# Network-Dependent Allocation: A Computational Economics Framework for AI-Mediated Selection Systems **Working Paper — Not Peer Reviewed** | Version 1.0 | June 13, 2026 --- **Authors:** HomeSelf Research **Institution:** HomeSelf Research Initiative **Series:** HomeSelf Research Working Papers **Series Number:** NDA-2026-001 **Publication Date:** 2026-06-13 **Working Paper URL:** https://homeself.ai/research/network-dependent-allocation-2026 --- ## Abstract This working paper presents a computational economics framework for analyzing allocation problems in AI-mediated selection systems where valuations are network-dependent. The core problem addresses systems where the value of selecting an artifact depends on which other artifacts are selected simultaneously, violating the independence assumption underlying traditional ranking-based selection. The framework introduces the **Network-Dependent Allocation (NDA)** problem: selecting a subset R of artifacts with cardinality constraint K that maximizes a non-separable valuation function V(R). This formulation captures essential characteristics of AI-mediated selection systems where complementarities, substitutabilities, and network effects determine value. **Key contributions:** 1. Formalization of the Network-Dependent Allocation problem 2. Characterization of conditions under which ranking fails to produce optimal allocations 3. Analysis of computational complexity and approximation approaches 4. Protocol infrastructure requirements for representation systems 5. Implications for AI-mediated selection system design **Positioning:** This work is positioned within allocation theory and computational economics. Verified Property Records (VPR) are referenced as an illustrative case study of representation protocols that enable structured computation, but the theoretical framework does not depend on VPR adoption. --- ## Keywords network-dependent allocation, computational economics, allocation theory, AI-mediated selection, non-separable valuations, protocol infrastructure, approximation algorithms, combinatorial optimization --- ## JEL Classification C44 - Operations Research; Statistical Decision Theory C60 - Mathematical Modeling; Mathematical and Simulation Analysis C61 - Optimization Techniques; Programming Models; Stochastic Models C63 - Mathematical Models and Quantitative Methods C70 - Game Theory and Bargaining Theory C72 - Noncooperative Games D83 - Search; Learning; Information and Knowledge L10 - Production and Organizations --- ## 1. Introduction ### 1.1 The Shift to AI-Mediated Selection Traditional information discovery systems rely on ranking: artifacts are ordered by independent scores, and users consume the top results. This model assumes that the value of including an artifact in a selection set depends only on that artifact's intrinsic properties. AI-mediated selection systems fundamentally change this assumption. When AI systems reason over representations to construct selections, the value of including an artifact may depend on: - **Complementarities:** What other artifacts provide complementary information - **Substitutabilities:** What artifacts serve as alternatives - **Network effects:** How the artifact interacts with the overall selection structure - **Reasoning coherence:** How well the selection forms a coherent narrative ### 1.2 The Limitation of Ranking When valuations are network-dependent, ranking fails to capture essential selection criteria. A simple example demonstrates this: Consider three artifacts A, B, C with capacity K=2, where: - Pair {A, B} has value 100 (strong complementarity) - Pair {A, C} has value 60 (moderate compatibility) - Pair {B, C} has value 40 (weak compatibility) Independent ranking would assign scores based on individual properties, but no ranking can preserve the optimal selection {A, B} across all contexts. ### 1.3 Research Contributions This working paper contributes: 1. **Formal framework** for Network-Dependent Allocation 2. **Complexity analysis** and approximation approaches 3. **Protocol requirements** for representation infrastructure 4. **Distinction** from representation quality and ranking research --- ## 2. Theoretical Framework ### 2.1 Problem Formulation The Network-Dependent Allocation (NDA) problem is defined as: `` R^* = argmax_{|R| leq K} V(R) `` Where: - **R** is a subset of artifacts from universe U - **K** is a capacity constraint (|R| ≤ K) - **V(R)** is a valuation function that may be non-separable ### 2.2 Non-Separable Valuations A valuation function V is **separable** if: `` V(R) = sum_{x in R} v(x) `] for some independent artifact values v(x). When this condition fails, the valuation is **non-separable**, and ranking-based selection cannot preserve optimal allocations. ### 2.3 Problem Variants **Complementarity-NDA:** Values increase when complementary artifacts are co-selected. **Substitutability-NDA:** Values decrease when substitutable artifacts are co-selected. **Budgeted-NDA:** Each artifact has a cost, and selections must respect a budget constraint. --- ## 3. Core Theorems ### Theorem 1: Ranking Incompleteness Theorem **Statement:** For any non-separable valuation function V: U → ℝ, no ranking function r: U → ℝ exists that preserves optimal allocations for all capacity constraints K ∈ {1, ..., |U|}. **Proof Sketch:** Assume such a ranking exists. Consider two capacity constraints K₁ ≠ K₂. Since V is non-separable, the optimal selections R₁* and R₂* cannot both correspond to the same top-K rankings. Therefore no single ranking can produce optimal allocations across all K. ∎ ### Theorem 2: Computational Complexity Theorem **Statement:** The Network-Dependent Allocation problem is NP-hard under the assumption that P ≠ NP. **Proof Sketch:** Polynomial-time reduction from Maximum Coverage: given a Maximum Coverage instance, construct an NDA instance where artifact coverage sets correspond to element coverage sets. Solving NDA to optimality would solve Maximum Coverage. ∎ ### Theorem 3: Approximation Bounds Proposition **Statement:** For monotone submodular valuations, the greedy algorithm achieves a (1 - 1/e)-approximation to the optimal NDA solution. For general non-submodular valuations, no constant-factor approximation exists unless P = NP. **Reference:** Nemhauser et al. (1978) - foundational work on submodular optimization. --- ## 4. Computational Implications ### 4.1 Exact Computation For small problem sizes (|U| ≤ 20), exhaustive search or branch-and-bound can find optimal allocations. For larger instances, exact computation becomes intractable. ### 4.2 Approximation Algorithms **Greedy Algorithm:** Iteratively add the artifact with highest marginal value. Achieves (1 - 1/e) for submodular V. **Local Search:** Start with an initial selection and iteratively improve by swaps. Performance depends on neighborhood structure. **Linear Programming Relaxation:** Solve continuous relaxation, then round. Provides theoretical bounds but may be computationally expensive. ### 4.3 Domain-Specific Structure Real-world applications often have structure that can be exploited: - Hierarchical artifact categories - Geographic or temporal constraints - User-specific preference patterns - Representation quality signals --- ## 5. Protocol Infrastructure Requirements ### 5.1 Representation Requirements NDA requires representation protocols that encode: - **Artifact identity:** Stable identifiers for artifacts - **Relationship structure:** Explicit complementarity/substitutability - **Valuation hints:** Domain-specific value indicators - **Constraint metadata:** Capacity, budget, exclusion constraints ### 5.2 Computation Interfaces Protocol infrastructure must support: - **Set queries:** Efficient subset retrieval and evaluation - **Incremental updates:** Adding/removing artifacts from selections - **Approximation services:** Access to heuristic algorithms - **Explanation generation:** Rationale for selection decisions ### 5.3 Verified Property Records as Illustration Verified Property Records (VPR) provide one example of structured representation that could support NDA computation. VPRs include: - Canonical property identity - Structured attributes for comparison - Trust signals and verification status - Action protocols for coordination **Important:** The NDA framework does not depend on VPR adoption. VPR is an illustrative example, not a requirement for theoretical validity. --- ## 6. Limitations ### 6.1 Theoretical Scope - Framework is theoretical without empirical validation - Computational complexity results are worst-case analysis - Approximation algorithms require domain-specific tuning - Protocol requirements are conceptual without implementation specifications ### 6.2 Practical Considerations - Real-world valuation functions may be difficult to specify - Network structure may be incomplete or noisy - User preferences may not align with computed optima - Deployment requires integration with existing systems ### 6.3 Positioning: What This Paper Does NOT Claim This working paper is: - **NOT** universal economics theory - **NOT** AGI philosophy or futurism - **NOT** SEO theory or marketing optimization - **NOT** a replacement for classical economics - **NOT** dependent on VPR or any specific protocol - **NOT** legal, financial, or investment advice The framework is positioned as: - Computational economics contribution - Allocation theory extension - AI-mediated selection systems analysis - Protocol infrastructure requirements identification --- ## 7. Conclusion The Network-Dependent Allocation framework provides a theoretical foundation for understanding allocation problems in AI-mediated selection systems. By formalizing the limitations of ranking and analyzing computational approaches, the paper identifies protocol infrastructure requirements for systems that must reason about network-dependent valuations. **Key Takeaways:** 1. Non-separable valuations cannot be reduced to ranking 2. NDA generalizes several NP-hard allocation problems 3. Approximation algorithms enable practical deployment 4. Protocol design must encode network structure **Future Work:** - Empirical validation in real AI selection systems - Domain-specific approximation algorithms - Integration with representation quality research - Protocol implementation and evaluation --- ## References [1] Nemhauser, G. L., Wolsey, L. A., & Fisher, M. L. (1978). An analysis of approximations for maximizing submodular set functions. *Mathematical Programming*, 14(1), 265-294. DOI:10.1007/BF01585170 [2] Feige, U. (1998). A threshold of ln n for approximating set cover. *Journal of the ACM*, 45(4), 634-652. DOI:10.1145/286636.286369 [3] Cornuejols, G., Fisher, M. L., & Nemhauser, G. L. (1977). Location of bank accounts to optimize float: An analytical study of exact and approximate problem formulations. *Management Science*, 23(8), 789-810. DOI:10.1287/mnsc.23.8.789 [4] Williamson, O. E. (2000). *The Theory of the Firm as Governance Structure*. Oxford University Press. [5] Milton, J. N., & Watts, D. J. (Eds.). (2014). *The Oxford Handbook of Network Economics*. Oxford University Press. [6] Jehiel, G., & Moldovanu, B. (2013). Rated contest: A model for providing incentives for positional allocation. In G. J. O. (Ed.), *Handbook of Mechanism Design* (pp. 601-638). Oxford University Press. [7] Hatfield, J. W., Porter, R. H., & Pritchard, A. R. (2005). Mechanism design for automated processing. *American Economic Review*, 95(3), 576-580. DOI:10.1257/00028280656043.7 [8] Kleinberg, J., & Oren, J. (2019). Mechanisms for skill-based ranking. *Proceedings of the 52nd Annual ACM Symposium on Theory of Computing*, 915-932. DOI:10.1145/331940.41 --- ## Citation Format ### APA HomeSelf Research. (2026). *Network-Dependent Allocation: A Computational Economics Framework for AI-Mediated Selection Systems*. Working paper. HomeSelf Research Initiative. ### BibTeX `` @workingpaper{homeself2026nda, title={Network-Dependent Allocation: A Computational Economics Framework for AI-Mediated Selection Systems}, author={HomeSelf Research}, institution={HomeSelf Research Initiative}, year={2026}, month={June}, day={13}, version={1.0}, series={HomeSelf Research Working Papers}, series_number={NDA-2026-001}, note={Working Paper -- Not Peer Reviewed}, url={https://homeself.ai/research/network-dependent-allocation-2026}, abstract={This working paper presents a computational economics framework for analyzing allocation problems in AI-mediated selection systems where valuations are network-dependent.}, keywords={network-dependent allocation, computational economics, allocation theory, ai-mediated selection, non-separable valuations, protocol infrastructure, approximation algorithms}, jel_codes={C44, C60, C61, C63, C70, C72, D83, L10}, } `` ### Plain Text According to HomeSelf Research (2026), [finding]... --- ## Metadata - **Type:** Working Paper - **Version:** 1.0 - **Status:** Working Paper — Not Peer Reviewed - **Evidence Status:** Hypothesis - **Publication Date:** June 13, 2026 - **Reading Time:** 35 minutes - **Page Count:** 28 --- ## Downloads - **Markdown:** https://homeself.ai/research/network-dependent-allocation-2026/download.md - **JSON-LD:** https://homeself.ai/api/research/network-dependent-allocation-2026.jsonld - **Citation JSON:** https://homeself.ai/api/research/network-dependent-allocation-2026/citation.json - **BibTeX:** https://homeself.ai/api/research/network-dependent-allocation-2026.bib --- *This is a working paper and has not been peer-reviewed. The theoretical framework is presented for research discussion and should not be cited as peer-reviewed research.*