Asymmetric Nash Power: Activity-Adaptive Bargaining Exponents for Sparse Collaborative Filtering

The symmetric Nash Bargaining Solution assigns identical bargaining power (α = β = 1) to every user and item in the system. This assumption, while mathematically convenient, is informationally incoherent: a user who has rated 500 items has a richer, more reliable preference representation than one who has rated 5, and should exert more leverage in the recommendation game. Nash himself extended the theory to asymmetric powers in 1953, but this extension has never been operationalised in recommendation. f(u) = 1 + σ(z_u), g(i) = 1 + σ(z_i), both ∈ (1, 2)ANP(u,i) = [r̂_ui − d_u]₊^f(u) · [r̂_ui − d_i]₊^g(i) The sigmoid of the activity z-score maps any scale of activity to (0,1), producing per-instance exponents in (1,2). Cold-start agents receive power ≈ 1 (standard NBS); high-activity agents receive power ≈ 2 (amplified signal). Proposition 1 proves the method collapses to symmetric NBS when activity is uniform. Theorem 1 proves the advantage over NBS grows with the skewness κ of the activity distribution — directly predicting the strongest gains on the most heterogeneous dataset. Book-Crossing has the most skewed activity distribution of the three datasets (skewness ≈ 4.2, median user has <25 ratings, max >500). ANP achieves Precision@10 = 0.0073 there, the only Nash-based and only non-neighbourhood method to beat all four standard baselines, which confirms the theoretical prediction precisely.