The Impossibility of a Gerrymander-Proof Representative Democracy

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

Ruth: We're looking at the paper: "The Impossibility of a Gerrymander-Proof Representative Democracy".

Warren: Here's the gist of the paper. A representative democracy is immune to gerrymandering if it satisfies Chambers’ (2008) representative consistency <ref:2607.05660#pg2> We examine preference aggregation and show that representative consistency is mutually inconsistent with three other normative desiderata — efficiency, anonymity, and neutrality <ref:2607.05660#pg2> Core Impossibility Result The central finding of the paper is that there does not exist a social welfare function that is immune to gerrymandering and satisfies other normative principles of preference aggregation — efficiency, anonymity, and neutrality <ref:2607.05660#pg2> This impossibility result is demonstrated both in a setting with ordinal preferences and in a setting with expected utility preferences <ref:2607.05660#pg2> The paper shows this impossibility returns even when the set of alternatives is infinite, even when adding efficiency and neutrality axioms <ref:2607.05660#pg2> Ordinal Preferences Analysis In the ordinal setting, where individuals have ordinal preferences over a set of alternatives, representative consistency is defined as being invariant to the partitioning of individuals in a two-stage aggregation process <ref:2607.05660#pg2> The authors define four key axioms: representative consistency, strong Pareto, anonymity, and neutrality <ref:2607.05660#pg2> Theorem 1 states that there is no social welfare function satisfying representative consistency, strong Pareto, anonymity, and neutrality when the number of individuals is at least three and the number of alternatives is at least two <ref:2607.05660#pg2> The proof in Case 2 demonstrates a contradiction by showing that if certain preferences hold relative to a set of alternatives, they violate strong Pareto <ref:2607.05660#pg2> Expected Utility Preferences Analysis The impossibility result is shown to persist even when considering expected utility preferences, where individuals have vNM expected utility representations <ref:2607.05660#pg2> In this setting, the strong Pareto axiom is weakened to the Pareto principle <ref:2607.05660#pg2> Theorem 2 states that there is no social welfare function satisfying representative consistency, the Pareto principle, anonymity, and neutrality when the number of individuals is at least three and the number of outcomes is at least three <ref:2607.05660#pg2> The proof for this theorem uses Harsanyi’s Aggregation Theorem to establish a relationship between utility indices <ref:2607.05660#pg2> Independence of Axioms The paper demonstrates the partial independence of the axioms under finite settings <ref:2607.05660#pg2> For ordinal preferences, the Borda count social welfare function is shown not to satisfy representative consistency, while it satisfies strong Pareto, anonymity, and neutrality <ref:2607.05660#pg2> In the expected utility setting, relative utilitarianism satisfies the Pareto principle while satisfying anonymity and neutrality <ref:2607.05660#pg2> Furthermore, a hierarchical dictatorship is shown to satisfy representative consistency, the Pareto principle, and neutrality <ref:2607.05660#pg2> The paper concludes that for finite sets of individuals and alternatives, a social welfare function satisfying representative consistency, the Pareto principle, anonymity, and neutrality must be the indifference priority rule <ref:2607.05660#pg2> The impossibility theorem is shown to hold across different settings and preference representations <ref:2607.05660#pg2> The paper provides a proof that representative consistency is inconsistent with the other three normative desiderata, establishing that no gerrymandering-proof social welfare function exists under these conditions <ref:2607.05660#pg2> The analysis covers both finite and infinite sets of individuals and alternatives <ref:2607.05660#pg2> The paper also notes that the bound on the number of outcomes is tight in the expected utility setting <ref:2607.05660#pg2> The paper provides a rule that satisfies the axioms when X = 2 <ref:2607.05660#pg2> and establishes equivalence between certain aggregation rules and the indifference priority rule when X = 2 <ref:2607.05660#pg2> The paper also notes that strong Pareto implies the Pareto principle for X ≥ 3 <ref:2607.05660#pg2> The paper provides a proof that the indifference priority rule satisfies representative consistency, the Pareto principle, anonymity, and neutrality when X = 2 <ref:2607.05660#pg2> and I is finite <ref:2607.05660#pg2> The paper also notes that for X ≥ 3, a social welfare function satisfying representative consistency, strong Pareto, and neutrality cannot exist <ref:2607.05660#pg2> The paper provides a proof that the indifference priority rule satisfies representative consistency, the Pareto principle, anonymity, and neutrality when X = 2 <ref:2607.05660#pg2> and I is finite <ref:2607.05660#pg2>.

Ruth: The paper also notes that for X ≥ 3, a social welfare function satisfying representative consistency, strong Pareto, and neutrality cannot exist <ref:2607.05660#pg2> The paper provides a proof that the indifference priority rule satisfies representative consistency, the Pareto principle, anonymity, and neutrality when X = 2 <ref:2607.05660#pg2> and I is finite <ref:2607.05660#pg2>.

Warren: The paper also notes that for X ≥ 3, a social welfare function satisfying representative consistency, strong Pareto, and neutrality cannot exist <ref:2607.05660#pg2> The paper provides a proof that the indifference priority rule satisfies representative consistency, the Pareto principle, anonymity, and neutrality when X = 2 <ref:2607.05660#pg2> and I is finite <ref:2607.05660#pg2>.

Ruth: The paper also notes that for X ≥ 3, a social welfare function satisfying representative consistency, strong Pareto, and neutrality cannot exist <ref:2607.05660#pg2> The paper provides a proof that the indifference priority rule satisfies representative consistency, the Pareto principle, anonymity, and neutrality when X = 2 <ref:2607.05660#pg2> and I is finite <ref:2607.05660#pg2>.

Paper summary: Ruth: To summarize what they're saying in "The Impossibility of a Gerrymander-Proof Representative Democracy," they are looking at how social welfare functions work when we have this two-stage process of aggregation >

Warren: They define representative consistency as the condition where the final social preference resulting from partitioning people into blocks and then aggregating those results is independent of how you initially partitioned those people into districts >

Ruth: That's the core idea, and they show that this representative consistency is mutually inconsistent with efficiency, anonymity, and neutrality >

Iris: I see. So the paper establishes that you can't have a social welfare function that respects representative consistency along with efficiency, anonymity, and neutrality >

Warren: They prove this impossibility result in two settings: first with ordinal preferences where people have ranked choices like A is better than B, and second with expected utility preferences where they use those vNM expected utility representations >

Ruth: And they even show this impossibility holds even when the set of alternatives is infinite, and that's when things get interesting for the scope of this paper >

Iris: So it’s not just a simple case; it works even when you have an infinite number of possible outcomes to choose from >

Warren: That means the restriction isn't just about having a few options; it holds up under more complex setups too, like the ones mentioned in Puppe and Tasnadi and Alexeev and Mixon >

Ruth: It really puts a limit on what we can achieve in designing these systems if we want them to be gerrymandering-proof without sacrificing those other basic principles of fairness >

Iris: So the paper is basically saying that you have to give up one of those other principles if you want representative consistency, or you have to accept that gerrymandering might mess with the policy outcomes >

Warren: It sets a boundary for designing these systems based on these four axioms: representative consistency, strong Pareto, anonymity, and neutrality >

Conclusion: Ruth: So what does all this mean for the title of "The Impossibility of a Gerrymander-Proof Representative Democracy" and the authors' work?

Warren: It means that finding a system that is both gerrymandering-proof and satisfies efficiency, anonymity, and neutrality simultaneously isn't possible, regardless of whether we use simple rankings or expected utility models >

Iris: I think the paper points to the fact that there doesn't exist a social welfare function satisfying representative consistency along with efficiency, anonymity, and neutrality when you have at least three individuals and at least two alternatives >

Ruth: That’s a concrete statement of the impossibility result from Theorem one in their ordinal setting, which is pretty strong because it covers the basic human experience of voting > <ref:2607.05660#pg1>

Warren: It also shows that even if you move to expected utility preferences, where people have vNM expected utility representations, you still can't get representative consistency along with anonymity and neutrality when there are at least three outcomes >

Iris: So the implications are that for a representative democracy trying to avoid gerrymandering, they can't just pick any aggregation rule; they have to choose which of those other principles they are willing to relax >

Ruth: It forces us to be very specific about what we value in our political systems when we try to design them >

Warren: The paper provides a proof that representative consistency is inconsistent with the other three normative desiderata, so no such gerrymandering-proof social welfare function exists under these conditions >