From Local Prediction Error to Final Outputs: A Theory of Expert Sharing in Multi-Step Reasoning

Mixture-of-experts (MoE) models can share experts across layers or computational steps, pooling training information from different contexts. This can reduce local prediction error, but multi-step systems are ultimately evaluated by their final outputs. We study when these local gains survive composition. In specified linear regression models, we derive exact sharing and merging criteria by balancing variance savings against bias from sharing across roles with different target behavior. Under a common fixed linear propagation map, we characterize expected squared final-output risk, retaining error directions and correlations. We construct cases where sharing lowers average local error yet increases final-output error. Within this setting, we also identify when orthogonal full pooling with isotropic noise preserves every local improvement at the final output. For general routed computations, we derive conditional end-to-end error bounds that account for distribution shifts, errors in surrounding components, and changes in expert selection. Separately, we show that correct predictions and route labels alone do not establish that individual experts implement the attributed operations. Controlled experiments realize a strict local-to-final risk reversal. In low-data symbolic settings, trained Transformers exhibit statistically supported teacher-forced one-step gains without statistically resolved free-running final-output gains. Separate intervention tests distinguish successful multi-layer actions from single-expert attribution and generalization to unseen transitions. Together, these results distinguish when sharing improves local estimation, when those gains survive to the final output, and what evidence is needed to attribute reusable operations to individual experts.