When better results hide missing care
Correct risk adjustment removes a missing-care pattern under the finite model’s payment and reward assumptions. Errors within groups can restore it—even when the average is right.
Healthcare should reward better decisions and better outcomes. But success among people who received care leaves a question unanswered: what happened to everyone else who was eligible?
VANTHEON is building the foundations of a new healthcare system, connecting care, evidence, learning and funding across patients, clinicians, institutions, AI agents and robots. Its proposed building blocks—its primitives—are intended to make responsibility traceable, results verifiable and improvements subject to shared governance. This research tests a requirement for that system: whether rewards could improve reported performance while care disappears.
A provider rewarded for successful outcomes may also have a reason to avoid harder cases. Our original model made that trade-off visible: knowing which opportunities could never be reviewed raised the reported success rate while reducing the care completed.
The risk-adjustment test changes the conclusion. Correctly adjusting for case difficulty removes the original missing-care pattern under this finite model’s payment and reward assumptions. Errors within case groups can restore it—even when the risk model is right on average.
These results put two requirements together: check whether the benchmark is appropriate for the case, and account for eligible care that never reaches the outcome record. The reward, its risk adjustment and what the provider knows about review all matter.
Exact expectations of an illustrative finite model. No patient data or insurer evaluation. The note applies established economics and has not undergone external peer review.
A better reported rate can hide less successful care.
The original benchmark, before risk adjustment. Both versions review 50% of opportunities under the same contract: λ = 1, K = 1/5. Only knowledge of review status changes. Units show expected counts per 40 opportunities, not individual patients; assumptions below.
How can the reported result improve while care falls?
The provider encounters equally likely easier and harder cases, then chooses acceptance with low or high effort, or withholding. Every opportunity is appropriate, willing and feasible. Legitimate refusal, patient choice and capacity shortages are outside this model.
Accepted care earns a fixed payment and an outcome-dependent payment. A score also affects the value of future opportunities. In the original comparison it uses the same benchmark for both groups, without adjusting for case difficulty. This is a research proxy, not VANTHEON’s Provider Trust Index formula.
Half of all opportunities enter review, independently of case group. Review can miss avoidable withholding or incorrectly flag appropriate care; a bounded consequence follows a flag. The provider enters a binding contract before learning the case or review status, with no cost-free exit afterward. The contract must be worthwhile before that information arrives.
We change only whether review status is known when the provider chooses. If it is known, an opportunity outside review has no possibility of later discovery. One of the four equally likely case-and-review combinations then changes the decision: the harder case outside review.
Per 40 eligible opportunities, the expected result changes from 40 completed and 31 successful to 30 completed and 25 successful. The reported rate rises because it divides 25 by 30. Successful care across all opportunities falls because it divides 25 by 40. No case becomes more likely to succeed.
Follow the four case-and-review combinations
| Case and actual review status | Status unknown at choice | Status known at choice |
|---|---|---|
| Easier · enters review | Accept, high effort | Accept, high effort |
| Easier · outside review | Accept, high effort | Accept, high effort |
| Harder · enters review | Accept, high effort | Accept, high effort |
| Harder · outside review | Accept, high effort | Withhold |
The unknown-status provider observes the case group but cannot distinguish review states when choosing. Realized actions can depend on revealed status, even though review assignment is exogenous.
Read the original exact values
| Measure | Review unknown | Review known |
|---|---|---|
| Completion per eligible opportunity | 1 (100%) | 3/4 (75%) |
| Success among completed cases | 31/40 (77.5%) | 5/6 (≈83.3%) |
| Successful completion per eligible opportunity | 31/40 (77.5%) | 5/8 (62.5%) |
| Current care-and-review spending per eligible opportunity | 29/50 (0.58 units) | 9/20 (0.45 units) |
Both regimes have the same budget cap of 3/4 model units per eligible opportunity. Actual spending differs because completing more care costs more. The model has no patient benefit-and-harm calculation; completion alone does not establish welfare.
Does correct risk adjustment remove the pattern?
Yes, in this model. Replace the common benchmark with each group’s true expected success under a prespecified high-effort standard: 95% for easier cases and 60% for harder cases. At the original λ = 1 and K = 1/5, both review regimes now complete all eligible care.
The benchmark is fixed before the provider chooses effort. It rewards performance relative to that standard, rather than adjusting away the effort actually chosen. These probabilities are known because we specified the model; no risk model was trained on records.
To test sensitivity, we move gradually from correct adjustment back to the original common benchmark. Figure 2 maps that error against λ, the strength of the future reward. It asks where some permitted consequence can make acceptance strictly preferable in every case and review state.
The access boundary moves with the risk error.
This is one specified error path, not a distribution of real risk-model errors. K may vary from 0 to 1/5; the plot shows feasibility, not the behavior of one fixed contract. Each boundary marks a tie for the corresponding review regime; strict acceptance requires staying below that regime’s curve. At correct adjustment, feasibility extends beyond the plotted range.
At the correct benchmark, positive current care margins make strict acceptance feasible under either review regime for every nonnegative λ. As the benchmark becomes biased against harder cases, the region where acceptance is feasible contracts. When review status is known, raising a consequence cannot change the decision on an opportunity that will never be reviewed.
At λ = 1, the original pattern returns only after this error path passes 80% of the way toward the original benchmark. At exactly 80%, acceptance and withholding tie. The direction and size of error matter; misspecification does not automatically create missing care.
Inspect the adjustment path and exact boundaries
Let θ run from 0 (correct adjustment) to 1 (the original common benchmark). Estimated reference success is 19/20 − θ/20 for easier cases and 3/5 + 3θ/10 for harder cases.
| Risk benchmark | Review unknown | Review known |
|---|---|---|
| Correct (θ = 0) | 100% | 100% |
| Halfway to original (θ = 1/2) | 100% | 100% |
| 90% toward original (θ = 9/10) | 100% | 75% |
| Original (θ = 1) | 100% | 75% |
For θ > 0, a full-acceptance best response is feasible through λ = 19/(15θ) with unknown review and λ = 4/(5θ) with known review. Strict acceptance excludes the endpoints. At θ = 1 these recover the original limits, 19/15 and 4/5. At θ = 0 both boundaries are unbounded. The proof checks participation, the current budget and the false-flag burden as well as acceptance.
This positive control depends on the specified payments, positive care margins and linear reward. Reducing the fixed payment supplies a counterexample even with correct adjustment. It is not a universal access guarantee, a PTI recommendation or a reason to conceal review rules.
What if the risk model is right on average?
A separate example keeps the population average exactly correct. The true reference success probabilities remain 95% and 60%; the estimates are 80% and 75%. Both averages are 77.5%, and easier cases still rank above harder cases. But expected success is understated for one group and overstated for the other.
This comparison uses λ = 2, a stronger future reward than Figure 1. Within the comparison, λ, K = 1/5 and every other contract term stay fixed. Revealing review status lowers completion from 100% to 75%, while the adjusted result among completed cases rises from meeting expectations to five percentage points above the benchmark.
A correct average can conceal the wrong incentives.
Exact counterexample with opposite 15-percentage-point subgroup errors. The +5-point residual is measured among the selected completed cases; it is not a gain in health. These figures are model expectations, not insurer ratings or patient observations.
When all care is completed, the two groups’ benchmark errors cancel. When some harder cases disappear, they no longer cancel within the completed group. The reported adjusted result improves because the group being measured changes.
Read the average-calibration counterexample
| Measure | Review unknown | Review known |
|---|---|---|
| Completion per eligible opportunity | 1 (100%) | 3/4 (75%) |
| Success among completed cases | 31/40 (77.5%) | 5/6 (≈83.3%) |
| Successful completion per eligible opportunity | 31/40 (77.5%) | 5/8 (62.5%) |
| Mean adjusted residual among completed cases | 0 | 1/20 (+5 percentage points) |
The adjusted residual is observed success minus the fixed group benchmark. The population contains equal proportions of the two groups before selection. The completed population changes to two-thirds easier cases and one-third harder cases when review status is known.
The lesson is specific: aggregate calibration and correct risk ordering do not rule out this failure. We also need to examine errors within relevant groups and account for omitted opportunities. Here the eligible denominator is known by construction; establishing it in real care is a separate measurement problem.
What does this tell us about five-star ratings?
A favorable quality rating and access to needed care answer different questions. CMS’s Medicare Advantage and Part D Star Ratings combine clinical, experience and access measures. They are assigned by CMS, and the 2027 ratings affect 2028 Medicare Advantage quality bonus payments. They are more than a success rate among treated cases. CMS’s current description.
The empirical evidence deserves a balanced reading. Abaluck and colleagues, studying plan exits in 2008–2011, found that historical ratings did not predict estimated causal mortality differences across plans. Meyers and colleagues, studying contract consolidations in 2014–2015, found some favorable outcomes after shifts to higher-rated contracts, including fewer 90-day readmissions. Neither study evaluates the current 2027 rating methodology.
Access problems also have direct evidence: an HHS OIG review estimated that 13% of sampled prior-authorization denials from 15 large Medicare Advantage organizations in June 2019 met Medicare coverage rules. The denominator is denied requests, not all requests or patients. That finding supports scrutiny of avoidable barriers; it does not establish that star ratings caused deaths.
Our model evaluates no insurer, profit mechanism or mortality effect. It identifies a narrower design question: can a reward look better when eligible care disappears from what is measured? Its answer depends on the risk benchmark, the reward contract and whether omissions remain discoverable.
What prior research established
Provider selection under performance reporting is well established. Chen and Sivey model quality competition, patient selection and future demand. Mak studies performance reports when initial demand and excluded consumers are not observed.
The noisy-review incentive term also has an exact antecedent in Kaplow and Shavell’s analysis of accuracy in liability determination. The timing result here applies a standard information-set argument. An equally informed conventional contract with the same instruments has the same feasible region.
The risk-adjustment extension tests the original mechanism rather than treating its failure as inevitable. Together, the negative result, positive control and counterexample expose conditions that an incentive claim must satisfy. It does not establish a new general economic theorem or a VANTHEON-specific advantage.
What this means for VANTHEON
Keep missing care visible. Care events must connect eligible needs, responsible participants and unresolved follow-up, distinguishing omissions from patient choice, clinical exceptions and capacity limits. The Pool and Treasury must connect care commitments to capacity and funding, preserving unpaid obligations and reserves.
Test rewards in context. Provider Trust Index (PTI) is intended to link human and machine providers’ service-relevant performance to stronger incentives and relevant Codex influence. Trust, payment and voting need separate evaluated rules that account for subgroup error and missing care. Patient rights remain independent.
Learn from omissions too. Codex connects knowledge, care decisions and coordination to proposals for improvement; those proposals must be evaluated for access and outcomes. THEON’s foundation-model, multi-agent and recursive-improvement research must likewise examine selective records before using them as training signals or rewards, with independent evaluation and authorized promotion.
These are requirements for VANTHEON’s proposed Learn → Verify → Settle system. This study establishes no PTI formula, validated incentive contract, clinical benefit or trained-model improvement.
Reproduce the result
The written proof and Python programs use the same finite model. Exact rational arithmetic preserves the values behind all three figures; independent constraint calculations and event-tree checks verify the results without sampling. The files below include the assumptions, proofs, source and generated outputs.
Model settings and verification
The provider is risk neutral. Easier and harder cases each have probability 1/2. In the original comparison, the raw score begins at 9/10 for either group and becomes the average of that prior and a binary outcome after acceptance; withholding leaves the score unchanged. The risk-adjustment extension replaces the continuation increment with λ/2 times (outcome − fixed group benchmark). The original model is recovered when both benchmarks equal 9/10. Current payments remain unchanged.
| Quantity | Value |
|---|---|
| Review coverage | 1/2 |
| True / false flag rates, conditional on review | 4/5 / 1/10 |
| Permitted consequence K | 0 to 1/5 |
| Fixed payment / outcome bonus | 2/5 / 1/5 |
| Low / high effort costs | 3/10 / 2/5 |
| Easier-case success, low / high effort | 4/5 / 19/20 |
| Harder-case success, low / high effort | 2/5 / 3/5 |
| Review cost per reviewed opportunity | 1/20 |
| Current budget per eligible opportunity | 3/4 |
| Expected false-flag burden cap before review status is known | 1/100 |
Figure 1 fixes λ = 1 and K = 1/5. Figure 2 varies λ and the specified benchmark error, allowing K to range from 0 to 1/5. Figure 3 fixes λ = 2 and K = 1/5 with a different error pattern. The false-flag cap applies to a predetermined compliant action before review status is known, not conditionally to the cases ultimately accepted.
The original analysis retains 313 distinct assertions. The risk-adjustment evaluator adds 1,026 parameterized checks, and a separate audit performs 1,726 consistency and provenance checks. These counts describe software verification, not independent findings, statistical confidence or clinical evidence. Written proofs establish the stated parameter families.
Read the original finite-model proof, review-timing proof and risk-adjustment propositions and limits. No Lean kernel verification is claimed.
Run the model
Download and extract the ZIP below. From that package directory, run these commands with Python 3.9 or later. No additional packages are required.
python3 -B exact_solver.py --output reproduced
python3 -B visible_capture.py --output reproduced
python3 -B risk-adjustment/risk_adjustment.py --output reproduced-risk
python3 -B risk-adjustment/independent_solver_audit.py \
--original exact_solver.py \
--risk-adjustment risk-adjustment/risk_adjustment.py \
--output reproduced-risk/independent-audit.jsonThe programs use Python’s standard-library rational arithmetic. Outputs go into reproduced/ and reproduced-risk/. The package README and risk-adjustment README explain the settings, checks and outputs.
The next test adds later discovery and lawful exit with fallback care. Estimated risk models also need tests for omitted variables and selection in their training data. Each comparison must give a conventional contract the same information and instruments and represent legitimate clinical exceptions. CMS data remains for a later, explicitly defined empirical study; no CMS dataset enters these figures. Before training an agent to optimize a reward, we want to understand which behavior that reward makes worthwhile.
For the wider architecture, read the VANTHEON whitepaper .