A Decision-Theoretic Take on Social Choice

Using L2 to solve the preference dilution problem

September 23, 2025

Voting seems simple. You count the votes, and the option with the most votes wins. Yet, as anyone who has followed an election knows, the reality is a messy, often paradoxical affair. The field that formally studies this mess is Social Choice Theory, and it’s full of surprising results that tell us there is no “perfect” voting system.

But what if we’ve been thinking about the problem from the wrong angle?

Traditional Social Choice Theory focuses on the paradoxes of ranking candidates. But we can reframe the problem: what if making a collective choice is not about ranking, but about approximating the best possible outcome from noisy, incomplete data?

This article will walk through that decision-theoretic framework. We’ll define the “best” outcome, diagnose a fundamental flaw in our standard methods, and show how allowing for the expression of negative preference is the key to a more robust system.

The Goal: Social Choice as an Optimization Problem

Let’s start with a powerful assumption. Imagine an oracle can tell us the true, numerical “utility” every person has for every option. Let $u_i(x)$ be the true utility person $i$ gets from outcome $x$.

If we had this, our task would be a straightforward optimization problem. But what are we optimizing?

  1. Utilitarian (or Benthamite) Welfare: We aim to maximize the total utility in society. The best outcome is the one that wins a simple sum:

    \[x^* = \underset{x}{\text{argmax}} \sum_i u_i(x), u \in \mathbb R\]

    A huge gain for one person can outweigh small losses for many others. The key mechanism here is cancellation: a strong negative preference (e.g., -50) can be cancelled out by a strong positive one (+50). This property is intuitive, powerful, and, as we’ll see, absent from most real-world voting systems.

  2. Nash Welfare: 1 We aim to maximize the product of everyone’s utility. This is inherently inequality-averse, as an outcome that gives one person zero utility causes the entire product to become zero, effectively giving that person a veto.

    \[x^* = \underset{x}{\text{argmax}} \prod_i u_i(x), u \in\]

For this post, we will focus on the Utilitarian goal, because its principle of cancellation is what we want to emulate. Our challenge is that we don’t have an oracle. We have to ask people for their preferences, and that’s where the trouble begins.

The Problem: Dilution and the Failure of Simple Normalization

If we just let people report a number, everyone has an incentive to report exaggeratedly high or low numbers to pull the outcome in their direction. The obvious fix is to give everyone an equal “budget” of preference. The standard way to do this is with L1 normalization, forcing each person’s preferences to sum to one: $\sum_x p_i(x) = 1$.

But this creates a devastating flaw: the preference dilution problem.

Consider an election with three candidates:

  • Candidate A & B: Two similar, center-ground candidates.
  • Candidate C: A very different, polarizing candidate.

Imagine the electorate is split:

  • 60% of voters are “Generalists.” They dislike C but find A and B both perfectly acceptable.
  • 40% of voters are “Specialists.” They are passionate, single-issue supporters of C.

Under L1 normalization, the Generalists must split their budget: $p(A)=0.5, p(B)=0.5, p(C)=0$. The Specialists can pour their entire budget into C: $p(A)=0, p(B)=0, p(C)=1.0$.

If we sum these declared preferences to find the utilitarian optimum, C wins in a landslide (40 to 30 for A and 30 for B). The 60% majority who preferred either A or B to C have their collective will thwarted because the normalization rule forced them to dilute their support.

This is a clear failure. Both A and B are Condorcet winners 2 (they would beat C in a head-to-head 60-40 vote), yet the system picks the Condorcet loser. The core issue is that this system gives voters no way to express cancellation; they can only express positive support.

A Better Way: The Crucial Role of Negative Preferences

The dilution problem arose because the Generalists could not express their opposition to C. What if we allow them to?

Let’s try a simple fix. We’ll keep the L1 norm, but allow for negative numbers. The rule is now that the sum of the absolute values of the preferences must be 1: $\sum_x p_i(x) = 1$.

How does this change our election?

  • The 40% Specialists: Unchanged. They use their whole budget on C. Their vector is $(0, 0, 1)$.
  • The 60% Generalists: They like A and B and dislike C. They could split their budget like this: $p(A)=0.4, p(B)=0.4, p(C)=-0.2$. (Check: $ 0.4 + 0.4 + -0.2 = 1$).

Now let’s sum the scores:

  • Score for A: $(40 \times 0) + (60 \times 0.4) = 24$
  • Score for B: $(40 \times 0) + (60 \times 0.4) = 24$
  • Score for C: $(40 \times 1) + (60 \times -0.2) = 40 - 12 = 28$

C still wins. Even with negative preferences, the dilution problem remains. Why? Because the L1 norm creates a zero-sum budget. To express more support for A, the Generalists must reduce their support for B or their opposition to C. Their preferences are still coupled.

The solution requires a different kind of normalization, one that decouples the intensity of preferences. Instead of the sum of values being 1, let’s demand that the sum of the squares of the preferences be 1. \(\sum_x p_i(x)^2 = 1\) This is L2 normalization. Its power is that it allows for multiple, independent, high-intensity preferences.

Let’s revisit the election one last time with this rule:

  • The 40% Specialists (pro-C): Their preference vector is $(0, 0, 1)$. (Check: $0^2 + 0^2 + 1^2 = 1$).
  • The 60% Generalists (anti-C, pro-A/B): They feel strongly and positively about A and B, and negatively about C. A plausible vector is $(0.7, 0.7, -0.141)$. (Check: $0.7^2 + 0.7^2 + (-0.141)^2 \approx 1$).

Now, let’s sum the preferences:

  • Score for A: $(40 \times 0) + (60 \times 0.7) = 42$
  • Score for B: $(40 \times 0) + (60 \times 0.7) = 42$
  • Score for C: $(40 \times 1) + (60 \times -0.141) = 40 - 8.5 = 31.5$

Candidates A and B are now the decisive winners. The L2 norm, by allowing for negative values and measuring “intensity” quadratically, correctly identified the Condorcet winners. It allows the majority to express independent support for multiple good options while simultaneously expressing opposition to the bad one, solving the dilution problem.

An Escape from Arrow’s Impossibility Theorem?

How does this system of cardinal utilities fare against the most famous result in social choice, Arrow’s Impossibility Theorem? The theorem proves that no voting system for three or more options can satisfy a few basic fairness criteria simultaneously. 3

The key is that Arrow’s theorem applies to systems that use ordinal preferences (rankings like A > B > C). Our proposed system uses cardinal preferences (scores like A=0.7, B=0.7, C=-0.141), which contain much more information. By using this richer information, we can bypass the theorem.

Specifically, cardinal voting systems like this one violate a core Arrow condition: Independence of Irrelevant Alternatives (IIA).

  • What is IIA? In simple terms, it states that the group’s preference between A and B should not change if a third option, C, is introduced or removed.
  • How does our system violate it? In our L2 system, the introduction of a new candidate C forces voters to re-distribute their preference “intensity.” A voter might lower their score for A to express a score for C, and this change can alter the societal outcome between A and B.

This is not a flaw; it’s a feature. The “irrelevant” alternative C is, in fact, relevant to a voter’s overall satisfaction. By allowing voters to express the magnitude of their preferences, we escape the paradoxes that arise when we only consider rankings. The trade-off is that we must assume that one person’s “0.7” is comparable to another’s, an idea known as “interpersonal utility comparison,” which many classical economists were skeptical of.

Conclusion: A Clearer Goal for Social Choice

By reframing social choice as an estimation problem, we move from a search for perfect rules to a search for the best approximation of a collective good. The key properties we need are:

  1. A Utilitarian Sum: To allow for the simple and powerful mechanism of preference cancellation.
  2. Negative Preferences: To allow voters to express both support and opposition.
  3. L2 Normalization: To decouple preferences and prevent the dilution that plagues simpler systems.

This framework correctly identifies the will of the majority in our test case and bypasses Arrow’s Impossibility Theorem by using richer, cardinal information.

However, this power is not without risk. A system that allows for negative preferences could formalize a “tyranny of the majority,” where a cohesive group uses negative scores to punish a minority’s preferred candidate. This reveals a deep trade-off: protecting the majority from dilution versus protecting the minority from majoritarianism.

The path to implementation raises its own challenges, but by first defining a more robust objective, we now have a clear goal. The L2 framework doesn’t promise a perfect system, but it gives us a clearer picture of what we are trying to optimize, and what trade-offs we must confront to get there.

References

  1. The Nash welfare function is derived from John Nash’s work on bargaining, specifically the Nash Bargaining Solution, which seeks outcomes that are both efficient and equitable. See: Nash, J. F. (1950). “The Bargaining Problem.” Econometrica, 18(2), 155–162. ↩

  2. The Marquis de Condorcet first described this paradox in the 18th century. A Condorcet winner is considered a very strong indicator of the collective will, and voting systems that can fail to elect an existing Condorcet winner are often seen as flawed. ↩

  3. Arrow, K. J. (1951). Social Choice and Individual Values. Wiley. For a more accessible overview, the Stanford Encyclopedia of Philosophy entry on “Arrow’s Theorem” is an excellent resource. ↩