Beyond Independent Values: Optimal Auctions with Correlated Valuations for LLM-Native Advertising

Shiman Sheng; Mingyu Guo

cs.GTOct 10, 2026

Beyond Independent Values: Optimal Auctions with Correlated Valuations for LLM-Native Advertising

Authors: Shiman Sheng, Mingyu Guo

Organizations: The University of Adelaide, Adelaide, Australia

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.

Figures & tables

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.

Figure 2. How conditioning moves money (illustrative two-bidder prior with means $45$ and $35$ cents, standard deviations $13$ and $12$ , $\rho=0.7$ ; 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 $t_{0}$ , which rises with the opponent’s bid ( $\partial t_{0}/\partial s_{j}>0$ in this bivariate case, Section 4.3 ). Right: the revenue difference at each bid pair ( AffMyerson minus IPV-Myerson). The ellipses enclose $1\sigma$ and $2\sigma$ of the prior, that is, where most bid pairs fall. Averaged over the prior the difference is $+1.9$ cents ( $+5.4\%$ ). Two-bidder example: the conditional distribution, the winner's payment surface, and the revenue gain over IPV-Myerson.

Mechanism	Revenue	Lead
Second price (SPA)	$28.39$	$+33.3\%$
Contextual-reserve SPA	$32.98\pm 0.18$	$+14.7\%$
IPV-Myerson (same prior)	$34.66\pm 0.19$	$+9.2\%$
Tuned-reserve SPA ( $r^{*}{=}32$ )	$35.62$	$+6.2\%$
Ronen lookahead ( 8 )	$35.67\pm 0.08$	$+6.1\%$
CAN, rank scores, no context ( 21 )	$37.43\pm 0.07$	$+1.1\%$

Table 1. Revenue on held-out LLM-agent bids (

n{=}2

, cents, mean

\pm

standard deviation over

5

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.

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

\rho

. The panel that tracks the fitted

\bar{\rho}

is Appendix S9. Revenue lift over IPV-Myerson and over the tuned-reserve second-price auction as the within-query correlation is varied.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

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(n^{2})

row-read of the precision matrix plus a

60

-step bisection prices each auction in

\sim

12\,\mu

s at

n{=}2

(pricing only; p50

0.3

ms end-to-end with the network forward pass, Section S14 ).

Prior	Test NLL	$R^{2}$	$\bar{\rho}$	$\rho_{01}$	Outside cone
A: global closed form (no context)	$18.56$	$-0.00$	$0.20$	$0.67$	$33\%$
B: unconstrained precision Cholesky	$20.75\pm 0.46$	$-0.03\pm 0.02$	$0.47\pm 0.06$	$0.61\pm 0.03$	$25\pm 3\%$
C: diagonally dominant M-matrix	$19.00\pm 0.62$	$+0.60\pm 0.06$	$0.22\pm 0.02$	$0.23\pm 0.03$	$0$
S: spectral M-matrix (deployed)	$18.37\pm 0.57$	$+0.01\pm 0.02$	$0.47\pm 0.08$	$0.58\pm 0.05$	$0$

Appendix

Table 6

Lemma / quantity	Check	Result
S2 : $R^{\prime}=zR-1$	finite differences, $z\in[-3,6]$	rel. error $3\times 10^{-8}$
S2 : $2-zR>1$	grid $z\in[-6,6]$	min $1.026$ ; $\to 1$ at $+\infty$
S2 (a): $\|R^{\prime}\|$ decreasing	grid	confirmed
S2 (b): $\max g=\sqrt{\pi/2}$	grid	$1.2533$ at $z=0$
S3 : $h(z_{i})\leq m_{i}/\sigma_{i}$	all active bidders, sold auctions	$0/4{,}339$ violations
$\kappa=\max_{i}m_{i}/\sigma_{i}$	test set, fitted prior (median $3.25$ )	$16.93$ ( $\geq 6.54$ ✓)

Appendix

Table 7

Prompt scheme	Gradient	Corr.	Median	$\leq 10$	P95	Top-3
Facts only, qualitative (v3)	$0.98$	$+0.78$	$30$	$13\%$	$94$	$27\%$
$+$ qualitative conversion funnel (v4)	$1.74$	$+0.74$	$30$	$6\%$	$85$	$32\%$
$+$ quantified conversion facts (v4.2)	$8.56$	$+0.41$	$12$	$46\%$	$80$	$25\%$
$+$ chain-of-thought (v4.2 $+$ CoT)	$6.71$	$+0.06$	$5$	$80\%$	$80$	$54\%$
$+$ quantified demand index $+$ decimals (v4.3)	$7.40$	$+0.09$	$4$	$84\%$	$100$	$27\%$
$+$ low temperature ( $T{=}0.2$ )	$8.94$	$-0.03$	$3$	$88\%$	$120$	$29\%$

Appendix

Table 8

Cell group (share of cells won by the normal)	Designed ( $n{=}4$ , $3{,}065$ cells)	Real queries ( $2{,}232$ cells)
All cells	$36\pm 2\%$	$31\pm 2\%$
Transactional queries	$57\pm 4\%$	$43\pm 4\%$
Niche-premium queries	$59\pm 6\%$	—
Home-turf queries (four brands)	$43$ – $61\%$	—
Comparison queries	$21\pm 3\%$	$34\pm 4\%$
Informational queries	$22\pm 3\%$	$23\pm 4\%$

Appendix

Table 10

Mechanism	Revenue	Rent vs. IPV	Share of rent	Welfare	Drop, $20\%$ shade
IPV-Myerson (no conditioning)	$34.83$	–	–	$84.2\%$	$10.1\%$
$c=0$ (scale only, no shift)	$35.16$	$+1.0\%$	$11\%$	$87.6\%$	$11.0\%$
$c=0.25$	$36.56$	$+5.0\%$	$56\%$	$90.8\%$	$11.4\%$
$c=0.5$	$37.21$	$+6.8\%$	$77\%$	$92.7\%$	$12.1\%$
$c=1$	$37.66$	$+8.1\%$	$91\%$	$93.9\%$	$13.1\%$
$c=2$	$37.87$	$+8.7\%$	$98\%$	$94.0\%$	$13.8\%$

Appendix

Table 11

Mechanism	Revenue (cents)
SPA	28.391
Contextual-reserve SPA	$32.979\pm 0.179$
IPV-Myerson	$34.661\pm 0.188$
Ronen lookahead	$35.667\pm 0.075$
SPA + tuned reserve	35.621
AffMyerson	$\mathbf{37.838\pm 0.056}$

Appendix

Table 2. Main fixed-split revenue, five optimization seeds.

Split	Lift vs. tuned reserve	Lift vs. IPV
0	$+5.80\%$	$+8.14\%$
1	$+6.78\%$	$+7.94\%$
2	$+5.80\%$	$+9.59\%$
3	$+4.60\%$	$+8.91\%$
4	$+5.97\%$	$+9.80\%$
Mean $\pm$ sd	$+5.79\%\pm 0.78\%$	$+8.88\%\pm 0.83\%$

Appendix

Table 3. Independent query-split reruns with the spectral M-matrix prior.

$n$	Market	Rent vs. IPV-Myerson	Lift vs. tuned reserve
2	Flagship $+$ Premium	$+5.6\%$ $[+2.5,+9.0]$	$+1.6\%$ $[-2.9,+6.8]$
2	Flagship $+$ Value	$+7.1\%$ $[+5.2,+9.0]$	$+1.3\%$ $[-1.0,+3.2]$
2	Flagship $+$ Clearance	$+0.5\%$ $[-2.0,+3.1]$	$+22.2\%$ $[+13.5,+31.8]$
2	Premium $+$ Value	$+8.4\%$ $[+6.7,+10.2]$	$+10.5\%$ $[+7.8,+13.2]$
2	Premium $+$ Clearance	$+1.5\%$ $[-0.3,+3.2]$	$+37.9\%$ $[+31.2,+44.9]$
2	Value $+$ Clearance	$+8.4\%$ $[+7.1,+9.7]$	$+12.4\%$ $[+10.8,+14.2]$

Appendix

Table 14

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