Power in Liquid Democracy

Theorem 4 is constructively computationally relevant, but both anchors rely on unilateral named-agent deviations and the paper's individual \(\mathrm{DB}_i\). The proposed \(\Pi_t\) samples type coalitions and forces every member of a cohort to share one action; \(\mu\) does not determine the individual delegation graph or a limit of \(\mathrm{DB}_i\). A nonatomic/Wardrop or coordinated-bloc reformulation would be a new strategic model, so neither named equilibrium theorem has a surviving continuous analogue.

The objection that survived

The opponent shows that identical \(\mu\), \(q\), and access data can correspond to different individual delegation graphs and different Banzhaf payoffs, while the proposed type-level policy changes unilateral deviations into bloc deviations.

What the mirror covers

The proposed construction targets only Theorems 2 and 4; it leaves the delegative Banzhaf characterization, comparative facts, imported Banzhaf hardness, and simulations without a faithful population analogue.

Open questions for a prover

The case FOR (proponent)

The strongest positive case is a high-multiplicity, type-level liquid-democracy game. I should be explicit about one limitation first: the paper contains no numbered theorem proving NP-hardness, membership in \( \mathrm{P} \), W[1]-hardness, or FPT. Its statement that Banzhaf computation is intractable is attributed to Matsui and is not itself a named result proved here. The closest computational anchors are therefore the named equilibrium-existence theorems.

Take a finite set \(T\) of complete agent types. A type records accuracy \(q_t\), admissible proxy types \(A(t)\), and the common power-sensitivity parameter \(\alpha\). The society is a mass vector \(\mu\), with \(\mu_t\ge 0\) and \(\sum_t\mu_t=1\). This represents, for example, a large civic platform with millions of users but only a small number of accuracy/access profiles: users in the same organizational role have the same expertise estimate and the same available proxies. The quota is a population fraction \(\beta\in(1/2,1]\).

For the pure-strategy mirror, the action is a type-level delegation policy \(\delta:T\to T\cup\{0\}\), with \(\delta(t)=t\) meaning that all mass of type \(t\) votes directly and \(0\) meaning abstention. This is the cohort version of a pure delegation profile. For \(S\subseteq T\), let \(\widehat D_\delta(S)\) be the types whose delegation paths remain inside \(S\) and terminate at a guru in \(S\). Define \(\nu_{\mu,\delta}(S)=1\) exactly when \(\sum_{t\in\widehat D_\delta(S)}\mu_t\ge\beta\). The natural type-cohort analogue of the paper’s delegative Banzhaf index is then \(\Pi_t(\mu,\delta)=2^{-(|T|-1)}\sum_{S\subseteq T\setminus\{t\}}(\nu_{\mu,\delta}(S\cup\{t\})-\nu_{\mu,\delta}(S))\). If \(t\) eventually reaches guru \(g_\delta(t)\), its utility is \(U_t(\delta)=\Pi_t(\mu,\delta)^\alpha q_{g_\delta(t)}\), with utility zero for cycles and abstention. When every type is a singleton, this is exactly the paper’s \(\DB\)-based model; with many agents per type, \(\mu\) replaces counted weights.

My lead anchor is Theorem 2, proved in this paper: “There are delegation games that have no (pure strategy) NE.” The paper gives a concrete six-agent witness with accuracies \(q=(0.51,0.7,0.9,0.6,0.7,0.9)\), quota \(4\), and directed edges \((1,3),(2,3),(4,6),(5,6)\). Each of agents \(1,2,4,5\) can either vote directly or delegate along its edge, while \(3\) and \(6\) must be gurus. The appendix checks all \(16\) profiles and exhibits a profitable deviation from every one.

The corresponding continuous problem is:

Continuous Constrained Delegation Equilibrium. Given rational \((T,\mu,q,A,\beta,\alpha)\), determine whether there exists a type-level policy \(\delta\) such that, for every type \(t\) and every admissible alternative \(a\in A(t)\cup\{t,0\}\), \(U_t(\delta)\ge U_t(\delta_{t\to a})\). If one exists, output \(\delta\); otherwise output a certificate consisting of a type and a profitable deviation for every candidate policy.

This is a genuine population mirror rather than a finite election with fractional outcomes: the input society is the distribution \(\mu\), quota is measured in population mass, and the delegation graph resolves mass through chains and cycles. The six-agent witness becomes a high-multiplicity society with six large cohorts, \(\mu_t=1/6\), and normalized quota \(\beta=2/3\). Thus the nonexistence phenomenon itself transfers to a population with arbitrarily many individuals. For variable \(|T|\), I would expect the general problem to become continuum-specifically hard: the headcount disappears, but exact pivotality, cycles, quota discontinuities, and the type graph remain. The finite theorem does not prove that complexity classification, but it gives a concrete negative instance for the continuous problem.

The second useful anchor is Theorem 4, also proved here: “In any delegation game … where \(G\) is a complete network, there exists at least one (pure strategy) NE.” Its construction selects a highest-accuracy agent \(i^\star\), lets the remaining agents sequentially choose between voting directly and delegating to \(i^\star\), freezes agents who delegate to \(i^\star\), and stops when no further delegation is profitable. The proof uses the paper’s power-monotonicity facts to show that \(i^\star\), delegators, and remaining gurus have no profitable deviations.

Its continuous counterpart is:

Complete-Network Mass Delegation Equilibrium. Given rational type masses \(\mu\), accuracies \(q\), quota \(\beta\), and a complete type-access graph, compute a policy \(\delta\) satisfying the same equilibrium inequalities above.

I expect this restricted problem to be tractable at fixed \(|T|\), and plausibly polynomial in \(|T|\) if the paper’s monotonicity argument extends to unequal masses. The natural algorithm repeatedly considers whether a type should move its mass to the highest-accuracy type; once mass has moved there, the paper’s direct-versus-indirect and power-monotonicity arguments should make the state monotone. The exact runtime is an open question, because computing \(\Pi_t\) naïvely still ranges over \(2^{|T|}\) type coalitions. The theorem itself is an existence result, not a claim that the authors already proved a polynomial algorithm.

The paper’s authors should recognize these as their problem: the mirror retains binary truth-tracking, accuracy, quota voting, delegation constraints, transitive delegation, cycles, the delegative power index, and power-seeking utilities. It covers the equilibrium-analysis part of the paper, not the axiomatization or the simulation section. The weighted-voting preliminaries do not create a novelty collision: the paper has finite named agents and finite delegation profiles, whereas the mirror makes the society itself a distribution over repeated agent types.

The weakest point is unavoidable. The paper’s Banzhaf index is defined for individual finite agents, and there is no canonical non-atomic Banzhaf index. Giving a whole homogeneous type one bloc-level pivotality \(\Pi_t\) is a defensible high-multiplicity co-modification, but it is not the only possible limit. A stricter individual-level mirror would allow mass \(x_{t,a}\) to split across proxy types and require a Wardrop-style condition: every destination receiving positive mass must maximize the deviating agent’s utility. Defining transitive delegation and power for that split flow is itself a substantial follow-up problem. If the authors insist that identical agents must remain individually rather than collectively strategic, this cohort mirror becomes substantially weaker.

The case AGAINST (opponent, writing after the proponent)

The negative case is strongest at the level of the strategic object: the paper’s power index is not an aggregative quantity, so its game does not descend to a society distribution \(\mu\).

There is also no genuine computational result in the paper to mirror. Theorem 2 and Theorem 4 are existence statements for finite pure-strategy equilibria, not complexity or algorithmic theorems. The paper’s claim that Banzhaf computation is intractable is imported from earlier work. Thus the proposed equilibrium-decision problems are new problems built around the paper, rather than continuous versions of a computational result it established.

More importantly, the proposed \(\Pi_t\) is not the high-multiplicity limit of the paper’s \(\mathrm{DB}_i\). Suppose each of the six agents in the Theorem 2 witness is replaced by \(k\) identical agents. The same population distribution has \(\mu_t=1/6\) for every \(k\), but the paper’s index is now computed over \(6k\) named agents, with denominator \(2^{6k-1}\). A random coalition of types is not a random coalition of agents. In particular, \(\Pi_t\), which samples subsets of six types, treats each cohort as one indivisible player and remains unchanged as \(k\) grows. It is therefore a Banzhaf index for a new six-player weighted game, not a continuous limit of the original individual-level index.

The problem is deeper than choosing the wrong normalization. A type-level delegation target does not specify the individual delegation graph: \(k\) agents of type \(t\) may delegate to one proxy of type \(t'\), distribute themselves among \(k\) proxies, or form different chains. These constructions have the same \((T,\mu,q,A)\) but different sets \(\widehat D(C)\), different pivotality, and different equilibria. The continuous input does not determine the paper’s payoff function.

This defeats the Theorem 2 anchor. The six “large cohorts” reproduce the finite witness only by making each cohort a single coordinated strategic bloc. But the theorem’s deviations are unilateral deviations by named agents. Moving an entire cohort is a coalition move; allowing one member to deviate leaves the type-level policy unchanged. Hence the claimed nonexistence result is not transferred to high multiplicity. It is restated for a different game whose players are cohorts.

Theorem 4 has the same defect. Its construction freezes individual agents who delegate to the highest-accuracy agent. In a high-multiplicity population, freezing an entire type is again a collective commitment, not a unilateral best response. If individual agents remain strategic, a single deviation has zero population mass and the aggregate state does not change; if type blocs act together, the model has changed its strategic units. The complete network does not resolve this mismatch.

The apparently natural rescue is to define a nonatomic influence density, perhaps by rescaling individual Banzhaf power, or to use a Wardrop condition for split delegation mass. But that requires replacing the paper’s random-coalition Banzhaf index with a new influence concept and specifying a law for random coalitions or infinitesimal pivotal coalitions. Different choices—individual influence, bloc influence, random type coalitions, or finite-mass perturbations—produce different games. This is not a forced co-modification like replacing a headcount quota by a population fraction; it changes the paper’s central object, namely what it means for an individual delegator to possess power.

A coordinated-cohort model could certainly be worthwhile, but it should be presented as a new collective liquid-democracy model, not as the continuous-population mirror of this paper. The negative case is not a proof that no imaginable reinterpretation could work; the universal claim is too strong for a modelling judgment. It does establish that neither equilibrium anchor survives as a genuine high-multiplicity mirror without replacing the paper’s agents, payoff, and strategic deviations.

The adversarial triple: the proponent anchors on up to three named results; the opponent sees that case and must defeat every anchor; the judge decides which case convinced it. These are the pipeline’s own outputs, generated by tools/triple_run.py — no human edited them. The paper’s own text is not reproduced here beyond the quoted statement above.