# A finite model of provider incentives and incomplete review 23 October 2026 · Methods and proof This is an illustrative mathematical model with exact rational computation. It uses no patient data. The propositions are applications of established economic principles, not claims of new general theory, clinical validation or an adopted policy. The analysis has not undergone external peer review; no Lean kernel verification was performed. [Supplement overview](model-and-proof.md) · [Reproduction instructions](README.md) · [Related research](REFERENCES.md) ## 1. Model and meaning of the statement The fixed clinical opportunity has group `g ∈ {E,H}`, equally likely, observed by the provider after enrollment. All cases in this first model are willing, clinically appropriate and feasible. Withholding means avoidable noncompletion of such a case. Patient refusal, justified escalation and genuine capacity infeasibility are not represented by this withholding action and must not be penalized by importing this model into practice. The provider enrolls before learning the group, then chooses withholding `W` or acceptance with low/high effort `A0,A1`. Enrollment permits no cost-free post-group exit that evades the contracted consequence. The provider is risk neutral. The research score proxy is `r'=(r+Y)/2`, with `r=9/10`; withholding retains `r`. Future continuation margin is `λ r'`, with `λ≥0`. It is an exogenous continuation-value assumption, not a solved second-period market. Normalize all utilities by subtracting the incumbent continuation value `λr`; the ex ante outside option is also normalized to zero. Thus participation compares the contract with retaining that incumbent continuation opportunity. This normalization is substantive: one cannot subtract `λr` from enrolled utilities while leaving an unrelated absolute outside option unchanged. Current payment is `b+sY`, where `b=2/5`, `s=1/5`; costs are `c0=3/10`, `c1=2/5`. Success probabilities are: | Group | Low effort | High effort | |---|---:|---:| | E | 4/5 | 19/20 | | H | 2/5 | 3/5 | An opportunity is assigned to review by an exogenous Bernoulli draw with probability `q∈[0,1]`, independent of case group and the randomness governing potential outcomes. The provider cannot change this assignment. **Its realized capture state is not known to the provider when choosing the action**; q is its conditional capture probability at that information set. If that state is revealed before the choice, the realized action and resulting observed outcomes can depend on it, despite exogenous assignment. The visible-capture comparison is proved in [the capture-timing limitation](capture-timing-limitation.md). Conditional on capture, an adverse flag occurs with probability `d` after avoidable withholding and `f` after appropriate acceptance, where `d,f∈[0,1]`. The rate `f` does not vary with group or effort. These are stipulated error rates, not consequences of immutable records. A bounded modeled utility consequence `K∈[0,Kmax]` follows an adverse flag. No operational authority to fine, withhold earned payment or reduce care follows from this mathematical instrument. Let `t=qK` and `h=d−f`. With no additional enrollment transfer (`F=0`), the action utilities are ``` U_g(W) = −dt U_g(Ae) = A_ge(λ) − ft A_ge(λ) = b + s p_ge − c_e + λ(p_ge−r)/2. ``` For a predetermined compliant action, before capture is realized or revealed, expected loss from a false flag is bounded by `ft≤ε`, with `ε≥0`. This bound does not establish safety for a legitimate refusal or escalation action, which is absent from the model. Review costs `a≥0` for each captured opportunity. The **current-period gross payer budget** is `B1` per eligible opportunity, covering earned current payment and capture/review cost. Consequence receipts are not credited against this budget. Future continuation rents, future care, funding of those rents, reserves and overall Pool solvency are outside this budget statement. There is sufficient current capacity for the single offered case. Define `A_g(λ)=max_e A_ge(λ)` and `m(λ)=(A_E(λ)+A_H(λ))/2`. A *weak full-access contract* means there exists a best-response policy that accepts both groups at optimal effort, satisfies ex ante participation `m−ft≥0`, respects protection and the `K` cap, and fits the current-period gross budget. Ties are exposed below; weak implementation is not a guarantee that every best response accepts. ## 2. Complete best responses The four acceptance utilities before the false-flag consequence are: ``` A_E0 = 13/50 − λ/20 A_E1 = 19/100 + λ/40 A_H0 = 9/50 − λ/4 A_H1 = 3/25 − 3λ/20. ``` Conditional on acceptance, H uses low effort for `λ<3/5`, both efforts are optimal at `λ=3/5`, and high effort is uniquely optimal for `λ>3/5`. E has the same cases with threshold `14/15`. For any given `K`, acceptance is uniquely optimal against withholding when `A_g+h t>0`; withholding is uniquely optimal when `A_g+h t<0`; both are optimal at equality. Combine this rule with the effort rule to obtain every pure best response. Any mixture over tied pure maximizers is also optimal. At an effort tie, acceptance utility and participation are unchanged by the mixture, but outcomes and gross payment differ. For `h≥0`, E always strictly prefers acceptance because `A_E≥16/75>0`. H's acceptance boundary is governed by ``` G(λ) = max(0,−A_H(λ)) = max(0,3λ/20−3/25). ``` The average acceptance surplus is continuous: ``` m(λ) = 11/50 − 3λ/20 for 0≤λ≤3/5 19/100 − λ/10 for 3/5≤λ≤14/15 31/200 − λ/16 for λ≥14/15. ``` ## 3. Exact implementability theorem **Theorem.** Suppose the model above holds, `d>f`, and select the lower-cost effort at any effort tie. Let `h=d−f`. A weak full-access contract exists at a given `λ≥0` if and only if all four conditions hold: ``` G(λ)/h ≤ q Kmax [bounded consequence/capture] f G(λ)/h ≤ ε [false-flag protection] f G(λ)/h ≤ m(λ) [ex ante participation] C_min(λ) ≤ B1 [current gross budget]. ``` Here ``` C_min(λ) = 13/25 + aq for 0≤λ≤3/5 27/50 + aq for 3/5<λ≤14/15 111/200 + aq for λ>14/15. ``` If `q>0`, a least-consequence contract is `K*=G/(qh)`. If `q=0`, the first condition forces `G=0`, and `K=0` suffices when the other conditions hold. The theorem still applies without dividing by `q`. **Proof.** Under full acceptance, each group's optimal effort maximizes `A_ge`, because the common acceptance penalty is independent of effort. Comparing the resulting acceptance action with withholding yields `A_g+h t≥0`. E's condition is automatically satisfied, and H's condition is equivalent to `h t≥G`. Hence every full-access contract requires `t≥G/h`. Since `f≥0`, both expected false-flag loss and the participation loss are weakly increasing in `t`; consequently protection and participation are easiest to satisfy at `t*=G/h`. The cap gives `t≤qKmax`, protection gives `ft≤ε`, and ex ante participation gives `ft≤m`. Thus their common interval is nonempty exactly when the first three displayed conditions hold. The current gross payment depends on the optimal effort but not `K`; among tied efforts, low effort minimizes `s E[Y]` and therefore minimizes current payment, yielding `C_min`. This proves necessity and sufficiency, with the stated tie convention. ∎ This is a restricted contract-class theorem: the designer chooses only `K`; base pay, outcome pay, information, error rates, capacity and continuation rule remain fixed. It is not an impossibility for all healthcare contracts. ## 4. Closed-form frontier For `d>f`, set ``` ΛK = 4/5 + (20/3)(d−f) q Kmax ΛP = ∞ if f=0 4/5 + (20/3)(d−f) ε/f if f>0 ΛIR = (19d−7f)/(10d+5f) if f/d≥29/35 (62d−14f)/(25d+35f) if f/d≤29/35. ``` Both expressions for `ΛIR` equal `14/15` at `f/d=29/35`, so the overlap is harmless. In particular, if `f=0`, `ΛIR=62/25`: even perfect specificity does not remove the fixed-contract participation constraint. Let `C0=13/25+aq`, `C1=27/50+aq`, `C2=111/200+aq`. Define the budget frontier ``` ΛB = ∞ if B1≥C2 14/15 if C1≤B14/5` can be restored in this contract class. The same conclusion applies when `q=0`, `Kmax=0`, or `f>0` and `ε=0`. With `f=0`, an `ε=0` protection limit does **not** preclude deterrence. ## 5. Strictness, ties and robust claims - For `λ<4/5`, `K=0` makes acceptance strictly preferable for both groups. Effort can still be tied at `3/5`. - At `λ=4/5`, H acceptance and withholding tie at `K=0`. Strict acceptance needs `d>f` and positive feasible exposure `t>0`. - For `λ>4/5`, the least-consequence contract `K*` makes H acceptance and withholding exactly tied. A result claiming realized access under `K*` must explicitly assume compliant tie-breaking; arbitrary optimal choice can still exclude H. - With `d>f` and `λ≥4/5`, strict acceptance for both groups exists exactly when there is `t>G/h` satisfying `t≤qKmax`, `ft≤ε`, and `ft≤m`. For `f>0`, this is `G/h < min(qKmax,ε/f,m/f)`. For `f=0`, it is `G/h0`; a participation subsidy would change the fixed-`F=0` theorem. - At effort thresholds `3/5` and `14/15`, every optimal effort mixture has the same utility but a different budget cost. The theorem's cheaper tie is existential. A budget guaranteed for **every** best response must cover the more expensive tied effort (`C1` at `3/5`, `C2` at `14/15`). This matters when `ΛB` determines the endpoint. - A robust incentive margin `η>0` changes `ht≥G` to `ht≥G+η` for the H comparison when `λ≥4/5`; uncertainty in `q,d,f` must then use jointly valid worst-case bounds. This robustness extension is not proven for arbitrary parameter ambiguity or strategic capture here. ## 6. Exact illustrative calibration Choose ``` q=1/2, d=4/5, f=1/10, Kmax=1/5, ε=1/100, a=1/20, B1=3/4. ``` Then `h=7/10`, `ΛK=ΛP=19/15`, `ΛIR=482/235>19/15`, and `B1>C2=29/50`. Hence weak full access is possible exactly for `0≤λ≤19/15`, and strict full access is possible for `0≤λ<19/15` (with suitable positive `K` at and above `4/5`). At the endpoint, `K=1/5` is the only feasible full-access consequence. | Quantity at λ=19/15 | Exact value | |---|---:| | E high-effort surplus `A_E` | 133/600 | | H high-effort surplus `A_H` | −7/100 | | Effective exposure `t=qK` | 1/10 | | False-flag expected loss `ft` | 1/100 | | Withholding utility `−dt` | −2/25 | | H accepted utility `A_H−ft` | −2/25 | | E accepted utility `A_E−ft` | 127/600 | | Mean enrollment utility | 79/1200 | | Gross current care and review cost | 29/50 | The contract has positive ex ante enrollment surplus, yet H has negative conditional utility and is indifferent between acceptance and withholding. This is why neither strong access nor post-type participation may be inferred from the weak frontier. At `λ=1`, choose `K=1/10`. Then `t=1/20`, `ft=1/200`, H acceptance utility is `−7/200` and withholding utility is `−1/25`, so H strictly prefers acceptance by `1/200`. E acceptance utility is `21/100`; enrollment utility is `7/80`; gross current cost remains `29/50`. Both efforts are high. Without the consequence, H withholds and E accepts: conditional success is `19/20`, eligible completion `1/2`, and successful completion per eligible opportunity `19/40`. Under the strict contract those values become `31/40`, `1`, and `31/40`. These are exact model outcomes, not clinical effect estimates or a welfare theorem. At `λ=3/2`, even the maximum permitted exposure leaves H acceptance worse than withholding by `7/200`; the constrained fixed contract cannot restore access. ## 7. Exit limitation **Corollary (post-group escape).** Add an action `X` available after learning the group, with normalized utility zero, that escapes the consequence without changing any other payoff. Whenever `A_H<0`, no nonnegative `K` can make H acceptance optimal, because `U_H(A*)=A_H−ft<0=U_H(X)`. For this model, that includes every `λ>4/5`. At `λ=4/5`, H acceptance is at best tied with escape if `ft=0`; if `ft>0`, escape is strictly preferred. Therefore ex ante participation alone does not implement the restored frontier if the contract permits such post-type avoidance. This corollary is a mathematical warning about timing and commitment, not a recommendation to prevent lawful clinician withdrawal or legitimate escalation. A feasible alternative must model lawful participation, transfer, fallback care and how responsibilities survive an exit. Additional compensation or a changed routing rule is a different contract class. ## 8. Interpretation, controls and scope The exact limitation is that incomplete capture reduces the maximum usable consequence, and false-positive protection plus enrollment constrain it further. At the calibrated frontier, false-positive protection and the consequence cap bind before enrollment or the current budget. A stronger score-to-routing margin cannot be offset simply by increasing punishment. The classical content is expected-utility comparison with bounded, imperfect enforcement. Kaplow and Shavell provide the exact antecedent of the differential-error incentive term; Chen and Sivey and Mak study provider selection under performance reporting. See [the model-level reference comparison](REFERENCES.md). This derivation exposes a conditional limit in the specified contract class; it does not establish economic novelty. A conventional contract with the identical information, consequence instrument and resources has the identical feasible frontier. Risk adjustment is a necessary positive control: if the continuation reward is centered at correct fixed-effort group risk, the particular below-prior-mean selection gradient can disappear. A probability-based routing implementation of centered residuals needs its own bounded mapping; a shifted score must not be allowed to exceed `[0,1]` or substituted through a nonlinear routing function without re-solving expected utility. The theorem does not include unknown eligibility, private capture manipulation, multidimensional clinically legitimate actions, risk-averse providers, a full patient queue, substitute providers, delayed outcome identification, group-dependent adjudication, total two-period finances or a dynamic equilibrium. Those omissions are explicit future tests, not implied guarantees. PTI is provider evidence, not a patient score. Score, training reward, care payment, routing and Codex voting remain separate mappings. The model uses a research proxy, not an operational PTI formula. Codex can assess proposed contract rules; Pool/Treasury must fund their actual obligations; Trust-Time needs a separate capacity model. Human, agent and robot providers have distinct clinical capabilities and authority. THEON foundation models, longitudinal and multimodal work, uncertainty/causal research, multi-agent learning and independently evaluated RSI remain the full program, with selective-data feedback as a later linked study. This calculation supplies no trained-model, clinical-performance, governance-optimality or systemwide-solvency result.