| paper | d4-conf-atal-2015-0389 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | yes |
other→P discrete result → its continuous mirror
Answer: Polynomial-time solvable; whenever unanimity is reachable, its earliest round satisfies \(0\leq t\leq\tau-1\), and this bound is sharp. · discrete side: other · adjudication: proved
full proof & adversarial review →
Proposition 4
Given a finite alternative set \(W\), supported types \(K\) with positive rational masses \(\mu_k\) summing to \(1\), an initial total-preorder preference \(\pi_k\) for each type, and a reliability order over types inducing a most-reliable successor \(f(k)\), decide whether some finite round makes all positive-mass types share the same preference, and return the earliest such round if it exists. At each round, type \(k\) adopts the current preference of type \(f(k)\).
A continuous society is a distribution \(\mu\) over deliberation types. Each type carries a preference over \(W\) and a reliability hierarchy over types; its successor is the type it most relies on. Preferences evolve by successor copying, while masses remain fixed. Exact unanimity holds when every type with positive mass has the same preference.
The paper orders named individuals by reliability, while the mirror orders cohort types; the role-based scenario makes this plausible but does not show that the type-level relation preserves every identity-level trust distinction.
fatal: False
Covers iterative simultaneous drastic upgrades and exact unanimity, including the earliest convergence round; it leaves the other upgrade policies, modal axiom systems, and broader deliberation framework untouched.
On the strict anchor rule, this paper has no named complexity result: it does not classify a problem as polynomial-time, NP-hard, or otherwise. The nearest computational anchor is Proposition 4, proved here. It says that, when agents’ initial preferences are all distinct, repeated simultaneous drastic upgrades reach unanimity exactly when their reliability streams eventually converge to the same agent. The proposition expresses this as reachability in a finite agent graph, using a PDL formula. It is a decision characterization, not a theorem that the problem is in P. Theorem 1 and Theorem 2 give sound and complete axiom systems; they are not computational complexity results.
The positive mirror I would defend is Cohort Drastic-Unanimity Reachability. Let \(W\) be a finite set of alternatives and \(\mathcal K\) a finite set of deliberation types. A type \(k\) has a total-preorder preference \(\pi_k\) over \(W\), a strict reliability order \(\rho_k\) over the types, and positive rational mass \(\mu_k\), with \(\sum_{k\in\mathcal K}\mu_k=1\). Write \(f(k)\) for the most reliable type under \(\rho_k\). Assume initial preferences are distinct across types, matching Proposition 4’s condition.
At round \(r+1\), every member of type \(k\) adopts the current preference of type \(f(k)\): \(\pi_k^{r+1}=\pi_{f(k)}^r\). The mass of each type stays fixed. Given \(W,\mathcal K,(\pi_k),(\rho_k),(\mu_k)\), decide whether there is a finite round \(r\) at which all positive-mass types have the same preference, and return the earliest such \(r\) if there is one. A “yes” solution is that round; a “no” solution certifies that no such round exists. This mirrors the underlying deliberation problem, rather than the paper’s logical encoding of it.
A plausible regime is a large online consultation on a fixed slate of projects: millions of participants, but relatively few cohorts with the same ranking and the same trust hierarchy over cohort channels or delegates. The type records the full description the dynamics use; mass is the fraction of participants in that cohort. The co-modification is that reliability is represented over cohorts rather than named people. In a role-based deliberation setting, where participants trust cohort representatives and each cohort remains preference-homogeneous, the authors could plausibly recognize this as their drastic-upgrade process at high multiplicity. The expected answer is tractability: with distinct initial rankings, unanimity is equivalent to the supported types’ trajectories under \(f\) coalescing. One can test each pair’s trajectories in the finite functional graph and report the latest pairwise meeting round, giving a polynomial algorithm in \(\tau=|\mathcal K|\), \(m=|W|\), and the input bit length \(L\).
The scope is deliberately narrow: it mirrors Proposition 4’s unanimity question under repeated drastic upgrades, not the paper’s general lexicographic upgrades, modal axioms, or broader deliberation framework. A natural follow-on would ask whether at least a fraction \(\theta\) of the population shares one preference by round \(r\); that makes the masses computationally relevant, but changes exact unanimity to a threshold-consensus question.
My weakest point is that the masses do not affect the exact unanimity answer beyond identifying which types are present. The underlying proposition already has a finite graph formulation, so the continuum adds a plausible population regime but may add little computational substance. That is a modest mirror, and Proposition 4 is not the kind of complexity anchor the programme ultimately seeks.
The paper offers no named computational result of the kind ChoCo is set up to continuize. Theorem 1 and Theorem 2 are axiomatic completeness results. Proposition 4 is the closest anchor: it characterizes unanimous agreement under drastic upgrades as eventual convergence in a finite functional graph. That gives a decision predicate, but the paper does not pose or classify its computational complexity. A continuous “mirror” would therefore start by inventing a computational problem around a structural characterization, rather than extending one of the paper’s computational results.
Proposition 4 is still the proponent’s strongest anchor, so the model mismatch matters. Its graph has one node per named agent, and an edge to that agent’s most reliable named individual. In the paper, reliability is a strict total order over the agents themselves. To describe a society by finitely many preference-and-reliability types, with each type pointing to another type, one must make reliability an order over cohort types or channels. That may describe a sensible role-based deliberation system, but it changes what agents trust and who supplies the preference they adopt. Alternatively, preserve the paper’s semantics and include each agent’s full ranking over the named population in their type. Those rankings depend on the identities in that population, so the proposed finite type space no longer captures the usual high-multiplicity setting of a fixed repertoire of interchangeable agents.
The role-based version is the best repair: cohorts share preferences and trust a hierarchy of cohort channels. But in that model, mass does not affect the dynamics or exact unanimity; it only says how much of the population occupies each node. A threshold question—whether at least mass \(\theta\) has a common preference at some round—is a plausible new deliberation statistic. It is not the paper’s unanimity result, though: it adds a population-weighted objective to a finite graph process whose transitions are unchanged by the population distribution. That may be worth studying on its own, but it is a thin basis for a continuous-computational-social-choice programme.
So the negative case is real but not airtight. A reader could reasonably accept the cohort-channel model as a recognizable extension of the paper and find the weighted consensus question useful. The strongest objection is narrower: the paper supplies no complexity result to mirror, and preserving its stated identity-level trust semantics makes the proposed type-level population model much less natural.
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.