eigenvalue, so ρ(B)<1ρ(B)<1. The factorization is the definition rearranged. Conversely, symmetry and ρ(B)<1ρ(B)<1 imply I−B≻0I-B 0; congruence by D1/2D^1/2 preserves positive definiteness, while the signs of the off-diagonals are nonpositive. ∎ The theorem concerns the abstract factorization. The implementation uses entrywise softplus weights, diagonal floor Dii>10−3D_i>10^-3, and strength (1−ϵ)sigmoid(g)(1-ε)sigmoid(g) with ϵ=10−3ε=10^-3, so its realized set is ϵ=D1/2(I−B)D1/2: _ε=\D^1/2(I-B)D^1/2: Dii>10−3,B>0entrywise off-diagonal, D_i>10^-3,\ B>0\ entrywise off-diagonal, ρ(B)<1−ϵ, ρ(B)<1-ε\, a strict subset of the cone. Four precise statements replace any “entire class” reading. (i) Every SPD M-matrix with Qii>10−3Q_i>10^-3 for all i, normalized coupling ρ(B)<1−ϵρ(B)<1-ε, and strictly negative off-diagonals is realized exactly. (i′) Since Qii=DiiQ_i=D_i in this factorization, the diagonal floor excludes matrices with a conditional precision below 10−310^-3, i.e. a conditional scale above 1/10−3=31.61/ 10^-3=31.6 cents, a hard cap on how diffuse a conditional can be rather than a spectral sliver; the fitted seed-00 prior touches it (miniQii=0.032 _i Q_i=0.032). Truthfulness is unaffected; the coverage claim is “the cone with ρ(B)<1−ϵρ(B)<1-ε and Qii>10−3Q_i>10^-3”, not “the cone up to an ϵε-sliver”. (i) M-matrices with zero off-diagonal entries (including diagonal Q) are not realized exactly but lie in the closure of ϵS_ε, since softplus weights can be made arbitrarily small. (i) M-matrices with ρ(B)∈(1−ϵ,1)ρ(B)∈(1-ε,1), within ϵε of singularity, are excluded, by design: the margin doubles as a conditioning guarantee for the online inversion-free pricing. Both exclusions, the spectral sliver and the diagonal floor, are dimension-independent; in particular no dimension-dependent partial-correlation budget of the diagonally dominant 1/(n−1)1/(n-1) kind applies at any n, which is the property the main text uses. (Strict diagonal dominance budgets partial correlations; marginal correlations can still approach 11, but only at vanishing dominance margin.) Relaxed-model audit. We also fit an SPD precision-Cholesky model and repair it by clipping positive off-diagonals, symmetrizing, and shifting the diagonal when the minimum eigenvalue is too small. This is not an exact nearest-cone projection. On the four-advertiser benchmark the repair is material and reserve-relative revenue is seed-unstable; these diagnostics are why the paper deploys the structured model instead. Four priors on the designed n=4n=4 data (main text, Section 7.2). Same split (6:1:36:1:3 over 780780 queries), five optimization seeds, test-set statistics; R2R^2 is the mean over the four brands of the context-mean prediction on unseen queries, ρ¯ ρ the mean fitted correlation over all six brand pairs, ρ01 _01 the fitted correlation of the flagship–premium pair, and “outside cone” the share of positive off-diagonal precision entries before any repair (fit_four_priors_n4b.py, fit_four_priors_n4b.json). Prior Test NLL R2R^2 ρ¯ ρ ρ01 _01 Outside cone A: global closed form (no context) 18.5618.56 −0.00-0.00 0.200.20 0.670.67 33%33\% B: unconstrained precision Cholesky 20.75±0.4620.75± 0.46 −0.03±0.02-0.03± 0.02 0.47±0.060.47± 0.06 0.61±0.030.61± 0.03 25±3%25± 3\% C: diagonally dominant M-matrix 19.00±0.6219.00± 0.62 +0.60±0.06+0.60± 0.06 0.22±0.020.22± 0.02 0.23±0.030.23± 0.03 00 S: spectral M-matrix (deployed) 18.37±0.5718.37± 0.57 +0.01±0.02+0.01± 0.02 0.47±0.080.47± 0.08 0.58±0.050.58± 0.05 00 The global fit A is included as the no-context reference: its ρ01 _01 of 0.670.67 is the raw correlation of the two brands’ bids, which the context-dependent fits redistribute between the context mean and the residual correlation. C is the only prior that learns an informative context mean, and it does so by giving up correlation; S keeps the correlation and, as Section 7.6 of the main text shows on real queries, that is the channel revenue rides. Appendix S6 Revenue stability under prior misspecification: proof and numerical certification Why the conditional coordinates, and a heuristic bridge to KL (moved from the main text). The ruler in Proposition 5.8 is the conditional coordinate, not ‖Q^−Q‖\| Q-Q\|: with σi|−i=Qii−1/2 _i -i=Q_i^-1/2, revenue is Lipschitz in the precision matrix only locally, with a constant that grows like Qii−3/2Q_i^-3/2 and is useless near the fitted priors (§S6.4). For Gaussians the conditional-coordinate error is, to leading order, the Fisher metric, which suggests Δ≲KLE KL, the quantity maximum likelihood controls, and the one for which MTP2MTP_2-constrained estimation is unusually well behaved (Lauritzen, Uhler and Zwiernik 2019). We record this as a heuristic bridge, not a theorem: a KL rate for the constrained MLE is what would close the chain sample size → KL error → revenue loss. The misspecification-robustness theorem of the main text guarantees that truthfulness survives any prior in the M-matrix cone. This section quantifies what misspecification costs in revenue, proves the main text’s revenue-stability proposition for arbitrary n, and certifies each step numerically on the fitted seed-00 prior and the held-out bids. Roadmap. Six lemmas, in the order they are used. Lemma S2 is Mills-ratio calculus and fixes the constant C(⋅)C(·) that bounds how fast a virtual value can move when its conditional moments move. Lemma S3 shows that being active (ψi≥0 _i≥ 0) by itself bounds a bidder’s standardized residual from below, so the constant is a property of the prior class rather than a measured quantity, which is what keeps it finite. Lemma S4 shows that moving a bidder’s conditional moments from one prior to the other, scale first and mean second, keeps its residual within an explicit slack of the pinned value; this is what lets a bidder that is active under only one of the two priors be controlled. Lemma S5 converts a parameter perturbation into a uniform perturbation of the threshold equation’s gap function, and is where the clamp max0,⋅ \0,·\ earns its keep by excluding inactive bidders. Lemma S6 converts that into a bound on the threshold itself, using only the fact that the gap function rises at unit rate; it replaces the implicit function theorem and is what makes the result hold at every n. Lemma S7 handles the profiles where the two priors disagree about who wins, or about whether to sell at all. Proposition S2 assembles them. Two facts that shape the result. First, the natural metric is not the precision matrix: since σi|−i=Qii−1/2 _i -i=Q_i^-1/2, revenue is not Lipschitz in ‖Q^−Q‖\| Q-Q\| near singular priors (§S6.4). Second, the bound is Lipschitz only up to a boundary term: sale and winner decisions flip, so pointwise payments jump, and a bounded signal density is what turns those jumps into a controlled contribution. S6.1. Setting and notation Bids are truthful for every prior in the cone (main text, misspecification-robustness theorem), so the true law P⋆P , supported on [0,s¯]n[0, s]^n with bounded density, is fixed while the prior varies. Fix a profile s with provisional winner w. For a prior θ=(,Q)θ=( μ,Q) write σi=Qii−1/2,βij=−Qij/Qii≥0, _i=Q_i^-1/2, _ij=-Q_ij/Q_i\ ≥ 0, mi(t)=μi+βiw(t−μw)+∑j≠i,wβij(sj−μj), m_i(t)= _i+ _iw(t- _w)+\!\! _j≠ i,w\!\! _ij(s_j- _j), so mi(t)m_i(t) is bidder i’s conditional mean when w’s report is set to t; mwm_w does not depend on t. Put zi(t)=(si−mi(t))/σiz_i(t)=(s_i-m_i(t))/ _i and ψi(t)=si−σiR(zi(t)) _i(t)=s_i- _iR(z_i(t)), with ψw(t)=t−σwR((t−mw)/σw) _w(t)=t- _wR ((t-m_w)/ _w ). The threshold solves Dθ(t)≜ψwθ(t)−max0,maxj≠wψjθ(t)= 0,D_θ(t)\; \;ψ^θ_w(t)- \0,\ _j≠ wψ^θ_j(t) \\;=\;0, and Rev(θ)=P⋆[t0(;θ) 1maxiψiθ≥0]Rev(θ)=E_P [t_0(s;θ)\,1\ _iψ^θ_i≥ 0\ ]. Call bidder j active at t under θ if ψjθ(t)≥0ψ^θ_j(t)≥ 0; otherwise the clamp removes it. With θ θ the second prior, Δ()=∑iΔi(),Δi()=|σi−σ^i|+supt∈[0,sw]|mi(t)−m^i(t)|, (s)= _i _i(s), _i(s)=| _i- σ_i|+ _t∈[0,s_w]|m_i(t)- m_i(t)|, the supremum over the bracket [0,sw][0,s_w] being attained at an endpoint because mim_i is affine in t; ‖Δ‖∞=supΔ()<∞\| \|_∞= _s (s)<∞ by compact support. Finally σmin=miniminσi,σ^i _ = _i \ _i, σ_i\ and κ is a bound on the conditional mean-to-scale ratio, mi()/σi≤κm_i(s)/ _i≤κ at every profile in the support, for both priors, an empirical regularity of the fitted priors (κ=16.93κ=16.93 on the seed-00 prior and held-out bids, §S6.5), not an assumption of the main text. Since mwm_w is free of t and every mj(t)m_j(t) is nondecreasing in t (βjw≥0 _jw≥ 0), the bound at the realized profile covers every t∈[0,sw]t∈[0,s_w], which is all the proof uses. Remark S1 (Why the supremum over t). Evaluating the conditional means at the realized profile only is not enough: the competitor’s conditional mean enters the threshold equation at t0t_0, not at sws_w, and when the two priors disagree on a regression weight β the discrepancy grows like |β−β^||sw−t0||β- β|\,|s_w-t_0|, which the realized-profile version does not cover. Taking the supremum over the bracket [0,sw][0,s_w] closes this and costs nothing computationally. S6.2. Six lemmas Lemma S2 (Mills-ratio calculus and the constant). R′(z)=zR(z)−1<0R (z)=zR(z)-1<0 and zR(z)<1zR(z)<1 for all z, hence ∂ψi∂si=2−ziR(zi)>1,∂ψi∂mi=R′(zi),∂ψi∂σi=−R(zi)+ziR′(zi). ∂ _i∂ s_i=2-z_iR(z_i)>1, ∂ _i∂ m_i=R (z_i), ∂ _i∂ _i=-R(z_i)+z_iR (z_i). Write g(z)=R(z)−zR′(z)=R(z)(1−z2)+zg(z)=R(z)-zR (z)=R(z)(1-z^2)+z. Then (a) z↦|R′(z)|=1−zR(z)z |R (z)|=1-zR(z) is strictly decreasing, so supz≥z0|R′|=1−z0R(z0) _z≥ z_0|R |=1-z_0R(z_0); (b) g′(z)=z[z−R(z)(1+z2)]g (z)=z [z-R(z)(1+z^2) ] has the sign of −z-z, so g is unimodal with maximum g(0)=π/2g(0)= π/2, and g(z)→0+g(z)→ 0^+ as z→+∞z→+∞, g(z)→−∞g(z)→-∞ as z→−∞z→-∞; (c) consequently, with C(z0)≜max1−z0R(z0),|g(z0)|,π/2,C(z_0) \1-z_0R(z_0),\ |g(z_0)|,\ π/2 \, one has supz≥z0max|R′(z)|,|g(z)|≤C(z0) _z≥ z_0 \|R (z)|,|g(z)|\≤ C(z_0), with equality whenever z0≤0z_0≤ 0; C is nonincreasing and C(z0)→∞C(z_0)→∞ as z0→−∞z_0→-∞. Proof. Differentiating (1−Φ)/ϕ(1- )/φ with ϕ′=−zϕφ =-zφ gives R′=(−ϕ2+zϕ(1−Φ))/ϕ2=zR−1R =(-φ^2+zφ(1- ))/φ^2=zR-1; zR<1zR<1 is the classical Mills bound. The three partials follow by the chain rule with ∂zi/∂mi=−1/σi∂ z_i/∂ m_i=-1/ _i and ∂zi/∂σi=−zi/σi∂ z_i/∂ _i=-z_i/ _i. (a) dz(1−zR)=z−R(1+z2)<0 ddz(1-zR)=z-R(1+z^2)<0 by the Gordon bound R(z)>z/(1+z2)R(z)>z/(1+z^2), trivial for z≤0z≤ 0. (b) g′=R′(1−z2)−2zR+1=z[z−R(1+z2)]g =R (1-z^2)-2zR+1=z[z-R(1+z^2)], whose bracket is negative by the same bound; g(0)=R(0)=π/2g(0)=R(0)= π/2. The tails use R(z)=z−1−z−3+O(z−5)R(z)=z^-1-z^-3+O(z^-5) at +∞+∞, giving g(z)=2/z+O(z−3)g(z)=2/z+O(z^-3), and R(z)∼2πez2/2R(z) 2πe^z^2/2 at −∞-∞ against (1−z2)→−∞(1-z^2)→-∞. (c) On [z0,∞)[z_0,∞) the supremum of g is π/2 π/2 when z0≤0z_0≤ 0 and g(z0)<π/2g(z_0)< π/2 otherwise; its infimum is ming(z0),0 \g(z_0),0\. Monotonicity of C follows from (a), (b). ∎ Remark S2 (All three terms of C are needed). Two of the three terms are easy to drop. The absolute value on g is needed because g<0g<0 for z0<−0.840z_0<-0.840 and |g||g| overtakes |R′||R | once z0<−1.586z_0<-1.586. The interior term π/2 π/2 is needed whenever z0≤0z_0≤ 0, since g peaks at z=0z=0 rather than at the left endpoint. At the residual floor measured at the threshold in our data (−1.297-1.297) the first term binds and the correction is invisible; at the derived floor z0=h−1(κ)=−1.908z_0=h^-1(κ)=-1.908 (§S6.5) the second term binds, and the corrected constant 41.641.6 replaces the 29.729.7 an uncorrected formula would print. Lemma S3 (Activity pins the residual). Let h(z)=R(z)−zh(z)=R(z)-z; then h′(z)=zR(z)−2<0h (z)=zR(z)-2<0, so h is a strictly decreasing bijection of ℝR. If bidder i is active at t under θ, i.e. ψiθ(t)≥0ψ^θ_i(t)≥ 0, then h(ziθ(t))≤miθ(t)σi,i.e.ziθ(t)≥h−1(miθ(t)σi)≥h−1(κ).h (z^θ_i(t) )≤ m^θ_i(t) _i, .e. z^θ_i(t)\ ≥\ h^-1\! ( m^θ_i(t) _i )\ ≥\ h^-1(κ). In particular z0≜minh−1(κ),0z_0 \h^-1(κ),0\ lower-bounds the residual of every bidder at every point where it is active, under either prior; this covers the winner at any t≥t0t≥ t_0 (where ψw(t)≥ψw(t0)=max0,⋅≥0 _w(t)≥ _w(t_0)= \0,·\≥ 0) and every active competitor. Proof. Substituting si=mi+σizis_i=m_i+ _iz_i into ψi=si−σiR(zi)≥0 _i=s_i- _iR(z_i)≥ 0 gives mi/σi≥R(zi)−zi=h(zi)m_i/ _i≥ R(z_i)-z_i=h(z_i); invert the decreasing h and use mi/σi≤κm_i/ _i≤κ. Monotonicity of h is the Mills bound again: h′=R′−1=zR−2<0h =R -1=zR-2<0; h(z)→±∞h(z)→±∞ as z→∓∞z→∓∞. ∎ This replaces a measured quantity by a derived one: z0z_0 follows from the single scalar κ. The bound is the companion, from above, of the main text’s positive-mean-dominance regularity, which constrains the same ratio from below. It also yields a consistency check reported in §S6.5. What Lemma S3 does not give is a bound on the residual of a bidder under a prior where it is inactive; the next lemma supplies it. Lemma S4 (Residual slack along the two-step path). Fix a bidder i, a point t, and suppose ziθ(t)≥z0z^θ_i(t)≥ z_0 with z0≤0z_0≤ 0 (for instance because i is active at t under θ). Move the parameters from (mi,σi)(m_i, _i) to (m^i,σ^i)( m_i, σ_i) in two monotone steps, scale first, then mean: (mi,σi)→(mi,σ^i)→(m^i,σ^i)(m_i, _i)→(m_i, σ_i)→( m_i, σ_i). Every intermediate residual satisfies z z ≥z0−|z0||σi−σ^i|+|mi(t)−m^i(t)|σmin ≥\ z_0- |z_0|\,| _i- σ_i|+|m_i(t)- m_i(t)| _ ≥z1≜z0−max1,|z0|‖Δ‖∞σmin, ≥\ z_1 z_0- \1,|z_0|\\,\| \|_∞ _ , and consequently, by Lemma S2(c), |ψiθ(t)−ψiθ^(t)|≤C(z1)Δi(). |ψ^θ_i(t)-ψ θ_i(t) |≤ C(z_1)\, _i(s). The same holds with the roles of θ and θ θ exchanged. Proof. Write z=ziθ(t)z=z^θ_i(t), δ=si−mi(t)=zσiδ=s_i-m_i(t)=z _i. Step 1 (σλ _λ from σi _i to σ^i σ_i, mean fixed): the residual is δ/σλ=zσi/σλδ/ _λ=z\, _i/ _λ. If z≥0z≥ 0 it stays nonnegative, hence ≥z0≥ z_0. If z<0z<0 it is smallest when σλ _λ is smallest, and zσi/σmin=z+z(σi−σmin)/σmin≥z−|z||σi−σ^i|/σmin≥z0−|z0||σi−σ^i|/σminz _i/ _ =z+z( _i- _ )/ _ ≥ z-|z|\,| _i- σ_i|/ _ ≥ z_0-|z_0|\,| _i- σ_i|/ _ , using z0≤z<0z_0≤ z<0. Step 2 (mean from mi(t)m_i(t) to m^i(t) m_i(t) at scale σ^i σ_i): the residual changes monotonically by at most |mi(t)−m^i(t)|/σ^i≤|mi(t)−m^i(t)|/σmin|m_i(t)- m_i(t)|/ σ_i≤|m_i(t)- m_i(t)|/ _ from its value at the end of step 1, which is ≥z0−|z0||σi−σ^i|/σmin≥ z_0-|z_0|| _i- σ_i|/ _ by the same argument (that value is either ≥0≥ 0 or the step-1 minimum). Combining gives the first display; bounding |mi(t)−m^i(t)|+|σi−σ^i|≤Δi()≤‖Δ‖∞|m_i(t)- m_i(t)|+| _i- σ_i|≤ _i(s)≤\| \|_∞ gives z1z_1. Along the path both parameter sensitivities of ψi _i are bounded by C(z1)C(z_1) (Lemma S2(c) with z1≤z0≤0z_1≤ z_0≤ 0), so integrating dψi=(∂ψi/∂σ)dσ+(∂ψi/∂m)dmd _i=(∂ _i/∂σ)\,dσ+(∂ _i/∂ m)\,dm along the two steps gives |ψiθ(t)−ψiθ^(t)|≤C(z1)(|σi−σ^i|+|mi(t)−m^i(t)|)≤C(z1)Δi()|ψ^θ_i(t)-ψ θ_i(t)|≤ C(z_1) (| _i- σ_i|+|m_i(t)- m_i(t)| )≤ C(z_1) _i(s). ∎ Remark S3 (Order of the moves matters). Moving the mean first and the scale second would pass through the point (m^i,σi)( m_i, _i), whose residual (si−m^i)/σi=ziθ^σ^i/σi(s_i- m_i)/ _i=z θ_i\, σ_i/ _i is a residual under neither prior and, when σ^i>σi σ_i> _i and ziθ^<0z θ_i<0, lies below both endpoint residuals by a factor rather than an additive slack. Scale first keeps the excursion additive and first-order in Δ . The statement “every intermediate residual lies between the endpoint values” is false for the mean-first order. Lemma S5 (Uniform perturbation of the gap function). At every t∈[0,sw]t∈[0,s_w] at which the winner w is active under at least one of the two priors, |Dθ(t)−Dθ^(t)|≤C(z1)Δ()≜η(). |D_θ(t)-D_ θ(t) |\;≤\;C(z_1)\, (s)\; \;η(s). Proof. Write aj=ψjθ(t)a_j=ψ^θ_j(t), a^j=ψjθ^(t) a_j=ψ θ_j(t) for j≠wj≠ w, A=maxjajA= _ja_j, A^=maxja^j A= _j a_j, so Dθ−Dθ^=(aw−a^w)−(max0,A−max0,A^)D_θ-D_ θ=(a_w- a_w)-( \0,A\- \0, A\). Winner. w is active at t under one prior, so Lemma S3 pins its residual there and Lemma S4 (started from that prior) gives |aw−a^w|≤C(z1)Δw()|a_w- a_w|≤ C(z_1) _w(s). Only active competitors enter. If A≤0A≤ 0 then max0,A−max0,A^≤0 \0,A\- \0, A\≤ 0. If A>0A>0 with argmax j⋆j , then max0,A=aj⋆ \0,A\=a_j and max0,A^≥a^j⋆ \0, A\≥ a_j , so max0,A−max0,A^≤aj⋆−a^j⋆ \0,A\- \0, A\≤ a_j - a_j , where j⋆j is active at t under θ; Lemmas S3 and S4 give aj⋆−a^j⋆≤C(z1)Δj⋆()a_j - a_j ≤ C(z_1) _j (s). Symmetrically, max0,A^−max0,A≤a^j^−aj^≤C(z1)Δj^() \0, A\- \0,A\≤ a_ j-a_ j≤ C(z_1) _ j(s) with j j the θ θ-argmax, active under θ θ, or ≤0≤ 0 if A^≤0 A≤ 0. Bidders inactive under both priors never appear. Adding the winner’s term to whichever competitor term applies and using Δw+Δj≤Δ _w+ _j≤ gives the claim. ∎ Remark S4 (No combinatorial factor). A factor 22 “for winner and active loser”, or 1+#active1+\#\active\ at general n, would double count: Δ() (s) already sums over bidders, and at any t the gap function involves the winner plus whichever single competitor attains the inner maximum. The constant in §S6.5 is correspondingly 13.613.6, not 21.321.3. Lemma S6 (Threshold stability, sandwich form). DθD_θ is continuous and strictly increasing, and Dθ(t)−Dθ(t0)≥t−t0D_θ(t)-D_θ(t_0)≥ t-t_0 for t≥t0t≥ t_0. Consequently, if |Dθ(t)−Dθ^(t)|≤η|D_θ(t)-D_ θ(t)|≤η at t=t0(θ)+ηt=t_0(θ)+η and at t=t0(θ^)+ηt=t_0( θ)+η, then |t0(θ)−t0(θ^)|≤η|t_0(θ)-t_0( θ)|≤η. Proof. ∂ψw/∂t=2−zwR(zw)>1∂ _w/∂ t=2-z_wR(z_w)>1 by Lemma S2, while each t↦ψj(t)t _j(t) is nonincreasing because ∂ψj/∂t=βjwR′(zj)≤0∂ _j/∂ t= _jwR (z_j)≤ 0 with βjw≥0 _jw≥ 0 and R′<0R <0; the clamp and the maximum preserve this. So DθD_θ rises at rate at least one, which is the displayed inequality. Then Dθ^(t0+η)≥Dθ(t0+η)−η≥η−η=0D_ θ(t_0+η)≥ D_θ(t_0+η)-η≥η-η=0, and since Dθ^D_ θ is strictly increasing with unique zero t^0 t_0, t^0≤t0+η t_0≤ t_0+η. Exchanging the roles of the priors at t=t^0+ηt= t_0+η gives t0≤t^0+ηt_0≤ t_0+η. ∎ Remark S5 (What this buys). This is the step that replaces the implicit function theorem. It requires no differentiability of maxj _j (which fails wherever the inner argmax switches) and no interpolation path inside the cone. The latter matters: linear interpolation of (βi,σi)( _i, _i) violates the symmetry Qij=QjiQ_ij=Q_ji, so the interpolated “prior” need not be admissible. The argument uses only σ>0σ>0 and β≥0β≥ 0, which the coordinatewise moves of Lemma S4 preserve, and it is indifferent to n. Note also that Lemma S5 is invoked only at t0+ηt_0+η and t^0+η t_0+η, points at which the winner is active under the corresponding prior, so the winner’s residual is never needed off its own active region. Lemma S7 (Boundary term). Let ηmax=C(z1)‖Δ‖∞ _ =C(z_1)\| \|_∞. The sale sets and winner sets under θ and θ θ differ only on |maxiψiθ|≤ηmax∪|ψ(1)θ−ψ(2)θ|≤ηmax \| _iψ^θ_i|≤ _ \∪ \|ψ^θ_(1)-ψ^θ_(2)|≤ _ \ (virtual values at the realized profile). If fs,fwf_s,f_w bound the densities of maxiψiθ _iψ^θ_i and of ψ(1)θ−ψ(2)θψ^θ_(1)-ψ^θ_(2) uniformly on [−ηmax,ηmax][- _ , _ ] (a bound at 00 alone does not control a strip of finite width), and t¯s,t¯w t_s, t_w bound the conditional mean of the larger of the two payments on the respective strip, the disagreement set contributes at most 2(t¯sfs+t¯wfw)C(z1)‖Δ‖∞2( t_sf_s+ t_wf_w)\,C(z_1)\| \|_∞ to |Rev(θ)−Rev(θ^)||Rev(θ)-Rev( θ)|. Proof. If maxiψiθ>ηmax _iψ^θ_i> _ , its argmax i is active under θ, so Lemmas S3–S4 give ψiθ^≥ψiθ−C(z1)Δi>0ψ θ_i≥ψ^θ_i-C(z_1) _i>0: both priors sell. If maxiψiθ<−ηmax _iψ^θ_i<- _ and some j were active under θ θ, the same lemmas started from θ θ would give ψjθ≥ψjθ^−C(z1)Δj≥−ηmaxψ^θ_j≥ψ θ_j-C(z_1) _j≥- _ , a contradiction; so neither sells. For the winner, suppose both sell and ψ(1)θ−ψ(2)θ>ηmaxψ^θ_(1)-ψ^θ_(2)> _ with θ-winner i: then ψiθ^≥ψiθ−C(z1)Δiψ θ_i≥ψ^θ_i-C(z_1) _i, and any j≠ij≠ i active under θ θ has ψjθ^≤ψjθ+C(z1)Δj≤ψ(2)θ+C(z1)Δjψ θ_j≤ψ^θ_j+C(z_1) _j≤ψ^θ_(2)+C(z_1) _j, so ψiθ^−ψjθ^≥(ψ(1)θ−ψ(2)θ)−C(z1)(Δi+Δj)>0ψ θ_i-ψ θ_j≥(ψ^θ_(1)-ψ^θ_(2))-C(z_1)( _i+ _j)>0; bidders inactive under θ θ cannot win. The probability of each strip is at most 2fηmax2f _ , and on it the revenue difference is at most the larger payment, whose conditional mean is bounded by t¯ t; payments never exceed the winner’s own report (Theorem S2), which is bounded on the support. ∎ Remark S6 (Why this term is L∞L^∞, not L1L^1). Applying the strip bound with the random radius η()η(s) and then taking expectations would be inadmissible: the density bound Pr(flip)≤2fδ (flip)≤ 2fδ requires a deterministic δ, and ‖Δ‖∞=∞\| \|_∞=∞ under unbounded support whenever βi≠β^i _i≠ β_i, because mi(⋅)m_i(·) is affine in the opponent’s report. Compact support, which is the deployed regime since bids are bounded in cents, repairs the step. Under unbounded support, truncating at radius M and optimizing M≍2log(1/ε)M 2 (1/ ) with ε=[Δ] =E[ ] gives O(εlog(1/ε))O ( (1/ ) ): still near-linear, at the cost of one logarithmic factor. S6.3. The proposition Proposition S2 (Revenue stability; arbitrary n). Let θ,θ^θ, θ lie in the cone with conditional mean-to-scale ratios bounded by κ on the support, let P⋆P be supported on [0,s¯]n[0, s]^n with bounded density, and let z0=minh−1(κ),0z_0= \h^-1(κ),0\, z1=z0−max1,|z0|‖Δ‖∞/σminz_1=z_0- \1,|z_0|\\| \|_∞/ _ . Then |Rev(θ)−Rev(θ^)|≤C(z1)P⋆[Δ()]+2(t¯sfs+t¯wfw)C(z1)‖Δ‖∞. |Rev(θ)-Rev( θ) |≤ C(z_1)\,E_P [ (s) ]+2 ( t_sf_s+ t_wf_w )C(z_1)\,\| \|_∞. As ‖Δ‖∞→0\| \|_∞→ 0, z1→z0z_1→ z_0 and C(z1)→C(z0)C(z_1)→ C(z_0): revenue is locally Lipschitz in the conditional coordinates with sharp constant C(h−1(κ))C(h^-1(κ)). Proof. Split on the event A that the sale decision and the winner identity agree under the two priors. On A, the winner w is common, it is active under θ at t0+ηt_0+η and under θ θ at t^0+η t_0+η, so Lemma S5 applies at both points (when t0+η≥swt_0+η≥ s_w the inequality t^0≤t0+η t_0≤ t_0+η is immediate from t^0≤sw t_0≤ s_w, Theorem S2, and likewise with the priors exchanged) and Lemma S6 gives |t0(θ)−t0(θ^)|≤C(z1)Δ()|t_0(θ)-t_0( θ)|≤ C(z_1) (s) pointwise; taking expectations yields the first term. On AcA^c, Lemma S7 yields the second. Continuity of C gives the limit statement. ∎ Remark S7 (What is derived and what is measured). Every ingredient of the constant is now read off the prior class and the perturbation size: κ and σmin _ from the priors, ‖Δ‖∞\| \|_∞ from the pair. Nothing about the realized residuals is assumed. The price of this is the slack z0→z1z_0→ z_1, which is first order in ‖Δ‖∞/σmin\| \|_∞/ _ and vanishes in the Lipschitz limit; quoting C(z0)C(z_0) for a finite perturbation would be correct only for bidders active under both priors and for the winner exactly at its threshold. The certification below reports both C(z0)C(z_0), the limit constant, and the realized residual floor. S6.4. Why not ‖Q^−Q‖\| Q-Q\| Locally around any fixed SPD prior, revenue is Lipschitz in the precision matrix: Q↦Qii−1/2Q Q_i^-1/2 and Q↦−Qij/QiiQ -Q_ij/Q_i are smooth on the SPD interior, and composing them with Proposition S2 gives a constant of order Qii−3/2Q_i^-3/2. What fails is a useful statement Rev(Q^)≥Rev(Q)−L‖Q^−Q‖Rev( Q) (Q)-L\| Q-Q\| with one L for the class: the constant explodes as Qii→0Q_i→ 0, and at the fitted priors it is already hopeless. The seed-00 spectral prior has miniQii=0.032 _i Q_i=0.032 (median 0.0640.064): conditional standard deviations reach ≈31≈ 31 cents, and σ=Qii−1/2σ=Q_i^-1/2 is violently nonlinear there. Perturbing Q additively in operator norm and replaying on the held-out bids (script regret_probe2.py), the smallest perturbation already moves revenue by 99 cents out of 37.937.9; the empirical ratio |ΔRev|/‖ΔQ‖| |/\| Q\| is 220220–650650 (far below the local constant, as it must be, and far above anything a platform could use), and neither a linear nor a quadratic fit has positive R2R^2 at the perturbation sizes tried (a large finite ratio is evidence of bad conditioning, not of non-Lipschitzness). Restricting to a sub-cone Qii≥cQ_i≥ c does not repair it (c=0.1c=0.1: ratio 223223). The conditional coordinates are the right ruler, since they are exactly what the precision-conditioning lemma reads off row i of Q, and for Gaussians their squared error is, to leading order, the Fisher metric, which suggests Δ≲KLE KL and so stability in KL KL: the quantity maximum likelihood controls, and the one under which MTP2MTP_2-constrained estimation is unusually well behaved (Lauritzen, Uhler and Zwiernik, 2019). We record this as a heuristic bridge rather than a theorem. What is missing is a KL rate for the constrained MLE, of which the cited work gives existence under very few samples; supplying it would close the chain sample size → KL error → revenue loss and make the class learnable in the revenue sense, not only in the sense that its MLE is convex. S6.5. Numerical certification Fitted seed-00 spectral prior, n=2n=2 held-out bids (180180 queries, 3,6003,600 auctions, 3,2413,241 sold); scripts prop_verify.py, prop_verify2.py, regret_probe3.py, and the lemma-level diagnostics mills_checks.py (Lemma S2), check_z0.py (Lemma S3, elementwise) and check_stability.py (Lemmas S3 and S5–S7 on a pair of priors), driven by run_certification.py; all in code/n4, numbers read from their JSON outputs and from prop_verify2.json and regret_probe3_results.json. The perturbed prior follows the relative scheme of regret_probe3.py (diagonal ×eεu× e u, off-diagonal ×(1+εu)×(1+ u) clipped to the cone, means shifted by εuσ uσ); the row quoted uses ε=5×10−4 =5× 10^-4, the smallest scale, as the Lipschitz-limit regime. Lemma / quantity Check Result S2: R′=zR−1R =zR-1 finite differences, z∈[−3,6]z∈[-3,6] rel. error 3×10−83× 10^-8 S2: 2−zR>12-zR>1 grid z∈[−6,6]z∈[-6,6] min 1.0261.026; →1→ 1 at +∞+∞ S2(a): |R′||R | decreasing grid confirmed S2(b): maxg=π/2 g= π/2 grid 1.25331.2533 at z=0z=0 S3: h(zi)≤mi/σih(z_i)≤ m_i/ _i all active bidders, sold auctions 0/4,3390/4,339 violations κ=maximi/σiκ= _im_i/ _i test set, fitted prior (median 3.253.25) 16.9316.93 (≥6.54≥ 6.54 ✓) z0=h−1(κ)z_0=h^-1(κ), derived closed form −1.908-1.908 C(z0)C(z_0), derived limit constant closed form (|g||g| binds) 41.641.6 σmin _ ; ‖Δ‖∞\| \|_∞ cone pair, ε=5×10−4 =5× 10^-4 2.892.89 cents; 0.400.40 cents z1z_1; C(z1)C(z_1) same pair −2.172-2.172; 99.399.3 S5: |Dθ−Dθ^|≤C(z1)Δ|D_θ-D_ θ|≤ C(z_1) at t0+η,t^0+ηt_0+η, t_0+η same pair 0/6,4760/6,476 violations S6: |t0−t^0|≤η|t_0- t_0|≤η same pair, agreement event 0/3,2380/3,238 violations S7: flips inside the strips same pair 0/3,2410/3,241 violations |t0−t^0|≤C(z0)Δ|t_0- t_0|≤ C(z_0) , limit constant same pair (diagnostic) 0/3,2380/3,238 violations measured |∂t0/∂m||∂ t_0/∂ m| median / max 0.700.70 / 0.900.90 measured |∂t0/∂σ||∂ t_0/∂σ| median / max 0.280.28 / 0.750.75 active competitors share of sold auctions 65.6%65.6\% (inactive 34.4%34.4\%) residual floor at t0t_0 (winner; active competitor) min, sold auctions −1.291-1.291; −1.297-1.297 residual floor at the realized profile (active competitor) min, sold auctions −1.834-1.834 C at the threshold floor closed form at −1.297-1.297 7.807.80 S7: fs,t¯sf_s, t_s |maxψ|<1| ψ|<1 cent 0.00810.0081/cent, 14.814.8 cents S7: fw,t¯wf_w, t_w |ψ1−ψ2|<1| _1- _2|<1 cent 0.00570.0057/cent, 43.643.6 cents aggregate C(1+2(t¯sfs+t¯wfw))C (1+2( t_sf_s+ t_wf_w) ), derived (C=41.6C=41.6) uniform bound 72.272.2 aggregate at the t0t_0 floor (C=7.80C=7.80) lower bound on the derived one 13.613.6 per-auction |∂D/∂m||∂ D/∂ m| mean / median / max 2.382.38 / 2.352.35 / 8.838.83 empirical |ΔRev|/| |/mean coordinate error cone perturbations, 88 scales (regret_probe3.py) 0.240.24 (linear fit R2R^2 0.930.93) empirical vs. KLKL same perturbations 1.65KL1.65\,KL (R2R^2 0.940.94); 2.36KL2.36 KL (0.910.91) Reading. The derived uniform constant (72.272.2) exceeds the measured average slope (0.240.24) by a factor of about 300300, an order-of-magnitude comparison only, since the constant multiplies ΔE plus an ‖Δ‖∞\| \|_∞ boundary term while the slope is per unit of mean coordinate error, as a worst-case-over-auctions, worst-direction bound must; the constant one would quote from the residual floor measured at the threshold (13.613.6) is a factor 55 smaller, and Lemma S3 explains why it is only a lower bound, since the floor is attained by a bidder at ψ≈0ψ≈ 0 with m/σ=6.54m/σ=6.54, whereas the class contains conditional means up to 16.916.9 scales. The analytic sensitivities average 2.42.4 against a measured ∂D/∂t≈3.4∂ D/∂ t≈ 3.4, so the typical threshold moves by ≈0.7≈ 0.7 cent per cent of conditional-mean error: sub-unit, the visible work of ∂ψ/∂s>1∂ψ/∂ s>1. Dropping the active-competitor refinement of Lemma S5 admits residuals down to z=−3.46z=-3.46, where C=1.09×104C=1.09× 10^4 and the aggregate constant is 1.9×1041.9× 10^4, so the clamp in the threshold equation buys more than two orders of magnitude and is what makes the bound informative at all. At finite perturbation size the slack z0→z1z_0→ z_1 is real: with σmin=2.9 _ =2.9 cents, a 0.40.4-cent perturbation already costs a factor 2.42.4 in the constant (99.399.3 vs. 41.641.6), and by ‖Δ‖∞≈1.5\| \|_∞≈ 1.5 cents the guaranteed constant exceeds 10310^3, while the limit constant C(z0)C(z_0) still bounds every realized threshold move (0/3,2380/3,238 violations at every scale tested). The gap between the two is the price of a proof that assumes nothing about where the perturbed residuals land. A support bound for κ. The tabulated κ=16.93κ=16.93 is the maximum of mi/σim_i/ _i over realized test profiles. Because mi()=μi+∑jβij(sj−μj)m_i(s)= _i+ _j _ij(s_j- _j) has βij=−Qij/Qii≥0 _ij=-Q_ij/Q_i≥ 0, its supremum over the box [0,s¯]n[0, s]^n is attained at the vertex sj=s¯s_j= s for all j, so a bound valid on the whole support, not a finite-sample maximum, is computable in closed form from the fitted prior: κbox=40.0 _box=40.0 on the seed-00 test contexts (s¯=200 s=200), giving z0=h−1(κbox)=−2.33z_0=h^-1( _box)=-2.33 and C(z0)=169C(z_0)=169 (kappa_box.py, kappa_box_results.json). The proposition may be read with either constant: the realized one describes the data, the vertex one is a certificate. Consistency check on the residual floor. Lemma S3 turns the measured residual floor into a test of the fitted priors: a residual of −1.297-1.297 among active bidders is possible only if maximi/σi≥h(−1.297)=6.54 _im_i/ _i≥ h(-1.297)=6.54 somewhere in the test set, so κ≥6.54κ≥ 6.54 and the derived constant C(h−1(κ))C(h^-1(κ)) is at least the 7.807.80 quoted above, with equality iff the floor is attained by a bidder exactly at ψ=0ψ=0. The realized κ=16.93κ=16.93 passes this check with room to spare, and the elementwise version of Lemma S3 holds at every one of the 4,3394,339 active-bidder records (minimum slack −3.6×10−15-3.6× 10^-15, i.e. equality at the binding records up to rounding). The two easy ways to fail the check (evaluating the winner’s residual at sws_w instead of t0t_0, or reading σ from Σii _i rather than 1/Qii1/Q_i) are what check_z0.py guards against. Note also that the competitor’s floor depends on where it is evaluated: −1.297-1.297 at the threshold t0t_0 (where Lemma S5 uses it) but −1.834-1.834 at the realized profile (where Lemma S7 uses it), because mj(t)m_j(t) is nondecreasing in t and t0≤swt_0≤ s_w; the derived z0=−1.908z_0=-1.908 lies below both, as it must. Empirical trace. The main text’s prior-perturbation study is the bound’s empirical trace: mean noise ε perturbs mim_i alone, and the rent decays monotonically (+6.4%→+2.5%→−4.3%+6.4\%→+2.5\%→-4.3\% at ε=0,10,20 =0,10,20). The successive drops (−3.9-3.9, −6.8-6.8) are mildly convex in ε , so the study is compatible with the bound rather than a confirmation of a linear law: an upper bound constrains the decay from above only. In the same twelve-regime study, mis-scaling the conditional variance by ×0.5× 0.5 and ×2× 2 leaves the rent at +5.9%+5.9\% and +2.4%+2.4\% (perturb_out.json). Appendix S7 Experimental protocol and reproducibility Within-query correlations on the four-brand data (main text, Section 7.1). Bids are demeaned within each query of the four-brand file (simulated_bids_n4b.pt; 780780 queries, 15,60015,600 rows per brand) and correlated across all rows. All six pairs are positive; the main-line pair (premium, value) is 0.580.58 on this data set and 0.660.66 on the n=2n=2 main-line data (ST_600.pt, 600600 queries), which differ in query set and scenario count. Script and results: pairwise_within_query_corr (.py and _n4b.json). Flagship Premium Value Clearance Flagship 11 0.630.63 0.280.28 0.200.20 Premium 0.630.63 11 0.580.58 0.430.43 Value 0.280.28 0.580.58 11 0.570.57 Clearance 0.200.20 0.430.43 0.570.57 11 Prompt-ablation table (main text, Section 7.7). Seven gpt-4o-mini prompt schemes and the gpt-4o replication. Gradient == ratio of the premium brand’s mean bid on transactional to informational queries (query means first); Corr. == within-query correlation of the two brands’ bids after removing query means. The last four columns describe the shape of the bid distribution (cents on a 00–200200 scale): median, share of bids at or below 1010 cents, 95th percentile, and the share of all bids carried by the three most frequent values. Explicit calculation piles bids at the floor; the judgment-based schemes, facts-only and behavioral alike, spread bids across the scale, and differ in whether the spread follows the query’s intent (gradient) or only the brand. Source: pilot*.json (the v3 row uses the full v3 run simulated_bids_n4_full.json, the data its gradient and correlation were computed on), statistics by bid_shape_by_scheme.py (code/n4/bid_shape_by_scheme.json). The v4.1 pilot (4848 calls) is omitted: its expected values exceeded the 200200-cent cap and the run was stopped. Prompt scheme Gradient Corr. Median ≤10≤ 10 P95 Top-3 Facts only, qualitative (v3) 0.980.98 +0.78+0.78 3030 13%13\% 9494 27%27\% ++qualitative conversion funnel (v4) 1.741.74 +0.74+0.74 3030 6%6\% 8585 32%32\% ++quantified conversion facts (v4.2) 8.568.56 +0.41+0.41 1212 46%46\% 8080 25%25\% ++chain-of-thought (v4.2++CoT) 6.716.71 +0.06+0.06 55 80%80\% 8080 54%54\% ++quantified demand index++decimals (v4.3) 7.407.40 +0.09+0.09 44 84%84\% 100100 27%27\% ++low temperature (T=0.2T=0.2) 8.948.94 −0.03-0.03 33 88%88\% 120120 29%29\% Behavioral traits (main-line design) 5.765.76 +0.66+0.66 2525 24%24\% 157157 27%27\% gpt-4o, v4.3 14.5214.52 +0.36+0.36 1.21.2 88%88\% 100100 20%20\% Is the demand signal the only common input? (main text, Sections 7.1 and 7.7). Given a query and a shock realization, each brand’s bid is an independent LLM call, so any co-movement beyond what the shared signal u induces would have to come from somewhere else in the pipeline. We check this on the main-line data of Table 1 (ST_600.pt: 600600 queries × 2020 scenarios, 12,00012,000 rows; partial_corr_u.py, partial_corr_u.json) by removing successively more of the common structure and recomputing the Pearson correlation of the two brands’ residual bids. The signal enters the prompts as one of five demand-level texts (very weak, below normal, normal, elevated, very strong), which defines the cells of the last layer. Bootstrap intervals over queries. Residualized on Correlation 95%95\% CI query means (the within-query correlation) +0.66+0.66 [+0.64,+0.67][+0.64,+0.67] query means ++ demand-level means +0.16+0.16 [+0.14,+0.19][+0.14,+0.19] (query, demand-level) cell means −0.01-0.01 [−0.03,+0.01][-0.03,+0.01] The demand level explains 88%88\% and 82%82\% of the two brands’ within-query bid variance, and once query and shock realization are both held fixed the residual correlation is indistinguishable from zero. The benchmark therefore contains no correlation of its own making: what the mechanism conditions on is the agents’ transmission of a common demand signal, the classical source of affiliation (Milgrom and Weber 1982), and the prompt study above measures how much of that signal different agents let through. (The prompt-ablation table above is computed on the pilot sets, and its bid-shape columns pool all four brands; the behavioral statistics in the main text, Sections 7.1 and 7.7, are computed on the two main-line brands in ST_600.pt: gradient 5.765.76, within-query correlation +0.66+0.66, median bid 3737 cents, 95th percentile 112112; behavior_mainline.py, behavior_mainline.json.) CAN baseline (main text, Table 1). We could not find a public implementation of CAN, so every number reported for it comes from our own reimplementation of the mechanism as described in the paper, not from the authors’ code. We reimplement the Conditional Auction Net of Huo, Zheng and Wu (AAAI 2025) as a context-free mechanism on the main line (can_baseline_fixed.py, results can_baseline_fixed_results.json). As in the original paper, CAN maps the bids alone to an allocation and a payment. We keep it that way rather than add a context input of our own design, which would make it a different mechanism. The value of context is measured separately, by the contextual-reserve baseline (ctxSPA) and by the fixed-opponent market below, where every context-free mechanism is capped by the no-context ceiling. Each bidder i has a rank-score function ri()=minzmaxq[wzq⋅(bi,−i)+czq]r_i(b)= _z _q[\,w_zq·(b_i,-b_-i)+c_zq\,], a min-max network (six groups of six affine units, nonnegative weights via softplus) over the inputs (bi,−i)(b_i,-b_-i), so ∂ri/∂bi≥0∂ r_i/∂ b_i≥ 0 and ∂rj/∂bi≤0∂ r_j/∂ b_i≤ 0 hold by construction (the single-crossing conditions, the paper’s eq. (4)). Following the paper’s eq. (5), a learned reserve score r0()=minzmaxq[wzq0⋅(−)+czq0]r_0(b)= _z _q[\,w^0_zq·(-b)+c^0_zq\,], nonincreasing in every bid, plays the role of an additional bidder: the item goes to argmaxiri() _ir_i(b) if that score is at least r0()r_0(b), otherwise it is unsold. The winner pays its critical bid, the smallest report at which it still wins, found by 4040 bisection steps (its own score is nondecreasing and every competing score, including r0r_0, nonincreasing in the report, so the root is unique). Training maximizes expected revenue on the 7,2007,200 training bids of the Table 1 split with a softmax-relaxed allocation over the n+1n+1 scores (temperature annealed from 0.30.3 to 0.020.02 over 3,0003,000 Adam steps, batch 256256, learning rate 2×10−32× 10^-3) and gradients through the critical bid by implicit differentiation at interior roots (a payment clipped at zero or at the bid contributes no gradient, as in the original training objective); the checkpoint with the highest validation revenue is kept. The mechanism receives no query context, as in the original paper, and no prior. Test revenue over five training seeds is 37.43±0.0737.43± 0.07 cents (37.4337.43, 37.4437.44, 37.4837.48, 37.4637.46, 37.3237.32); four seeds learn a reserve low enough to trade on every test profile (trade rate 100%100\%, welfare 54.654.6–54.954.9), the fifth trades 85%85\% (welfare 53.253.2). An incentive-compatibility probe (100 test profiles, 25 deviations per bidder) finds 0/5,0000/5,000 violations for every seed. Against the other baselines CAN is clearly ahead: on seed 00 it earns +5.1%+5.1\% over the tuned reserve and +8.0%+8.0\% over IPV-Myerson. AffMyerson’s five-seed revenue on the same test rows is 37.84±0.0637.84± 0.06 against CAN’s 37.43±0.0737.43± 0.07. The two seed distributions are disjoint (the smallest value the AffMyerson mean and sd admit, 37.7437.74, exceeds CAN’s largest seed, 37.4837.48), and a Welch test on the seed means gives t=10.3t=10.3, p<0.001p<0.001. The original paper trains CAN on 100,000100,000 samples per market, against 7,2007,200 here. When the conversation matters to one bidder only (main text, Section 7.3). 200200 pairs of conversations (400400 in all) share the same product words. One member of each pair carries a high-value cue (an upcoming trip, a broken item, a group purchase, a gift with a budget) and the other a low-value cue (writing an article, a school project, an item already bought, a same-name non-product). Each conversation has 2020 rows (8,0008,000 rows), embedded with the same frozen sentence encoder as the main line, and split 6:1:36:1:3 by conversation with the main-line split seed (240/40/120240/40/120 conversations; 2,4002,400 test rows, 46%46\% high-cue). Bidder A (“marketplace”) does not read the conversation, and its value is (30,32)N(30,3^2) cents on every row. Bidder B (“specialist”) reads it, with value (200,252)N(200,25^2) on a high cue and (100,252)N(100,25^2) on a low cue, clipped at zero. The two are independent by construction, so this experiment isolates what reading the conversation is worth. It is not a measurement of the affiliation rent, and AffMyerson and IPV-Myerson with the same prior coincide here. The prior is the spectral parameterization of Section 6 fitted on the training rows (five seeds), and CAN is the full model of the paragraph above, trained on the same rows (five seeds). These CAN runs use the corrected trainer described above. One reference quantity: the no-context ceiling is Myerson’s optimal auction (with ironing) for the empirical marginal bid distributions of the training set, which is the most any mechanism that sees only bids can earn. Code ctx_branch/fixed_opponent.py, results fixed_opponent_results.json. Mechanism Revenue (cents) High-cue rows Low-cue rows Second price (SPA) 29.9329.93 Tuned-reserve SPA (r∗=163.5r^*=163.5) 71.2671.26 No-context ceiling (ironed Myerson) 84.3984.39 155.0155.0 24.624.6 CAN (five seeds) 83.95±0.1783.95± 0.17 153.5153.5 25.125.1 Lookahead (same prior) 100.33±0.28100.33± 0.28 Contextual-reserve SPA 103.92±0.21103.92± 0.21 IPV-Myerson (same prior) 104.79±0.12104.79± 0.12 AffMyerson (five seeds) 104.78±0.22104.78± 0.22 149.7149.7 66.866.8 The fitted prior reads the specialist’s mean as 192.5192.5 on high-cue and 106.3106.3 on low-cue conversations (true values 200200 and 100100) with conditional scale 3131 (true 2525). Prompt scheme × mechanism: conditioning on calculation-type bids. To ask what conditioning is worth if bids looked like the calculation-type schemes, the v4.2++CoT scheme was rerun at main-line scale (600600 queries × 2020 scenarios × 22 brands =24,000=24,000 calls; run_full_cot.py, data ST_cot.pt) and passed through the same fitting and auction pipeline (five seeds, split 00). The bid shape matches the pilot (median 55 cents, 82%82\% of bids at or below 1010 cents, three values carrying 55%55\%). The fitted spectral prior has ρ¯=0.17 ρ=0.17–0.190.19 on the test set; AffMyerson earns 4.995±0.1844.995± 0.184 cents, +2.50%±1.01%+2.50\%± 1.01\% over IPV-Myerson and −7.24%±3.42%-7.24\%± 3.42\% against the tuned-reserve SPA (r⋆=10r =10), both directions stable across the five seeds (cot_auc_output.txt). This is the low-correlation regime of the shuffling scan (main text, Section 7.4), in which the two-tier criterion tells a platform not to condition. Note the two correlation measures differ: the table’s Corr. is the raw within-query correlation of pilot bids; ρ¯ ρ is the fitted prior’s conditional correlation on held-out queries. We requested access to the Yahoo! Webscope A1 advertiser bidding data, which records competing bids on the same keywords; two months after the request, access had not been granted. Queries, rather than individual scenarios, are assigned to train, validation, and test partitions. Reserves are tuned only on the training partition. Reported fixed-split uncertainty uses five optimization seeds. A separate five-split audit repeats preprocessing, fitting, reserve tuning, and testing for independent query partitions. Test-set comparisons use paired bootstrap resampling of query clusters. Context-predictability diagnostics on real queries (the R2R^2 figures for the diagnostic classes and the encoder/feature ablations) are measured with the unconstrained class under the legacy fitting pipeline; they are an optimistic estimate of what a context mean could recover and are reported as diagnostics, not properties of the deployed prior. The designed-data figure they are compared with in the main text (R2=0.60R^2=0.60) is the diagonally dominant fit of the four-prior table (Section S5), the best context mean any class attains there; the unconstrained class itself reaches R2≈0R^2≈ 0 on the designed data. The accompanying code records the exact data file, split seed, optimization seed, per-split revenue, and paired intervals in machine-readable JSON. Seven unit tests cover SPD construction, M-matrix feasibility, score monotonicity, cross-monotonicity, exact reduction to IPV pricing for diagonal Q, round-trip spectral representation, and the critical-payment score equality and bracket; the lemma-level diagnostics of Section S6 (check_z0.py, check_stability.py) ship alongside them. Appendix S8 Log-link revenue comparison (controlled, legacy pipeline) Same test set, same held-out real bids, same reserve grid; the two blocks differ only in the prior’s link function (both fit by the legacy diagonally dominant pipeline, which predates the spectral network and budgets partial correlations at roughly 1/(n−1)1/(n-1), so the rent columns are not comparable to the main line’s). Mechanism Identity link Log link (c0=0.45c_0=0.45) SPA 28.39 28.39 SPA + tuned reserve (r∗=32r^*=32) 35.62 35.62 IPV-Myerson 33.31 36.81 AffMyerson 36.15 37.35 The better-fitting marginal is revenue-positive for both prior-based mechanisms (the SPA rows are link-free sanity checks); the measured rent narrows because IPV-Myerson gains more, an effect confounded by the legacy pipeline’s correlation cap. Constructive regularity: the diagonal floor c0=0.45>1/σ⋆2≈0.4342c_0=0.45>1/σ 2≈ 0.4342 guarantees σ<σ⋆σ<σ for every network output. Appendix S9 Shuffling scan: prior-tracking panel Companion panel to the main text’s shuffling-scan figure: the fitted mean correlation ρ¯ ρ of the retrained prior tracks the injected within-query correlation at every level. Fixed-policy variant (main text, Section 7.4). The retrained protocol refits the prior and re-tunes the reserve at every shuffling level. The fixed-policy variant keeps the prior fitted on the original data (fitted mean correlation ρ¯=0.735 ρ=0.735) and the original tuned reserve (r⋆=32r =32), and re-pairs only the test set. With no retraining the rent still rises monotonically with the within-query correlation, so the response in the main text is not a retraining artifact; the negative endpoint at ρ≈0ρ≈ 0 is the cost of conditioning on a correlation that is no longer in the data. Bootstrap intervals over test queries; script rho_scan_fixed.py, results rho_fixed_results.json. Shuffle fraction Test ρ Lift vs. IPV-Myerson Lift vs. tuned reserve 1.001.00 0.0090.009 −2.2%-2.2\% [−3.7,−0.5][-3.7,-0.5] −2.2%-2.2\% [−3.7,−0.8][-3.7,-0.8] 0.750.75 0.1830.183 +0.7%+0.7\% [−0.8,+2.4][-0.8,+2.4] +0.3%+0.3\% [−1.3,+1.9][-1.3,+1.9] 0.500.50 0.3650.365 +4.4%+4.4\% [+2.8,+6.2][+2.8,+6.2] +3.2%+3.2\% [+1.6,+4.7][+1.6,+4.7] 0.250.25 0.5210.521 +6.9%+6.9\% [+5.3,+8.8][+5.3,+8.8] +5.0%+5.0\% [+3.3,+6.6][+3.3,+6.6] 0.000.00 0.6540.654 +8.9%+8.9\% [+7.2,+10.9][+7.2,+10.9] +6.5%+6.5\% [+4.7,+8.3][+4.7,+8.3] Appendix S10 Link choice: where each marginal fits better Two home turfs: where each marginal fits better (main text, Section 7.2). For every (query, brand) cell with at least ten scenario bids we fit a normal and a log-normal by maximum likelihood and compare their negative log-likelihoods on the original bid scale (the log-normal includes the Jacobian term ∑logxΣ x; both have two parameters, so the comparison is a likelihood ratio). The table reports the share of cells the normal wins, with 95%95\% binomial half-widths (link_home_turf.py, link_home_turf_out/results.json). Overall the log-normal is ahead (36%36\% normal wins on the designed data, 31%31\% on real queries), but the normal is on par or ahead exactly where revenue concentrates: transactional and niche-premium queries, home-turf queries, and the value and clearance brands whose bids sit in the high-value range with additive rather than multiplicative spread. Cell group (share of cells won by the normal) Designed (n=4n=4, 3,0653,065 cells) Real queries (2,2322,232 cells) All cells 36±2%36± 2\% 31±2%31± 2\% Transactional queries 57±4%57± 4\% 43±4%43± 4\% Niche-premium queries 59±6%59± 6\% — Home-turf queries (four brands) 4343–61%61\% — Comparison queries 21±3%21± 3\% 34±4%34± 4\% Informational queries 22±3%22± 3\% 23±4%23± 4\% Adjacent queries 26±3%26± 3\% 23±3%23± 3\% Clearance brand (Basecamp) 48±4%48± 4\% 56±4%56± 4\% Value brand (Trail) 47±4%47± 4\% 45±4%45± 4\% Premium brand (Summit) 27±3%27± 3\% 13±3%13± 3\% Flagship brand (Apex) 23±3%23± 3\% 8±2%8± 2\% Appendix S11 Is the network necessary? Convex fits in the same cone Real-query benchmark, same split and replay as the channel decomposition (cents; deterministic fits, network rows are 5-seed means). Prior AffMyerson revenue SPA + tuned reserve (r∗=25r^*=25) 41.54 Embedding-context network 43.48 Global closed form + cone projection (paper) 43.65 Global MTP2MTP_2-constrained MLE (convex) 43.65 Diagonal ++ rank-one single factor 43.63 Intent-bucketed MTP2MTP_2-MLE (4 buckets) 44.29 Intent-feature network 44.33 The convex MLE and the clip-projection agree to 0.0010.001; the bucketed convex fit ties the intent-feature network; the single-factor prior loses only 0.020.02 to the full M-matrix fit here. On the designed four-brand data the ordering changes (seed-0 configuration, split 00, cents): unconstrained network repaired into the cone 49.2949.29; projected global 48.3348.33; convex global MLE 48.1748.17; SPA + tuned reserve 47.0947.09 (r⋆=30r =30); diagonal-plus-rank-one 46.3146.31 (convex_compare_n4b.py, convex_compare_n4b.json; the network row is the unconstrained precision network of proj_cost.py after projection into the cone, proj_cost_out.json). The projected global fit and the convex MLE differ here because clipping the sample precision matrix distorts the marginal variances (its implied IPV revenue is 43.6443.64 against 45.9545.95 for the MLE, which preserves them), while their conditioning revenues stay within 0.160.16 cents. Where context is predictable the repaired unconstrained network is worth +0.96+0.96 over the best convex fit, and one Gaussian factor is not enough, the applicability boundary reported in the main text. Appendix S12 Collusion sensitivity and the conditioning clip Single-shot DSIC covers unilateral deviations only. The main text reports that joint shading by the winner’s opponents moves the winner’s payment more under conditioning than under IPV pricing, because the same channel that earns the rent transmits coordinated misreports. The exposure is tunable. Define the clipped conditional mean by capping the shift that competitors’ reports induce, mi(c)=μi+clip(∑j≠iβij(sj−μj),−cσi|−i,cσi|−i),m^(c)_i= _i+clip ( _j≠ i _ij(s_j- _j),\ -c\, _i -i,\ c\, _i -i ), and run the mechanism with ψi(c)=si−σi|−iR((si−mi(c))/σi|−i)ψ^(c)_i=s_i- _i -iR ((s_i-m^(c)_i)/ _i -i ). The clip is monotone in every sjs_j, so own-monotonicity and cross-monotonicity (main text, Lemmas 5.1–5.2) are preserved and the mechanism stays EPIC and EPIR for every c∈[0,∞]c∈[0,∞]; c=∞c=∞ is AffMyerson, c=0c=0 prices with the conditional scale but no conditional shift. Seed-00 spectral prior, held-out n=2n=2 bids, script spectral_unify.py (results in spectral_unify_results.json); “drop” is the winner’s payment reduction when all opponents shade their reports by 20%20\%; welfare is relative to first best; allocations are unchanged by the shading in every row; single seed, hence the small offsets from the five-seed means of the main text. Mechanism Revenue Rent vs. IPV Share of rent Welfare Drop, 20%20\% shade IPV-Myerson (no conditioning) 34.8334.83 – – 84.2%84.2\% 10.1%10.1\% c=0c=0 (scale only, no shift) 35.1635.16 +1.0%+1.0\% 11%11\% 87.6%87.6\% 11.0%11.0\% c=0.25c=0.25 36.5636.56 +5.0%+5.0\% 56%56\% 90.8%90.8\% 11.4%11.4\% c=0.5c=0.5 37.2137.21 +6.8%+6.8\% 77%77\% 92.7%92.7\% 12.1%12.1\% c=1c=1 37.6637.66 +8.1%+8.1\% 91%91\% 93.9%93.9\% 13.1%13.1\% c=2c=2 37.8737.87 +8.7%+8.7\% 98%98\% 94.0%94.0\% 13.8%13.8\% c=∞c=∞ (full conditioning) 37.9337.93 +8.9%+8.9\% 100%100\% 93.7%93.7\% 14.5%14.5\% Full conditioning earns +8.9%+8.9\% over IPV and loses 14.5%14.5\% of the winner’s payment to a 20%20\% joint shade, against 10.1%10.1\% for IPV; the extra exposure is 4.44.4 points. At c=0.5c=0.5 the mechanism keeps 77%77\% of the rent while the extra exposure falls to 2.02.0 points; at c=0.25c=0.25, 56%56\% of the rent for 1.31.3 points. The trade-off is monotone in c, so a platform can set the clip from its own estimate of collusion risk; nothing else in the pipeline changes. Every clipped variant is EPIC/EPIR by the main text’s Theorem 5.7; only c=∞c=∞ is the revenue-optimal rule of Theorem 5.5 for a Gaussian prior. (Numbers here are the spectral main line and supersede any legacy-pipeline figures.) Appendix S13 Prior-poisoning attack and mitigation (full numbers) Shading one advertiser’s bid history by δ=10%/20%δ=10\%/20\% before the refit (period-one cost ignored) raises that advertiser’s next-period utility by +14%/+37%+14\%/+37\%, while platform revenue falls by 0.6%/3.0%0.6\%/3.0\% (the IPV baseline loses 1.6%1.6\% under the same attack). Mitigation by prior averaging: replacing the current prior with a convex combination of the current and previous-period priors cuts the δ=20%δ=20\% utility gain to +17%+17\% at λ=12λ= 12 and +8%+8\% at λ=14λ= 14, at no revenue cost when no attack is present. Full dynamic-IC treatment remains open; single-shot EPIC is unaffected. Appendix S14 Complete rerun summary Table 2. Main fixed-split revenue, five optimization seeds. Mechanism Revenue (cents) SPA 28.391 Contextual-reserve SPA 32.979±0.17932.979± 0.179 IPV-Myerson 34.661±0.18834.661± 0.188 Ronen lookahead 35.667±0.07535.667± 0.075 SPA + tuned reserve 35.621 AffMyerson 37.838±0.05637.838± 0.056 Table 3. Independent query-split reruns with the spectral M-matrix prior. Split Lift vs. tuned reserve Lift vs. IPV 0 +5.80%+5.80\% +8.14%+8.14\% 1 +6.78%+6.78\% +7.94%+7.94\% 2 +5.80%+5.80\% +9.59%+9.59\% 3 +4.60%+4.60\% +8.91%+8.91\% 4 +5.97%+5.97\% +9.80%+9.80\% Mean± +5.79%±0.78%+5.79\%± 0.78\% +8.88%±0.83%+8.88\%± 0.83\% On the full n=4n=4 market, AffMyerson earns 49.047±0.43149.047± 0.431 cents, 4.16%±0.92%4.16\%± 0.92\% above the tuned reserve and 6.17%±0.37%6.17\%± 0.37\% above matched IPV. Sweeping the share of special (niche/home-turf) queries from 00 to 23%23\% moves the rent between +5.5%+5.5\% and +6.8%+6.8\% and the margin over the tuned reserve between +4.3%+4.3\% and +7.6%+7.6\%, all confidence intervals positive. On the n=2n=2 main line AffMyerson also trades more often than the tuned-reserve SPA (89.8%89.8\% vs. 83.3%83.3\% of auctions) at equal welfare (53.853.8). On real-user queries with LLM-agent valuations, it earns 43.480±0.14643.480± 0.146 cents, with corresponding lifts of 4.67%±0.35%4.67\%± 0.35\% and 4.48%±0.07%4.48\%± 0.07\%. Brand submarkets (main text, Section 7.5). Because each brand bids without seeing the others, every subset of the four brands is a valid market. The table lists all eleven submarkets with the spectral prior (seed 00; bootstrap intervals over test queries; script rho_scan_b.py, results n_sweep_out_b/results_spectral.json). Fitted ρ¯ ρ alone does not rank the markets: the three lowest rents (+0.4+0.4, +0.5+0.5, +1.5%+1.5\%, none significant) are the pairings of a high-end brand with the clearance brand, whose near-constant bids carry no usable signal for conditioning, while the same clearance-containing markets show the largest margins over the tuned reserve because a single reserve fits them poorly. One three-brand market loses to the tuned reserve (−1.5%-1.5\% [−2.6,−0.2][-2.6,-0.2]). The across-submarket median rent is +6.3/+6.5/+6.4%+6.3/+6.5/+6.4\% for n=2/3/4n=2/3/4; a symmetric synthetic market (μ=50μ=50, σ=15σ=15 cents, equicorrelation ρ=0.6ρ=0.6, 5,0005,000 auctions per n; mech_aff_n.py) declines instead, +4.0/+3.3/+3.1/+2.2%+4.0/+3.3/+3.1/+2.2\% for n=2/3/4/5n=2/3/4/5, so in the stratified market the conditioning information that an added competitor brings offsets the dilution of the rent. n Market Rent vs. IPV-Myerson Lift vs. tuned reserve 2 Flagship++Premium +5.6%+5.6\% [+2.5,+9.0][+2.5,+9.0] +1.6%+1.6\% [−2.9,+6.8][-2.9,+6.8] 2 Flagship++Value +7.1%+7.1\% [+5.2,+9.0][+5.2,+9.0] +1.3%+1.3\% [−1.0,+3.2][-1.0,+3.2] 2 Flagship++Clearance +0.5%+0.5\% [−2.0,+3.1][-2.0,+3.1] +22.2%+22.2\% [+13.5,+31.8][+13.5,+31.8] 2 Premium++Value +8.4%+8.4\% [+6.7,+10.2][+6.7,+10.2] +10.5%+10.5\% [+7.8,+13.2][+7.8,+13.2] 2 Premium++Clearance +1.5%+1.5\% [−0.3,+3.2][-0.3,+3.2] +37.9%+37.9\% [+31.2,+44.9][+31.2,+44.9] 2 Value++Clearance +8.4%+8.4\% [+7.1,+9.7][+7.1,+9.7] +12.4%+12.4\% [+10.8,+14.2][+10.8,+14.2] 3 Flagship++Premium++Value +6.4%+6.4\% [+4.3,+8.8][+4.3,+8.8] −1.5%-1.5\% [−2.6,−0.2][-2.6,-0.2] 3 Flagship++Premium++Clearance +0.4%+0.4\% [−1.5,+2.5][-1.5,+2.5] +3.4%+3.4\% [+0.8,+6.4][+0.8,+6.4] 3 Flagship++Value++Clearance +8.9%+8.9\% [+7.4,+10.5][+7.4,+10.5] +19.4%+19.4\% [+15.2,+23.8][+15.2,+23.8] 3 Premium++Value++Clearance +6.7%+6.7\% [+4.9,+8.5][+4.9,+8.5] +15.4%+15.4\% [+11.9,+19.3][+11.9,+19.3] 4 All four +6.4%+6.4\% [+4.8,+7.9][+4.8,+7.9] +4.7%+4.7\% [+3.5,+5.9][+3.5,+5.9] Parameterization audit. On n=4n=4, relaxed precision learning plus feasibility repair clips 2424–27%27\% of off-diagonal entries, removes 1616–21%21\% of their L1L_1 mass, and shifts the diagonal of 3636–43%43\% of matrices; its reserve-relative effect changes sign. The diagonally dominant M-matrix ablation is stable but fits the flagship–premium pair correlation near 0.230.23 (all-pairs mean 0.220.22). The spectral model fits 0.58±0.050.58± 0.05 for the same pair (all-pairs mean 0.470.47; four-prior table, Section S5) and gives stable revenue effects. Distributional misspecification audit. On the main n=2n=2 split, we refit the same spectral M-matrix network after monotone transforms and compare test NLL on the original bid scale using the exact Jacobian. Across five optimization seeds, the original-scale Gaussian has NLL 9.410±0.0519.410± 0.051 and whitened-residual standard deviation, skewness, and kurtosis 1.410±0.0221.410± 0.022, 0.846±0.1250.846± 0.125, and 7.996±0.7267.996± 0.726. The log(s+0.1) (s+0.1) transform has NLL 8.435±0.0168.435± 0.016, an improvement of 0.976±0.0570.976± 0.057 nats, with residual diagnostics 1.088±0.0321.088± 0.032, −0.066±0.022-0.066± 0.022, and 3.124±0.0793.124± 0.079. This is evidence against treating the Gaussian as the literal agent-value distribution. We retain it as a working prior because the closed-form regularity and incentive proofs are for the original-scale Gaussian; the transformed fit is a diagnostic, not an auction baseline. Accordingly, the experiment supports structural truthfulness and replay revenue under misspecification, not the correct-prior optimality premise. Latency. With 60 bisection iterations, the final reference pipeline (spectral network, tensor transfer, and pricing; sentence encoder excluded) has p50/p95 latency 0.308/0.4500.308/0.450 ms for n=2n=2, 0.676/0.9460.676/0.946 ms for n=4n=4, and 0.887/1.1950.887/1.195 ms for n=5n=5 on the reported AMD CPU. Unseen-shock audit. The prior and reserve are trained only on the central 80%80\% of latent demand shocks; evaluation uses held-out queries in the unseen lower and upper 10%10\% tails. The latent shock is used only to define this audit and is not supplied to the network or mechanism. Across five optimization seeds, combined-tail revenue is 37.046±0.27137.046± 0.271 cents, a 9.52%±0.49%9.52\%± 0.49\% lift over matched IPV and 14.08%±0.84%14.08\%± 0.84\% over the training-tuned reserve. Seed-0 paired query-cluster intervals are [6.21,12.32]%[6.21,12.32]\% and [8.89,16.82]%[8.89,16.82]\%. As a deliberately optimistic diagnostic, we additionally select one constant reserve directly on the combined tail test sample. This post-hoc oracle reserve is not deployable and has an information advantage over AffMyerson; nevertheless, AffMyerson exceeds it by 8.54%±0.80%8.54\%± 0.80\% across seeds, with a seed-0 paired query-cluster interval of [3.26,11.81]%[3.26,11.81]\%. Thus the combined-tail result is not explained only by carrying a central-regime reserve into the tails. If the diagnostic is given the hidden tail label and may tune a different reserve separately in each regime, those two test-tuned oracles exceed AffMyerson by 31.21%±2.93%31.21\%± 2.93\% in the low tail and 3.29%±0.50%3.29\%± 0.50\% in the high tail. This is not a deployable shock-blind baseline, but it identifies a remaining limitation: the present context model cannot adapt to an unobserved regime label. Appendix S15 Use of generative AI Generative AI (Claude, Anthropic, via the Claude desktop app) was used to edit prose, to write and review code, to check proofs and propose fixes for gaps the authors then verified, and to help refine experimental designs proposed by the authors. All mathematical statements, citations, code changes, and numerical results were checked by the authors. A short note on which research stages received AI assistance, with the tool version, will accompany the released code. The LLM agents that act as bidders in the experiments are part of the experimental setup, not writing assistance, and are documented with their prompts and sampling settings. Appendix S16 Real-query prediction and channel diagnostics This section provides the per-source and channel decompositions supporting Section 7.6 of the main paper. Construction of the real-query benchmark (main text, Section 7.6). Queries come from two public corpora of genuine user queries, the Amazon ESCI shopping-query examples and the MS MARCO query set. Each corpus is first screened by outdoor-gear keywords, then ranked by cosine similarity of its MiniLM embedding to five seed phrases (“hiking and camping gear”, “outdoor equipment for trekking”, “mountaineering and climbing equipment”, “ski and snowboard gear”, “camping tent sleeping bag backpack”), keeping the top 2,0002,000 ESCI and 3,2003,200 MARCO queries. Two gpt-4o-mini passes follow. The first labels each query with one of the four intent tiers used for the designed queries (transactional, comparison, informational, adjacent), with the same definitions. The second is a topic gate that answers yes or no to whether the query is genuinely about outdoor recreation gear or activities, and rejects gym equipment, toys, baby gear, school bags, grills, RV dealers, road-cycling brands, fashion items, events and stores, and anything that merely contains outdoor-sounding words. From the queries that pass the gate we draw up to 150150 per tier, filling transactional and comparison tiers from ESCI first and informational and adjacent tiers from MARCO first, which yields the 580580 queries (274274 ESCI, 306306 MARCO) of the main text. Neither pass sees any bid or revenue. Script build_real_queries.py; both prompts are in the script and the released artifact. Per-source decomposition. The two sources are independent sub-benchmarks; the split localizes the prediction failure: R2R^2 is mildly positive on MARCO (+0.13+0.13) but negative on ESCI (−0.45-0.45); within concrete shopping queries the value differences ride on product-tier and budget details invisible to sentence embeddings, so the model’s variation is spurious; a modern encoder (bge-base, C-Pack) doubles overall R2R^2 to 0.210.21 without flipping ESCI; a 44-dimensional intent label alone reaches 0.180.18; a split-half noise ceiling puts attainable R2R^2 at ≈0.81≈ 0.81 (split-half estimate). Crucially, the rent persists on exactly the source where text prediction is counterproductive. Channel decomposition. A 2×22× 2 ablation (context on/off × conditioning on/off, same held-out bids) makes “the rent flows through conditioning” quantitative: with a global prior (zero text prediction) conditioning alone is worth +3.03+3.03 cents, more than the entire rent (+1.86±0.03+1.86± 0.03), and already beats the tuned-reserve SPA by +5.1%+5.1\%; the embedding context channel adds nothing (−0.17±0.14-0.17± 0.14), while the 44-dim intent feature adds +0.68±0.03+0.68± 0.03. The same 2×22× 2 on the designed data completes the contrast: conditioning is worth +4.48+4.48 cents there (vs. rent +3.18±0.16+3.18± 0.16) while the context channel is worth ≈+1.9≈+1.9. Conditioning is the engine in both regimes. Revenue is insensitive to μ-accuracy beyond a coarse level: the engine is conditioning on competitors’ reports (main-paper Theorem 5.7).