NER: Nash Entropy-Regularised Scoring for Diversity-Aware Collaborative Filtering

NBS has a structural popularity bias: popular items, by virtue of their high average rating dᵢ, produce larger item surplus (r̂ᵤi − dᵢ) for any well-matched user, systematically dominating top-K lists. The paper proves this formally in Lemma 1 and then argues that the correct remedy is not to penalise popularity per se but to reward audience diversity — items whose rating mass is spread across heterogeneous users represent broader, more robust relevance. Hᵢ = −Σ pₔ'₈ log pₔ'₈ (Shannon entropy of item i's rating distribution) NER (u, i;γ) = MinMax (NBS (u, i) ) + γ · MinMax (Hᵢ) The additive (not multiplicative) entropy bonus ensures that high-entropy items with low NBS can still rise — a deliberate design choice. Proposition 1 proves a smooth redistribution law: items with above-average entropy gain recommendation frequency linearly in γ at the expense of below-average items. Theorem 2 proves the non-monotone NDCG optimum — there exists a γ* > 0 strictly interior, because beyond it the entropy signal overwhelms the relevance signal. The theorem condition (Corr (Hᵢ, r̂ₔ*₈) > 0) is satisfied on dense datasets where diverse items accumulate sufficient training signal, correctly predicting γ* = 0. 3 on ML-100K. NER is the strongest Nash variant on accuracy: P@10 = 0. 0827 and NDCG@10 = 0. 0369 on ML-100K, a 5. 6× improvement over symmetric NBS in Precision.