| paper | d4-journals-geb-2008-0068 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | no |
Theorem 5.1 is the closest foothold, but it presents a dividend representation of the allocation rather than a named computational result. The repeated-community and cohort ideas are described as extensions, and no concrete continuous computational question survives from them.
fails bit a — no named computational result to mirror
No mirror survives; Theorem 5.1 is the closest foothold, while the axiomatic characterizations and stability results remain unmirrored.
There is no qualifying computational anchor in this paper, so I can’t honestly give an anchor-based continuous mirror or predict its place in the A/B/C trichotomy. The numbered results concern solution properties: Theorem 3.4 and Corollary 3.8 establish uniqueness from axioms; Theorem 3.7 verifies those axioms; Theorem 4.1 establishes stability; and Theorem 5.1 gives a payoff formula. None asserts that a computational problem is in P, hard, or belongs to another complexity class.
The closest foothold is Theorem 5.1, proved here. For each player \(i\), it expresses the average-tree payoff as
\[
AT_i(v,L)=\sum_{\substack{S\in CL(N)\\i\in S}}
\frac{1+p_S^L(i)}{|S|+\sum_{j\in S}p_S^L(j)}\,\Delta_S(v^L),
\]
where \(CL(N)\) is the set of connected coalitions, \(p_S^L(i)\) counts the players outside \(S\) represented by \(i\), and \(\Delta_S(v^L)\) is the Harsanyi dividend in the restricted game. This is an explicit allocation formula, but it gives no complexity bound or input model; its sum ranges over connected coalitions. The paper’s statement that the solution is “easy to calculate” from \(n\) marginal vectors is not a numbered complexity result.
A natural extension might put mass on roles in repeated tree-organized cooperative groups and ask for a mass-weighted average-tree allocation. That could be a sensible continuous-population question, but it would be a new extension of the paper’s value, not a continuous mirror anchored in one of its named computational results.
The proponent offers no computational anchor to defeat: the numbered results characterize the average tree solution, establish its properties, or give a payoff formula. None classifies the complexity of a computational problem. The paper’s claim that the solution is “easy to calculate” is the closest candidate, but it is not a complexity result.
The strongest attempt to turn Theorem 5.1 into an anchor would be to ask for the complexity of computing the average tree payoff. With the characteristic function given explicitly, however, the input already lists values for \(2^n\) coalitions. The paper also gives a calculation from \(n\) marginal vectors; on a tree, those can be obtained with polynomially many lookups in that table. If \(v\) is instead given succinctly, the difficulty depends on that encoding. Either way, the theorem identifies no computational question whose difficulty comes from making the population continuous.
There are possible continuous extensions, but they do not rescue the mirror. One could put mass on repeated copies of the same finite tree game. Each copy would receive its average tree allocation independently, and the aggregate payoff would be a weighted sum. The mass distribution does not affect the calculation because the paper’s graph game has no cooperation across disconnected components. Or one could make the nodes of a tree into cohorts and replace component averages with mass-weighted averages. That is a coherent model change, and the tree’s component-efficiency and fairness equations would still be directly solvable. But it would extend an axiomatic allocation result, not open the kind of computational question ChoCo is meant to chart.
For a single large tree, the obstacle is sharper: a player’s role includes their position and connections in the tree, and the paper’s allocation depends on that structure outside the coalition as well. A distribution over player types alone loses the network information needed to determine payoffs. Adding enough structure to preserve it can make nearly every player a distinct type; aggregating players into cohorts instead changes the graph model. Either route may motivate new cooperative-game theory, but neither yields a computational mirror of a result in this paper.
So the negative case is strong on the programme’s stated gate: there is no named computational result here to continuize, and the most plausible extensions are either aggregation across independent games or a new weighted axiomatic model. I would not claim universal impossibility. Repeated tree-organized communities could support a sensible continuous study. The weaker point is that such a study would need its own computational problem and motivation; this paper supplies neither.
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.