| paper | d4-journals-jet-2011-0035 |
| 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: The mass Weak Sequential Core is necessarily nonempty, and exact search is polynomial-time solvable in \(m,\tau,L\), with running time \(O(m\tau\,\operatorname{poly}(L))\). A rational schedule suffices. · discrete side: other · adjudication: proved
full proof & adversarial review →
Theorem 5.3
statement extracted from the paper’s text layer
Given rational type masses \(\mu_t\), finitely many states, rational piecewise-linear worth functions \(V_s(q)\) for coalition masses \(q\in[0,\mu]\), and type-specific increasing state-separable utilities, find a state-contingent per-capita payoff schedule in the mass analogue of the Weak Sequential Core. Define that analogue by the paper's characterization: each state's allocation is in the full-population core, and no positive-mass coalition \(q\) can improve every participating type's utility using allocations in the cores of its restricted games in every state. Require each \(V_s\) to have increasing differences, meaning \(V_s(q+\delta)-V_s(q)\le V_s(r+\delta)-V_s(r)\) whenever \(q\le r\) and \(r+\delta\le\mu\).
A nonatomic TUU game with finitely many complete types carrying rational masses, state-dependent coalition worths \(V_s(q)\), and per-capita state-contingent payoffs; stability is the mass version of the paper's characterized no-commitment core condition.
The finite recursion defining credible deviations does not directly carry over to a continuum, and the case does not specify a finite encoding for the mass-based worth functions or utilities.
fatal: False
The mirror covers the constructive nonemptiness result in Theorem 5.3 under convex ex-post games; it leaves Theorem 4.4's characterization and the examples establishing emptiness or limits of weaker convexity assumptions outside the computational mirror.
The strongest honest positive case is a nonatomic extension of the TUU model. Imagine a large risk-sharing pool with many participants in a small number of recurring contractual classes. A type would include a participant’s state-contingent utility and all the features relevant to how coalitions generate surplus. The mass \(\mu_t\) of type \(t\) is its share of the pool; the number of participants is large while the number of types \(\tau\) stays modest. Uncertainty and the paper’s no-commitment setting remain intact.
A candidate problem would take state-dependent coalition-worth functions \(V_s(q)\), where \(q\) records the masses of each type in a coalition, and ask whether there is a state-contingent allocation in the continuum Weak Sequential Core. In that extension, each state’s allocation must be in the core, and no coalition may improve every member’s utility using an allocation that lies in the core of that coalition’s restricted game in every state. The paper’s Theorem 4.4 gives exactly this characterization for finite games. The computational question would be to decide whether such an allocation exists, or to find one, for finitely encoded worth and utility functions.
That is a recognizable extension of the authors’ subject: it retains their TUU games, their solution concept, and their stability question while replacing finite player counts with type masses. But it is a new computational question, not a result this paper establishes. A large risk-sharing pool with repeated participant classes is a plausible regime; the paper itself studies neither such a population limit nor its computation.
There are no eligible computational anchors in the paper. Theorem 4.4, proved here, says that an allocation is in the Weak Sequential Core exactly when it is in the core of every ex-post game and no coalition can improve all its members’ utilities using statewise core allocations of its restricted games. Theorem 5.3, also proved here, says that the Weak Sequential Core is nonempty if every ex-post TU game is convex. Both are substantive mathematical results, but neither asserts that a computational problem is in P, hard, fixed-parameter tractable, or otherwise classifies its complexity. The paper gives no algorithmic complexity result to mirror, so I would not present either theorem as the required computational anchor.
The convexity result makes the proposed extension a promising place to ask whether a tractability story survives: the finite theorem even constructs a stable allocation from marginal vectors. But that construction does not itself give an algorithm for the continuum input, and the paper supplies no basis for predicting the general complexity class. My weakest point is therefore also decisive for the programme’s rubric: this is a plausible continuous-population research direction, but the positive case cannot be anchored in a named computational result of this paper.
The strongest case against a mirror is that the paper has no computational result to mirror. Theorem 4.4 characterizes the Weak Sequential Core; Theorem 5.3 proves it is nonempty when every ex post game is convex. Neither states an algorithm, a running time, or a hardness result. The examples show that the core can be empty, but they do not classify the complexity of deciding that. Those are substantive results, just not computational ones.
The risk-sharing pool is a plausible high-multiplicity setting. But its proposed question—whether a continuum Weak Sequential Core exists, or how to find one—would require new primitives: a worth function \(V_s(q)\) over coalition masses, a representation of those functions as finite input, and corresponding feasibility and utility rules. The paper’s arbitrary characteristic functions \(v_s:2^N\to\mathbb{R}\) do not specify a canonical mass-based version. Once those choices are made, the complexity may turn on the chosen representation of \(V_s\) and the utilities. That could be a worthwhile project, but it would originate a computational problem in a new nonatomic TUU model; Theorems 4.4 and 5.3 do not supply its computational anchor.
So the negative case is strong for the programme’s stated requirement of mirroring computational results, but weak as a universal claim that no useful continuous-population project could grow from this paper. The repeated-type risk-sharing scenario is credible; the honest objection is that it is a new direction, not a continuous mirror of a 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.