| paper | d4-journals-mss-2012-0025 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | no |
Theorem 3.1 constructively gives equilibrium strategies and an outcome, so I count it as computational despite the absence of a runtime bound. The proposed continuous object is a distribution over bargaining pairs, each of which still plays its own two-player game; the aggregate is only a post hoc average. That does not yield a population-level computational question that survives the mirror test.
fails bit b-none — no population with multiplicity to relax
Each type represents a separate bargaining pair, so computing its equilibrium and averaging the results does not make the population part of the strategic problem.
fatal: True
No result survives as a continuous-population mirror; the candidate uses Theorem 3.1's equilibrium construction, while Theorem 4.1's conditional uniqueness and the comparative-statics results remain outside it.
The strongest affirmative angle I can defend is a large batch of standardized bilateral negotiations. A type \(t\) would describe both bargainers’ loss-aversion coefficients and continuation probability, \((\lambda_1^t,\lambda_2^t,\delta^t)\); \(\mu_t\) would be the fraction of cases of that type. This fits settings such as standardized insurance claims, where the number of negotiations is large but the number of distinct bargaining environments is small.
The candidate problem is: given finite support \(T\), rational masses \(\mu_t\), and rational parameters for each type, compute the stationary Markov equilibrium proposals and acceptance strategies for each type’s unit-pie bargaining game, then return the mass-weighted agreement \(\bar{x}=\sum_{t\in T}\mu_t x_t\). A solution is the type-by-type strategy table and aggregate division. Each bargaining pair maximizes its own expected loss-averse utility; there is no separate social objective. This is likely tractable: applying the paper’s nine-region formulas to each type takes work proportional to \(|T|\), with exact rational arithmetic. It covers the equilibrium-construction result, not the paper’s comparative-statics results as separate problems.
The closest named result is Theorem 3.1, proved in the paper: for the alternating-offers game with loss-averse players, the authors construct a subgame perfect equilibrium whose strategies depend on the current reference points through nine regions; with initial reference points zero, its outcome is
\[
x_{3,\mathrm{III}}=
\left(
\frac{\mu_1(\mu_2-\delta)}{\mu_1\mu_2-\delta^2},
\frac{\delta(\mu_1-\delta)}{\mu_1\mu_2-\delta^2}
\right),
\qquad \mu_i=1+\lambda_i(1-\delta).
\]
Theorem 4.1, also proved here, says this is the unique SPE satisfying stationary Markov strategies, immediate acceptance of equilibrium offers, and accept-reject indifference. These provide a concrete equilibrium-computation foothold, but neither theorem asserts a complexity classification. The paper contains no named result asserting that a computational problem is in P, hard, FPT, or otherwise classifying its complexity. So under your anchor rule, neither is an eligible anchor, and I cannot honestly present the cohort problem as a properly anchored positive case.
That is the weakness as well as the boundary of the case: the continuous object here is a population of bargaining *cases*—effectively pairs—not a society of agents bargaining together. The cases do not interact, so aggregation adds little computational content, and the authors may see it as batching copies of their game rather than a continuous form of their problem. The theorem supplies a useful lead, but not the computational result this programme needs.
The strongest objection is that this paper gives no named computational result to continuize. Theorem 3.1 constructs an equilibrium, and Theorem 4.1 proves uniqueness only among equilibria satisfying three stated conditions. Neither asks how hard it is to compute an outcome or classifies any algorithmic problem. The other results concern comparative statics and a limiting relation to Nash bargaining. A new computation task could be inspired by the paper, but it would not mirror a computational result the paper establishes.
The proposed batch of negotiations is the best candidate, but its continuous object is a distribution over *cases*, not a society whose members jointly make a choice. Each bargaining pair keeps its own offers, reference points, and equilibrium; one pair’s play has no effect on another’s. The type-by-type strategy table and mass-weighted division therefore combine independent instances after solving them. Calling the weights population shares does not give the bargaining model a population-level decision problem.
A stronger version could distribute initial reference points as well as loss aversion and continuation parameters, or study how the aggregate division changes when the type distribution changes. That remains a mixture of separate games: the nine-region equilibrium is computed for each case, then averaged. It may be useful for describing a portfolio of negotiations, but it does not turn Theorem 3.1’s equilibrium construction or Theorem 4.1’s conditional uniqueness result into computational questions about a continuous society.
To create a genuinely coupled population problem, one would need to add something like endogenous partner matching, shared resources, or bargaining among groups. Those could raise worthwhile computational questions, but they change the paper’s two-player alternating-offers model. The negative case is strongest against a *mirror of this paper*: repeated negotiations make the aggregate statistic plausible, but the paper supplies no computational result to anchor it, and the proposed population never enters the strategic problem. The broader claim that no useful extension could be devised is harder to establish; the objection is to treating such an extension as this paper’s continuous mirror.
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.