Loss aversion in strategy-proof school-choice mechanisms

Lemma 3 characterizes an essentially unique cutoff equilibrium but does not state that the equilibrium or cutoff can be computed, nor provide a complexity or hardness claim. The proposed tiered high-multiplicity model is coherent, but it cannot satisfy the programme’s computational-anchor gate. Moreover, under the paper’s atomless priority-score limit, admission probabilities become \(0\) or \(1\) almost everywhere, eliminating the central loss-aversion effect.

The objection that survived

With atomless priorities and capacity scaled with population, \(f(\omega)\) converges to \(0\) or \(1\) almost everywhere, so the \(\Lambda\)-dependent cutoff disappears; tier lotteries introduce a different source of uncertainty.

What the mirror covers

The proposed mirror covers only the elite-school, common-priority cutoff analysis behind Lemma 3; it leaves the individual characterization, incentive bounds, equilibrium existence, mechanism impossibility, and sequential-mechanism results untouched.

Open questions for a prover

The case FOR (proponent)

The strongest honest positive case is narrow: a high-multiplicity equilibrium-computation mirror of the paper’s elite-school DA model. I would use only Lemma 3 as the anchor.

There is an important qualification. The paper contains no numbered theorem, lemma, corollary, or proposition proving NP-hardness, membership in \( \mathrm P \), FPT, approximation, or any other computational classification. Lemma 3 is a structural equilibrium result, not a complexity result. Thus this is a positive case only if the anchor rule admits named equilibrium characterizations as algorithmic anchors.

Lemma 3, proved in this paper, states the following in full: in the elite-school problem with common priorities, deferred acceptance has an essentially unique choice-acclimating Bayesian Nash equilibrium, where “essentially unique” means unique after fixing how agents break ties between equally good reports. In that equilibrium, a student with loss dominance \( \Lambda \) applies to the elite school if and only if her priority score exceeds a cutoff \( \overline{\omega}(\Lambda)\in(0,1) \), and this cutoff is increasing in \( \Lambda \). More loss-averse students therefore require better admission prospects before applying.

A natural continuous mirror is Continuum-EBLA Elite-DA Equilibrium.

The instance has one elite school \(e\), an unlimited-capacity district school \(d\), elite capacity \( \kappa\in(0,1) \), and finitely many complete student types
\[ \theta=(\ell,j)\in [K]\times[L]. \]
Here \( \ell \) is one of \(K\) priority tiers, ordered from highest to lowest priority, and \(j\) identifies one of \(L\) loss-dominance values \( \Lambda_j \). Every type has the same true ranking \(e\succ d\), value \(v_e=v>0\), and \(v_d=0\). The society is a rational mass distribution
\[ \mu=(\mu_{\ell j}),\qquad \mu_{\ell j}\ge 0,\qquad \sum_{\ell,j}\mu_{\ell j}=1. \]

A type-\((\ell,j)\) student applies to \(e\) with population fraction \( \sigma_{\ell j}\in[0,1] \). Thus the applying mass is
\[ x_{\ell j}=\mu_{\ell j}\sigma_{\ell j}. \]
Let \(x_\ell=\sum_jx_{\ell j}\) and let \(H_\ell=\sum_{r<\ell}x_r\) be the mass of higher-priority applicants. Under uniform tie-breaking within a priority tier, an applicant in tier \(\ell\) is accepted with probability
\[ f_\ell= \begin{cases} 1, & H_\ell+x_\ell\le \kappa,\\[2mm] \dfrac{\kappa-H_\ell}{x_\ell}, & H_\ell<\kappa<H_\ell+x_\ell,\\[3mm] 0, & H_\ell\ge \kappa. \end{cases} \]
The EBLA utility of applying is
\[ v\bigl(f_\ell-\Lambda_j f_\ell(1-f_\ell)\bigr), \]
whereas applying to the district school gives utility \(0\). A solution is a vector \( \sigma \) satisfying the CPE best-response conditions
\[ \sigma_{\ell j}>0 \Rightarrow f_\ell-\Lambda_jf_\ell(1-f_\ell)\ge0, \]
and
\[ \sigma_{\ell j}<1 \Rightarrow f_\ell-\Lambda_jf_\ell(1-f_\ell)\le0. \]
The output is an equilibrium application profile together with the elite assignment masses
\[ a_{\ell j}=x_{\ell j}f_\ell \]
and the district assignment masses \( \mu_{\ell j}-a_{\ell j} \).

This is a genuine population problem, not a fixed-belief restatement. Aggregate application mass determines admission probabilities, which determine each type’s loss-averse incentive to apply. With rational \( \mu \) and \( \kappa \), denominator clearing gives a finite high-multiplicity interpretation: replicate each type proportionally and scale the elite capacity by the same population factor.

The regime is plausible in a large centralized school district or university-admissions market with tens or hundreds of thousands of applicants, a small number of standardized priority bands, and a modest number of empirically estimated loss-aversion categories. The type count \(KL\) can remain small while the population is enormous. Priority-band tie-breaking is a natural mechanism convention; it does not make the tie-breaking lottery an additional student-specific type.

I would expect this problem to be Class A. For fixed \(K\) and \(L\), the equilibrium conditions are piecewise linear. Within each priority tier, students apply in threshold order according to
\[ f_\ell\ge 1-\frac1{\Lambda_j}, \]
with at most one marginal loss-aversion type mixing. A scan through priority tiers and loss-aversion thresholds should therefore compute an equilibrium in time polynomial in \(K\), \(L\), and the input bit length. That algorithm is not proved by this paper; it is the downstream ChoCo question.

The mirror is recognizable because it preserves the paper’s central objects: DA, elite versus district schools, common priorities, application reports, stochastic admission prospects, and expectation-based loss aversion. It also preserves the paper’s main comparative-static claim: stronger loss aversion produces a higher application cutoff.

The weakest point is serious. In the paper, uncertainty comes from individually varying priority scores. A pure atomless limit with deterministic priorities could make each student’s admission probability effectively certain and eliminate the very uncertainty that drives loss-averse misreporting. The finite priority tiers and stochastic tie-breaking repair this, but they are a co-modification of the model. If that repair is rejected, the proposed problem becomes a new mean-field school-choice model rather than a faithful continuization.

So the best positive conclusion is conditional: the paper supports a plausible, population-sensitive continuous equilibrium problem based on Lemma 3, likely tractable and worth studying. But under the programme’s strict requirement that an anchor be a named computational result, this paper has no qualifying anchor.

The case AGAINST (opponent, writing after the proponent)

The decisive objection is that this paper has no qualifying computational anchor. It contains no theorem, lemma, proposition, or corollary about worst-case complexity, exact computation, approximation, parameterized algorithms, or computational hardness. Lemma 3 is an equilibrium characterization in a finite Bayesian game: it says that a cutoff exists and is essentially unique. That is a structural economics result, not a computational result of the kind the ChoCo programme is designed to continuize. Proposition 3 is merely an existence result for finite type spaces; Proposition 2 gives individual incentive bounds. None supplies a named computational problem to mirror.

Even if Lemma 3 is admitted as an anchor, its central phenomenon does not survive the natural high-multiplicity limit. In the paper, a student’s admission probability is nontrivial because the number of competing higher-priority students is random. Let \(q_n/n\to\kappa\) while priority scores are drawn from the paper’s atomless distribution \(G\). For a fixed score \(\omega\), the aggregate mass of higher-priority applicants converges by the law of large numbers to a deterministic quantity. Away from the capacity threshold, the admission probability therefore converges to either \(0\) or \(1\). Consequently,
\[ \Lambda f(\omega)(1-f(\omega))\longrightarrow 0, \]
so the loss-aversion term disappears for almost every type. The limiting decision is simply to apply when admission is certain and not apply when it is impossible. The paper’s \(\Lambda\)-dependent cutoff, and hence its district-school bias, vanish except at a measure-zero boundary type. If capacity is kept fixed instead of scaled with population, the elite school itself becomes a zero-mass object, which is another degeneration.

The proposed repair—finitely many priority tiers with random tie-breaking—does preserve fractional admission probabilities, but it is not the high-multiplicity limit of the paper’s stated model. It replaces the atomless individual score, which is the source of the paper’s uncertainty, by an atomic priority structure and an added lottery within each atom. That may define a sensible new anonymous matching model, but it is not a continuization of Lemma 3’s mechanism; it changes the source of uncertainty that EBLA is meant to explain.

Nor does the repair rescue a computational contribution. Once priority tiers and loss-aversion categories are finite, the equilibrium is governed by residual capacity and scalar acceptance probabilities. The application profile can be found by scanning the finitely many thresholds \(1-1/\Lambda_j\), with at most a marginal category mixing. This could be a useful applied model, but it is a new finite-type equilibrium calculation, not a computational landscape exposed by continuizing this paper.

The negative case is therefore strong under ChoCo’s stated gate: there is no named computational result here to mirror, and the natural continuous limit removes the paper’s substantive behavioral mechanism. But the universal claim is weaker than the programme’s gate-based rejection. A large school-choice market with discrete priority bands and lottery tie-breaking is genuinely plausible, and the proponent’s model is a coherent population game. I cannot honestly claim that no worthwhile continuous-population question can be built from this paper; only that doing so requires introducing a different priority regime rather than continuizing one of the paper’s 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.