Knowing When to Ask for Help: Bayesian Self-Escalation in Hierarchical LLM Agents

Current LLM agent systems decide delegation before reasoning begins (a router picks a model) or after a response is complete (a verifier scores it and may retry). We study a third regime: an agent that recognises, during its own reasoning, that it is unlikely to succeed and transfers control to a stronger model. We formulate intra-generation delegation as a Bayesian optimal-stopping problem over a learned competence posterior -- an online estimate of the agent's eventual task success whose sufficient statistics are learned from labelled trajectories, not read off raw entropy. We derive the myopic escalation threshold in closed form, characterise the optimal policy via dynamic programming, and prove that the optimal policy is a time-varying threshold with no shape assumption on the raw signal. We further prove exponential separation of the oracle belief at the Chernoff-information rate of the signal, a regret bound governed by the calibration of the posterior, and a finite-sample guarantee: with n labelled calibration trajectories the deployed plug-in policy's regret decays as 1/sqrt(n). A controlled simulation study confirms each prediction of the theory, including the predicted 1/sqrt(n) rate. We additionally report a real-model validation on a Qwen2.5-Coder 1.5B->7B code cascade (MBPP, 257 tasks), confirming two of three pre-registered predictions: the escalation frontier dominates post-hoc routing at equal cost, and the cumulative competence belief's discrimination rises over generation.

Self-Certification of Representation Adequacy: Sequential Certification at Minimum Task Loss

Agents that act on a compressed representation of their history face a structural risk: if the representation aliases histories with different optimal actions, no rule measurable with respect to the representation can avoid an irreducible per-round loss, and the agent may be unable to detect this from its own transcript. This paper develops a four-layer theory of self-certification of representation adequacy. The static layer defines decision-theoretic adequacy through a Bayes-risk grouping identity and prices a one-shot external verification by an exact total-variation threshold. The sequential layer poses certification as an optimal-stopping problem in the currency of task loss: we define an environment-wise certification complexity constant through a covering linear program, prove an information-task-loss lower bound for every delta-correct strategy, and give a Certification Track-and-Stop policy whose cost matches the bound asymptotically. A final boundary layer gives an explicit kernel-switching example and identifies the open theorem needed to cover policy switching or representation repair; it does not claim that the fixed-kernel guarantees extend to representation revision. The proofs of the two main theorems are given in full in the appendices.

LLM-OSDA: An Optimal-Stopping Dynamic Auction for Native Advertising in Multi-Turn LLM Conversations

LLM-native advertising embeds sponsored content directly into model-generated responses, shifting the unit of sale from a fixed slot to a moment within an evolving conversation. Existing LLM ad-auction mechanisms primarily operate within a single response, settling the winner but not the timing. The extension is nontrivial: with one native insertion opportunity per session, the stopping time depends on bids, coupling timing with allocation, so static truthfulness arguments no longer apply. We propose the LLM-based Optimal Stopping Dynamic Auction (LLM-OSDA), a dynamic cost-per-click auction that integrates Bellman optimal stopping, winner allocation, and envelope pricing. A bid-independent LLM layer estimates contextual click quality and seamlessly renders the winning ad, while bids enter only the committed auction mechanism. Under an exact Bellman oracle, the expected discounted-click allocation is monotone in each advertiser's bid, and the corresponding envelope payment makes truthful bidding weakly dominant in expectation. For practical deployment, a learned StopNet approximates the Bellman action values. We show that its decisions differ from the optimal policy only near the stopping boundary and bound the resulting incentive loss in terms of its approximation error. Experiments on a simulated conversational advertising corpus show that LLM-OSDA improves net revenue by 11 percent over the strongest fixed-timing baseline while maintaining comparable user retention. Code is at https://github.com/2025Fang2025/llm-osda.