The Complementarity Trap in Cost-Aware Multi-Index Retrieval: Recovering Abandoned Indices with an Information-Gain Reward

Retrieval-augmented generation (RAG) increasingly fuses heterogeneous indices—dense, lexical (BM25), and structured (graph)—and increasingly delegates which index to call to a cost-aware reinforcement-learning router that trades answer accuracy against retrieval cost. We report a negative result that complicates this intuition: although the indices are genuinely complementary at the recall level, under cost-aware RL routing the complementarity fails to cash out end-to-end. We diagnose the cause and propose a corrective signal. The diagnosis localizes the harm not to "bad indices" but to low-signal marginal retrieval: the policy learns to take one more retrieval round, admitting lexical noise and diluting or exhausting the answer context. A natural fix—an indiscriminate redundancy penalty—fails, because it meters retrieval quantity without judging its value. Our fix is an information-gain reward that credits only retrievals surfacing genuinely new gold-relevant evidence, with built-in diminishing returns. It eliminates the multi-index regression across all datasets and scales monotonically with the reward weight; paired-bootstrap tests show the gain is specific—adding indices alone is not significantly better than single-index, whereas the information-gain reward is significantly better than naive multi-index. The central message: reward retrieval quality, do not merely penalize cost—otherwise cost-aware RAG abandons exactly the cheap complementary evidence it was built to exploit.