Behavioral Stable Marriage Problems

Abstract

The stable marriage problem (SMP) is a mathematical abstraction of two-sided matching markets with many practical applications including matching resident doctors to hospitals and students to schools. Several preference models have been considered in the context of SMPs including orders with ties, incomplete orders, and orders with uncertainty, but none have yet captured behavioral aspects of human decision making, e.g., contextual effects of choice. We introduce Behavioral Stable Marriage Problems (BSMPs), bringing together the formalism of matching with cognitive models of decision making to account for multi-attribute, non-deterministic preferences and to study the impact of well known behavioral deviations from rationality on two core notions of SMPs: stability and fairness. We analyze the computational complexity of BSMPs and show that proposal-based approaches are affected by contextual effects. We then propose and evaluate novel ILP and local-search-based methods to efficiently find optimally stable and fair matchings for BSMPs.

Appendices

A Proofs of Theorems

Theorem 1. For BSMPs, StabilityProbability is polynomially solvable.

Proof

This result derives directly from Theorem 1 in [1]. Given the pairwise probabilities induced by the BSMPs, we can compute the probability that a un-matched pair is not blocking in constant time. We then take the product over such pairs which are quadratic in number.

Theorem 2. ExistPossiblyStableMatching is NP-complete even if one side of the market has linear preferences and the other side has weakly stochastic transitive (WST) pairwise probabilities.

Proof

This results strengthens the statement of Theorem 2 in [1] by further restricting the preferences of one side of the market. There the authors reduce from ExistCompleteStableMatching in Stable Matching with Ties and Incompleteness (SMTIs) [19] to ExistPossiblyStableMatching when men have linear preferences and by leveraging the ability to define a cycle of length three of certainly preferred relations in the women’s preferences. WST pairwise probabilities do not allow for cycles of length three comprised of certainly preferred relations. However, they do allow for cycles of length four as the one shown in Fig. 5. This observation allows the proof to proceed in a very similar way as that of Theorem 2 in [1].

For the reader’s convenience, we provide the complete proof below incorporating the extended cycle and associated modifications.

Given Theorem 1, we know that computing StabilityProbability for BSMPs is polynomially solvable. This implies that checking if a matching has a non-zero probability of being stable can be done in polynomial time, and thus the problem is in NP.

To prove NP-hardness, we follow the proof of Theorem 2 in [1] and we reduce from the problem of deciding whether an instance of SMTI admits a complete stable matching. This problem was shown to be NP-complete even if the ties appear only on the women’s side, and each woman’s preference list is either strictly ordered or consists entirely of a tie of size two [19].

Let $M=\{m_1, m_2, \ldots , m_n\}$ and $W=\{w_1, w_2, \ldots , w_n\}$ be the set of men and women in SMTI I. We create an instance of the pairwise probability model $I'$ where women’s preferences are WST as follows. We add 4 men and women: $m_{n+1}, m_{n+2}, m_{n+3}$ and $m_{n+4}$ and $w_{n+1}, w_{n+2}, w_{n+3}$ and $w_{n+4}$. As in [1] we call acceptable partners in I proper partners in $I'$. For each man $m_{i}$, $i\in \{1, ...,n\}$, in the original instance I, we extend his strict preference ordering on his proper partners arbitrarily, by appending the four new women and his unacceptable partners in I in some arbitrary order. For every woman $w_{i}$, $i \in \{1,...n\}$, in I, we create the pairwise preferences as follows. Firstly, $w_i$ prefers every proper partner of hers to every new or unacceptable man. Secondly, $w_i$ prefers each of the 4 new men to unacceptable men in I.

The pairwise preferences of $w_i$ over her proper partners are defined in the same way as in [1]: $w_i$ certainly prefers $m_k$ to $m_l$ if $w_i$ strictly prefers $m_k$ to $m_l$ in I, and if $w_i$ is indifferent between $m_k$ and $m_l$ in I then the corresponding pairwise probability is 0.5 in $I'$.

We then define the pairwise preferences of $w_i$ over the 4 new men as in Fig. 5. More in detail, $w_i$ certainly prefers $m_{n+1}$ to $m_{n+2}$, $m_{n+2}$ to $m_{n+3}$, and $m_{n+3}$ to $m_{n+4}$ and $m_{n+4}$ to $m_{n+1}$, while she is indifferent between $m_{n+1}$ and $m_{n+3}$, and $m_{n+2}$ and $m_{n+4}$. We note that these preferences form a cycle of length 4 and respect WST.

The preferences of $w_i$ over the unacceptable original candidates are arbitrary. Similarly to [1], we let each of the four new men have all the original women at the top of his preference list ordered according to their indices, followed with new women $w_{n+1}$, $w_{n+2}$, $w_{n+3}$ and $w_{n+4}$ (in this order). Moreover, the four new women have $m_{n+1}$, $m_{n+2}$, $m_{n+3}$ and $m_{n+4}$ at the top of their strict preference lists, followed by the original men in an arbitrary order. (Note that every complete linear order implies pairwise probability preferences and satisfies WST). At this point we can show that there exists a complete weakly stable matching in I if and only if there is a matching with positive stability probability in $I'$ following the exact same reasoning as in [1]. To see the first direction, let $\mu$ be a complete weakly stable matching in I. It is easy to see that if we extend $\mu$ with pairs $(m_{n+1}, w_{n+1})$, $(m_{n+2}, w_{n+2})$, $(m_{n+3}, w_{n+3})$ and $(m_{n+4}, w_{n+4})$ then the resulting matching $\mu '$ has positive probability of being stable in $I'$. This is because there is no pair which would be certainly blocking for $\mu '$. Conversely, suppose that $\mu '$ is a complete matching in $I'$ with positive probability of being stable (i.e., it has no certainly blocking pair). It can be shown that every original woman has to be matched with a proper partner. Suppose for a contradiction that $w_i$ is the woman with the smallest index who is not matched to a proper partner. If $w_i$ is matched to an original man who was unacceptable to her in I then $w_{i}$ would form a certainly blocking pair with any of the four new men. In fact, note that $w_i$ certainly prefers either of the four new men to her partner. Moreover, as none of the new men are matched to a original woman with index smaller than i, hence they all certainly prefer $w_{i}$ to their partners. Suppose now that $w_{i}$ is matched to one of the four new men. Then $w_{i}$ would form a certainly blocking pair with the subsequent new man according to her cyclical preference. (For instance, if $w_{i}$ is matched to $m_{n+1}$ in $\mu '$ then she forms a certainly blocking pair with $m_{n+4}$.) This is because the subsequent new man cannot have any better partner, since all the women with smaller indices than i are matched to a proper partner. So we arrive at the conclusion that every original woman is matched with a proper partner. Since $\mu '$ does not admit a certainly blocking pair and all original women are matched with proper partners, the restriction of $\mu '$ to the original agents is a stable and complete matching in I.

Fig. 5.

Example of WST preferences with a cycle of length four. Directed edge represents dominance of source node on target node and the edges are annotated with the probabilities.

Theorem 3. For s -BSMPs, ExistPossiblyStableMatching is polynomially solvable.

Proof

Consider BSMP B where all agents have s-MDFTs. For each man and woman, we extract a linear order from the pairwise probabilities induced by their s-MDFT thus obtaining an stable matching problem I. We then show that a matching is stable in I if and only if it is $\alpha$-B-stable with $\alpha >0$ in B. We illustrate the linearization for man $m_i$ denoting with $Q_i$ his s-MDFT model.

• 1. For every pair such that $P^{Q_i}_{\{w_{k},w_{j}\}}(w_k)>0.5$ we set $w_{k}>_{m_i} w_{j}$ in I. Note that, since the pairwise probabilities induced by an s-MDFt are MST, by doing this we cannot create any cycles in $>_{m_i}$ in I.

• 2. We perform the transitive closure adding all induced order relations.

• 3. At this point the only pairs that may still be not ordered in $m_i$’s preferences in I must be such that $P^{Q_i}_{\{w_{k},w_{j}\}}(w_k)=0.5$. We order such remaining pairs (for example lexicographically) and we proceed in this order to pick one pair, order it in a random way, and then perform transitive closure.

It is easy to see that MST ensures that at the end of this process we obtain a linear order. Moreover each linearization requires polynomial time since in the worst case it performs a transitive closure $O(n^2)$ for each pair linearized in step 3, that is $O(n^2)$ times. Let $\mu$ be a complete stable matching in I. We know one exists [19]. Let’s assume that $\mu$ is 0-B-stable in B. Then it must have a certainly blocking pair (m, w), where m prefers w to $\mu (m)$ and w prefers m to $\mu (w)$ with probability of 1 in B. If $P^{m}_{\{w,\mu (m)\}}(w) =1$ in B then $w >_m \mu (m)$ in I. If $P^{w}_{\{m,\mu (w)\}}(m) =1$ in B then $m >_w\mu (w)$ in I. That is, (m, w) is also a blocking pair in I, thus $\mu$ cannot be stable in I. This is a contradiction.

Theorem 5

MatchingWithHighestStabilityProbability

is NP-hard, even if the certainly preferred relation is transitive for one side of the market and the other side has WST preferences.

Proof

In [1] it is shown that this problem is NP-hard even if the certainly preferred relation is transitive for one side of the market and the other side has linear orders. Our result for WST preferences is orthogonal, as WST does not imply transitive certainly preferred relation and vice-versa. The proof is adaptation of the one of Theorem 3 in [1] and leverages the same cycle described in Fig. 5. Indeed, replacing the cycles of length three with cycles of length four, as the one depicted in Fig. 5, does not affect the reasoning described in the proof. For the reader’s convenience we provide the full details below, clarifying why the key steps still hold.

Following a similar reasoning as in [1], we derive the result by modifiyng the proof of Theorem 2. Let SMTI I and pairwise probability model with WST preferences $I'$ be defined as in Theorem 2. We denote a new instance of a pairwise probability model with WST preferences $I''$ as follows. Whenever some women have cyclic certainly preferred relations in $I'$, we modify the probabilities in these pairwise comparisons by a small value $\epsilon$. That is, whenever a woman $w_i$ certainly prefers man $m_k$ to man $m_l$ within a cycle in $I'$, we modify the probability of the relation to $1-\epsilon$ in $I''$. For example, the probabilities of the perimeter edges in Fig. 5 would be set to $1-\epsilon$. We note that, given how $I'$ is defined, this modification will not cause violations of WST. Thus, we have no certainly preferred relations in any cycle in $I''$. However, as in [1] when considering the matching with the highest stability probability in $I''$, we can still articulate our reasoning along two cases with respect to the original NP-complete problem for I. Let’s first assume that we have a complete stable matching for I. In this case this matching, extended with the four new pairs in $I'$, will have a probability of being stable at least $\frac{1}{2^{n}}$ in both $I'$ and $I''$. This is because every woman who is indifferent between some men has at most one tie of length two in her preference list in I by definition, and so if this woman is matched to one of the men in her tie then only the other man in this tie may block, which happens with 0.5 probability. Note that this step is not affected by the fact that we are using cycles of length four involving only the new men. On the other hand, if there exists no complete stable matching for I then we know from the proof of Theorem 2 that there always existed a certain blocking pair in $I'$. This certain blocking pair will now have a probability of $1-\epsilon$ to be blocking, implying that any matching in this case has less than $\epsilon$ probability of being stable. Therefore, if we choose $\epsilon$ to be $0< \epsilon < \frac{1}{2^{n}}$ we can use an algorithm which solves MatchingWithHighestStabilityProbability to decide the existence of a complete stable matching for SMTI efficiently.

B Algorithms

Algorithms B-GS and EB-GS. As we mentioned in the paper, the Gale Shapley procedure can be extended in a straightforward way to BSMPs by invoking the relevant MDFT models when a proposal or an acceptance has to be made. When man m is proposing, model $Q_m$ will be run to select the woman to propose to among the set of women to whom m has not proposed yet. In fact, an MDFT model can be run on any subset of options by simply removing irrelevant rows from the personal evaluation matrix and resizing the other matrices (contrast and feedback). Similarly, when woman w, currently matched with man $\sigma (w)$ receives a proposal from m, the choice will be picked by running $Q_w$ on the set $\{m, \sigma (w)\}$. We call this variant of GS, Behavioral Gale Shapley, denoted with B-GS. While it is clear that B-GS still converges, since the sets of available candidates shrink by one every time a proposal is made, it is no longer deterministic and may return different matchings when run on the same BSMP. This is, of course, a consequence of the non-determinism of the underlying MDFT models.

We can also define another variant of GS that we call Expected Behavioral Gale Shapley (EB-GS). We first note that, given a man, we can extract a linear order from the expected positions of the women according to his MDFT model (breaking ties if needed). EB-GS corresponds to running GS on the profile of linear orders obtained in this fashion.

Algorithm FB-ILP. For each combination of $m_i \in M$ and $w_j \in W$, $|M| = |W| = n$, we introduce a binary variable $m_iw_j$ that takes value 1 if $m_i$ is matched with $w_j$ and 0 otherwise. We assume that for FB-ILP we have access to an $n \times n$ matrix $pos_M[i,j]$ where entry i, j gives us the expected position of $w_j$ in the ranking of $m_i$, and the same matrix is available for the women, denoted $pos_W$.

Recall that finding the solution with lowest sex equality cost requires minimizing $SEC = | \sum _{i,j \in n} pos_M[i,j] \cdot m_iw_j - \sum _{i,j \in n} pos_W[j,i] \cdot m_iw_j |$. We cannot implement this absolute value directly as the optimization objective in Gurobi [13] as it is non-linear due to the presence of the absolute value. Since the SECs are always $\ge 0$ we can overcome this using a standard trick in ILPs using indicator variables [3]. The SEC objective can be viewed as adding up the total man cost and the total woman cost, so we add indicator variables $tmc \ge 0$ and $twc \ge 0$ and minimize the difference between these two quantities. Hence, our full FB-ILP can be written as follows.

min	ind, s.t.,
(1)	$\sum _{j \in n} m_iw_j = 1$	$\forall i \in n$
(2)	$\sum _{i \in n} m_iw_j = 1$	$\forall j \in n$
(3)	$\sum _{i,j \in n} m_iw_j = n$
(4)	$\sum _{i,j \in n} pos_M[i,j] \cdot m_iw_j = tmc$
(5)	$\sum _{i,j \in n} pr_W[j,i] \cdot m_iw_j = twc$
(6)	$twc \ge 0$
(7)	$twc \ge 0$
(8)	$twc - tmc = ind$

In the constraints above (1) and (2) ensures that every man $m_i$ has exactly one match across all possible women and every woman $w_j$ has one match across all possible men. The redundant constraint (3) ensures that we have exactly n matches, i.e., everyone is matched. Constraint (4) captures the total cost to the men by multiplying the expected position by the indicator variables for the matches. Likewise constraint (5) captures the total woman cost. Constraint (8) is necessary to ensure that Gurobi handles our absolute value constraint correctly. We know that both $tmc \ge 0$ and $twc \ge 0$ from constraints (6) and (7), hence when Gurobi uses the Simplex Algorithm to solve, it will set $tmc = ind$ and $twc = 0$ if $ind > 0$ and otherwise we will have $tmc = 0$ and $tmc = -ind$. In either case we have a bounded objective function and we can find a solution if one exists.

Algorithm B-ILP. To find the optimal $\alpha$-B-Stable solution with B-ILP, we begin with the same setup. For each $m_i \in M$ and $w_j \in W$ we introduce a binary variable $m_iw_j$ defined as above. In addition, for B-ILP we assume that for each man and each woman we are given an $n \times n$ matrix $Pr_{m_i}$ where entry $Pr_{m_i}[j,k]$ gives the probability that man $m_i$ prefers $w_j$ to $w_k$. This matrix can be computed by running the BSMP of man $m_i$ a sufficiently large number of times.

There are two interrelated complications with formulating this probabilistic matching problem as an ILP: first we need the product of the probabilities which is a convex not linear function, and, second, stability is a pairwise notion over a given matching. To deal with both of these issues we introduce $\forall ((i,j),(k,l)) \in \left( {\begin{array}{c}\left( {\begin{array}{c}n\\ 2\end{array}}\right) \\ 2\end{array}}\right)$ possible combinations of pairs of pairs, an indicator variable $m_iw_j+m_kw_l$ to indicate that both $m_iw_j$ is matched and $m_kw_l$ is also matched. This allows us to compute the blocking probability of $m_i$ and $w_l$ as well as of $m_k$ and $w_j$. Given the formulation in [2], we know that we want to maximize the probability that no blocking pair exists. Hence for every pair of possible marriages $m_iw_j+m_kw_l$ we can compute the probability that these four individuals are not involved in blocking pairs by taking the likelihood that they swap partners, formally let $block[(ij),(kl)] = (1 - Pr_{m_i}[l,j] * Pr_{w_l}[i,k]) * (1 - Pr_{m_k}[j,l] * Pr_{w_j}[k,i])$. To handle the convex constraint we simply take the $\log$ of this quantity and maximize using an indicator variable we which we implement using the Gurobi And constraint. We can write the full program as follows.

max	$\sum _{ \forall ((i,j),(k,l)) \in \left( {\begin{array}{c}\left( {\begin{array}{c}n\\ 2\end{array}}\right) \\ 2\end{array}}\right) } pair_{m_iw_j+m_kw_l} * log(block[(ij),(kl)]), s.t.,$
(1)	$\sum _{j \in n} m_iw_j = 1$	$\forall i \in n$
(2)	$\sum _{i \in n} m_iw_j = 1$	$\forall j \in n$
(3)	$\sum _{i,j \in n} m_iw_j = n$
(4)	$AND(m_iw_j,m_kw_l) = pair_{m_iw_j+m_kw_l}$	$\forall ((i,j),(k,l)) \in \left( {\begin{array}{c}\left( {\begin{array}{c}n\\ 2\end{array}}\right) \\ 2\end{array}}\right)$

In the constraints above (1) and (2) ensures that every man $m_i$ has exactly one match across all possible women and every woman $w_j$ has one match across all possible men. The redundant constraint (3) ensures that we have exactly n matches, i.e., everyone is matched. Constraint (4) uses the Gurobi [13] AND constraint to set the value of $pair\_{m_iw_j+m_kw_l}$ to be 1 if and only if both $m_iw_j$ and $m_kw_l$ are both 1. This allows us to capture all possible pairs of man/woman pairs and maximize the probability that no blocking pair occurs.

C Convergence Analysis for B-LS

The convergence analysis performed for $n=16$ is shown in Fig. 6. While we depict the results of for seven runs we performed a total of 50 runs. The results indicated that B-LS plateaus after 300 iterations, corresponding to approximately 340 s. B-LS does so around 88% of the time and returns a matching $1.006*10^{-6}$ far from optimal otherwise.

Fig. 6.

Convergence of B-LS algorithm implementation with respect to $\alpha$-B-Stability when $n = 16$.