| paper | Strategic Abstention based on Preference Extensions: Positive Results and Computer-Generated Impossibilities |
| authors | F. Brandl, F. Brandt, Christian Geist, Johannes Hofbauer |
| venue | JAIR 2019 |
| filed under | voting · manipulation |
| judged by | gpt-5.6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | unclear |
Theorems 1 and 2 are named results, but they quantify over properties of SCFs and do not assert that any computational problem is tractable or hard. The proposed positive-mass threshold problem fixes a rule and introduces a new optimization objective, so it is a new problem rather than a continuous analogue of either theorem. Therefore no eligible paper result supplies both required bits.
fails bit a — no named computational result to mirror
The proposed threshold changes a global existence theorem into fixed-rule optimization and replaces one-agent abstention with coordinated abstention by positive mass.
fatal: True
The proposed model covers only mass abstention for a fixed rule; the paper's Fishburn impossibilities and Kelly-participation implication remain axiomatic and receive no qualifying computational mirror.
The strongest honest case is a model-level yes, but not a qualifying ChoCo case under the programme’s anchor rule.
The paper contains no named computational result of the required kind. Its central results are axiomatic impossibilities and implications, not claims that a problem is in \(P\), NP-hard, FPT, or similar. The SAT solver is used to prove those theorems; that does not turn them into complexity results. The unnumbered remark that computing the bipartisan set is polynomial-time, and the results cited in Table 2, cannot serve as anchors.
The closest possible anchor is Theorem 1, proved in this paper:
There is no majoritarian and Pareto-optimal social choice function satisfying Fishburn-participation when \(|A|\ge4\) and \(|N|\ge6\).
Theorem 2, also proved here, strengthens this to strict preferences:
There is no majoritarian and Pareto-optimal social choice function satisfying Fishburn-participation when \(|A|\ge5\) and \(|N|\ge7\), even when preferences are strict.
These are genuine results about strategic abstention, and they provide a natural continuous-population question, although not a computational anchor.
A plausible mirror would be Continuous Fishburn-Abstention Threshold. An instance consists of a finite candidate set \(A\), a finite set \(T\) of complete preference types, a rational population distribution \(\mu\in\Delta(T)\), a specified majoritarian social choice function \(f\), and a rational budget \(B\). A type \(t\) may abstain with mass \(a\), where \(0<a\le\mu_t\). The remaining society is
\[ \mu^{(-t,a)}_u= \begin{cases} \dfrac{\mu_t-a}{1-a} & u=t,\\[4pt] \dfrac{\mu_u}{1-a} & u\neq t. \end{cases} \]
The question is whether there exist \(t\in T\) and \(a\le B\) such that
\[ f(\mu^{(-t,a)})\succ_F^t f(\mu), \]
where \(\succ_F^t\) is Fishburn’s preference extension for type \(t\). A solution is a witnessing pair \((t,a)\); an optimization version asks for the minimum profitable abstention mass.
This is a credible population continuization. A type is a complete ranking, mass is the fraction of a large electorate with that ranking, and the outcome remains an integral choice set \(f(\mu)\subseteq A\). Nothing about the alternatives, choice sets, or Fishburn comparison is fractionalized. The regime is a large recurring electorate—say, employees voting on institutional policies—with millions of voters but relatively few repeated preference rankings. Abstention by a positive mass is the natural continuum replacement for one named voter’s abstention; otherwise an individual has measure zero and cannot change the majority relation.
The paper’s authors would recognize the question: it preserves their majority relation, majoritarian SCFs, Pareto-optimality, Fishburn’s extension, and the strategic-abstention action. Theorem 1 supplies the motivation: every sufficiently rich finite environment forces some profitable Fishburn abstention for every majoritarian Pareto-optimal rule. Rational finite profiles embed directly by setting \(\mu_t=n_t/n\).
The expected computational direction is plausibly Class A for fixed rules with efficient winner determination. For fixed \(t\),
\[ M_{\mu^{(-t,a)}}(x,y) = \frac{M_\mu(x,y)-a\,d_t(x,y)}{1-a}, \]
so each majority comparison changes only at a computable rational breakpoint. There are only \(O(m^2)\) relevant breakpoints per type. For rules such as the top cycle, or rules whose outcome is obtained by an LP such as the bipartisan set, one could search these intervals and optimize the minimum profitable abstention mass. Hardness may instead transfer for a rule whose winner problem is hard.
The weakest point is decisive: this is a new computational problem inspired by Theorems 1 and 2, not a computational result established by them. Moreover, the paper studies abstention by one agent, whereas the nondegenerate continuum model requires abstention by positive mass. The original impossibility thresholds are driven by small finite electorates and alternative structure, not by population multiplicity. Thus the model is plausible and potentially valuable, but under the strict ChoCo grading rule I would report:
A natural continuous-population mirror exists, centred on Continuous Fishburn-Abstention Threshold, but the paper has no eligible named computational anchor. The case should not receive a positive anchor-based verdict.
The decisive objection is that this paper has no eligible computational anchor. Theorems 1 and 2 are axiomatic impossibilities: they quantify over all majoritarian Pareto-optimal SCFs and show that none can satisfy Fishburn-participation beyond certain finite thresholds. The SAT solver is a proof method, not a complexity result about an input problem. Theorem 3 is likewise an axiomatic implication. The remark that computing the bipartisan set is polynomial-time is neither a named computational contribution nor the paper’s object of study.
That defeats both proposed anchors. Theorem 1 and Theorem 2 differ only in whether weak or strict preferences are allowed, and in their finite thresholds. Neither asserts that some computational problem is hard, tractable, approximable, or parameterized. Thus there is no paper-level computational question whose population axis can be continued.
The proposed Continuous Fishburn-Abstention Threshold is a reasonable new problem, but it is not the continuous version of either theorem. The theorems have the logical shape
\[ \neg\exists f\;\bigl(\text{majoritarian}(f)\wedge\text{Pareto}(f)\wedge \text{Fishburn-participation}(f)\bigr), \]
whereas the proposed problem fixes \(f\), fixes \(\mu\), and asks whether a type can remove some mass. It replaces a global existence theorem about social choice functions by an optimization problem about one specified rule and one specified society. The budget \(B\), the objective of minimizing \(a\), and the positive-mass action are all newly supplied. That may be worthwhile research, but it is an invention inspired by the paper, not a mirror of a computational result in it.
The more faithful continuous formulation would impose participation directly on the simplex: for every \(\mu\), type \(t\), and admissible \(a>0\),
\[ f(\mu^{(-t,a)})\not\succ_F^t f(\mu). \]
But that is simply the paper’s axiom rewritten over distributions. It is continuization of the profile domain, precisely the kind of axiomatic continuization excluded by the programme. Turning verification of this property into an algorithmic problem again requires adding an encoding for \(f\), a finite representation of the domain, and a computational objective absent from the paper.
There is also a substantive mismatch in the action itself. In the finite theorems, one named agent abstains. Preserving that action in a high-multiplicity sequence means removing mass \(1/n\). For any society whose pairwise margins are bounded away from zero, that perturbation eventually changes no majority comparison at all; the individual’s effect vanishes. The finite constructions work by carefully arranging margins of one or two votes, and those normalized margins disappear in the continuum. The phenomenon survives only on exact majority-boundary hyperplanes.
Replacing \(1/n\) by a positive \(a\) avoids that degeneration, but then the actor is no longer an individual. It is a coordinated block of agents of one type. Allowing arbitrary type mixtures produces a coalition-abstention or mass-manipulation problem, not the paper’s strategic-abstention problem. Theorem 2’s restriction to strict preferences does not repair this change of object.
One cannot honestly object that voting lacks a high-multiplicity regime: repeated ballots in large electorates are an entirely sensible continuous society, and existing high-multiplicity work would support that modelling choice. Nor is the proposed question mathematically meaningless. A fixed rule’s outcome along a type-abstention path can indeed be studied through affine changes in majority margins and breakpoint geometry. The negative case is therefore not that no related continuous voting problem could ever be interesting.
It is that this paper supplies no computational result to mirror, while its actual results either become an out-of-scope axiom on the profile simplex or lose their individual-abstention content when passed to positive mass. The proposed threshold problem could launch a new project, but it cannot qualify as a worthwhile ChoCo mirror of 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.