# Entrepreneurial Decision-Making Framework: Mathematical Relationships | Component | Mathematical Form | Primal Interpretation | Dual Interpretation | Sublime Implementation (STRAP success case) | Segway Implementation (STRAP fail case) | | ----------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------- | -------------------------------------------- | ------------------------------------------- | ------------------------------------------------ | | **Objective Function** | $W_d \cdot U_d + W_s \cdot U_s + W_i \cdot U_i$ | Minimize weighted uncertainty | Maximize stakeholder satisfaction likelihood | Balanced stakeholder needs | Prioritized technical perfection | | **Uncertainty State** | $S = [U_d, U_s, U_i]$ | Current knowledge gaps | Entropy in belief distributions | Targeted reduction in key dimensions | Slow reduction in high-dimensional space | | **Resource Constraint** | $\sum_{j} c_j a_j \leq R$ | Budget limitation | Shadow price (γ) of resources | Optimized returns per resource unit | Consumed high resources with low returns | | **State Transition** | $D(S,A) = S'$ | Experimental outcomes | Belief updating mechanism | Progressed to deterministic transitions | Unpredictable transitions remained variable | | **Action Selection** | $a_j^* = \begin{cases} 1 & \text{if } w_j[\lambda_j + \beta_j^T \mu_j(1) - \log Z_j(\beta_j)] > \gamma c_j \\ 0 & \text{otherwise} \end{cases}$ | Choose experiments that break bottlenecks | Maximize expected information gain | Parallel actions prioritized by bottlenecks | Sequential action selection in silos | | **Stakeholder Weights** | $W = [W_d, W_s, W_i]$ | Relative importance of stakeholders | Optimal allocation of attention | Adaptive weights based on venture stage | Overweighted technical considerations | | **Lambda (λ)** | $\lambda_j$ in $w_j[\lambda_j + \beta_j^T \mu_j(a_j) - \log Z_j(\beta_j)]$ | Uncertainty penalty | Belief precision requirement | High λ persisted across venture stages | λ methodically reduced across stages | | **Gamma (γ)** | $\gamma$ | Resource scarcity penalty | Marginal value of capital | Remained resource-constrained | Reduced capital dependency over time | | **Expectation Term** | $\mu_j(a)$ | Expected impact of action | Belief alignment mechanism | Misaligned expectations across stakeholders | Coordinated expectations via parallel engagement | ## Key Insights from the Mathematical Framework 1. The primal-dual transformation converts uncertainty minimization to likelihood maximization 2. Stakeholder expectations need coordination to avoid circular dependencies 3. Breaking bottlenecks requires targeting actions with highest uncertainty reduction per dollar 4. Resource allocation efficiency determines venture success more than total funding 5. Successful ventures progress from stochastic to deterministic state transitions # color coding