Whose Turn Is It? and agentic auctions: routing becomes a sequential, market-priced decision

Date: 2026-10-16 · Source: X following feed (DAIR.AI picks, @dair_ai) · Papers: Whose Turn Is It? (Microsoft), arXiv 2610.19292 · Price of Anarchy of First-Price Auctions in Agentic Markets (Google DeepMind), arXiv 2610.20587 Raw: raw/twitter/feed/2026-10-16-morning-ranked.json (enriched abstracts)

TL;DR

Whose Turn Is It? studies zero-shot, cost-aware routing for stateful agents at the user-turn boundary. A turn can involve several model and tool calls, its difficulty depends on earlier turns, and switching models throws away the prompt cache. Tested on multi-turn SheetCopilot and SpreadsheetBench 2 with model pairs that differ in size, reasoning effort or provider:

At a matched offloading rate, a router that reads the session history passes 3.3 points more tasks than one that sees only the current request, gains for every model pair, and stays within one point of always using the expensive model.
Cost depends on execution, not token price. The cheap model's per-task advantage shrinks from 6.8x to 3.2x on longer sessions (it takes more calls).
A closed-form cache analysis: switching down (expensive to cheap) recovers its cache cost within two calls; switching up never broke even in their conditions.

Agentic auctions (DeepMind) models a decentralized market where agents bid for each prompt, bidding expected reward minus inference cost. First-price auctions are simple but not truthful, so a prompt can go to the wrong agent; the loss is the price of anarchy. Two market features shrink it: a competitive floor (many agents are near the best, e.g. because strong models are distilled into cheap ones) drives the loss toward zero, and near-independence (value mostly set by prompt difficulty) gives efficiency approaching 1 - 1/e² (~86%) of optimal versus the general 1 - 1/e (~63%). Simulations on the EmbedLLM routing dataset match the theory.

The switch has a price, and only one direction pays it back

A history-aware router decides per turn; the cache cost makes down-switches cheap and up-switches costly.

flowchart LR
  H["Session history<br/><small>earlier turns, workbook state</small>"] --> R["Turn router<br/><small>zero-shot, cost-aware</small>"]
  U["New user turn<br/><small>may need many calls</small>"] --> R
  R -->|stay or go up| E["Expensive model<br/><small>warm cache</small>"]
  R -->|go down| C["Cheap model<br/><small>pays back in 2 calls</small>"]
  E --> O["Turn result<br/><small>pass rate within 1 pt</small>"]
  C --> O
  E -.->|cache lost| X["Up-switch cost<br/><small>never broke even</small>"]
  classDef input fill:#d0ebff,stroke:#1971c2,color:#1b1b1b,stroke-width:2px
  classDef core fill:#e5dbff,stroke:#6741d9,color:#1b1b1b,stroke-width:2px
  classDef loop fill:#fff3bf,stroke:#f08c00,color:#1b1b1b,stroke-width:2px
  classDef exit fill:#d3f9d8,stroke:#2f9e44,color:#1b1b1b,stroke-width:2px
  classDef err fill:#ffe3e3,stroke:#e03131,color:#1b1b1b,stroke-width:2px
  class H,U input
  class R loop
  class E,C core
  class O exit
  class X err
  linkStyle 3 stroke:#2f9e44,stroke-width:2px
  linkStyle 6 stroke:#e03131,stroke-width:2px

Blue is input, amber is the routing decision, purple is a model, green is the result, red is the switching cost.

Relation to prior wiki pages

Directly extends the 10-14 session-routing entry on the LLM routing page: Arena's Max router picks one model per session (+14.7% at $3.27/task) and Cohere's study routes by role. Whose Turn goes one level finer, per turn, and adds the missing piece: the cache-switch cost. This is the first paper in the wiki to put prompt-cache economics inside the routing objective.
Gemini 4 Argon (10-14) prices cached input 95% off. At that discount, an up-switch that drops a warm cache is expensive; Whose Turn's asymmetry result is the theory for why.
Fewer Bits, More Tries (10-15) found a small model with verified retries can replace routing up on code. Whose Turn's 6.8x to 3.2x shrinkage is the counterweight: a cheap model's extra calls eat its price advantage on long sessions.
The auction paper gives the economic justification for distillation markets: the more teacher capability is distilled into cheap students, the higher the competitive floor and the more efficient a decentralized routing market gets.

Gaps

Whose Turn: spreadsheet agents only; zero-shot routers, no trained router compared. The auction paper is theory plus simulation on EmbedLLM; no live market.

Research angle

A turn-level router should carry a cache-state feature and an asymmetric switching penalty. Nobody has trained one with that objective. The obvious test bed is Arena's Max router data.