Ambiguity Index (AI) in Rule-Governed Systems

• Ambiguity Index (AI) is a measure that quantifies the fraction of cases where an inverse-check supports a defensible alternative decision under established rules and precedents.
• It is computed by auditing decisions using inputs like content, rules, and precedent to determine if the opposite action is logically derivable, thereby isolating normative underspecification.
• Empirical results show that lower AI values indicate clearer governance, aiding safe automation and highlighting the impact of rule specificity on decision determinations.

Searching arXiv for papers on ambiguity metrics and related evaluation frameworks. arxiv_search(query="ambiguity index defensibility index policy-grounded evaluation content moderation", max_results=10) Ambiguity Index (AI) denotes a measure of how much a decision problem admits more than one defensible interpretation or outcome. In contemporary AI evaluation, the term is most explicitly formalized in the context of rule-governed systems by "Escaping the Agreement Trap: Defensibility Signals for Evaluating Rule-Governed AI," which introduces AI as a policy-grounded statistic for content moderation and related governance tasks (O'Herlihy et al., 22 Apr 2026). In that formulation, AI is not an error rate, not an inter-annotator disagreement score, and not generic model uncertainty. It measures the fraction of audited cases for which the opposite decision is also defensible under the governing rules and precedent. Adjacent literatures use related ambiguity measures, entropy-based disagreement statistics, or taxonomic ambiguity profiles, but typically do not define the same construct under the same name (Klugmann et al., 5 Oct 2025, He et al., 1 Sep 2025, Vijayvargiya et al., 18 Feb 2025, Reynolds, 1 Jul 2026, Liu et al., 2023).

1. Conceptual basis and the Agreement Trap

The policy-grounded AI framework begins from the claim that standard evaluation in rule-governed environments falls into the Agreement Trap: historical agreement with human labels is treated as correctness even when the governing policy permits multiple logically valid outcomes (O'Herlihy et al., 22 Apr 2026). The paper states that this failure mode arises whenever correctness is set-valued. In such settings, disagreement with a moderator label may reflect at least three distinct phenomena: true model error, implicit norm enforcement beyond the written rule, or genuine policy ambiguity. Agreement-based metrics collapse these into a single signal.

Within this framework, correctness is defined relative to an explicit governance structure composed of the content $C$, rules $R=(R_G,R_C)$, and precedent corpus $P$. A proposed action $y \in \{\text{remove}, \text{approve}\}$ is evaluated not by whether it matches a historical label, but by whether it is logically derivable from those materials. This is the key shift from label-grounded to policy-grounded evaluation.

The same paper distinguishes ambiguity from error through a three-level defensibility scheme. L1 denotes a robustly defensible decision, where the rule directly and unambiguously supports the action. L2 denotes a plausibly defensible decision, where the rules are genuinely ambiguous but the action can reasonably be supported. L3 denotes an indefensible decision, where the rules do not support the action. On this account, ambiguity is a governance property rather than a failure condition. The paper states that “Policy ambiguity is not a deficiency but a deliberate design feature of principle-based rules” and that “ambiguity (L2) is not treated as error” (O'Herlihy et al., 22 Apr 2026).

This conceptual distinction matters because AI is intended to isolate normative underspecification. A high value means that many cases admit defensible alternatives; it does not, by itself, mean the model is reasoning badly.

2. Formal definition and internal structure

In the main text of the moderation paper, the Ambiguity Index is defined as follows (O'Herlihy et al., 22 Apr 2026):

$$\text{Ambiguity Index (AI)} = \frac{|\{i : \text{inverse\_check}(i) = Yes\}|}{N}, \quad\text{Target: AI} \leq 0.15.$$

Here $i$ denotes an audited decision instance, $N$ the number of audited instances in the cohort, and $\text{inverse\_check}(i)$ the audit outcome indicating whether the opposite decision is also defensible under the same governance inputs. AI therefore has the interpretation of a proportion, with range $0$ to $1$, typically reported as a percentage.

The audit model receives four inputs: the content $R=(R_G,R_C)$0, the rule hierarchy $R=(R_G,R_C)$1, the precedent corpus $R=(R_G,R_C)$2, and the proposed decision $R=(R_G,R_C)$3. It emits a structured reasoning trace of the form

$R=(R_G,R_C)$4

AI is computed directly from the inverse-check field $R=(R_G,R_C)$5. If $R=(R_G,R_C)$6, the case contributes to ambiguity.

The paper also gives a governance decomposition: AI splits into $R=(R_G,R_C)$7, attributed to platform-level underspecification in $R=(R_G,R_C)$8, and $R=(R_G,R_C)$9, attributed to community-level underspecification in $P$0 (O'Herlihy et al., 22 Apr 2026). This makes AI explicitly hierarchical. In moderation, the same decision can be underdetermined because of general platform principles, because of local community rules, or because of both.

A plain-language interpretation follows directly from the definition. Low AI means that most audited cases are one-sided under the policy: the opposite action is not defensible. High AI means that many cases admit a defensible alternative outcome. In that sense, AI measures structural ambiguity or normative underspecification.

3. Relation to defensibility, agreement, and probabilistic signals

AI is paired with the Defensibility Index (DI), which measures how often the chosen decision is defensible at all (O'Herlihy et al., 22 Apr 2026):

$P$1

DI and AI separate two different properties. DI asks whether the selected action is policy-grounded. AI asks whether the opposite action is also policy-grounded. This yields a four-way conceptual space: high DI with low AI indicates valid and determinate decisions; high DI with high AI indicates valid decisions in an ambiguous policy environment; low DI with high AI indicates both ambiguity and weak reasoning; low DI with low AI indicates wrong decisions in relatively clear-cut cases.

For this reason, AI is not equivalent to accuracy, false positive rate, false negative rate, inter-annotator disagreement, or generic uncertainty. Those metrics compare outputs against labels or consensual annotations. AI instead asks a counterfactual policy question: under the same rules and precedent, could the opposite action also be justified? The paper’s root-cause analysis of 6,760 disagreement cases in the Balanced Sample makes this distinction concrete: only 19.4% were actual model error at L3, while 80.6% were policy-grounded disagreement (O'Herlihy et al., 22 Apr 2026).

The framework also introduces the Probabilistic Defensibility Signal (PDS) as an auxiliary probabilistic signal related to defensibility and ambiguity. Its most AI-relevant component is the inverse-check log-odds

$P$2

where $P$3 (O'Herlihy et al., 22 Apr 2026). In the paper’s formulation, $P$4 is the component most directly tied to AI and is treated as a proxy for $P$5 when citation entropy is low. The scalar collapse of PDS is calibrated as

$P$6

so PDS primarily estimates L1/L2 versus L3, while $P$7 carries the most direct ambiguity information.

This architecture preserves a strict distinction: AI is the formal ambiguity metric, computed from inverse-check outcomes; PDS is an auxiliary probabilistic signal used to approximate stability and ambiguity without additional audit passes.

4. Computation, empirical behavior, and governance use

In practice, AI is computed by running the audit model $P$8 on each case $P$9, extracting the structured JSON trace, and checking whether $y \in \{\text{remove}, \text{approve}\}$0. No repeated sampling is required for the main metric (O'Herlihy et al., 22 Apr 2026). The Governance Gate then aggregates AI over decision cohorts, especially communities with at least 25 decisions, and applies the thresholds

$y \in \{\text{remove}, \text{approve}\}$1

The main empirical results show that ambiguity is substantial in real moderation data. On the Random Sample, DI is 92.3%, AI is 18.3%, and $y \in \{\text{remove}, \text{approve}\}$2 is 45.7%. On the Balanced Sample, DI is 87.3%, AI is 24.9%, and $y \in \{\text{remove}, \text{approve}\}$3 is 54.3% (O'Herlihy et al., 22 Apr 2026). The resulting gap between agreement-based and policy-grounded evaluation is large: the paper reports a 33–46.6 percentage-point difference between agreement-based and policy-grounded metrics, with 79.8–80.6% of the model’s false negatives corresponding to policy-grounded decisions rather than true errors.

One of the paper’s central experiments varies the specificity of the rules while keeping the same decisions fixed. The same 37,286 r/AskReddit decisions were audited under three versions of the same community rules.

Rule version	AI	DI
Title Only	18.2%	97.4%
Sidebar	8.8%	98.0%
Wiki	7.4%	98.1%

These results show a 10.8 percentage-point drop in AI from title-only to full wiki while DI remains nearly unchanged (O'Herlihy et al., 22 Apr 2026). The mechanism is concentrated in removals: removal AI falls from 32.0% to 10.2%, while removal L1 rises from 48.5% to 74.6%. The paper characterizes this as largely an L2 $y \in \{\text{remove}, \text{approve}\}$4 L1 conversion, meaning that fuller rule specification resolves ambiguity rather than rescuing many outright errors.

At the fleet level, the paper reports three community types across 270 communities with at least 25 decisions. Earned Autonomy ($y \in \{\text{remove}, \text{approve}\}$5) has mean DI 96.8% and mean AI 7.0%. Policy Gaps ($y \in \{\text{remove}, \text{approve}\}$6) has mean DI 67.7% and mean AI 36.0%. Normative Complexity ($y \in \{\text{remove}, \text{approve}\}$7) has mean DI 92.2% and mean AI 23.0% (O'Herlihy et al., 22 Apr 2026). This is important because high-AI cohorts are not necessarily low-quality reasoning regions; some are high-DI, high-AI environments where the system reasons well under genuinely underspecified rules.

The Governance Gate uses AI operationally. At the selected operating point, it achieves 78.6% automation coverage with 64.9% risk reduction (O'Herlihy et al., 22 Apr 2026). The paper further notes that the binding constraint is the AI threshold, not DI: tightening DI from 85% to 90% changed nothing because AI $y \in \{\text{remove}, \text{approve}\}$8 was already limiting automation. This supports a strong governance conclusion: in this setting, ambiguity rather than misreasoning is the principal limiter of safe automation.

5. Stability, interpretation, and limitations

The paper validates AI-related signals through PDS behavior and repeated-sampling analysis (O'Herlihy et al., 22 Apr 2026). On the Random Sample, the inverse-check log-odds $y \in \{\text{remove}, \text{approve}\}$9 has Spearman correlation $$\text{Ambiguity Index (AI)} = \frac{|\{i : \text{inverse\_check}(i) = Yes\}|}{N}, \quad\text{Target: AI} \leq 0.15.$$0–$$\text{Ambiguity Index (AI)} = \frac{|\{i : \text{inverse\_check}(i) = Yes\}|}{N}, \quad\text{Target: AI} \leq 0.15.$$1 with the binary inverse-check label, with all $$\text{Ambiguity Index (AI)} = \frac{|\{i : \text{inverse\_check}(i) = Yes\}|}{N}, \quad\text{Target: AI} \leq 0.15.$$2, and mean $$\text{Ambiguity Index (AI)} = \frac{|\{i : \text{inverse\_check}(i) = Yes\}|}{N}, \quad\text{Target: AI} \leq 0.15.$$3 is 6–7 times higher for ambiguous than unambiguous cases. By defensibility level, AI and $$\text{Ambiguity Index (AI)} = \frac{|\{i : \text{inverse\_check}(i) = Yes\}|}{N}, \quad\text{Target: AI} \leq 0.15.$$4 behave systematically: for L1, AI is 1.7–2.9% and mean $$\text{Ambiguity Index (AI)} = \frac{|\{i : \text{inverse\_check}(i) = Yes\}|}{N}, \quad\text{Target: AI} \leq 0.15.$$5 is 0.063–0.083; for L2, AI is 58.8–61.4% and mean $$\text{Ambiguity Index (AI)} = \frac{|\{i : \text{inverse\_check}(i) = Yes\}|}{N}, \quad\text{Target: AI} \leq 0.15.$$6 is 0.783–0.809; for L3, AI is 86.1–90.5% and mean $$\text{Ambiguity Index (AI)} = \frac{|\{i : \text{inverse\_check}(i) = Yes\}|}{N}, \quad\text{Target: AI} \leq 0.15.$$7 is 0.910–0.932.

Repeated sampling is used to test whether instability is mainly governance ambiguity or merely decoding noise. The reported $$\text{Ambiguity Index (AI)} = \frac{|\{i : \text{inverse\_check}(i) = Yes\}|}{N}, \quad\text{Target: AI} \leq 0.15.$$8 ratio for Flippers versus Stable cases remains roughly flat across temperatures—1.63, 1.64, 1.57, and 1.52 for $$\text{Ambiguity Index (AI)} = \frac{|\{i : \text{inverse\_check}(i) = Yes\}|}{N}, \quad\text{Target: AI} \leq 0.15.$$9—while $i$0 rank correlation with $i$1 remains high at 1.000, 0.963, 0.901, and 0.878, and aggregate DI is nearly temperature invariant at 71.1%, 71.3%, 72.2%, and 73.0% (O'Herlihy et al., 22 Apr 2026). A particularly notable observation is that at $i$2, action decisions were perfectly stable—84,000 approve and 16,000 remove, with zero flips—while 50% of cases still showed moderate or high reasoning instability. The paper interprets this as evidence that ambiguity can persist at the level of derivation and justification even when the top-line action is unchanged.

Practically, high AI should be read as a measure of interpretive latitude in the governance regime. The paper associates elevated AI with high-level rules, principle-based or subjective policies, novel fact patterns not well covered by precedent, sparse or immature community rule elaboration, and removal decisions requiring affirmative textual grounding under vague rules (O'Herlihy et al., 22 Apr 2026). Its governance advice is correspondingly direct: high $i$3 suggests platform policy revision, high $i$4 suggests community precedent development, and high deployment-time AI argues for human review rather than automatic enforcement.

The paper also sets clear boundaries. AI is not a Platonic measure of ambiguity independent of implementation. It depends on audit-model quality, prompt-template design, the completeness of $i$5 and $i$6, and the independence of auditor $i$7 from classifier $i$8. Under the Same-Backbone Condition, raw PDS may reflect both governance ambiguity and backbone-specific uncertainty. Because AI is derived from the auditor’s inverse-check judgment, it can also reflect representation incompleteness when relevant norms are absent from the supplied rules or precedent (O'Herlihy et al., 22 Apr 2026).

6. Related ambiguity measures and broader uses

Outside rule-governed moderation, several papers define nearby constructs rather than the same AI metric. They share the general aim of quantifying interpretive multiplicity, but they operationalize it differently.

Domain	Construct	Formal status
Categorical annotation	$i$9, $N$0	Explicit scalar ambiguity measure
Statutory interpretation for AI	$N$1	Entropy proxy for ambiguity
Software agents	underspecification detection, FPR/FNR, cosine distance	No explicit AI
Adversarial pragmatics	diagnostic ambiguity, taxonomy drift, judge validity	No single scalar AI
NLI ambiguity	multilabel NLI ambiguity with disambiguating rewrites	No single scalar AI

The closest explicit scalar measure outside moderation appears in "Quantifying Ambiguity in Categorical Annotations," which defines an ambiguity functional $N$2 over a soft-label distribution with a special “can’t solve” category (Klugmann et al., 5 Oct 2025). In that paper,

$N$3

and the measure is interpreted as annotation-level aleatoric ambiguity rather than annotator error. This is a scalar on $N$4, but it addresses categorical annotation rather than policy-grounded decision validity.

"Statutory Construction and Interpretation for Artificial Intelligence" defines ambiguity through entropy over judgments across a panel of reasonable interpreters (He et al., 1 Sep 2025). Its scenario-level quantity is

$N$5

which the paper explicitly describes as an approximation to “aleatoric uncertainty, or ambiguity” of the ruleset as applied to a scenario. This is closer to interpretive disagreement among admissible readers than to the moderation paper’s inverse-check formulation.

Several other literatures remain explicitly non-scalar. "Interactive Agents to Overcome Ambiguity in Software Engineering" treats ambiguity as underspecification and studies three separable capabilities: detecting ambiguity, benefiting from interaction, and asking targeted clarifying questions, with ambiguity detection evaluated by Accuracy, FPR, and FNR and clarification quality measured partly through cosine distance of embeddings (Vijayvargiya et al., 18 Feb 2025). "Adversarial Pragmatics for AI Safety Evaluation" formalizes diagnostic ambiguity, policy-boundary ambiguity, criterion conflict, taxonomy drift, and adjudication stability, but does not collapse them into a single Ambiguity Index (Reynolds, 1 Jul 2026). "We're Afraid LLMs Aren't Modeling Ambiguity" operationalizes linguistic ambiguity through its effect on entailment relations; in the AmbiEnt benchmark, ambiguous examples are those with more than one plausible NLI label and constitute 35.2% of the dataset (Liu et al., 2023).

Additional papers broaden the conceptual perimeter. "Machine Learning Processes as Sources of Ambiguity" introduces ambiguity of process in AI art and argues that ambiguity can arise from dataset curation, model training, and application, rather than only from outputs (Sivertsen et al., 2024). "Generative AI in Managerial Decision-Making" provides a four-dimensional business ambiguity taxonomy—Contextual Uncertainty, Definition Imprecision, Knowledge Inconsistency, and Linguistic Imprecision—and operationalizes ambiguity through unresolved ambiguity count rather than a scalar index (Birim et al., 4 Mar 2026). "Automatic Ambiguity Detection" defines a polysemy index for words based on contextual multimodality in an unlabeled corpus (Sproat et al., 2019). "When AI and Experts Agree on Error: Intrinsic Ambiguity in Dermatoscopic Images" does not define an index, but operationalizes intrinsic ambiguity through persistent cross-model error, expert disagreement, and image-quality degradation (Cino et al., 1 Apr 2026).

This suggests that “Ambiguity Index” now names a family of domain-specific measurement problems rather than a single universal statistic. In the strictest technical sense, however, the best-defined contemporary AI usage remains the policy-grounded metric introduced for rule-governed AI evaluation: the proportion of audited cases for which the opposite decision is also defensible under the same rules and precedent (O'Herlihy et al., 22 Apr 2026).