d4-journals-geb-2012-0161

Theorem 1 is the strongest candidate because it gives an explicit implementation mechanism. But the proposed mirror replaces pivotal individuals with a nonatomic population, where one report cannot change the aggregate outcome; it also replaces at least one honest individual with a positive honest mass. That is a different strategic problem, so the anchor does not survive.

The objection that survived

The proposed model substitutes a positive mass of honest individuals for the paper's at-least-one unknown honest individual, with no result showing that this change preserves the implementation question.

What the mirror covers

The candidate concerns only Theorem 1's many-person implementation result; the two-person characterizations and the Bayesian extension are left aside.

Open questions for a prover

The case FOR (proponent)

I can’t honestly give this paper a qualifying ChoCo anchor: it has no named result classifying an algorithmic problem as polynomial-time solvable, NP-hard, FPT, or similar. Its named results concern mechanism existence and characterization. The strongest nearby result is Theorem 1, proved in this paper: when \(n\ge3\), at least one individual is partially honest, and the planner knows this but not the individual’s identity, every social choice correspondence satisfying No Veto Power is Nash implementable. That is a constructive implementation theorem, but it is not a computational-complexity result under the programme’s anchor rule.

The most credible *candidate mirror*, if that broader theorem were admitted, would be nonatomic Nash implementation with a positive mass of partially honest individuals. Take a finite alternative set \(A\) and a finite set \(T\) of preference rankings. A society has a rational distribution \(\mu\in\Delta(T)\) with support size \(\tau\); think of many participants in a public consultation whose ballots fall into relatively few preference types, with each person’s full ranking and honesty disposition defining their type. Assume a known positive share \(\varepsilon\) is partially honest, but the planner does not know who they are. Everyone knows the population state \(\mu\), as in the paper’s complete-information model. The planner’s target is a correspondence \(F(\mu)\subseteq A\), with the continuous analogue of No Veto Power: if every type with positive mass ranks \(a\) first, then \(a\in F(\mu)\).

The question is: does there exist an anonymous mechanism whose outcome depends on the distribution of reported messages and whose Nash equilibrium outcomes, for every such \(\mu\) and every placement of the honest mass, are exactly \(F(\mu)\)? A solution would be such a mechanism; an impossibility result would show that no such mechanism exists. The objective is implementation of \(F\), not optimization. This is close to Theorem 1’s underlying problem, but it is a theorem-prover question rather than a complexity question, so it does not repair the missing ChoCo anchor.

The regime is at least imaginable: a large, publicly informed population with many participants per ballot type and an intrinsic truth-telling norm among a positive share. The weak point is substantial. Theorem 1’s mechanism relies on named individuals being able to change the outcome through the integer game. In an anonymous nonatomic population, one person’s report has zero mass and cannot change an aggregate outcome; strategic reports by the remaining population may therefore create equilibria that honest reports cannot rule out. I would expect the universal No Veto Power conclusion to fail for continuum-specific reasons, though characterizing the surviving class would be a sensible follow-up. The paper’s two-person characterizations and its Bayesian extension are also implementation results, not complexity anchors, so adding them would not strengthen the case.

The case AGAINST (opponent, writing after the proponent)

The proponent has identified the paper’s strongest nearby result, but their own account concedes the central problem: Theorem 1 is an existence theorem about Nash implementation, not a result about computational complexity. The same is true of Theorem 4, the two-person characterizations in Theorems 2 and 3, and the impossibility propositions. The proposed question—whether an anonymous mechanism exists—is still an implementation question, not a computational one. To turn it into a complexity problem, one would first need a finite encoding of the target correspondence over the continuum of societies, and of the mechanisms being considered. The paper supplies neither.

The best candidate to rescue is Theorem 1, but its mechanism relies on a finite individual’s unilateral deviation changing the outcome through the integer game. In a nonatomic population whose reports affect outcomes through their aggregate distribution, one person’s deviation has zero mass. Self-interested agents are then indifferent among reports that leave the aggregate unchanged, while honest agents may prefer truth-telling. That does not reproduce the paper’s mechanism or its implementation claim. Treating each preference type as a strategic bloc could restore pivotality, but it changes the players: a type is a description of individuals, not a coordinating coalition. Keeping pivotal individuals instead preserves an atomic, finite-agent game, even if many agents share each type.

The proposed positive share of honest agents is also a substantive change to the premise. Theorem 4 concerns one particular individual who has some probability of being honest; it does not concern a positive mass of honest agents. A positive honest share may be a sensible new model, but the remaining self-interested mass can still choose reports freely in the nonatomic game. And the two-person results concern exactly two strategic individuals. Repeating that problem across many markets would aggregate separate bilateral games, rather than continuize the population inside the implementation problem.

So I would reject this paper as a ChoCo anchor. The honest limit of the negative case is that someone could define a computational mechanism-synthesis problem for a continuous society, with an explicit encoding of the target rule and allowed mechanisms. That could be worthwhile on its own. But it would need new computational inputs and a new strategic model; it is not a mirror of a named computational result in this paper.

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.