Price of Anarchy of First-Price Auctions in Agentic Markets

Abstract: An agentic market is a decentralized serving-time market where AI agents bid to claim each user prompt. An agent's bid is based on its expected value for the prompt, which is the expectation of the reward for its response minus its inference costs. First-price auctions (FPAs), in which the highest bidder pays its bid and receives the prompt, are a natural way to allocate prompts in such a market, since they are simple and transparent to run, and do not require central knowledge of private capabilities and costs. However, FPAs are not truthful: participants may bid less than their true value in the hope of a larger profit, and the prompt may then be allocated to someone other than the bidder who values it the most. This welfare loss is quantified by the $\textit{price-of-anarchy}$ (PoA). In this paper, we argue that agentic markets are likely to possess features that reduce the price-of-anarchy. The first is a $\textit{competitive floor}$, which occurs when multiple bidders usually value the prompt above some baseline share of the optimal welfare (e.g., because the best models are often distilled into cheaper alternatives, which perform similarly). We show that competition among them then limits the welfare loss, which can be driven to zero as the floor rises. The second feature is $\textit{near-independence}$ of values after conditioning on the prompt, which occurs when most of the predictable variation in the expected value is determined by the difficulty of the prompt. As the value distribution approaches conditional independence, we can derive an efficiency guarantee that approaches a factor of $1 - 1/e^2$ of the optimal welfare. Thus, both features give rise to welfare lower bounds that are significantly better than the well-known general case of $1 - 1/e$. We validate these results empirically using auctions with strategic bidders simulated on the EmbedLLM prompt routing dataset.

• The paper derives distribution-sensitive welfare guarantees showing that competitive floors improve the standard 0.632 price-of-anarchy bound, reaching about 0.856 with 16 credible competitors when α=0.9 and ε=0.1.
• The paper proves that values conditionally near-independent given prompt information approach the 0.865 independent-private-values guarantee, while platform reserves and careful filtering can strengthen competition without excluding prompt-specific winners.
• The paper evaluates simulated strategic bidding on more than 30,000 prompts and 110 LLMs, finding first-price auctions achieve 0.992 welfare in typical equilibria and 0.988 under adversarial approximate equilibria.

The paper studies first-price auctions (FPAs) as an allocation mechanism for decentralized “agentic markets,” in which multiple AI systems compete at serving time for individual prompts. Its central claim is that the standard worst-case price-of-anarchy guarantee for FPAs is substantially too pessimistic for value distributions plausibly induced by model specialization, distillation, and prompt-dependent performance. The analysis identifies two structural conditions—competitive floors and conditional near-independence—that improve welfare guarantees, develops platform interventions that can promote these conditions, and evaluates the resulting claims using simulated strategic bidding over the EmbedLLM dataset (2610.20587).

Market model and welfare objective

For a prompt, bidder $i$ has value equal to expected response reward minus inference cost. The auctioneer observes only bids, while each bidder privately observes or estimates its own prompt-specific value. In a sealed-bid FPA, the highest bidder wins and pays its own bid. The paper models strategic behavior as a Bayesian game with an arbitrary joint prior over bidder values and considers every non-overbidding Bayes–Nash equilibrium (BNE).

The welfare objective is the value of the winning bidder. If $V_{(1)}$ denotes the highest realized value, the first-best welfare is $V_{(1)}$, whereas the auction welfare is the value of the highest bidder. The price of anarchy is the infimum, over value distributions and BNE, of expected auction welfare divided by expected first-best welfare.

This formulation isolates the allocative cost of strategic shading. Revenue is included in the standard social-welfare accounting, but because the winner’s payment is transferred to the platform, welfare is equivalent to the winner’s value. The analysis assumes nonnegative values and no overbidding, $0 \leq b_i \leq v_i$, which is standard in the FPA literature and is essential for the stated guarantees.

Under arbitrary correlated values, the known welfare guarantee is $1-1/e \approx 0.632$. Under independent private values, the tight guarantee is $1-1/e^2 \approx 0.865$ [Jin and Lu, arXiv ID not supplied in the manuscript]. The paper’s objective is not to improve these universal constants directly, but to derive distribution-sensitive bounds that exploit empirically meaningful structure in agentic markets.

Competitive floors

The first structural condition is a competitive floor. For a fraction $\alpha$ of the highest value, $m$ bidders, and failure probability $\epsilon$, the market has an $(\alpha,m,\epsilon)$-competitive floor when

$V_{(1)}$0

Thus, with high probability, at least $V_{(1)}$1 bidders value the prompt at least $V_{(1)}$2 times as much as the best bidder. This condition captures markets in which several models are credible substitutes on most prompts, even if some models are weak or highly specialized.

The analysis begins with a deterministic floor. If two bidders always have values at least $V_{(1)}$3, then every BNE achieves

$V_{(1)}$4

This interpolates between the universal $V_{(1)}$5 guarantee at $V_{(1)}$6 and full efficiency at $V_{(1)}$7. For example, $V_{(1)}$8 yields a guarantee of approximately $V_{(1)}$9, compared with $V_{(1)}$0 without the floor. The mechanism behind the result is direct: a credible runner-up prevents the winning bid from falling below the floor, while the usual smoothness argument recovers a fraction of the best bidder’s value above that floor.

The main theorem allows the floor to fail on an $V_{(1)}$1 fraction of prompts and generalizes from two competitors to $V_{(1)}$2 competitors:

$V_{(1)}$3

where

$V_{(1)}$4

The penalty $V_{(1)}$5 decreases as the number of credible competitors grows. It equals zero when the floor never fails and approaches $V_{(1)}$6 as $V_{(1)}$7 increases. The result is stronger than simply multiplying the deterministic guarantee by the probability that the floor holds: the equilibrium bidding incentives themselves constrain the expected shortfall of the winning bid below the floor.

The numerical consequences are substantial. If $V_{(1)}$8 and the floor fails with probability $V_{(1)}$9, the bound is approximately $0 \leq b_i \leq v_i$0 for $0 \leq b_i \leq v_i$1, approximately $0 \leq b_i \leq v_i$2 for $0 \leq b_i \leq v_i$3, and approximately $0 \leq b_i \leq v_i$4 for $0 \leq b_i \leq v_i$5 according to the plotted values. The implication is that reliability and multiplicity of competition are distinct: two strong competitors may be insufficient if they disappear together on difficult prompts, whereas a larger set of credible alternatives can preserve welfare despite a comparable failure rate.

The paper also derives a query-conditional form of the result. If public prompt information $0 \leq b_i \leq v_i$6 is available before bidding, the floor can be defined relative to $0 \leq b_i \leq v_i$7 rather than a pooled market-wide benchmark. This matters because a global floor can be too stringent on difficult prompts and too weak on easy ones. Conditioning allows the guarantee to adapt to prompt difficulty and is never weaker than applying a single pooled floor.

Near-independent values

The second structural condition concerns the dependence among bidder values after conditioning on public prompt information. Values are generally correlated across models because models tend to succeed on easy prompts and fail on difficult prompts. However, if the prompt itself explains most of this common variation, residual values may be close to independent conditional on $0 \leq b_i \leq v_i$8.

For each public state $0 \leq b_i \leq v_i$9, the paper compares the conditional joint distribution $1-1/e \approx 0.632$0 with the product of its conditional marginals,

$1-1/e \approx 0.632$1

and measures their discrepancy using total variation distance $1-1/e \approx 0.632$2. The principal theorem establishes that, for bounded values, every BNE satisfies a conditional welfare guarantee of the form

$1-1/e \approx 0.632$3

where $1-1/e \approx 0.632$4 is nondecreasing, $1-1/e \approx 0.632$5, and $1-1/e \approx 0.632$6 as $1-1/e \approx 0.632$7.

When values are exactly conditionally independent, the additive error disappears and the $1-1/e \approx 0.632$8 guarantee is recovered. Thus arbitrary dependence across prompts is compatible with the independent-values guarantee if the dependence is explained by publicly observed prompt information. For small departures from conditional independence, the result degrades continuously, although the general theorem is qualitative rather than explicitly rate-optimal.

The proof is technically nontrivial because changing the prior affects both welfare expectations and equilibrium incentives. An equilibrium under the original joint distribution need not remain an equilibrium under the product distribution. The paper addresses this by showing that total variation distance bounds the induced regret under the product prior, then proving that independent-value FPAs with vanishing regret cannot maintain a fixed welfare gap below the Jin–Lu guarantee. A compactness argument handles limiting bid distributions, mixed strategies, and bid-only tie-breaking rules.

Under additional regularity assumptions—bounded and continuous marginal densities and a sufficiently smooth likelihood-ratio perturbation from independence—the paper obtains an explicit linear refinement:

$1-1/e \approx 0.632$9

where $1-1/e^2 \approx 0.865$0 depends on the number of bidders and the regularity parameters. This quantitative result is more restrictive than the qualitative theorem and should not be interpreted as a general Lipschitz property without those assumptions.

Platform design implications

The theory identifies two interventions through which a platform can improve the conditions supporting FPA efficiency.

Pre-auction filtering

A platform may restrict participation using public prompt information, admitting only models whose predicted cluster-level value is within a fraction $1-1/e^2 \approx 0.865$1 of the strongest model. Filtering weak bidders raises a direct lower bound based on the minimum value among admitted bidders. However, removing the strongest model on a particular prompt lowers the attainable welfare ceiling.

The theoretical result therefore has an important asymmetry: removing weak bidders can improve strategic competition, but removing strong bidders creates an allocation loss that no bidding mechanism can recover. The platform must retain sufficiently strong models while eliminating low-value participants.

Platform-owned bidding

The platform may enter an in-house model that publicly bids its own value $1-1/e^2 \approx 0.865$2 on every prompt. This bid acts as a query-specific reserve price. The resulting guarantee is

$1-1/e^2 \approx 0.865$3

Unlike the deterministic two-bidder floor, this result requires only an average lower bound on the platform model’s value. A platform model that consistently supplies a substantial fraction of first-best welfare therefore constrains bid shading even when no external competitor clears the corresponding floor on every prompt.

This intervention also illustrates a potential conflict between mechanism simplicity and platform power. The same model that improves welfare by acting as a reserve price may alter market access and competitive conditions. The paper analyzes its allocative role but does not study strategic platform objectives, model cannibalization, or revenue effects.

Empirical evaluation on EmbedLLM

The empirical study uses more than 30,000 prompts and 110 LLMs from EmbedLLM [Zhuang et al., arXiv ID not supplied in the manuscript]. Prompt-specific model values are estimated as predicted accuracy minus a parameter-count-based inference cost. Predictions use $1-1/e^2 \approx 0.865$4-nearest neighbors over prompt embeddings, with $1-1/e^2 \approx 0.865$5. Prompts are partitioned into 20 embedding clusters, and bidders condition their strategies on the cluster and their own value.

Approximate BNE are computed using damped fictitious play over discretized value bins and bid-shading factors. The authors evaluate both a typical equilibrium and an adversarial approximate equilibrium. In the latter, bidders choose responses that preserve at least 95% of their maximum surplus while minimizing welfare. This is intended to stress-test the bounds, although the solver does not have a general convergence guarantee for correlated-value FPAs and starts from truthful bidding, potentially selecting only equilibria reachable by downward shading.

On the full market, the FPA achieves very high welfare:

Allocation rule	Welfare relative to oracle
Random allocation	0.542
Best single model	0.781
Best model per prompt cluster	0.894
FPA, typical equilibrium	0.992
FPA, adversarial equilibrium	0.988

The worst-cluster ratios are also strong: $1-1/e^2 \approx 0.865$6 for the typical FPA equilibrium and $1-1/e^2 \approx 0.865$7 for the adversarial equilibrium, compared with $1-1/e^2 \approx 0.865$8 for the best-per-cluster router and $1-1/e^2 \approx 0.865$9 for the best single model. The auction benefits from conditioning on each model’s private prompt-specific value: specialists can win precisely when the usual cluster leader is weak.

The empirical competitive-floor bound on the full market is at least $\alpha$0 overall and $\alpha$1 on the worst cluster. These guarantees are conservative relative to the realized values of $\alpha$2 and $\alpha$3, but they improve materially over the distribution-free $\alpha$4 bound. The gap is expected because the competitive-floor theorem ignores the additional efficiency contributed by near-independence and uses only a limited summary of the value distribution.

Across 100,000 randomly sampled submarkets, realized welfare is above $\alpha$5 for the overwhelming majority of auctions. In the lowest competitive-floor bound bin, the bottom $\alpha$6 percentile of welfare is approximately $\alpha$7 at typical equilibria and $\alpha$8 at adversarial equilibria. In the highest bound bin, these values rise to approximately $\alpha$9 and $m$0, respectively. Thus, the empirical lower tail tracks the market-specific competitive-floor bound rather than the universal worst-case constant.

The hand-constructed four-model cohorts further isolate the role of competition near the top of the value distribution:

Cohort	Runner-up gap	Adversarial shading	Adversarial welfare loss
Similar generalists	7.7%	12.3%	0.4%
Diverse generalists	17.1%	19.8%	1.2%
One generalist, three niche models	35.6%	29.1%	5.1%
Two generalists, two niche models	15.9%	22.3%	0.8%

These results support the paper’s specific mechanism: the level of shading is not itself the welfare problem; welfare loss arises when bidders shade by different amounts and the ranking of bids diverges from the ranking of values. A close runner-up reduces the incentive for the top bidder to shade aggressively and limits misallocation.

Evidence for conditional near-independence

The supplementary empirical analysis regresses each model’s value on prompt embeddings and compares this with prediction from a rival model’s value. Prompt features alone explain an average $m$1 of $m$2, while a rival’s value alone explains $m$3. Adding the rival after conditioning on the prompt increases $m$4 by only $m$5 on average and $m$6 at the 90th percentile.

The paper reports that the prompt explains approximately 90% of the predictive information supplied by a rival’s value, and is the better predictor for 84% of ordered model pairs. This is consistent with prompt-conditioned residual dependence being weaker than unconditional dependence.

The authors correctly state that these regression statistics do not establish the total-variation condition required by the theorem. Small incremental $m$7 does not, in general, imply small multivariate total variation distance. The manuscript offers a Gaussian residual calculation under a simplifying assumption, relating incremental $m$8 to pairwise correlations and then to total variation. That calculation is diagnostic rather than a validated estimate of $m$9 for the actual bounded, non-Gaussian value distribution.

Platform interventions in the experiments

The platform-owned bidder experiments use markets containing the strongest model, random rivals, and a platform model chosen at different percentiles of average model quality. Because these markets are already efficient, average welfare changes little when the platform model is weak or average. The effect is concentrated in the lower tail of adversarial equilibria.

For markets with five rivals, the adversarial $\epsilon$0 percentile increases from approximately $\epsilon$1 for a weak platform model to $\epsilon$2 for a platform model near the median and to $\epsilon$3 when the platform model matches the strongest model. With ten rivals, the corresponding values are approximately $\epsilon$4, $\epsilon$5, and $\epsilon$6. The marginal benefit declines with market size because additional competitors already maintain high bid floors.

Filtering produces a similar pattern. Eligibility cutoffs up to approximately $\epsilon$7 remove weak models without materially reducing the filtered oracle, while slightly improving adversarial lower-tail welfare. More aggressive filtering begins to exclude models that are best on some prompts. At $\epsilon$8, when only the cluster leader is admitted, mean welfare falls to approximately $\epsilon$9 for five-model markets and $(\alpha,m,\epsilon)$0 for ten-model markets, with adversarial $(\alpha,m,\epsilon)$1 percentiles near $(\alpha,m,\epsilon)$2 and $(\alpha,m,\epsilon)$3. The result directly validates the theoretical warning that participation filtering is beneficial only while it preserves the strongest prompt-specific responders.

Limitations and open questions

The analysis assumes that agents are aligned with user welfare and derive utility only from response reward minus inference cost. Advertising, strategic data acquisition, model exposure, or other externalities could make the welfare objective differ from the winner’s value.

The guarantees apply to exact BNE, while the experiments use approximate equilibria generated by a heuristic fictitious-play solver. The solver has no general convergence guarantee in the studied correlated-value setting, restricts strategies to five value bins and multiplicative shading factors, and starts from truthful bids. The adversarial construction is therefore a stress test within a restricted strategy class, not a proof that it finds the worst equilibrium.

The two principal structural conditions are analyzed separately. Taking the maximum of the competitive-floor and near-independence guarantees is immediate, but a theorem that combines both conditions could be stronger. The experiments likewise compute the competitive-floor bound rather than a combined bound, despite empirical evidence for both competitive floors and prompt-conditioned predictive structure.

The quantitative near-independence theorem requires regularity assumptions that may fail for discrete, highly concentrated, or irregular value distributions. The empirical analysis does not directly estimate conditional total variation distance, so it supports but does not verify the theorem’s key premise. Finally, the paper studies welfare rather than platform revenue. Revenue may respond differently to competition, reserve bids, and bidder participation, and the welfare guarantees do not imply analogous revenue guarantees.

Conclusion

The paper establishes that the welfare cost of strategic shading in first-price agentic markets depends strongly on the structure of model values. A competitive floor among several credible bidders improves the universal $(\alpha,m,\epsilon)$4 guarantee, with the improvement increasing in both the floor level and the number of competitors. Conditional near-independence yields an approximate $(\alpha,m,\epsilon)$5 guarantee, although the general robustness result is qualitative and its empirical premise is only indirectly tested.

On EmbedLLM, simulated FPAs attain approximately $(\alpha,m,\epsilon)$6 of oracle welfare on the full market and remain highly efficient in adversarial approximate equilibria. The empirical results indicate that close competition among top-valued models, rather than merely a large number of participants, is the main determinant of robustness. Platform-owned bids and conservative pre-auction filtering can reinforce this structure, but aggressive filtering can reduce welfare by excluding prompt-specific winners. The paper therefore provides a distribution-sensitive justification for FPAs in agentic markets while leaving equilibrium learning dynamics, combined structural guarantees, and revenue analysis unresolved.