| paper | d4-conf-aaai-2015-0787 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | yes |
P→P discrete result → its continuous mirror
Answer: Polynomial-time solvable in \(m,\tau,L\), with bit running time \(O((m+1)(m+\tau)L^2)\). · discrete side: polynomial · adjudication: proved
full proof & adversarial review →
Theorem 1
statement extracted from the paper’s text layer
Given a finite set of types \(T\), rational masses \(\mu_t \ge 0\) summing to \(1\), and for each type a set \(A_t \subseteq [0,1]\) of attendance fractions at which attending is preferred to the outside option, encoded as a finite union of rational intervals with specified endpoint inclusion, choose invited masses \(y_t \in [0,\mu_t]\) maximizing \(a=\sum_t y_t\). Require \(y_t>0\) to imply \(a\in A_t\), and require that whenever \(y_t<\mu_t\), no right-neighborhood of \(a\) lies in \(A_t\). Find a maximum stable invitation or certify that none exists.
A continuum of event attendees with finitely many preference types: type \(t\) has population mass \(\mu_t\) and accepts attendance fractions in \(A_t\); the organizer chooses invited mass \(y_t\), and maximizes total attendance subject to individual rationality and no profitable infinitesimal entry.
The paper defines preferences over integer attendance counts, so the continuous acceptance sets are a chosen preference domain rather than a canonical interpolation; the finite-to-continuous correspondence remains to be established.
fatal: False
The mirror covers the truthful anonymous maximum stable invitation result, Theorem 1. It leaves the strategic results and the general stable invitation results, including Theorem 4, untreated.
My strongest mirror is the paper’s anonymous, truthful problem. Picture a large event population with \(n\) in the hundreds of thousands but only \(\tau\) distinct crowd-size preference types: for example, recurring donor or participant cohorts with the same response to event size. A type is a complete preference over crowd sizes and the outside option; \(\mu_t\) is its fraction of the population. The organizer chooses how much mass of each type to invite and maximizes total attendance.
The anchor is Theorem 1, proved by Lee and Shoham, with its formal proof deferred in this version to the long version. It says that for any ASIP instance, Algorithm 1 runs in polynomial time and either returns a maximum stable invitation, if one exists, or determines that none exists. This is the right result to mirror because the anonymous problem depends on crowd size rather than on named relationships among invitees.
Call the continuous problem Maximum Stable Invitation over a Continuum (MSIC). An instance gives a finite support \(T\), rational masses \(\mu_t\geq 0\) summing to \(1\), and for each type \(t\) a set \(A_t\subseteq[0,1]\) of attendance fractions at which attending is strictly preferred to the outside option. Encode each \(A_t\) as a finite union of rational intervals with specified endpoint inclusion. A solution is a vector \(y_t\in[0,\mu_t]\), where \(y_t\) is invited mass of type \(t\); its total attendance is \(a=\sum_t y_t\). It is stable when every invited type accepts the current crowd, \(y_t>0\Rightarrow a\in A_t\), and no uninvited agent wants to join. For the latter condition, use the right-hand limit \(a+0^+\): an infinitesimal entrant increases attendance by an infinitesimal amount, just as a single outsider increases a finite crowd from \(k\) to \(k+1\). Thus \(y_t<\mu_t\) requires that no right-neighborhood of \(a\) lies in \(A_t\). The task is to return a stable vector maximizing \(a\), or certify that none exists.
I expect MSIC to be tractable. Between the finitely many preference endpoints, each type’s acceptance and right-hand-limit status is fixed. At any such attendance level, these statuses determine which types must be absent, which must be fully invited, and which may be invited partially. A sweep over the resulting intervals and endpoints should therefore find the maximum stable mass in time polynomial in \(\tau\), the interval-description size, and the input bit length. The model also prompts a useful bridge question: under what endpoint conditions does the optimum approximate the maximum stable invitation in a large finite population, and with what rounding error?
This case covers only the paper’s truthful ASIP result; it does not claim a mirror for the strategic theorems or GSIP. The more striking hardness result, Theorem 4, concerns \((1,1)\)-instances of GSIP with simple preferences, where deciding whether a stable invitation of size at least \(k\) exists is NP-hard. I would not use it as my anchor: its acceptance and rejection sets refer to named agents, and a high-multiplicity mirror would need to replace those relationships with dependencies among types or cohorts. That may be a sensible further problem, but it is a model change whose hardness this paper does not establish.
The weak point is that MSIC is a modest mirror: it extends an existing polynomial result and is unlikely to expose a new hardness boundary. Its limit convention at preference thresholds also needs a careful finite-to-continuous dictionary. Still, it is a direct population continuization of the authors’ own anonymous problem, with an explicit mass allocation, stability test, and objective.
The strongest objection to this mirror is about its likely payoff, not whether it is a legitimate continuization. Theorem 1 is a named computational result, and a large recurring event population with a few preference cohorts is a plausible high-multiplicity setting. Cohort members can share a complete preference over attendance levels, while the organizer chooses invited mass. That makes the anonymous problem a natural mirror; the objection that invitations are really about named people does not apply here.
The proposed MSIC also has a workable finite-population dictionary. For a population of size \(n\), type \(t\) has mass \(n_t/n\); at attendance fraction \(a=k/n\), invited members accept if \(a\in A_t\), and an outsider’s entry is tested at \((k+1)/n\). With fixed finite unions of intervals, the infinitesimal-entry convention captures that test in the large-population limit. So the endpoint issue calls for care in stating the bridge, but does not defeat it.
There is a more substantial limitation: Theorem 1’s anonymous structure reduces the continuous problem to a one-dimensional sweep. At each attendance level, a type must be absent, fully invited, or available for partial invitation; a sweep over interval endpoints can check which totals are feasible. That is a modest algorithmic result, and it may add little beyond the paper’s existing polynomial algorithm. Also, arbitrary finite rankings over attendee counts do not have a unique interpolation to fractional attendance, so a bridge needs a specified family of preference curves. These points support a low-priority assessment, not the claim that no worthwhile mirror exists.
I cannot honestly defeat this anchor. A recurring population with a bounded set of response curves gives a coherent continuous computational problem, and the proposed sweep is a credible starting result. The negative case is therefore weak exactly where it must be strongest: a better-specified MSIC survives the objections.
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.