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 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 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 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/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.
Figures & tables
Figure 1: Price-of-anarchy bounds under competitive floors, for various settings of (α,m,ϵ) . Each entry represents the bound for any distribution with P(V(m)<αE[V(1)])≤ϵ . At ϵ=0 , we recover the deterministic floor bound from Equation 4 ; higher ϵ start to degrade the efficiency. Larger numbers of credible competitors ( m≥4 ) stabilize the market, and preserve efficiency even under high ϵ .
Oracle
Random
Best single
Best per cluster
FPA
FPA (adv)
Full dataset
Empirical welfare
0.768
0.416
0.600
0.686
0.762
0.758
Empirical PoA
1.000
0.542
0.781
0.894
0.992
0.988
Worst cluster
Empirical PoA
1.000
0.083
0.060
0.640
0.960
0.930
Table 1: Realized welfare on the full EmbedLLM market under different allocation rules, raw and as a fraction of the optimal welfare. As baselines, Random allocates queries randomly, while Best single and Best per cluster use the model with the highest average value overall or within each cluster, respectively; the latter is a router that sees only the prompt’s cluster, the same public state the bidders condition on. FPA and FPA (adv) use a typical and an adversarially chosen (approximate) equilibrium (§ 6.1 ). Worst-cluster welfare is averaged on each rule’s lowest-welfare prompt cluster (out of 20 ).
Figure 2: Left: Empirical distributions of the Theorem 3 bound (dotted) and the empirical PoA (solid) across 100k random n -model subsets of EmbedLLM, for n∈{2,5,10} . Both distributions tighten as n grows, and are well above the distribution-free baseline (dashed). Right: Bottom 0.1% of empirical PoA across sub-markets grouped by bound bin ( 100k samples per bin), for both typical equilibria (solid green) and approximate, adversarial equilibria (hatched red).
Cohort
E[maxiVi]
1−E[V(2)]/E[V(1)]
Loss bound
Shading
Welfare loss (adv)
Generalists (similar)
0.557
7.7%
24.2%
12.3%
0.4%
Generalists (diverse)
0.553
17.1%
27.4%
19.8%
1.2%
Generalists (1) + niche (3)
0.552
35.6%
29.9%
29.1%
5.1%
Generalists (2) + niche (2)
0.555
15.9%
27.9%
22.3%
0.8%
Table 2: Four cohorts constructed to have comparable E[maxiVi] , but different value structure. We report welfare as a loss, i.e., as the fraction of optimal welfare that is lost. Loss bound is one minus the competitive-floor bound of Theorem 3 (worst case is 1/e≈36.8% ), and Welfare loss is one minus the realized PoA (at the adversarially-chosen equilibrium). Shading is 1−E[Bwinner/Vwinner] .
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 4: Empirical PoA with a platform-owned bidder, as a function of where the platform model sits in the ranking of EmbedLLM models by average value. Each point averages 10,000 markets made of the strongest model, n random rivals, and the platform model. Solid lines are means and dashed lines are the 0.1 percentile across markets, for typical (blue) and adversarial (orange) equilibria. Gray lines mark the cluster router on the same markets (mean solid, 0.1 percentile dash-dotted).
Figure 5: Empirical PoA under pre-auction filtering, as a function of the eligibility cutoff β . Each point averages 10,000 random markets of n models. Solid lines are means and dashed lines are the 0.1 percentile across markets, for typical (blue) and adversarial (orange) equilibria. The dotted line is the filtered oracle, the welfare of always allocating to the best admitted model. Gray lines mark the cluster router, which sends every prompt to its cluster’s best model (mean solid, 0.1 percentile dash-dotted).
We present a scalable framework for solving multi-step tasks through a decentralized market of self-interested agents. Prior theoretical work demonstrated that agents organized through sequential auctions can converge to a globally optimal policy in a Markov decision process (MDP), but relied on the restrictive assumption of the existence of perfect clones as a model of perfect competition. In reality, exact clones drive agent profit to zero, violating the natural risk-reward incentives of participants. Conversely, we empirically demonstrate that the opposite extreme, absence of competition, also leads to market failure and value extraction collapse. Between these two extremes, we investigate how heterogeneous agents can survive in markets that span the critical middle ground of imperfect competition. To be profitable, agents learn to strategically shade bids to secure positive individual profit margins, while still ensuring that the marketplace functions effectively enough so that collaborative tasks posed by market consumers can still be solved. We formulate and evaluate four bidding algorithms (Mirror Descent, Thompson Sampling, Bayes-Optimal, and PPO) across three auction protocols (1st-Price, 2nd-Price, and Credit-Conserving Vickrey). Across a diverse set of domains, we demonstrate that imperfect competition successfully sustains necessary agent profit margins while maintaining high task success rates at low costs.
Ian Gemp, Shubhendu Trivedi, Adam Fisch +7
Google DeepMind · Google Research · University of Waterloo
Agentic commerce is moving from concept to deployed infrastructure: payment networks, retailers, and AI platforms are setting the stage for agents to transact on behalf of merchants and consumers. Yet whether the LLMs behind these agents can price competently in real markets, where customer preferences are hidden, competitors adapt in real time, and demand can shift without warning, has not been systematically tested. We introduce Bazaar, a dynamic sealed-bid benchmark for multi-attribute auction under these conditions. Despite its dynamics, the benchmark is grounded in closed-form customer utilities, enabling exact evaluation. Across 11 frontier LLMs from four providers, the leading agents on customer acquisition (e.g. Gemini 3.1 Pro) are often not the leading agents on profit (e.g. Opus 4.6). The ranking shifts again under demand shocks: agents that learned fastest pre-shock are typically the slowest to revise their beliefs afterwards, while Gemini 3.1 Pro recovers fastest despite not leading on profit. However, even the strongest agent captures less than a third of hindsight-optimal profit, suggesting current LLMs are progressing in agentic commerce but leave substantial headroom.
Markets are a promising way to coordinate AI agent activity for similar reasons to those used to justify markets more broadly. In order to effectively participate in markets, agents need to have informative signals of their own ability to successfully complete a task and the cost of doing so. We propose MarketBench, a benchmark for assessing whether AI agents have these capabilities. We use a 93-task subset of SWE-bench Lite, a software engineering benchmark, with six recently released LLMs as a demonstration. These LLMs are miscalibrated on both success probability and token usage, and auctions built from these self-reports diverge from a full-information allocation. A follow-up intervention where we add information about capabilities from prior experiments to the context improves calibration, but only modestly narrows the gap to a full-information benchmark. We also document the performance of a market-based scaffolding with these LLMs. Our results point to self-assessment as a key bottleneck for market-style coordination of AI agents.
Andrey Fradkin, Rohit Krishnan
Boston University and the MIT Initiative on the Digital Economy. · Independent Researcher.