Trivedi et al. analyze first-price auction efficiency in AI markets

Efficiency bounds in agentic markets reach 1 - 1/e²

The paper defines agentic markets as decentralized serving-time environments where agents bid to win user prompts. While the general price-of-anarchy bound for first-price auctions reaches 1 - 1/e of optimal welfare, efficiency improves to 1 - 1/e² when value variation depends on prompt difficulty.

Welfare loss in these markets can reach zero when multiple agents value a prompt above a competitive floor.

Competitive floors and prompt difficulty limit welfare loss

A competitive floor occurs when several bidders value a prompt above a baseline share of optimal welfare, often because developers distill top models into cheaper alternatives with similar performance. Competition among these bidders limits welfare loss, which the authors show decreases as the floor rises.

The researchers also identify a near-independence of values after conditioning on the prompt, a condition where most predictable variation in expected value stems from prompt difficulty.

Strategic bidding impacts auction transparency

First-price auctions are simple and transparent, but they lack truthfulness because participants bid less than their true value to increase profit. Agents calculate bids based on their expected value, which is the anticipated reward for a response minus inference costs. This strategy can cause a prompt to go to a bidder who does not value it most, resulting in welfare loss quantified by the price of anarchy.

In a first-price sealed-bid auction, all bidders submit secret bids, and the highest bidder wins and pays that submitted price. Lack of transparency in similar systems has led to accusations of under-quoting against estate agents in Victoria and New South Wales.

For affine costs, the exceptional sets become finite unions of lower-dimensional polyhedral cells linked by an exact scaling law. The maximum price of anarchy over a cell closure is attained on the cell boundary, which generalizes the one-dimensional endpoint result.

Empirical validation via EmbedLLM simulations

The authors validated their theoretical bounds using simulations of strategic bidders on the EmbedLLM prompt routing dataset. This validation relies on simulation rather than production traffic, and the abstract does not publish specific efficiency numbers for these runs.

The nondifferentiability set of the price of anarchy has Lebesgue measure zero. When every edge cost is strictly increasing or constant, the nondifferentiability set is covered by countably many codimension-one submanifolds. Under these conditions, all equilibrium objects are cellwise as smooth as the edge costs.

Google transitions to first-price auction model

Google is migrating its AdExchange to a first-price auction model to reduce complexity and increase competition for each impression. The company previously used a second-price auction, which it viewed as a clearer evaluation of market price for publisher inventory.

Google lost impressions after competitors like Rubicon, Index, and Pubmatic introduced first-price auctions, as Google only passed a second-price bid into those auctions. In a first-price auction, the highest bidder wins and pays the bid price.

Bid shading and market revenue growth

Buyers use bid shading to analyze bid history by site or ad position to determine winning prices, acting as a compromise between first and second-price auctions. In a second-price auction, a buyer bidding $15 against a $10 bid wins but pays only $10.01.

Publift reported an average 54% net increase in publisher revenue 6 months after clients moved to its fuse platform, which uses header bidding. In the housing market of Victoria and New South Wales, homes were selling for significantly more than reserve prices and advertised ranges.