📈 economics

Centralization and Stability in Formal Constitutions

This paper establishes a framework for analyzing the stability of social-choice functions against self-replacement, demonstrating that while standard assumptions often lead to centralized dictatorships, incorporating forward-looking voters or wisdom-of-crowd effects can sustain less centralized, stable institutional designs relevant to systems like DAOs.

Original authors: Yotam Gafni

Published 2026-05-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yotam Gafni

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 a group of friends trying to decide where to eat dinner. They have a set of rules (a "constitution") for how to vote: maybe it's a simple majority, maybe it requires everyone to agree, or maybe one person gets to decide.

Now, imagine that the friends can also vote on changing the rules themselves. They can propose, "Let's stop voting by majority and start letting the person with the most money decide."

This paper asks a simple but deep question: Are there any voting rules that are so strong that the group will never vote to change them? The author calls these "self-maintaining" rules. If a rule is self-maintaining, the group will keep using it forever, no matter what other rules they consider.

The paper uses this idea to study DAOs (Decentralized Autonomous Organizations), which are digital groups on the blockchain that manage huge amounts of money. The author wants to know: Will these digital groups naturally stay democratic, or will they inevitably drift toward being controlled by a single person or a tiny elite?

Here is the breakdown of the findings using simple analogies:

1. The "Selfish Voter" Scenario (The Baseline)

First, the author assumes everyone is purely selfish. They only care about winning the vote so they can get the resources (like pizza or money). They don't care if the decision is "good" for the group; they just want their voice to count.

The "Optimistic" View: If the group is hopeful that things will work out, almost any rule can survive. Even a weird rule where two people have to agree can stick around if the voters believe it will help them win.
The "Pessimistic" View: If the group is worried and assumes the worst, the results get dark.
- The "Dictator" Rule: The only rule that is perfectly safe from being changed is Dictatorship (one person decides). Why? Because the dictator always gets their way, so they will never vote to change the rules.
- The "Tiny Club" Rule: The only other stable rules are small groups of 2 or 3 people who have to agree. If you have a rule where a large group (like 50% of the people) decides, the author shows that a smaller, tighter group can usually conspire to change the rules to give them more power.
- The "Unanimity" Trap: You might think "Everyone must agree" (Unanimity) is the safest, most democratic rule. But the paper shows it's actually very unstable. If the group is stuck and can't make a decision, they will eventually vote to change the rule to something easier (like a simple majority), because being stuck is worse than being less democratic.

The Big Takeaway: If everyone is just looking out for themselves, democracy (majority rule) is unstable. The system naturally drifts toward centralization (one person or a tiny group holding all the power).

2. The "Forward-Looking" Scenario (Thinking Ahead)

What if the voters are smart and think ahead? They might think, "If we change the rule to let a small group decide, that small group might later change the rule again to let one person decide. So, let's not change the rule at all."

The Result: This thinking helps save some slightly larger groups. Instead of just a 2-person club, a group of 4 or 5 might be able to hold onto power if they realize that changing the rules will eventually lead to a dictatorship.
The Limit: Even with this smart thinking, the system still tends to shrink. The "stable" groups get smaller as the voters become more selfish.

3. The "Wisdom of the Crowd" Scenario (The Twist)

The author adds a second ingredient: Wisdom. Imagine the voters aren't just fighting for pizza; they are trying to solve a puzzle. They believe that if more people vote, the group is more likely to get the right answer (this is based on the "Condorcet Jury Theorem").

The Trade-off: Now, voters have two goals:
1. Power: I want to be the one deciding (Extractive).
2. Truth: I want the group to make the right decision (Participative).
The Result:
- If the voters care mostly about Power, the group shrinks to a Dictatorship (one person).
- If the voters care mostly about Truth, the group expands to Full Democracy (everyone votes).
- The Sweet Spot: In the middle, you get a Committee. The size of this committee depends on how smart the voters are and how much they care about power.
  - If voters are very smart (high "wisdom"), the stable group can be quite large.
  - If voters are less smart or very greedy, the stable group shrinks to a tiny elite.

4. What This Means for Real-World Systems (Like DAOs)

The paper uses these findings to explain why many digital organizations (DAOs) are becoming centralized.

The "Airdrop" Lesson: To keep a system decentralized (a large committee), you need to start with a very fair distribution of power (tokens). If you start with a few rich people holding most of the power, the system will naturally evolve toward a dictatorship.
The "Shock" Lesson: If the quality of decision-making drops (e.g., if voters stop caring about the "truth" and only care about "power"), the system will naturally collapse into a smaller, more centralized group.

Summary Analogy

Think of a voting system as a boat.

Selfish voters are like passengers trying to grab the steering wheel. If everyone is fighting for the wheel, the boat naturally ends up with one person (the strongest) holding it, because that's the only way to stop the fighting.
Wise voters are like passengers who realize that if they all hold the wheel together, the boat sails better. If they trust each other enough, they can keep a large group steering.
The Paper's Warning: If the passengers start caring more about grabbing the wheel than sailing the boat, the boat will inevitably end up with a single captain, no matter how much they tried to start as a democracy.

The paper concludes that centralization is a natural gravitational pull in systems where people act in their own self-interest, unless there is a strong, shared belief that "more voices make better decisions."

Technical Summary: Centralization and Stability in Formal Constitutions

1. Problem Statement

The paper investigates the dynamics of decision-making procedures within formal systems, specifically focusing on whether a voting method (Social Choice Function or SCF) can be stable against replacement by another method. This is motivated by the observation that in systems like Decentralized Autonomous Organizations (DAOs), the voting mechanism itself is subject to change via proposals. The central question is: Which voting rules are "self-maintaining"?

An SCF is defined as self-maintaining if, when agents vote according to their ex-ante preferences between the incumbent SCF and any potential replacement SCF, the incumbent SCF aggregates those votes to reject the replacement. The paper explores how the stability of these rules correlates with centralization, specifically asking whether democratic or oligarchic structures tend to evolve toward dictatorship under various belief structures and tie-breaking rules.

2. Methodology

2.1 Formal Model

Agents and Beliefs: The model considers $n \ge 3$ agents. Agents hold beliefs over the distribution of future voting vectors (decisions). These beliefs are modeled as probability distributions $F$ over $\{0, 1\}^n$ .
Utility: An agent's ex-ante utility for an SCF $f$ is the probability that $f$ 's outcome matches the agent's preference for a decision drawn from their belief distribution: $u_i(f, F) = P_{v \sim F}[f(v) = v_i]$ .
Self-Maintenance Condition: An SCF $f$ is self-maintaining if, for any alternative $f'$ , the vote to replace $f$ with $f'$ (where agents vote based on comparing $u_i(f)$ and $u_i(f')$ ) fails when aggregated by $f$ .
Tie-Breaking: The paper analyzes two tie-breaking rules for agents indifferent between $f$ $f$ and $f'$ $f^{'}$ :
1. Arbitrary: Agents may vote either way.
2. Status-Quo Bias (SQB): Indifferent agents vote to keep the current rule ( $f$ ).

2.2 Approaches to Beliefs

The authors evaluate stability under three distinct epistemic approaches:

Optimistic: An SCF is self-maintaining if there exists a common belief $F$ supporting it.
Pessimistic: An SCF is self-maintaining only if it is supported by all common beliefs $F$ .
I.I.D. (Independent and Identically Distributed): Agents' preferences are modeled as Bernoulli variables with parameter $p$ . The paper focuses on the unbiased case ( $p=0.5$ ) and biased cases.

2.3 Domains

The analysis is conducted across three domains of SCFs:

General Boolean Functions ( $D_G$ ): All mappings $\{0, 1\}^n \to \{0, 1\}$ .
Simple Games ( $D_S$ ): Monotone, non-trivial SCFs (incentive-compatible).
Weighted Majority Games ( $D_W$ ): SCFs defined by weights and a threshold (e.g., token-based voting).

2.4 Extensions

Forward-Looking Agents: Agents consider the long-term trajectory of changes (e.g., if changing to $f'$ leads to $f''$ , they might prefer staying at $f$ ).
Common Value (Condorcet Jury Theorem): Agents derive utility not just from winning (extractive) but from making the "correct" decision based on aggregated signals (participative).

3. Key Contributions and Results

3.1 Optimistic Approach

Arbitrary Tie-Breaking: An SCF is self-maintaining if and only if it is proper (it never outputs 1 for both a vector $v$ and its negation $\neg v$ ). In the domain of Weighted Majority Games, all rules are proper, meaning the entire domain is self-maintaining.
Status-Quo Bias (SQB): An SCF is self-maintaining if it is either proper or null-preserving (outputs 0 when all vote 0). Since all simple games and weighted majority games are null-preserving, the entire domain is self-maintaining under SQB.

3.2 Pessimistic Approach

Arbitrary Tie-Breaking: In the general domain $D_G$ , only the constant-0 function is self-maintaining. In $D_S$ and $D_W$ , no SCF is self-maintaining.
Status-Quo Bias (SQB): The paper provides a tight characterization for $D_S$ $D_{S}$ and $D_W$ $D_{W}$ . The only self-maintaining rules are:
1. Dictatorship: A single agent decides.
2. 3-Oligopoly: A rule where a majority of a specific set of 3 agents decides.
3. Poly-Duopolies (PDPs): Rules where minimal winning coalitions are pairs, and one specific agent appears in all pairs (e.g., a "veto player" combined with a partner).
- Note: The paper proves that any rule with a minimal winning coalition of size $\ge 3$ (other than the 3-oligopoly) is unstable under pessimistic beliefs.

3.3 I.I.D. Approach (Unbiased)

General Boolean Functions: The only self-maintaining rules are Dictatorships, Anti-Dictatorships, and the Constant-0 function.
Simple Games: Only Dictatorships are self-maintaining.
Weighted Majority Games: While dictatorships are stable, the paper identifies a gap where certain PDPs (specifically those with a veto player and a threshold) can be self-maintaining under unbiased i.i.d. beliefs, unlike in the general simple game domain.

3.4 Forward-Looking Agents

When agents anticipate future changes (modeled as a directed acyclic graph of transitions):

For $n=3$ , the set of forward-looking self-maintaining rules is identical to the pessimistic SQB set.
For $n=4$ and $n=5$ , the set strictly expands. Rules that are not stable under one-step analysis become stable because agents realize that moving to a "better" alternative might lead to a worse outcome later. These new stable rules often have higher Nash welfare and social welfare than the standard dictatorships or oligarchies.

3.5 Common Value (Extractive vs. Participative)

In a model where utility is a mix of extractive power ( $\lambda$ ) and decision accuracy ( $p$ ):

High Extractive ( $\lambda \to 1$ ): The system converges to Dictatorship or small oligarchies (Consensus-Duopoly).
High Participative ( $\lambda \to 0$ ): The system converges to Full Democracy (large committees).
Mixed Case: There is a monotone relationship where increasing the extractive nature of the economy or the accuracy of individual signals ( $p$ ) leads to smaller stable oligarchies.
Dynamics: The system exhibits sensitivity to initial conditions. Starting with a highly distributed token allocation (e.g., via airdrops) can lead to a stable large committee, whereas concentrated starting points lead to dictatorship.

4. Significance and Claims

The paper claims to provide a "basic framework and body of results for centralization dynamics and stability." Its primary contributions are:

Theoretical Foundation for DAO Governance: It offers a micro-foundation for the observed centralization in DAOs, suggesting that without specific constraints, the dynamics of self-referential voting rules naturally drive systems toward centralization (dictatorship or small oligarchies).
Refutation of Simple Majority Stability: Contrasting with prior work (Barbera & Jackson, 2004) which found simple majority stable under anonymity, this paper shows that removing the anonymity constraint (allowing for weighted voting or non-neutral rules) leads to the collapse of democratic stability into dictatorship under pessimistic or i.i.d. assumptions.
Role of Tie-Breaking and Beliefs: The results highlight that stability is highly sensitive to tie-breaking rules (SQB vs. Arbitrary) and the nature of agent beliefs (Optimistic vs. Pessimistic).
Policy Implications: The authors suggest that achieving inclusive, decentralized steady-states in formal systems (like DAOs) requires specific initial conditions, such as distributive token allocations (airdrops/ICOs), to bootstrap the system into a basin of attraction for large committees rather than dictatorships.
Extension of Social Choice Theory: The paper extends the concept of "self-selectivity" (Koray, 2000) by relaxing the neutrality axiom, arguing that constitutions naturally differentiate between status-quo and change, and showing how this affects stability outcomes.

The paper concludes that while "wisdom of the crowd" (Condorcet Jury Theorem effects) can support larger stable committees, the extractive nature of resource allocation in many systems creates a strong gravitational pull toward centralization.