# BI-01-RA1: risk adjustment changes the missing-care result

23 October 2026. Derived results in a stipulated finite model. This is a sensitivity analysis of BI-01-L1, not a new general economic theorem, empirical insurer evaluation, clinical study or adopted PTI rule.

## Model and reference risk

Preserve the two equiprobable case groups, actions, payments, costs, review process, information regimes, binding ex-ante enrollment, outside option, consequence cap, protection bound and current budget of [the original model](../finite-model-and-proof.md). All opportunities are willing, appropriate and feasible; valid refusal, escalation, capacity shortages and lawful exit remain unmodeled. There is zero later discovery outside review.

For an accepted case, replace only the normalized continuation increment by

\[
\Delta V_g(Y)=\frac{\lambda}{2}(Y-\widehat m_g),\qquad \lambda\geq0.
\]

Withholding has zero continuation increment. The benchmark \(\widehat m_g\in[0,1]\) is fixed before the acceptance/effort decision. It represents expected success under a **prespecified reference effort**, not under whatever effort the provider subsequently chooses. The primary positive control uses high-effort success: \(m_E=19/20,m_H=3/5\). These are oracle probabilities stipulated by the model, not estimated from selectively completed cases.

The raw model is exactly nested by \(\widehat m_E=\widehat m_H=9/10\): its normalized expected utilities are identical. Current payments remain unadjusted; this experiment changes the continuation reward, not both instruments at once. For action \(e\), write

\[
A_{ge}=b+s p_{ge}-c_e+\frac{\lambda}{2}(p_{ge}-\widehat m_g).
\]

At a choice information state with review probability \(c\), acceptance utility is \(A_{ge}-cfK\) and withholding utility is \(-cdK\). Unknown review has \(c=q=1/2\); known review has \(c=0\) or \(1\), each with positive probability. Review assignment is exogenous; realized actions may depend on revealed assignment.

Because the same benchmark is subtracted from both accepted actions, effort thresholds remain \(3/5\) for H and \(14/15\) for E. The benchmark is not an operational PTI probability. Adding the centered residual to the old score can leave its domain; a bounded/nonlinear score-to-opportunity rule would need a new analysis. Nor is changing the benchmark fiscally neutral by definition: it changes conditional continuation compensation. Future obligations are outside the current-period budget, as in the original model.

## Proposition 1: the correct-reference positive control

Under the original parameters, with \(\widehat m_g=m_g=p_{g,high}\), strict acceptance in every information state is feasible for every \(\lambda\geq0\), under either review regime. In particular \(K=0\) suffices. The feasible region is unbounded in this abstract continuation multiplier.

**Proof.** The available high-effort surpluses are the strictly positive constants \(A_{E,high}=19/100\) and \(A_{H,high}=3/25\). The best accepted action is at least as good. At \(K=0\), withholding has zero utility, so acceptance is strict regardless of review information. Mean enrollment surplus is at least \(31/200>0\); the protection cost is zero. Current payment plus review cost is at most \(111/200+1/40=29/50<3/4\). Thus every nonnegative multiplier is feasible. Under the fixed comparison contract \(K=1/5\), the acceptance–withholding difference adds \(c(d-f)K\geq0\); mean utility remains at least \(31/200-1/100=29/200>0\). ∎

This result depends on positive current care margins and the affine, risk-neutral reward. It is not a universal access guarantee. If base payment is reduced from \(2/5\) to \(1/5\), correct high-effort centering gives H low-effort surplus \(-1/50-\lambda/10\) and high-effort surplus \(-2/25\). In the known-unreviewed state every accepted action is then worse than withholding for every nonnegative multiplier.

## Proposition 2: a path from correct adjustment to the original bias

Let \(0\leq\theta\leq1\), and define

\[
\widehat m_E=\frac{19}{20}-\frac{\theta}{20},\qquad
\widehat m_H=\frac35+\frac{3\theta}{10}.
\]

Zero is correct high-effort centering; one is the original common-prior score. This is a controlled error path, not an estimated distribution of real-world model errors.

For \(\theta>0\), weak full acceptance under the allowed choice of \(K\) exists exactly through

\[
\lambda_{unknown}\leq\frac{19}{15\theta},\qquad
\lambda_{known}\leq\frac{4}{5\theta}.
\]

Strict acceptance in every information state is attainable below, but not at, these endpoints. At \(\theta=0\), Proposition 1 applies; the boundary is unbounded, not an undefined or zero value.

**Proof.** High-effort surpluses are \(A_{E1}=19/100+\theta\lambda/40\), \(A_{H1}=3/25-3\theta\lambda/20\). H's acceptance deficit first appears when \(\theta\lambda>4/5\), a region where H already chooses high effort because \(\lambda>4/5>3/5\). Its deficit is \(G=\max\{0,3\theta\lambda/20-3/25\}\). E's high-effort surplus is always positive. In unknown review, the largest permitted offset is \(q(d-f)K_{max}=7/100\), giving \(\theta\lambda\leq19/15\). The protection constraint has the same cap. At the upper endpoint both groups use high effort; mean enrollment utility with \(K=1/5\) is \(79/1200>0\), and gross current spending is \(29/50\). More generally, accepted surplus is bounded below by the high-effort mean \(31/200-\theta\lambda/16\), so protection/participation and the budget do not tighten this frontier. Taking the least feasible consequence supplies weak acceptance; a slightly larger feasible consequence supplies strict acceptance in the interior when needed.

In known review, the state \((H,c=0)\) cannot be affected by K. Above \(\theta\lambda=4/5\), its best accepted surplus is negative; below the boundary all groups have a positive accepted action, and K=0 suffices. At equality H acceptance and withholding tie in this state. The unknown endpoint also forces a tie because its full offset is exhausted. ∎

At the article's fixed \(\lambda=1,K=1/5\), unknown review completes all cases throughout this error path. Known review also completes all cases for \(\theta<4/5\); at \(\theta=4/5\) all-accepting and selectively withholding pure policies tie; for \(\theta>4/5\) completion is \(3/4\). Thus misspecification does not automatically reproduce the original pattern: the size and direction of the error and the reward strength matter.

## Proposition 3: average calibration can coexist with missing care

Consider the distinct error family

\[
\widehat m_E=19/20-\delta,\qquad \widehat m_H=3/5+\delta,
\quad0<\delta\leq7/40.
\]

The average benchmark remains exactly \(31/40\), the true high-effort success probability over all eligible opportunities. The risk ordering E≥H is preserved. Nevertheless, the benchmarks are wrong within groups. The weak full-acceptance frontiers are \(19/(50\delta)\) under unknown review and \(6/(25\delta)\) under known review; strict endpoints are excluded.

**Proof.** In the relevant high-effort range, \(A_{E1}=19/100+\lambda\delta/2\) and \(A_{H1}=3/25-\lambda\delta/2\). The two limits follow by allowing offsets of \(7/100\) and zero respectively in H's binding state. The specified delta range puts both limits above the effort thresholds. The full-acceptance mean is exactly \(31/200\), so participation remains positive at every feasible K; current spending and protection satisfy the original bounds. ∎

For a concrete strict example take \(\delta=3/20\), giving \(\widehat m=(4/5,3/4)\), and \(\lambda=2,K=1/5\). High-effort surpluses are \(17/50\) and \(-3/100\). The unknown-review H acceptance advantage is \(-3/100+7/100=1/25>0\). In known review it is \(-3/100<0\) without review and \(-3/100+7/50=11/100>0\) with review. Thus only H without review withholds.

| Exact expectation | Review unknown | Review known |
|---|---:|---:|
| Completion per eligible opportunity | 1 | 3/4 |
| Success among completed | 31/40 | 5/6 |
| Successful completion per eligible opportunity | 31/40 | 5/8 |
| Mean adjusted residual among completed, E[Y−m̂] | 0 | 1/20 |
| Residual summed per eligible opportunity | 0 | 3/80 |
| Provider enrollment utility | 29/200 | 61/400 |
| Current care/review spending | 29/50 | 9/20 |

Under full acceptance the two group residuals, +δ and −δ, cancel. After half of H disappears, the selected residual is \(((1/2)\delta-(1/4)\delta)/(3/4)=\delta/3=1/20\). This is an apparent five-percentage-point advantage over the benchmark among completed cases, **not a five-point improvement in health**. No potential success probability changed. Missing outcomes are known only because this is a fully specified model; the table does not establish that they are identifiable in real records.

## Diagnostic: adjusting for chosen effort changes the mechanism

If the benchmark is instead set equal to each action's own true success probability, \(\widehat m_{ge}=p_{ge}\), every accepted action's expected continuation increment is zero. Current utility then chooses low effort for both groups, since \(13/50>19/100\) and \(9/50>3/25\). Completion remains one, but success per eligible opportunity is \(3/5\), not \(31/40\). This is a deliberately different mechanism demonstrating why a reference policy must be declared. It is not a valid replacement for a benchmark fixed before the provider chooses effort.

## What is and is not established

The original missing-care pattern is contingent on the reward's risk specification. Correct reference-risk adjustment removes it in this calibration. Error by subgroup can restore it even with exact aggregate calibration. These are reproducible applications of known selection, monitoring and calibration principles; no new general-theory priority is claimed. Written arguments cover the stated parameter families; finite arithmetic checks verify implementations and witnesses rather than replacing those arguments.

Nothing here estimates insurer profit, mortality, CMS Star performance, patient welfare, real clinical behavior or trained-agent improvement. Current budget feasibility does not finance future continuation promises or establish Pool solvency. Neither reviewed records nor exact arithmetic ensure that the true eligible denominator is observed. Other error patterns, nonlinear mappings, risk aversion, later discovery, lawful exit and justified clinical exceptions require their own solutions.

PTI remains provider trust for humans and machines. Patient rights, Codex proposal/voting rules, payment, routing, training reward and trust estimation remain distinct. THEON's healthcare foundation models, multimodal/longitudinal work, multi-agent research and independently evaluated RSI remain the full program. This test motivates calibration and selective-learning checks; it adopts no protocol rule and alters no corrected whitepaper proof.