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Econ-241 · economics · computer science · Computational economics

Stable-roommates robustness to preference perturbations

LLM status Open unverified

Statement

MM-20 |  Matching markets and allocation

Research agenda formalization; corrected scope

Definitions and assumptions

A complete strict stable-roommates instance $I$ consists of an even number of agents, each strictly ranking all other agents. A matching partitions the agents into pairs and is stable if no non-matched pair mutually prefers one another to its assigned partners. An adjacent swap interchanges consecutive names in one agent's preference list. Let $\mathcal B_d(I)$ be all instances obtainable using at most $d$ such swaps across all lists. A matching is $d$-robust if \[ M\in\bigcap_{J\in\mathcal B_d(I)}\mathcal S(J), \] where $\mathcal S(J)$ is the set of stable matchings for $J$. For a separately specified distribution $\nu$ over perturbations, define average-case robustness as $r_\nu(M)=\Pr_{J\sim\nu}(M\in\mathcal S(J))$, with $M\in\mathcal S(I)$ required.

Formal question

Develop existence criteria and computational classifications for $d$-robust stable-roommates matchings, for verification of a specified matching, and for maximizing $r_\nu(M)$. Specify the perturbation representation, swap budget encoding, and any structural restrictions. For weighted objectives, determine the complexity of finding a minimum-cost matching among those satisfying the chosen robustness condition.

Scope and status

The source's unstudied-roommates statement concerns robustness to changes in preferences, in the senses of Mai--Vazirani and Chen--Skowron--Sorge. Pair-failure robustness in roommates was already studied by Genc and coauthors and is covered separately in MM-18--MM-19. The original catalog's blanket "robust roommates had not been studied" wording is therefore corrected. The formal questions above instantiate a published research agenda, rather than assert that every specified perturbation variant is currently open. Worst-case robustness and stability probability are distinct objectives; $d=0$ reduces to ordinary stable-roommates solvability, and any input lacking an ordinary stable matching is immediately infeasible.

References for the source of the problem

Katarína Cechlárová, Ágnes Cseh, and David F. Manlove (2019), Selected open problems in Matching Under Preferences, Section 3.3.2, final paragraph and preceding perturbation definitions: https://hpi.de/friedrich/docs/publications/2019/BEATCS.pdf. Jiehua Chen, Piotr Skowron, and Manuel Sorge (2019), Matchings under Preferences: Strength of Stability and Trade-offs, EC 2019, robust-stability definition: https://arxiv.org/abs/1902.10535. Begum Genc, Mohamed Siala, Gilles Simonin, and Barry O'Sullivan (2019), An Approach to Robustness in the Stable Roommates Problem and Its Comparison with the Stable Marriage Problem, provides the distinct pair-failure theory: https://siala.github.io/preprints/19-CPAIOR.pdf.

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-241-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-241

Stable-roommates robustness to preference perturbations: An exact swap radius and its sharp universal bound

Worst-case robustness

There are $N\ge4$ agents, with $N$ even, complete strict preference lists, and a fixed perfect matching $M$. The canonical robustness condition is \[ M\in\bigcap_{J:\,\text{at most }d\text{ adjacent swaps from }I} \mathcal S(J). \tag{1} \] The perturbation agenda is distinguished from pair failures in [survey, Section 3.3.2]; quantitative swap robustness for marriage is studied in [strength]. We analyze (1) directly for roommates. The resulting verification formula and sharp radius bound do not solve weighted search over all robust matchings or stability probabilities under arbitrary distributions of perturbations.

Let $r_i(j)\in\{1,\ldots,N-1\}$ denote the original rank of $j$ in $i$'s list, with smaller ranks better. For an unmatched pair $\{i,j\}$, define \[ c_i(j;M)=\max\{0,r_i(j)-r_i(M(i))\},\qquad s_I(M)=\min_{\{i,j\}\notin M} \{c_i(j;M)+c_j(i;M)\}. \tag{2} \]

A complete verification certificate

Theorem: The minimum total number of adjacent swaps needed to make $M$ unstable is exactly $s_I(M)$. Consequently $M$ is $d$-robust if and only if $s_I(M)>d$. After rank preprocessing, verification and a shortest destabilizing swap sequence take $O(N^2)$ time plus the output length. The formula also detects original instability through $s_I(M)=0$.
Proof: Consider one list and two distinct names $j$ and $M(i)$. If $j$ is already above $M(i)$, no change is necessary. Otherwise their rank distance is $r_i(j)-r_i(M(i))$. To reverse their relative order, they must cross by an adjacent interchange. Before that crossing, the names between them have to be moved past one of the two, requiring at least their distance minus one swaps; the crossing is one more. Equivalently, in the final order every intermediate name has changed its relative order with at least one of the endpoints, in addition to the endpoints' own inversion. Each adjacent swap changes exactly one pairwise order. The lower bound is attained by moving $j$ upward to just above $M(i)$. The required cost is therefore $c_i(j;M)$.

A blocking pair requires the two corresponding strict comparisons, in two different agents' lists. Their swap costs add, and swaps in other lists cannot reduce either requirement. Thus any perturbation creating this blocker costs at least $c_i(j;M)+c_j(i;M)$, and the two upward moves attain that cost simultaneously. Every instability has some blocking pair; minimizing over those pairs proves (2). Scan all $O(N^2)$ pairs and retain a minimizer to obtain the algorithm and its explicit certificate. ∎

The largest possible radius

Theorem: For every strict complete profile and every perfect matching, \[ s_I(M)\le N-1. \tag{3} \] For each even $N\ge4$, there is a profile and a matching attaining $s_I(M)=N-1$. Hence the largest attainable integer robustness budget is $N-2$, and no matching can satisfy (1) when $d\ge N-1$.
Proof: For fixed $i$, writing $r=r_i(M(i))$ gives \[ \sum_{j\ne i,M(i)}c_i(j;M) =\frac{(N-1-r)(N-r)}2 \le\frac{(N-2)(N-1)}2. \] Summing over agents counts the total cost in (2) over the $N(N-2)/2$ unmatched unordered pairs. Their average is at most $N-1$, so their minimum is at most $N-1$.

For sharpness, fix any perfect matching $M$ and let $G$ be the complete graph with its matching edges deleted. It is regular of even degree $N-2=2k$. Decompose its edges into $k$ disjoint spanning unions of cycles. Here is a proof that such a decomposition exists. Orient an Euler tour in each connected component of this even-degree graph, giving every vertex $k$ incoming and $k$ outgoing edges. Form a bipartite graph with a left and right copy of every vertex and an edge from the left tail to the right head of each oriented edge. It is $k$-regular. For every left subset $S$, its $k|S|$ incident edges enter at most $k$ times each right neighbor, so $|\Gamma(S)|\ge|S|$. Hall's theorem gives a perfect matching. Remove it and repeat to obtain $k$ perfect matchings. Each corresponds to directed cycles covering every vertex in the original graph, as required.

Number these cycle factors $h=1,\ldots,k$. Rank $M(i)$ first for every agent. In factor $h$, give $i$'s outgoing neighbor cost $h$ and its incoming neighbor cost $N-1-h$, meaning rank one plus that cost. These $2k$ costs are precisely $1,\ldots,N-2$, so they define a strict complete list. Every unmatched edge is outgoing at one endpoint and incoming at the other within one factor; its two costs sum to $N-1$. Equation (2) gives $s_I(M)=N-1$. In particular $M$ is originally stable because everyone has its first choice. ∎

A finite search formulation, without a polynomial-time claim

The radius formula also gives a polynomial-size integer feasibility formulation. Introduce $x_{ij}=x_{ji}\in\{0,1\}$ for $i\ne j$ with $\sum_{j\ne i}x_{ij}=1$. For distinct prospective blockers $i,j$ and prospective partners $k\notin\{i,j\}$, $l\notin\{i,j\}$, define \[ c_i(j;k)=\max\{0,r_i(j)-r_i(k)\}. \] Whenever $c_i(j;k)+c_j(i;l)\le d$, impose \[ x_{ik}+x_{jl}\le1. \tag{4} \] There are $O(N^4)$ such inequalities. Inconsistent partner choices, such as $k=l$, are already excluded by the matching equations.
Proposition: The integer solutions of these constraints are exactly the $d$-robust perfect matchings. Adding any rational linear edge-cost objective gives an exact formulation of the weighted robust search problem.
Proof: If a matching violates robustness, its minimizing pair in (2), together with the two assigned partners, yields a violated inequality (4). Conversely, a violated inequality selects two partners whose corresponding non-matched pair has destabilization cost at most $d$, contradicting (1). The edge objective simply evaluates the selected matching. ∎
This is an integer formulation, not an integrality theorem or an efficient algorithm for solving it. Average-case stability depends on overlaps between blocking events and on the declared perturbation distribution; the minimum distance alone does not determine that probability. These search and probability questions remain open in this attempt.

Completion Estimate

60%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt gives an exact polynomial verification formula for swap robustness, a sharp universal radius bound and an exact weighted integer formulation. Existence and computational classifications for finding robust roommates matchings, and optimization of stability probabilities under specified perturbation laws, are still missing.

References

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