d4-journals-jet-2018-0051

Theorem 1 supplies equilibrium formulas and uniquely determined cutoff and mixing conditions, but no algorithmic claim, complexity bound, hardness result, or computation-oriented formulation; it therefore fails the named-computational-result gate. Its natural population limit also changes the finite Bayesian contest into a nonatomic threshold game, which Section 6.3 analyzes as an extension rather than a computational result of Theorem 1.

The objection that survived

The opponent points out that the continuum limit removes individual influence and replaces the finite-player Bayesian contest with a nonatomic game, so Theorem 1 does not transfer.

What the mirror covers

Section 6.3 derives continuum threshold equilibria and identical CCA and DCA allocations; this does not cover the finite-player equilibrium results, welfare comparisons in Propositions 2 and 3, or the experiments.

Open questions for a prover

The case FOR (proponent)

I can make a plausible population mirror, but I cannot honestly supply a qualifying anchor: the paper has no numbered result about a problem being in P, NP-hard, FPT, or otherwise computationally classified. Its results solve Bayesian equilibria and compare welfare. The closest is Theorem 1, proved here: it characterizes the unique symmetric monotone equilibrium of the finite-student DCA game. Types below a cutoff mix between colleges; higher types choose college 2; the equilibrium effort function is strictly increasing and the cutoff and mixing probabilities are uniquely determined by indifference conditions. That is an equilibrium characterization, not a computational-complexity result.

The natural mirror would take ability \(a\) as the type, mass \(dF(a)\) as the population, and effort cost \(e/a\), with college capacities scaled to population shares. In DCA, each type chooses a college and effort; in CCA, each chooses effort. The question would be to find the equilibrium strategies and the resulting assignments. A plausible regime is a large annual admissions cohort with many students sharing the same preparation cost and college values.

But this mirror is already substantially in the paper. Section 6.3 explicitly studies a continuum of students, gives each college a measure of seats, and derives the equilibrium thresholds. With \(c_1=F^{-1}(1-\mu_1-\mu_2)\) and \(c_2=F^{-1}(1-\mu_2)\), the effort thresholds are \(e_1=v_1c_1\) and \(e_2=v_1c_1+(v_2-v_1)c_2\): lower types do not apply, middle types attend college 1, and higher types attend college 2. The continuum allocations coincide under CCA and DCA. The paper conjectures numerical convergence of its finite-market equilibria to this continuum outcome, but does not prove that convergence.

So the positive case is that admissions contests have a natural population-level formulation, and the authors themselves make it. For ChoCo’s computational programme, though, that is not enough: there is no named computational anchor, and the direct continuum-equilibrium question is already treated analytically in the paper. A new complexity question for a finite-support population of ability-cost types could be posed, but it would be an extension without a qualifying result to anchor it to.

The case AGAINST (opponent, writing after the proponent)

The strongest objection is that this paper supplies no computational result to mirror. Theorem 1 characterizes the unique symmetric monotone equilibrium of the finite-student DCA game through effort functions, a cutoff, and indifference conditions. It gives no computational problem with an encoded input, algorithm, or complexity claim. The other propositions compare welfare; the experiments report behavior. This is the programme’s decisive objection: the paper’s results are game theoretic and empirical, not computational.

That is not because a population model would be implausible. Ability can be a type when it fully determines effort cost, and large admissions cohorts provide a credible high multiplicity regime. The best such mirror—a continuum of students, with capacity measured as population mass—is already modeled in Section 6.3. There the authors derive the continuum equilibrium thresholds. I would not call that a collision with prior computational work: it is an analytic treatment, and the paper has no computational result for it to preempt. It does show that the natural continuous equilibrium question is already part of this paper.

The strongest rescue is to discretize ability and costs, give types rational masses, and ask for an algorithm to compute the CCA or DCA equilibrium or welfare. That is a coherent possible research question, but it adds a computational task the paper never poses. In the continuum population model, individual influence vanishes and the paper derives mass thresholds; if we retain the finite market’s order statistic lotteries, we have retained a finite population. Neither choice turns Theorem 1 into a computational result. The numerical conjecture about convergence could motivate further analysis, but it is likewise a new analytic question, not a complexity mirror.

So the negative case is strongest as a programme judgment: there is no worthwhile *computational mirror of this paper’s results* to take forward. It is weaker as a universal claim that no valuable algorithmic work could arise from the model. Such work may be worthwhile, but its value would come from adding a computational question to the paper, not continuizing one of its computational results.

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.