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Econ-468 · economics · computer science · Computational economics

Rank-maximality probability under uncertain preferences

LLM status Open unverified

Statement

MM-07 |  Matching markets and allocation

Research agenda formalization

Definitions and assumptions

Let $I$ be a house-allocation instance with applicants $A$, houses $H$, and fixed acceptable pairs $E$. Houses have no preferences. A random strict profile $P$ determines each applicant's ranks on acceptable houses; rank one is best. Write $r_P(a,h)$ for the rank of an acceptable pair and $R=\max_a|\{h:(a,h)\in E\}|$, taking $R=0$ if $E$ is empty. For a matching $M$, define its padded rank vector \[ v_P(M)=(v_1,\ldots,v_R),\qquad v_j=|\{(a,h)\in M:r_P(a,h)=j\}|. \] Matching $M$ is rank-maximal if $v_P(M)$ is lexicographically maximal over all feasible matchings. Unmatched applicants contribute no coordinate; there is no independent preliminary cardinality maximization.

Formal question

Using each of independent explicit lotteries, independent uniform refinements of weak orders, and explicit joint-profile lotteries, classify exact computation and rational-threshold comparison of \[ \eta_I(M)=\Pr_P\left(v_P(M)=\max_{\mathrm{lex},\,N\in\mathcal M(I)}v_P(N)\right). \] Hold acceptability fixed and measure complexity in the chosen uncertainty encoding.

Scope and status

The source explicitly suggests rank-maximal matchings as a further solution concept to study under uncertain preferences, without separately classifying these probability problems. The present question is a precise instance of that research direction. Padding rank vectors to a common length removes ambiguity when comparing matchings of different sizes. The ranks are evaluated after random strict refinement, so the event is not rank maximality under the initial tied profile. For explicitly listed joint support, rank-maximality can be checked profile by profile, giving a polynomial evaluation baseline; independent compact support can be exponentially larger than its description.

References for the source of the problem

Katarína Cechlárová, Ágnes Cseh, and David F. Manlove (2019), Selected open problems in Matching Under Preferences, Section 3.1.2, final paragraph, and Section 2.1 for the rank-maximality definition: https://hpi.de/friedrich/docs/publications/2019/BEATCS.pdf. Haris Aziz, Péter Biró, Serge Gaspers, Ronald de Haan, Nicholas Mattei, and Baharak Rastegari (2020), Stable Matching with Uncertain Linear Preferences, model definitions: https://doi.org/10.1007/s00453-019-00650-0.

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-468-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-468

Rank-maximality probability: counting bounds and an exact path algorithm

Problem ID: C78-464.

The probability event and the progress obtained

Acceptability is fixed; a random strict profile determines the ranks. A given feasible matching $M$ is successful exactly when its padded rank vector is lexicographically maximal over all matchings, without a preliminary cardinality objective. We establish polynomial evaluation for listed joint support, exact counting and threshold upper bounds for both independent encodings, and a polynomial exact algorithm on a nontrivial unbounded family of acceptability paths. A matching is assumed feasible; if it is not, its success probability is zero by convention.

The source defines rank maximality in Section 2.1 and proposes uncertain-preference versions in the final paragraph of Section 3.1.2 [ccm]. It does not supply the hardness classification asked here. In particular, hardness results for stability or Pareto optimality are not transferred to this event.

A polynomial verification predicate

Let $n=|A|$ and $R$ be the largest acceptable degree. For $n\geq1$, put $B=n+1$ and \[ w_P(a,h)=B^{R-r_P(a,h)},\qquad W_P(N)=\sum_{(a,h)\in N}w_P(a,h). \] For $E=\varnothing$ every feasible matching is empty and the answer is one, so assume $R\geq1$.
Lemma: For every profile $P$, $W_P$ orders matchings exactly as their padded rank vectors do. Rank maximality of a specified matching is decidable in time polynomial in the deterministic profile's bit length.
Proof: Suppose rank vectors first differ at coordinate $j$ and the first matching has at least one more rank-$j$ edge. The corresponding weight advantage is at least $B^{R-j}$. The absolute possible loss from lower coordinates is at most \[ n\sum_{\ell=j+1}^R B^{R-\ell} =B^{R-j}-1. \] The first difference therefore determines the sign. Equal vectors give equal weights.

Each weight has at most $1+R\log_2 B$ bits. A maximum-weight bipartite matching can be found in polynomial bit time, for example by a min-cost-flow algorithm with unit capacities, allowing unmatched vertices via zero-weight dummy edges. Comparing its weight with $W_P(M)$ verifies the event. This allows arbitrary matching cardinalities exactly as required. ∎

Proposition (Listed joint-profile support): For an explicitly listed joint lottery with rational masses $p_s$, exact probability and comparison with a rational threshold are polynomial-time computable.
Proof: Run Lemma Lemma on each listed profile and sum its mass if $M$ is successful. The number of checks is the input support size. A common denominator formed by multiplying the input denominators has bit length at most their total bit length; exact rational summation and comparison are therefore polynomial. ∎

Independent encodings: exact counting upper bounds

Theorem: For independent explicit rational lotteries, there is a polynomial-bit computable positive integer $D$ such that $D\eta_I(M)$ is a #P function of the encoded instance. The same assertion holds for independent uniform refinements of weak orders, with $D=\prod_{a}\prod_{T\text{ a tie of }a}|T|!$. In either encoding, exact evaluation is in $\mathrm{FP}^{\#\mathrm P}$ and rational-threshold comparison is in PP, including nonstrict comparison $\eta_I(M)\geq q$.
Proof: For explicit lotteries, choose for each applicant a common denominator $d_a$, the product of its listed denominators. Write its masses as nonnegative integers $w_{a,s}/d_a$, so $\sum_s w_{a,s}=d_a$. A nondeterministic machine guesses a fixed-length binary integer $b_a$ for each applicant; reject if $b_a\notin\{0,\ldots,d_a-1\}$. Partition that interval into consecutive blocks of lengths $w_{a,s}$ and assign preference $s$ when $b_a$ belongs to its block. Then each profile has exactly $\prod_a w_{a,s_a}$ surviving branches. Accept precisely those satisfying Lemma Lemma. The number of accepting branches is $N=D\eta_I(M)$ with $D=\prod_a d_a$. All branch lengths and computations are polynomial, since $\log D$ is bounded by input bit length.

For weak orders, guess within each tie a full list of its distinct elements, using a fixed binary representation and rejecting repeated or invalid indices. Every permutation has exactly one valid encoding; concatenating ties gives each strict refinement exactly one valid branch. The denominator is the product of factorials, whose logarithm is polynomial in the stated lists' length. Apply the same verification predicate.

The numerator can be obtained from a #P oracle and divided by $D$. For a rational threshold $q=u/v$, $v>0$, the desired inequality is $vN-uD\geq0$. Counts multiplied by a polynomial-bit nonnegative integer remain #P counts by adding a bounded integer guess; polynomial-bit integers themselves are such counts. Thus $vN-uD$ is a difference of #P functions (a GapP function), with signed $u$ handled by moving its sign. The condition is equivalent to strict positivity of the GapP function $2(vN-uD)+1$. The sign characterization of PP gives membership. Thresholds outside $[0,1]$ may alternatively be handled directly. ∎

These are upper bounds, not completeness claims. The proof does not produce a parsimonious reduction from a known hard counting problem, and no such reduction is asserted.

An exact algorithm on acceptability paths

Consider a connected instance with houses $h_0,\ldots,h_m$ and applicants $a_1,\ldots,a_m$, where $a_i$ accepts exactly $h_{i-1}$ and $h_i$. Preferences are independent and may have arbitrary rational probabilities of the two possible orders. This covers independent explicit lotteries, after consolidating duplicate orders, and weak-order refinements (probability zero, one, or one half). The path has unbounded length and exponentially many possible strict profiles.

Here $R=2$. Use $B=m+1$, so each applicant's pair of weights is either $(B,1)$ or $(1,B)$. For a realized prefix of $i$ applicants, let $D_i(0)$ be the largest weight of a matching in that prefix with $h_i$ unused; let $D_i(1)$ be the largest weight with $h_i$ used. At $i=0$ set $D_0(0)=0$ and $D_0(1)=-\infty$. Let $S_i$ be the weight contributed by the specified matching $M$ on applicants $a_1,\ldots,a_i$.

Lemma (Prefix recursion): Writing $L_i=w(a_i,h_{i-1})$ and $R_i=w(a_i,h_i)$, \begin{align*} D_i(0)&=\max\{D_{i-1}(0)+L_i,D_{i-1}(1)\},\\ D_i(1)&=\max\{D_{i-1}(0),D_{i-1}(1)\}+R_i,\\ S_i&=S_{i-1}+s_i, \end{align*} where $s_i$ is $L_i$, $R_i$, or zero according as $M$ assigns $a_i$ left, right, or not at all.
Proof: If $h_i$ is used, it must be used by $a_i$, which takes its right edge; all previous matching choices remain available. This gives the second equation. If $h_i$ is unused, $a_i$ is either unmatched or takes its left edge. The latter requires $h_{i-1}$ unused and yields $D_{i-1}(0)+L_i$. Leaving $a_i$ unmatched gives the maximum of the two previous states; its $D_{i-1}(0)$ term is dominated by $D_{i-1}(0)+L_i$ because $L_i>0$. This proves the first equation. The third is the definition of the candidate's score. ∎
Theorem (Polynomial exact path evaluation): For the path family just described, exact rank-maximality probability and rational-threshold comparison are polynomial in $m$ and the total rational input length, even though joint support may have size $2^m$.
Proof: Maintain a probability table on triples $(D_i(0),D_i(1),S_i)$. Initialize the table with mass one at $(0,-\infty,0)$. For each applicant and each of its two possible orders, apply Lemma Lemma (Prefix recursion) to each current triple, multiply its mass by that order's probability, and combine equal resulting triples. Independence makes this transition exact: the new order distribution is the same conditional on every previous triple. Induction therefore proves that the table is the joint distribution of the three indicated quantities.

All finite coordinates are integers in $[0,mB]$. There are at most $(mB+2)^3=O(m^6)$ triples, where the extra value permits the initialization symbol $-\infty$. Each of the $m$ stages has at most two transitions per triple, so $O(m^7)$ arithmetic operations suffice. The bound is deliberately conservative; no efficient-state-compression claim is needed.

All probability denominators divide the product of the applicants' input common denominators. That product has polynomial bit length. Numerators are bounded by the denominator because table masses are probabilities. Exact operations therefore have polynomial bit complexity. Finally sum the masses of triples with $S_m=\max\{D_m(0),D_m(1)\}$. Lemma Lemma proves that this is exactly the target probability. ∎

Proposition (Independent connected components): For either independent uncertainty encoding, if acceptability decomposes into components $I_1,\ldots,I_s$, then $\eta_I(M)=\prod_{t=1}^s\eta_{I_t}(M\cap E(I_t))$. Thus disjoint unions of the paths in Theorem Theorem (Polynomial exact path evaluation) also have polynomial exact evaluation.
Proof: A matching is rank-maximal globally exactly when its restriction is rank-maximal in every component. Necessity follows by improving one component while retaining all others. For sufficiency, compare to an arbitrary matching: every component's rank vector is lexicographically no better. If any differs, take the earliest coordinate where any component differs. At that coordinate each differing component contributes a negative difference and no component contributes a positive one, so their sum is negative. The global matching cannot improve. Component events depend on disjoint independent sets of applicants, so their probabilities multiply. Products of rational probabilities of polynomial total encoding length can be computed in polynomial bit time. ∎

Independent finite implementation check

The distribution recursion was checked against a separate exhaustive matching and profile enumeration for every path length $m=1,\ldots,6$, every feasible candidate matching, and all $2^m$ preference profiles. The numbers of candidate matchings were respectively $3,8,21,55,144,377$. The probability that applicant $i$ prefers its left house was $i/(i+2)$ for $i=1,\ldots,m$, and all arithmetic used exact fractions. Every computed probability agreed. This is a check of the implementation over the stated finite scope, separate from the proof for all path lengths.

Remaining gap and source audit

The path recursion works because the boundary has only one house and the two-rank scalar weights are polynomially bounded in $m$. General bounded-rank graphs do not automatically inherit this recursion: a larger separator carries many possible matching states. Conversely, large scalar weights alone do not imply counting hardness, since their bit lengths are small. Establishing #P-hardness or a general polynomial algorithm for either independent encoding, and the corresponding threshold lower bounds, remains the classification gap.

The original PDF [ccm] was inspected on 9 October 2026, including the rank-vector definition on PDF page 5 and the three uncertainty models and final research suggestion on PDF pages 8--9. Those are definition and agenda sources. The verification, counting construction and path recursion above are proved explicitly; neither a source's open table for other solution concepts nor finite tests are used as lower-bound evidence.

Completion Estimate

60%

This is a post hoc, heuristic estimate for the full catalogue target. Listed joint lotteries are solved in polynomial time, independent encodings receive counting and threshold upper bounds, and an exact polynomial algorithm handles unbounded acceptability paths. Hardness reductions or a general polynomial algorithm, together with matching threshold lower bounds, are still needed to classify the two compact independent encodings.

References

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