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Econ-558 · economics · Economics

Realizability of every admissible input-block shape

LLM status Solved unverified

Statement

ID: GT-077. Category: Stable matching decomposition.  Kind: P. Status: Conjecture in the cited source.

Definitions and mathematical statement

Let $I$ have proposer and receiver sets $P,R$ of size $n\geq 1$, with strict complete rankings. An input block is a pair $(A,B)$, where $\varnothing\ne A\subseteq P$, $B\subseteq R$, and $|A|=|B|=k$, such that \[ \begin{aligned} \{\text{the first }k\text{ receivers in }p\text{'s ranking}\} &=B &&(p\in A),\\ \{\text{the first }k\text{ proposers in }r\text{'s ranking}\} &=A &&(r\in B). \end{aligned} \] Let $\mathcal B(I)$ be the set of all input blocks, including $(P,R)$. Order it by $(A,B)\leq(A',B')$ iff $A\subseteq A'$ and $B\subseteq B'$. Blocks are laminar: intersecting blocks are comparable. The containment Hasse diagram is therefore a rooted tree after the whole-instance root is included. Label a block by its size $k$.

An admissible shape of order $n$ is a finite rooted unordered tree $T$, with positive integer labels $s(v)$, satisfying \[ s(\mathrm{root})=n,\qquad s(w)<s(v)\text{ for each child }w\text{ of }v,\qquad \sum_{w:\,w\text{ child of }v}s(w)\leq s(v). \]

Conjecture.
For every $n\geq1$ and every admissible shape $T$ of order $n$, there is a strict complete order-$n$ instance $I$ such that the containment Hasse diagram of $\mathcal B(I)$ is isomorphic to $T$, with the isomorphism preserving the root and all size labels.

Short English statement

Can every shape satisfying the block-size constraints occur without creating additional blocks?

Variants and scope boundaries

All input blocks must be accounted for; exhibiting only selected blocks does not prove realizability. A proper block need not have a complementary block, so a block is not the same as a factor of an input-independent partition. With the whole-instance block included, the source's block forest is a rooted tree.

Source relationship and status

This is [S1, Section 7, problem D]. The top-rank definition is the source's input-block notion, and the admissibility constraints are those in that problem. The source reports small-order realizations, while accidental extra blocks remain the obstruction to a general construction.

References

[S1] Yoshiteru Ishida, Prime Factorisation of Stable-Matching Instances: Uniqueness, Simultaneous Products, and an Exact Census, 2026, arXiv:2610.05617v1. https://arxiv.org/html/2610.05617v1. Locators: Section 3, Theorem 9.4 (block condition) and Proposition 9.5 (laminarity); Section 7, problem D.

Common conventions: Source mathematical conventions

Finite games. For a finite set $A$, $\Delta(A)$ is its probability simplex. Players choose independent mixed strategies in Nash equilibrium; correlation is allowed only when explicitly part of the solution concept. In a game with payoff functions $u_i$ and strategy profile $x$, an additive $\varepsilon$ Nash equilibrium satisfies \[ u_i(x)\geq u_i(x_i',x_{-i})-\varepsilon \quad\text{for every player $i$ and every admissible deviation $x_i'$}. \] For numerical approximation, payoffs are normalized as locally specified. An $\varepsilon$ payoff residual is distinct from distance at most $\varepsilon$ to the set of exact equilibria. Point convergence, convergence of empirical play, and convergence in distance to an equilibrium set are also distinct.

Computational representations. Unless stated otherwise, a complexity question takes rational data with binary encoding; polynomial time is measured in total bit length. A fixed-accuracy polynomial algorithm is not necessarily polynomial in $1/\varepsilon$, and neither is necessarily polynomial in $\log(1/\varepsilon)$. Succinct games and value, demand, or sampling oracles must declare their interfaces. Exact real solutions require an explicit representation; an unspecified list of arbitrary real numbers is not a finite computational output. A claimed algorithmic barrier is meaningful only for its specified model and reductions.

Dynamic games. Histories, information, observation, and allowable randomization are part of the model. A stationary strategy depends only on the current state; a positional strategy in a deterministic graph game chooses one action at each controlled vertex. Nash equilibrium at an initial state and equilibrium after every history are different requirements. An expectation of a pathwise $\liminf$ payoff, a $\liminf$ of expected averages, a limiting expected average, and a uniform finite-horizon equilibrium are different criteria. None is silently substituted for another.

Allocation and mechanisms. A complete allocation assigns every item exactly once unless otherwise stated. Zero-valued goods, negative values, capacity constraints, feasibility, lotteries, and copies of goods affect fairness definitions and are specified locally. Dominant-strategy incentive compatibility, Bayesian incentive compatibility, universal truthfulness, and truthfulness in expectation impose different inequalities. Whenever an objective ratio has zero denominator, its convention is stated or those instances are excluded.

Infinite-dimensional models. Expectations, integrals, stochastic equations, and optimization objectives require their local regularity, integrability, filtrations, and admissible domains. A backward--forward mean-field system is a boundary-value problem; it is not an initial-value semiflow without an additional construction. Long-time, large-population, and vanishing-noise limits require an order of limits and a convergence topology. If a minimum need not be attained, the objective is its infimum or supremum and the approximation requirement is stated separately.

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Econ-558-A2 · Full solution claimed · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-558

Realizability of every admissible input-block shape: A cyclic construction excluding every additional block

The exact target

An input block in a strict complete marriage instance is a pair $(A,B)$ of equally large nonempty proposer and receiver sets satisfying \[ \operatorname{Top}_{|B|}(p)=B\quad(p\in A),\qquad \operatorname{Top}_{|A|}(r)=A\quad(r\in B). \] The whole instance counts as a block. The question in [ishida, Section 7, problem D] asks whether every admissible size-labelled rooted tree is the containment tree of all input blocks. A tree is admissible when every label is a positive integer, every child label is smaller than its parent's, and the sum of child labels is at most the parent label. We give a construction for the exact target. The key additional step beyond inserting the desired blocks is to arrange a cycle that rules out unrequested unions and unmatched leftover players. No condition on the eventual stable matching is used.

Laminarity on both sides

The following elementary fact is also the input-block laminarity property in [ishida, Proposition 9.5]; its proof is included because the exclusion argument needs its two-sided form.
Lemma: Let $(A,B)$ and $(C,D)$ be input blocks of sizes $k\le l$. If they share a proposer or a receiver, then $A\subseteq C$ and $B\subseteq D$. Otherwise both sides are disjoint.
Proof: If $p\in A\cap C$, its top $k$ receivers are a subset of its top $l$, so $B\subseteq D$. Choose $r\in B$, which is possible since $k>0$. Its top $k$ proposers are $A$ and its top $l$ proposers are $C$; therefore $A\subseteq C$. Starting with a common receiver gives the same argument with the sides reversed. Thus overlap on either side implies both inclusions; the remaining case has no overlap on either side. ∎

Recursive preferences

Associate with each node $v$ two sets $P_v,R_v$, each of size $s_v$, the node's label. For each child $w$, choose disjoint subsets $P_w\subset P_v$ and $R_w\subset R_v$ of size $s_w$. The capacity condition permits this. Pair the remaining proposers and receivers arbitrarily as $(p_j,r_j)$. These are bookkeeping pairs, not necessarily mutually preferred or matched pairs.

The atoms at $v$ are its child pairs $(P_w,R_w)$ and these free singleton pairs. Put the atoms in any cyclic order $1,\ldots,m$. If $s_v>1$, then $m\ge2$: a sole child has smaller size and hence leaves a free pair, while a node with no children has $s_v$ free pairs. For each atom $j$, fix a receiver $r_j^*$ in it. At a free atom this is its unique receiver. The successor index is understood modulo $m$.

Construct local complete lists within $(P_v,R_v)$ from the children's already constructed lists as follows. Every player in a child keeps all partners in that child first, in the inherited order. Every proposer in child atom $j$ ranks $r_{j+1}^*$ first after its child, then the other receivers in $R_v$ outside the child in arbitrary strict order. A free proposer $p_j$ ranks $r_{j+1}^*$ first and all remaining receivers in $R_v$ afterward. A free receiver $r_j$ ranks $p_j$ first and all other proposers afterward. A receiver in a child appends the proposers outside that child in any strict order.

For $s_v=1$, there are no children; give the unique two players their unique local lists. Apply the construction up the tree. The root lists are the required complete rankings. Appending outside partners never changes a descendant's internal ranking or top-set property.

Theorem: For every finite admissible size-labelled rooted tree with root label $n$, the construction above produces a strict complete marriage instance with $n$ players on each side whose input blocks are exactly the prescribed node pairs $(P_v,R_v)$. Consequently every admissible shape in the stated problem is realizable.
Proof: Every node pair is a block: each player ranks its own child internally first, then the remaining partners of the parent, and only then partners outside that parent. Induction toward the root shows that the first $s_v$ partners of every player at node $v$ are exactly the opposite side of $v$.

Suppose $(A,B)$ is an additional block, and choose the smallest prescribed node $v$ containing both its sides. Such a node exists because the root contains everyone; uniqueness follows from the nested-or-disjoint construction. The additional block is not contained in any child. By the lemma it therefore either contains an entire child block or is disjoint from that child on both sides. Moreover it is strictly larger than every child it contains: equality would make it that prescribed block.

Call an atom reached if its whole child block belongs to $(A,B)$, or, for a free atom, if its proposer belongs to $A$. At least one atom is reached. Indeed, any member of the nonempty set $A$ is in a child, forcing that whole child by the lemma, or is a free proposer. We claim that reaching atom $j$ reaches atom $j+1$. For a child atom of size $s_j$, the block size $|A|$ exceeds $s_j$. Every proposer in that child must have its first outside receiver, $r_{j+1}^*$, in its top $|A|$ set $B$. For a free atom the same receiver is the proposer's first choice and again lies in $B$. If the successor is a child, overlap on the receiver side and the lemma force its whole block into $(A,B)$. If it is free, its receiver's first proposer is its own $p_{j+1}$, which must therefore lie in $A$. This proves the claim.

The successor cycle reaches every atom. Thus all child proposers and all free proposers lie in $A$, so $A=P_v$. Since $B\subseteq R_v$ and $|B|=|A|=s_v$, also $B=R_v$. This contradicts the premise that the block is additional. When $s_v=1$ the equality is immediate and no cycle argument is needed. Hence there are no additional blocks. ∎

An illustrative nonpartition shape

Take a root of size four with a sole child of size two, and give that child no children. Within it use $p_1:r_2,r_1$, $p_2:r_1,r_2$, $r_1:p_1,p_2$, and $r_2:p_2,p_1$. Its two free atoms form the two-cycle that prevents size-one blocks. At the root order the three atoms as the child, $(p_3,r_3)$, $(p_4,r_4)$. The child proposers' first outside receiver is $r_3$; $p_3$ starts with $r_4$ and $p_4$ with $r_1$; $r_3,r_4$ start with $p_3,p_4$ respectively. Complete the lists as above. The only blocks are the specified size-two child and the size-four root. In particular, the child's complement is not forced to be a block.

Scope of the conclusion

The proof supplies existence for every admissible shape and explicitly excludes accidental blocks; it does not enumerate how many preference profiles realize a shape, classify their stable matchings, or impose extra restrictions such as a common master list. Necessity of the admissibility conditions follows from laminarity, strict inclusion, and disjoint child sets, so together these conditions characterize the possible size-labelled shapes. This is a claim about the precise input-block question, without a priority claim about related factorization questions.

Completion Estimate

100%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt constructs preferences recursively and uses two-sided laminarity and successor cycles to exclude every accidental input block. Every admissible labelled tree is therefore realized with exactly its prescribed blocks, establishing the complete canonical existence target without needing additional matching or enumeration conclusions.

References

  • [ishida] Y. Ishida (2026), Prime factorisation of stable-matching instances: uniqueness, simultaneous products, and an exact census, arXiv:2610.05617v1, Section 3 and Section 7, problem D. https://arxiv.org/html/2610.05617v1.

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Econ-558-A1 · Full solution claimed · LLM-generated · gpt 6 astra pro ·

GPT 6 Astra Pro attempt on Econ-558, version 2

Realizability of every admissible input-block shape: A cyclic construction excluding every additional block

Problem ID: C78-554. Model: GPT 6 Astra Pro. Version: 2. Status: Solved.

The exact target

An input block in a strict complete marriage instance is a pair $(A,B)$ of equally large nonempty proposer and receiver sets satisfying \[ \operatorname{Top}_{|B|}(p)=B\quad(p\in A),\qquad \operatorname{Top}_{|A|}(r)=A\quad(r\in B). \] The whole instance counts as a block. The question in [ishida, Section 7, problem D] asks whether every admissible size-labelled rooted tree is the containment tree of all input blocks. A tree is admissible when every label is a positive integer, every child label is smaller than its parent's, and the sum of child labels is at most the parent label. We give a construction for the exact target. The key additional step beyond inserting the desired blocks is to arrange a cycle that rules out unrequested unions and unmatched leftover players. No condition on the eventual stable matching is used.

Pro revision. Version 2 preserves the complete recursive construction, laminarity lemma, universal realizability theorem, and exclusion proof. It adds the supplied independently written finite audit [audit], which tests exact equality with every prescribed block set at small orders rather than merely verifying that selected desired blocks occur.

Laminarity on both sides

The following elementary fact is also the input-block laminarity property in [ishida, Proposition 9.5]; its proof is included because the exclusion argument needs its two-sided form.
Lemma: Let $(A,B)$ and $(C,D)$ be input blocks of sizes $k\le l$. If they share a proposer or a receiver, then $A\subseteq C$ and $B\subseteq D$. Otherwise both sides are disjoint.
Proof: If $p\in A\cap C$, its top $k$ receivers are a subset of its top $l$, so $B\subseteq D$. Choose $r\in B$, which is possible since $k>0$. Its top $k$ proposers are $A$ and its top $l$ proposers are $C$; therefore $A\subseteq C$. Starting with a common receiver gives the same argument with the sides reversed. Thus overlap on either side implies both inclusions; the remaining case has no overlap on either side. ∎

Recursive preferences

Associate with each node $v$ two sets $P_v,R_v$, each of size $s_v$, the node's label. For each child $w$, choose disjoint subsets $P_w\subset P_v$ and $R_w\subset R_v$ of size $s_w$. The capacity condition permits this. Pair the remaining proposers and receivers arbitrarily as $(p_j,r_j)$. These are bookkeeping pairs, not necessarily mutually preferred or matched pairs.

The atoms at $v$ are its child pairs $(P_w,R_w)$ and these free singleton pairs. Put the atoms in any cyclic order $1,\ldots,m$. If $s_v>1$, then $m\ge2$: a sole child has smaller size and hence leaves a free pair, while a node with no children has $s_v$ free pairs. For each atom $j$, fix a receiver $r_j^*$ in it. At a free atom this is its unique receiver. The successor index is understood modulo $m$.

Construct local complete lists within $(P_v,R_v)$ from the children's already constructed lists as follows. Every player in a child keeps all partners in that child first, in the inherited order. Every proposer in child atom $j$ ranks $r_{j+1}^*$ first after its child, then the other receivers in $R_v$ outside the child in arbitrary strict order. A free proposer $p_j$ ranks $r_{j+1}^*$ first and all remaining receivers in $R_v$ afterward. A free receiver $r_j$ ranks $p_j$ first and all other proposers afterward. A receiver in a child appends the proposers outside that child in any strict order.

For $s_v=1$, there are no children; give the unique two players their unique local lists. Apply the construction up the tree. The root lists are the required complete rankings. Appending outside partners never changes a descendant's internal ranking or top-set property.

Theorem: For every finite admissible size-labelled rooted tree with root label $n$, the construction above produces a strict complete marriage instance with $n$ players on each side whose input blocks are exactly the prescribed node pairs $(P_v,R_v)$. Consequently every admissible shape in the stated problem is realizable.
Proof: Every node pair is a block: each player ranks its own child internally first, then the remaining partners of the parent, and only then partners outside that parent. Induction toward the root shows that the first $s_v$ partners of every player at node $v$ are exactly the opposite side of $v$.

Suppose $(A,B)$ is an additional block, and choose the smallest prescribed node $v$ containing both its sides. Such a node exists because the root contains everyone; uniqueness follows from the nested-or-disjoint construction. The additional block is not contained in any child. By the lemma it therefore either contains an entire child block or is disjoint from that child on both sides. Moreover it is strictly larger than every child it contains: equality would make it that prescribed block.

Call an atom reached if its whole child block belongs to $(A,B)$, or, for a free atom, if its proposer belongs to $A$. At least one atom is reached. Indeed, any member of the nonempty set $A$ is in a child, forcing that whole child by the lemma, or is a free proposer. We claim that reaching atom $j$ reaches atom $j+1$. For a child atom of size $s_j$, the block size $|A|$ exceeds $s_j$. Every proposer in that child must have its first outside receiver, $r_{j+1}^*$, in its top $|A|$ set $B$. For a free atom the same receiver is the proposer's first choice and again lies in $B$. If the successor is a child, overlap on the receiver side and the lemma force its whole block into $(A,B)$. If it is free, its receiver's first proposer is its own $p_{j+1}$, which must therefore lie in $A$. This proves the claim.

The successor cycle reaches every atom. Thus all child proposers and all free proposers lie in $A$, so $A=P_v$. Since $B\subseteq R_v$ and $|B|=|A|=s_v$, also $B=R_v$. This contradicts the premise that the block is additional. When $s_v=1$ the equality is immediate and no cycle argument is needed. Hence there are no additional blocks. ∎

An illustrative nonpartition shape

Take a root of size four with a sole child of size two, and give that child no children. Within it use $p_1:r_2,r_1$, $p_2:r_1,r_2$, $r_1:p_1,p_2$, and $r_2:p_2,p_1$. Its two free atoms form the two-cycle that prevents size-one blocks. At the root order the three atoms as the child, $(p_3,r_3)$, $(p_4,r_4)$. The child proposers' first outside receiver is $r_3$; $p_3$ starts with $r_4$ and $p_4$ with $r_1$; $r_3,r_4$ start with $p_3,p_4$ respectively. Complete the lists as above. The only blocks are the specified size-two child and the size-four root. In particular, the child's complement is not forced to be a block.

Scope of the conclusion

The proof supplies existence for every admissible shape and explicitly excludes accidental blocks; it does not enumerate how many preference profiles realize a shape, classify their stable matchings, or impose extra restrictions such as a common master list. Necessity of the admissibility conditions follows from laminarity, strict inclusion, and disjoint child sets, so together these conditions characterize the possible size-labelled shapes. This is a claim about the precise input-block question, without a priority claim about related factorization questions.

Independent finite verification. The supplied audit [audit] enumerates every admissible rooted unordered size-labelled tree with root size $n=1,\ldots,7$. Their numbers by root size are \[ 1,3,10,40,170,785,3770, \qquad 1+3+10+40+170+785+3770=4779. \] For each of these 4,779 shapes, verify_matching.py reconstructs the child atoms and free singleton atoms, completes the lists by the stated successor-cycle rule, and records every prescribed node pair. It then enumerates every nonempty proposer subset $A$: its size $k$ and one member's first $k$ receivers determine the only possible opposite set $B$, after which all proposer and receiver top-$k$ equalities are checked. This enumerates all actual input blocks. In every case the resulting block set is exactly the recorded prescribed set, including the root. The supplied matching_results.json reports this exact-set equality for all 4,779 constructions.

These are finite checks of one deterministic labeling and list completion for each unordered shape; they do not enumerate all profiles realizing a shape. Their role is to detect accidental blocks in the tested construction instances. They do not establish the theorem by extrapolation, replace the two-sided laminarity and cycle-exclusion proofs, or constitute proof-assistant formalization. The proof above establishes realizability at every finite order.

Completion Estimate

100%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt constructs preferences recursively and uses two-sided laminarity and successor cycles to exclude every accidental input block. Every admissible labelled tree is therefore realized with exactly its prescribed blocks, establishing the complete canonical existence target without needing additional matching or enumeration conclusions.

References

  • [ishida] Y. Ishida (2026), Prime factorisation of stable-matching instances: uniqueness, simultaneous products, and an exact census, arXiv:2610.05617v1, Section 3 and Section 7, problem D. https://arxiv.org/html/2610.05617v1.
  • [audit] Independent checks for the economics "solved" audit (9 October 2026), user-supplied audit attachments README.md, verify_matching.py, and matching_results.json. Reviewed repository snapshot: ccccd798864807c1f3fb8fe36727fe35ee875aeb. Its Python standard-library program checks all admissible shapes through root size seven by exact equality of prescribed and enumerated block sets; this is supplementary finite evidence.

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