# 🟧🟩G1920 - Special Cases of Sigmoid Model **Four Propositions: Closed-Form Solutions Under Tractable Limits** The general asymmetric sigmoid model yields four special cases with elegant closed-form solutions that bundle costs and match bonus into single "net-penalty" log-ratios, enabling rapid analytical insights for distinct stakeholder configurations. **Proposition 2.3.1 (Symmetric Responsiveness)** establishes that when $\beta_c = \beta_r = \beta$, optimal quality follows $q^* = \frac{1}{\beta}\ln\left(\frac{C_u + V}{C_o + V}\right)$, balancing net under-commitment penalties against net over-commitment penalties with equal stakeholder steepness. **Proposition 2.3.2 (Customer-Dominant Responsiveness)** demonstrates that when $\beta_c \gg \beta_r$, the solution becomes $q^* = \frac{1}{\beta_r}\ln\left(\frac{C_o + V}{C_u}\right)$, where steep customer responses create all-or-nothing dynamics that shift optimization focus to partner-side trade-offs. **Proposition 2.3.3 (Partner-Dominant Responsiveness)** shows the inverse case where $\beta_r \gg \beta_c$ yields $q^* = \frac{1}{\beta_c}\ln\left(\frac{C_o}{C_u + V}\right)$, with razor-sharp partner acceptance thresholds that pin down quality through customer-side balancing. **Proposition 2.3.4 (High-Match-Value Limit)** reveals that when $V \gg C_u, C_o$, the solution simplifies to $q^* = \frac{1}{\beta_c + \beta_r}\ln\left(\frac{\beta_r}{\beta_c}\right)$, where overwhelming match bonuses drive pure probability maximization through marginal gain equalization across sigmoid functions. **Strategic Implications: Net-Penalty Framework and Stakeholder Dominance Patterns** These special cases illuminate fundamental strategic principles where stakeholder responsiveness asymmetries can override pure cost considerations, creating context-dependent optimization patterns that require entrepreneurs to calibrate quality decisions based on observable stakeholder sensitivity rather than assuming symmetric responses. The net-penalty log-ratio structure reveals that match bonuses $V$ systematically compress cost differentials in symmetric cases while amplifying dominant stakeholder effects in asymmetric scenarios—when customers are highly responsive ($\beta_c \gg \beta_r$), quality optimization becomes primarily driven by partner economics $(C_o + V)/C_u$, while partner dominance ($\beta_r \gg \beta_c$) shifts focus to customer economics $C_o/(C_u + V)$. The high-match-value limit demonstrates that sufficiently large collaboration rewards transform stakeholder management from cost-driven optimization into pure responsiveness ratio balancing $\beta_r/\beta_c$, suggesting that ventures with substantial network effects or platform dynamics should prioritize stakeholder sensitivity calibration over traditional cost structure analysis. This framework enables entrepreneurs to rapidly identify which stakeholder dominates their optimization landscape and adjust quality strategies accordingly, consistent with [[📜🟦_sarasvathy01_leverage(contingencies, uncertainty)]] effectuation principles where entrepreneurs leverage stakeholder heterogeneity as strategic resources while maintaining mathematical rigor for systematic decision-making under uncertainty and resource constraints characteristic of entrepreneurial environments.