Liquid Democracy: An Algorithmic Perspective

The strongest case is a population-continuous version of the paper’s delegation problem, with Theorem 1 as the lead anchor and Theorem 2 as a secondary positive anchor. The paper contains no named NP-hardness or tractability theorem; its named results are asymptotic impossibility and mechanism-guarantee theorems.

The continuous society should retain the paper’s network structure. Let \(\mathcal T\) be a finite set of complete voter-role types. A type includes competence \(p_t\), the voter’s local neighborhood profile, the approved-neighbor set determined by \(p_{t'}>p_t+\alpha\), and any ordering information visible to a local mechanism. A society is a mass vector \(\mu\in\Delta(\mathcal T)\). Thus \(\mu_t\) is the fraction of ordinary voters of role \(t\), while the number of voters is large and \(|\mathcal T|\) is fixed or small.

A delegation decision is a mass flow along approved edges. If a fraction \(x_{t,t'}\) of type \(t\) delegates to type \(t'\), the row sums of \(x\) are bounded by \(\mu_t\). Delegation remains acyclic because competence strictly increases. The objective is still the probability that weighted majority selects the correct binary alternative; only the electorate has become continuous. For a positive-mass sink, independent votes concentrate by the law of large numbers. It is also important to permit finitely many distinguished hub voters of zero limiting population mass but positive delegated load: the paper’s central phenomenon is precisely that a few voters can accumulate a macroscopic share of the votes. If hubs are forbidden, the mirror removes the main liquid-democracy effect.

The high-multiplicity regime is plausible in a large party, union, or online civic platform with millions of members but recurring competence bands and recurring network roles. Many voters may have the same competence estimate, the same number and types of neighbors, and the same visible ordering of approved delegates. The type is therefore a genuine complete description for this problem, not an arbitrary grouping of voters with different prices or abilities.

My lead anchor is Theorem 1, proved in Section 3 of the paper, although the technical feasibility Claim 1 is deferred to the full version. As printed, it states: for every \(\alpha_0\in[0,1)\), no local delegation mechanism satisfies both positive gain and do no harm. Positive gain means that there exist \(\gamma>0\) and \(n_0\) such that, for every \(n\ge n_0\), some \(n\)-voter graph \(G_n\) satisfies
\[ \operatorname{gain}(M,G_n)\ge \gamma. \]
Do no harm means that, for every \(\varepsilon>0\), there is \(n_1\) such that every graph with \(n\ge n_1\) voters satisfies
\[ \operatorname{gain}(M,G_n)\ge -\varepsilon. \]
The theorem therefore rules out every local mechanism that sometimes gives a constant improvement while never suffering a nonvanishing asymptotic loss, even though voters may delegate only to strictly more competent neighbors.

The corresponding continuous problem is Local-Continuum Liquid Democracy. An instance consists of \(\alpha_0\), a finite catalogue of voter-role types, an admissible family of mass distributions \(\mu\), their local neighborhood profiles, and any finite set of distinguished hubs. A local policy is a kernel \(M\) assigning, to each local view, probabilities of retaining the vote or delegating to each approved neighbor slot. Its limiting delegation probabilities induce a mass flow \(x\). If ordinary sink type \(t\) receives mass \(w_t\), its limiting correct contribution is \(w_t p_t\). If a distinguished hub \(h\) receives mass \(w_h\), its single vote contributes \(w_hY_h\), where \(Y_h\sim\operatorname{Bernoulli}(p_h)\). Thus the continuous success probability is
\[ P_M(\mu) = \Pr\!\left[ \sum_t w_t p_t+\sum_h w_hY_h>\frac12 \right], \]
with direct voting \(P_D(\mu)\) obtained from the no-delegation flow. The gain is \(P_M(\mu)-P_D(\mu)\).

The question is whether there exists a local kernel \(M\) such that, across arbitrarily fine high-multiplicity realizations of these mass instances, it has continuous positive gain and continuous do-no-harm. A solution is either such a local kernel, together with its gain guarantees, or a continuum counterexample.

I expect the answer to remain negative for the natural class that allows sparse hubs. The paper’s \(k\)-center \(m\)-uniform star becomes particularly transparent: a positive mass of leaves has a local view inducing delegation probability \(\rho>0\), while \(k\) hub agents have vanishing native mass but receive a positive total delegated mass. Direct voting concentrates correctly, whereas the delegated outcome can be wrong whenever all \(k\) hubs vote incorrectly. The limiting loss remains bounded away from zero, essentially the paper’s \((1-p_c)^k\). This is a genuine mass-flow formulation of their correlation argument, not a different voting problem.

The further questions are computationally meaningful: can one recognize from a finite role catalogue whether a local rule has a dangerous hub pattern; can one optimize local delegation probabilities subject to a guaranteed loss bound; and what changes if neighborhoods have uniformly bounded degree or if all sinks must have positive population mass?

The secondary anchor is Theorem 2, a result of the paper, with its technical proof relegated to the full version and only sketched in the supplied text. It assumes that there is a \(\beta\in(0,1/2)\) such that every competence level lies in \([\beta,1-\beta]\). For every \(\alpha\in(0,1-2\beta)\), GREEDYCAP with a cap \(C:\mathbb N\to\mathbb N\) satisfying
\[ C(n)\in\omega(1) \qquad\text{and}\qquad C(n)\in o(\sqrt{\log n}) \]
satisfies both positive gain and do no harm. GREEDYCAP repeatedly chooses a voter with the largest number of approved predecessors, delegates to that voter as many currently available votes as permitted by the cap, and removes the voter and the delegated voters from further consideration. Delegations are therefore one-hop, and no voter has weight greater than \(C(n)\).

Its continuous counterpart is Capped Continuum Delegation. An instance consists of \(\beta,\alpha\), a finite network-role catalogue, a mass vector \(\mu\), and the approval relation. The decision variable is a one-hop mass flow \(x\) generated by the greedy rule: select the currently available recipient with maximum approved incoming mass and route approved voter mass to it up to the scaled capacity. At resolution \(N\), a per-voter cap \(C(N)\) corresponds to a mass capacity \(C(N)/N\); the continuous question is therefore about the family of flows with \(C(N)\to\infty\) and \(C(N)=o(\sqrt{\log N})\), rather than about a fixed unscaled integer cap.

The problem asks whether, for every admissible \(\mu\), the limiting gain of this capped flow is nonnegative, and whether there are admissible mass distributions for which the limiting gain is bounded below by a positive constant. A solution is the induced mass flow and its correctness guarantee, or a mass distribution witnessing failure.

I expect this version to be tractable, at least for a fixed finite role catalogue: the mass constraints are linear, and the bounded-competence assumption prevents a vanishing set of perfectly informed or perfectly misinformed voters from dominating the variance. The finite-type formulation also exposes a natural continuous-optimization problem: choose a capped delegation flow maximizing limiting correctness subject to approval and network constraints. This is recognizably GREEDYCAP’s question, not merely a generic mean-field voting model.

The weakest point is that the continuous mirror must decide how to treat sparse hubs and cap scaling. If a referee insists on a completely atomless population in which no individual can receive macroscopic delegated mass, Theorem 1’s central counterexample disappears and Theorem 2 becomes largely a law-of-large-numbers statement. Conversely, allowing zero-mass hubs is a substantive modelling choice. I think it is justified because preserving the possibility of a few high-weight delegates is necessary to remain faithful to the paper’s underlying problem. The mirror therefore covers the paper’s local-impossibility and capped-delegation results, not its discussion of more than two alternatives or the unrelated literature on continuous outcome spaces.