| paper | d4-conf-sagt-2021-0007 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | no |
None of the named results asserts an algorithm, computational hardness, or a formulation used to compute; they establish equilibrium outcomes, comparisons, and existence. The proposed mass delegation game would also remove individual pivotality, changing the strategic question behind the paper’s results. The computational-result requirement fails regardless of whether that new continuous game is worth studying.
fails bit a — no named computational result to mirror
The proposed mirror concerns the equilibrium thresholds in Theorem 1; it does not mirror Proposition 1’s probability comparison or Proposition 2’s equilibrium-existence interval. The numerical comparisons provide no stated computational guarantee.
The strongest positive case is that the paper has a natural high-multiplicity setting: large blockchain stakeholder pools with two complete voter types, well-behaving stakeholders with voting cost \(c\) and bad alternative preference, and misbehaving stakeholders who vote for the bad alternative. The paper assumes agents within each type are interchangeable; the number of distinct types can stay at \(\tau=2\) as the population grows. That makes a population-level model plausible.
But the paper has no named computational-complexity result to anchor a ChoCo mirror. Its numbered results concern equilibrium existence and outcomes, not whether a computational problem is in P, NP-hard, FPT, or similar. In particular, Theorem 1, proved in the paper, says that for any \(c>0\), there are thresholds \(f^*(c)\in\Theta(\delta^2/c^2)\) and \(n^*(f)\in\Theta(f^2/\delta^2)\), for any \(\delta\in\mathbb N\), such that when \(f\ge f^*(c)\) and \(n\ge n^*(f)\), well-behaving agents lose. The paper explains this as the absence of a solution to its mixed-equilibrium indifference equation, so all well-behaving agents delegate and the misbehaving voters win. This is a substantive analytic result, but it is not a computational classification. Proposition 1 likewise compares winning probabilities, and Proposition 2 gives an equilibrium-existence interval for \(f=1\); neither supplies such a classification. The numerical equilibrium calculations do not add a named complexity result.
A plausible *new* continuous research question would be Mass Delegation Equilibrium. Its instance is rational masses \(\mu_G,\mu_M\) with \(\mu_G+\mu_M=1\), \(\mu_M>0\), and a voting cost \(c>0\). Good mass chooses a delegation share \(\gamma\in[0,1]\); bad mass votes; delegated mass is distributed among active voters in proportion to their voting masses. Ask for every symmetric equilibrium delegation share and the resulting winner. This is grounded in the paper’s two types, costly voting, and uniform delegation, in a credible large-stakeholder regime.
The weakness is decisive for mirroring Theorem 1. In this non-atomic model, an individual deviation changes no aggregate vote total, so voting’s pivotality benefit disappears. Since voting costs \(c>0\), all good mass delegates; with \(\mu_M>0\), bad voters receive the delegated mass and win. The result is immediate and no longer has the paper’s thresholds in \(f\) and \(n\). The authors could recognize this as a nonatomic limit of their game, but it strips away the finite-agent pivotal incentives that drive their analysis. Keeping those incentives requires retaining atomic voters and Poisson count fluctuations, which leaves a stochastic finite-population model rather than the programme’s continuous society.
So the positive case is a credible population interpretation and a precise, tractable extension question. It does not furnish a continuous computational mirror of a named computational result in this paper; under the stated anchor requirement, there is no eligible anchor.
The positive case correctly identifies a plausible high-multiplicity setting: a large stakeholder pool with two complete types. But that establishes that the population story is sensible; it does not give ChoCo a computational result to mirror. Theorem 1 is an equilibrium threshold result, Proposition 1 compares winning probabilities, and Proposition 2 gives an equilibrium-existence interval. None states the complexity of a computational problem. The paper’s scalar indifference equation could motivate a new task—computing equilibria—but that would be a task we add, not a mirror of one of these results.
Theorem 1 is the strongest candidate for a substantive mirror, and its key object exposes the difficulty. The voting incentive comes from an individual’s chance of being pivotal, balanced against the cost \(c\). In a nonatomic population, one individual’s vote changes aggregate vote mass by zero, so that pivotality term disappears. The equilibrium question at the heart of the theorem has lost its strategic margin. Preserving it requires retaining atomic voters or finite-population fluctuations. The theorem’s interesting threshold regime is especially sparse: with fixed \(c\), \(f\) is on the order of \(1/c^2\) while \(n\) is on the order of \(f^2\), so the misbehaving share \(f/(n+f)\) tends to zero as \(f\) grows. A type distribution over normalized mass then erases the attackers; keeping their finite count also keeps the atom size and the Poisson game.
Proposition 1’s comparison relies on probabilities generated by those finite counts. A population model with fixed positive type shares has deterministic aggregate masses; recovering fluctuation-sensitive probabilities requires carrying a finite-size or stochastic correction. Proposition 2 is even more exposed: its \(f=1\) case depends on one atomic misbehaving voter. That voter has vanishing normalized mass as the electorate grows. Giving the misbehaving type positive mass changes the regime; giving it zero mass removes the one-voter case. Neither proposition supplies a distinct computational anchor.
The proposed Mass Delegation Equilibrium is the best version of the mirror to consider. It is a coherent continuous-population game, and its nonatomic limit makes the loss of pivotality explicit. One could enrich it with a scaling parameter for voter size, rare attackers, or Poisson fluctuations. That might be worthwhile as an asymptotic game-theory study, but it restores precisely the finite-population structure the mass-only model removed. With the paper’s two types and one delegation share, the computational variant would still be a low-dimensional equilibrium calculation, not a complexity landscape unlocked by continuization.
So the case against is narrow but strong: this paper has no named computational result for ChoCo to mirror, and its best equilibrium mirror loses the individual pivotality that drives the analysis. A stochastic, two-scale extension remains possible, but it is a new finite-game asymptotic question rather than a continuous-society computational mirror of the paper’s results.
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.