Price of Anarchy of First-Price Auctions in Agentic Markets

An agentic marketplace is a decentralized, serving-time market where AI agents bid to claim each user query. An agent's bid is based on its expected value for the query, which is the expected reward for producing a high-quality response, less the agent's expected inference costs. First-price auctions (FPAs), in which the highest bidder wins and pays its bid, are a natural mechanism for this, since they are simple and transparent to run and require no central knowledge of private capabilities and costs. But a potential weakness is that FPAs are not truthful. Participants may bid less than their true value in the hope of a larger profit, so the query can be allocated to someone other than the bidder who values it most highly. This welfare loss is quantified by the _price-of-anarchy_. In this paper, however, we argue that agentic markets are likely to possess special structural features that reduce it. The first is _value concentration_. If the gap between the top two values is usually small (e.g., because the best models are often distilled into cheaper alternatives which perform similarly), then the welfare gap can go to zero. The second is _weak dependence_ between values after conditioning on the prompt, in which case the efficiency guarantee approaches $1-1/e^2$. In both cases, we derive bounds on the price of anarchy that are significantly better than the well-known factor $1-1/e$ from the general case, and validate them empirically on prompt routing in the EmbedLLM dataset.