📈 economics

A Dominance Argument Against Incompleteness

This paper argues that a combination of a weak negative dominance principle and a weak ex ante Pareto principle, supported by modest auxiliary assumptions, effectively rules out many forms of incompleteness in both individual prudential rankings and moral evaluations of outcomes and lotteries.

Original authors: Christian Tarsney, Harvey Lederman, Dean Spears

Published 2026-04-21

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Christian Tarsney, Harvey Lederman, Dean Spears

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

The Big Question: Can Some Things Just Be "Incomparable"?

Imagine you are deciding between two very different careers:

The Artist: You have total creative freedom, but you might struggle to pay rent.
The Banker: You have a steady, high salary, but you feel bored and trapped.

Many philosophers argue that these two lives are incomparable. This means:

The Artist life isn't better than the Banker life.
The Banker life isn't better than the Artist life.
They aren't equal either.

It's like comparing an apple to a symphony. They are just too different to rank. If you add $1,000 to the Banker's salary, it still doesn't make the Banker life "better" than the Artist life; they just remain in a state of "no comparison."

The authors of this paper say: "Stop. This doesn't make sense." They argue that if you accept two very common-sense rules about how we make choices, you cannot say that lives (or outcomes) are incomparable. You must be able to rank them.

The Two Rules of the Game

To prove their point, the authors use two principles that sound like common sense but have powerful consequences.

Rule #1: The "No Magic" Rule (Negative Dominance)

The Idea: You can't say Lottery A is better than Lottery B unless Lottery A has at least one possible outcome that is better than one possible outcome of Lottery B.

The Analogy: Imagine two boxes of mystery gifts.

Box A contains: A broken toaster, a wet sock, and a stale cracker.
Box B contains: A rusty nail, a used tissue, and a rock.

If you open Box A and Box B, and every single item in Box A is worse than or equal to every single item in Box B, you cannot claim Box A is the "better" box. There is no "magic" where the combination of bad things makes it good. If Box A has nothing better to offer than Box B, it can't be the winner.

Rule #2: The "Everyone Wins" Rule (Personal Good)

The Idea: If you have a choice between two lotteries, and Lottery A gives everyone involved a better (or equal) chance at a good life than Lottery B, then Lottery A is the better choice.

The Analogy: Imagine you are a parent choosing a school for your two kids.

School A: Kid 1 has a 50% chance of getting an A and 50% of getting a B. Kid 2 has a 50% chance of an A and 50% of a B.
School B: Kid 1 has a 50% chance of getting an A and 50% of getting a B. Kid 2 has a 50% chance of getting an A and 50% of getting a B.
School C (The Winner): Kid 1 has a 50% chance of getting an A+ and 50% of a B. Kid 2 has the exact same chances as in School A.

Even if we don't know who will get the A+, School C is clearly better because it offers a strictly better chance for at least one kid and doesn't hurt the other. It's a "no-brainer" improvement.

The Trap: The "Sweetened" Lottery

The authors set a trap for the idea of "incomparability" using a coin flip.

The Setup:
Imagine two people, P1 and P2.

Life A (Artist) and Life B (Banker) are incomparable.
Life A+ is the Artist life, but with a tiny bonus (e.g., an extra $1,000). It is still incomparable with Life B.

Now, you have to choose between two lotteries (coin flips):

Lottery 1:

Heads: P1 gets Life A, P2 gets Life B.
Tails: P1 gets Life A, P2 gets Life B.
(Basically, they both get the same mix every time.)

Lottery 2:

Heads: P1 gets Life A+ (The slightly better Artist life), P2 gets Life B.
Tails: P1 gets Life B, P2 gets Life A.

The Conflict:

The "Incomparable" Argument:
If Life A and Life B are truly incomparable, then every possible outcome of Lottery 1 is incomparable to every possible outcome of Lottery 2.
- (A, B) is incomparable to (A+, B).
- (A, B) is incomparable to (B, A).
- Since no outcome in Lottery 2 is "better" than any outcome in Lottery 1, Rule #1 (Negative Dominance) says: Lottery 2 cannot be better than Lottery 1.
The "Common Sense" Argument:
Look at the people!
- P1 gets a 50% chance of Life A and 50% of Life B in Lottery 1.
- P1 gets a 50% chance of Life A+ and 50% of Life B in Lottery 2.
- Life A+ is better than Life A. So, Lottery 2 is strictly better for P1.
- P2 gets the exact same chances in both lotteries.
- Since Lottery 2 is better for P1 and equal for P2, Rule #2 (Personal Good) says: Lottery 2 must be better than Lottery 1.

The Result:
You have a contradiction.

Rule #1 says: Lottery 2 is not better.
Rule #2 says: Lottery 2 is better.

The only way to fix this contradiction is to admit that Life A and Life B were never actually incomparable. They must be rankable. If they were rankable, then (A+, B) would be better than (A, B), and the contradiction disappears.

Why This Matters

The authors argue that if you believe in these two simple rules (which almost everyone does), you have to give up the idea that some lives or outcomes are "incomparable."

For Personal Choices: You can't say "Artist vs. Banker" is a tie that can't be broken. There must be a way to say one is better, or they are equal.
For Population Ethics: This applies to big questions too, like "Is it better to have a world with 1 billion happy people or 10 billion slightly less happy people?" The paper argues you can't say these are "incomparable." You must be able to rank them.

The "Infinite" Loophole (And Why It Doesn't Help)

The paper admits there is a weird exception. If you deal with infinity (infinite numbers of people or infinite outcomes), these two rules can fight each other in a way that allows incomparability to survive.

The Analogy: Imagine a hotel with infinite rooms. You can move guests around in ways that break normal math rules.

However, the authors say: "We are talking about real life, which is finite. In the real world, with a finite number of people, the trap works. You can't hide behind infinity to say things are incomparable."

The Bottom Line

The paper is a "Gotcha!" moment for philosophers who love the idea of incomparability. It says:

"You can't have your cake and eat it too. You can't claim that some things are unrankable while also believing that 'if everyone wins, it's a win' and 'if nothing is better, it's not better.' You have to pick one. And if you pick the common-sense rules, you have to admit that everything can be ranked."

1. Problem Statement

The paper addresses the problem of value incompleteness (or incomparability) in ethics and decision theory. In many philosophical frameworks, it is argued that certain alternatives cannot be ranked as "better than," "worse than," or "equal to" one another.

Personal Incompleteness: Two lives (e.g., being an artist vs. a banker) may be incomparable from an individual's perspective.
Impersonal Incompleteness: Two social outcomes (e.g., different population sizes or distributions of welfare) may be incomparable from an impersonal (social welfare) perspective.
The Challenge: Standard decision-theoretic principles (like Stochastic Dominance) often assume a complete ordering. Defenders of incompleteness argue that these principles must be weakened or rejected to accommodate incomparability.

The authors aim to demonstrate that accepting Negative Dominance and Personal Good (two weak, intuitive principles) renders the existence of incomparable lives and outcomes logically impossible in finite contexts.

2. Methodology and Framework

The authors employ a formal decision-theoretic framework applied to population ethics.

Formal Objects:
- Outcomes: Functions mapping possible individuals to lives (including non-existence).
- Lotteries: Discrete probability distributions over finite sets of outcomes.
- Relations: "Better than," "equally good," and "incomparable" (neither better, worse, nor equal).
Key Principles:
1. Negative Dominance (ND): If no possible outcome of Lottery $L_1$ $L_{1}$ is better than any possible outcome of Lottery $L_2$ $L_{2}$ , then $L_1$ $L_{1}$ is not better than $L_2$ $L_{2}$ .
  - Motivation: The value of a lottery is derivative of the final value of its potential outcomes. If no single outcome of $L_1$ offers a reason to prefer it over $L_2$ , the lottery itself cannot be better.
2. Personal Good (PG): A weak extension of the Ex Ante Pareto principle. If two lotteries have the same population probabilities, and $L_1$ gives every individual a personal lottery that stochastically dominates (or equals) the personal lottery in $L_2$ , with strict dominance for at least one individual, then $L_1$ is impersonally better than $L_2$ .
Auxiliary Assumptions:
- Richness: The domain of outcomes includes any finite configuration of individuals and lives.
- Incomparability Transmission: If lives $a$ and $b$ are personally incomparable, then outcomes differing only by these lives are impersonally incomparable.
- Small Improvements: If an outcome is incomparable to others, it can be slightly improved (or worsened) while remaining incomparable to those same outcomes.
- Strong Incomparability (for outcomes): A generalization where distributions of lives are incomparable regardless of the specific individuals realizing them.

3. Key Contributions and Arguments

A. The "Opaque Sweetening" Argument (Proposition 1)

The core argument uses a reductio ad absurdum based on a specific lottery comparison (Table 3 in the paper):

Setup: Assume lives $a$ and $b$ are incomparable. Let $a^+$ be a slight improvement of $a$ (still incomparable to $b$ ).
Lottery 1: 50% chance of $(a, a)$ , 50% chance of $(b, b)$ .
Lottery 2: 50% chance of $(a^+, b)$ , 50% chance of $(b, a)$ .
The Conflict:
1. From ND: Every outcome of Lottery 2 (e.g., $a^+, b$ ) is incomparable to every outcome of Lottery 1 (e.g., $a, a$ ). By ND, Lottery 2 cannot be better than Lottery 1.
2. From PG: Lottery 2 gives individual $P_1$ a stochastically dominant personal lottery ( $a^+$ vs $a$ ) and $P_2$ an equivalent lottery. By PG, Lottery 2 must be better than Lottery 1.
Result: The assumption that $a$ and $b$ are incomparable leads to a contradiction. Therefore, personal betterness must be complete.

B. Extension to Variable Populations (Proposition 2)

The argument is extended to cases where population size changes (e.g., existence vs. non-existence).

If existence with life $a$ is incomparable to non-existence, the same lottery structure applies (where $b$ = non-existence).
Result: Views like "Critical-Range Utilitarianism" (which posits incomparability between adding a slightly positive life and non-existence) are incompatible with ND and PG.

C. General Incompleteness in Outcomes (Proposition 3)

The authors generalize the argument to cases where incomparability arises from impersonal trade-offs (e.g., total welfare vs. equality) rather than just personal incomparability.

They define Strong Incomparability between distributions of lives.
Using subpopulations as proxies for individuals, they construct a lottery analogue to Proposition 1.
Result: Any form of impersonal incompleteness that satisfies a separability condition (analogous to Incomparability Transmission) is ruled out.

D. Completeness of Lotteries (Proposition 4)

The paper argues that if outcomes are completely ordered, then finite rational lotteries must also be completely ordered.

Method: Using Substitution (replacing individuals with fresh ones to avoid overlap) and Transitivity, they show that any finite rational lottery can be transformed into a "stacked" outcome where all probability mass is concentrated in a single state.
Result: If outcomes are complete, the ranking of lotteries supervenes on the ranking of these stacked outcomes, ensuring completeness for lotteries.

4. Results

The paper establishes four main propositions:

Proposition 1: ND + PG + Incomparability Transmission + Small Improvements $\implies$ No personal incomparability between lives.
Proposition 2: ND + PG + Small Improvements $\implies$ No variable-population incompleteness (e.g., existence vs. non-existence).
Proposition 3: ND + PG + Small Improvements $\implies$ No strong incomparability between outcomes (general impersonal incompleteness).
Proposition 4: If outcomes are complete, then finite rational lotteries are complete (given PG, Stochastic Dominance, Substitution, and Transitivity).

5. Significance and Implications

Challenge to Incompleteness: The paper provides a novel, structural argument against value incompleteness that does not rely on expected utility maximization or specific risk attitudes. It forces proponents of incompleteness to reject either:
- Negative Dominance: The idea that lottery value is grounded in the value of potential outcomes.
- Personal Good: The minimal claim that if a lottery is better for everyone (stochastically), it is better overall.
Rejection of Statewise Dominance: The authors distinguish their principle from Caspar Hare's "Statewise Negative Dominance." They argue that while Statewise Dominance is incompatible with Stochastic Dominance in the presence of incompleteness, their weaker Negative Dominance is compatible with Stochastic Dominance. Thus, defenders of incompleteness cannot simply reject Statewise Dominance to save their view; they must reject the more fundamental Negative Dominance.
Finite vs. Infinite: The authors acknowledge that ND and PG can conflict in infinite contexts (e.g., infinite populations with unbounded welfare). However, they argue that this does not undermine the arguments for finite contexts, which are the primary domain of standard ethical dilemmas. They suggest that if one accepts these principles for finite cases, one must accept completeness for finite lives and outcomes.
Philosophical Impact: The argument suggests that the intuition of "incomparability" (e.g., between an artist and a banker) is structurally unstable when combined with standard decision-theoretic constraints on risk and aggregation. It implies that if we accept that lotteries are evaluated based on their outcomes and that individual improvements matter, we must accept that all lives and outcomes are, in principle, comparable.

6. Response to Objections

The authors preemptively address three main objections:

Fundamentality: Critics might argue that principles about lotteries shouldn't constrain lives. The authors counter that structural principles provide evidence for the structure of value; just as galaxy behavior informs particle physics, lottery constraints inform the structure of personal value.
Infinite Contexts: Critics note ND and PG conflict in infinite domains. The authors argue this is a known limitation of many principles (like Pareto) in infinite settings and does not invalidate their force in finite, realistic scenarios.
Supervaluational Negative Dominance: Critics might propose a weaker principle based on "admissible completions" of value. The authors reject this, arguing it relies on disjunctive reasoning about hypothetical completions, which fails to capture the intuition that actual value (and thus actual reasons for preference) must be grounded in specific, realized outcomes.