Self-Equivalent Voting Rules

arXiv:2506.15310 · econ.TH · Submitted 2025-06-18 · Read on arXiv

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Introduction to the show: ident: Economics. The day in economics: theory, mechanisms and the evidence beneath them.

Ruth: We're looking at the paper: "Self-Equivalent Voting Rules".

Warren: Here's the gist of the paper. In this paper, I introduce a novel stability axiom for stochastic voting rules, called self-equivalence, by which a society considering whether to replace its voting rule using itself will choose not to do so The result of this paper (Theorem 1) states that the unique anonymous, efficient, monotonic, neutral, and self-equivalent voting rule is the uniform random dictatorship Thus, this paper suggests one reason why most democratic societies normally choose whether to replace their voting rule using a voting rule other than itself, or in binary elections against just one other voting rule.

Ruth: The paper introduces the concept of self-equivalence as a novel stability axiom for stochastic voting rules, where a society will not choose to replace its existing rule if it is self-equivalent.

Warren: This axiom adds to the four democratic axioms—anonymity, efficiency, monotonicity, and neutrality—to characterize the uniform random dictatorship.

Ruth: The gist of this novel axiomatic characterization is the addition of self-equivalence to four democratic axioms: anonymity, efficiency, monotonicity, and neutrality.

Title and authors: Ruth: Now, looking at the title "Self-Equivalent Voting Rules," I think it points toward how a rule behaves when you consider replacing itself with another rule. It's about that decision process.

Warren: It’s interesting because they define self-equivalence by two conditions: first, there has to be this finite lottery over neutral and deterministic voting rules that produces a convex combination identical to the original voting rule.

Iris: And the second condition is more complex: for any finite set of those rules that includes the support of that lottery, there must always be some compatible linear order profile where the original rule recovers that lottery.

Ruth: That second part sounds like they're ensuring that even if you try to switch to a different rule, there’s always a way—a profile—where the original rule still acts like it's following the initial lottery structure.

Warren: It basically means the system is robust against replacement because it always retains some structural connection back to that initial random setup, which is what they call self-equivalence.

Iris: So, if a rule satisfies these conditions, it suggests a kind of internal consistency that makes replacing it with itself unappealing.

Ruth: It’s about characterizing the uniform random dictatorship using this principle as an extra constraint on the democratic axioms we already know.

The paper's summary: Warren: So to summarize, the main result of "Self-Equivalent Voting Rules" is Theorem one, which states that there is only one anonymous, efficient, monotonic, neutral, and self-equivalent voting rule: the uniform random dictatorship.

Ruth: That’s a huge statement because it says that under these five conditions together you narrow down all the possibilities to just this one highly randomized scenario.

Iris: What I find interesting from the summary is how they prove this uniqueness by showing that any rule satisfying those five axioms must be that uniform random dictatorship.

Warren: They do it by introducing two extra axioms, independence of irrelevant alternatives and regularity, which help bridge the gap between the set of anonymous, efficient, monotonic, neutral rules and those satisfying self-equivalence.

Ruth: So they essentially say that if you want to understand why a society chooses to stick with its current rule or switch to another one in a binary election, you need this five-axiom framework plus self-equivalence.

Iris: It suggests that the stability of the system isn't just about whether the rule is fair on its own terms, but also about its internal relationship to other possible rules.

The paper's improvements: Ruth: The improvements suggested by this paper aren't necessarily new voting rules, but rather a better way to analyze stability in stochastic environments, using self-equivalence as the key tool.

Warren: That means the application here is that AI systems could potentially select voting rules that meet all five criteria—anonymity through neutrality and efficiency plus this new self-equivalence axiom.

Iris: If an AI can verify if a given stochastic voting mechanism meets all four democratic axioms plus this self-equivalence, it gives us a definitive test for identifying the uniform random dictatorship.

Ruth: That’s the idea: you use these tools to check if a complex system is actually behaving like that specific, highly randomized dictatorship they identified.

Warren: It helps explain why societies might choose to replace their voting rule or run binary elections against just one other rule by showing what mathematical constraints lead to that choice being avoided.

Iris: The paper points out a limitation in the literature review: extending self-equivalence to nondeterministic voting rules doesn't bring much new insight without much more advanced mathematical machinery.

Conclusion: Ruth: So to wrap up, the main implication of "Self-Equivalent Voting Rules" is that it provides a unique characterization for a very specific type of randomized dictatorship within democratic principles.

Warren: It suggests that the reason most societies don't just randomly switch their rules around is because they are avoiding those self-equivalent configurations.

Iris: I think the paper gives us a solid mathematical foundation for understanding rule selection stability by incorporating this idea of self-equivalence alongside anonymity, efficiency, monotonicity, and neutrality.

Ruth: It’s about using these tools to understand how people might choose to replace their voting rule with another or engage in binary elections against just one other rule.

Warren: This work provides a way for AI to select voting rules that align with this uniform random dictatorship by checking if they satisfy all five conditions.

Iris: It’s a precise test for identifying that specific model in complex systems where you need to know what's actually going on underneath the surface of the choice.

Budapest University of Technology and Economics

econ.TH

Submitted: 2025-06-18

Updated: 2026-10-28

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 76/100

The gist: In this paper, I introduce a novel stability axiom for stochastic voting rules, called self-equivalence, by which a society considering whether to replace its voting rule using itself will choose not

Key concepts

Self-Equivalence: This novel axiom means a society will not choose to replace its current voting rule if that rule is self-equivalent. Formally, it requires the rule to induce a convex combination of neutral and deterministic rules identical to itself, ensuring stability against replacement.
Anonymity: Anonymity requires that the voting rule's outcome must be independent of the specific identities of the voters. This means that knowing who voted does not change the final result, focusing solely on the aggregated preferences.
Efficiency: Efficiency dictates that a voting rule must assign zero probability to any alternative that is strictly worse than another alternative for every voter. In simple terms, it prevents rules from accidentally favoring outcomes that are clearly dominated by others.
Uniform Random Dictatorship: This is the unique voting rule identified as the solution. It represents a scenario where the outcome is determined randomly, mimicking a dictatorship where one person's preference dictates the result with uniform randomness.

Terminology

Axioms — 16× in this paper · explained in 2 episodes
Monotonicity — 15× in this paper · explained in 2 episodes
Consistency — 3× in this paper · explained in 3 episodes
Core — 1× in this paper · explained in 2 episodes
Nash Equilibrium — 1× in this paper · explained in 2 episodes

Summary

In this paper, I introduce a novel stability axiom for stochastic voting rules, called self-equivalence, by which a society considering whether to replace its voting rule using itself will choose not to do so The result of this paper (Theorem 1) states that the unique anonymous, efficient, monotonic, neutral, and self-equivalent voting rule is the uniform random dictatorship Thus, this paper suggests one reason why most democratic societies normally choose whether to replace their voting rule using a voting rule other than itself, or in binary elections against just one other voting rule.

Introduction and Core Concepts

The paper introduces the concept of self-equivalence as a novel stability axiom for stochastic voting rules, where a society will not choose to replace its existing rule if it is self-equivalent. This axiom adds to the four democratic axioms—anonymity, efficiency, monotonicity, and neutrality—to characterize the uniform random dictatorship. The gist of this novel axiomatic characterization is the addition of self-equivalence to four democratic axioms: anonymity, efficiency, monotonicity, and neutrality.

Axiomatic Characterization

The paper defines several axioms for a voting rule within a stochastic environment (N, Aτ, P(Aτ), Δ(Aτ), f). These include:

Anonymity:

Anonymity requires that the voting rule is independent of voters’ identities, formally if and only if it is independent of voters’ identities.

Efficiency:

Efficiency means that the voting rule assigns null probability to every strictly Pareto dominated alternative.

Monotonicity:

Monotonicity states that the probability of an alternative does not decrease whenever said alternative does not fall in any voter’s linear order.

Self-Equivalence Formalization

The key axiom, self-equivalence, requires two conditions to be met for a neutral voting rule. First, there must exist a finite lottery over neutral and deterministic voting rules inducing a convex combination identical to the voting rule itself. Second, for every finite set of such voting rules containing the support of said lottery, there must always exist some compatible linear order profile at which the voting rule recovers said lottery.

Result and Proof Structure

Theorem 1 establishes that the unique anonymous, efficient, monotonic, neutral, and self-equivalent voting rule is the uniform random dictatorship. The proof proceeds in two main statements: showing that the uniform random dictatorship satisfies all five axioms (Statement 1), and showing that any rule satisfying these five axioms must be the uniform random dictatorship (Statement 2). Statement 2 involves introducing two new axioms, independence of irrelevant alternatives and regularity, to prove that the intersection of anonymous, efficient, monotonic, neutral rules is contained within those satisfying self-equivalence.

Examples and Intuition

The paper provides examples illustrating the intuition underpinning the results. Examples include the plurality rule, Borda rule, and Condorcet rule, which are shown to not be self-equivalent. The uniform random dictatorship is shown to satisfy the predicate of self-equivalence at a given profile.

Literature and Extensions

The literature review discusses prior work by Koray (2000) and others who have extended the analysis of self-selectivity. The paper also notes that extending self-equivalence to accommodate nondeterministic voting rules does not yield substantial new insights. The authors suggest that allowing the lottery inducing an equivalent convex combination to contain nondeterministic voting rules would require notably richer mathematical machinery.

Conclusion

The paper concludes that the uniform random dictatorship is uniquely characterized by the intersection of anonymity, efficiency, monotonicity, neutrality, and self-equivalence. The result suggests a reason why most democratic societies choose to replace their voting rule or hold binary elections against another rule.

References

Arrow, K. J., A Difficulty in the Concept of Social Welfare, Journal of Political Economy 58.4, pp. 328–346

Bandhu, S., A Lahiri, & A Pramanik (2024), Stochastic Same-Sidedness in the Random Voting Model, Social Choice and Welfare 62.1, pp. 167–196

Gibbard, A., Manipulation of Voting Schemes: A General Result, Econometrica 41.4, pp. 587–601

Hermida-Rivera, H., & T. T. Kerman (2025), Binary Self-Selective Voting Rules, Journal of Public Economic Theory 27.3, pp. 1–7

Koray, S., Self-Selective Social Choice Functions Verify Arrow and Gibbard-Satterthwaite Theorems, Econometrica 68.4, pp. 981–996

Pattanaik, P. K., & B Peleg (1986), Distribution of Power Under Stochastic Social Choice Rules, Econometrica 54.4, pp. 909–921

Satterthwaite, M. A., Strategy-Proofness and Arrow’s Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions, Journal of Economic Theory 10.2, pp. 187–217

von Neumann, J., & O Morgenstern (1944), Theory of Games and Economic Behavior, Princeton University Press [p. 12]

Improvements for AI systems

Improved Voting Rule Selection for Democratic Systems: AI systems can now select voting rules that are anonymous, efficient, monotonic, neutral, and self-equivalent by ensuring they align with the uniform random dictatorship. This allows for a mathematically rigorous foundation in choosing between replacing a voting rule with itself or switching to another rule.
Characterization of Uniform Random Dictatorship: AI can verify if a given stochastic voting mechanism is the uniform random dictatorship by checking if it satisfies all four democratic axioms plus self-equivalence. This provides a definitive test for identifying this specific, highly randomized dictatorship model in complex systems.
Stability Analysis of Stochastic Environments: By utilizing the concept of self-equivalence, AI can analyze why societies choose to replace their voting rule with another or engage in binary elections against one other rule. This helps explain observed political behavior based on the mathematical stability axiom described as a voting rule must always induce a convex combination of neutral and deterministic voting rules that coincides with itself.

Abstract

In this paper, I introduce a novel stability axiom for stochastic voting rules, called self-equivalence, by which a society considering whether to replace its voting rule using itself will choose not to do so. I then show that the unique voting rule satisfying the democratic principles of anonymity, efficiency, and neutrality as well as the stability principle of self-equivalence must assign to every voter equal probability of being a dictator (i.e., uniform random dictatorship). Thus, this paper suggests one reason why most democratic societies normally choose whether to replace their voting rule using a voting rule other than itself, or in binary elections against just one other voting rule.