House exchanges along longer social-network cycles

MM-09 |  Matching markets and allocation

Research agenda formalization

Definitions and assumptions

Let $A,H$ have the same finite size, $G=(A,F)$ be an undirected social network, and $M_0:A\to H$ an initial bijective endowment. Each applicant strictly ranks a fixed set of acceptable houses containing her endowment. A feasible allocation is a bijection assigning each applicant an acceptable house. Fix $k\ge3$. A legal cyclic trade from $M$ involves distinct applicants $a_1,\ldots,a_\ell$, where $2\le\ell\le k$, consecutive applicants are adjacent in $G$ (indices modulo $\ell$), and \[ M(a_{i+1})\succ_{a_i}M(a_i)\quad(1\le i\le\ell). \] The resulting allocation gives $a_i$ the house $M(a_{i+1})$ and leaves all others unchanged. This explicitly chooses network-edge feasibility; requiring all participants to form a clique defines another model.

Formal question

Classify the complexity of deciding reachability of a specified allocation, or a specified applicant--house assignment, from $M_0$ under these trades. Also determine the complexity of finding $M\in\mathcal R_k(M_0)$ that is Pareto optimal within the reachable set: \[ \nexists N\in\mathcal R_k(M_0): [\forall a,\ N(a)\succeq_a M(a)]\ \land\ [\exists a,\ N(a)\succ_a M(a)]. \] Distinguish fixed $k$, input $k$, and unbounded cycle length.

Scope and status

The source highlights longer-cycle exchanges as an open direction beyond swaps. It does not specify a unique feasibility convention for longer cycles; the convention above is therefore part of this formalization. Strict improvement implies termination of every trade sequence, but terminality is only absence of a further legal local trade. Pareto optimality in the entire set reachable from $M_0$ is a stronger requirement. The known swap results do not automatically classify longer-cycle models, especially when network classes contain no simple cycles of length at least three.

References for the source of the problem

Katarína Cechlárová, Ágnes Cseh, and David F. Manlove (2019), Selected open problems in Matching Under Preferences, Section 3.1.3, final paragraph: https://hpi.de/friedrich/docs/publications/2019/BEATCS.pdf. Laurent Gourvès, Julien Lesca, and Anaëlle Wilczynski (2017), Object Allocation via Swaps along a Social Network, IJCAI 2017, pp. 213--219, defines the motivating swap model: https://doi.org/10.24963/ijcai.2017/31.

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.