A derivation / motivation of Liquid democracy via Vector Quantisation
January 3, 2026
We often talk about politics in terms of ideology, history, or tribalism. But at its core, governance is an geometry problem. It is the challenge of aggregating the distinct, high-dimensional preferences of millions of people into a single output: policy.
If we strip away the campaign slogans and look at the math, we find that many of our frustrations with modern democracy—polarization, the “lesser of two evils,” and the feeling of being unheard—are not bugs. They are geometric inevitabilities of the mechanisms we use.
In this post, I want to explore a formal framework for voting, show why the “Two-Party System” is mathematically equivalent to a lossy compression algorithm, and explore how we might break the “Curse of Dimensionality.”
Figure 1: A 1-dimensional, two-party system is incapable of representing 75% (6/8) of possible preference combinations across just 3 binary issues. As the number of issues voters care about increases, their ability to express true preferences diminishes. Voters are effectively forced to choose the “lesser of two evils”—the party centroid that minimizes distance to their own views, even if the fit is poor.
1. The Setup: Voters as Vectors
Let’s model the political compass not as a flat square, but as a high-dimensional space.
- The Space: Imagine a space with $d$ dimensions. Each dimension represents a specific policy issue (e.g., Tax Rate, AI Regulation, Zoning Laws).
- The Voter ($v$): Every citizen is a point in this space, represented by a vector $v \in \mathbb{R}^d$. This vector encodes their ideal preferences on every issue.
- The Society: The electorate is a cloud of $m$ points distributed throughout this space.
The goal of any governance mechanism is to select a policy outcome—let’s call it vector $P$—that minimizes the collective unhappiness of the society. In geometry, “unhappiness” is simply distance. The further the outcome $P$ is from your position $v$, the more dissatisfied you are.
Ideally, we want to find the Geometric Median: the point $P$ that minimizes the sum of distances to all voters:
\[\min_P \sum_{i=1}^m \| v_i - P \|\]If we could achieve this, we would have the “most representative” compromise for society. So, how do we get there?
2. The Two-Party Trap ($k=2$)
In most modern democracies, we don’t calculate the median directly. We use intermediaries: Political Parties.
A party is essentially a pre-packaged bundle of policies—a single “codeword” vector $c$ in that $d$-dimensional space. In a two-party system, the voter is presented with exactly two options: $c_A$ and $c_B$.
The voter’s job is simple: vote for the one closest to them. The system’s job is effectively k-means clustering (with $k=2$). It tries to find the two centroids that minimize the error for their respective halves of the population.
The Dimensionality Collapse
While this sounds efficient, it introduces a catastrophic mathematical flaw: Dimensionality Collapse.
Any two points ($c_A$ and $c_B$) define a straight line. By forcing voters to choose between them, the system projects the complex, $d$-dimensional variance of the population onto a 1-dimensional axis.
- If you care about an issue that aligns with this axis (e.g., “Left vs. Right”), you are represented.
- If you care about an issue orthogonal (perpendicular) to this axis—say, a specific tech policy that neither party talks about—your preference is mathematically invisible.
To the system, your vote is just a single bit of information (0 or 1). It cannot capture the nuance of a $d$-dimensional vector. This leads to high Distortion—the gap between what voters want and what they get is mathematically guaranteed to be high.
We can think of this distortion as “forced strategic voting.” Thorburn et al. (1) quantify this error using Kendall tau distance—the number of “swaps” required to make a voter’s true preferences fit the model. If you prefer Party A, but the geometric map forces you closer to Party B, the system has effectively swapped your preference. In the real world, we call this the “lesser of two evils”—a mathematical artifact of projecting high-dimensional citizens onto a 1-dimensional ballot.
3. Why not just add more parties? ($k=2^d$)
If $k=2$ is too crude, the natural intuition is to increase $k$. Why not have 5, 10, or 20 parties?
We see this in systems using Mixed-Member Proportional (MMP) representation or pure proportional representation (like in Germany, New Zealand, or the Netherlands). These parliaments often have 10+ parties, offering voters a wider menu than the binary choice in the US.
As we add more centroids (parties), the representation error does decrease (this is known as Zador’s Bound in quantization theory). However, even with 15 parties, we run into a hard limit: The Curse of Dimensionality.
Let’s assume distinct policy issues are binary (Yes/No). Let’s take just three issues:
- Progressive Wealth Tax? (Y/N)
- Nuclear Power Expansion? (Y/N)
- Universal Basic Income? (Y/N)
- To represent a voter who wants [Yes, Yes, Yes], you need a party with that exact platform.
- To represent a voter who wants [Yes, No, Yes], you need a completely different party.
If a Green party supports the Tax and UBI but opposes Nuclear, the pro-Nuclear environmentalist is stranded. They must compromise.
To guarantee that every voter can find a party that represents their views on just $d=30$ binary issues, we would need a party for every possible combination:
\[2^{30} \approx 1 \text{ Billion Parties}\]This intuition is backed by recent results in social choice theory. Thorburn et al. (2023) (1) prove that to losslessly represent arbitrary preference rankings over $A$ alternatives, the spatial model must have at least $A-1$ dimensions.
In our context, this implies a hard geometric limit: you cannot compress the complex, high-dimensional will of the electorate into a low-dimensional party system without introducing “embedding error.” This error isn’t just noise; it is the mathematical definition of a disenfranchised voter.
The “Independent Voter” Problem
This brings us to a fundamental philosophical question about how we structure governance:
Should every citizen have the right to express their preferences losslessly?
Critics might argue that real-world preferences are correlated—that ‘Left’ and ‘Right’ bundles naturally exist, and thus we only need to represent those specific corners of the hypercube. But this possibly confuses cause and effect, and ignores the fundamental measure of information: Entropy.
If 95% of the population truly fell into two neat ideological buckets, the entropy of the electorate would be low. A low-dimensional system (like two parties) would suffice, yielding low distortion. But if voters are even slightly independent—if they have unique combinations of views—the entropy of the distribution increases.
When the system’s capacity (1 bit: Red vs. Blue) is lower than the entropy of the electorate, information is inevitably lost. By aligning our political dimensions to yield artificially low entropy, we are effectively disenfranchising minorities—forcing them to choose between two “evils,” neither of which they particularly want. They are not allowed to express themselves.
This loss manifests as voter disenfranchisement. When a voter’s complex, high-entropy preference vector is compressed into a single low-entropy bit, they are not being “represented”; they are being silenced. It is plausible that this disconnect contributes to voter apathy. If the menu of options (parties) fails to cover the support of the distribution of voters, rational actors may simply choose not to participate.
The “Impartial Culture” assumption in voting theory models a population of independent thinkers. The math shows that as voters become more independent (higher entropy), the error of low-dimensional systems explodes. A rigid party system effectively relies on artificial voter conformity to function. If we want a system that respects independent thought (e.g., a Pro-Gun Environmentalist), we face a combinatorial explosion that fixed parties cannot solve.
This creates a paradox:
- Few Parties ($k=2$): Low choice cost, but terrible representation accuracy (high error).
- Many Parties ($k \to \infty$): Perfect representation, but impossible cognitive load (infinite search cost).
We cannot “party” our way out of this problem.
4. The Information Bottleneck
The fundamental problem here is Information Theory.
Traditional voting treats governance as a Vector Quantization problem. We are trying to compress the infinite complexity of human preference into a discrete, finite set of “buckets” (parties).
In a high-dimensional world (where $d$ is large), bundling issues together is an incredibly inefficient way to transmit information.
- Bundling: “You must buy the whole menu. If you want the salad, you have to eat the steak.”
- Result: You throw away the steak. That is waste (or in our model, “dissatisfaction”).
We need a mechanism that breaks the bundle.
5. The Solution: Unbundling (Liquid Democracy)
To break the curse of dimensionality, we must switch from selecting a vector to constructing one. Recent research confirms that low-dimensional embeddings (like 2D political compasses) cannot accurately capture high-dimensional data without significant error.
By unbundling the issues, Liquid Democracy avoids the “Curse of Dimensionality” entirely. Instead of trying to fit complex voters into a simple low-dimensional map (and failing), we simply conduct the vote in the native high-dimensional space of the issues themselves. We don’t compress the voter; we expand the ballot.
In this framework:
- The system doesn’t solve one massive clustering problem in $d$-dimensional space.
- It solves $d$ tiny clustering problems in 1-dimensional space.
If you care about Dimension 1 (Environment), you vote (or delegate) on that. If you are indifferent about Dimension 2 (Tax), you abstain or delegate to someone else.
Mathematically, this changes the scaling of the error. We no longer need $2^d$ parties to cover the space. By treating the policies as an orthogonal basis—building the policy vector element-by-element—we allow the voter to express a precise location in high-dimensional space without requiring an infinite menu of options.
Summary
- Preferences are high-dimensional vectors.
- Two-Party Systems compress these vectors onto a single line, discarding most of the information.
- Multi-Party Systems fail to scale because the volume of the space grows exponentially ($2^d$).
- Optimal Governance requires unbundling: treating dimensions independently.
The math suggests that as long as we rely on bundling (rigid parties), we will always suffer from high distortion. To align governance with voter will, we don’t need better parties; we need a mechanism that transcends them.