| paper | d4-journals-jair-2025-0030 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | unclear |
Theorem 4 is a clear computational result and matches the proposed edge-addition decrease operation. But coalitions and permutations in the proposed model range over the finite set of indivisible blocs, while each mass is only a normalized voting weight; scaling the weights recovers an ordinary finite-player game. The party interpretation is plausible, but it interprets the paper’s existing weighted game rather than continuizing its population.
fails bit b-none — no population with multiplicity to relax
The proposed model treats each bloc as one indivisible player, so its members never enter coalitions or permutations individually; their population mass affects only the normalized voting weight.
fatal: True
The proposal focuses on Theorem 4’s edge-addition decrease results for PBI and SSI. It leaves the paper’s deletion, increase, maintain, comparison, and upper-bound results aside; the proposed formulation does not yield a continuous-population mirror of the covered result.
The strongest mirror I believe in is a cohesive-bloc version of the paper’s edge-control problem. Consider a federation with many members—such as shareholder associations or party caucuses—but only \(\tau\) voting blocs. A type is a bloc whose members share voting rights, preferences, and communication relationships, and whose caucus votes as a unit. Its mass \(\mu_t\) is its fraction of the federation’s total voting weight. The regime is \(N\gg\tau\): many represented members, relatively few blocs.
That bloc assumption matters. These are organized groups that enter coalitions together, not independent individuals whose votes can be split. It fits the paper’s own examples, where parties or shareholder groups are the players.
Here is the continuous question, Cohesive-Bloc Edge-Addition Decrease Control, my lead mirror. An instance consists of a finite type set \(T\), positive binary-rational masses \((\mu_t)_{t\in T}\) summing to \(1\), a quota \(q\in(1/2,1]\), a communication graph \(H=(T,E)\), a distinguished bloc \(p\), a nonnegative integer \(k\), and the allowed nonedges not incident to \(p\). A coalition \(S\subseteq T\) wins if it contains some \(U\subseteq S\) that is connected in \(H\) and has mass \(\sum_{t\in U}\mu_t\ge q\). Define the bloc power indices by the paper’s PBI and SSI formulas over coalitions or permutations of \(T\), treating each bloc as one actor. Given either index \(\mathrm{PI}\in\{\mathrm{PBI},\mathrm{SSI}\}\), ask whether there is a nonempty set \(F\) of at most \(k\) allowed edges such that \(\mathrm{PI}_{H+F}(p)<\mathrm{PI}_H(p)\). A solution is such an edge set \(F\).
My anchor is Theorem 4, proved in this paper: for either PBI or SSI, control by adding edges to decrease the distinguished player’s power is PP-hard both when additions are restricted to edges not incident to that player and in the unrestricted case. The reduction already uses a limit of one added edge. Thus the theorem covers the precise control goal and restriction in this mirror; it is not merely a result about computing an index or an encoding.
I expect the mirror to be hardness-transferring (Class B). Given a discrete instance with integer weights \(w_i\), let \(W=\sum_i w_i\) and make type \(i\) a bloc of mass \(\mu_i=w_i/W\), with quota \(q/W\). Keep the graph and distinguished player. Every coalition has the same win/loss status, both power indices are unchanged, and the possible edge additions correspond exactly. This embeds the discrete problem in the rational-mass one, including the one-edge restriction. It also gives the intended high-multiplicity regime: the represented population can be very large compared with the number of blocs, while its distribution is specified succinctly by \(\tau\) rational masses.
I think the authors would recognize this as their problem in a plausible regime: their players already stand for parties, shareholders, or other collective actors, and the graph still controls which coalitions can form. The mirror changes the weight description to population shares while preserving bloc-level coalition formation, power, and edge control. It covers the paper’s edge-addition-to-decrease results, not its deletion, increase, or maintain variants.
The weak point is that the mirror keeps blocs indivisible. It does not define the power of an individual member who can join a coalition independently; that would require a different continuous power-index model, and Theorem 4 would not automatically settle it. I think the narrower case survives because cohesive parties and shareholder associations are credible decision-making actors in the paper’s own settings. Natural next questions are whether hardness persists with equal bloc masses or restricted graph structure, and whether the problem admits useful algorithms parameterized by \(k\), \(\tau\), or graph structure.
Theorem 4 is a strong computational anchor, but the proponent’s mirror has a real mismatch with the programme’s population axis. In the paper, a weight belongs to one player, and coalitions and permutations are formed from those individual players. In the proposed model, they are formed from the \(\tau\) blocs. A bloc’s members never join separately, and the indices never measure any member’s power. Its mass acts as a voting weight.
That distinction matters for high multiplicity. Replacing a weight-\(w_i\) player by \(w_i\) ordinary players would change which coalitions can form and who can be pivotal. To preserve the paper’s game exactly, the members have to act as one indivisible player. The mirror therefore treats a large group as a single actor precisely where the paper’s power indices count individual actors. And since every rational mass and quota can be scaled to integers, the proposed input is also a rational-weight version of the same finite-player game; the player set and its coalition structure remain finite.
A more faithful population model would let members of each type join coalitions individually, with a type-level graph describing their communication. But then an edge between types must specify what relationships it creates among their members, and PBI and SSI must be replaced or carefully limited: those indices assign power to individual players through finite coalitions and permutations. In an atomless population, an individual has zero mass, so the ordinary pivotality question no longer captures the bloc’s influence. A limiting power density or a new type-level index might give a worthwhile question, but Theorem 4 does not establish its relevance or settle it.
The negative case is weakest against the proposed party interpretation. The paper explicitly treats parties as players, and parties whose members act through a common leadership could sensibly be modeled as blocs with population-share weights and group-level communication edges. That is a recognizable edge-control problem, and the authors might well accept it. I therefore cannot honestly rule out every worthwhile scenario. My narrower objection is that this anchor, as stated, changes the units of the players’ weights without making the population itself continuous; the more genuinely population-based version needs a different model of coalitions and power.
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.