Individual-Pieces Set¶
In fair cake-cutting, the set of all utility vectors attainable by partitions of the cake among named agents; its boundary exposes feasible and Pareto-efficient allocations under declared valuations.
Core Idea¶
The Individual-Pieces Set is the set of all utility vectors attainable by partitions of a cake among fixed agents under declared valuations and allocation rules. It represents feasibility geometrically: points are outcomes, the northeast boundary is Pareto-efficient, and standard convexity/compactness results depend on divisibility and valuation assumptions. The set does not itself choose a fair allocation. The resulting set represents feasibility before a fairness rule chooses an outcome.
Scope of Application¶
Individual-Pieces Set applies in fair division and related work only when its carrier, rules, and evidence boundary are explicit. Use it in fair division, bargaining, convex analysis, algorithms, and mechanism design with agents, resource, valuations, normalization, partition/disposal rules, divisibility, coordinate meaning, and topology explicit. Separate the full feasible set from its Pareto frontier and from the richer FIPS value-matrix extension.
- Fair division. Describes attainable welfare combinations.
- Bargaining theory. Locates efficient tradeoffs.
- Convex analysis. Uses supporting hyperplanes/frontiers.
- Algorithm design. Searches allocations producing target vectors.
- Mechanism design. Separates feasibility from selection rules.
Clarity¶
State resource model, agents/order of coordinates, valuation assumptions and normalization, partition/disposal rules, divisibility or atomic pieces, vector versus matrix output, and topology. Do not infer fairness from feasibility or compare utilities interpersonally without justification.
Manages Complexity¶
Enumerating partitions quickly becomes combinatorial; the IPS compresses them into an outcome region. A point answers what utility vector is attainable, while the frontier shows where improving one agent necessarily harms another. The set does not choose among frontier points, and convexity does not make every division ethically acceptable. With divisible nonatomic cake, mixing/splitting arguments support convexity; with indivisible items, the attainable set may be discrete and its convex hull adds lotteries or fractional allocations not literally available. Normalizing each agent's total value makes geometric comparisons convenient but does not equate cardinal scales across people. In a two-agent piecewise-constant example, ordering pieces by marginal-value ratios traces an efficient boundary; ties and zero values create segments or degeneracy. The FIPS records every agent's value for every assigned piece, retaining envy-related information that the own-utility IPS discards. Model declarations therefore determine what the geometry means. The central compact representation–lost allocation detail tradeoff is this: Many partitions can map to one utility vector.
Abstract Reasoning¶
Use three linked moves: fix agents, resource, valuations, and allocation rules; map every admissible partition to its utility vector; establish compactness/convexity only under stated assumptions. As a collapse test, the object changes class when the mapping no longer ranges over all admissible partitions or the output is not the declared utility-vector image.
Knowledge Transfer¶
The attainable-vector representation transfers to other divisible-resource and bargaining models when allocation and valuation roles remain explicit. It stops at settings where outcomes cannot be represented by the declared utility coordinates or where convexification introduces unavailable lotteries. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG.
Relationships to Other Abstractions¶
Current abstraction Individual-Pieces Set Domain-specific
Parents (1) — more general patterns this builds on
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Individual-Pieces Set is a kind of Representation Prime
The Individual-Pieces Set is a strict kind of Representation: it maps every admissible cake partition to its agents' own-piece utility vector.
Hierarchy path (1) — routes to 1 parentless root
- Individual-Pieces Set → Representation → Abstraction
Neighborhood in Abstraction Space¶
Individual-Pieces Set sits in a moderately populated region (41st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Allocation, Ranking & Bargaining Models (11 abstractions)
Nearest neighbors
- Exchange economy — 0.89
- Participation constraint (mechanism design) — 0.88
- 3SUM — 0.87
- Rubinstein bargaining model — 0.87
- Strong Nash equilibrium — 0.86
Computed from structural-signature embeddings · 2026-10-08