Pooled Replacement-Token RAPM

This diagnostic estimates a replacement-player group effect directly in the RAPM design matrix. It corrects a failed first attempt that averaged the separately ridged coefficients of low-exposure players; those individual coefficients were mechanically shrunk toward zero and cannot estimate the group's impact.

Token Design

Within each completed regular season, identify cataloged player IDs with a realized team-possession share below a cutoff. Replace every such individual player token with one shared replacement token. For stint \(t\), its signed feature is

where \(n\) is the number of low-exposure players in the lineup. All other players retain their ordinary signed RAPM token. The fitted equation is

Thus \(\beta_{replacement}\) is the per-player RAPM coefficient applied once for each replacement token in a lineup. It pools the group's possession exposure instead of regularizing hundreds of sparse individual columns independently toward zero.

The model reuses the canonical season-specific RAPM lambda. This isolates the tokenization change; it does not claim that the existing lambda is optimal for the lower-dimensional token design.

Historical Result

The primary immutable run is artifacts/models/replacement_token/2025-26/replacement-token-2025-26-20260806T010704Z-4863a48f/. It fits one full-season token model for each season from 1996-97 through 2025-26, using a 5% realized exposure cutoff.

At the 5% cutoff, the mean of the original separately ridged player coefficients was only -0.247. The pooled token therefore demonstrates the expected shrinkage failure directly: low-exposure individual estimates are not an estimate of low-exposure group quality.

The pooled estimate is more negative than the usual informal -2 replacement benchmark. It is a valid consequence of this exact retrospective group and this RAPM scale, but it should not yet be promoted as the project-wide replacement level. The cutoff is based on realized same-season exposure, and the low-minute pool can contain injury cases alongside two-way, 10-day, and fringe-roster players.

What This Establishes

This is the correct measurement strategy for the proposed group: represent replacement candidates with a shared feature and let the lineup model estimate their pooled effect. It is not a preseason cold-start model because group membership uses future, realized exposure.

The next predictive slice is to learn a preseason-only gate that assigns a replacement token before the season, then evaluate that gate on a complete future-season holdout. Contract, transaction, and roster-status data would make the historical group definition more precise.

Artifacts