# 🟧G14 - Asymmetric Model **Proposition 3: Optimal Quality Under Asymmetric Sigmoid Commitment Probabilities** Given asymmetric sigmoid commitment functions $P_c(q) = \frac{e^{\beta_c q}}{1 + e^{\beta_c q}}$ and $P_r(q) = \frac{e^{-\beta_r q}}{1 + e^{-\beta_r q}}$ where $\beta_c, \beta_r > 0$, the optimal quality $q^*$ satisfies the first-order condition: $P_c'(q)[C_u(1-P_r) - C_oP_r - V(1-2P_r)] + P_r'(q)[C_o(1-P_c) - C_uP_c + V(1-2P_c)] = 0$ where $P_c'(q) = \beta_c P_c(1-P_c)$ and $P_r'(q) = -\beta_r P_r(1-P_r)$. No closed-form solution exists; numerical methods required. The asymmetric sigmoid model achieves maximum theoretical generality by incorporating distinct responsiveness parameters $β_c$ and $β_r$ that capture heterogeneous stakeholder sensitivity to quality investments: customer commitment follows $P_c(q) = e^{β_c q}/(1 + e^{β_c q})$ while resource partner commitment follows $P_r(q) = e^{-β_r q}/(1 + e^{-β_r q})$, where $β_c, β_r > 0$ represent independent responsiveness intensities that enable the model to accommodate varying stakeholder types across different venture contexts. The expected loss function $L(q) = C_o[1-P_c(q)]P_r(q) + C_u P_c(q)[1-P_r(q)] - V P_c(q)P_r(q)$ creates a complex optimization landscape where first-order conditions yield the implicit solution $P_c'(q)[C_u(1-P_r) - C_oP_r - V(1-2P_r)] + P_r'(q)[C_o(1-P_c) - C_uP_c + V(1-2P_c)] = 0$, requiring numerical solution methods due to the nonlinear interaction between asymmetric responsiveness parameters and cost structures. Unlike the symmetric case where closed-form solutions illuminate clear cost-priority relationships, asymmetric responsiveness creates path-dependent optimization where the relative magnitude of $β_c$ versus $β_r$ fundamentally alters the decision landscape and can override pure cost considerations in determining optimal quality. The mathematical complexity of this general formulation reflects real entrepreneurial environments where stakeholders exhibit fundamentally different decision-making processes, psychological triggers, and sensitivity patterns to quality signals, consistent with [[📜🟩_mitchell97_identify(stakeholders, salience)]] stakeholder heterogeneity framework across venture contexts and industry characteristics. This theoretical advancement establishes the mathematical foundation for practical stakeholder prioritization algorithms while maintaining sufficient generality to accommodate diverse entrepreneurial scenarios documented by [[📜🟧_anderson13_model(startup, integration-decisions)]] in adaptive optimization approaches for sparse data environments. **Business Intuition: Responsiveness Asymmetries Override Cost-Priority Logic** The asymmetric model reveals that stakeholder responsiveness differences can fundamentally override the cost-priority principle established in symmetric cases, creating strategic complexity where behavioral parameters dominate pure economic considerations in optimal quality determination. When customers exhibit high responsiveness (large $β_c$) while resource partners respond gradually (small $β_r$), even slight quality improvements generate dramatic customer commitment gains while producing minimal partner resistance, potentially justifying high-quality strategies even when underage costs slightly exceed overage costs ($C_u > C_o$). Conversely, when partners are highly sensitive (large $β_r$) but customers are gradual responders (small $β_c$), optimal quality may shift toward partner accommodation despite economic incentives favoring customer focus, demonstrating how behavioral asymmetries reshape fundamental strategic trade-offs. The mathematical condition governing this phenomenon involves the interplay $β_c P_c(1-P_c)P_r + β_r P_r(1-P_r)P_c$, which determines whether responsiveness effects amplify or counteract cost-driven quality adjustments, requiring entrepreneurs to calibrate strategies based on empirically observable stakeholder sensitivity patterns rather than relying solely on cost structure analysis. This behavioral complexity transforms stakeholder management from mechanical cost optimization into dynamic psychological navigation, where entrepreneurs must balance economic efficiency against stakeholder responsiveness asymmetries, supporting [[📜🟪_busenitz97_recognize(entrepreneurs, biases)]] findings on the limitations of purely rational decision-making frameworks in entrepreneurial contexts and emphasizing the need for behaviorally-informed optimization approaches that accommodate real-world stakeholder heterogeneity in commitment mechanisms and quality sensitivity patterns.