# Figure: Parameter Space Evolution ![[🖼️eff_prof_eff_acc(p,p,pp) 2025-06-24-5.svg]] %%[[🖼️eff_prof_eff_acc(p,p,pp) 2025-06-24-5|🖋 Edit in Excalidraw]]%% This figure visualizes how the three approaches (prediction, prescription, prediction-prescription) evolve in the 3D parameter space (q, βr, βc). ### **G1 Panel (Symmetric Case: βr = βc = β)** In the symmetric responsiveness case where resource partners and customers exhibit equal sensitivity to quality (βr = βc = β), the parameter space collapses to two dimensions (q, β). Assuming, (q, βr, βc) changes from (0,1,1) to (q*, 2,2), the 🟦 **Prediction (Dual)** approach moves vertically from (0,1) to (0,2), achieving high **prediction accuracy** by correctly learning β=2, but suffers from zero **prescription effectiveness** as it remains stuck at q=0, unable to leverage its knowledge. The 🟫 **Prescription (Primal)** approach moves horizontally from (0,1) to (ln(3/2),1), achieving moderate **prescription effectiveness** by reaching the newsvendor-optimal q* for its assumed β=1, but with poor **prescription profitability** due to using incorrect responsiveness parameters. The 🟥 **Integrated** approach moves diagonally from (0,1) to (ln(3/2),2), simultaneously achieving both high **prediction effectiveness** (correctly learning β=2) and optimal **prescription effectiveness** (reaching the true optimal q*), while demonstrating superior **update efficiency** by converging along the optimal curve q* = 1/β ln((2Co+V)/(2Cu+V)) rather than requiring sequential steps. ### **G2 Panel (Asymmetric Case: βr << βc)** In the asymmetric case where resource partner responsiveness dominates (βr << βc), the full three-dimensional parameter space reveals how different approaches handle complexity. Assuming, (q, βr, βc) changes from (0,1,1) to (q*, 2,5), the 🟦 **Prediction (Dual)** approach moves from (0,1,1) to (0,2,5), achieving high **prediction accuracy** for both parameters but wasting computational resources learning the irrelevant βc=5, resulting in poor **update efficiency** and zero **prescription effectiveness**. The 🟫 **Prescription (Primal)** approach moves from (0,1,1) to (ln(4),1,1), reaching a suboptimal q* based on incorrect β assumptions, yielding moderate **prescription effectiveness** but potentially catastrophic **prescription profitability** when the true βr=2 differs substantially from the assumed βr=1. The 🟥 **Integrated** approach demonstrates intelligent dimensional reduction by moving from (0,1,1) to (ln(4),2,1), anchoring βc=1 upon recognizing its minimal impact, achieving optimal **prescription effectiveness** while maximizing **update efficiency** through selective learning - it ignores the irrelevant βc dimension and focuses computational resources on the critical βr parameter, converging along the reduced-dimension optimal surface q* = 1/βr ln((Co+V)/Cu).