Deliberation as Evidence Disclosure: A Tale of Two Protocol Types

The strongest honest case is a two-alternative, high-volume deliberation mirror, with the simultaneous protocol as the lead. One qualification matters: this paper contains no named theorem asserting NP-hardness, membership in \( \mathrm{P} \), W[1]-hardness, or FPT status. Its named results are protocol-correctness theorems. Thus, under a strict complexity-only reading, it supplies no eligible computational-complexity anchor. Under the broader and more useful reading of “computational result”, two of its theorems support a genuine population-continuous mirror.

The plausible regime is a large deliberative platform: perhaps \(10^5\) reviewers evaluating two competing proposals, with only a few dozen recurring epistemic profiles. A type records an agent’s initial evidence quantities for each alternative and whether the agent is keen or lazy. Agents of one type have the same evidence counts and behaviour; their evidence packets are distinct but exchangeable, since the paper uses only their quantities and public/private status. The population is represented by masses \( \mu_t \), with \( \sum_t\mu_t=1 \). A scale \(M\) records total population intensity, so type \(t\) has mass \(M\mu_t\). This scale is necessary because the original model counts evidence items rather than normalising them by population size.

My lead problem is \(\textsc{Continuous-Simultaneous-Deliberation}\), mirroring Theorem 2, proved in this paper.

An instance consists of \(A=\{a,b\}\), a finite type set \(T\), rational masses \( \mu_t \), a population scale \(M\), rational initial evidence quantities \(q_t(a),q_t(b)\), and a target \(a\). The total objective evidence is

\[ E(x)=M\sum_{t\in T}\mu_t q_t(x). \]

The instance satisfies completeness and disjointness in the natural packet sense: all objective evidence is held by the population, and each packet is initially private to one agent. At each round, every type computes its evidence order from its remaining private evidence plus the public evidence \(K(a),K(b)\). Plurality is computed by mass:

\[ p(x)=\sum_{t\in T}\mu_t\,\mathbf{1}[x\in\operatorname{top}_t]. \]

The simultaneous protocol is then applied exactly as in the paper: every positive-mass type discloses all remaining evidence for every alternative in its favoured set; the corresponding public evidence is added; all rankings and plurality masses are updated; the process stops when no positive-mass type dissents.

The computational question is: output the terminal plurality winner set, or decide whether \(a\) is a final winner. A solution is the exact terminal winner together with the finite disclosure trace.

This is a genuine continuous-population problem. The alternatives, evidence logic, behavioural assumptions, plurality rule, and stopping condition are unchanged. Only the electorate is represented by a distribution over finitely many types, and disclosure is aggregated by mass. It is also likely tractable. Under the simultaneous protocol, a type can fully disclose its remaining evidence for a given alternative at most once, so there are at most \(O(|A||T|)\) substantive type-level disclosure events. Exact winner evaluation should therefore be polynomial in \( |A| \), \( |T| \), and the encoding length of the rational data.

The named anchor is Theorem 2:

For \(A=\{a,b\}\), with \(a\) optimal and the initial distribution complete and disjoint, if \(a\notin F_{\mathrm{PL}}(\succ^0)\), then \(P_{\mathrm{sim}}\) is full-disclosure equivalent for both keen and lazy agents.

The same argument extends to the high-multiplicity type representation when every occupied type has mass at least one agent, \(M\mu_t\ge1\). If the initially underdog \(a\) has any supporters, their disclosure creates public evidence for \(a\). If some \(b\)-supporting agent still preferred \(b\) afterwards, that agent’s private surplus for \(b\) would be large enough to outweigh all disclosed evidence for \(a\); because there is at least one such agent, total evidence for \(b\) would exceed total evidence for \(a\), contradicting \(a\)’s optimality. Thus the continuous problem is Class A on this restricted domain: the type-level simulation is tractable and the theorem supplies a correctness guarantee.

The second, more intervention-oriented mirror is \(\textsc{Continuous-Speaker-Order-Control}\), anchored on Theorem 3, also proved in this paper, though the printed text gives a proof sketch.

Here the instance again has two alternatives and a mass distribution over non-tied initial evidence-order types. The organizer knows that \(a\) is objectively optimal and knows which agents initially rank \(a\) or \(b\) first, but does not know the exact distribution of evidence quantities. The decision variable is a measurable speaking schedule. In a finite representation, this is a sequence of type blocks with rational masses; a block \( (t,\lambda) \) means that mass \( \lambda \) of type \(t\) speaks next under \(P_{\mathrm{seq\text{-}con}}\). The objective is to find a schedule that guarantees \(a\) as the final winner for every complete and disjoint evidence allocation consistent with the known initial rankings.

A solution is a schedule together with the guarantee that every admissible evidence allocation leads to \(a\). Theorem 3 supplies an explicit solution: schedule all agents initially putting \(b\) first before all agents initially putting \(a\) first. In the continuous mirror this is simply a two-block mass schedule,

\[ \text{b-top mass first} \;\longrightarrow\; \text{a-top mass second}. \]

This is a tractable Class A problem on the theorem’s domain: the schedule is compactly represented by the two blocks, and checking the hypotheses is polynomial in the number of types. It is also recognisably the authors’ question, not a softened substitute: the paper explicitly frames the organizer as choosing the order in which people speak.

This mirror naturally generates harder follow-up problems: minimise the cost of moving mass between speaking blocks; impose per-round speaking capacities; allow uncertainty about which alternative is optimal; include neutral mass; or move to \(m\ge3\). The paper’s Proposition 4 shows why these extensions should not be waved away: neutral agents can defeat the simple \(b\)-first ordering even with two alternatives. That is a useful boundary, not a reason to reject the mirror.

I would not anchor on the simulations, or on Theorem 1’s \(n\)-dependent evidence threshold. The simulations are not named computational results, and the factor \(n\) in Theorem 1 is precisely where a continuum limit requires care. A bare model with only normalised \( \mu \) and no population scale \(M\) can make public evidence too small relative to one agent’s private evidence, destroying the theorem. The mirror must therefore carry the evidence-intensity convention explicitly.

That is also the weakest point of the positive case. In the simultaneous mirror, exchangeable private evidence packets and the scale \(M\) are natural in a large reviewing or audit population, but they are an additional modelling decision. The sequential mirror is more fragile still: “one evidence item per speaker” has to become a mass-flow or high-multiplicity limit, and an arbitrary ordering of an atomless population is not automatically meaningful. For that reason, I regard \(\textsc{Continuous-Simultaneous-Deliberation}\), anchored on Theorem 2, as the strongest case. It preserves the paper’s actual epistemic dynamics while avoiding the main pathology of continuousising a sequential speaking order.

The paper does not cover the whole ChoCo programme: it offers no NP-hardness or algorithmic-complexity classification, and I would not claim one. But it does contain a credible Class A entry point: computing and guaranteeing the truth-tracking outcome of deliberation when a large society is given as a finite distribution over epistemic types.