# Price of Anarchy of First-Price Auctions in Agentic Markets

arXiv: https://arxiv.org/abs/2610.20587 · HTML: https://mlsanity.com/paper/2610.20587/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 43 references that are in the map.
- First posted on arXiv 14 Oct 2026.

## Placed in

- LLM agents and tool use

## Builds on (most cited first, 25 of 25)

- [FrugalGPT: How to Use Large Language Models While Reducing Cost and Improving Performance](https://mlsanity.com/paper/2305.05176/) (2023)
- [RouteLLM: Learning to Route LLMs with Preference Data](https://mlsanity.com/paper/2406.18665/) (2024)
- [Hybrid LLM: Cost-Efficient and Quality-Aware Query Routing](https://mlsanity.com/paper/2404.14618/) (2024)
- [RouterBench: A Benchmark for Multi-LLM Routing System](https://mlsanity.com/paper/2403.12031/) (2024)
- [Universal Model Routing for Efficient LLM Inference](https://mlsanity.com/paper/2502.08773/) (2025)
- [EmbedLLM: Learning Compact Representations of Large Language Models](https://mlsanity.com/paper/2410.02223/) (2024)
- [Fluid Language Model Benchmarking](https://mlsanity.com/paper/2509.11106/) (2025)
- [Multi-Item Mechanisms without Item-Independence: Learnability via Robustness](https://mlsanity.com/paper/1911.02146/) (2019)
- [State of AI: An Empirical 100 Trillion Token Study with OpenRouter](https://mlsanity.com/paper/2601.10088/) (2026)
- [Decentralized Reinforcement Learning: Global Decision-Making via Local Economic Transactions](https://mlsanity.com/paper/2007.02382/) (2020)
- [The Price of Anarchy in Auctions](https://mlsanity.com/paper/1607.07684/) (2016)
- [Gemini Embedding 2: A Native Multimodal Embedding Model from Gemini](https://mlsanity.com/paper/2605.27295/) (2026)
- [From Benchmarks to Skills: Low-Rank Factors for LLM Evaluation](https://mlsanity.com/paper/2507.20208/) (2025)
- [Why Do Competitive Markets Converge to First-Price Auctions?](https://mlsanity.com/paper/2001.06074/) (2020)
- [Pay for The Second-Best Service: A Game-Theoretic Approach Against Dishonest LLM Providers](https://mlsanity.com/paper/2511.00847/) (2025)
- [Economy of Minds: Emerging Multi-Agent Intelligence with Economic Interactions](https://mlsanity.com/paper/2606.02859/) (2026)
- [Computing Bayes-Nash Equilibria in Combinatorial Auctions with Verification](https://mlsanity.com/paper/1812.01955/) (2018)
- [Test-Time Compute Games](https://mlsanity.com/paper/2601.21839/) (2026)
- [The Economics of No-regret Learning Algorithms](https://mlsanity.com/paper/2601.22079/) (2026)
- [Price of Anarchy for Auction Revenue](https://mlsanity.com/paper/1404.5943/) (2014)
- [Markets, Not Planners: Decentralized Orchestration of LLM Agents with Private Information](https://mlsanity.com/paper/2608.23867/) (2026)
- [Tight Bounds for the Price of Anarchy of Simultaneous First Price Auctions](https://mlsanity.com/paper/1312.2371/) (2013)
- [The Price of Stability for First Price Auction](https://mlsanity.com/paper/2207.04455/) (2022)
- [Marketplace Operators Can Induce Competitive Pricing](https://mlsanity.com/paper/2503.06582/) (2025)
- [An empirical study of Fictitious Play for estimating Nash equilibria in first-price auctions with correlated values](https://mlsanity.com/paper/2606.16389/) (2026)

## Abstract (arXiv)

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 textitprice-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 textitcompetitive 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 textitnear-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/e2 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.