d4-journals-jet-2023-0106

The strongest positive case is for the paper’s self-evaluation winner-selection problem. There are no named computational-complexity results here: the paper makes no claim that a problem is in P, NP-hard, or similarly classified. Its closest anchors are named results characterizing and evaluating an optimization problem. I use those as anchors without presenting them as complexity theorems.

The lead is Theorem 1, proved in the paper’s Appendix. It characterizes every optimal anonymous BIC mechanism for the finite-\(n\) self-evaluation model. The mechanism selects from \(M\), the agents with the highest referee report. If everyone in \(M\) gives the same self-report, it selects uniformly from \(M\). If \(M\) contains both low and high self-reports, the tie-breaking depends on the referee precision. When \(r<r^*\), a high-referee shortlist selects a low-self-report member with probability \(\alpha_z/|M_0|\) per member and a high-self-report member with probability \((1-\alpha_z)/|M_1|\); a low-referee shortlist selects uniformly from \(M_0\). When \(r\ge r^*\), a high-referee shortlist selects uniformly from \(M_1\); a low-referee shortlist uses the corresponding probabilities \(\beta_z/|M_0|\) and \((1-\beta_z)/|M_1|\). Here \(M_0,M_1\) are the low- and high-self-report members of \(M\), and \(z\) records counts of the report pairs. The vectors \(\alpha_z,\beta_z\) lie in \([0,1]\) and satisfy the balance equations (6) and (7). The threshold is determined by \(Q=pr+(1-p)(1-r)\): if \(x_n\) is the unique root of \(\varphi_n(x)=2^n\), then \(r^*=(x_n+p-1)/(2p-1)\). The theorem’s main structural message is that referee reports establish the shortlist and self-reports break ties within it.

Here is a continuous version I would be willing to hand to a prover. Call it Continuous BIC Winner Selection. Its instance has a unit-mass population, a common binary suitability and signal environment with rational \(p,r\) as in the paper, and two private-signal types \(x\in\{0,1\}\), each with mass \(\mu_x=1/2\). Suitability \(\Theta\) remains hidden; each agent knows their own signal \(x\), and an external referee produces \(y\). Write \(\nu_{xy}=\mu_x\Pr(Y=y\mid X=x)\) for the mass in stratum \((x,y)\), and \(q_{xy}=\Pr(\Theta=1\mid X=x,Y=y)\). A mechanism chooses nonnegative allocation densities \(a_{b,y}\), where \(b\) is the agent’s reported self-signal. The densities must allocate exactly one prize, so \(\sum_{x,y}\nu_{xy}a_{x,y}=1\). They must satisfy BIC for each private signal type: \(\sum_y\Pr(Y=y\mid X=x)a_{x,y}\geq\sum_y\Pr(Y=y\mid X=x)a_{1-x,y}\), for \(x=0,1\). The objective is to maximize \(\sum_{x,y}\nu_{xy}q_{xy}a_{x,y}\), the probability the selected candidate is suitable. A solution is an optimal density vector and its value. The density is selection probability per unit of population mass; for a finite population it corresponds to scaling an individual’s selection probability by \(n\).

This is a direct atomless version of their question: who should receive a single award when applicants’ self-reports are incentivized and external reports are informative? A credible regime is a large annual fellowship or grant competition with standardized applicant cohorts: millions of applicants, but only two incentive-relevant self-signal types in the benchmark, sharing the same signal technology and selection payoff. The report outcomes remain what the planner observes; they need not define permanent agent types. This is a strong high-multiplicity story, and the paper itself discusses grant and conference selection.

I expect this mirror to be tractable: it is a rational linear program with four allocation variables, two BIC constraints, and one normalization constraint. Uniform selection among \(y=1\) candidates is feasible and has value \(r\). The theorem’s shortlist structure and the paper’s finite-\(n\) value comparison make it plausible that this is also optimal in the atomless model, with the BIC advantage disappearing. The key proof question is whether any allocation to \(y=0\) candidates can compensate for greater reliance on self-reports among \(y=1\) candidates enough to improve the objective. This is a concrete optimization question about the underlying selection problem, not about the theorem’s LP representation.

A second useful anchor is Proposition 1, whose formulas are proved in Online Appendix 1. It gives the finite-population optimal values. With \(Q=pr+(1-p)(1-r)\), the optimal anonymous BIC value is
\(\Pi_B(p,r)=r-\frac{2r-1}{2^n}\left(1-\frac{r(1-r)(2p-1)^2}{Q(1-Q)}\min\left\{1-(1-Q)^{n-1},\frac{2^n(1-Q)-(2-Q)^n+Q}{Q}\right\}\right)\),
while the optimal anonymous DSIC value is
\(\Pi_D(p,r)=r-\frac{2r-1}{2^n}\).
For interior parameters the BIC value is strictly higher; the paper identifies equality at the uninformative and perfectly informative referee limits. The continuous question corresponding to this proposition is to compute the BIC and DSIC optima in Continuous BIC Winner Selection and decide whether their gap is positive. My prediction is again tractable, with gap zero: the finite-population gain is an \(O(2^{-n})\) correction, and the paper explicitly observes that it becomes negligible as \(n\) grows.

I would keep the scope to these self-evaluation BIC and DSIC results. The continuous mirror does not cover the peer-evaluation or EPIC sections, and it does not continuize the outcome, time, or information space. Follow-up questions include whether heterogeneous cohorts with different \(p\) and \(r\) still admit a compact LP; whether BIC can strictly beat DSIC in any atomless selection regime; and whether a nonvanishing gap requires some feature absent from this homogeneous population.

The weak point is substantial: Theorem 1 and Proposition 1 are economic optimization results, not computational-complexity results, so this paper does not offer the usual hardness anchor for a continuization programme. Also, the finite-\(n\) BIC advantage appears to vanish in the limit that motivates the mirror. Finally, a single person in an atomless population has zero probability of winning; the per-capita density convention is the natural high-multiplicity scaling of the finite model, but authors or a referee might reject it as the wrong incentive notion. The case survives if they accept that scaling: the underlying question remains their winner-selection problem, and a continuous analysis could establish exactly where the finite-population incentive gain stops helping.