📈 economics

Sufficientarian Grading Rules and Rankings: Characterizations and Implementation

This paper characterizes sufficientarian grading rules and the total preorders they induce based on binary grading functions, while also defining additional rankings using sufficiency-gap information and demonstrating that specific inclusive and strategy-proof protocols, such as simple majority, can be used to select a particular grading rule.

Original authors: Marcello Basili, Ernesto Savaglio, Stefano Vannucci

Published 2026-07-10

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Marcello Basili, Ernesto Savaglio, Stefano Vannucci

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 running a massive, futuristic video game where every player starts with a unique "capability backpack." Inside these backpacks are different levels of skills, tools, and resources—let's call them "affordances." Some backpacks have a rusty spoon and a torn map; others have a golden compass and a full water filter. The big question the authors of this paper are asking is: How do we decide who has "enough" to play the game properly, and how do we rank players when they don't?

The "Enough" Checklist

The paper suggests that instead of comparing players to each other (like saying "Player A is richer than Player B"), we should use an absolute checklist. Think of it like a "Pass/Fail" test for life. The authors propose a system called a Binary Grading Function (BGF).

Imagine a giant, invisible referee holding a checklist of "thresholds." These thresholds are like a set of minimum requirements: "Must have at least a compass," "Must have at least a water filter." If a player's backpack meets or exceeds any of these minimums, they get a 1 (Pass/Yes). If they fall short of all of them, they get a 0 (Fail/No).

The paper proves that for this "Pass/Fail" system to be fair and logical, it must follow three simple rules:

Isotony: If you upgrade your backpack (get a better compass), you should never get a lower grade. Better stuff always equals a better or equal score.
Separability: Your grade depends only on your own backpack. It doesn't matter if your neighbor has a super-cool backpack; your score is based entirely on your own gear.
Symmetry: The rules apply to everyone equally. It doesn't matter if you are Player 1 or Player 100; the same checklist is used for all.

If a grading system follows these three rules, the authors show it is mathematically guaranteed to be a "sufficientarian" rule. It's like finding the secret code that makes the game fair without needing to know exactly what the "perfect" backpack looks like, just what the minimum looks like.

Ranking the Players: Who is "Worse Off"?

Once we know who passed (1) and who failed (0), how do we rank the players? The paper explores a few ways to do this, but it highlights a major debate regarding one popular idea.

Some people think we should just count how many people passed. If 90% of the class passes, that's a "good" class. The authors call this the Sufficiency-Count ranking. However, they point out a major flaw often cited by critics: this method treats a class where everyone barely passed the same as a class where everyone passed with flying colors. It also fails to tell us how far behind the failing students are.

The paper notes that many advocates of sufficientarianism actually reject using the Sufficiency-Count ranking to guide rescue policies because it doesn't measure the depth of the problem. Instead, the authors argue that for fixing problems (like helping the failing students), we need to know the gap. How far is a player's backpack from the "Pass" line?

The "Gap" Idea: Imagine the "Pass" line is a cliff edge. If you are standing right on the edge, you are safe. If you are 10 feet back, you are in trouble. If you are 100 feet back, you are in deep trouble.
The authors define a "distance" metric. They calculate how many "steps" a player is away from the minimum requirements.
They propose two main ways to rank the whole group based on these gaps:
1. The "Average Gap" (Min-Average): We look at the total distance of all failing players and divide by the number of failing players. We want to minimize this average distance. This is like a utilitarian approach: "Let's help the group get as close to the cliff edge as possible on average."
2. The "Worst-Case Gap" (Min-Leximax): We look at the player who is furthest away from the edge. We try to help that specific person first, then the next furthest, and so on. This is like an egalitarian approach: "We must bring the person in the deepest hole out first, before we worry about the others."

The paper does not explicitly reject the Sufficiency-Count ranking as a tool for describing a society, but it emphasizes that relying on it to guide remedial policies is highly controversial and often rejected by sufficientarian advocates because it ignores the severity of insufficiency.

The "Who Decides the Rules?" Problem

Here is the trickiest part. Who gets to pick the checklist? Who decides what "enough" means? Is it a compass? A water filter? Or maybe a golden compass?

The authors note that in the real world, we can't just have a dictator pick the rules. We need a democratic way to agree on the "thresholds." But here's the catch: if people know the rules will be decided by a vote, they might lie and say, "I want the threshold to be super low so I pass!" or "I want it super high so my friends fail!"

The paper suggests a clever solution using Mechanism Design. They show that if we treat the "thresholds" like points on a map, and if everyone's opinion is "single-peaked" (meaning everyone has one favorite threshold, and they prefer thresholds that are closer to their favorite over ones that are further away), we can use a Simple Majority Vote to pick the rules.

Imagine a line of possible thresholds. If everyone has a favorite spot on that line, and they prefer spots closer to their favorite, then voting on the middle spot (the median) is a "strategy-proof" way to decide. This means no one can cheat the system by lying about what they want. If you have an odd number of voters, a simple majority vote will always pick a fair, honest threshold that no one can manipulate.

What the Paper Does NOT Say

It is important to know what this paper doesn't do:

It does not tell you exactly what the "enough" threshold should be (e.g., it doesn't say "enough is $50 a day"). It only explains how to pick one fairly.
It does not claim that "counting heads" (Sufficiency-Count) is a bad way to just describe a society. It acknowledges that while many sufficientarian advocates reject it for guiding rescue policies, the paper itself focuses on providing alternative "gap" rankings for those specific policy decisions.
It does not solve the problem of "burdens" (like having to carry heavy rocks) or "public goods" (like clean air) in this specific version, though the authors suggest their math could be stretched to include those in the future.

The Bottom Line

The authors have built a mathematical toolkit that says:

Fairness in checking if someone has "enough" comes from three simple rules (Isotony, Separability, Symmetry).
Fixing problems requires looking at how far people are from the goal, not just how many people failed.
Choosing the goal can be done democratically without cheating, as long as we use a simple majority vote on a specific type of "map" of options.

It's a way to turn the fuzzy idea of "everyone should have enough" into a clear, fair, and mathematically sound game plan.

Technical Summary: Sufficientarian Grading Rules and Rankings

Problem Statement
The paper addresses a gap in the literature on sufficientarianism, a distributive principle asserting that "everyone should have enough." While existing literature extensively characterizes sufficientarian rankings (total preorders over social states), it often relies on exogenously given thresholds or focuses solely on the resulting orderings rather than the underlying ratings (binary assessments of individual sufficiency). Furthermore, the literature largely neglects the procedural mechanism for selecting the specific threshold system required to operationalize these rules. The authors aim to:

Characterize sufficientarian rating rules (binary grading functions) independently of explicit threshold references.
Derive and characterize specific ranking rules (including sufficiency-gap rankings) from these ratings.
Provide a mechanism-design solution for the collective selection of the threshold system itself.

Methodology and Model
The authors construct a formal model based on a finite capability-type space $X$ , defined as the Cartesian product of finite, linearly ordered sets of affordances/achievements.

Binary Grading Functions (BGFs): A sufficientarian rule is modeled as a function $g: X^n \to \{0, 1\}^n$ that assigns a binary grade (1 for "sufficient," 0 for "insufficient") to each agent's capability type.
Threshold Systems: A threshold system $X^*$ is defined as an antichain (a set of pairwise incomparable elements) in $X$ . An agent is rated 1 if their capability type dominates at least one element in $X^*$ .
Metric Structure: The paper exploits the lattice structure of $X$ (a finite distributive lattice) to define a natural metric $d$ based on the length of intervals (shortest paths in the covering graph). This metric allows for the calculation of "sufficiency gaps" (distances from the threshold system).
Aggregation: The selection of a threshold system is treated as a social choice problem over the set of antichains $AX$, which forms a bounded distributive lattice. Preferences over these systems are modeled as single-peaked with respect to a metric betweenness relation.

Key Contributions and Results

1. Characterization of Sufficientarian Grading Rules (Rating Rules)
The paper establishes that a non-trivial Binary Grading Function is a sufficientarian grading rule if and only if it satisfies three independent properties:

Isotony: Higher capability levels never result in lower grades.
Separability: An agent's grade depends only on their own capability type, not on the assignments of others.
Symmetry: The grading rule is agent-neutral (universal).
Significance: This characterization is unique because it defines sufficientarian rules without any explicit or implicit reference to thresholds. It proves a one-to-one correspondence between such rules and threshold systems (antichains).

2. Characterization of Induced Rankings
The authors characterize the total preorders induced by these rating rules:

Simple Sufficientarian Preorder: A coarse ranking with two indifference classes (all sufficient vs. at least one insufficient). It is characterized as the coarsest "top-faithful" extension of the partial order induced by the grading rule.
Sufficiency-Count Preorder: A ranking based on the number of agents who are sufficient. It is characterized by Anonymity and Strict Monotonicity with respect to the grading function.
Sufficiency-Gap Preorders: To address policy design for remedial actions (minimizing insufficiency), the paper introduces rankings based on the distance of capability types from the threshold system. Two specific rules are characterized:
- Min-Average Sufficiency-Gap: Characterized by Anonymity, Strong Antitony, and Restricted Translation Invariance. This aligns with generalized utilitarian principles.
- Min-Leximax Sufficiency-Gap: Characterized by Strong Antitony and Restricted Hammond Equity. This aligns with egalitarian principles.

3. Mechanism Design for Threshold Selection
Addressing the "open issue" of how to select a specific threshold system, the paper proposes a protocol based on mechanism design:

Domain: The set of all antichains (threshold systems) is shown to form a bounded distributive lattice.
Preferences: Agents' preferences over threshold systems are modeled as single-peaked with respect to the metric betweenness on this lattice.
Result: The paper proves the existence of aggregation rules (protocols) that are Anonymous, Idempotent (unanimity-respecting), and Strategy-Proof on this domain.
Specific Implementation: If the number of agents is odd, the Simple Majority Rule is a valid strategy-proof protocol for selecting a unique threshold system. The paper also notes that sub-protocols can be used to select committees or handle even numbers of agents.

Significance and Claims
The authors claim their work provides a "sound minimal common core" for sufficientarian stances by:

Bridging Literature: Connecting the literature on ratings (benchmarks) and rankings (orderings) by deriving the latter from the former.
Threshold Independence: Offering characterizations of sufficientarian rules that do not rely on exogenous threshold definitions, thereby clarifying the structural properties of the rules themselves.
Operationalizing Selection: Moving beyond theoretical discussions of "impartial observers" or "compassion" to provide a concrete, strategy-proof democratic mechanism for selecting the necessary threshold systems.
Policy Relevance: Providing specific ranking tools (sufficiency-gap rules) that can guide remedial policies aimed at minimizing insufficiency, distinguishing between utilitarian (average gap) and egalitarian (leximax gap) approaches.

The paper concludes that while the sufficiency threshold is often treated as a given, the proposed framework allows for its rigorous, collective, and strategic selection, making the implementation of sufficientarian principles more robust and applicable to real-world social choice scenarios.