Is Escalation Worth It? A Decision-Theoretic Characterization of LLM Cascades

Model cascades, in which a cheap LLM defers to an expensive one on low-confidence queries, are widely used to navigate the cost-quality tradeoff at deployment. Existing approaches largely treat the deferral threshold as an empirical hyperparameter, giving limited guidance on the geometry of the resulting cost-quality frontier over a model pool. We develop a decision-theoretic characterization of deterministic threshold cascades over a model pool: two-model frontiers are piecewise concave on decreasing-benefit regions, the two-model pool frontier is the pointwise envelope over pairwise cascades, and fixed $k$-model cascades satisfy stagewise first-order conditions that equalize marginal quality-per-cost. Across five benchmarks and eight models, full fixed chains underperform the pairwise envelope and optimized subsequence cascades add little on held-out data. A lightweight pre-generation router beats the best cascade policy on four of five datasets, suggesting that cascade performance is limited more by paying the cheap model before escalation than by a shortage of intermediate stages.