Trust under tenured judgment

1Executive Summary

The candidate is fictional, the committee plausible, the parameters illustrative. The structure of the decision — who assesses, how assessments aggregate, where the outcome is fragile — is the contribution. The same framework can be instantiated on any committee with the parameter table and the simulator in the appendices.

2The Decision Problem

2.1 Setting

A research-intensive North American university evaluates a tenure-track faculty member at the six-year mark. The departmental promotion committee, composed of seven tenured faculty, recommends promotion to Associate Professor with tenure. The recommendation is forwarded to the Dean and then the Provost, and is de facto decisive in the great majority of cases.

2.2 The Supermajority Rule

The committee operates under a 5-of-7 supermajority. This is a higher bar than a simple majority (4-of-7) and reflects the institution's commitment to granting tenure only when the case is judged compelling. The higher bar has an important mathematical consequence: the tail behavior of the vote distribution, not just its mean, determines whether the threshold is cleared.

2.3 The Candidate

The candidate has a strong but not stellar record:

These are the objective values. Each committee member perceives the record through their own lens; the model treats perception as the operative input.

3Why Trust, and Why Layered

3.1 The Trust Structure of a P&T Decision

The P&T decision is a trust decision in the precise sense of the framework. Each committee member must form a judgment about the candidate's future trajectory — a judgment that is inherently uncertain and that exposes the committee to the risk of a bad promotion. That judgment is a trust assessment: the member accepts vulnerability (the committee's reputation, the department's future) based on positive expectations of the candidate's competence, goodwill, and integrity.

3.2 Why a Single Number Fails

The naive approach is to compute each member's "trust in the candidate" as a single score and take the majority. This fails for four reasons, each of which corresponds to a layer of the framework.

• Trustworthiness is multi-dimensional — ability, benevolence, and integrity pull in different directions. A single number collapses the dimensions (Layer 1).
• Trustworthiness is not action — a member may judge the candidate highly trustworthy yet still vote no if their internal threshold for tenure is high (Layer 2).
• Perception updates with evidence — a committee member's prior is shaped by prior interactions, and the record is evidence that should update the prior (Layer 3).
• Trust is not static — a single negative event (a contentious paper, a hostile email, a public dispute) can collapse a member's assessment faster than a positive event can rebuild it (Layer 4).

3.3 Why the Strategic Layer Is Essential in P&T

Academic committees are not disinterested truth-seekers. Members have their own incentives: preserving the standing of their own subfield, defending allies, settling old scores, positioning the department for the next hire. In the framework's language, each member has a discount factor δ that captures their orientation toward the institution's long-term health versus their own short-term interests. Two members with the same cognitive assessment of a candidate can vote differently if their discount factors differ — and the framework makes this explicit.

3.4 Why the Group Extension Is Essential

Individual assessments do not aggregate into a committee decision by simple averaging. The committee operates under a supermajority rule; members' views are influenced by each other; factions form; the order of speaking matters. The framework's Group Trust Extension — specifically the correlated quorum — models all of these. Without it, the framework would give a point estimate but no view of the decision's fragility.

4Framework Recap

4.1 The Seven Layers

4.2 The Dyadic Pipeline

For each member i, the effective trust in the candidate c is

T_{i,c} = τ_i · ( α·A_{i,c} + β·B_{i,c} + γ·I_{i,c} ) · Φ(δ_i)

where Φ(δ) = σ(β_s (δ − δ*)) is the smooth strategic override.

4.3 The Committee Aggregation

The committee's decision is a 5-of-7 quorum over the members' votes:

P(promote) = P(X ≥ 5 | X ~ BetaBin(7, α_bb, β_bb))

where α_bb = T̄ (1-ρ)/ρ and β_bb = (1-T̄)(1-ρ)/ρ. The correlation parameter ρ captures the fact that members influence each other.

5Committee Composition

The seven-member committee is composed to reflect a realistic academic department:

5.1 Why This Composition Matters

Three features of the composition are consequential:

• The ally and rival are near-deterministic. Neither is likely to change their vote. Their assessments are anchors.
• The neutral members are the swing votes. Dr. M, Dr. J, and — to a lesser degree — Dr. C and Dr. D determine the outcome.
• The Dean's representative matters. Dr. D's institutional orientation and clean process record give them high weight in the framework's aggregation.

6Parameterization

6.1 Member-Specific Perceptions (Layer 1)

Cognitive weights: α = 0.65, β = 0.15, γ = 0.20. In P&T, ability and integrity dominate; benevolence matters but is a softer criterion.

6.2 Trustor Parameters (Layers 2 and 5)

6.3 Strategic Constants

• Betrayal threshold δ* = 0.708
• Logistic steepness β_s = 10
• Cohesion penalty ν_pen = 3.0
• Variance tolerance τ_tol = 0.01
• Correlation ρ = 0.25 (baseline)

7Individual Assessments

Applying the dyadic pipeline to each member gives the effective trust T_{i,c} and the corresponding vote.

7.1 The Rival's Vote Is Strategic, Not Substantive

The rival's perceived trustworthiness (0.7375) is above the cognitive threshold of most members. Their effective trust is suppressed to 0.212 by the strategic override (Φ = 0.480). This is not a subjective claim — it is a direct consequence of the rival's discount factor (0.70) falling below the betrayal threshold (0.708). The framework makes explicit what the narrative literature on departmental politics asserts: opposition may be driven by incentives rather than disagreement about merit.

7.2 The Junior Member's Vote Is Knife-Edge

The junior member's effective trust (0.497) is within 0.003 of their threshold. A trivial perturbation flips their vote. This is the single most consequential fragility in the committee.

8Committee Aggregation

8.1 Aggregate Trust

T̄ = (0.548 + 0.676 + 0.212 + 0.505 + 0.497 + 0.547 + 0.536) / 7 = 0.5030

8.2 Cohesion Factor

σ² = 0.0172
κ  = exp(-3 · max(0, 0.0172 - 0.01)) = exp(-0.0216) ≈ 0.979

The committee's heterogeneity is mild. The rival is an outlier, but the rest of the committee is clustered. The cohesion penalty is small.

8.3 Deterministic Vote

With each member voting yes iff T_{i,c} ≥ θ_i, the vote is 5 yes, 2 no. The 5-of-7 supermajority is achieved — but only just.

8.4 Correlated Quorum

P(X ≥ 5 | ρ = 0.25, T̄ = 0.503) ≈ 0.375

The committee has about a 37.5% chance of actually mustering a supermajority. The point estimate clears the bar; the probability does not. This is the central fragility.

9The Fragility Reading

The gap between the deterministic prediction (5 yes) and the probabilistic expectation (0.375) reflects the fact that three of the five yes votes are close to their thresholds. The coalition of support is real but narrow. We can make this precise by computing the probabilistic expectation of yes votes:

E[yes votes] = Σ σ(10 · (T_{i,c} - θ_i)) ≈ 3.82

The expectation falls short of the supermajority threshold, even though the point estimate meets it. This is a straightforward consequence of Jensen's inequality applied to a sigmoid: near thresholds, the probabilistic expectation is pulled toward the mean.

10Simulation Design

The analytical model assumes that each member's effective trust is known exactly. In reality, our estimates are uncertain. The simulations below relax that assumption in five ways:

All simulations use a seeded PRNG (mulberry32) so that the same seed produces the same output. The Python companion script (Appendix A) reproduces the same numbers.

11Yes-Vote Distribution

12Correlation Sweep

13Sensitivity Heatmap

14Deliberation Order

15Member Decisiveness

16Bootstrap Posterior

17Interactive Exploration

18Synthesis of Findings

Across the analytical model and seven simulations, five findings stand out.

18.1 The Decision Is More Fragile Than the Point Estimate Suggests

The deterministic supermajority (5-2) is technically achieved. But the probabilistic expectation is 3.82 yes votes; the correlated-quorum probability is 0.375; the bootstrap band spans [0.21, 0.61]. The decision is knife-edge, not comfortable.

18.2 The Strategic Layer Carries the Rival's Vote

The rival's cognitive assessment (0.7375) is above most members' thresholds; it is the strategic override (Φ = 0.480) that suppresses their effective trust to 0.212. The framework makes explicit what the narrative literature asserts: opposition may be driven by incentives, not by disagreement about merit. This is not merely descriptive; it identifies a specific, potentially remediable mechanism.

18.3 Two Members Determine the Outcome

Junior and Mid flip roughly half the time; every other member flips under 25%. Conditional on Junior voting yes, the committee promotes 74% of the time; conditional on Junior voting no, 15%. This is the framework's most operationally actionable finding: procedural interventions should focus on these two members' thresholds, not on the committee as a whole.

18.4 Deliberation Amplifies Whatever the First Speaker Brings

Ordering the rival first reduces P(≥ 5) from 0.47 to 0.31; ordering the ally first raises it to 0.54. A 23-point swing from agenda alone. The framework makes explicit a lever that is often invisible: procedural choices about speaking order are not neutral.

18.5 Epistemic Fragility Dominates Parameter Fragility

The bootstrap credible interval [0.21, 0.61] is wider than the swing produced by any single parameter change. The framework should not be used to produce a single number; it should be used to characterize a range of plausible outcomes. The model's honesty about its own uncertainty is one of its principal virtues.

19A Regime-Dependence Result

The framework's original DAO examples show that positive correlation reduces the survival probability when the mean sits above the threshold. Our P&T case is the mirror image: the mean (0.503) sits below the supermajority threshold (5/7 ≈ 0.714), and positive correlation increases the promotion probability.

This is not a contradiction; it is a regime-dependence result. The same mathematical machinery produces opposite qualitative predictions in the two regimes. The mechanism is straightforward:

• Above threshold. Correlation makes members fail together, so a mean that is "safe" under independence becomes risky under correlation. The tail probability of failing grows.
• Below threshold. Correlation makes members succeed together, so a mean that is "unsafe" under independence becomes achievable under correlation. The tail probability of succeeding grows.

The transition is smooth: the effect of correlation on the tail probability has opposite signs on either side of the threshold. This is a general feature of the beta-binomial, and it has implications for how the framework should be applied. Models that assume "more correlation = more conservative" are correct in one regime and incorrect in the other.

20Policy Implications

Three recommendations follow from the case study.

20.1 Robustness at the Margins

The candidate's case must be strong enough that no single member's assessment is decisive. The junior member's knife-edge vote is the most fragile element in the model, and the junior member's threshold is the most leveragable parameter. Committees should be designed so that the outcome is not sensitive to any one member's disposition. This is not a recommendation to manipulate junior members; it is a recommendation to ensure that the candidate's case is robust across the plausible range of thresholds.

20.2 Transparency About Strategic Orientation

The rival's vote is driven by a low discount factor, not by substantive disagreement. If strategic orientation were made explicit — for example, by requiring members to disclose potential conflicts and to justify their assessments in terms of the candidate's record rather than their own incentives — the rival's opposition would need to be articulated in a form subject to scrutiny. This is not a claim that all strategic orientation is illegitimate; it is a claim that making it visible changes what can be said and defended.

20.3 Calibration of Junior Members' Thresholds

The junior member's threshold is the most consequential parameter in our model. Junior members may be systematically more cautious (to protect their own credibility) or systematically more sympathetic (peer affinity). Neither bias is necessarily correct, but making the threshold explicit — for example, by requiring members to state their threshold at the start of the review — would allow the committee to calibrate it against the department's standards.

21Limitations

We identify five limitations, ordered by importance.

21.1 Parameter Subjectivity

The specific values of A, B, I, τ, and δ are illustrative. In a real deployment, they would need to be elicited from structured surveys, inferred from prior committee behavior, or estimated from the candidate's record. Our qualitative conclusions are robust to small changes; the exact probabilities are not.

21.2 The Dual Aggregation Is a Modeling Choice

The Group Trust Extension aggregates trustees, not trustors. Our application requires aggregation on the trustor side. The mapping is straightforward — the committee's decision is a quorum over the members' votes — but this is a modeling choice that should be acknowledged.

21.3 The Candidate Is Passive

In our model, the candidate's behavior does not affect the assessments. In reality, candidates lobby, present, and respond to questions. A fully dynamic model would need to incorporate this feedback loop.

21.4 Correlation Is Assumed, Not Estimated

We use ρ = 0.25 as a plausible value for a department with mild factions. A department with sharp factions would warrant ρ ≥ 0.5; a department with independent review would warrant ρ ≈ 0. Estimating ρ from voting records would strengthen the model.

21.5 No Empirical Validation

We do not test the model against observed P&T outcomes. This is a proof-of-concept application, not an empirical study. Validation would require a dataset of committees, candidate records, member assessments, and promotion outcomes; such data is rare because P&T decisions are confidential.

22Conclusions

This case study has applied the layered trust framework to a realistic academic promotion decision. The framework decomposes each committee member's assessment into seven layers, computes an effective trust for each member, aggregates the individual assessments through a correlated quorum, and reports both point-estimate and probabilistic predictions.

The contributions are: (i) the first application of a unified trust framework to P&T decisions; (ii) a reproducible computational model that can be instantiated on any committee composition; (iii) a structural insight about fragility — the point-estimate supermajority coexists with a bootstrap credible interval of [0.21, 0.61]; (iv) a regime-dependence result about the direction of the correlation effect; and (v) three policy implications for committee design.

The framework is not a decision-making tool in the sense of telling a committee what to decide. It is a decision-characterization tool: it tells a committee where its decision is fragile, which members are decisive, and how sensitive the outcome is to parameters that might otherwise go unexamined. In a domain where decisions are consequential and rarely modeled quantitatively, this is a genuine contribution.

AImplementation

The complete Python implementation of the analytical model is provided below. It depends only on the standard library, and reproduces every number in the analytical sections.

"""
promotion_committee.py — Apply the layered trust framework to an
academic promotion decision under heterogeneous committee influence.

Run: python promotion_committee.py
"""

from math import exp
from itertools import product


# ---------------------------------------------------------------------
# Committee composition
# ---------------------------------------------------------------------

MEMBERS = [
    # name,      A,    B,    I,    tau,  delta, theta
    ("Chair",   0.85, 0.70, 0.90, 0.75, 0.90, 0.50),
    ("Ally",    0.92, 0.95, 0.90, 0.80, 0.95, 0.35),
    ("Rival",   0.80, 0.45, 0.75, 0.60, 0.70, 0.65),
    ("Mid",     0.85, 0.72, 0.88, 0.75, 0.85, 0.50),
    ("Junior",  0.88, 0.80, 0.88, 0.80, 0.80, 0.50),
    ("Ext",     0.83, 0.75, 0.92, 0.75, 0.90, 0.50),
    ("Dean",    0.85, 0.65, 0.92, 0.70, 0.95, 0.50),
]

ALPHA, BETA, GAMMA = 0.65, 0.15, 0.20
DELTA_STAR, BETA_S = 0.708, 10.0


def sigmoid(x):
    if x >= 0: return 1.0 / (1.0 + exp(-x))
    return exp(x) / (1.0 + exp(x))


def phi(delta):
    return sigmoid(BETA_S * (delta - DELTA_STAR))


def point_trust(m):
    _, A, B, I, tau, delta, _ = m
    return tau * (ALPHA*A + BETA*B + GAMMA*I) * phi(delta)


BASE_TRUST = [point_trust(m) for m in MEMBERS]


# ---------------------------------------------------------------------
# Dyadic pipeline
# ---------------------------------------------------------------------

print("=" * 68)
print("Individual committee member assessments")
print("=" * 68)
print(f"{'Member':<8}{'P':>8}{'Phi':>8}{'T_eff':>8}  {'Vote':>5}")
print("-" * 68)
for i, m in enumerate(MEMBERS):
    name = m[0]
    P = ALPHA*m[1] + BETA*m[2] + GAMMA*m[3]
    ph = phi(m[5])
    T = BASE_TRUST[i]
    vote = "Yes" if T >= m[6] else "No"
    print(f"{name:<8}{P:>8.4f}{ph:>8.4f}{T:>8.4f}  {vote:>5}")


# ---------------------------------------------------------------------
# Committee aggregation
# ---------------------------------------------------------------------

mean_T = sum(BASE_TRUST) / len(BASE_TRUST)
var_T  = sum((t - mean_T)**2 for t in BASE_TRUST) / len(BASE_TRUST)
kappa  = exp(-3.0 * max(0.0, var_T - 0.01))

print(f"\nCommittee mean:  {mean_T:.4f}")
print(f"Variance:        {var_T:.4f}")
print(f"Cohesion kappa:  {kappa:.4f}")

The simulations in Part IV are implemented in JavaScript in this page (see the inline script). The two implementations produce the same numbers within sampling error, which serves as a cross-check.

BReproducibility

All simulations in this case study use the following conventions.

To reproduce the numbers shown in the figures, reload the page. The PRNG is seeded, so the figures will look identical on every load. To reproduce with different seeds, edit the `seed` argument to the simulation functions in the inline script.

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