Axiomatic Choice and the Decision-Evaluation Paradox

Abstract

Axiomatic Choice provides a foundational theoretical framework for formalizing human values in decision-making systems. Moving beyond traditional Social Choice Theory, this work addresses how to ensure systems follow not just the right outcomes, but the right procedures. The authors identify the Decision-Evaluation Paradox, where committing to a rule derived from a specific value can counterintuitively result in a decision that violates that very value.

Introduction

In fields like AI alignment and Reinforcement Learning from Human Feedback (RLHF), translating complex human values into formal constraints remains a significant challenge. Traditional Social Choice Theory often focuses narrowly on "blackbox" outcomes—for instance, ensuring the winner of an election is the most popular candidate. However, humans care deeply about the process of decision-making, demanding fairness, transparency, and auditability. If an AI selects a "correct" outcome through a biased or deceptive process, it is often rejected. This paper addresses the lack of a formal, domain-agnostic language to describe these procedural values and the logical pitfalls that occur when enforcing them.

Formal Framework

The authors propose a mathematical model where a Decision is defined as a triplet $(x, f, y)$:

• Profile ($x$): The input or context (e.g., voter preferences).
• Rule ($f$): The procedure used to make the choice.
• Outcome ($y$): The result of applying the rule to the profile ($y = f(x)$).

Expand

Source, p. 2

Taxonomy of Axioms

Using this triplet, the authors develop a taxonomy of axioms based on which components of the decision they constrain. These range from simple constraints on outcomes to complex dependencies between the rule and the result.

Expand

Source, p. 3

As shown in the taxonomy, axioms can be classified into several types:

1. Blackbox: Constraints only on the input and output (the "what").
2. Procedural: Constraints only on the rule (the "how").
3. Caudal/Exigent: Complex constraints depending on the relationship between the rule and the outcome (e.g., "Use Rule A, unless it produces Outcome B").

The Decision-Evaluation Paradox

The central finding of the paper is the Decision-Evaluation Paradox. This occurs when a complex value (an "exigent" axiom) implies a specific rule for implementation. However, once that rule is fixed and applied, the resulting decision may violate the original axiom. This happens because axioms may require a rule to change based on the outcome, but a rule, by definition, is a fixed function.

Theorem 1 formally proves that making a decision using the rule implied by a "forcing" axiom does not guarantee the decision obeys that axiom. Conversely, Theorem 2 shows this paradox cannot occur when dealing only with simple "Blackbox" or "Procedural" axioms. The danger lies in complex, interdependent values.

Implications for AI Alignment

The framework also formalizes Counterfactual Deception, explaining why AI "explanations" can be misleading. If an observer cares about the procedure but can only see the outcome, a system can deceive them by claiming it used a specific rule when it actually used a different one that happened to produce the same result.

Because the framework treats $x$ and $y$ as arbitrary sets, it is domain-agnostic. It can be applied to diverse fields, from LLM prompt-engineering to autonomous vehicle ethics, providing a rigorous way to analyze how combining simple values (using AND/OR logic) can inadvertently create complex "Exigent" axioms vulnerable to the paradox.

Limitations and Future Work

While foundational, the work faces several hurdles for practical application:

• Elicitation: Extracting formal axioms from natural language remains an unsolved challenge.
• Complexity: Evaluating whether a decision obeys an "Exigent" axiom is computationally more demanding than checking simple outcomes.
• Cardinality: The current model is binary; future work could explore "graded" satisfaction of axioms.
• Causality: Deception definitions depend on the observer's causal model, which varies among stakeholders.

@misc{abramowitz2025axiomatic,
  author = {Ben Abramowitz and Nicholas Mattei},
  title = {{Axiomatic Choice and the Decision-Evaluation Paradox}},
  year = {2025},
  eprint = {2509.21836},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2509.21836},
}