| paper | d4-journals-mss-2012-0038 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | yes |
Theorem 3.1
statement extracted from the paper’s text layer
Given a finite type set \(T\), rational population shares \(\mu_t\), a rational multilinear worth function \(W\), and two rationally specified coalition structures \(C^0,C^1\), form replica societies of size \(n\) along common denominator multiples, with \(v_n(S)=nW((|S_t|/n)_{t\in T})\), where \(S_t\) counts active members of type \(t\). Assuming each replica has an all-proposals-accepted stationary SPE, do the per-member type payoffs \(\zeta^{(n,C)}\) from Theorem 3.1 converge, and what is the complexity of computing their limits and deciding whether the specified merger raises its members’ payoffs?
Types \(t\in T\) capture all payoff-relevant member attributes. The input gives rational type masses, type shares within each coalition under two specified structures, and a rational multilinear worth function whose replica-game worth is \(v_n(S)=nW((|S_t|/n)_{t\in T})\). The output is the limiting per-member payoff by type under Theorem 3.1’s recurrence.
Theorem 3.1 is conditional on an all-proposals-accepted stationary SPE, and neither the paper nor the cases establishes that condition uniformly across replicas; it can be imposed as a promise on the instance class.
fatal: False
The mirror covers Theorem 3.1’s payoff recurrence and coalition-merger comparisons for TU games with the promised equilibrium; it leaves NTU games, equilibrium existence, and the axiomatic discussion untreated.
This paper contains no named computational result of the required kind. Proposition 3.1 says that, when proposals are accepted, each player has equal probability of making the final proposal. Theorem 3.1 characterizes the resulting equilibrium payoff vector \(\zeta_S\) by an induction over active-player sets. Neither states a complexity classification or gives a computational formulation that the paper uses to solve an underlying problem. The Harsanyi-paradox comparison is an example, not a numbered computational result.
So I cannot honestly supply the required named anchor or a continuous problem *per anchor*. A possible research extension would be to ask how to compute equilibrium payoffs in a continuum bargaining model, with mass representing agents of each bargaining type. But that would be a new problem, not a continuous mirror of a computational result in this paper. It would also need extra structure: the paper’s game specifies \(v(S)\) for every finite coalition \(S\), and its mechanism recursively removes individual players or whole coalitions. Passing directly to a continuum does not preserve that recursion in an obvious way.
Under the requested anchor standard, the positive case is therefore limited: this paper may inspire a continuous bargaining project, but it does not provide a named computational result to mirror.
The “no named computational result” objection is accurate in a narrow sense: the paper gives no complexity classification. But it does have a named, constructive anchor in Theorem 3.1. The theorem specifies the equilibrium payoff vector \(\zeta_S\) by a recurrence, and the paper uses that object to assess whether coalition formation benefits its members. A natural computational question is therefore: given a succinctly represented game and coalition structure, how hard is it to compute those payoffs—or decide whether forming a coalition improves its members’ payoffs? The paper does not answer that question, but its theorem makes it concrete.
There is a plausible high-multiplicity setting. Think of a large union or federation whose members fall into a small number of bargaining types, with each type sharing the same relevant contribution to coalition worth, outside option, and bargaining role. Represent the society by rational masses of those types and give coalition worth by a finite description of how it depends on type masses, scaled with population. Then compute per-capita \(\zeta\) and test whether a merger helps its members. This preserves the paper’s central issue—how coalition size and the right to make proposals affect expected payoffs—while replacing headcounts with population shares.
The continuum should be defined through a sequence of finite replica societies, not by deleting one atomless individual in a single step. In Theorem 3.1, individual deletion changes the active set; along the replica sequence, that change becomes infinitesimal, while a coalition’s exit can still remove positive mass. The limit might yield an integral or differential characterization, or it might fail to exist for some games. Establishing which happens, and the complexity of computing the limit from a finite description, is part of the research question—not a reason the question lacks an object.
The weak point is that the paper supplies neither a succinct game representation nor evidence that its payoff recurrence has a tractable or interesting limit. Those choices need careful modeling. But the proposed mirror is more than a generic continuum-bargaining idea: it is tied to Theorem 3.1’s payoff characterization and the paper’s coalition-benefit question, in a setting where type masses and per-capita worth have a credible interpretation.
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.