Strategic Ranking: source clarifications and theorem scope

Source: Strategic Ranking, AISTATS 2022. Write $p$ for effort cost, $g$ for production, $f$ for the skill quantile, $0<\rho<1$ for capacity, and $0<c\le1-\rho$ for a two-level cutoff. The high admission reward is $q(c)=\rho/(1-c)$.

Stated model: known priorities and continuum best response

• Ties and equilibrium: applicants know their fixed tie priority before choosing effort. A unilateral deviation holds priorities and the population score distribution fixed. One full-measure set of applicants must each best respond against every feasible deviation; uniqueness is up to a null set. Differentiability is read on domain interiors, with endpoint continuity.

Proposition 3.1

• Pure-randomization optimum: weight conditional band means by their population shares. Welfare is admission reward $\rho$ minus nonnegative effort cost, and pure randomization attains $\rho$. The proof’s extra uniqueness assertion is not needed for the stated optimum.
• Cutoff monotonicity: sufficient additional conditions are $$x\mapsto\log f(1-e^{-x})\text{ concave on }(0,\infty),\qquad z\mapsto\log C(e^z)\text{ concave where }g(0)<e^z\le g(E),$$ where $E=p^{-1}(1)$ and $C=p\circ g^{-1}$. They imply $W(d)\le W(c)$ for $0<c\le d\le1-\rho$, holding the market fixed. They apply only to monotonicity, not the pure-randomization optimum.
• Why a restriction is needed: with $p(e)=e^2$, $g(e)=e$, $f(t)=(1-t^2/4)^{-1/2}$, and $\rho=3/10$, actual equilibrium welfare satisfies $W(1/10)=18/665<14/495=W(1/5)$.
• Welfare derivative: if $A(c)=\int_c^1H(c,t)\,dt$ is total effort cost, the boundary term in the printed derivative changes sign: $W'(c)=H(c,c)-\int_c^1\partial_cH(c,t)\,dt$.

For differentiable positive interior skill, the sufficient skill condition is that $(1-t)f'(t)/f(t)$ is nonincreasing. It is weaker than log concavity: $f(t)=1/(2-t)$ separates them. Nondegenerate uniform skill on a nonnegative interval satisfies it; all $p(e)=e^r$, $g(e)=e^s$ with $r>1$, $0<s\le1$ satisfy the cost condition. These are sufficient conditions, not a necessity claim.

Associated normalization and access corrections

• Proposition 3.2: private utility is the unnormalized $\mathbb E[vZ]=\rho\mathbb E[v\mid Z=1]$. Admitted score is $\max\{g(p^{-1}(q(c)))f(c),g(0)f(t)\}$, retaining baseline production. These corrections to the proof preserve the monotonicity and deterministic optimum.
• Proposition 3.3: the proof’s condition $g(p^{-1}(x))=x$ conflicts with strict convexity of cost and concavity of production. Use $p(e)=e^2$, $g(e)=e$, $f(t)=t^3$, $\rho=1/2$, cutoffs $(0,1/4,3/4,1)$, and rewards $(0,1/2,1)$ instead. Private utility exceeds $9/128$, versus deterministic utility $1/16$, proving the stated existential improvement.
• Proposition 3.4: for $p(e)=e^2$, $g(e)=\sqrt e$, $f(t)=t$, societal utility is $\rho^{1/4}c(1-c)^{3/4}$, globally maximized at $c=4/7$. Interior feasibility requires $\rho<3/7$, replacing the proof’s $\rho<5/9$; this proves its existential conclusion.
• Proposition 4.1: use the mixture’s generalized quantile and clip group thresholds to $[0,1]$, extending CDFs by zero and one outside support. This handles disjoint and touching supports without a smooth inverse.
• Proposition 4.4: pure-randomization access is $\rho$, replacing zero in the proof, and $\theta_B(c)\ge c$, strictly at positive cutoffs. At exhausted support the threshold is one and access is zero. Finite differences prove monotonicity without differentiating at support joins.

Propositions 4.2 and 4.3: exact under clarified weak/strict interpretation

• Welfare-gap strictness: $\Delta=W^A-W^B\ge0$ almost everywhere, with zero gap under pure randomization. Among types admitted in both groups, $\Delta(c,t)>0$ exactly when disadvantaged effort $e^B_c(t)>0$. Positive baseline production can permit admission with zero effort.
• Cutoff comparisons: for $0<c\le d\le1-\rho$, almost every type admitted in both groups at $d$ has $\Delta(d,t)\ge\Delta(c,t)$, strictly if $c<d$ and $e^B_c(t)>0$.
• Derivatives: the same signs apply where derivatives along feasible cutoffs exist; deterministic admission uses the one-sided derivative. Smooth primitives can still yield kinks at mixture-support junctions. For each fixed type, the canonical gap is differentiable almost everywhere on the interior common-admission domain, not necessarily at every cutoff for every representative of the equilibrium family.

Appendix effort comparative statics and Proposition B.1

• Corollary A.1: use the stated $i<k$ range, replacing $i<k-1$ in the proof, and retain baseline clipping and the above-baseline condition for strict comparisons.
• Proposition B.1: with $X=\max_i\alpha_i f_i(t_i)$ and population CDF $F_X$, rank preservation means $\lambda(\theta_{post})=\lambda(F_X(X))$ almost everywhere. Different ranks can receive the same reward within a policy band. Effort is allocated to a maximizing coordinate, unique almost everywhere under independent continuous skills; total effort is endogenous, with no fixed budget.

Proposition B.2

• Budget: impose the stated constraint $e^M,e^U\ge0$, $e^M+e^U=B$ directly. The extra penalty in the displayed payoff is zero on this domain and does not itself enforce the constraint.
• Global optimum: sufficient additional conditions are affine skill $f_M(t)=a+bt$ with $a\ge0$, $b>0$; $g(0)=0$; $B\ge E=p^{-1}(1)$; concavity of $g^2$ on $(0,B)$; and convexity of $x/C(x)$ on $[g(p^{-1}(\rho)),g(E)]$, where $C=p\circ g^{-1}$. Cost is twice differentiable on $(0,E)$, production twice differentiable on $(0,B)$ with continuous $g''$, and source endpoint continuity is retained. Unmeasurable skill is independent and regular, with positive finite mean.

Under these conditions, each $c\in(0,1-\rho)$ maximizes a weighted utility $\beta M+(1-\beta)U$ for some $\beta\in(0,1)$, over all $0<d\le1-\rho$ and for every equilibrium, holding the market fixed. The weight does not depend on equilibrium selection. The domain includes deterministic admission; pure randomization is a separate one-level policy. Square-root production, quadratic cost, affine measurable skill, and $B\ge1$ give an example satisfying the conditions.

• Utility calculation: include conditional normalization and the independent skill factor $\kappa_U=\mathbb E[f_U(t_U)]$: $$M(c)=g(p^{-1}(q(c)))f_M(c),\qquad U(c)=\frac{\kappa_U}{1-c}\int_c^1 g\!\left(B-g^{-1}\!\left(M(c)/f_M(t)\right)\right)\,dt.$$ Differentiation includes the changing-population term. Curvature establishes global optimality; stationarity alone does not.
• Counterexample without the added conditions: with $p(e)=e^2$, $g(e)=e$, $f_M(t)=1/(2-t)$, $f_U(t)=t$, $B=1$, and $\rho=1/16$, the interior cutoff $7/16$ is beaten for every positive weight by $9/25$ or $5/9$ in actual equilibria.