MRet: Matching for Retention in Platforms

• The paper introduces MRet, a dynamic learning-to-rank algorithm that maximizes incremental retention gain through personalized retention curves and concavity-based lower bounds.
• It leverages user-specific retention models learned from profile and interaction history to allocate matching opportunities where they yield the highest retention gain.
• Empirical evaluations on synthetic and real-world platforms show that MRet achieves higher retention with fewer matches compared to traditional match maximization and fairness-based methods.

Matching for Retention (MRet) is a dynamic learning-to-rank algorithm introduced for two-sided matching platforms such as online dating and recruiting, where the optimization target is user retention rather than the total number of matches or an axiomatic fairness criterion. Its central premise is that platforms should learn how many matches each user needs to remain on the platform, then allocate limited matching opportunities where they produce the largest expected retention gain for both the user receiving recommendations and the users being recommended. In this formulation, retention is treated as the primary objective and fairness as, at most, an indirect proxy (Kishimoto et al., 17 Feb 2026).

1. Problem setting and objective

MRet is defined for a platform with two user sets, $\mathcal{X}$ and $\mathcal{Y}$. At each time step $\tau$, a user arrives from one side, say $x \in \mathcal{X}$, and the platform recommends an ordered list of $K$ users from the opposite side,

$$\sigma_\tau = [\sigma_{\tau,1}, \sigma_{\tau,2}, \dots, \sigma_{\tau,K}], \qquad \sigma_{\tau,k} \in \mathcal{Y},$$

with position-dependent visibility weights

$$1=\alpha_1 \ge \alpha_2 \ge \dots \ge \alpha_K \ge 0.$$

If $r(x,y)\in[0,1]$ denotes the match probability between $x$ and $y$, then the expected matches obtained by the receiver at time $\mathcal{Y}$0 are

$\mathcal{Y}$1

and cumulative matches up to time $\mathcal{Y}$2 are

$\mathcal{Y}$3

The defining departure from conventional reciprocal recommendation is that MRet does not optimize $\mathcal{Y}$4 directly. Instead, it maximizes the incremental retention probability induced by the recommendation on both sides of the interaction:

$\mathcal{Y}$5

Here $\mathcal{Y}$6 is the probability that user $\mathcal{Y}$7 stays on the platform after receiving $\mathcal{Y}$8 matches. The objective formalizes the intuition that a recommendation affects retention on both sides: the receiver may remain active if they obtain enough matches, while the recommended user may also benefit from reciprocated interaction (Kishimoto et al., 17 Feb 2026).

This problem statement is motivated by a specific critique of existing platform objectives. Match maximization tends to concentrate exposure on already popular users, while many others receive too few matches and eventually abandon the platform. Fairness methods can reduce imbalance, but fairness is not itself the ultimate objective for most platforms, because equalizing exposure does not guarantee that the right users receive the right number of matches needed to stay.

Paradigm	Optimization target	Limitation emphasized in the literature
Match maximization	Short-term expected matches	Exposure concentrates on already popular users
Fairness methods	Exposure parity or exposure-to-merit parity	Retention is optimized only indirectly
MRet	Expected retention gain on both sides	Requires learned retention curves and match estimates

2. Personalized retention curves

The retention component of MRet is user-specific. Rather than assuming a single global relationship between matches and churn, the method learns a personalized retention curve for each user from profile and interaction history. In the paper’s notation, this is the estimation of $\mathcal{Y}$9 from data. The training data are triples

$\tau$0

where $\tau$1 denotes user features or context, $\tau$2 is cumulative match count, and $\tau$3 is a retention label (Kishimoto et al., 17 Feb 2026).

A key theoretical assumption is that, for every user $\tau$4, the retention function $\tau$5 is concave in the number of matches:

$\tau$6

This encodes diminishing marginal retention gains: the first match matters more than the tenth. The assumption is not merely behavioral; it is the condition that makes the subsequent ranking approximation tractable.

In synthetic experiments, the retention model is generated from a user-specific concave function with a satisfactory match count $\tau$7:

$\tau$8

This construction makes retention saturate once the user has received a satisfactory number of matches.

In the real-world setting, exact retention labels for arbitrary match counts are unavailable for the users in the matching matrix, so the paper estimates retention curves through a multi-stage procedure. Users are clustered by profile features using k-means; empirical average retention is computed for each cluster and match-count level; this is treated as the cluster-specific retention function; and an XGBoost predictor is then trained to generalize from $\tau$9 to retention probability. The appendix specifies XGBClassifier with 200 trees, max depth 6, and learning rate 0.05. The resulting retention model is therefore a supervised reward regression or classification component embedded inside the ranking pipeline (Kishimoto et al., 17 Feb 2026).

3. Concavity-based ranking approximation

The exact joint optimization problem is NP-hard, because the receiver-side term couples all positions in the slate through cumulative exposure, and the recommended-user term introduces an additional cross-user dependency. MRet addresses this by deriving lower bounds under the concavity assumption, then converting the problem into a decomposable sorting rule (Kishimoto et al., 17 Feb 2026).

For the receiver side, the paper proves a Jensen lower bound:

$x \in \mathcal{X}$0

where

$x \in \mathcal{X}$1

For the candidate side, it proves a linear lower bound:

$x \in \mathcal{X}$2

with

$x \in \mathcal{X}$3

Combining these bounds yields an objective of the form

$x \in \mathcal{X}$4

where $x \in \mathcal{X}$5 is a per-candidate quantity with a receiver-side retention-benefit term and a recommended-user retention-benefit term. The ranking rule is then

$x \in \mathcal{X}$6

By the rearrangement inequality, this maximizes the lower bound.

The computational consequence is that the combinatorial ranking problem is reduced to per-candidate score computation followed by sorting, with complexity

$x \in \mathcal{X}$7

for $x \in \mathcal{X}$8 candidates. This is the core algorithmic reason MRet is presented as a dynamic LTR method rather than as an intractable joint slate optimizer.

4. Dynamic operation and empirical behavior

MRet is dynamic in the literal sequential sense. At each round, a user arrives; candidate scores are recomputed from current cumulative match counts and the current retention model; a ranking is produced; observed interactions update cumulative matches; and retention estimates evolve over time through the changed state variables. The method is therefore not a static re-ranker but a sequential policy whose scores depend on current user state (Kishimoto et al., 17 Feb 2026).

The synthetic evaluation uses

$x \in \mathcal{X}$9

with popularity skew parameter $K$0, recommendation ratio $K$1, and visibility weights $K$2. Match probabilities are generated by

$K$3

where

$K$4

Baselines include Max Match, Uniform, FairCo, and an oracle-style MRet (best) that uses the ground-truth retention function.

The real-world evaluation uses interaction data from a major Japanese online dating platform, with 1,000 male and 1,000 female users for the matching matrix and a separate dataset of 60,000 records for estimating retention curves. The retention label is whether a user logs in again in the following month, and missing values in the match matrix are imputed using ALS (Kishimoto et al., 17 Feb 2026).

Across synthetic and real-world experiments, the reported pattern is consistent. Max Match obtains the most matches, but not the best retention. In synthetic data, Uniform and Max Match can have similar retention, which shows that maximizing matches alone does not solve churn. FairCo can reduce extreme imbalance, but it does not align exposure with the satisfactory match threshold each user actually needs for retention. MRet achieves the highest user retention across synthetic and real-world experiments, often with fewer matches than Max Match. The paper further reports robustness to stronger popularity skew, noise in match probabilities, changing popularity over time, different examination or position-bias profiles, and different group recommendation ratios. In a small-scale setting, MRet nearly matches the exhaustive-search optimal policy, which supports the quality of the lower-bound approximation (Kishimoto et al., 17 Feb 2026).

A central empirical interpretation is that MRet allocates matches closer to each user’s satisfactory level $K$5 rather than merely spreading exposure or concentrating it on already popular users. The case-study analysis uses the gap $K$6 to show that many users under MRet receive at least the number of matches needed to stabilize retention.

5. Relation to adjacent retention-aware recommendation

MRet belongs to a broader line of work in which retention is optimized directly rather than treated as a by-product of click or match objectives, but its specific formulation is tied to two-sided matching. In adjacent recommender-system work, retention is modeled as a session-level terminal reward rather than as a function of cumulative reciprocal matches. A notable example is GFN4Retention, which regards retention as an overall estimate of end-of-session satisfaction and uses detailed-balance flow matching to back-propagate the retention reward toward each recommendation decision in the session (Liu et al., 2024).

The difference in granularity is substantial. In GFN4Retention, the sample is organized as

$K$7

where $K$8 is a terminal retention reward. The integrated reward is

$K$9

and the method learns a trajectory distribution whose probability mass matches the retention reward through a detailed-balance objective. That framework is a retention-aware policy-learning method for session-wise recommendation. MRet, by contrast, is a dynamic LTR method for two-sided matching, with per-user retention curves $$\sigma_\tau = [\sigma_{\tau,1}, \sigma_{\tau,2}, \dots, \sigma_{\tau,K}], \qquad \sigma_{\tau,k} \in \mathcal{Y},$$0, cumulative match counts as the state variable, and a ranking objective that jointly values retention gains on both sides of a potential match.

This suggests a useful taxonomy. Some retention-aware methods allocate credit backward across recommendation trajectories, while MRet allocates limited reciprocal recommendation opportunities according to predicted marginal retention gains. The shared theme is direct retention optimization; the operational object differs between a sequential recommendation session and a two-sided match slate.

6. Scope, limitations, and terminological ambiguity

MRet is explicitly a matching algorithm, but the term “retention” is used very differently across research areas, and this has created predictable ambiguity. In survival analysis on teacher retention, for example, retention is modeled through expected remaining tenure and equilibrium processes. The paper on Starshaped Mean Residual Life models is fundamentally about teacher retention in rural STEM districts, and it states explicitly that it does not present a matching method, retention matching, propensity-score matching, pair matching, or any algorithm that would normally be called MRet (Sepehrifar, 17 May 2026).

In LLM and continual-learning research, “retention” often refers to memory persistence rather than platform survival. The “Retention Layer” paper uses retention to denote a persistent, writable memory module for Transformer-based architectures, with memory attention, episodic buffers, retrieval, and update across sessions (Yaslioglu, 15 Jan 2025). The Multi-Layer Memory Framework likewise studies long-term context retention through working, episodic, and semantic memory, adaptive retrieval gating, and a retention regularization term

$$\sigma_\tau = [\sigma_{\tau,1}, \sigma_{\tau,2}, \dots, \sigma_{\tau,K}], \qquad \sigma_{\tau,k} \in \mathcal{Y},$$1

to stabilize semantic memory across sessions (Tiwari et al., 31 Mar 2026). ATLAS uses the term in the continual-learning sense of mitigating catastrophic forgetting through local support, sparse gradients, and distal orthogonality (Deventer et al., 2022).

Within two-sided matching itself, MRet carries several explicit assumptions. Match probabilities $$\sigma_\tau = [\sigma_{\tau,1}, \sigma_{\tau,2}, \dots, \sigma_{\tau,K}], \qquad \sigma_{\tau,k} \in \mathcal{Y},$$2 are assumed known or estimated upstream. Retention functions are learned offline rather than adaptively online in the current formulation. The main theory assumes concavity of retention in matches. The objective treats both sides symmetrically, even though real platforms may care more about one side than the other. The paper therefore points to a practical extension: side-specific weights in the retention objective for settings where retaining paying users, employers, or other scarce-side participants matters more than symmetric retention maximization (Kishimoto et al., 17 Feb 2026).

Under this narrower and explicit meaning, Matching for Retention denotes a retention-optimized ranking method for two-sided platforms: it learns personalized retention curves, estimates the incremental retention gain induced by each candidate recommendation on both sides of the market, approximates an NP-hard slate objective through concavity-based lower bounds, and ranks candidates accordingly. Its significance lies in making retention, rather than matches or fairness, the direct object of optimization.