3.1💭Theorize solution(🧬)

Approximate POMDP value function using LP; exploit structure where actions are experiments revealing hidden states | | gpt link | | ------------------------------------------------------------ | -------------------------------------------------------------- | | [[# matroid exchange property]] | https://chatgpt.com/c/6818b760-4a50-8002-9272-a4f6877af1c5 | | [[# three matroid policy example to max gamma min lambda d]] | | | [[# relevant papers from 📝]] | | | | https://chatgpt.com/c/68176d70-1670-8002-b36a-d719ae913f09 | | [[# Investment Case Types]] | https://chatgpt.com/share/681c9c4c-7f0c-8002-aaad-42d48d9902a4 | # Investment Case Types ![[Pasted image 20250508075902.png]] | Type | Characteristics | Example | | -------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | **Smooth Progress** | - Well-understood technology with clear market applications- Incremental innovation rather than radical disruption- Strong founding team with relevant domain expertise- Clear milestones that can be validated sequentially | A SaaS company building enterprise software for a familiar industry where the team has connections. Each customer interview and feature development directly increases confidence in product-market fit, while simultaneously tightening understanding of the maximum potential market. | | **Slow Guarantee(Lower bound improves slowly)** | - Hard-to-quantify value proposition- Complex regulatory environment with uncertain outcomes- Multiple interdependent stakeholders- Difficult to conduct small experiments that validate the core thesis | A pharmaceutical startup developing a novel treatment. Clinical trials take years, making it difficult to establish a lower bound on efficacy or approval probability. Even after significant investment, the minimum guaranteed outcome remains uncertain until late-stage trials conclude. | | **Slow Solution(Upper bound improves slowly)** | - Large addressable market with unclear boundaries- Network effects that are difficult to model- "Winner-take-all" dynamics where maximum upside is ambiguous- Difficult to determine what "success" looks like | An early social media platform where the upper bound on growth and revenue remains highly uncertain for years. Even after achieving product-market fit, the maximum potential value keeps shifting as new monetization channels and user behaviors emerge. | | **Slow Overall Progress(Both bounds improve very late)** | - "Moonshot" technologies with binary outcomes- Very long development cycles before meaningful validation- High scientific uncertainty combined with market uncertainty- Few intermediate signals of progress | A nuclear fusion energy startup. For years, both the minimum viability (lower bound) and maximum potential (upper bound) remain highly uncertain. Only after achieving a critical scientific breakthrough do both bounds rapidly tighten, revealing whether the venture is viable and its potential scale. | # matroid exchange property In mathematics, the concept of **exchangeability** is central to the definition of a **matroid**, meaning that given two independent subsets of a matroid, if one subset is larger, you can always swap an element from the larger subset into the smaller one without losing independence. Specifically, for any two independent sets AA and BB, if ∣A∣<∣B∣|A| < |B|, there exists an element x∈B∖Ax \in B \setminus A such that A∪{x}A \cup \{x\} remains independent. This exchange property ensures flexibility and adaptability within the matroid structure, as decisions (elements chosen) are not irreversibly committed to—allowing continuous reallocation of resources and enabling robust, parallel decision-making pathways that avoid lock-in scenarios. **One-sentence summary**: The matroid exchange property mathematically ensures decision flexibility by allowing independent options to be swapped without losing structural independence, thus preventing path-dependent lock-ins. # three matroid policy example to max gamma min lambda d [[compact and matroid society]] 1. **Maximizes γ (Marginal Resource Value)** - Matroid structures yield higher marginal benefit per resource unit invested, as each action directly reduces a specific uncertainty without interdependencies, leading to optimal resource allocation in the dual interpretation of the entrepreneurial decision problem. 2. **Minimizes λ (Stakeholder Uncertainty Penalty)** - By decomposing complex problems into independent components, matroid structures allow stakeholders to update their beliefs with minimal conflicting information, reducing the dual variables associated with stakeholder alignment constraints. 3. **Optimizes D(S,a) → S' (State Transition Dynamics)** - Matroid innovation creates deterministic, measurable state transitions after each action, enabling the social planner to create clear incentives around predictable state changes rather than managing complex, potentially non-monotonic belief updates seen in non-matroid settings. | Policy | Pros | Cons | Best For | Main Challenge | Primal-Dual Relation | | -------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------- | ----------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | | **Innovation Vouchers**($25K-100K for quick tests) | • Fast results• Low overhead • Maximum learning per dollar • Rapid decisions | • Doesn't fix market barriers • May create orphaned innovations • Limited scale | Early testing of specific material properties | Creating fair metrics to compare different innovations | **γ (Gamma) Optimization**Directly maximizes the marginal value of resource allocation by funding activities where uncertainty reduction per dollar (∂H/∂R) is highest - corresponds to optimizing the resource shadow price γ in the dual formulation | | **Certification Consortium**(Standardize acceptance criteria) | • Reduces regulatory barriers • Builds market confidence • Helps all innovators • Lowers insurance costs | • Slow to establish • Expensive to set up • Political challenges • Bureaucracy risk | Materials that work technically but face adoption hurdles | Preventing established companies from blocking new entrants | **λ (Lambda) Optimization**Minimizes the uncertainty penalty by aligning stakeholder constraints - corresponds to optimizing λj term in the dual which represents the value of resolving constraint mismatches across the probability space | | **Predictable Progress**(Tiered funding: 50K→50K→50K→250K→$1M) | • Clear path to market • Funding grows with success • Guaranteed contracts at end • Investor-friendly | • Rigid structure • Slower initial progress • Hard to change course • Complex administration | Materials needing systematic, long-term development | Setting milestones that are both achievable and meaningful | **D(S,a)→S' Optimization**Explicitly maps state transition dynamics by defining the required actions a to move between states S→S' - corresponds to optimizing the action selection a_j in the primal and the expected state after action μj(a) | # Key Theoretical Innovations 1. ⭐️**Matroid-Nonmatroid Boundary**: The thesis explores when entrepreneurial decision problems exhibit matroid properties (allowing greedy optimization) versus non-matroid properties (requiring dynamic programming) - along with [[📜Terwiesch09_innov_tourn]] 2. **Dual Value Function Decomposition**: Identifies how marginal resource values (γ) differ between sectors despite similar sustainability goals 3. **State Transition Matrix Calibration**: Uses empirical data to calibrate theoretical transition matrices (D) based on actual entrepreneurial experiences 4. --- # relevant papers from 📝 [[📝moon24_csv_ai_cofounder]] | Paper Title | Reason for Classification (Federated Learning & Spatio-Temporal Complexity) | Optimization Component (3.1 Theoretical) | | ------------------------------------------------------------------------------------ | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | **Probabilistic Programming for Entrepreneurial Decision Support Under Uncertainty** | Offers a unified probabilistic programming framework that integrates distributed insights across stakeholders and time, systematically reducing spatio-temporal complexity by synthesizing multiple information sourcesoaicite:2. | Establated state-transition functions and uncertainty weights, explicitly structuring collective uncertainty reduction across spatio-temporal dimensions ($U_d, U_s, U_i$)oaicite:3. | | **Mining Test Quantities with Exchangeability: Bayesian Reversibility** | Introduces altructures ensuring reversible learning from sequential experiments, transforming path-dependent complexity into cumulative federated knowledge that reduces spatio-temporal uncertainty . | Clarifies federated learning strategies via reversible test quantity metrics, systematically shaping spatio-temporal state transitions ($D(S,A)=S'$) into a coordinated federated learning process . | | **Complexity Analysis of Entrepreneurial Decision-Making (NP-Hardness Proof)** | Provides foundational complexity theory justifying the necessity for federated approximation strategies by demonstrating intrinsic computational complexity of comprehensive spatio-temporal optimization, motivating federated learning as necessary due to NP-hardness . | Establishes theoretical justification for federated learning approaches due to computational complexity, defining why exact spatio-temporal uncertainty optimization ($U_d, U_s, U_i$) is intractable without federated strategies . | | **Calibrated Federated Learning via Entrepreneur–Social Planner Coordination** | Extends federated learning to ecosystem-level coordination, defining collaborative milestone-driven benchmarks that federate spatio-temporal insights from individual ventures and social planners, systematically reducing ecosystem-wide complexityoaicite:4. | Clarifies theoretical structures for collaborative, federated optimization across ventures, explicitly aligning multi-venture state transitions ($D(S,A)=S'$) into unified spatio-temporal uncertainty reduction strategies ($U_d, U_s, U_i$)oaicite:5. | [[federated_learning]] [[📜gans20_choose(tech)]]