Source Clarifications

Source: Kleinberg and Raghavan, Algorithmic Monoculture and Social Welfare, arXiv v2.

Noise normalizations and ranking laws

• Section 3.1 Gumbel weights $e^{\theta x_i}$ → $e^{(\pi/\sqrt6)\theta x_i}$ under unit-variance noise. Here $x_i$ is candidate value and $\theta$ ranking accuracy. A Gumbel scale $s$ gives weights $e^{\theta x_i/s}$; unit variance requires $s=\sqrt6/\pi$. The ranking family is unchanged after rescaling accuracy, but equal numerical parameters otherwise denote different noise levels (source lines 497–540). The unit-variance specialization assumes the standard identity $\operatorname{Var}(-\log T)=\pi^2/6$ for $T\sim\mathrm{Exp}(1)$; that special-value identity is not proved here.

Strictness and candidate-domain readings

• Theorem 3 strict welfare conclusion → at least three candidates. With two firms and two candidates of values $(1,0)$, both are hired and total welfare is always $1$, precluding strict welfare loss from shared rankings.
• Theorem 5 differentiable full-support density → global absolute continuity with an integrable almost-everywhere derivative ($f\in W^{1,1}$). This sufficient replacement permits differentiation of translated densities in $L^1$ and covers Gaussian and Laplace noise, including Laplace’s corner (source lines 784–817). Smooth full support alone permits narrow paired density peaks whose difference distribution has an infinite derivative at a positive gap; the explicit counterexample has mean zero and variance one. It is a mathematical argument, not a Lean-checked counterexample. Necessity of the sufficient $W^{1,1}$ condition remains unresolved.
• Appendix C Lemma 1: the global strict Laplace comparison becomes weak globally and strict on overlap; the four-point calculation gives the exact change and replacement argument.
• Appendix F Lemma 4 weakly decreasing likelihood ratio → a strict ratio comparison at some ordered pair for the strict expectation inequality $\sum_i p_iy_i<\sum_i q_iy_i$, with increasing outcomes $y_i$ and positive normalized weights $p_i,q_i$. If “decreasing” is weak, $p=q$ gives equality; if strictness was intended, this makes its witness explicit.

Appendix algebra and sequential experiments

• Appendix C.1: $h'\ge0$ everywhere and $h'>0$ somewhere, with $h(a)\to L$, imply $h(a)\le L$, not $h(a)<L$ at every finite cutoff. A function may reach its limit and then stay constant. The Laplace proof establishes each finite-cutoff strict gap directly; the Gaussian proof establishes $h'>0$ at every cutoff (source lines 1134–1148).
• Theorem 4’s sequential experiment: each firm chooses before candidates are exhausted ($k<n$ for $k$ firms and $n$ candidates), with a fresh independent human ranking drawn after the remaining set is fixed. This is the source proof’s setting; its strict comparison requires a choice of candidates (source lines 620–631, 3176–3180).
• Appendix C Gaussian product comparison → Definition 4’s direction: the log-product cross difference is positive for $a>b,c>d$.
• Appendix C.9: $g(t)\delta\sqrt\pi+1/g(t+\delta)-1$ → $g(t)[\delta\sqrt\pi+1/g(t+\delta)]-1$. Here $g(t)=(1+\operatorname{erf}(t))e^{t^2}$, $\operatorname{erf}$ is the Gaussian error function, and $\delta=x_i-x_j>0$ in the source normalized coordinates. The missing multiplication changes the finite expression.
• Appendix D.1 cross inequality → the direction of its preceding increasing likelihood-ratio calculation, with ranking mass proportional to $\phi^{-\mathrm{distance}}$ and inverse-accuracy parameter $q=\phi^{-1}$.
• Appendix E.3 swapped term: $-\phi^{-(|i-j|+1)}\Pr[\tau_1=x_i]$ → the same term with a plus sign, as in the next display. Here $\tau_1$ is the first-ranked candidate.

Representation boundaries

• Finite-support and simulation remarks → no unrestricted formal theorem. Finite computations and the multi-firm numerical example establish neither all finite-support noise extensions nor a general complexity guarantee.

• Theorem 2’s “any candidate distribution $D$” → an outer law with ordered support and defined ranking/payoff expectations. The checked Gaussian and Laplace result uses candidate values that are strictly ordered almost surely, have finite first moments, and give measurable ranking probabilities and well-defined conditional and payoff expectations. These conditions make the source’s Definitions 2–3 meaningful for an outer distribution; the source does not enumerate them in its Theorem 2 statement.

• Equation (6) regularity on ordered source support → regularity of the supplied ranking family on every value profile in the current endpoint. This is a current proof restriction outside source-supported profiles. Necessity of the stronger total-family input is unresolved.

Theorem 1: strict-improvement domain

• Definition 1 full-set strictness → the standing strictly ordered candidate values. The source fixes $x_1>\cdots>x_n$ before stating Definition 1 and Theorem 1 (lines 133–138 and 244–246). Its strict comparison is therefore evaluated on those ordered candidate profiles, including when values are drawn from $D$; it is not a condition on profiles with tied candidate values. The formalized Theorem 1 uses that same source domain.