# Pandora's AI Model Routing Box: Efficient Allocation with Costly Value Estimation

arXiv: https://arxiv.org/abs/2608.20316 · HTML: https://mlsanity.com/paper/2608.20316/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 52 references that are in the map.
- First posted on arXiv 20 Aug 2026.

## Builds on (most cited first, 30 of 37)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [ReAct: Synergizing Reasoning and Acting in Language Models](https://mlsanity.com/paper/2210.03629/) (2022)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [When Not to Trust Language Models: Investigating Effectiveness of Parametric and Non-Parametric Memories](https://mlsanity.com/paper/2212.10511/) (2022)
- [FrugalGPT: How to Use Large Language Models While Reducing Cost and Improving Performance](https://mlsanity.com/paper/2305.05176/) (2023)
- [RouteLLM: Learning to Route LLMs with Preference Data](https://mlsanity.com/paper/2406.18665/) (2024)
- [Adaptive-RAG: Learning to Adapt Retrieval-Augmented Large Language Models through Question Complexity](https://mlsanity.com/paper/2403.14403/) (2024)
- [Omni-MATH: A Universal Olympiad Level Mathematic Benchmark For Large Language Models](https://mlsanity.com/paper/2410.07985/) (2024)
- [RouterBench: A Benchmark for Multi-LLM Routing System](https://mlsanity.com/paper/2403.12031/) (2024)
- [Large Language Model Routing with Benchmark Datasets](https://mlsanity.com/paper/2309.15789/) (2023)
- [Head-to-Tail: How Knowledgeable are Large Language Models (LLMs)? A.K.A. Will LLMs Replace Knowledge Graphs?](https://mlsanity.com/paper/2308.10168/) (2023)
- [Routing to the Expert: Efficient Reward-guided Ensemble of Large Language Models](https://mlsanity.com/paper/2311.08692/) (2023)
- [Simple Entity-Centric Questions Challenge Dense Retrievers](https://mlsanity.com/paper/2109.08535/) (2021)
- [Gemini Embedding: Generalizable Embeddings from Gemini](https://mlsanity.com/paper/2503.07891/) (2025)
- [AgentNet: Decentralized Evolutionary Coordination for LLM-based Multi-Agent Systems](https://mlsanity.com/paper/2504.00587/) (2025)
- [Universal Model Routing for Efficient LLM Inference](https://mlsanity.com/paper/2502.08773/) (2025)
- [EmbedLLM: Learning Compact Representations of Large Language Models](https://mlsanity.com/paper/2410.02223/) (2024)
- [Adaptive Retrieval Without Self-Knowledge? Bringing Uncertainty Back Home](https://mlsanity.com/paper/2501.12835/) (2025)
- [Intelligent AI Delegation](https://mlsanity.com/paper/2602.11865/) (2026)
- [Calibrate-Then-Act: Cost-Aware Exploration in LLM Agents](https://mlsanity.com/paper/2602.16699/) (2026)
- [Cost-aware Bayesian Optimization via the Pandora's Box Gittins Index](https://mlsanity.com/paper/2406.20062/) (2024)
- [Decentralized Reinforcement Learning: Global Decision-Making via Local Economic Transactions](https://mlsanity.com/paper/2007.02382/) (2020)
- [Don't lie to your friends: Learning what you know from collaborative self-play](https://mlsanity.com/paper/2503.14481/) (2025)
- [LLM-Auction: Generative Auction towards LLM-Native Advertising](https://mlsanity.com/paper/2512.10551/) (2025)
- [Pandora's Problem with Nonobligatory Inspection](https://mlsanity.com/paper/1905.01428/) (2019)
- [Weitzman's Rule for Pandora's Box with Correlations](https://mlsanity.com/paper/2301.13534/) (2023)

## Built on by (most cited first, 2 of 2)

- [COVER: Identifiable Evaluation of Coalition Routing](https://mlsanity.com/paper/2608.28475/) (2026)
- [Collaborative Learning and Reasoning over Model Slices](https://mlsanity.com/paper/2610.12927/) (2026)

## Abstract (arXiv)

Heterogeneous AI systems composed of multiple models, architectures, harnesses, or inference-time settings can improve quality and efficiency by routing queries to the specialist who can answer most effectively at the lowest cost. Routing requires estimating each specialist's expected return, but this value estimation has a cost. Cheap estimators (e.g., embedding-based predictors) are fast but noisy, while accurate estimators (e.g., fine-tuned models with access to retrieval results or partial reasoning traces) are expensive. We formalize this tradeoff as an instance of Pandora's Box, the classical problem of optimal search with costly inspection. Under a Gaussian signal model, the resulting policies have closed-form value-of-information expressions that determine, for each specialist and input, whether refining the value estimate is worth its cost. We call the centralized policy Pandora's Router. We extend this to a decentralized setting, Pandora's Bidder, where specialists independently decide whether to invest in self-assessment before accepting an offered price to claim a query. Experiments across three domains---a standard multi-LLM benchmark, retrieval-augmented specialists, and LLMs with variable inference-time reasoning---show that Pandora's Router matches the routing quality of exhaustive estimation, while querying the expensive estimator far less often. In the decentralized setting, value-of-information reasoning improves allocative efficiency when competing estimates are accurate; when competing estimates are noisy, however, it can increase the strategic specialist's utility at the expense of others.