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Axiomatic Choice

This paper introduces "Axiomatic Choice," a formal framework that uses normative axioms to capture diverse decision-making concerns—such as ethics and procedural fairness—to analyze decision evaluation, transparency, and the inherent paradoxes in justifying choices.

Original authors: Ben Abramowitz, Nicholas Mattei

Published 2026-02-11

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Ben Abramowitz, Nicholas Mattei

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are at a restaurant with a group of friends. You are trying to decide what to eat.

Usually, when we make decisions, we focus on the outcome: "I want pizza." This is what social scientists call "Social Choice." But in real life, we also care deeply about how the decision is made: "I want us to vote on it," or "I want the person who pays to decide," or "I want the decision to be fair to the vegetarians."

This paper, "Axiomatic Choice," is a new mathematical way to talk about both the what and the how at the same time.

1. The "Rules of the Game" (The Taxonomy)

The authors argue that our "rules" (which they call axioms) aren't all the same. They categorize them like different types of "vibes" in a decision:

The Consequentialist (The "Result" Person): This person only cares about the final result. "I don't care if we flipped a coin or held a massive debate; as long as we get pizza, I'm happy."
The Procedural (The "Process" Person): This person cares about the method. "I don't care if we get pizza or salad, but we MUST vote democratically to do it."
The Structural (The "Context" Person): This person cares about the situation. "If it's a Friday, we should eat out; if it's a Monday, we should cook."

2. The "Decision-Evaluation Paradox" (The Broken Promise)

This is the most mind-bending part of the paper. The authors discovered a "glitch in the matrix" of decision-making.

Imagine you make a solemn promise: "I promise that every time we eat, we will use a democratic vote." This is your "Rule."

Now, imagine a specific situation (a "profile") comes up where a vote is impossible—maybe everyone is too tired to talk. To keep the peace, you just pick pizza.

Later, your friend looks at the decision and says, "Wait! You promised we would always vote! You broke your rule!"

You argue, "But look! The result is pizza, which is what we would have voted for anyway! The outcome is fine!"

The Paradox is this: You can follow a rule to pick an outcome, but the very act of using that rule might technically violate the rule itself. It’s like promising to "always be spontaneous"—the moment you follow a rule to be spontaneous, you aren't being spontaneous anymore.

3. Transparency and Deception (The "Fake It 'Til You Make It" Problem)

Because of this paradox, the paper warns about Deception.

Imagine a politician says, "I am using the 'Fairness Rule' to make this law." But in reality, they are using a secret, complex formula that usually looks fair but actually favors them.

If you only look at the outcome (the law), it looks fine. You can't tell they cheated. But if you could see the process (the formula), you’d see they lied. The authors show that when we try to explain decisions to people, we often use "shortcuts" that are technically lies, even if the result is what people wanted. They call this "Counterfactual Deception"—telling someone a decision was made one way, when it was actually made another way.

Why does this matter?

As we move into a world where AI (like ChatGPT) makes decisions for us—from who gets a loan to how a self-driving car reacts in an emergency—we can't just tell the AI "be fair."

"Fairness" is a messy mix of results, processes, and contexts. This paper provides the mathematical "grammar" to help us tell AI exactly what we mean, while warning us that even with perfect math, we might still end up accidentally deceiving ourselves or breaking our own promises.

Technical Summary: Axiomatic Choice

Title: Axiomatic Choice
Authors: Ben Abramowitz and Nicholas Mattei (Tulane University, San Diego State University)

1. Problem Statement

Current formalisms for decision-making, primarily rooted in Social Choice Theory, focus heavily on aggregating agent preferences over outcomes. However, human decision-making is driven by a broader range of normative concerns that include how a decision is made (e.g., transparency, fairness, procedural justice, or ethical constraints).

Existing methods struggle to:

Formally capture diverse normative positions (e.g., "the rule must be simple" vs. "the outcome must be equal").
Bridge the gap between making a decision (selecting a rule) and evaluating a decision (checking if a specific outcome/rule pair satisfies a norm).
Address the complexities of human-in-the-loop systems, such as Reinforcement Learning from Human Feedback (RLHF), where feedback can be unconstrained (natural language) and potentially conflicting.

2. Methodology

The authors propose Axiomatic Choice, a domain-agnostic framework that abstracts decision-making above Social Choice.

Formal Model:

Components: They define a decision as a tuple $(x, f, y)$ , where $x$ is a profile (input), $f$ is a rule (function), and $y$ is an outcome (output).
Axioms: An axiom $A$ is defined as a partition of the set of all possible decisions $D$ , where $A(x, f, y) = 1$ if the decision obeys the axiom and $0$ otherwise.
Taxonomy: They categorize axioms based on their structural properties into several classes:
- Structural: Depend on the profile $x$ .
- Procedural: Depend on the rule $f$ .
- Consequentialist: Depend on the outcome $y$ .
- Blackbox: Depend on the profile-outcome pair $(x, y)$ .
- Caudal: Depend on the rule and outcome $(f, y)$ .
- Exigent: Do not fit the above categories.

3. Key Contributions & Results

A. The Decision-Evaluation Paradox
The paper’s most significant theoretical contribution is the formalization of the Decision-Evaluation Paradox.

The Paradox: If an axiom is "forcing" (it implies a unique outcome for every profile), one can derive an "implied rule" to make decisions. However, using that implied rule does not guarantee that the resulting decisions will obey the original axiom.
Condition: This paradox occurs specifically when an axiom is an Arrovian impossibility (no single rule can satisfy it for all profiles) but is still implementable (it doesn't lead to an impasse).

B. Transparency and Deception
The authors provide a formal framework for evaluating the ethics of decision explanations:

Transparency: A decision is transparent if an agent has sufficient information (e.g., $x, f,$ or $y$ ) to determine if the axiom was obeyed.
Deception: They define Counterfactual Deception, which occurs when a statement implies a different rule $f'$ was used than the one actually applied. They show that if an agent's normative position is procedural, a "blackbox" explanation (only providing $x$ and $y$ ) can be materially deceptive.

C. Limitations of "Voting by Axioms"
The paper critiques existing methods like "Voting by Axioms" (which combines multiple procedural axioms). They prove that even if all constituent axioms are intraprofile (meaning they only require knowledge of the current profile to evaluate), the combined procedure can still fall victim to the Decision-Evaluation Paradox.

D. Redefinition of Intraprofile Axioms
They provide a new, domain-agnostic, and language-independent definition of intraprofile axioms, characterizing them as those that are "procedural extensions" of blackbox axioms.

4. Significance

The paper provides a rigorous mathematical foundation for the "normative layer" of AI and democratic decision-making. Its significance lies in three areas:

AI Alignment: It offers a way to formalize complex human values in RLHF and Constitutional AI, moving beyond simple reward functions to complex axiomatic constraints.
Algorithmic Accountability: By formalizing deception and transparency, it provides tools to audit whether AI explanations are merely "justifications" that mask non-compliant underlying rules.
Social Choice Theory: It expands the scope of Social Choice from "what should the winner be?" to "what should the process look like?", identifying fundamental limits on how we can combine different social values without creating logical contradictions or deceptive implementations.